Research Article Response Time Analysis of Distributed Web...

Research ArticleResponse Time Analysis of DistributedWeb Systems Using QPNs

Tomasz Rak

Faculty of Electrical and Computer Engineering Rzeszow University of Technology Aleja Powstancow Warszawy 1235-959 Rzeszow Poland

Correspondence should be addressed to Tomasz Rak trakkiaprzedupl

Received 5 March 2015 Revised 15 May 2015 Accepted 24 May 2015

Academic Editor GuanJun Liu

Copyright copy 2015 Tomasz Rak This 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

A performance model is used for studying distributed Web systems Performance evaluation is done by obtaining load testmeasurements Queueing Petri Nets formalism supports modeling and performance analysis of distributed World Wide Webenvironments The proposed distributed Web systems modeling and design methodology have been applied in the evaluation ofseveral system architectures under different external loads Furthermore performance analysis is done to determine the systemresponse time

1 Introduction

Distributed Web Systems (DWS) development assumes thatthe systems consist of a set of distributed nodes Thesesystems provide up-to-date data within the set time framesGroups of nodes (clusters) are organized in layers conductingpredefined servicesThis approach makes it possible to checkthe response time and easily scale the system An example of aDWS is a stock trading system [1] (as one class of Internetsystems) In this system it may be a requirement for certainpositions to be bought or sold when market events occur Acertain amount of system latency may be acceptable but theevent must still be reacted to within a deterministic periodof time If a lot of system responses exceed the time limitthe system will not be often used by users The response timespecified as an average value is normally dictated by businessnot by the environment

Modeling and design of DWS are developed in two ways(Figure 1) On one hand formal models which can be usedto analyze performance parameters are proposed [2ndash7] Todescribe such systems formal methods like Queuing Netsand Petri Nets are used Sometimes elements of the controltheory are used to manage the movement of packages in webservers [1] Experiments connected with simulation modelsare the second way especially when there are many nodes

[1 8 9] Applying experiments and models greatly influencesthe validity of the systems being developed The convergenceof simulation results with the real systems results confirmsthe correctness of the modeling methods

Our earlier works [1 10] are based on Queuing Nets andTimed Coloured Petri Nets A distributed Internet systemmodel initially described in compliance with Queuing Netrules was mapped onto Timed Coloured Petri Net structureby means of queueing system templates We have used twotypes of formal models that have been exploited in theindustry We created some separate system models usingQueuing Nets and Petri Nets which allow the performanceanalysis The final Timed Coloured Petri Net based modelcan be executed and used for modeled system performanceprediction

In our solution we propose alternative Queueing PetriNet models [11] The models have been used as a backgroundfor developing a programming tool which is able to maptimed behavior of QueueingNets bymeans of simulationWedeveloped [12 13] our individual method of modeling andanalysis of DWS The well-known software toolkits such asQueueing Petri net Modeling Environment (QPME) [3] canbe naturally used for ourmodels simulation and performanceanalysis We develop QPN models of DWS that allow theperformance evaluation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 490835 10 pageshttpdxdoiorg1011552015490835

2 Mathematical Problems in Engineering

Internetsystem

Experiments with Experiments with

Mathematicalmodel

Analiticalsolution Simulation

the real system a formal model ofthe system

Figure 1 Performance modeling possibilities

The remaining work is organized as follows Section 2describes QPN Section 3 presents DWS architecture anddescribes the modeling approach Section 4 presents perfor-mance analysis results The final section contains concludingremarks

2 Queueing Petri Nets

In our solution we propose a popular formal methodmdashQueueing Petri Net [11] This method is based on QueueingNets and Petri Nets Queuing Theory deals with modelingand optimizing different types of service units QueueingNet usually consists of a set of connected queuing systemsThe various queue systems represent computer componentsQueueing Nets (QN) are very popular for the quantitativeanalysis [14] QNs have a queue scheduling discipline andare suitable for modeling competition of equipment Toanalyze any queue system it is necessary to determine arrivalprocess service distribution service discipline and waitingroom (scheduling strategies) Petri Nets (PN) are used tospecify and analyze the concurrence in systems The systemdynamics is described by the rules of tokens flow The netscheme can be subjected to a formal analysis in order tocarry out a qualitative analysis based on determining itslogical validity PN have tokens representing the tasks andare suitable for modeling software PN are referred to asthe connection between the engineering description and thetheoretical approach Petri Nets are well-known models usedto describe and analyze service units PN cannot be used fora quantitative analysis due to lack of time aspectsThe studiesfocus on incoming load measuring for example measureof the response time or presentation of an overall modelingplanQNmdashquantitative analysismdashhas a queue and schedulingdiscipline and are suitable formodeling competition of equip-ment PNmdashqualitative analysismdashhave tokens representingthe tasks and are suitable for modeling software

Queueing Petri Net (QPN) formalism is a very popularformal method of functional and performance modeling(performance analysis) These nets provide sufficient power

