Research ArticleSpeed-Density Model of Interrupted Traffic FlowBased on Coil Data

Chen Yu1 Jiajie Zhang1 Dezhong Yao1 Ruiguo Zhang2 and Hai Jin1

1Services Computing Technology and System Lab Big Data Technology and System Lab Cluster and Grid Computing LabSchool of Computer Science and Technology Huazhong University of Science and Technology Wuhan 430074 China2Siemens Ltd ChinaCorporate TechnologyWireless Technology andWeb of SystemWuhan InnovationCenterWuhan 430074 China

Correspondence should be addressed to Chen Yu yuchenhusteducn

Received 2 September 2016 Accepted 13 November 2016

Academic Editor Beniamino Di Martino

Copyright copy 2016 Chen Yu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As a fundamental traffic diagram the speed-density relationship can provide a solid foundation for traffic flow analysis and efficienttraffic management Because of the change in modern travel modes the dramatic increase in the number of vehicles and trafficdensity and the impact of traffic signals and other factors vehicles change velocity frequently which means that a speed-densitymodel based on uninterrupted traffic flow is not suitable for interrupted traffic flow Based on the coil data of urban roads inWuhan China a new method which can accurately describe the speed-density relation of interrupted traffic flow is proposed forspeed fluctuation characteristics The model of upper and lower bounds of critical values obtained by fitting the data of the coilson urban roads can accurately and intuitively describe the state of urban road traffic and the physical meaning of each parameterplays an important role in the prediction and analysis of such traffic

1 Introduction

Flow speed and density are known as the basic elements oftraffic flow theory Flow can measure the number of vehiclesand the demand for traffic infrastructure Speed is an impor-tant control index in road planning and it is also an evalua-tion index of vehicle operation efficiency Density reflects theintensity of the vehicles on the road and determines trafficmanagement and controlmeasuresThe relationships betweenflow speed and density called fundamental diagrams playa very important role in traffic flow theory and trafficengineering For example the speed-flow relationship can beused in highway capacity analysis in order to determine thehighway service quality and the speed-density relationshipcan reflect dynamic change in traffic flow which can be usedto study the disturbance propagation between vehiclesThere-fore sound mathematical models provide a solid foundationfor traffic flow analysis and efficient traffic management Therelationship between speed and density which can reflectthe quality of service received from the road is attractingconsiderable research attention

The earliest speed-density model was a linear model pro-posed byGreenshields et al [1] in 1935The linearmodel over-laps and classifies the observed data groups which is provedto be unreasonable and observation time is a holiday with anarrow range of representations so there are some deviationsbetween the derived speed-density relation and the actualsituation Later the relationship between speed and densitywas studied in greater depth and the Greenberg logarithmicmodel Edie model Underwood exponent model Pipes-Munjal model modified Greenshields model Newell modeland so forth emerged in turn [2 3] Heydecker and Addison[4] studied the relationship between speed and density undervarious speed limits and found that zero speed induces trafficjams not the other way aroundMa et al [5] derived a generallogistic model of traffic flow characteristics which includesseveral traffic flow parameters with clear physical meaningsand analyzed the effects of the parameters on speed-densitylogistic curves The experimental results showed that thismodel can well describe the traffic flow characteristics in dif-ferent states Shao et al [6] proposed a speed-density model

Hindawi Publishing CorporationMobile Information SystemsVolume 2016 Article ID 7968108 12 pageshttpdxdoiorg10115520167968108

2 Mobile Information Systems

under congested traffic conditions combined with the mini-mum safety spacing constraint and the experimental resultsshowed that the absolute error of this model was smallerthan that of other models fitting the traffic data of twofreeways Wang et al [7] proposed a family of speed-densitymodels with different numbers of parameters with importantphysical significance and got good performance in the finalexperiment

All of the above studies are based on continuous trafficflow data These data also called uninterrupted traffic floware traffic flow with no effect of external fixation factorssuch as freeway urban expressway and so forth Discon-tinuous traffic flow referred to as interrupted traffic flow isperiodically influenced by external fixation factors The mostcommon interrupted traffic flow is originated by signal lampsof urban intersections Because of the variety of vehicle typesthe periodic effect of signal lamps shunts in the canal sectionand other factors the characteristics of interrupted trafficflow are very complex compared with uninterrupted trafficflow In addition the city is still in a rapid increase in popula-tion and with the development of economy people are moreinclined to self-driving travel thus more and more vehiclesand more and more congestion in the city which leads to theincrease of travel time the growth of fuel consumption [8]the aggravation of environmental pollution and other awfulissues [9 10] Compared with the highway the urban roadhas a strong influence on the individual society and theenvironment Therefore further study of the characteristicsof interrupted traffic flow to provide support formanagementdecisions is particularly important

Research on interrupted traffic flow has attracted a lot ofattention [11ndash15] Many scholars see traffic flow located at acertain distance from the intersection as continuous trafficflow believing that it can be described by continuous trafficflowmodels Some of the literature [16 17] suggests howeverthat because of the short distance between intersections in thecity and the influence of signal lamps there are differencesbetween traffic flow located at a certain distance from theintersection and the traffic flow of freeways Because trafficdata are difficult to obtain and for other objective reasonsonly a few scholars focus on the speed-density model ofdiscontinuous traffic flow Wang et al [18] introduced a four-parameter logit model for complete data fitting and estab-lished a speed-density logit model for left-turning straightand right-turning trafficflowHowever the experimental datawere obtained by VISSIM simulation and the simulationparameters were not accurate enough to depict the complexcity road environment so the experimental results havecertain limitationsWang et al [19] thought that the stochasticmodel would contain more traffic information and putforward the stochastic speed-density model This stochasticmodel can generate a probabilistic traffic flowmodel and canachieve real-time traffic prediction

In order to provide favorable data analysis and presen-tation for city traffic thus to provide decision support forintelligent transportation characterizing the speed-densityrelationship of interrupted traffic flow more accurately is fullof importance By analyzing a large amount of data we pro-pose a description method for a speed-density relationship

model which is suitable for discontinuous traffic flow usingthe upper and lower curves to describe the upper and lowerbounds of velocity values Because of the discrepant charac-teristics of the traffic flow in the outer and inner lanes the coildata of the outer and inner lanes are analyzed and verified

2 Speed-Density Model

Three basic parameters (flow 119902 speed 119906 and density 119896) arethe core content of the traffic flow model The three have thefollowing relationship

119902 = 119896 times 119906 (1)

that is flow is the product of density and speed Therelationship between two parameters of the three is of greatsignificance in traffic flow and the relationship betweenspeed and density has received a lot of research attentionGreenshields et al was an early researcher who proposed thespeed-density linear relationship [1]

119906 = 119906119891 times (1 minus 119896119896119895) (2)

where 119906119891 is the speed of free flow that is the speed of vehiclesunimpeded when the traffic density tends to zero and 119896119895 isthe density of block flow that is the density when the trafficflow is blocked and cannot move As shown in Figure 1 when119896 = 0 the speed can reach the theoretical maximum valuenamely the free flow velocity 119906119891 The area surrounded by theabscissa the ordinate of any point on the line and the coordi-nate origin is the traffic flow

Equation (2) can change to

119896 = 119896119895 times (1 minus 119906119906119891) (3)

Respectively introduce (2) and (3) into (1) and we get

119902 = 119906119891 times (119896 minus 1198962119896119895) 119902 = 119896119895 times (119906 minus 1199062119906119891)

(4)

Equations (4) illustrate that 119902-119896 and 119902-119906 are quadraticfunction relations as shown in Figure 1

The linear model is too simple and there are manydeficiencies In order to improve the model scholars haveproposedmodels based on the linear model but with a higherdegree of accuracy Table 1 lists results for the speed-densitymodel including the Greenberg model Underwood modelNorthwestern model Newellrsquos model Pipes-Munjal modelDrew model Modified Greenshields model Del Castillo andBenitez model Van Aerde model MacNicholas modelThesemodels with the parameters of important physical meaningprovide good results

Wang et al [19] established a speed-density logit proba-bility model with four parameters Wang et al used VISSIM

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Table 1 Speed-density models

Model Function Parameters

Greenshields model (1935) 119906 = 119906119891 times (1 minus 119896119896119895) 119906119891 119896119895Greenberg model (1959) 119906 = 119906119898 times ln(119896119895119896 ) 119906119898 119896119895Underwood model (1961) 119906 = 119906119891 times exp(minus 119896119896119898 ) 119906119891 119896119898Newellrsquos model (1961) 119906 = 119906119891 times 1 minus exp[minus 120582119906119891 times (

1119896 minus 1119896119895)] 119906119891 120582 119896119895Northwestern model (1967) 119906 = 119906119891 times exp[minus12 times ( 1198961198960 )

2] 119906119891 1198960Pipes-Munjal model (1967) 119906 = 119906119891 times [1 minus ( 119896119896119895)

119899] 119906119891 119896119895Drew model (1968) 119906 = 119906119891 times [1 minus ( 119896119896119895)

119899+12] 119906119891 119896119895Modified Greenshields model (1995) 119906 = 1199060 + (119906119891 minus 1199060) times (1 minus 119896119896119895)

120572 1199060 119906119891 119896119895Del Castillo and Benitez model (1995) 119906 = 119906119891 times 1 minus exp[

1003816100381610038161003816100381611986211989510038161003816100381610038161003816119906119891 times (1 minus119896119895119896 )] 119906119891 119862119895 119896119895

Van Aerde model (1995) 119896 = 11198881 + 1198882 (119906119891 minus 119906) + 1198883 times 119906 1198881 1198882 1198883 119906119891

MacNicholas model (2008) 119906 = 119906119891 times 119896119899119895 minus 119896119899119896119899119895 + 119898 times 119896119899 119906119891 119896119895 119899119898

0 0 Q

Q

0

u

k

Qm

Qm

uf

um

km kj

Figure 1 The mapping of speed-density flow-density and speed-flow

simulation software to set up and change six parameters ofroad traffic including section length 119871 stretch section length119897 cart rate 120572 signal period 119862 the ratio of the time span ofleft-turn green signal to signal period 120582119897 and the ratio of thetime span of the straight green signal to signal period 120582119904 and

established 22 groups of parameters The simulation resultsshowed that the relationship between speed and densitypresents an inverse S curveTherefore a four-parameter logitmodel is proposed here to describe the speed-density inverseS curve and its expression is as follows

119906 = 119906min times 119906max minus 119906min1 + exp ((119873 minus 119873119908) 120579) (5)

where 119906min is the mean value of the minimum speed 119906max isthe mean value of the maximum speed119873 is the flow value ofa section 119873119908 is the flow value at the inflection point of thecurve and 120579 is a parameter determining curve shape

Then the data obtained from the 22 groups of simulationparameters were fitted The four parameters (119906min 119906max119873119908and 120579) were calculated for each simulation environment119873119908and 120579were respectively fitted in left-turn straight and right-turn cases and the fitting results are as follows119873119908

=

07146 left-turning

minus02231120572 + 01989 straight

minus000011119871 + 00078119897 minus 02113120572 + 02282 right-turning

120579 =

00664 left-turning

minus0000032119871 + 0085119897 straight

0045119897 right-turning

(6)

4 Mobile Information Systems

One-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(a) Graph of one-day data

0

20

40

60

80

Spee

d (k

mh

)

Three-day data

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(b) Graph of three-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250

Seven-day data

Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(c) Graph of seven-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

Fourteen-day data

(d) Graph of fourteen-day data

Figure 2 Comparison of six speed-density models

3 The Description Method of the Speed-Density Model for Interrupted Traffic Flow

31The Characteristics of the Data of Interrupted Traffic Coildata for one day three days seven days and fourteen dayswere selected to compare and analyze the discontinuous flow

data and the existing six speed-density models as shown inFigure 2 We found the following

(1) The six modelsrsquo performance was poor when the coildata of interrupted traffic flow were fitted illustratingthat although suitable for uninterrupted traffic flow

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0 5 10 15

0

50

100

150

200

Occupancy ()

Flow

(pcu

h)

Figure 3 Flow-occupancy graph of small density

they are unsuitable for describing the speed-densityrelation of interrupted traffic flow because of diversedata sources different traffic environments or otherfactors In contrast the logarithmic model gave thebest performance and the linear model gave the worstperformance

(2) The interval value of critical densities 119896119898 of one-daythree-day seven-day and fourteen-day data sets was[6256 pcukm 7123 pcukm] and most of the datawere located in 119896 lt 119896119898 range meaning unimpededflow data accounted for the absolute proportion sothe traffic flow of the location coil was in a state offlow most of the time

(3) When 119896 lt 119896119898 with the increase of density thevelocity decreased sharply when 119896 gt 119896119898 as thedensity increased the velocity decreased slowly andthe speed variation amplitude was very small

(4) When the density was small the speed had a largerange of values of which the largest was [23 kmh72 kmh] We filtered out the small-density data toobtain a scatter diagram of flow and occupancywhich were directly collected by a loop detector asshown in Figure 3 In Figure 3 it is obvious thatthe loop detector acquires large-range flow valuesfor the same occupancy value and the largest rangecan reach 100 pcuh Hence after calculating speedand density by density formula and velocity formulaspeed accordingly has a large range of values for thesame density in speed-density diagram

(5) In addition the density values were found to be neara number of points and the difference between adja-cent pointswas approximately equal to a certain value

From the above analysis we found that because of thebig differences between uninterrupted and interrupted trafficflow existing models suitable for uninterrupted traffic floware unsuited for describing the speed-density relation of

interrupted traffic flow What is more the flow collected bya loop detector has a large range of values Therefore for thespeed-density relation of interrupted trafficflowwemust finda new descriptive method

32 Description Method of Speed-Density Relationship forInterrupted Traffic Flow Because of the difference betweenthe uninterrupted and interrupted traffic flow and thevolatility of speed the speed-density relationship cannot beadequately described by a single model so we use two curves119906upper and 119906lower to describe the supremum and infimum ofvelocity values

119906upper = 119892upper (119880upper) 119906lower = 119892lower (119880lower) (7)

where119880upper and119880lower are respectively the upper and lowerbounds of velocity and 119892upper and 119892lower are fitting functions

Divide the density interval [119896min 119896max] into 119899 connectedintervals 1198961 1198962 119896119899 Partition data 119863 as 1198631 1198632 119863119899by density intervals and correspondingly get speed sets1198801 1198802 119880119899 causing that for any 119894 isin (1 2 119899) we have

119882(119880119894) gt 119882120572 (8)

where 119882(119880119894) is used test for 119880119894 with the Shapiro-Wilknormal test method Sort 119898 independent observations in119880119894 by nondescending order recorded as 1199091 1199092 119909119898 andconstruct the119882-test statistic

119882 = [sum119898119894=1 119886119894 times (119909119898+1minus119894 minus 119909119894)]2sum119898119894=1 119886119894 times (119909119894 minus 119909)2 (9)

where 119886119894 is the coefficient when sample size is 119898 When thepopulation distribution is normal distribution the value of119882 should be close to one 120572 quantile119882120572 of statistic119882 can beobtained by the look-up table method When119882 le 119882120572 theoriginal hypothesis should be rejected at the significant levelindicating that 119880119894 does not obey normal distribution when119882 gt 119882120572 the original hypothesis cannot be rejected and 119880119894satisfies normal distribution

Under the conditions of (8) for every 119894 isin (1 2 119899)extract the upper quantile 119906upper119894 and lower quantile 119906lower119894as the upper and lower critical values of speed for densityinterval 119896119894

119906upper119894 = 119902norm (uppermean (119880119894) sd (119880119894)) 119906lower119894 = 119902norm (lowermean (119880119894) sd (119880119894)) (10)

where 119902norm( ) is quantile function mean( ) calculates themean value of 119880119894 and sd( ) calculates the variance of 119880119894

Get the upper bound and lower bound sets

119880upper = 119906upper119894 119894 = 1 2 119899 119880lower = 119906lower119894 119894 = 1 2 119899 (11)

Fit 119880upper and 119880lower using the nonlinear least squaremethod The tabulated function 119906119894 = 119906(119896119894) 119894 = 1 2 119899 is

6 Mobile Information Systems

available by (10) Then we need to obtain the fitting function119892(119896) = 1198860 + 1198861 times 1198921(119896) + sdot sdot sdot + 119886119901 times 119892119901(119896) making the sum ofsquared deviations

119878 (1198860 1198861 119886119901) = 119899sum119894=1

[119892 (119896119894) minus 119906119894]2

= 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]2 (12)

Take theminimum of which 1198921(119896) 1198922(119896) 119892119901(119896) are 119901nonmergeable monomials of variable 119896 and 1198860 1198861 119886119901 arethe coefficients of monomials 119878 is a nonnegative polynomialof 1198860 1198861 119886119901 so there must be a minimum value Respec-tively calculate partial derivatives of 119878 for 1198860 1198861 119886119901 andmake them equal to zero

