Vehicular Traffic Flow Dynamics on a Bus Route

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

MULTISCALE MODEL. SIMUL. c© 2013 Society for Industrial and Applied MathematicsVol. 11, No. 3, pp. 925–942

VEHICULAR TRAFFIC FLOW DYNAMICS ON A BUS ROUTE∗

INGENUIN GASSER† , CORRADO LATTANZIO‡ , AND AMELIO MAURIZI‡

Abstract. We propose a new model in the analysis of car traffic flow phenomena which inparticular refers to a bus route, namely, a closed path embedded in the urban network of roads.The model consists in a scalar balance law for the (macroscopic) density of cars, with source termsrepresenting incoming and outgoing roads (with respect to the bus route), and a coupled systemof ODEs describing the dynamics of (microscopic) buses along the route. The coupling takes intoaccount both the buses as moving bottlenecks in the car flow and the effect of the car flow on thebuses’ ODEs as microscopic vehicles. The resulting fully coupled micro–macro model promises togive an appropriate description for the car flow as well as for the buses. Finally, we study the modelnumerically.

Key words. traffic flow, moving bottlenecks, micro–macro coupled systems

AMS subject classifications. 35L65, 34A34

DOI. 10.1137/130906350

1. Introduction. Due to its strong impact on everyday experience, vehiculartraffic flow is widely studied by researchers from all areas, ranging from engineeringto mathematics, and using different techniques and models. Many different questionsare of interest, requiring different methods and models.

Our main concern is the study of the dynamics of the vehicular traffic flow along abus route, namely, a circular path embedded in an urban traffic network, where bothcars and buses are traveling, and with mutual interaction between them. In addition,we assume the presence of bus stops on the bus route, and we assume that the buseson the bus route know about the position of the bus in front. Therefore, we need todescribe the regular vehicular traffic, the bus traffic, and the interaction between thebuses and the regular traffic.

As far as the modeling of the above-mentioned situation is concerned, we can relyon a huge number of available models and extensive literature. In what follows, weshall concentrate mainly on two kinds of models: macroscopic models, describing thedynamics of macroscopic quantities as density and velocity of the whole flow, andmicroscopic models.

The former were introduced in the 1950s by Lighthill and Whitham [25] andRichards [30] and have been extensively developed in recent years in different scientificcommunities ranging from mathematicians to traffic engineers.

On the other side, when we deal with microscopic models, we describe the dynam-ics of each single car (or bus) in terms of ODEs for its position and velocity. Amongothers, we recall here at the very beginning the follow-the-leader model [16] and theso-called optimal velocity model, introduced by Bando et al. (see, for instance, [1]),and adapted to bus routes in [17].

∗Received by the editors January 17, 2013; accepted for publication (in revised form) June 21,2013; published electronically September 19, 2013.

http://www.siam.org/journals/mms/11-3/90635.html†Center for Differential Equations and Dynamical Systems, University of Hamburg, D-20146

Hamburg, Germany ([email protected]).‡Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Universita degli Studi

dell’Aquila, Via Vetoio, I-67010 Coppito (L’Aquila) AQ, Italy ([email protected], [email protected]).

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926 I. GASSER, C. LATTANZIO, AND A. MAURIZI

Aiming to describe the above-mentioned situation of a bus route embedded in atraffic network, we have to face different problems:

1. To describe the regular traffic on a network, a macroscopic approach seemsto be more appropriate. A macroscopic model is clearly an additional simplification,because it gives a description of the phenomenon in terms of average quantities butallows much faster simulations.

2. Due to the low number of buses on a bus route, a microscopic description isadequate. It also allows a good description of the interaction between the buses.

3. To include simple bus stop modeling approaches known from the microscopictraffic world.

4. To properly couple macro and micro models.5. To take care that the bus route is embedded in a bigger network of regular

vehicular traffic.All of this asks for a combination of a microscopic and a macroscopic model, giving riseto a fully coupled multiscale model—the microscopic for buses and the macroscopicfor the surrounding vehicular traffic on the closed bus route.

As far as 2 and 3 are concerned, it turns out that the aforementioned Bando-type follow-the-leader discrete models correctly describe the dynamics of the discretevehicles (buses) traveling on the route, because they allow the buses to regulate theirspeed with respect to the headway. In case of additional bus stops the behavior isslightly different with respect to the classical follow-the-leader model (see, for instance,[29]). A correction of the optimal velocity function (made by Huijberts [17]) allowsthe inclusion of bus stops, thus giving an optimal dynamics in terms of waiting timeof passengers at bus stops (not too long) and in terms of efficiency of the bus service(not too short waiting time). Related discussions about headways in a bus route havebeen carried out by Daganzo, Pilachowski, Bartholdi, III, and Eisenstein in [8, 10, 2].

As far as 1 is concerned, we take a Lighthill–Whitham–Richards (LWR)–typemodel. In order to handle 4, we have to fully couple this model to the Bando-type bus route model. The buses are included in the LWR-type model in terms ofmoving bottlenecks, that is, a nonnegligible dropping of flux capacity caused by thepresence of the buses. Moreover, the coupling of the buses with the whole flow istaken into account by making the optimal velocity function dependent also on theaverage velocity of cars (which is itself dependent on the density) and by defining thisvelocity as a threshold for admissible velocities for buses.

There is a long tradition in describing moving bottlenecks with the kinematicwave theory. A very first step was done by Gazis and Herman in [15]. In [28] Newellconsidered a moving convoy with a given movement, and its influence on the regularvehicular traffic is studied in a stationary homogeneous setting. In [23] Lebacque,Lesort, and Giorgi studied how to include a bus in a first order model of LWR type:the bus, whose path is assigned a priori, is lowering the equilibrium speed function. Inthis setting, some numerical results are provided via a Godunov-type scheme. More-over, Munoz and Daganzo [27] found a quite complete (empirical) theory based onexperiments: they introduced a (velocity-dependent) passing rate, treating the mov-ing bottleneck as a boundary condition which can be integrated with kinematic wavetheory. Daganzo and Laval, in [9], presented a numerical method to model kinematicwave traffic streams containing slow vehicles. Slow discrete vehicles are representedwith moving boundaries and can affect the traffic stream, but not vice versa. Thesame authors discussed moving bottlenecks in [21, 22]: again, the moving bottlenecksatisfies a microscopic model, with a given (even geometry-dependent) acceleration,and including a feedback from the regular vehicular traffic on the bottleneck. This

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VEHICULAR TRAFFIC FLOW DYNAMICS ON A BUS ROUTE 927

also includes the case of more than one (independent) bottleneck. A unified frame-work of moving bottleneck models based on the LWR theory is discussed by Leclercq,Chanut, and Lesort in [24]. Moreover, the dynamic traffic assignment model proposedby Juran et al. in [18] can evaluate the effects of moving bottlenecks on network per-formance; however, the movement of the bottleneck is predefined, e.g., for plannedconvoys. Therefore, the effect of the traffic stream on the slow vehicle is not con-sidered. On the other hand, in applied mathematical literature moving bottleneckswere considered theoretically and numerically by Lattanzio, Maurizi, and Piccoli in[20, 19].

In summary, our model promises to give an accurate and reasonable descriptionof the bus route, and it includes in particular the following features:

1. It is a continuous model based on a scalar balance law for the macroscopicdensity and a system of ODEs for the microscopic positions of the buses.

2. It is fully coupled, thus taking into account both the effect of the bottle-necks on the surrounding flow and the effect of the whole density on the bottlenecks’dynamics.

3. In addition, the buses’ dynamics are governed by a follow-the-leader–typemodel which properly takes into account buses’ headway dynamics and bus stops.

4. The bus route is easily embedded in a network of roads. Indeed, the macro-scopic modeling choice for the global density allows us to treat nodes of the urbannetwork as source terms, hence giving advantages in terms of numerical simulations.To the best of our knowledge, up to now the aforementioned features were not includedall together in a single model. As a result we obtain very interesting dynamics forboth the vehicular traffic and the buses, e.g., referring to 3, we detect reasonablebus headway dynamics (see Examples 2 and 4), oscillations in buses’ headways, and“periodic-like” solutions (see Example 5).

The remaining part of this paper is organized as follows. In the next sectionwe describe our model, starting from the macroscopic part. Then in section 2.3 weintroduce the microscopic part of our model, describing in particular the couplingwith the macroscopic part and the modeling of overtaking buses. Section 3 is devotedto the description of the numerical algorithm [3]. Finally, numerical simulations arepresented in the last section.

2. Description of the model.

2.1. Macroscopic model. Let us consider a constant L > 0 representing thelength of a circular bus route, where we shall study the density of cars in the presenceof N buses, which will be responsible for moving bottlenecks for the traffic flow [20,19]. This situation can be described in terms of a coupled macroscopic–microscopicmodel for the total density of cars and the dynamics of discrete vehicles. For themacroscopic model we choose a fluidodynamical description by using a Lighthill–Whitham–Richards-type model [25, 30], namely, a balance law with a source termmodeling the incoming and outgoing roads along the closed bus route:

(2.1)

⎧⎪⎨⎪⎩ρt + f(x, y1(t), . . . , yN (t), ρ)x = g(t, x, ρ), t > 0, N ∈ N,

ρ(0, x) = ρ0(x), x ∈ [0, L],

ρ(t, 0) = ρ(t, L), t ≥ 0,

where ρ = ρ(t, x) ∈ [0, ρmax] is the density of cars and yi = yi(t) is the distance theith discrete vehicle has covered at time t. The flux function f is given by

(2.2) f(x, y1(t), . . . , yN (t), ρ) = ρ · v(ρ) · Φ(x, y1(t), . . . , yN(t)),

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where v(ρ) is the average speed of cars, and, as is customary for traffic flow models,it is assumed to satisfy

(i) v : [0, ρmax] → [0, vmax];(ii) v(ρmax) = 0 and v(0) = vmax;(iii) v(ρ) is smooth and decreasing;

(iv) d2(ρv)dρ2 < 0.

In (2.2), Φ is the function responsible for the coupling with the ODEs describing theposition yi of the buses (see section 2.3 for the description of the related dynamics).It is based on a special choice of cut-off functions, leading to a capacity-drop effectin the model (see section 2.4). In this way, we obtain a mathematical model of themoving bottleneck phenomenon: the flux capacity of the road is indeed reduced wherediscrete moving vehicles are located.