120597119878120597119886119894 = 0 119894 = 0 1 119901 (13)

Equation (13) is expanded as follows1205971198781205971198860 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]= 0

1205971198781205971198861 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 1198921 (119896119894) = 0

120597119878120597119886119901 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 119892119901 (119896119894) = 0

(14)

Continue to expand (14)

1198860 times 119899 + 1198861 times 119899sum119894=1

1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119899sum119894=1

119892119901 (119896119894) = 119899sum119894=1

119906119894

1198860 times 119899sum119894=1

1198921 (119896119894) + 1198861 times 119899sum119894=1

[1198921 (119896119894)]2 + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)] = 119899sum119894=1

[119906119894 times 1198921 (119896119894)]

1198860 times 119899sum119894=1

119892119901 (119896119894) + 1198861 times 119899sum119894=1

(1198921 (119896119894) times 119892119901 (119896119894)) + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894)]2 = 119899sum119894=1

[119906119894 times 119892119901 (119896119894)](15)

and get its matrix form

[[[[[[[[[[[[[[

119899 119899sum119894=1

1198921 (119896119894) sdot sdot sdot 119899sum119894=1

119892119901 (119896119894)119899sum119894=1

1198921 (119896119894) 119899sum119894=1

[1198921 (119896119894)]2 sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)]

119899sum119894=1

119892119901 (119896119894) 119899sum119894=1

[1198921 (119896119894) times 119892119901 (119896119894)] sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894)]2

]]]]]]]]]]]]]]

times[[[[[[[[

11988601198861119886119901

]]]]]]]]=

[[[[[[[[[[[[[[

119899sum119894=1

119906119894119899sum119894=1

[119906119894 times 1198921 (119896119894)]

119899sum119894=1

[119906119894 times 119892119901 (119896119894)]

]]]]]]]]]]]]]]

(16)

Solve (16) and 1198860 1198861 119886119901 are availableUsing the above least square method to fit 119880upper and119880lower respectively obtains

119906upper (119896) = 119892upper (119896 1198860 1198861 119886119901) 119906lower (119896) = 119892lower (119896 1198860 1198861 119886119901) (17)

and upper and lower curves 119906upper(119896) and 119906lower(119896) which isthe speed-density relation description model

4 Experiment and Analysis

The experiment data is collected by the coil detectors under-ground closed to Optical Valley Walking Street in WuhanChina Coil detectors collect data every 15minutes recordingtime flow occupancy and so forth as shown in Table 2

Use the method in [20 21] to calculate speed and densityand the ratio of the amount of data between twomodel curvesto the total amount of experiment data is used to describe theperformance of model The loop detector in the outer lanemeasures the traffic flow of straight and right-turning lanesand the loop detector in the inner lane measures the trafficflow of the left-turning laneThe traffic flow characteristics oftwo loop detectors must have certain differences Thereforeanalyze the coil data of both the outer lane and the inner laneto find the diversity of their speed-density relationship

41 Coil Data Analysis of the Outer Lane The experimentalsteps are as follows

Step 1 Analyze coil data of the outer lane andfind that densityvalues are clustered at a number of points 1198961 1198962 119896119899

Mobile Information Systems 7

Table 2 The data example

Date Week Flow Occupancy Minute id Hour20141120 4 132 7 00000 41751051 020141120 4 91 5 01500 41751051 020141120 4 98 7 03001 41751051 020141120 4 103 5 04501 41751051 020141120 4 77 6 10000 41751051 120141120 4 71 4 11500 41751051 120141120 4 64 3 13001 41751051 120141120 4 40 75 14501 41751051 1

where the mean value of the difference between the adjacentpoints is about 25 pcukm Divide density 119896 into a number ofintervals with length 25 pcukm by 1198961 1198962 119896119899Step 2 Correspondingly split data 119863 into small data sets1198631 1198632 119863119899 according to density segmentations and getdata sets of speed 1198801 1198802 119880119899Step 3 Execute a distribution test for 119880119894 where the resultshows that one data set is too small to meet the requirementsof the test Merge the adjacent density segments in Step 2to enlarge the amount of the small data set Redo thedistribution test for the new data set more than 80 ofwhich meets the normal distribution with totally 95 of thetotal data satisfying the normal distribution which makesit reasonable to consider all the small data set satisfying thenormal distribution

Step 4 Get two quantiles 119906upper119894 and 119906lower119894 of speed set 119880119894 asupper and lower critical values of velocity for density 119896119894Step 5 Then have upper and lower critical value set 119880upper =sum119899119894=1 119906upper119894 and 119880lower = sum119899119894=1 119906lower119894

Step 6 (fit 119906upper and 119906lower) Because the loop detector islocated near commercial street which has heavy trafficwe usethe logarithmic model to formulize the data

119906095 = 14204 times ln(216412119896 ) 119906005 = 7169 times ln(254497119896 )

(18)

Figure 4 shows the validation result of the speed-densitylogarithmic model of the outer lane when upper value =095 and lower value = 005 Equations (18) correspondinglyare the green and blue curves in Figure 4 which is thespeed-density model of interrupted traffic flow created bythe new description method Significant test results indicatethat 119875 values of two regression coefficients of two curves areminima (119875 lt 2119890 minus 16) which means that coefficients aresignificant and two log models constructed with density asthe independent variable are applied to estimate velocity asthe dependent variable

The coil data of the outer lane for two weeks fourweeks six weeks and eight weeks are respectively selected

095 quantile fractileFitting logarithmic model (upper)005 quantile fractileFitting logarithmic model (lower)

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Figure 4 Speed-density logarithmic model of the outside lane

and four groups of parameters are established for modelvalidation Table 3 gives the ratio of the data between twologarithmic curves to the total amount of data in each caseMake a longitudinal observation it is obvious that with uppervalue increasing and lower value decreasing the proportionincreases accordingly where amplitudes are obvious respec-tively 72 62 and 69 On the other hand the maintransverse trend is that the proportion increases along withthe increase of experiment data loosely where however six-week data has the best performance The above suggests thatthe two logarithmic models are able to describe the speed-density relation of the outer lane Figure 5 shows the fourgroupsrsquo validation results when upper value = 095 and lowervalue = 005

42 Coil Data Analysis of the Inner Lane We select coildata of the inner lane and follow Steps 1 to 5 as for theouter lane When fitting sets 119906upper and 119906lower at Step 6 wefind that the speed-density models proposed by scholars allhave poor performance with goodness of fit of less than 05

8 Mobile Information Systems

Table 3 Validation results of the model of the outer lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 619 658 662 661 650Lower value = 020Upper value = 085 689 731 736 733 722Lower value = 015Upper value = 090 745 788 804 797 784Lower value = 010Upper value = 095 842 851 867 853 853Lower value = 005

Two-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(a) Validation result for two-week data

Four-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(b) Validation result for four-week data

Six-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 5 Validation result of the speed-density logarithmic model of the outside lane

Mobile Information Systems 9

Table 4 Validation results of the model of the inner lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 800 779 813 788 795Lower value = 020Upper value = 085 820 823 852 831 832Lower value = 015Upper value = 090 838 849 864 853 851Lower value = 010Upper value = 095 890 903 889 898 895Lower value = 005

which suggests that a singlemodel cannot accurately describethe quantile set of the coil data Thus we consider using asegmentation model

In the density-flow curve there is a critical density 119896119898which is the density of maximum traffic flow as shown inFigure 1 When the density 119896 lt 119896119898 the traffic is in a stateof flow when 119896 gt 119896119898 the traffic flow gradually becomescrowded Therefore consider using 119896119898 as the critical value ofthe subsection

A density-flow curve is obtained by local polynomialregression fitting and the density value at the curve vertexis just 119896119898 Take 119896119898 as the critical value and piecewise analyze119906upper119896lt119896119898

119906lower119896lt119896119898

119906upper119896gt119896119898

and 119906lower119896gt119896119898

The analysis shows that thequantile set 119906upper

119896lt119896119898

and 119906lower119896lt119896119898

agrees with the exponentialmodel and the quantile set 119906upper

119896gt119896119898

and 119906lower119896gt119896119898

has goodagreement with the logarithmic model

119906095 = 69647 times 119890minus11989614449 + 12716 119896 lt 1198961198988227 times ln(254971119896 ) 119896 ge 119896119898

119906005 = 5108 times 119890minus11989610010 minus 2245 119896 lt 1198961198985337 times ln(243306119896 ) 119896 ge 119896119898

(19)

Figure 6 shows the fitting result of a segmentation modelof the outer lane when upper value = 095 and lower value= 005 and (19) are the models corresponding to the greencurve and blue curve in Figure 6 which is the speed-densitymodel of interrupted traffic flow via the new descriptionmethod 119875 value of each parameter is very small suggestingthe coefficient is very significant

The coil data of the inner lane for two weeks four weekssix weeks and eight weeks are respectively selected and fourgroups of parameters are established for themodel validationthe same as that for the outer lane Table 4 gives the ratio ofthe data between two logarithmic curves to the total amountof data in each case Comparing the result with that of theouter lane we find that the validation results of the model ofthe inner lane are better with greater ratio

Take a longitudinal observation similarly it is obviousthat with upper value increasing and lower value decreasingthe proportion increases accordingly where amplitudes aresmaller than that of outer lane respectively 37 19 and43 The main transverse trend is the same as outer lane

km

095 quantile fractileMultisession model (upper)005 quantile fractileMultisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

Figure 6 Speed-density multisession model of the inside lane

except the case of upper value = 095 and lower value = 005The result indicates that the two segmentation models aresuitable for describing the speed-density relation of the innerlane Figure 7 shows the four groupsrsquo validation results whenupper value = 095 and lower value = 005

43 Experimental Result Analysis

431 Difference between the Models of the Outer Lane andthe Inner Lane The loop detector of the outer lane measuresright-turning and straight lanes and the coil is located in aroad adjacent to a commercial pedestrian street with a heavyflow of people and traffic A logarithmic model is appliedto describe traffic flow with large density and therefore itis accepted that the coil data of the outer lane satisfy thelogarithmic model

The loop detector of the inner lane measures the left-turning lane which also has heavy traffic flow The speed-density relation of the inner lane does not satisfy the singlelog model but is suitable for the segmentation model The

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

2 Mobile Information Systems

under congested traffic conditions combined with the mini-mum safety spacing constraint and the experimental resultsshowed that the absolute error of this model was smallerthan that of other models fitting the traffic data of twofreeways Wang et al [7] proposed a family of speed-densitymodels with different numbers of parameters with importantphysical significance and got good performance in the finalexperiment

All of the above studies are based on continuous trafficflow data These data also called uninterrupted traffic floware traffic flow with no effect of external fixation factorssuch as freeway urban expressway and so forth Discon-tinuous traffic flow referred to as interrupted traffic flow isperiodically influenced by external fixation factors The mostcommon interrupted traffic flow is originated by signal lampsof urban intersections Because of the variety of vehicle typesthe periodic effect of signal lamps shunts in the canal sectionand other factors the characteristics of interrupted trafficflow are very complex compared with uninterrupted trafficflow In addition the city is still in a rapid increase in popula-tion and with the development of economy people are moreinclined to self-driving travel thus more and more vehiclesand more and more congestion in the city which leads to theincrease of travel time the growth of fuel consumption [8]the aggravation of environmental pollution and other awfulissues [9 10] Compared with the highway the urban roadhas a strong influence on the individual society and theenvironment Therefore further study of the characteristicsof interrupted traffic flow to provide support formanagementdecisions is particularly important

Research on interrupted traffic flow has attracted a lot ofattention [11ndash15] Many scholars see traffic flow located at acertain distance from the intersection as continuous trafficflow believing that it can be described by continuous trafficflowmodels Some of the literature [16 17] suggests howeverthat because of the short distance between intersections in thecity and the influence of signal lamps there are differencesbetween traffic flow located at a certain distance from theintersection and the traffic flow of freeways Because trafficdata are difficult to obtain and for other objective reasonsonly a few scholars focus on the speed-density model ofdiscontinuous traffic flow Wang et al [18] introduced a four-parameter logit model for complete data fitting and estab-lished a speed-density logit model for left-turning straightand right-turning trafficflowHowever the experimental datawere obtained by VISSIM simulation and the simulationparameters were not accurate enough to depict the complexcity road environment so the experimental results havecertain limitationsWang et al [19] thought that the stochasticmodel would contain more traffic information and putforward the stochastic speed-density model This stochasticmodel can generate a probabilistic traffic flowmodel and canachieve real-time traffic prediction

In order to provide favorable data analysis and presen-tation for city traffic thus to provide decision support forintelligent transportation characterizing the speed-densityrelationship of interrupted traffic flow more accurately is fullof importance By analyzing a large amount of data we pro-pose a description method for a speed-density relationship

model which is suitable for discontinuous traffic flow usingthe upper and lower curves to describe the upper and lowerbounds of velocity values Because of the discrepant charac-teristics of the traffic flow in the outer and inner lanes the coildata of the outer and inner lanes are analyzed and verified

2 Speed-Density Model

Three basic parameters (flow 119902 speed 119906 and density 119896) arethe core content of the traffic flow model The three have thefollowing relationship

119902 = 119896 times 119906 (1)

that is flow is the product of density and speed Therelationship between two parameters of the three is of greatsignificance in traffic flow and the relationship betweenspeed and density has received a lot of research attentionGreenshields et al was an early researcher who proposed thespeed-density linear relationship [1]

119906 = 119906119891 times (1 minus 119896119896119895) (2)

where 119906119891 is the speed of free flow that is the speed of vehiclesunimpeded when the traffic density tends to zero and 119896119895 isthe density of block flow that is the density when the trafficflow is blocked and cannot move As shown in Figure 1 when119896 = 0 the speed can reach the theoretical maximum valuenamely the free flow velocity 119906119891 The area surrounded by theabscissa the ordinate of any point on the line and the coordi-nate origin is the traffic flow

Equation (2) can change to

119896 = 119896119895 times (1 minus 119906119906119891) (3)

Respectively introduce (2) and (3) into (1) and we get

119902 = 119906119891 times (119896 minus 1198962119896119895) 119902 = 119896119895 times (119906 minus 1199062119906119891)

(4)

Equations (4) illustrate that 119902-119896 and 119902-119906 are quadraticfunction relations as shown in Figure 1

The linear model is too simple and there are manydeficiencies In order to improve the model scholars haveproposedmodels based on the linear model but with a higherdegree of accuracy Table 1 lists results for the speed-densitymodel including the Greenberg model Underwood modelNorthwestern model Newellrsquos model Pipes-Munjal modelDrew model Modified Greenshields model Del Castillo andBenitez model Van Aerde model MacNicholas modelThesemodels with the parameters of important physical meaningprovide good results

Wang et al [19] established a speed-density logit proba-bility model with four parameters Wang et al used VISSIM

Mobile Information Systems 3

Table 1 Speed-density models

Model Function Parameters

Greenshields model (1935) 119906 = 119906119891 times (1 minus 119896119896119895) 119906119891 119896119895Greenberg model (1959) 119906 = 119906119898 times ln(119896119895119896 ) 119906119898 119896119895Underwood model (1961) 119906 = 119906119891 times exp(minus 119896119896119898 ) 119906119891 119896119898Newellrsquos model (1961) 119906 = 119906119891 times 1 minus exp[minus 120582119906119891 times (

1119896 minus 1119896119895)] 119906119891 120582 119896119895Northwestern model (1967) 119906 = 119906119891 times exp[minus12 times ( 1198961198960 )

2] 119906119891 1198960Pipes-Munjal model (1967) 119906 = 119906119891 times [1 minus ( 119896119896119895)

119899] 119906119891 119896119895Drew model (1968) 119906 = 119906119891 times [1 minus ( 119896119896119895)

119899+12] 119906119891 119896119895Modified Greenshields model (1995) 119906 = 1199060 + (119906119891 minus 1199060) times (1 minus 119896119896119895)

120572 1199060 119906119891 119896119895Del Castillo and Benitez model (1995) 119906 = 119906119891 times 1 minus exp[

1003816100381610038161003816100381611986211989510038161003816100381610038161003816119906119891 times (1 minus119896119895119896 )] 119906119891 119862119895 119896119895

Van Aerde model (1995) 119896 = 11198881 + 1198882 (119906119891 minus 119906) + 1198883 times 119906 1198881 1198882 1198883 119906119891

MacNicholas model (2008) 119906 = 119906119891 times 119896119899119895 minus 119896119899119896119899119895 + 119898 times 119896119899 119906119891 119896119895 119899119898

0 0 Q

Q

0

u

k

Qm

Qm

uf

um

km kj

Figure 1 The mapping of speed-density flow-density and speed-flow

simulation software to set up and change six parameters ofroad traffic including section length 119871 stretch section length119897 cart rate 120572 signal period 119862 the ratio of the time span ofleft-turn green signal to signal period 120582119897 and the ratio of thetime span of the straight green signal to signal period 120582119904 and

established 22 groups of parameters The simulation resultsshowed that the relationship between speed and densitypresents an inverse S curveTherefore a four-parameter logitmodel is proposed here to describe the speed-density inverseS curve and its expression is as follows

119906 = 119906min times 119906max minus 119906min1 + exp ((119873 minus 119873119908) 120579) (5)

where 119906min is the mean value of the minimum speed 119906max isthe mean value of the maximum speed119873 is the flow value ofa section 119873119908 is the flow value at the inflection point of thecurve and 120579 is a parameter determining curve shape

Then the data obtained from the 22 groups of simulationparameters were fitted The four parameters (119906min 119906max119873119908and 120579) were calculated for each simulation environment119873119908and 120579were respectively fitted in left-turn straight and right-turn cases and the fitting results are as follows119873119908

=

07146 left-turning

minus02231120572 + 01989 straight

minus000011119871 + 00078119897 minus 02113120572 + 02282 right-turning

120579 =

00664 left-turning

minus0000032119871 + 0085119897 straight

0045119897 right-turning

(6)

4 Mobile Information Systems

One-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(a) Graph of one-day data

0

20

40

60

80

Spee

d (k

mh

)

Three-day data

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(b) Graph of three-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250

Seven-day data

Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(c) Graph of seven-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

Fourteen-day data

(d) Graph of fourteen-day data

Figure 2 Comparison of six speed-density models

3 The Description Method of the Speed-Density Model for Interrupted Traffic Flow

31The Characteristics of the Data of Interrupted Traffic Coildata for one day three days seven days and fourteen dayswere selected to compare and analyze the discontinuous flow

data and the existing six speed-density models as shown inFigure 2 We found the following

(1) The six modelsrsquo performance was poor when the coildata of interrupted traffic flow were fitted illustratingthat although suitable for uninterrupted traffic flow

Mobile Information Systems 5

0 5 10 15

0

50

100

150

200

Occupancy ()

Flow

(pcu

h)

Figure 3 Flow-occupancy graph of small density

they are unsuitable for describing the speed-densityrelation of interrupted traffic flow because of diversedata sources different traffic environments or otherfactors In contrast the logarithmic model gave thebest performance and the linear model gave the worstperformance

(2) The interval value of critical densities 119896119898 of one-daythree-day seven-day and fourteen-day data sets was[6256 pcukm 7123 pcukm] and most of the datawere located in 119896 lt 119896119898 range meaning unimpededflow data accounted for the absolute proportion sothe traffic flow of the location coil was in a state offlow most of the time

(3) When 119896 lt 119896119898 with the increase of density thevelocity decreased sharply when 119896 gt 119896119898 as thedensity increased the velocity decreased slowly andthe speed variation amplitude was very small

(4) When the density was small the speed had a largerange of values of which the largest was [23 kmh72 kmh] We filtered out the small-density data toobtain a scatter diagram of flow and occupancywhich were directly collected by a loop detector asshown in Figure 3 In Figure 3 it is obvious thatthe loop detector acquires large-range flow valuesfor the same occupancy value and the largest rangecan reach 100 pcuh Hence after calculating speedand density by density formula and velocity formulaspeed accordingly has a large range of values for thesame density in speed-density diagram

(5) In addition the density values were found to be neara number of points and the difference between adja-cent pointswas approximately equal to a certain value

From the above analysis we found that because of thebig differences between uninterrupted and interrupted trafficflow existing models suitable for uninterrupted traffic floware unsuited for describing the speed-density relation of

interrupted traffic flow What is more the flow collected bya loop detector has a large range of values Therefore for thespeed-density relation of interrupted trafficflowwemust finda new descriptive method

32 Description Method of Speed-Density Relationship forInterrupted Traffic Flow Because of the difference betweenthe uninterrupted and interrupted traffic flow and thevolatility of speed the speed-density relationship cannot beadequately described by a single model so we use two curves119906upper and 119906lower to describe the supremum and infimum ofvelocity values

119906upper = 119892upper (119880upper) 119906lower = 119892lower (119880lower) (7)

where119880upper and119880lower are respectively the upper and lowerbounds of velocity and 119892upper and 119892lower are fitting functions

Divide the density interval [119896min 119896max] into 119899 connectedintervals 1198961 1198962 119896119899 Partition data 119863 as 1198631 1198632 119863119899by density intervals and correspondingly get speed sets1198801 1198802 119880119899 causing that for any 119894 isin (1 2 119899) we have

119882(119880119894) gt 119882120572 (8)

where 119882(119880119894) is used test for 119880119894 with the Shapiro-Wilknormal test method Sort 119898 independent observations in119880119894 by nondescending order recorded as 1199091 1199092 119909119898 andconstruct the119882-test statistic

119882 = [sum119898119894=1 119886119894 times (119909119898+1minus119894 minus 119909119894)]2sum119898119894=1 119886119894 times (119909119894 minus 119909)2 (9)

where 119886119894 is the coefficient when sample size is 119898 When thepopulation distribution is normal distribution the value of119882 should be close to one 120572 quantile119882120572 of statistic119882 can beobtained by the look-up table method When119882 le 119882120572 theoriginal hypothesis should be rejected at the significant levelindicating that 119880119894 does not obey normal distribution when119882 gt 119882120572 the original hypothesis cannot be rejected and 119880119894satisfies normal distribution

Under the conditions of (8) for every 119894 isin (1 2 119899)extract the upper quantile 119906upper119894 and lower quantile 119906lower119894as the upper and lower critical values of speed for densityinterval 119896119894

119906upper119894 = 119902norm (uppermean (119880119894) sd (119880119894)) 119906lower119894 = 119902norm (lowermean (119880119894) sd (119880119894)) (10)

where 119902norm( ) is quantile function mean( ) calculates themean value of 119880119894 and sd( ) calculates the variance of 119880119894

Get the upper bound and lower bound sets

119880upper = 119906upper119894 119894 = 1 2 119899 119880lower = 119906lower119894 119894 = 1 2 119899 (11)

Fit 119880upper and 119880lower using the nonlinear least squaremethod The tabulated function 119906119894 = 119906(119896119894) 119894 = 1 2 119899 is

6 Mobile Information Systems

available by (10) Then we need to obtain the fitting function119892(119896) = 1198860 + 1198861 times 1198921(119896) + sdot sdot sdot + 119886119901 times 119892119901(119896) making the sum ofsquared deviations

119878 (1198860 1198861 119886119901) = 119899sum119894=1

[119892 (119896119894) minus 119906119894]2

= 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]2 (12)

Take theminimum of which 1198921(119896) 1198922(119896) 119892119901(119896) are 119901nonmergeable monomials of variable 119896 and 1198860 1198861 119886119901 arethe coefficients of monomials 119878 is a nonnegative polynomialof 1198860 1198861 119886119901 so there must be a minimum value Respec-tively calculate partial derivatives of 119878 for 1198860 1198861 119886119901 andmake them equal to zero

120597119878120597119886119894 = 0 119894 = 0 1 119901 (13)

Equation (13) is expanded as follows1205971198781205971198860 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]= 0

1205971198781205971198861 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 1198921 (119896119894) = 0

120597119878120597119886119901 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 119892119901 (119896119894) = 0

(14)

Continue to expand (14)

1198860 times 119899 + 1198861 times 119899sum119894=1

1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119899sum119894=1

119892119901 (119896119894) = 119899sum119894=1

119906119894

1198860 times 119899sum119894=1

1198921 (119896119894) + 1198861 times 119899sum119894=1

[1198921 (119896119894)]2 + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)] = 119899sum119894=1

[119906119894 times 1198921 (119896119894)]

1198860 times 119899sum119894=1

119892119901 (119896119894) + 1198861 times 119899sum119894=1

(1198921 (119896119894) times 119892119901 (119896119894)) + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894)]2 = 119899sum119894=1

[119906119894 times 119892119901 (119896119894)](15)

and get its matrix form

[[[[[[[[[[[[[[

119899 119899sum119894=1

1198921 (119896119894) sdot sdot sdot 119899sum119894=1

119892119901 (119896119894)119899sum119894=1

1198921 (119896119894) 119899sum119894=1

[1198921 (119896119894)]2 sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)]

119899sum119894=1

119892119901 (119896119894) 119899sum119894=1

[1198921 (119896119894) times 119892119901 (119896119894)] sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894)]2

]]]]]]]]]]]]]]

times[[[[[[[[

11988601198861119886119901

]]]]]]]]=

[[[[[[[[[[[[[[

119899sum119894=1

119906119894119899sum119894=1

[119906119894 times 1198921 (119896119894)]

119899sum119894=1

[119906119894 times 119892119901 (119896119894)]

]]]]]]]]]]]]]]

(16)

Solve (16) and 1198860 1198861 119886119901 are availableUsing the above least square method to fit 119880upper and119880lower respectively obtains

119906upper (119896) = 119892upper (119896 1198860 1198861 119886119901) 119906lower (119896) = 119892lower (119896 1198860 1198861 119886119901) (17)

and upper and lower curves 119906upper(119896) and 119906lower(119896) which isthe speed-density relation description model

4 Experiment and Analysis

The experiment data is collected by the coil detectors under-ground closed to Optical Valley Walking Street in WuhanChina Coil detectors collect data every 15minutes recordingtime flow occupancy and so forth as shown in Table 2

Use the method in [20 21] to calculate speed and densityand the ratio of the amount of data between twomodel curvesto the total amount of experiment data is used to describe theperformance of model The loop detector in the outer lanemeasures the traffic flow of straight and right-turning lanesand the loop detector in the inner lane measures the trafficflow of the left-turning laneThe traffic flow characteristics oftwo loop detectors must have certain differences Thereforeanalyze the coil data of both the outer lane and the inner laneto find the diversity of their speed-density relationship

41 Coil Data Analysis of the Outer Lane The experimentalsteps are as follows

Step 1 Analyze coil data of the outer lane andfind that densityvalues are clustered at a number of points 1198961 1198962 119896119899

Mobile Information Systems 7

Table 2 The data example

Date Week Flow Occupancy Minute id Hour20141120 4 132 7 00000 41751051 020141120 4 91 5 01500 41751051 020141120 4 98 7 03001 41751051 020141120 4 103 5 04501 41751051 020141120 4 77 6 10000 41751051 120141120 4 71 4 11500 41751051 120141120 4 64 3 13001 41751051 120141120 4 40 75 14501 41751051 1

where the mean value of the difference between the adjacentpoints is about 25 pcukm Divide density 119896 into a number ofintervals with length 25 pcukm by 1198961 1198962 119896119899Step 2 Correspondingly split data 119863 into small data sets1198631 1198632 119863119899 according to density segmentations and getdata sets of speed 1198801 1198802 119880119899Step 3 Execute a distribution test for 119880119894 where the resultshows that one data set is too small to meet the requirementsof the test Merge the adjacent density segments in Step 2to enlarge the amount of the small data set Redo thedistribution test for the new data set more than 80 ofwhich meets the normal distribution with totally 95 of thetotal data satisfying the normal distribution which makesit reasonable to consider all the small data set satisfying thenormal distribution

Step 4 Get two quantiles 119906upper119894 and 119906lower119894 of speed set 119880119894 asupper and lower critical values of velocity for density 119896119894Step 5 Then have upper and lower critical value set 119880upper =sum119899119894=1 119906upper119894 and 119880lower = sum119899119894=1 119906lower119894

Step 6 (fit 119906upper and 119906lower) Because the loop detector islocated near commercial street which has heavy trafficwe usethe logarithmic model to formulize the data

119906095 = 14204 times ln(216412119896 ) 119906005 = 7169 times ln(254497119896 )

(18)

Figure 4 shows the validation result of the speed-densitylogarithmic model of the outer lane when upper value =095 and lower value = 005 Equations (18) correspondinglyare the green and blue curves in Figure 4 which is thespeed-density model of interrupted traffic flow created bythe new description method Significant test results indicatethat 119875 values of two regression coefficients of two curves areminima (119875 lt 2119890 minus 16) which means that coefficients aresignificant and two log models constructed with density asthe independent variable are applied to estimate velocity asthe dependent variable

The coil data of the outer lane for two weeks fourweeks six weeks and eight weeks are respectively selected

095 quantile fractileFitting logarithmic model (upper)005 quantile fractileFitting logarithmic model (lower)

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Figure 4 Speed-density logarithmic model of the outside lane

and four groups of parameters are established for modelvalidation Table 3 gives the ratio of the data between twologarithmic curves to the total amount of data in each caseMake a longitudinal observation it is obvious that with uppervalue increasing and lower value decreasing the proportionincreases accordingly where amplitudes are obvious respec-tively 72 62 and 69 On the other hand the maintransverse trend is that the proportion increases along withthe increase of experiment data loosely where however six-week data has the best performance The above suggests thatthe two logarithmic models are able to describe the speed-density relation of the outer lane Figure 5 shows the fourgroupsrsquo validation results when upper value = 095 and lowervalue = 005

42 Coil Data Analysis of the Inner Lane We select coildata of the inner lane and follow Steps 1 to 5 as for theouter lane When fitting sets 119906upper and 119906lower at Step 6 wefind that the speed-density models proposed by scholars allhave poor performance with goodness of fit of less than 05

8 Mobile Information Systems

Table 3 Validation results of the model of the outer lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 619 658 662 661 650Lower value = 020Upper value = 085 689 731 736 733 722Lower value = 015Upper value = 090 745 788 804 797 784Lower value = 010Upper value = 095 842 851 867 853 853Lower value = 005

Two-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(a) Validation result for two-week data

Four-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(b) Validation result for four-week data

Six-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 5 Validation result of the speed-density logarithmic model of the outside lane

Mobile Information Systems 9

Table 4 Validation results of the model of the inner lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 800 779 813 788 795Lower value = 020Upper value = 085 820 823 852 831 832Lower value = 015Upper value = 090 838 849 864 853 851Lower value = 010Upper value = 095 890 903 889 898 895Lower value = 005

which suggests that a singlemodel cannot accurately describethe quantile set of the coil data Thus we consider using asegmentation model

In the density-flow curve there is a critical density 119896119898which is the density of maximum traffic flow as shown inFigure 1 When the density 119896 lt 119896119898 the traffic is in a stateof flow when 119896 gt 119896119898 the traffic flow gradually becomescrowded Therefore consider using 119896119898 as the critical value ofthe subsection

A density-flow curve is obtained by local polynomialregression fitting and the density value at the curve vertexis just 119896119898 Take 119896119898 as the critical value and piecewise analyze119906upper119896lt119896119898

119906lower119896lt119896119898

119906upper119896gt119896119898

and 119906lower119896gt119896119898

The analysis shows that thequantile set 119906upper

119896lt119896119898

and 119906lower119896lt119896119898

agrees with the exponentialmodel and the quantile set 119906upper

119896gt119896119898

and 119906lower119896gt119896119898

has goodagreement with the logarithmic model

119906095 = 69647 times 119890minus11989614449 + 12716 119896 lt 1198961198988227 times ln(254971119896 ) 119896 ge 119896119898

119906005 = 5108 times 119890minus11989610010 minus 2245 119896 lt 1198961198985337 times ln(243306119896 ) 119896 ge 119896119898

(19)

Figure 6 shows the fitting result of a segmentation modelof the outer lane when upper value = 095 and lower value= 005 and (19) are the models corresponding to the greencurve and blue curve in Figure 6 which is the speed-densitymodel of interrupted traffic flow via the new descriptionmethod 119875 value of each parameter is very small suggestingthe coefficient is very significant

The coil data of the inner lane for two weeks four weekssix weeks and eight weeks are respectively selected and fourgroups of parameters are established for themodel validationthe same as that for the outer lane Table 4 gives the ratio ofthe data between two logarithmic curves to the total amountof data in each case Comparing the result with that of theouter lane we find that the validation results of the model ofthe inner lane are better with greater ratio

Take a longitudinal observation similarly it is obviousthat with upper value increasing and lower value decreasingthe proportion increases accordingly where amplitudes aresmaller than that of outer lane respectively 37 19 and43 The main transverse trend is the same as outer lane

km

095 quantile fractileMultisession model (upper)005 quantile fractileMultisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

Figure 6 Speed-density multisession model of the inside lane

except the case of upper value = 095 and lower value = 005The result indicates that the two segmentation models aresuitable for describing the speed-density relation of the innerlane Figure 7 shows the four groupsrsquo validation results whenupper value = 095 and lower value = 005

43 Experimental Result Analysis

431 Difference between the Models of the Outer Lane andthe Inner Lane The loop detector of the outer lane measuresright-turning and straight lanes and the coil is located in aroad adjacent to a commercial pedestrian street with a heavyflow of people and traffic A logarithmic model is appliedto describe traffic flow with large density and therefore itis accepted that the coil data of the outer lane satisfy thelogarithmic model