The function ρ0 gives the initial density of the cars on the closed bus route. Thelast relation in (2.1) presents periodic boundary conditions for the car density in orderto model the closed bus route. Following [11], we describe the source term g(t, x, ρ).

2.2. Source term. The bus route is a closed path covered by a bus into anurban road network; it is crossed by other roads having incoming and outgoing lanes.Thus we can model such a situation in terms of pointwise sources, which will indeedbe approximated by a simple function (for a more complete mathematical treatmentof vehicular traffic flow on urban networks of roads, see [12]). Therefore, if xk isthe central point of the kth crossing road, and 2Δx is its width, we define special“window” functions

δ+k (x) =

{1

ρmaxΔx, xk < x < xk +Δx,

0 otherwise

for incoming lanes and

δ−k (x) =

{1

ρmaxΔx, xk −Δx < x < xk,

0 otherwise

for outgoing lanes, with 0 < Δx � L, k = 1, . . . ,K.Hence we can represent the source term using the incoming rates ν+k (t, ρ) and

the outgoing rates ν−k (t, ρ) of cars as follows (rates are weighted with the “window”functions):

g(t, x, ρ) =

K∑k=1

(δ+k (x)ν

+k (t, ρ)− δ−k (x)ν

−k (t, ρ)

).

The incoming rate ν+k (t, ρ) at time t is related to the number of cars occupying theroad between the kth junction and the following one at that time, which is

ρk(t) =1

xk+1 − xk

∫ xk+1

xk

ρ(t, x)

ρmaxdx.

Let us now fix a maximal constant incoming density for the kth cross road ρinc,k.Then we introduce a function r(ρ) ∈ C1([0, 1]) representing the probability that a carenters the bus route at a junction with the following properties:

1. r(0) = 1;

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2. drdρ < 0 for any 0 < ρ < λ0, λ0 ∈ (0, 1);

3. drdρ > 0 for any λ0 < ρ < 1;

4. r(1) = 12 .

The first property follows from the fact that when ρ = 0 every incoming car canenter the bus route, while the last one describes the following behavior: wheneverboth the route and the incoming road are full of cars, then cars enter the junctionalternately with the same (1/2) probability. Between the two extreme density valuesthere is a minimum in the entering probability reflecting the known effect that it ismore difficult to enter at an intermediate density. A possible choice for this functionis

r(ρ) = 2ρ2 − 5

2ρ+ 1;

see Figure 2.1.In order to get the rate of the incoming cars, we must focus now on their velocities.

Let vbound,k ∈ (0, vmax) be a bound on the possible velocities due to the physicalcharacteristic of the kth junction. Then cars enter the bus route with speed

vjunc,k(ρ) = min{vbound,k, v(ρ)Φ

}.

Then we are ready to give the expression of the incoming rate of cars at the kthjunction:

ν+k (ρ) = ρinc,k · vjunc,k(ρ) · ρmax · r(ρk), k = 1, . . . ,K,

where 1/ρmax is the length of a car.

vjunk,k(ρ) · pk · ρ(t, x) · ρmax

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r()

Fig. 2.1. On the left: The schematic probability of outgoing cars. On the right: The graph ofthe incoming probability functions r(ρ).

Finally, for the kth outgoing road it is sufficient to observe that a percentage ofthe total density of cars is leaving the road, so that

ν−k (ρ) = vjunc,k(ρ) · pk · ρ(t, x) · ρmax, k = 1, . . . ,K,

where pk ∈ (0, 1) is given, and the source term is completely described.From now on, we shall rescale the variables so that ρmax = 1 and vmax = 1.Remark 2.1. It is worthwhile to mention that our source term g(t, x, ρ) prevents

the numerical realization of the density from going out of the physical interval [0, 1],

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being that g(t, x, 0) ≥ 0 and g(t, x, 1) ≤ 0. This clearly excludes any blow-up for ρtime asymptotically. Further studies about long time behavior for scalar balance lawsare clearly connected with our results, but their analytical treatment is far beyondthe scope of this paper. For a discussion on existing results about long time behaviorfor scalar balance laws, we refer the reader to Chapter 11, and in particular section11.12, in [7].

2.3. Microscopic model. The microscopic model is a follow-the-leader–typemodel [1, 16], where the behavior of the drivers of the vehicles is influenced by thebehavior of the drivers ahead. In particular, we choose for t > 0 the following “secondorder” model, which is an adaptation of those considered in [1, 17]:(2.3)⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

yi(t) =min{νi(t),

(v(ρ(x, t))Φ(x, y1(t), . . . , yN (t))

)∣∣x=yi(t)

},

νi(t) =− 1

τi

(νi(t)

−min{V i0

(yi+1(t)− yi(t)

),(v(ρ(x, t))Φ(x, y1(t), . . . , yN(t))

)∣∣x=yi(t)

}),

yi(0) = y0i ,

yi(0) = ν0i

for any i = 1, . . . , N . In (2.3) the buses’ positions yi have to be intended modulo L,and moreover we have to define yN+1 in the Nth equation. Being that the bus routeis circular, we have the first car in front of the Nth car. In order to obtain the positiveheadway we define yN+1 = y1 +L; i.e., the (fictive) (N +1)th car is the first car withone more turn-around.