The loop detector of the inner lane measures the left-turning lane which also has heavy traffic flow The speed-density relation of the inner lane does not satisfy the singlelog model but is suitable for the segmentation model The

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Advances in

FuzzySystems

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Volume 2014

International Journal of

ReconfigurableComputing

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

Biomedical Imaging

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ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 3

Table 1 Speed-density models

Model Function Parameters

Greenshields model (1935) 119906 = 119906119891 times (1 minus 119896119896119895) 119906119891 119896119895Greenberg model (1959) 119906 = 119906119898 times ln(119896119895119896 ) 119906119898 119896119895Underwood model (1961) 119906 = 119906119891 times exp(minus 119896119896119898 ) 119906119891 119896119898Newellrsquos model (1961) 119906 = 119906119891 times 1 minus exp[minus 120582119906119891 times (

1119896 minus 1119896119895)] 119906119891 120582 119896119895Northwestern model (1967) 119906 = 119906119891 times exp[minus12 times ( 1198961198960 )

2] 119906119891 1198960Pipes-Munjal model (1967) 119906 = 119906119891 times [1 minus ( 119896119896119895)

119899] 119906119891 119896119895Drew model (1968) 119906 = 119906119891 times [1 minus ( 119896119896119895)

119899+12] 119906119891 119896119895Modified Greenshields model (1995) 119906 = 1199060 + (119906119891 minus 1199060) times (1 minus 119896119896119895)

120572 1199060 119906119891 119896119895Del Castillo and Benitez model (1995) 119906 = 119906119891 times 1 minus exp[

1003816100381610038161003816100381611986211989510038161003816100381610038161003816119906119891 times (1 minus119896119895119896 )] 119906119891 119862119895 119896119895

Van Aerde model (1995) 119896 = 11198881 + 1198882 (119906119891 minus 119906) + 1198883 times 119906 1198881 1198882 1198883 119906119891

MacNicholas model (2008) 119906 = 119906119891 times 119896119899119895 minus 119896119899119896119899119895 + 119898 times 119896119899 119906119891 119896119895 119899119898

0 0 Q

Q

0

u

k

Qm

Qm

uf

um

km kj

Figure 1 The mapping of speed-density flow-density and speed-flow

simulation software to set up and change six parameters ofroad traffic including section length 119871 stretch section length119897 cart rate 120572 signal period 119862 the ratio of the time span ofleft-turn green signal to signal period 120582119897 and the ratio of thetime span of the straight green signal to signal period 120582119904 and

established 22 groups of parameters The simulation resultsshowed that the relationship between speed and densitypresents an inverse S curveTherefore a four-parameter logitmodel is proposed here to describe the speed-density inverseS curve and its expression is as follows

119906 = 119906min times 119906max minus 119906min1 + exp ((119873 minus 119873119908) 120579) (5)

where 119906min is the mean value of the minimum speed 119906max isthe mean value of the maximum speed119873 is the flow value ofa section 119873119908 is the flow value at the inflection point of thecurve and 120579 is a parameter determining curve shape

Then the data obtained from the 22 groups of simulationparameters were fitted The four parameters (119906min 119906max119873119908and 120579) were calculated for each simulation environment119873119908and 120579were respectively fitted in left-turn straight and right-turn cases and the fitting results are as follows119873119908

=

07146 left-turning

minus02231120572 + 01989 straight

minus000011119871 + 00078119897 minus 02113120572 + 02282 right-turning

120579 =

00664 left-turning

minus0000032119871 + 0085119897 straight

0045119897 right-turning

(6)

4 Mobile Information Systems

One-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(a) Graph of one-day data

0

20

40

60

80

Spee

d (k

mh

)

Three-day data

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(b) Graph of three-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250

Seven-day data

Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(c) Graph of seven-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

Fourteen-day data

(d) Graph of fourteen-day data

Figure 2 Comparison of six speed-density models

3 The Description Method of the Speed-Density Model for Interrupted Traffic Flow

31The Characteristics of the Data of Interrupted Traffic Coildata for one day three days seven days and fourteen dayswere selected to compare and analyze the discontinuous flow

data and the existing six speed-density models as shown inFigure 2 We found the following

(1) The six modelsrsquo performance was poor when the coildata of interrupted traffic flow were fitted illustratingthat although suitable for uninterrupted traffic flow

Mobile Information Systems 5

0 5 10 15

0

50

100

150

200

Occupancy ()

Flow

(pcu

h)

Figure 3 Flow-occupancy graph of small density

they are unsuitable for describing the speed-densityrelation of interrupted traffic flow because of diversedata sources different traffic environments or otherfactors In contrast the logarithmic model gave thebest performance and the linear model gave the worstperformance

(2) The interval value of critical densities 119896119898 of one-daythree-day seven-day and fourteen-day data sets was[6256 pcukm 7123 pcukm] and most of the datawere located in 119896 lt 119896119898 range meaning unimpededflow data accounted for the absolute proportion sothe traffic flow of the location coil was in a state offlow most of the time

(3) When 119896 lt 119896119898 with the increase of density thevelocity decreased sharply when 119896 gt 119896119898 as thedensity increased the velocity decreased slowly andthe speed variation amplitude was very small

(4) When the density was small the speed had a largerange of values of which the largest was [23 kmh72 kmh] We filtered out the small-density data toobtain a scatter diagram of flow and occupancywhich were directly collected by a loop detector asshown in Figure 3 In Figure 3 it is obvious thatthe loop detector acquires large-range flow valuesfor the same occupancy value and the largest rangecan reach 100 pcuh Hence after calculating speedand density by density formula and velocity formulaspeed accordingly has a large range of values for thesame density in speed-density diagram

(5) In addition the density values were found to be neara number of points and the difference between adja-cent pointswas approximately equal to a certain value

From the above analysis we found that because of thebig differences between uninterrupted and interrupted trafficflow existing models suitable for uninterrupted traffic floware unsuited for describing the speed-density relation of

interrupted traffic flow What is more the flow collected bya loop detector has a large range of values Therefore for thespeed-density relation of interrupted trafficflowwemust finda new descriptive method

32 Description Method of Speed-Density Relationship forInterrupted Traffic Flow Because of the difference betweenthe uninterrupted and interrupted traffic flow and thevolatility of speed the speed-density relationship cannot beadequately described by a single model so we use two curves119906upper and 119906lower to describe the supremum and infimum ofvelocity values

119906upper = 119892upper (119880upper) 119906lower = 119892lower (119880lower) (7)

where119880upper and119880lower are respectively the upper and lowerbounds of velocity and 119892upper and 119892lower are fitting functions

Divide the density interval [119896min 119896max] into 119899 connectedintervals 1198961 1198962 119896119899 Partition data 119863 as 1198631 1198632 119863119899by density intervals and correspondingly get speed sets1198801 1198802 119880119899 causing that for any 119894 isin (1 2 119899) we have

119882(119880119894) gt 119882120572 (8)

where 119882(119880119894) is used test for 119880119894 with the Shapiro-Wilknormal test method Sort 119898 independent observations in119880119894 by nondescending order recorded as 1199091 1199092 119909119898 andconstruct the119882-test statistic

119882 = [sum119898119894=1 119886119894 times (119909119898+1minus119894 minus 119909119894)]2sum119898119894=1 119886119894 times (119909119894 minus 119909)2 (9)

where 119886119894 is the coefficient when sample size is 119898 When thepopulation distribution is normal distribution the value of119882 should be close to one 120572 quantile119882120572 of statistic119882 can beobtained by the look-up table method When119882 le 119882120572 theoriginal hypothesis should be rejected at the significant levelindicating that 119880119894 does not obey normal distribution when119882 gt 119882120572 the original hypothesis cannot be rejected and 119880119894satisfies normal distribution

Under the conditions of (8) for every 119894 isin (1 2 119899)extract the upper quantile 119906upper119894 and lower quantile 119906lower119894as the upper and lower critical values of speed for densityinterval 119896119894

119906upper119894 = 119902norm (uppermean (119880119894) sd (119880119894)) 119906lower119894 = 119902norm (lowermean (119880119894) sd (119880119894)) (10)

where 119902norm( ) is quantile function mean( ) calculates themean value of 119880119894 and sd( ) calculates the variance of 119880119894

Get the upper bound and lower bound sets

119880upper = 119906upper119894 119894 = 1 2 119899 119880lower = 119906lower119894 119894 = 1 2 119899 (11)

Fit 119880upper and 119880lower using the nonlinear least squaremethod The tabulated function 119906119894 = 119906(119896119894) 119894 = 1 2 119899 is

6 Mobile Information Systems

available by (10) Then we need to obtain the fitting function119892(119896) = 1198860 + 1198861 times 1198921(119896) + sdot sdot sdot + 119886119901 times 119892119901(119896) making the sum ofsquared deviations

119878 (1198860 1198861 119886119901) = 119899sum119894=1

[119892 (119896119894) minus 119906119894]2

= 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]2 (12)

Take theminimum of which 1198921(119896) 1198922(119896) 119892119901(119896) are 119901nonmergeable monomials of variable 119896 and 1198860 1198861 119886119901 arethe coefficients of monomials 119878 is a nonnegative polynomialof 1198860 1198861 119886119901 so there must be a minimum value Respec-tively calculate partial derivatives of 119878 for 1198860 1198861 119886119901 andmake them equal to zero

120597119878120597119886119894 = 0 119894 = 0 1 119901 (13)

Equation (13) is expanded as follows1205971198781205971198860 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]= 0

1205971198781205971198861 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 1198921 (119896119894) = 0

120597119878120597119886119901 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 119892119901 (119896119894) = 0

(14)

Continue to expand (14)

1198860 times 119899 + 1198861 times 119899sum119894=1

1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119899sum119894=1

119892119901 (119896119894) = 119899sum119894=1

119906119894

1198860 times 119899sum119894=1

1198921 (119896119894) + 1198861 times 119899sum119894=1

[1198921 (119896119894)]2 + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)] = 119899sum119894=1

[119906119894 times 1198921 (119896119894)]

1198860 times 119899sum119894=1

119892119901 (119896119894) + 1198861 times 119899sum119894=1

(1198921 (119896119894) times 119892119901 (119896119894)) + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894)]2 = 119899sum119894=1

[119906119894 times 119892119901 (119896119894)](15)

and get its matrix form

[[[[[[[[[[[[[[

119899 119899sum119894=1

1198921 (119896119894) sdot sdot sdot 119899sum119894=1

119892119901 (119896119894)119899sum119894=1

1198921 (119896119894) 119899sum119894=1

[1198921 (119896119894)]2 sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)]

119899sum119894=1

119892119901 (119896119894) 119899sum119894=1

[1198921 (119896119894) times 119892119901 (119896119894)] sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894)]2

]]]]]]]]]]]]]]

times[[[[[[[[

11988601198861119886119901

]]]]]]]]=

[[[[[[[[[[[[[[

119899sum119894=1

119906119894119899sum119894=1

[119906119894 times 1198921 (119896119894)]

119899sum119894=1

[119906119894 times 119892119901 (119896119894)]

]]]]]]]]]]]]]]

(16)

Solve (16) and 1198860 1198861 119886119901 are availableUsing the above least square method to fit 119880upper and119880lower respectively obtains

119906upper (119896) = 119892upper (119896 1198860 1198861 119886119901) 119906lower (119896) = 119892lower (119896 1198860 1198861 119886119901) (17)

and upper and lower curves 119906upper(119896) and 119906lower(119896) which isthe speed-density relation description model

4 Experiment and Analysis

The experiment data is collected by the coil detectors under-ground closed to Optical Valley Walking Street in WuhanChina Coil detectors collect data every 15minutes recordingtime flow occupancy and so forth as shown in Table 2

Use the method in [20 21] to calculate speed and densityand the ratio of the amount of data between twomodel curvesto the total amount of experiment data is used to describe theperformance of model The loop detector in the outer lanemeasures the traffic flow of straight and right-turning lanesand the loop detector in the inner lane measures the trafficflow of the left-turning laneThe traffic flow characteristics oftwo loop detectors must have certain differences Thereforeanalyze the coil data of both the outer lane and the inner laneto find the diversity of their speed-density relationship

41 Coil Data Analysis of the Outer Lane The experimentalsteps are as follows

Step 1 Analyze coil data of the outer lane andfind that densityvalues are clustered at a number of points 1198961 1198962 119896119899

Mobile Information Systems 7

Table 2 The data example

Date Week Flow Occupancy Minute id Hour20141120 4 132 7 00000 41751051 020141120 4 91 5 01500 41751051 020141120 4 98 7 03001 41751051 020141120 4 103 5 04501 41751051 020141120 4 77 6 10000 41751051 120141120 4 71 4 11500 41751051 120141120 4 64 3 13001 41751051 120141120 4 40 75 14501 41751051 1

where the mean value of the difference between the adjacentpoints is about 25 pcukm Divide density 119896 into a number ofintervals with length 25 pcukm by 1198961 1198962 119896119899Step 2 Correspondingly split data 119863 into small data sets1198631 1198632 119863119899 according to density segmentations and getdata sets of speed 1198801 1198802 119880119899Step 3 Execute a distribution test for 119880119894 where the resultshows that one data set is too small to meet the requirementsof the test Merge the adjacent density segments in Step 2to enlarge the amount of the small data set Redo thedistribution test for the new data set more than 80 ofwhich meets the normal distribution with totally 95 of thetotal data satisfying the normal distribution which makesit reasonable to consider all the small data set satisfying thenormal distribution

Step 4 Get two quantiles 119906upper119894 and 119906lower119894 of speed set 119880119894 asupper and lower critical values of velocity for density 119896119894Step 5 Then have upper and lower critical value set 119880upper =sum119899119894=1 119906upper119894 and 119880lower = sum119899119894=1 119906lower119894

Step 6 (fit 119906upper and 119906lower) Because the loop detector islocated near commercial street which has heavy trafficwe usethe logarithmic model to formulize the data

119906095 = 14204 times ln(216412119896 ) 119906005 = 7169 times ln(254497119896 )

(18)

Figure 4 shows the validation result of the speed-densitylogarithmic model of the outer lane when upper value =095 and lower value = 005 Equations (18) correspondinglyare the green and blue curves in Figure 4 which is thespeed-density model of interrupted traffic flow created bythe new description method Significant test results indicatethat 119875 values of two regression coefficients of two curves areminima (119875 lt 2119890 minus 16) which means that coefficients aresignificant and two log models constructed with density asthe independent variable are applied to estimate velocity asthe dependent variable

The coil data of the outer lane for two weeks fourweeks six weeks and eight weeks are respectively selected

095 quantile fractileFitting logarithmic model (upper)005 quantile fractileFitting logarithmic model (lower)

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Figure 4 Speed-density logarithmic model of the outside lane

and four groups of parameters are established for modelvalidation Table 3 gives the ratio of the data between twologarithmic curves to the total amount of data in each caseMake a longitudinal observation it is obvious that with uppervalue increasing and lower value decreasing the proportionincreases accordingly where amplitudes are obvious respec-tively 72 62 and 69 On the other hand the maintransverse trend is that the proportion increases along withthe increase of experiment data loosely where however six-week data has the best performance The above suggests thatthe two logarithmic models are able to describe the speed-density relation of the outer lane Figure 5 shows the fourgroupsrsquo validation results when upper value = 095 and lowervalue = 005

42 Coil Data Analysis of the Inner Lane We select coildata of the inner lane and follow Steps 1 to 5 as for theouter lane When fitting sets 119906upper and 119906lower at Step 6 wefind that the speed-density models proposed by scholars allhave poor performance with goodness of fit of less than 05

8 Mobile Information Systems

Table 3 Validation results of the model of the outer lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 619 658 662 661 650Lower value = 020Upper value = 085 689 731 736 733 722Lower value = 015Upper value = 090 745 788 804 797 784Lower value = 010Upper value = 095 842 851 867 853 853Lower value = 005

Two-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(a) Validation result for two-week data

Four-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(b) Validation result for four-week data

Six-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 5 Validation result of the speed-density logarithmic model of the outside lane

Mobile Information Systems 9

Table 4 Validation results of the model of the inner lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 800 779 813 788 795Lower value = 020Upper value = 085 820 823 852 831 832Lower value = 015Upper value = 090 838 849 864 853 851Lower value = 010Upper value = 095 890 903 889 898 895Lower value = 005

which suggests that a singlemodel cannot accurately describethe quantile set of the coil data Thus we consider using asegmentation model

In the density-flow curve there is a critical density 119896119898which is the density of maximum traffic flow as shown inFigure 1 When the density 119896 lt 119896119898 the traffic is in a stateof flow when 119896 gt 119896119898 the traffic flow gradually becomescrowded Therefore consider using 119896119898 as the critical value ofthe subsection