In (2.3), τi > 0 is the relaxation time of each driver, and the function V i0 : R → R

stands for the velocity adaption function, describing the reaction of the ith driver tothe distance from the vehicle ahead. To take into account the effect of the density ofcars along the bus route, we introduce the coupling with the fluid dynamic part interms of the velocity of the related flow, namely, v(ρ)Φ. As in [21, 22], the coupling in(2.3)1 guarantees that the velocity of the buses is at most the flow velocity, dependingthus on the surrounding density, which becomes the maximal possible velocity whenthe effects of vehicles ahead are negligible. Moreover, the same kind of threshold isintroduced in (2.3)2, so that the drivers of discrete vehicles shall indeed adapt theirvelocities to the minimum between V i

0 and v(ρ)Φ. It is worthwhile to observe thatin (2.3) the term yi represents the real velocity of the ith bus, while νi stands forits ideal velocity, namely, in the absence of cars along the route. More precisely, yicoincides with νi below the threshold defined by v(ρ)Φ, while it is fixed to be the flowvelocity v(ρ)Φ whenever νi ≥ v(ρ)Φ, without further microscopic dynamics. Finally,in the former case, the dynamics of νi, and hence of yi, is completed according to thesecond equation, which takes into account the velocity of the whole flow and also theposition of the bus ahead, giving rise to the desired follow-the-leader effect.

Denoting by di(t) = yi+1(t)−yi(t) the headway, we can define the optimal velocityfunction as follows [26]:

Vopt(di) = d2i /(σ + d2i )

for a constant σ > 0. This is a typical monotone increasing in headway choice forfollow-the-leader car models. In the case of a bus route we want to include the effectof bus stops. An important effect of bus stops is that they reduce the (optimal)

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velocity of a bus the bigger the distance to the bus in front is. Therefore, a so-called velocity adaption function is introduced to correct the optimal velocity Vopt

from above. A possible expression for the velocity adaption function, with a suitablechoice of parameters, is [17]

(2.4) V i0 (di) = [γi + (1− γi)e

−λidi ] · Vopt(di),

where λi > 0 is a parameter depending on the physical properties of the route, and0 < γi < 1 is the desired speed of the ith driver, namely, the speed when the vehicleahead is sufficiently far away. If all vehicles (and drivers) have the same characteristics,we can assume that V i

0 = V0, and the (smooth) function V0 verifies the followingtypical properties:

1. V0(0) = 0;2. the (smooth) function V0 is strictly increasing on (0, d0) and strictly decreas-

ing on (d0,+∞), and the unique maximum at d0 > 0 is at most equal to the maximalvelocity, fixed to be 1;

3. limd→+∞ V0(d) = γ, where 0 < γ < 1;see Figure 2.2. In what follows, we shall refer to this case as the homogeneous case.As we can see, bus stops are modeled via the optimal velocity function. They arenot modeled by introducing local space dependent modifications of input data orfunctions.

0 10 20 30 40 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

yi+1−yi

V0

Fig. 2.2. The typical graph (solid line) of the function V0(d), given in (2.4) with an appropriatechoice of the parameters. Dotted lines represent the (increasing) function Vopt and the (exponentiallydecreasing) coefficient in the definition of V0(d).

Remark 2.2. The two optimal velocity functions for cars and buses are differentin this paper. The buses adhere to V0(d), given in (2.4), and the cars adhere toa macroscopic average velocity v(ρ), which corresponds to a microscopic speed forcars vcar(d) = v(1/d), d ∈ [1,∞). A maximal density ρmax = 1 corresponds to aminimal headway dmin equal to the length of a car. This effect is typically ignored inmicroscopic models by assuming that single vehicles are points.

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Remark 2.3. Even if the velocity adaption function V i0 is defined in principle

in the whole R, it cannot be considered for di < 0 under the modeling point ofview. Indeed, this will lead to a situation for which the drivers adapt their velocitiesaccording to the vehicle behind, losing the follow-the-leader flavor of the model, andgiving rise to nonoptimized dynamics for buses. However, as is well known in theliterature [5, 6], the discrete model considered here does admit overtaking, which leadsto negative values for di. Thus, we should also expect overtaking for our coupledmodel: for instance, for small values of density ρ, the dynamic of discrete vehiclesbasically uncouples with the fluid dynamic part.

Since (2.3) admits overtaking, let us assume there exist a time to and an indexio such that yio(to) = yio+1(to) and clearly νio(to) ≥ νio+1(to), because yio is initiallybehind yio+1. Then the model shall evolve according to the (continuous) dynamicsdefined by the ODEs (2.3) only in the time interval (0, to], and we shall define arestarting procedure for t = to according to the relabeling of indices (following theideas of [5, 6]). Indeed, we introduce a new index j, such that, for t ≥ to, and up tothe next overtake, jo = io + 1, jo + 1 = io, and j = i for any i �∈ {io, io + 1}. Then,we shall consider the continuous dynamics defined by (2.3) for the “new” variablesyj , νj with initial conditions at t = to, and up to the next overtake. In this contextwe mention the papers [5, 6] with analytical results on overtaking.

2.4. Cut-off in the flux term. To conclude the description of the model, wenow define the function Φ in the flux of the balance law, taking into account the bot-tleneck behavior introduced in the fluid dynamic part. In [19] the authors presentedtwo possible definitions. The first possibility is based on the product among cut-offfunctions ϕi(x − yi(t)), and, to avoid a nonrealistic reduction of flux capacity at thepoint where eventually more than one bus is located, overtakes are indeed excludedby a suitable definition of velocities of discrete vehicles. However, we are treating thecase of possible overtaking among buses, and thus we use here the second modelingchoice, which is

Φ(x, y1(t), . . . , yN (t)) = mini=1,...,N

{ϕi(x− yi(t))

}.

In this case, when two buses are occupying the same position, we do not get anexcessive drop of the flux capacity of the road: this can describe the realistic situation,for instance, in which one bus is stopping at a bus stop and the other one is overtakingit.

3. Numerical algorithm. Numerical simulations can be done with a slightmodification of the algorithm proposed in [4]. A splitting method is used for themacroscopic model, solving alternatively the homogeneous conservation law and theODE corresponding to the source term, by keeping the positions of the buses constant.Then, we can trace the trajectory of each bus using a modified explicit (first order)Euler scheme. Hence, each time step of the algorithm is composed of the followingtwo steps:Step 1. The splitting method computes the solution in two substeps. In the first,

the density value satisfying the homogeneous version of (2.1) is computed bya Godunov scheme, and, in the second, the ODE for the source term is solvedwith the explicit Euler scheme. In this step the buses’ positions are fixed.

Step 2. The buses’ positions are determined solving (2.3) by means of a trackingalgorithm which determines the positions of the buses using the density ob-tained in Step 1.

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3.1. Step 1: Numerical algorithm for density of cars. In order to describethe Godunov algorithm for the (homogeneous) conservation law

ρt + f(x, y1(t), . . . , yN (t), ρ)x = 0,

we need to introduce a grid with the following notation:1. x is the space grid size on the road [0, L];2. t is the time grid size on the time interval [0, T ];3. (tp, xm) = (pt,mx) are the grid points for p = 0, . . . , P andm = 0, . . . ,M

with P and M , respectively, the number of time and space nodes of the grid, Mx =L, Pt = T .For a function η defined on the grid we write ηpm = η(tp, xm), p, m integers. Piecewiseconstant approximation of the initial datum is derived by averaging on the interval[xm−1/2, xm+1/2], and it is used as the initial datum of the Riemann problems. As in

[3], the CFL condition in our case reduces to t ≤ 12x.

It is possible to define a unique solution to the (homogeneous) conservation lawin the strip (tp, tp+1)×R by piecing together the solutions obtained in each cell. Theexact solution obtained in this way is then projected on a piecewise constant functionagain by averaging on the aforementioned intervals.

Finally, the scheme can be expressed as a three-point scheme as follows:

ρp+1/2m = ρpm − t

x(F (xm+1/2, y

p1 , . . . , y

pN , ρpm, ρpm+1)

− F (xm−1/2, yp1 , . . . , y

pN , ρpm−1, ρ

pm)).

The numerical flux F is defined for any m and p as

F (·, yp1 , . . . , ypN , ρpm−1, ρpm) = f(·, yp1 , . . . , ypN ,R (

0; ρpm−1, ρpm

)),

where

R(x− xm−1/2

Δt; ρpm−1, ρ

pm

)

stands for the solution to the Riemann problem centered at xm−1/2 and with left

and right values, respectively, ρpm−1 and ρpm. The reconstructed value ρp+1/2m is the

intermediate state before computing the source term, and we set ypi = yi(tp), hencekeeping the buses’ positions fixed at the previous time step.

The source term is included by solving the ODE

ρt = g(t, x, ρ)

with the explicit Euler scheme, using the density previously obtained in the Godunovscheme. So we can write

ρp+1m = ρp+1/2

m +t · g(tp, xm, ρp+1/2m ).

We use the density computed in this way as the initial datum for the next step.

3.2. Step 2: Bus path. Without loss of generality we suppose

y1(0) < y2(0) < · · · < yN(0),

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with yN (0) < L.Grid points divide the road into subintervals Cmp = [(mp − 1)x,mpx[ of

length x. In [4], at each time tp the authors determine the new position yp+1i of the

generic microscopic vehicle by studying interactions of the trajectory of a bus withwaves within a fixed cell of the numerical grid, possibly updating the cell index. Weevaluate the new position yp+1

i of the generic microscopic vehicle in a simplified way.We start by defining ρ(tp, x), x ∈ [0, L], as a piecewise linear approximation of thedensity at time tp via a linear interpolation between ρ(tp, xm−1) and ρ(tp, xm) for

x ∈ [xm−1, xm]. Then, we update the value of ypi to yp+1i by an explicit Euler method

as follows:

(3.1)

{yp+1i = ypi +min

{νpi ; v(ρ(tp, y

pi ))Φ(x, y

p1 , . . . , y

pN )

∣∣x=yp

i

} · t,

νp+1i = νpi +

[− 1τ

(νpi −min

{V p0 ; v(ρ(tp, y

pi ))Φ(x, y

p1 , . . . , y

pN )

∣∣x=yp

i

})] · t,

where V p0 = V0(y

pi+1 − ypi ) (homogeneous case) and

v(ρ(tp, ypi ))Φ(x, y

p1 , . . . , y

pN)

∣∣x=yp

i

= (1 − ρ(tp, ypi )) · min

k=1,...,N{ϕk(y

pi − ypk)}.