A density-flow curve is obtained by local polynomialregression fitting and the density value at the curve vertexis just 119896119898 Take 119896119898 as the critical value and piecewise analyze119906upper119896lt119896119898

119906lower119896lt119896119898

119906upper119896gt119896119898

and 119906lower119896gt119896119898

The analysis shows that thequantile set 119906upper

119896lt119896119898

and 119906lower119896lt119896119898

agrees with the exponentialmodel and the quantile set 119906upper

119896gt119896119898

and 119906lower119896gt119896119898

has goodagreement with the logarithmic model

119906095 = 69647 times 119890minus11989614449 + 12716 119896 lt 1198961198988227 times ln(254971119896 ) 119896 ge 119896119898

119906005 = 5108 times 119890minus11989610010 minus 2245 119896 lt 1198961198985337 times ln(243306119896 ) 119896 ge 119896119898

(19)

Figure 6 shows the fitting result of a segmentation modelof the outer lane when upper value = 095 and lower value= 005 and (19) are the models corresponding to the greencurve and blue curve in Figure 6 which is the speed-densitymodel of interrupted traffic flow via the new descriptionmethod 119875 value of each parameter is very small suggestingthe coefficient is very significant

The coil data of the inner lane for two weeks four weekssix weeks and eight weeks are respectively selected and fourgroups of parameters are established for themodel validationthe same as that for the outer lane Table 4 gives the ratio ofthe data between two logarithmic curves to the total amountof data in each case Comparing the result with that of theouter lane we find that the validation results of the model ofthe inner lane are better with greater ratio

Take a longitudinal observation similarly it is obviousthat with upper value increasing and lower value decreasingthe proportion increases accordingly where amplitudes aresmaller than that of outer lane respectively 37 19 and43 The main transverse trend is the same as outer lane

km

095 quantile fractileMultisession model (upper)005 quantile fractileMultisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

Figure 6 Speed-density multisession model of the inside lane

except the case of upper value = 095 and lower value = 005The result indicates that the two segmentation models aresuitable for describing the speed-density relation of the innerlane Figure 7 shows the four groupsrsquo validation results whenupper value = 095 and lower value = 005

43 Experimental Result Analysis

431 Difference between the Models of the Outer Lane andthe Inner Lane The loop detector of the outer lane measuresright-turning and straight lanes and the coil is located in aroad adjacent to a commercial pedestrian street with a heavyflow of people and traffic A logarithmic model is appliedto describe traffic flow with large density and therefore itis accepted that the coil data of the outer lane satisfy thelogarithmic model

The loop detector of the inner lane measures the left-turning lane which also has heavy traffic flow The speed-density relation of the inner lane does not satisfy the singlelog model but is suitable for the segmentation model The

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Electrical and Computer Engineering

Journal of

Journal of

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httpwwwhindawicom Volume 2014

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International Journal of

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Advances in

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RoboticsJournal of

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Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Mobile Information Systems

One-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(a) Graph of one-day data

0

20

40

60

80

Spee

d (k

mh

)

Three-day data

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(b) Graph of three-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250

Seven-day data

Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

(c) Graph of seven-day data

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Linear modelLogarithmic modelExponential model

Northwestern modelDrew modelPipes-Munjal model

Fourteen-day data

(d) Graph of fourteen-day data

Figure 2 Comparison of six speed-density models

3 The Description Method of the Speed-Density Model for Interrupted Traffic Flow

31The Characteristics of the Data of Interrupted Traffic Coildata for one day three days seven days and fourteen dayswere selected to compare and analyze the discontinuous flow

data and the existing six speed-density models as shown inFigure 2 We found the following

(1) The six modelsrsquo performance was poor when the coildata of interrupted traffic flow were fitted illustratingthat although suitable for uninterrupted traffic flow

Mobile Information Systems 5

0 5 10 15

0

50

100

150

200

Occupancy ()

Flow

(pcu

h)

Figure 3 Flow-occupancy graph of small density

they are unsuitable for describing the speed-densityrelation of interrupted traffic flow because of diversedata sources different traffic environments or otherfactors In contrast the logarithmic model gave thebest performance and the linear model gave the worstperformance

(2) The interval value of critical densities 119896119898 of one-daythree-day seven-day and fourteen-day data sets was[6256 pcukm 7123 pcukm] and most of the datawere located in 119896 lt 119896119898 range meaning unimpededflow data accounted for the absolute proportion sothe traffic flow of the location coil was in a state offlow most of the time

(3) When 119896 lt 119896119898 with the increase of density thevelocity decreased sharply when 119896 gt 119896119898 as thedensity increased the velocity decreased slowly andthe speed variation amplitude was very small

(4) When the density was small the speed had a largerange of values of which the largest was [23 kmh72 kmh] We filtered out the small-density data toobtain a scatter diagram of flow and occupancywhich were directly collected by a loop detector asshown in Figure 3 In Figure 3 it is obvious thatthe loop detector acquires large-range flow valuesfor the same occupancy value and the largest rangecan reach 100 pcuh Hence after calculating speedand density by density formula and velocity formulaspeed accordingly has a large range of values for thesame density in speed-density diagram

(5) In addition the density values were found to be neara number of points and the difference between adja-cent pointswas approximately equal to a certain value

From the above analysis we found that because of thebig differences between uninterrupted and interrupted trafficflow existing models suitable for uninterrupted traffic floware unsuited for describing the speed-density relation of

interrupted traffic flow What is more the flow collected bya loop detector has a large range of values Therefore for thespeed-density relation of interrupted trafficflowwemust finda new descriptive method

32 Description Method of Speed-Density Relationship forInterrupted Traffic Flow Because of the difference betweenthe uninterrupted and interrupted traffic flow and thevolatility of speed the speed-density relationship cannot beadequately described by a single model so we use two curves119906upper and 119906lower to describe the supremum and infimum ofvelocity values

119906upper = 119892upper (119880upper) 119906lower = 119892lower (119880lower) (7)

where119880upper and119880lower are respectively the upper and lowerbounds of velocity and 119892upper and 119892lower are fitting functions

Divide the density interval [119896min 119896max] into 119899 connectedintervals 1198961 1198962 119896119899 Partition data 119863 as 1198631 1198632 119863119899by density intervals and correspondingly get speed sets1198801 1198802 119880119899 causing that for any 119894 isin (1 2 119899) we have

119882(119880119894) gt 119882120572 (8)

where 119882(119880119894) is used test for 119880119894 with the Shapiro-Wilknormal test method Sort 119898 independent observations in119880119894 by nondescending order recorded as 1199091 1199092 119909119898 andconstruct the119882-test statistic

119882 = [sum119898119894=1 119886119894 times (119909119898+1minus119894 minus 119909119894)]2sum119898119894=1 119886119894 times (119909119894 minus 119909)2 (9)

where 119886119894 is the coefficient when sample size is 119898 When thepopulation distribution is normal distribution the value of119882 should be close to one 120572 quantile119882120572 of statistic119882 can beobtained by the look-up table method When119882 le 119882120572 theoriginal hypothesis should be rejected at the significant levelindicating that 119880119894 does not obey normal distribution when119882 gt 119882120572 the original hypothesis cannot be rejected and 119880119894satisfies normal distribution

Under the conditions of (8) for every 119894 isin (1 2 119899)extract the upper quantile 119906upper119894 and lower quantile 119906lower119894as the upper and lower critical values of speed for densityinterval 119896119894

119906upper119894 = 119902norm (uppermean (119880119894) sd (119880119894)) 119906lower119894 = 119902norm (lowermean (119880119894) sd (119880119894)) (10)

where 119902norm( ) is quantile function mean( ) calculates themean value of 119880119894 and sd( ) calculates the variance of 119880119894

Get the upper bound and lower bound sets

119880upper = 119906upper119894 119894 = 1 2 119899 119880lower = 119906lower119894 119894 = 1 2 119899 (11)

Fit 119880upper and 119880lower using the nonlinear least squaremethod The tabulated function 119906119894 = 119906(119896119894) 119894 = 1 2 119899 is

6 Mobile Information Systems

available by (10) Then we need to obtain the fitting function119892(119896) = 1198860 + 1198861 times 1198921(119896) + sdot sdot sdot + 119886119901 times 119892119901(119896) making the sum ofsquared deviations

119878 (1198860 1198861 119886119901) = 119899sum119894=1

[119892 (119896119894) minus 119906119894]2

= 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]2 (12)

Take theminimum of which 1198921(119896) 1198922(119896) 119892119901(119896) are 119901nonmergeable monomials of variable 119896 and 1198860 1198861 119886119901 arethe coefficients of monomials 119878 is a nonnegative polynomialof 1198860 1198861 119886119901 so there must be a minimum value Respec-tively calculate partial derivatives of 119878 for 1198860 1198861 119886119901 andmake them equal to zero

120597119878120597119886119894 = 0 119894 = 0 1 119901 (13)

Equation (13) is expanded as follows1205971198781205971198860 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]= 0

1205971198781205971198861 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 1198921 (119896119894) = 0

120597119878120597119886119901 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 119892119901 (119896119894) = 0

(14)

Continue to expand (14)

1198860 times 119899 + 1198861 times 119899sum119894=1

1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119899sum119894=1

119892119901 (119896119894) = 119899sum119894=1

119906119894

1198860 times 119899sum119894=1

1198921 (119896119894) + 1198861 times 119899sum119894=1

[1198921 (119896119894)]2 + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)] = 119899sum119894=1

[119906119894 times 1198921 (119896119894)]

1198860 times 119899sum119894=1

119892119901 (119896119894) + 1198861 times 119899sum119894=1

(1198921 (119896119894) times 119892119901 (119896119894)) + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894)]2 = 119899sum119894=1

[119906119894 times 119892119901 (119896119894)](15)

and get its matrix form

[[[[[[[[[[[[[[

119899 119899sum119894=1

1198921 (119896119894) sdot sdot sdot 119899sum119894=1

119892119901 (119896119894)119899sum119894=1

1198921 (119896119894) 119899sum119894=1

[1198921 (119896119894)]2 sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)]

119899sum119894=1

119892119901 (119896119894) 119899sum119894=1

[1198921 (119896119894) times 119892119901 (119896119894)] sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894)]2

]]]]]]]]]]]]]]

times[[[[[[[[

11988601198861119886119901

]]]]]]]]=

[[[[[[[[[[[[[[

119899sum119894=1

119906119894119899sum119894=1

[119906119894 times 1198921 (119896119894)]

119899sum119894=1

[119906119894 times 119892119901 (119896119894)]

]]]]]]]]]]]]]]

(16)

Solve (16) and 1198860 1198861 119886119901 are availableUsing the above least square method to fit 119880upper and119880lower respectively obtains

119906upper (119896) = 119892upper (119896 1198860 1198861 119886119901) 119906lower (119896) = 119892lower (119896 1198860 1198861 119886119901) (17)

and upper and lower curves 119906upper(119896) and 119906lower(119896) which isthe speed-density relation description model

4 Experiment and Analysis

The experiment data is collected by the coil detectors under-ground closed to Optical Valley Walking Street in WuhanChina Coil detectors collect data every 15minutes recordingtime flow occupancy and so forth as shown in Table 2

Use the method in [20 21] to calculate speed and densityand the ratio of the amount of data between twomodel curvesto the total amount of experiment data is used to describe theperformance of model The loop detector in the outer lanemeasures the traffic flow of straight and right-turning lanesand the loop detector in the inner lane measures the trafficflow of the left-turning laneThe traffic flow characteristics oftwo loop detectors must have certain differences Thereforeanalyze the coil data of both the outer lane and the inner laneto find the diversity of their speed-density relationship

41 Coil Data Analysis of the Outer Lane The experimentalsteps are as follows

Step 1 Analyze coil data of the outer lane andfind that densityvalues are clustered at a number of points 1198961 1198962 119896119899

Mobile Information Systems 7

Table 2 The data example

Date Week Flow Occupancy Minute id Hour20141120 4 132 7 00000 41751051 020141120 4 91 5 01500 41751051 020141120 4 98 7 03001 41751051 020141120 4 103 5 04501 41751051 020141120 4 77 6 10000 41751051 120141120 4 71 4 11500 41751051 120141120 4 64 3 13001 41751051 120141120 4 40 75 14501 41751051 1

where the mean value of the difference between the adjacentpoints is about 25 pcukm Divide density 119896 into a number ofintervals with length 25 pcukm by 1198961 1198962 119896119899Step 2 Correspondingly split data 119863 into small data sets1198631 1198632 119863119899 according to density segmentations and getdata sets of speed 1198801 1198802 119880119899Step 3 Execute a distribution test for 119880119894 where the resultshows that one data set is too small to meet the requirementsof the test Merge the adjacent density segments in Step 2to enlarge the amount of the small data set Redo thedistribution test for the new data set more than 80 ofwhich meets the normal distribution with totally 95 of thetotal data satisfying the normal distribution which makesit reasonable to consider all the small data set satisfying thenormal distribution

Step 4 Get two quantiles 119906upper119894 and 119906lower119894 of speed set 119880119894 asupper and lower critical values of velocity for density 119896119894Step 5 Then have upper and lower critical value set 119880upper =sum119899119894=1 119906upper119894 and 119880lower = sum119899119894=1 119906lower119894

Step 6 (fit 119906upper and 119906lower) Because the loop detector islocated near commercial street which has heavy trafficwe usethe logarithmic model to formulize the data

119906095 = 14204 times ln(216412119896 ) 119906005 = 7169 times ln(254497119896 )

(18)

Figure 4 shows the validation result of the speed-densitylogarithmic model of the outer lane when upper value =095 and lower value = 005 Equations (18) correspondinglyare the green and blue curves in Figure 4 which is thespeed-density model of interrupted traffic flow created bythe new description method Significant test results indicatethat 119875 values of two regression coefficients of two curves areminima (119875 lt 2119890 minus 16) which means that coefficients aresignificant and two log models constructed with density asthe independent variable are applied to estimate velocity asthe dependent variable

The coil data of the outer lane for two weeks fourweeks six weeks and eight weeks are respectively selected

095 quantile fractileFitting logarithmic model (upper)005 quantile fractileFitting logarithmic model (lower)

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Figure 4 Speed-density logarithmic model of the outside lane

and four groups of parameters are established for modelvalidation Table 3 gives the ratio of the data between twologarithmic curves to the total amount of data in each caseMake a longitudinal observation it is obvious that with uppervalue increasing and lower value decreasing the proportionincreases accordingly where amplitudes are obvious respec-tively 72 62 and 69 On the other hand the maintransverse trend is that the proportion increases along withthe increase of experiment data loosely where however six-week data has the best performance The above suggests thatthe two logarithmic models are able to describe the speed-density relation of the outer lane Figure 5 shows the fourgroupsrsquo validation results when upper value = 095 and lowervalue = 005

42 Coil Data Analysis of the Inner Lane We select coildata of the inner lane and follow Steps 1 to 5 as for theouter lane When fitting sets 119906upper and 119906lower at Step 6 wefind that the speed-density models proposed by scholars allhave poor performance with goodness of fit of less than 05

8 Mobile Information Systems

Table 3 Validation results of the model of the outer lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 619 658 662 661 650Lower value = 020Upper value = 085 689 731 736 733 722Lower value = 015Upper value = 090 745 788 804 797 784Lower value = 010Upper value = 095 842 851 867 853 853Lower value = 005

Two-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(a) Validation result for two-week data

Four-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(b) Validation result for four-week data

Six-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 5 Validation result of the speed-density logarithmic model of the outside lane

Mobile Information Systems 9

Table 4 Validation results of the model of the inner lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 800 779 813 788 795Lower value = 020Upper value = 085 820 823 852 831 832Lower value = 015Upper value = 090 838 849 864 853 851Lower value = 010Upper value = 095 890 903 889 898 895Lower value = 005

which suggests that a singlemodel cannot accurately describethe quantile set of the coil data Thus we consider using asegmentation model

In the density-flow curve there is a critical density 119896119898which is the density of maximum traffic flow as shown inFigure 1 When the density 119896 lt 119896119898 the traffic is in a stateof flow when 119896 gt 119896119898 the traffic flow gradually becomescrowded Therefore consider using 119896119898 as the critical value ofthe subsection

A density-flow curve is obtained by local polynomialregression fitting and the density value at the curve vertexis just 119896119898 Take 119896119898 as the critical value and piecewise analyze119906upper119896lt119896119898

119906lower119896lt119896119898

119906upper119896gt119896119898

and 119906lower119896gt119896119898

The analysis shows that thequantile set 119906upper

119896lt119896119898

and 119906lower119896lt119896119898

agrees with the exponentialmodel and the quantile set 119906upper

119896gt119896119898

and 119906lower119896gt119896119898

has goodagreement with the logarithmic model

119906095 = 69647 times 119890minus11989614449 + 12716 119896 lt 1198961198988227 times ln(254971119896 ) 119896 ge 119896119898