In this way we avoid interactions among waves and trajectories; however, we takeinto account overtakes between buses. A refined algorithm, which takes into accountinteraction with waves, as in [4], is in preparation [3].

Let us now consider the case of an interaction between the trajectories of twobuses, namely, an overtake between yio and yio+1, say, at time tp < to < tp+1. Then,at this time to, the ioth and the (io + 1)th buses are in the same position, that is,

ypio +min{νpio ; v(ρ(tp, y

pio))Φ(x, yp1 , . . . , y

pN )

∣∣x=yp

io

} · (to − tp)

= ypio+1 +min{νpio+1; v(ρ(tp, y

pio+1))Φ(x, y

p1 , . . . , y

pN)

∣∣x=yp

io+1

} · (to − tp)

= yo,

which determines the time when the overtake occurs. Moreover, using the Eulerscheme (3.1) up to this time, we indeed evaluate also the velocities νoio and νoio+1. Inthis case, each bus involved in the overtake adjusts its velocity according to its “newleader,” and thus we formalize this situation by means of the relabeling procedure ofindices already mentioned, that is, jo = io+1, jo+1 = io, j = i for any i �∈ {io, io+1}.Therefore, the Euler scheme for the joth vehicle becomes(3.2){yp+1jo

= yo +min{νpjo ; v(ρ(tp, y

o))Φ(x, yp1 , . . . , ypN )

∣∣x=yo

} · (tp+1 − to),

νp+1jo

= νojo +[− 1

τ

(νpjo −min

{V p0 ; v(ρ(tp, y

o))Φ(x, yp1 , . . . , ypN)

∣∣x=yo

})] · (tp+1 − to).

Remark 3.1. The above procedure basically requires an adaptive mesh size t inthe presence of overtakes for the bus path described in Step 2. Indeed, if an overtakeoccurs at to, we must divide the time interval [tp, tp+1] into two subintervals to use(3.1) in the first one (for a time step to− tp instead of the full time step t) and (3.2)in the second one. Actually, one could restart the whole fractional step algorithm, byupdating the density ρ at this time, using Step 1, thus defining an adaptive mesh sizealso for the Godunov scheme.

However, in our simulations, we shall simplify Step 2 of the above algorithm inthe following way. We evaluate the new position of yp+1

i and yp+1i+1 using (3.1), and

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Table 1

Parameter settings.

Macro Microρmax = 1 vmax = 1 τi = 2 σ = 3δi = 1 βi = 0.5 γ = 0.5 λ = 0.2

Table 2

Parameter settings for the source term.

Sourcex1 = L/4 x2 = L/2 x3 = 3L/4Δx = 1 ρinc,k = 0.6 vbound,k = 1pk = 0.7

then we evaluate dp+1i = yp+1

i+1 −yp+1i . If dp+1

i < 0, then an overtake occurred for someto ∈ (tp, tp+1), and, in that case, we simply relabel the indices before starting with anew iteration of the algorithm, thus introducing an extra small error, which indeedvanishes for t � 1.

4. Results of numerical simulations. We now fix L = 50, T = 50, and thenumber of buses N = 3. We further fix the number of the cross roads to be K = 3.

For the macroscopic model, we choose, as usual, a linear constitutive law for theaverage velocity of cars, v(ρ) = (1− ρ), with normalized quantities ρmax = vmax = 1.Moreover, let the functions ϕi be equal for any i; that is, each bus gives the samedrop in the capacity of the road. The cut-off functions are defined as

ϕ(ξ) = ϕ(ξ ± L) = [1− (1− β)e(−10 ξ2

δ2)]

· [1− (1− β)e(−10 (ξ+L)2

δ2)]

· [1− (1− β)e(−10 (ξ−L)2

δ2)],

with β = 0.5 and δ = 1.In Table 1 we show our choice of the other parameters for both the macroscopic

and microscopic models; in Table 2 parameters for the source term are shown.Example 1. In Figure 4.1(a) we show the result of a simulation. Buses are

starting from points far enough apart. Each bus tends to increase its speed in orderto approach the vehicle in front. A jump in the initial density gives rise to a shockat x = 30, while in x = 20 a rarefaction wave is centered. We see very clearly theinteraction of the bus routes with the car density waves. In addition, at the crossingpoints x1, x2, and x3, indicated in Table 2, we note the effect of the source terms onthe evolution of the traffic.

In Figure 4.1(b) different colors are used to depict the time evolution of theaverage speed v(ρ)Φ and of the velocity adaption function Vopt(d) for each bus. A reddotted line shows the minimum between them, i.e., the velocity the buses are aimingto reach.

In Figure 4.2 we show two frames of the time evolution of the bus route. Thedensity is represented by a blue line. The green line shows the evolution of the sourceterm; it is always zero except at junctions. A square, a triangle, and a circle arerepresenting, respectively, the positions of the three buses. Junctions are depictedwith red vertical lines, delimiting the incoming and outgoing roads.

Example 2. In Figure 4.3(a) we show the result of another simulation. It is

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Space

Tim

e

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

densitybusses

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(a) The contour plot.

0 10 20 30 40 500

0.5

1

time

min

(V0(d

1),v)

0 10 20 30 40 500

0.5

1

time

min

(V0(d

2),v)

0 10 20 30 40 500

0.5

1

time

min

(V0(d

3),v)

(b) The velocity adaption function (blue) versusaverage speed of cars (black).

Density Positions Velocities

0.99 0 ≤ x ≤ 20.0 y01 = 15 ν01 = 0.5

0.3 20.0 < x ≤ 30.0 y02 = 19 ν02 = 0.5

0.99 30.0 < x < L y03 = 23 ν03 = 0.5

(c) Initial data.

Fig. 4.1. In (a) a contour plot of car density and buses’ positions. It is possible to see theinfluence of the traffic waves on buses. In particular, the acceleration of the last bus on the leftwhen interacting with the rarefaction wave in front of it is manifest. In (b) the velocity adaptionfunction versus average speed of cars. The red circled line represents the minimum between them.In the table we set initial data for this simulation.

(a) The evolution at t = 8.33. (b) The evolution at t = 25.

Fig. 4.2. Two frames of the time evolution of Figure 4.1(a), corresponding, respectively, tot = 8.33 and t = 25. The source term (green line) is always equal to zero except for points ofintersection between the bus route and the incoming (where it is positive) and outgoing (where it isnegative) roads, represented by dotted vertical lines.

possible to see how bus trajectories starting from very closed points tend to moveaway. Finally, they seem to approach a constant distance. It is also interesting to seehow the source term influences the dynamics. In particular, in x3 = 3

4L, it gives raiseto a triangular shape due to the peak of density, becoming larger as time increases.The time evolution of the average speed v(ρ)Φ and of the velocity adaption functionVopt(d) for each bus is depicted in Figure 4.3(b).

Example 3. In Figure 4.4 we simulate the particular case of only two buses onthe bus route, one of which is not moving for some reason (we can imagine it hasdamages to the engine or is waiting at the bus stop). The initial position of the firstbus is y01 = 10, and the initial position of the second one, kept constant for the whole

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Space

Tim

e

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50

densitybusses

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


0 10 20 30 40 500

0.5

1

time

min

(V0(d

1),v)

0 10 20 30 40 500

0.5

1

time

min

(V0(d

2),v)

0 10 20 30 40 500

0.5

1

time

min

(V0(d

3),v)

(b) Velocity adaption function (blue) versus av-erage speed of cars (black).


0.9 0 ≤ x ≤ 20.0 y01 = 21.5 ν01 = 0.5

0.3 20.0 < x ≤ 30.0 y02 = 22 ν02 = 0.5

0.9 30.0 < x < L y03 = 22.5 ν03 = 0.5

(c) Initial data.

Fig. 4.3. Another example of the contour plot of car density and buses’ positions. Startingfrom very close initial positions, here is clearly shown how the distance between buses increases;this behavior follows from the definition of the function V0. In the table we set initial data for thissimulation.

Space

Tim

e

0 10 20 30 40 500

5

10

15

20

25

30

35

40

45

50densitybuses

0.1

0.2

0.3

0.4

0.5

0.6

0.7


0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

time

min

(V0(d

1),v)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

time

min

(V0(d

2),v)

(b) The velocity adaption function (blue) and av-erage speed of cars (black).


0.01 0 ≤ x ≤ L y01 = 10 ν01 = 1

y02 = 20 ν02 = 0

(c) Initial data.

Fig. 4.4. A contour plot of density and positions of two buses, in the case of an overtake, anda velocity adaption function versus the average speed of cars. In the table we set initial data for thissimulation.

simulation, is y02 = 20 (i.e., y2 is not satisfying a differential equation). Since weare looking for an overtake, we choose the maximal initial velocity for the first bus,ν01 = 1, while ν02 = 0. In order to have an overtake in a short time interval and to

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underline the adaptation of the velocity to the headway, we choose the relaxation timeτ = 4. In addition, the initial density is ρ0 = 0.01 for any x ∈ [0, L]. Note that thesource term is very big due to the fact that whenever there are few cars on the streetmany other cars can enter the road. The incoming density on each road is fixed atρink,k = 0.6, and the source term has almost the same amplitude. In our result, itis easy to note that the first bus decelerates when it meets the crossway in x = 12.5,and it decelerates again before it reaches the vehicle in front of it. After the overtake,at around to = 20, it accelerates, changing the convexity of the trajectory, since thenew vehicle in front of it (the second bus) is far away, and it can move to reach itsdesired speed γ = 0.5.