119906005 = 5108 times 119890minus11989610010 minus 2245 119896 lt 1198961198985337 times ln(243306119896 ) 119896 ge 119896119898

(19)

Figure 6 shows the fitting result of a segmentation modelof the outer lane when upper value = 095 and lower value= 005 and (19) are the models corresponding to the greencurve and blue curve in Figure 6 which is the speed-densitymodel of interrupted traffic flow via the new descriptionmethod 119875 value of each parameter is very small suggestingthe coefficient is very significant

The coil data of the inner lane for two weeks four weekssix weeks and eight weeks are respectively selected and fourgroups of parameters are established for themodel validationthe same as that for the outer lane Table 4 gives the ratio ofthe data between two logarithmic curves to the total amountof data in each case Comparing the result with that of theouter lane we find that the validation results of the model ofthe inner lane are better with greater ratio

Take a longitudinal observation similarly it is obviousthat with upper value increasing and lower value decreasingthe proportion increases accordingly where amplitudes aresmaller than that of outer lane respectively 37 19 and43 The main transverse trend is the same as outer lane

km

095 quantile fractileMultisession model (upper)005 quantile fractileMultisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

Figure 6 Speed-density multisession model of the inside lane

except the case of upper value = 095 and lower value = 005The result indicates that the two segmentation models aresuitable for describing the speed-density relation of the innerlane Figure 7 shows the four groupsrsquo validation results whenupper value = 095 and lower value = 005

43 Experimental Result Analysis

431 Difference between the Models of the Outer Lane andthe Inner Lane The loop detector of the outer lane measuresright-turning and straight lanes and the coil is located in aroad adjacent to a commercial pedestrian street with a heavyflow of people and traffic A logarithmic model is appliedto describe traffic flow with large density and therefore itis accepted that the coil data of the outer lane satisfy thelogarithmic model

The loop detector of the inner lane measures the left-turning lane which also has heavy traffic flow The speed-density relation of the inner lane does not satisfy the singlelog model but is suitable for the segmentation model The

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 5

0 5 10 15

0

50

100

150

200

Occupancy ()

Flow

(pcu

h)

Figure 3 Flow-occupancy graph of small density

they are unsuitable for describing the speed-densityrelation of interrupted traffic flow because of diversedata sources different traffic environments or otherfactors In contrast the logarithmic model gave thebest performance and the linear model gave the worstperformance

(2) The interval value of critical densities 119896119898 of one-daythree-day seven-day and fourteen-day data sets was[6256 pcukm 7123 pcukm] and most of the datawere located in 119896 lt 119896119898 range meaning unimpededflow data accounted for the absolute proportion sothe traffic flow of the location coil was in a state offlow most of the time

(3) When 119896 lt 119896119898 with the increase of density thevelocity decreased sharply when 119896 gt 119896119898 as thedensity increased the velocity decreased slowly andthe speed variation amplitude was very small

(4) When the density was small the speed had a largerange of values of which the largest was [23 kmh72 kmh] We filtered out the small-density data toobtain a scatter diagram of flow and occupancywhich were directly collected by a loop detector asshown in Figure 3 In Figure 3 it is obvious thatthe loop detector acquires large-range flow valuesfor the same occupancy value and the largest rangecan reach 100 pcuh Hence after calculating speedand density by density formula and velocity formulaspeed accordingly has a large range of values for thesame density in speed-density diagram

(5) In addition the density values were found to be neara number of points and the difference between adja-cent pointswas approximately equal to a certain value

From the above analysis we found that because of thebig differences between uninterrupted and interrupted trafficflow existing models suitable for uninterrupted traffic floware unsuited for describing the speed-density relation of

interrupted traffic flow What is more the flow collected bya loop detector has a large range of values Therefore for thespeed-density relation of interrupted trafficflowwemust finda new descriptive method

32 Description Method of Speed-Density Relationship forInterrupted Traffic Flow Because of the difference betweenthe uninterrupted and interrupted traffic flow and thevolatility of speed the speed-density relationship cannot beadequately described by a single model so we use two curves119906upper and 119906lower to describe the supremum and infimum ofvelocity values

119906upper = 119892upper (119880upper) 119906lower = 119892lower (119880lower) (7)

where119880upper and119880lower are respectively the upper and lowerbounds of velocity and 119892upper and 119892lower are fitting functions

Divide the density interval [119896min 119896max] into 119899 connectedintervals 1198961 1198962 119896119899 Partition data 119863 as 1198631 1198632 119863119899by density intervals and correspondingly get speed sets1198801 1198802 119880119899 causing that for any 119894 isin (1 2 119899) we have

119882(119880119894) gt 119882120572 (8)

where 119882(119880119894) is used test for 119880119894 with the Shapiro-Wilknormal test method Sort 119898 independent observations in119880119894 by nondescending order recorded as 1199091 1199092 119909119898 andconstruct the119882-test statistic

119882 = [sum119898119894=1 119886119894 times (119909119898+1minus119894 minus 119909119894)]2sum119898119894=1 119886119894 times (119909119894 minus 119909)2 (9)

where 119886119894 is the coefficient when sample size is 119898 When thepopulation distribution is normal distribution the value of119882 should be close to one 120572 quantile119882120572 of statistic119882 can beobtained by the look-up table method When119882 le 119882120572 theoriginal hypothesis should be rejected at the significant levelindicating that 119880119894 does not obey normal distribution when119882 gt 119882120572 the original hypothesis cannot be rejected and 119880119894satisfies normal distribution

Under the conditions of (8) for every 119894 isin (1 2 119899)extract the upper quantile 119906upper119894 and lower quantile 119906lower119894as the upper and lower critical values of speed for densityinterval 119896119894

119906upper119894 = 119902norm (uppermean (119880119894) sd (119880119894)) 119906lower119894 = 119902norm (lowermean (119880119894) sd (119880119894)) (10)

where 119902norm( ) is quantile function mean( ) calculates themean value of 119880119894 and sd( ) calculates the variance of 119880119894

Get the upper bound and lower bound sets

119880upper = 119906upper119894 119894 = 1 2 119899 119880lower = 119906lower119894 119894 = 1 2 119899 (11)

Fit 119880upper and 119880lower using the nonlinear least squaremethod The tabulated function 119906119894 = 119906(119896119894) 119894 = 1 2 119899 is

6 Mobile Information Systems

available by (10) Then we need to obtain the fitting function119892(119896) = 1198860 + 1198861 times 1198921(119896) + sdot sdot sdot + 119886119901 times 119892119901(119896) making the sum ofsquared deviations

119878 (1198860 1198861 119886119901) = 119899sum119894=1

[119892 (119896119894) minus 119906119894]2

= 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]2 (12)

Take theminimum of which 1198921(119896) 1198922(119896) 119892119901(119896) are 119901nonmergeable monomials of variable 119896 and 1198860 1198861 119886119901 arethe coefficients of monomials 119878 is a nonnegative polynomialof 1198860 1198861 119886119901 so there must be a minimum value Respec-tively calculate partial derivatives of 119878 for 1198860 1198861 119886119901 andmake them equal to zero

120597119878120597119886119894 = 0 119894 = 0 1 119901 (13)

Equation (13) is expanded as follows1205971198781205971198860 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]= 0

1205971198781205971198861 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 1198921 (119896119894) = 0

120597119878120597119886119901 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 119892119901 (119896119894) = 0

(14)

Continue to expand (14)

1198860 times 119899 + 1198861 times 119899sum119894=1

1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119899sum119894=1

119892119901 (119896119894) = 119899sum119894=1

119906119894

1198860 times 119899sum119894=1

1198921 (119896119894) + 1198861 times 119899sum119894=1

[1198921 (119896119894)]2 + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)] = 119899sum119894=1

[119906119894 times 1198921 (119896119894)]

1198860 times 119899sum119894=1

119892119901 (119896119894) + 1198861 times 119899sum119894=1

(1198921 (119896119894) times 119892119901 (119896119894)) + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894)]2 = 119899sum119894=1

[119906119894 times 119892119901 (119896119894)](15)

and get its matrix form

[[[[[[[[[[[[[[

119899 119899sum119894=1

1198921 (119896119894) sdot sdot sdot 119899sum119894=1

119892119901 (119896119894)119899sum119894=1

1198921 (119896119894) 119899sum119894=1

[1198921 (119896119894)]2 sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)]

119899sum119894=1

119892119901 (119896119894) 119899sum119894=1

[1198921 (119896119894) times 119892119901 (119896119894)] sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894)]2

]]]]]]]]]]]]]]

times[[[[[[[[

11988601198861119886119901

]]]]]]]]=

[[[[[[[[[[[[[[

119899sum119894=1

119906119894119899sum119894=1

[119906119894 times 1198921 (119896119894)]

119899sum119894=1

[119906119894 times 119892119901 (119896119894)]

]]]]]]]]]]]]]]

(16)

Solve (16) and 1198860 1198861 119886119901 are availableUsing the above least square method to fit 119880upper and119880lower respectively obtains

119906upper (119896) = 119892upper (119896 1198860 1198861 119886119901) 119906lower (119896) = 119892lower (119896 1198860 1198861 119886119901) (17)

and upper and lower curves 119906upper(119896) and 119906lower(119896) which isthe speed-density relation description model

4 Experiment and Analysis

The experiment data is collected by the coil detectors under-ground closed to Optical Valley Walking Street in WuhanChina Coil detectors collect data every 15minutes recordingtime flow occupancy and so forth as shown in Table 2

Use the method in [20 21] to calculate speed and densityand the ratio of the amount of data between twomodel curvesto the total amount of experiment data is used to describe theperformance of model The loop detector in the outer lanemeasures the traffic flow of straight and right-turning lanesand the loop detector in the inner lane measures the trafficflow of the left-turning laneThe traffic flow characteristics oftwo loop detectors must have certain differences Thereforeanalyze the coil data of both the outer lane and the inner laneto find the diversity of their speed-density relationship

41 Coil Data Analysis of the Outer Lane The experimentalsteps are as follows

Step 1 Analyze coil data of the outer lane andfind that densityvalues are clustered at a number of points 1198961 1198962 119896119899

Mobile Information Systems 7

Table 2 The data example

Date Week Flow Occupancy Minute id Hour20141120 4 132 7 00000 41751051 020141120 4 91 5 01500 41751051 020141120 4 98 7 03001 41751051 020141120 4 103 5 04501 41751051 020141120 4 77 6 10000 41751051 120141120 4 71 4 11500 41751051 120141120 4 64 3 13001 41751051 120141120 4 40 75 14501 41751051 1

where the mean value of the difference between the adjacentpoints is about 25 pcukm Divide density 119896 into a number ofintervals with length 25 pcukm by 1198961 1198962 119896119899Step 2 Correspondingly split data 119863 into small data sets1198631 1198632 119863119899 according to density segmentations and getdata sets of speed 1198801 1198802 119880119899Step 3 Execute a distribution test for 119880119894 where the resultshows that one data set is too small to meet the requirementsof the test Merge the adjacent density segments in Step 2to enlarge the amount of the small data set Redo thedistribution test for the new data set more than 80 ofwhich meets the normal distribution with totally 95 of thetotal data satisfying the normal distribution which makesit reasonable to consider all the small data set satisfying thenormal distribution

Step 4 Get two quantiles 119906upper119894 and 119906lower119894 of speed set 119880119894 asupper and lower critical values of velocity for density 119896119894Step 5 Then have upper and lower critical value set 119880upper =sum119899119894=1 119906upper119894 and 119880lower = sum119899119894=1 119906lower119894

Step 6 (fit 119906upper and 119906lower) Because the loop detector islocated near commercial street which has heavy trafficwe usethe logarithmic model to formulize the data

119906095 = 14204 times ln(216412119896 ) 119906005 = 7169 times ln(254497119896 )

(18)

Figure 4 shows the validation result of the speed-densitylogarithmic model of the outer lane when upper value =095 and lower value = 005 Equations (18) correspondinglyare the green and blue curves in Figure 4 which is thespeed-density model of interrupted traffic flow created bythe new description method Significant test results indicatethat 119875 values of two regression coefficients of two curves areminima (119875 lt 2119890 minus 16) which means that coefficients aresignificant and two log models constructed with density asthe independent variable are applied to estimate velocity asthe dependent variable

The coil data of the outer lane for two weeks fourweeks six weeks and eight weeks are respectively selected

095 quantile fractileFitting logarithmic model (upper)005 quantile fractileFitting logarithmic model (lower)

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Figure 4 Speed-density logarithmic model of the outside lane

and four groups of parameters are established for modelvalidation Table 3 gives the ratio of the data between twologarithmic curves to the total amount of data in each caseMake a longitudinal observation it is obvious that with uppervalue increasing and lower value decreasing the proportionincreases accordingly where amplitudes are obvious respec-tively 72 62 and 69 On the other hand the maintransverse trend is that the proportion increases along withthe increase of experiment data loosely where however six-week data has the best performance The above suggests thatthe two logarithmic models are able to describe the speed-density relation of the outer lane Figure 5 shows the fourgroupsrsquo validation results when upper value = 095 and lowervalue = 005

42 Coil Data Analysis of the Inner Lane We select coildata of the inner lane and follow Steps 1 to 5 as for theouter lane When fitting sets 119906upper and 119906lower at Step 6 wefind that the speed-density models proposed by scholars allhave poor performance with goodness of fit of less than 05

8 Mobile Information Systems

Table 3 Validation results of the model of the outer lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 619 658 662 661 650Lower value = 020Upper value = 085 689 731 736 733 722Lower value = 015Upper value = 090 745 788 804 797 784Lower value = 010Upper value = 095 842 851 867 853 853Lower value = 005

Two-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(a) Validation result for two-week data

Four-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(b) Validation result for four-week data

Six-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 5 Validation result of the speed-density logarithmic model of the outside lane

Mobile Information Systems 9

Table 4 Validation results of the model of the inner lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 800 779 813 788 795Lower value = 020Upper value = 085 820 823 852 831 832Lower value = 015Upper value = 090 838 849 864 853 851Lower value = 010Upper value = 095 890 903 889 898 895Lower value = 005

which suggests that a singlemodel cannot accurately describethe quantile set of the coil data Thus we consider using asegmentation model

In the density-flow curve there is a critical density 119896119898which is the density of maximum traffic flow as shown inFigure 1 When the density 119896 lt 119896119898 the traffic is in a stateof flow when 119896 gt 119896119898 the traffic flow gradually becomescrowded Therefore consider using 119896119898 as the critical value ofthe subsection

A density-flow curve is obtained by local polynomialregression fitting and the density value at the curve vertexis just 119896119898 Take 119896119898 as the critical value and piecewise analyze119906upper119896lt119896119898

119906lower119896lt119896119898

119906upper119896gt119896119898

and 119906lower119896gt119896119898

The analysis shows that thequantile set 119906upper

119896lt119896119898

and 119906lower119896lt119896119898

agrees with the exponentialmodel and the quantile set 119906upper

119896gt119896119898

and 119906lower119896gt119896119898

has goodagreement with the logarithmic model

119906095 = 69647 times 119890minus11989614449 + 12716 119896 lt 1198961198988227 times ln(254971119896 ) 119896 ge 119896119898

119906005 = 5108 times 119890minus11989610010 minus 2245 119896 lt 1198961198985337 times ln(243306119896 ) 119896 ge 119896119898

(19)

Figure 6 shows the fitting result of a segmentation modelof the outer lane when upper value = 095 and lower value= 005 and (19) are the models corresponding to the greencurve and blue curve in Figure 6 which is the speed-densitymodel of interrupted traffic flow via the new descriptionmethod 119875 value of each parameter is very small suggestingthe coefficient is very significant

The coil data of the inner lane for two weeks four weekssix weeks and eight weeks are respectively selected and fourgroups of parameters are established for themodel validationthe same as that for the outer lane Table 4 gives the ratio ofthe data between two logarithmic curves to the total amountof data in each case Comparing the result with that of theouter lane we find that the validation results of the model ofthe inner lane are better with greater ratio

Take a longitudinal observation similarly it is obviousthat with upper value increasing and lower value decreasingthe proportion increases accordingly where amplitudes aresmaller than that of outer lane respectively 37 19 and43 The main transverse trend is the same as outer lane

km

095 quantile fractileMultisession model (upper)005 quantile fractileMultisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

Figure 6 Speed-density multisession model of the inside lane

except the case of upper value = 095 and lower value = 005The result indicates that the two segmentation models aresuitable for describing the speed-density relation of the innerlane Figure 7 shows the four groupsrsquo validation results whenupper value = 095 and lower value = 005

43 Experimental Result Analysis

431 Difference between the Models of the Outer Lane andthe Inner Lane The loop detector of the outer lane measuresright-turning and straight lanes and the coil is located in aroad adjacent to a commercial pedestrian street with a heavyflow of people and traffic A logarithmic model is appliedto describe traffic flow with large density and therefore itis accepted that the coil data of the outer lane satisfy thelogarithmic model