Example 4. Microscopic models proposed by Bando et al. [1] and Huijberts [17]admit stable solutions, namely, when the distances among buses and the velocitiesapproach fixed values. In our case this situation is in general perturbed due to theinteraction with the car flow. However, it is still possible to observe “bus stable-like”solutions, when their reciprocal distances and velocities are nearly constant. Indeed,in Figure 4.5(a), we choose a constant value for the initial density, ρ = 0.1, and thenumber of buses N = 4, together with initial positions and velocities. It is shown inthis example that distances between buses are not constant, because of the interactionwith the car flux, but eventually have small variations, giving rise to a “bus stable-like”behavior; see Figure 4.5(b).

Space

Tim

e

0 10 20 30 40 500

10

20

30

40

50

60

70

80

90

100

densitybusses 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(a) Contour plot. (b) Headways.


0.1 0 ≤ x ≤ L y01 = 15 ν01 = 0.5

y02 = 17 ν02 = 0.5

y03 = 19 ν03 = 0.5

y04 = 21 ν04 = 0.5

(c) Initial data.

Fig. 4.5. Even if the existence of stable solutions is not proved, in our simulations bus stable-like solutions can be still observed, as shown, setting T = 100 and τ = 2. In the table we set initialdata for this simulation.

Example 5. Microscopic models proposed by Bando et al. [1] and Huijberts [17]are known to admit periodic solutions (periodicity in the headways and the velocities)[13, 31, 14]. Here, we would like to give an example where “bus periodic-like” solutionscan be observed. To show such solutions, a particular setting is needed, as is clearfrom [13]. We keep the length of the road L = 30. Then we need to increase thenumber of buses, say, up to N = 20, and choose a steeper velocity adaption function

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0 100 200 300 400 500 600 700 800 900 10000

2

4

7−−

8

0 100 200 300 400 500 600 700 800 900 10000

2

4

8−−

9

0 100 200 300 400 500 600 700 800 900 10000

2

4

9−−

10

0 100 200 300 400 500 600 700 800 900 10000

2

4

10−

−11

Time

(a) The evolution in time of headways.

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Headway

Spe

ed

(b) The evolution of a headway versus its speed.

Space

Tim

e

4 8 12 16 20 24 28 30

955

960

965

970

975

980

985

990

995

1000

0.2

0.3

0.4

0.5

0.6

0.7

(c) Density on the road.

Fig. 4.6. (a) The evolution of four headways in the case of a “bus periodic-like” solution. Itis achieved by setting N = 20, T = 1000, L = 30, τ = 1, λ = 0.05, σ = 50, ρ0(x) = 0.3 for any0 < x ≤ L; the velocity adaption function is defined as in (4.1). (b) The evolution of a headwayplotted versus its speed. (c) A contour plot of the density for 950 ≤ T ≤ 1000; sources are notconsidered.

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0 200 400 600 800 10000

2

4

7−−

80 200 400 600 800 1000

0

2

4

8−−

9

0 200 400 600 800 10000

2

4

9−−

10

0 200 400 600 800 10000

2

4

10−

−11

Time

(a) The evolution in time of headways.

0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Headway

Spe

ed

(b) The evolution of a headway versus its speed.

Space

Tim

e

4 8 12 16 20 24 28 30

960

970

980

990

1000

0.2

0.3

0.4

0.5

0.6

0.7

(c) Density on the road.

Fig. 4.7. Source terms defined in Table 2 are considered in this simulation. (a) The evolutionof four headways in the same setting as in Figure 4.6. (b) The evolution of a headway plotted versusits speed. (c) A contour plot of the density for 950 ≤ T ≤ 1000.

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by setting λ = 0.05, σ = 50, and

(4.1) V 1opt(d) = d8/(σ + d8)

in the definition of V0. We do not consider source terms in the model. By fixing aconstant initial density for the macroscopic model at ρ0(x) = 0.3 for any 0 < x ≤ L,a time evolution T = 1000 and τ = 1, zero initial velocities, and equidistant initialpositions, with a small perturbation we are able to show a “bus periodic-like” solutionin Figure 4.6. The seemingly periodic behavior of the buses can be seen in Figure4.6(a). In Figure 4.6(b) this periodicity is plotted in the phase space (di, yi) of achosen bus, where we exclude the first 100 time instants to avoid any transient state.In addition, in Figure 4.6(c) we see that the bus periodicity implies a certain periodic-like behavior for the car flow dynamics (e.g., for any fixed position x along the route,as a function of the time t). In Figure 4.7 we consider source terms in the model,as defined in Table 2. Again we show the evolution in time of four headways, theevolution versus its speed of one of them excluding the first 100 time instants, anda contour plot of the density on the road for 950 ≤ T ≤ 1000. In this case theperiodicity appears to be more destroyed by sources, as expected.

5. Conclusions. We present a coupled micro–macro model for the descriptionof a bus route in a regular traffic network. It takes into account the mutual interac-tion between regular vehicular traffic and buses, and between buses themselves, thusgiving a meaningful description of buses’ headway dynamics, including bus stops andovertaking of buses. Among the limited literature on the modeling of a completebus route, our approach includes all relevant aspects involved and easily gives a goodqualitative description of the full dynamics, which can be then used as a startingpoint for future considerations such as optimization of the numbers of buses along theroute, of the buses’ paths, and of the waiting time at bus stops.

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Vehicular Traffic Flow Dynamics on a Bus Route

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