The loop detector of the inner lane measures the left-turning lane which also has heavy traffic flow The speed-density relation of the inner lane does not satisfy the singlelog model but is suitable for the segmentation model The

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Mobile Information Systems

available by (10) Then we need to obtain the fitting function119892(119896) = 1198860 + 1198861 times 1198921(119896) + sdot sdot sdot + 119886119901 times 119892119901(119896) making the sum ofsquared deviations

119878 (1198860 1198861 119886119901) = 119899sum119894=1

[119892 (119896119894) minus 119906119894]2

= 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]2 (12)

Take theminimum of which 1198921(119896) 1198922(119896) 119892119901(119896) are 119901nonmergeable monomials of variable 119896 and 1198860 1198861 119886119901 arethe coefficients of monomials 119878 is a nonnegative polynomialof 1198860 1198861 119886119901 so there must be a minimum value Respec-tively calculate partial derivatives of 119878 for 1198860 1198861 119886119901 andmake them equal to zero

120597119878120597119886119894 = 0 119894 = 0 1 119901 (13)

Equation (13) is expanded as follows1205971198781205971198860 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]= 0

1205971198781205971198861 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 1198921 (119896119894) = 0

120597119878120597119886119901 = 2times 119899sum119894=1

[1198860 + 1198861 times 1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119892119901 (119896119894) minus 119906119894]times 119892119901 (119896119894) = 0

(14)

Continue to expand (14)

1198860 times 119899 + 1198861 times 119899sum119894=1

1198921 (119896119894) + sdot sdot sdot + 119886119901 times 119899sum119894=1

119892119901 (119896119894) = 119899sum119894=1

119906119894

1198860 times 119899sum119894=1

1198921 (119896119894) + 1198861 times 119899sum119894=1

[1198921 (119896119894)]2 + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)] = 119899sum119894=1

[119906119894 times 1198921 (119896119894)]

1198860 times 119899sum119894=1

119892119901 (119896119894) + 1198861 times 119899sum119894=1

(1198921 (119896119894) times 119892119901 (119896119894)) + sdot sdot sdot + 119886119901times 119899sum119894=1

[119892119901 (119896119894)]2 = 119899sum119894=1

[119906119894 times 119892119901 (119896119894)](15)

and get its matrix form

[[[[[[[[[[[[[[

119899 119899sum119894=1

1198921 (119896119894) sdot sdot sdot 119899sum119894=1

119892119901 (119896119894)119899sum119894=1

1198921 (119896119894) 119899sum119894=1

[1198921 (119896119894)]2 sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894) times 1198921 (119896119894)]

119899sum119894=1

119892119901 (119896119894) 119899sum119894=1

[1198921 (119896119894) times 119892119901 (119896119894)] sdot sdot sdot 119899sum119894=1

[119892119901 (119896119894)]2

]]]]]]]]]]]]]]

times[[[[[[[[

11988601198861119886119901

]]]]]]]]=

[[[[[[[[[[[[[[

119899sum119894=1

119906119894119899sum119894=1

[119906119894 times 1198921 (119896119894)]

119899sum119894=1

[119906119894 times 119892119901 (119896119894)]

]]]]]]]]]]]]]]

(16)

Solve (16) and 1198860 1198861 119886119901 are availableUsing the above least square method to fit 119880upper and119880lower respectively obtains

119906upper (119896) = 119892upper (119896 1198860 1198861 119886119901) 119906lower (119896) = 119892lower (119896 1198860 1198861 119886119901) (17)

and upper and lower curves 119906upper(119896) and 119906lower(119896) which isthe speed-density relation description model

4 Experiment and Analysis

The experiment data is collected by the coil detectors under-ground closed to Optical Valley Walking Street in WuhanChina Coil detectors collect data every 15minutes recordingtime flow occupancy and so forth as shown in Table 2

Use the method in [20 21] to calculate speed and densityand the ratio of the amount of data between twomodel curvesto the total amount of experiment data is used to describe theperformance of model The loop detector in the outer lanemeasures the traffic flow of straight and right-turning lanesand the loop detector in the inner lane measures the trafficflow of the left-turning laneThe traffic flow characteristics oftwo loop detectors must have certain differences Thereforeanalyze the coil data of both the outer lane and the inner laneto find the diversity of their speed-density relationship

41 Coil Data Analysis of the Outer Lane The experimentalsteps are as follows

Step 1 Analyze coil data of the outer lane andfind that densityvalues are clustered at a number of points 1198961 1198962 119896119899

Mobile Information Systems 7

Table 2 The data example

Date Week Flow Occupancy Minute id Hour20141120 4 132 7 00000 41751051 020141120 4 91 5 01500 41751051 020141120 4 98 7 03001 41751051 020141120 4 103 5 04501 41751051 020141120 4 77 6 10000 41751051 120141120 4 71 4 11500 41751051 120141120 4 64 3 13001 41751051 120141120 4 40 75 14501 41751051 1

where the mean value of the difference between the adjacentpoints is about 25 pcukm Divide density 119896 into a number ofintervals with length 25 pcukm by 1198961 1198962 119896119899Step 2 Correspondingly split data 119863 into small data sets1198631 1198632 119863119899 according to density segmentations and getdata sets of speed 1198801 1198802 119880119899Step 3 Execute a distribution test for 119880119894 where the resultshows that one data set is too small to meet the requirementsof the test Merge the adjacent density segments in Step 2to enlarge the amount of the small data set Redo thedistribution test for the new data set more than 80 ofwhich meets the normal distribution with totally 95 of thetotal data satisfying the normal distribution which makesit reasonable to consider all the small data set satisfying thenormal distribution

Step 4 Get two quantiles 119906upper119894 and 119906lower119894 of speed set 119880119894 asupper and lower critical values of velocity for density 119896119894Step 5 Then have upper and lower critical value set 119880upper =sum119899119894=1 119906upper119894 and 119880lower = sum119899119894=1 119906lower119894

Step 6 (fit 119906upper and 119906lower) Because the loop detector islocated near commercial street which has heavy trafficwe usethe logarithmic model to formulize the data

119906095 = 14204 times ln(216412119896 ) 119906005 = 7169 times ln(254497119896 )

(18)

Figure 4 shows the validation result of the speed-densitylogarithmic model of the outer lane when upper value =095 and lower value = 005 Equations (18) correspondinglyare the green and blue curves in Figure 4 which is thespeed-density model of interrupted traffic flow created bythe new description method Significant test results indicatethat 119875 values of two regression coefficients of two curves areminima (119875 lt 2119890 minus 16) which means that coefficients aresignificant and two log models constructed with density asthe independent variable are applied to estimate velocity asthe dependent variable

The coil data of the outer lane for two weeks fourweeks six weeks and eight weeks are respectively selected

095 quantile fractileFitting logarithmic model (upper)005 quantile fractileFitting logarithmic model (lower)

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Figure 4 Speed-density logarithmic model of the outside lane

and four groups of parameters are established for modelvalidation Table 3 gives the ratio of the data between twologarithmic curves to the total amount of data in each caseMake a longitudinal observation it is obvious that with uppervalue increasing and lower value decreasing the proportionincreases accordingly where amplitudes are obvious respec-tively 72 62 and 69 On the other hand the maintransverse trend is that the proportion increases along withthe increase of experiment data loosely where however six-week data has the best performance The above suggests thatthe two logarithmic models are able to describe the speed-density relation of the outer lane Figure 5 shows the fourgroupsrsquo validation results when upper value = 095 and lowervalue = 005

42 Coil Data Analysis of the Inner Lane We select coildata of the inner lane and follow Steps 1 to 5 as for theouter lane When fitting sets 119906upper and 119906lower at Step 6 wefind that the speed-density models proposed by scholars allhave poor performance with goodness of fit of less than 05

8 Mobile Information Systems

Table 3 Validation results of the model of the outer lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 619 658 662 661 650Lower value = 020Upper value = 085 689 731 736 733 722Lower value = 015Upper value = 090 745 788 804 797 784Lower value = 010Upper value = 095 842 851 867 853 853Lower value = 005

Two-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(a) Validation result for two-week data

Four-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(b) Validation result for four-week data

Six-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 5 Validation result of the speed-density logarithmic model of the outside lane

Mobile Information Systems 9

Table 4 Validation results of the model of the inner lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 800 779 813 788 795Lower value = 020Upper value = 085 820 823 852 831 832Lower value = 015Upper value = 090 838 849 864 853 851Lower value = 010Upper value = 095 890 903 889 898 895Lower value = 005

which suggests that a singlemodel cannot accurately describethe quantile set of the coil data Thus we consider using asegmentation model

In the density-flow curve there is a critical density 119896119898which is the density of maximum traffic flow as shown inFigure 1 When the density 119896 lt 119896119898 the traffic is in a stateof flow when 119896 gt 119896119898 the traffic flow gradually becomescrowded Therefore consider using 119896119898 as the critical value ofthe subsection

A density-flow curve is obtained by local polynomialregression fitting and the density value at the curve vertexis just 119896119898 Take 119896119898 as the critical value and piecewise analyze119906upper119896lt119896119898

119906lower119896lt119896119898

119906upper119896gt119896119898

and 119906lower119896gt119896119898

The analysis shows that thequantile set 119906upper

119896lt119896119898

and 119906lower119896lt119896119898

agrees with the exponentialmodel and the quantile set 119906upper

119896gt119896119898

and 119906lower119896gt119896119898

has goodagreement with the logarithmic model

119906095 = 69647 times 119890minus11989614449 + 12716 119896 lt 1198961198988227 times ln(254971119896 ) 119896 ge 119896119898

119906005 = 5108 times 119890minus11989610010 minus 2245 119896 lt 1198961198985337 times ln(243306119896 ) 119896 ge 119896119898

(19)

Figure 6 shows the fitting result of a segmentation modelof the outer lane when upper value = 095 and lower value= 005 and (19) are the models corresponding to the greencurve and blue curve in Figure 6 which is the speed-densitymodel of interrupted traffic flow via the new descriptionmethod 119875 value of each parameter is very small suggestingthe coefficient is very significant

The coil data of the inner lane for two weeks four weekssix weeks and eight weeks are respectively selected and fourgroups of parameters are established for themodel validationthe same as that for the outer lane Table 4 gives the ratio ofthe data between two logarithmic curves to the total amountof data in each case Comparing the result with that of theouter lane we find that the validation results of the model ofthe inner lane are better with greater ratio

Take a longitudinal observation similarly it is obviousthat with upper value increasing and lower value decreasingthe proportion increases accordingly where amplitudes aresmaller than that of outer lane respectively 37 19 and43 The main transverse trend is the same as outer lane

km

095 quantile fractileMultisession model (upper)005 quantile fractileMultisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

Figure 6 Speed-density multisession model of the inside lane

except the case of upper value = 095 and lower value = 005The result indicates that the two segmentation models aresuitable for describing the speed-density relation of the innerlane Figure 7 shows the four groupsrsquo validation results whenupper value = 095 and lower value = 005

43 Experimental Result Analysis

431 Difference between the Models of the Outer Lane andthe Inner Lane The loop detector of the outer lane measuresright-turning and straight lanes and the coil is located in aroad adjacent to a commercial pedestrian street with a heavyflow of people and traffic A logarithmic model is appliedto describe traffic flow with large density and therefore itis accepted that the coil data of the outer lane satisfy thelogarithmic model

The loop detector of the inner lane measures the left-turning lane which also has heavy traffic flow The speed-density relation of the inner lane does not satisfy the singlelog model but is suitable for the segmentation model The

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 7

Table 2 The data example

Date Week Flow Occupancy Minute id Hour20141120 4 132 7 00000 41751051 020141120 4 91 5 01500 41751051 020141120 4 98 7 03001 41751051 020141120 4 103 5 04501 41751051 020141120 4 77 6 10000 41751051 120141120 4 71 4 11500 41751051 120141120 4 64 3 13001 41751051 120141120 4 40 75 14501 41751051 1

where the mean value of the difference between the adjacentpoints is about 25 pcukm Divide density 119896 into a number ofintervals with length 25 pcukm by 1198961 1198962 119896119899Step 2 Correspondingly split data 119863 into small data sets1198631 1198632 119863119899 according to density segmentations and getdata sets of speed 1198801 1198802 119880119899Step 3 Execute a distribution test for 119880119894 where the resultshows that one data set is too small to meet the requirementsof the test Merge the adjacent density segments in Step 2to enlarge the amount of the small data set Redo thedistribution test for the new data set more than 80 ofwhich meets the normal distribution with totally 95 of thetotal data satisfying the normal distribution which makesit reasonable to consider all the small data set satisfying thenormal distribution

Step 4 Get two quantiles 119906upper119894 and 119906lower119894 of speed set 119880119894 asupper and lower critical values of velocity for density 119896119894Step 5 Then have upper and lower critical value set 119880upper =sum119899119894=1 119906upper119894 and 119880lower = sum119899119894=1 119906lower119894

Step 6 (fit 119906upper and 119906lower) Because the loop detector islocated near commercial street which has heavy trafficwe usethe logarithmic model to formulize the data

119906095 = 14204 times ln(216412119896 ) 119906005 = 7169 times ln(254497119896 )

(18)

Figure 4 shows the validation result of the speed-densitylogarithmic model of the outer lane when upper value =095 and lower value = 005 Equations (18) correspondinglyare the green and blue curves in Figure 4 which is thespeed-density model of interrupted traffic flow created bythe new description method Significant test results indicatethat 119875 values of two regression coefficients of two curves areminima (119875 lt 2119890 minus 16) which means that coefficients aresignificant and two log models constructed with density asthe independent variable are applied to estimate velocity asthe dependent variable

The coil data of the outer lane for two weeks fourweeks six weeks and eight weeks are respectively selected

095 quantile fractileFitting logarithmic model (upper)005 quantile fractileFitting logarithmic model (lower)

0

20

40

60

80

Spee

d (k

mh

)

0 50 100 150 200 250Density (pcukm)

Figure 4 Speed-density logarithmic model of the outside lane

and four groups of parameters are established for modelvalidation Table 3 gives the ratio of the data between twologarithmic curves to the total amount of data in each caseMake a longitudinal observation it is obvious that with uppervalue increasing and lower value decreasing the proportionincreases accordingly where amplitudes are obvious respec-tively 72 62 and 69 On the other hand the maintransverse trend is that the proportion increases along withthe increase of experiment data loosely where however six-week data has the best performance The above suggests thatthe two logarithmic models are able to describe the speed-density relation of the outer lane Figure 5 shows the fourgroupsrsquo validation results when upper value = 095 and lowervalue = 005

42 Coil Data Analysis of the Inner Lane We select coildata of the inner lane and follow Steps 1 to 5 as for theouter lane When fitting sets 119906upper and 119906lower at Step 6 wefind that the speed-density models proposed by scholars allhave poor performance with goodness of fit of less than 05

8 Mobile Information Systems

Table 3 Validation results of the model of the outer lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 619 658 662 661 650Lower value = 020Upper value = 085 689 731 736 733 722Lower value = 015Upper value = 090 745 788 804 797 784Lower value = 010Upper value = 095 842 851 867 853 853Lower value = 005

Two-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(a) Validation result for two-week data

Four-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(b) Validation result for four-week data

Six-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 5 Validation result of the speed-density logarithmic model of the outside lane

Mobile Information Systems 9

Table 4 Validation results of the model of the inner lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 800 779 813 788 795Lower value = 020Upper value = 085 820 823 852 831 832Lower value = 015Upper value = 090 838 849 864 853 851Lower value = 010Upper value = 095 890 903 889 898 895Lower value = 005

which suggests that a singlemodel cannot accurately describethe quantile set of the coil data Thus we consider using asegmentation model

In the density-flow curve there is a critical density 119896119898which is the density of maximum traffic flow as shown inFigure 1 When the density 119896 lt 119896119898 the traffic is in a stateof flow when 119896 gt 119896119898 the traffic flow gradually becomescrowded Therefore consider using 119896119898 as the critical value ofthe subsection

A density-flow curve is obtained by local polynomialregression fitting and the density value at the curve vertexis just 119896119898 Take 119896119898 as the critical value and piecewise analyze119906upper119896lt119896119898

119906lower119896lt119896119898

119906upper119896gt119896119898

and 119906lower119896gt119896119898

The analysis shows that thequantile set 119906upper

119896lt119896119898

and 119906lower119896lt119896119898

agrees with the exponentialmodel and the quantile set 119906upper

119896gt119896119898

and 119906lower119896gt119896119898

has goodagreement with the logarithmic model

119906095 = 69647 times 119890minus11989614449 + 12716 119896 lt 1198961198988227 times ln(254971119896 ) 119896 ge 119896119898

119906005 = 5108 times 119890minus11989610010 minus 2245 119896 lt 1198961198985337 times ln(243306119896 ) 119896 ge 119896119898

(19)

Figure 6 shows the fitting result of a segmentation modelof the outer lane when upper value = 095 and lower value= 005 and (19) are the models corresponding to the greencurve and blue curve in Figure 6 which is the speed-densitymodel of interrupted traffic flow via the new descriptionmethod 119875 value of each parameter is very small suggestingthe coefficient is very significant

The coil data of the inner lane for two weeks four weekssix weeks and eight weeks are respectively selected and fourgroups of parameters are established for themodel validationthe same as that for the outer lane Table 4 gives the ratio ofthe data between two logarithmic curves to the total amountof data in each case Comparing the result with that of theouter lane we find that the validation results of the model ofthe inner lane are better with greater ratio

Take a longitudinal observation similarly it is obviousthat with upper value increasing and lower value decreasingthe proportion increases accordingly where amplitudes aresmaller than that of outer lane respectively 37 19 and43 The main transverse trend is the same as outer lane

km

095 quantile fractileMultisession model (upper)005 quantile fractileMultisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

Figure 6 Speed-density multisession model of the inside lane

except the case of upper value = 095 and lower value = 005The result indicates that the two segmentation models aresuitable for describing the speed-density relation of the innerlane Figure 7 shows the four groupsrsquo validation results whenupper value = 095 and lower value = 005

43 Experimental Result Analysis

431 Difference between the Models of the Outer Lane andthe Inner Lane The loop detector of the outer lane measuresright-turning and straight lanes and the coil is located in aroad adjacent to a commercial pedestrian street with a heavyflow of people and traffic A logarithmic model is appliedto describe traffic flow with large density and therefore itis accepted that the coil data of the outer lane satisfy thelogarithmic model

The loop detector of the inner lane measures the left-turning lane which also has heavy traffic flow The speed-density relation of the inner lane does not satisfy the singlelog model but is suitable for the segmentation model The

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Mobile Information Systems

Table 3 Validation results of the model of the outer lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 619 658 662 661 650Lower value = 020Upper value = 085 689 731 736 733 722Lower value = 015Upper value = 090 745 788 804 797 784Lower value = 010Upper value = 095 842 851 867 853 853Lower value = 005

Two-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(a) Validation result for two-week data

Four-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(b) Validation result for four-week data

Six-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Logarithmic model (upper)Logarithmic model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 5 Validation result of the speed-density logarithmic model of the outside lane

Mobile Information Systems 9

Table 4 Validation results of the model of the inner lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 800 779 813 788 795Lower value = 020Upper value = 085 820 823 852 831 832Lower value = 015Upper value = 090 838 849 864 853 851Lower value = 010Upper value = 095 890 903 889 898 895Lower value = 005

which suggests that a singlemodel cannot accurately describethe quantile set of the coil data Thus we consider using asegmentation model

In the density-flow curve there is a critical density 119896119898which is the density of maximum traffic flow as shown inFigure 1 When the density 119896 lt 119896119898 the traffic is in a stateof flow when 119896 gt 119896119898 the traffic flow gradually becomescrowded Therefore consider using 119896119898 as the critical value ofthe subsection

A density-flow curve is obtained by local polynomialregression fitting and the density value at the curve vertexis just 119896119898 Take 119896119898 as the critical value and piecewise analyze119906upper119896lt119896119898

119906lower119896lt119896119898

119906upper119896gt119896119898

and 119906lower119896gt119896119898

The analysis shows that thequantile set 119906upper

119896lt119896119898

and 119906lower119896lt119896119898

agrees with the exponentialmodel and the quantile set 119906upper

119896gt119896119898

and 119906lower119896gt119896119898

has goodagreement with the logarithmic model

119906095 = 69647 times 119890minus11989614449 + 12716 119896 lt 1198961198988227 times ln(254971119896 ) 119896 ge 119896119898

119906005 = 5108 times 119890minus11989610010 minus 2245 119896 lt 1198961198985337 times ln(243306119896 ) 119896 ge 119896119898

(19)

Figure 6 shows the fitting result of a segmentation modelof the outer lane when upper value = 095 and lower value= 005 and (19) are the models corresponding to the greencurve and blue curve in Figure 6 which is the speed-densitymodel of interrupted traffic flow via the new descriptionmethod 119875 value of each parameter is very small suggestingthe coefficient is very significant

The coil data of the inner lane for two weeks four weekssix weeks and eight weeks are respectively selected and fourgroups of parameters are established for themodel validationthe same as that for the outer lane Table 4 gives the ratio ofthe data between two logarithmic curves to the total amountof data in each case Comparing the result with that of theouter lane we find that the validation results of the model ofthe inner lane are better with greater ratio

Take a longitudinal observation similarly it is obviousthat with upper value increasing and lower value decreasingthe proportion increases accordingly where amplitudes aresmaller than that of outer lane respectively 37 19 and43 The main transverse trend is the same as outer lane

km

095 quantile fractileMultisession model (upper)005 quantile fractileMultisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

Figure 6 Speed-density multisession model of the inside lane

except the case of upper value = 095 and lower value = 005The result indicates that the two segmentation models aresuitable for describing the speed-density relation of the innerlane Figure 7 shows the four groupsrsquo validation results whenupper value = 095 and lower value = 005

43 Experimental Result Analysis

431 Difference between the Models of the Outer Lane andthe Inner Lane The loop detector of the outer lane measuresright-turning and straight lanes and the coil is located in aroad adjacent to a commercial pedestrian street with a heavyflow of people and traffic A logarithmic model is appliedto describe traffic flow with large density and therefore itis accepted that the coil data of the outer lane satisfy thelogarithmic model

The loop detector of the inner lane measures the left-turning lane which also has heavy traffic flow The speed-density relation of the inner lane does not satisfy the singlelog model but is suitable for the segmentation model The

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 9

Table 4 Validation results of the model of the inner lane

Parameters Two-week data Four-week data Six-week data Eight-week data AverageUpper value = 080 800 779 813 788 795Lower value = 020Upper value = 085 820 823 852 831 832Lower value = 015Upper value = 090 838 849 864 853 851Lower value = 010Upper value = 095 890 903 889 898 895Lower value = 005

which suggests that a singlemodel cannot accurately describethe quantile set of the coil data Thus we consider using asegmentation model

In the density-flow curve there is a critical density 119896119898which is the density of maximum traffic flow as shown inFigure 1 When the density 119896 lt 119896119898 the traffic is in a stateof flow when 119896 gt 119896119898 the traffic flow gradually becomescrowded Therefore consider using 119896119898 as the critical value ofthe subsection

A density-flow curve is obtained by local polynomialregression fitting and the density value at the curve vertexis just 119896119898 Take 119896119898 as the critical value and piecewise analyze119906upper119896lt119896119898

119906lower119896lt119896119898

119906upper119896gt119896119898

and 119906lower119896gt119896119898

The analysis shows that thequantile set 119906upper

119896lt119896119898

and 119906lower119896lt119896119898

agrees with the exponentialmodel and the quantile set 119906upper

119896gt119896119898

and 119906lower119896gt119896119898

has goodagreement with the logarithmic model

119906095 = 69647 times 119890minus11989614449 + 12716 119896 lt 1198961198988227 times ln(254971119896 ) 119896 ge 119896119898

119906005 = 5108 times 119890minus11989610010 minus 2245 119896 lt 1198961198985337 times ln(243306119896 ) 119896 ge 119896119898

(19)

Figure 6 shows the fitting result of a segmentation modelof the outer lane when upper value = 095 and lower value= 005 and (19) are the models corresponding to the greencurve and blue curve in Figure 6 which is the speed-densitymodel of interrupted traffic flow via the new descriptionmethod 119875 value of each parameter is very small suggestingthe coefficient is very significant

The coil data of the inner lane for two weeks four weekssix weeks and eight weeks are respectively selected and fourgroups of parameters are established for themodel validationthe same as that for the outer lane Table 4 gives the ratio ofthe data between two logarithmic curves to the total amountof data in each case Comparing the result with that of theouter lane we find that the validation results of the model ofthe inner lane are better with greater ratio

Take a longitudinal observation similarly it is obviousthat with upper value increasing and lower value decreasingthe proportion increases accordingly where amplitudes aresmaller than that of outer lane respectively 37 19 and43 The main transverse trend is the same as outer lane

km

095 quantile fractileMultisession model (upper)005 quantile fractileMultisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

Figure 6 Speed-density multisession model of the inside lane

except the case of upper value = 095 and lower value = 005The result indicates that the two segmentation models aresuitable for describing the speed-density relation of the innerlane Figure 7 shows the four groupsrsquo validation results whenupper value = 095 and lower value = 005

43 Experimental Result Analysis

431 Difference between the Models of the Outer Lane andthe Inner Lane The loop detector of the outer lane measuresright-turning and straight lanes and the coil is located in aroad adjacent to a commercial pedestrian street with a heavyflow of people and traffic A logarithmic model is appliedto describe traffic flow with large density and therefore itis accepted that the coil data of the outer lane satisfy thelogarithmic model

The loop detector of the inner lane measures the left-turning lane which also has heavy traffic flow The speed-density relation of the inner lane does not satisfy the singlelog model but is suitable for the segmentation model The

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Mobile Information Systems

Two-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80Sp

eed

(km

h)

(a) Validation result for two-week data

Four-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)(b) Validation result for four-week data

Six-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(c) Validation result for six-week data

Eight-week data

Multisession model (upper)Multisession model (lower)

0 50 100 150 200 250Density (pcukm)

0

20

40

60

80

Spee

d (k

mh

)

(d) Validation result for eight-week data

Figure 7 The validation result of the speed-density multisession model of the inside lane

exponentialmodels of (19) are two-item typeswith interceptsinstead of Underwoodrsquos monomial exponential model Theysuggest that the traffic flow of the inner lane differs from thetraffic flow of the freeway and outer lane

432 Characteristic Analysis of Traffic Flow The criticaldensity 119896119898 of the outer lane in Figure 6 is 536 pcukm andmost of the density values are less than 119896119898 or in a smallrange of 119896119898 similarly for the inner lane most of the densityvalues are smaller than 119896119898 and the data in the range 119896 gt 119896119898

are sparse This illustrates that (1) most of the time the roadsegmentwhere loop detectors located is unblocked where thedata with big density values which may lead to congestionis just a small proportion and (2) compared with the outerlane the proportion of the density 119896 lt 119896119898 of the inner laneis greater illustrating that the inner lane is more unimpededthan the outer lane

In summary the new description method can satisfacto-rily describe the speed-density relation of interrupted trafficwhere the speed-density relation of the outer lane meets the

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mobile Information Systems 11

logarithmic model and the inner lane meets the segmentmodel What is more the road segment where loop detectorslocated is unblocked at most of the time the inner lane ismore unimpeded than the outer lane

5 Conclusion

In this paper the characteristics of urban interrupted flowdata were analyzed and it was found that they differ fromthe data of uninterrupted flow Since the existing classicalmodels cannot describe them very well a descriptionmethodof speed-density relation for interrupted traffic flow wasproposed where the upper and lower curves were used asthe upper and lower bounds of the predicted speed In thismethod the speed was divided into small data sets whichsatisfied the normal distribution and two quantiles of normaldistribution were obtained as the predicted values Then twoquantile sets were fitted to get two curves as the speed-densityrelation model of the interrupted traffic flow Finally the coildata of the outer and inner lanes were applied for modelvalidation The results showed that the new method can givea good description of the speed-density relationship of inter-rupted traffic flow and get differentmodel results for the outerlane and inner lane whereby the speed-density relation of theouter lane satisfies the logarithmic model and the inner lanesatisfies the segmentmodel instead of the singlemodel wherewhen the density is less than critical density it conforms tothe exponential model and otherwise the logarithmic modelThe fitting results of the internal and external lanes were ana-lyzed in combination with the actual local road environmentand traffic flow theory So this model can provide favorabledata analysis and presentation for city traffic thus to providedecision support for intelligent transportation

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The work is partly supported by NSFC (no 61472149) theFundamental Research Funds for the Central Universities(2015QN67) the Wuhan Youth Science and Technology Plan(2016070204010132) and the National 863 Hi-Tech Researchand Development Program under Grant 2015AA01A203

References

[1] D B Greenshields R J Biddins S W Channing et al ldquoA studyin highway capacityrdquo Highway Research Board Proceedings vol14 no 1 pp 448ndash477 1935

[2] H Z Wang D h Ni Q-Y Chen and J Li ldquoStochastic mod-eling of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

[3] A K Gupta S Sharma and P Redhu ldquoAnalyses of latticetraffic flow model on a gradient highwayrdquo Communications inTheoretical Physics vol 62 no 3 pp 393ndash404 2014

[4] B G Heydecker and J D Addison ldquoAnalysis and modelling oftrafficflowunder variable speed limitsrdquoTransportationResearchPart C Emerging Technologies vol 19 no 2 pp 206ndash217 2011

[5] X-LMaD-FMaD-HWang and S Lin ldquoModeling of speed-density relationship in traffic flow based on logistic curverdquoChina Journal of Highway and Transport vol 28 no 4 pp 94ndash100 2015

[6] C-F Shao C-Z Xiao B-B Wang and M Meng ldquoSpeed-density relationmodel of congested trafficflowunderminimumsafety distance constraintrdquo Journal of Traffic and TransportationEngineering vol 15 no 1 pp 92ndash99 2015

[7] H Z Wang J Li Q-Y Chen and D Ni ldquoLogistic modelingof the equilibrium speed-density relationshiprdquo TransportationResearch Part A Policy and Practice vol 45 no 6 pp 554ndash5662011

[8] V CorcobaMagana andMMunoz-Organero ldquoWATI warningof traffic incidents for fuel savingrdquoMobile Information Systemsvol 2016 Article ID 3091516 16 pages 2016

[9] M-W Li W-C Hong and H-G Kang ldquoUrban traffic flowforecasting using GaussndashSVR with cat mapping cloud modeland PSO hybrid algorithmrdquo Neurocomputing vol 99 no 1 pp230ndash240 2013

[10] F Ahmad I Khan S A Mahmud et al ldquoReal time evaluationof shortest remaining processing time based schedulers fortraffic congestion control using wireless sensor networksrdquoin Proceedings of the International Conference on ConnectedVehicles and Expo (ICCVE rsquo13) pp 381ndash391 Las Vegas NevUSA 2013

[11] H Y Shang and Y Peng ldquoA new cellular automaton modelfor traffic flow considering realistic turn signal effectrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1624ndash1630 2012

[12] K Jung M Do J Lee and Y Lee ldquoVehicle running characteris-tics for interrupted traffic flow by using cellular automatardquoTheJournal of The Korea Institute of Intelligent Transport Systemsvol 11 no 6 pp 31ndash39 2012

[13] J M del Castillo ldquoThree new models for the flow-densityrelationship derivation and testing for freeway and urban datardquoTransportmetrica vol 8 no 6 pp 443ndash465 2012

[14] T-Q Tang L Caccetta Y-HWu H-J Huang and X-B YangldquoA macro model for traffic flow on road networks with varyingroad conditionsrdquo Journal of Advanced Transportation vol 48no 4 pp 304ndash317 2014

[15] X B Yang Z Y Gao H W Guo and M Huan ldquoSurvivalanalysis of car travel time near a bus stop in developingcountriesrdquo Science China Technological Sciences vol 55 no 8pp 2355ndash2361 2012

[16] R Akcelik ldquoRelating flow density speed and travel timemodelsfor uninterrupted and interrupted trafficrdquo Traffic Engineeringand Control vol 37 no 9 pp 511ndash516 1996

[17] R Jiang and Q-S Wu ldquoThe traffic flow controlled by thetraffic lights in the speed gradient continuum modelrdquo PhysicaA Statistical Mechanics and Its Applications vol 355 no 2ndash4pp 551ndash564 2005

[18] F J Wang W Wei H S Qi et al ldquoResearch on discontinuousflow speed-density relation modelrdquo Journal of Highway andTransportation Research andDevelopment vol 31 no 7 pp 108ndash114 2014

[19] H Wang D Ni Q-Y Chen and J Li ldquoStochastic model-ing of the equilibrium speed-density relationshiprdquo Journal ofAdvanced Transportation vol 47 no 1 pp 126ndash150 2013

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

12 Mobile Information Systems

[20] F Soriguera and F Robuste ldquoEstimation of traffic stream spacemean speed from time aggregations of double loop detectordatardquo Transportation Research Part C Emerging Technologiesvol 19 no 1 pp 115ndash129 2011

[21] Y Lao G Zhang J Corey and Y Wang ldquoGaussian mixturemodel-based speed estimation and vehicle classification usingsingle-loopmeasurementsrdquo Journal of Intelligent TransportationSystems Technology Planning and Operations vol 16 no 4 pp184ndash196 2012

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014