Place Queueingplace

Subnetplace

Queue Depository

Input OutputUser

subnet

Actualpopulation

Figure 2 Elements of QPN

to express modeling and analyzing of complex online sys-tems The choice of QPN was caused by a possibility ofobtaining different character information The main idea ofQPN is to add queueing and timing aspects to the net placesQPN have the advantages of QN (eg evaluation of thesystemperformance and the network efficiency) and PN (eglogical assessment of the system correctness) QN consistsof a collection of service stations and clients The servicestations (queues) represent system resources while the clientsrepresent users or transactions A service station is composedof one or more servers and a waiting area (Figure 2) Tokensenter the queueing place through the firing of input transi-tions as in other PN When a request arrives at a servicestation it is immediately serviced if a free server is availableOtherwise the request has to wait in the waiting area Dif-ferent scheduling strategies can be used to serve the requestswaiting in the waiting area Places (QPN) are of two typesordinary and queued A queueing place (resource or state)is composed of a queue (service station) and a depositoryfor tokens that completed their service at a queue (Figure 2)After being served by the service station (coloured) tokensare placed onto a depository Input transitions are fired andthen tokens are inserted into a queueing place accordingto the queuersquos scheduling strategy Queueing places canhave variable scheduling strategies and service distributions(timedqueueing places) Tokens in the queue are not availablefor output transitionswhile tokens in the depository are avail-able to all output transitions of the queued place Immediatequeueing places impose a scheduling discipline on arrivingtokens without a delay [11]The last place on Figure 2 is calleda subnet place Hierarchical QPN (HQPN) has a dedicatedinput and output place which are ordinary places of a PNTokens being inserted into a subnet place after a transition fir-ing are added to the input place of the corresponding HQPNsubnet The semantics of the output place of a subnet place issimilar to the semantics of the depository of a queueing placeEvery subnet contains actual population place used to keeptrack of the total number of tokens fired into the subnet place

QPN is a tuple (2) where CPN is Coloured Petri Net (1)[11 15]

CPN = (119875 119879 119862 119868119872) (1)

Mathematical Problems in Engineering 3

where

(i) 119875 = 1199011 1199012 119901119894 is a finite and nonempty set ofplaces

(ii) 119879 = 1199051 1199052 119905119895 is a finite and nonempty set oftransitions

(iii) 119875 cap 119879 = (iv) 119862 is a colour function defined from 119875 cup 119879 into finite

and nonempty sets (specify the types of tokens thatcan reside in the place and allow transitions to fire indifferent modes)

(v) 119868(119901 119905) are the backward and forward incidence func-tions defined on 119875 times 119879 such that 119868(119901 119905) isin [119862(119905) rarr

119862(119901)] forall(119901 119905) isin 119875 times 119879 (specify the interconnectionsbetween places and transitions)

(vi) 119872(119901) is a initial marking defined on 119875 such that119872(119901) isin 119862(119901) forall119901 isin 119875 (specify how many tokens arecontained in each place)

QPN = (CPN 119876119882) (2)

where

(i) 119876 = (1198761 1198762 (1199021 119902|119875|)) where

(a) 1198761 sube 119875 is a set of timed queueing places(b) 1198762 sube 119875 is a set of immediate queueing places(c) 1198761 cap 1198762 = (d) (1199021 119902|119875|) is an array with description of

places (if 119901119894is a queueing place 119902

119894denotes the

description of a queue with all colors of 119862(119901119894)

into consideration or if 119901119894is the ordinary place

(119901119894) equals 119899119906119897119897)

(ii) 119882 = (11988211198822 (1199081 119908|119879|)) where

(a) 1198821 sube 119879 is a set of timed transitions(b) 1198822 sube 119879 is a set of immediate transitions(c) 1198821 cap1198822 = 1198821 cup1198822 = 119879(d) (1199081 119908|119879|) is an array (entry 119908

119895isin [119862(119905

119895) 997891rarr

R+] such that forall119888 isin 119862(119905119895) 119908119895(119888) isin R+) of

(1) rate of a negative exponential distributionspecifying the firing delay due to colour if119905119895isin 1198821

(2) firing weight specifying the relative firingfrequency due to colour if 119905

119895isin 1198822

QPN have been recently applied in the performanceevaluation ofDWS databases [16] and grid environments [17]because they are more expressive to represent simultaneousresource possession and blocking Here QPN models areused to predict DWS performance

3 Distributed Web System

Among many Internet systems we can indicate DWS

31 Distributed Web System Architecture Distributed Inter-net system architecture is made up of several layers

(i) Layer 1 (Web serversmdashTier 1) presents information(system offer) for clients in the web pages form andcontains clusters

(ii) Layer 2 (Application serversmdashTier 2) manages trans-actions (clients requests) and provides the clusteringfunctionality that allows load balancing

(iii) Layer 3 (Database serversmdashTier 3) controls the trans-actions as a single element of this layer or multipleservers with database replication

(iv) Layer 4 (Data nodesmdashTier 4) is the data storagesystem

In our approach the presented architecture has been simpli-fied to two layers

(i) Front-End (FE) layer is based on the presentation andprocessing mechanisms (Tiers 1 and 2) These twofunctions are realized by software

(ii) Back-End (BE) layer contains one or moremdashin caseof replication [1]mdashseveral databases It consists of twopresented Internet system layers (Tiers 3 and 4) Thislayer keeps the system data

An architecture composed of these layers is used for e-busines systemsThe presented double-layer system architec-ture realizes Internet system functions These simplificationshave no influence on the modeling process which has beenshown repeatedly for example [3] Proposed in the paperapproach may be treated as an extension and continuation ofsolutions presented in [1] Clustering mechanism was used inboth layers We used 1 3 6 and 9 nodes

32 Client of DistributedWeb System An access to the systemis realized through transactions Activities related to therequests processing are as follows

(i) Searching the Internet and sending a request by theInternet client

(ii) Sending information to the Internet client or commu-nicating with the application server by the presenta-tion server

(iii) Sending a request to the database by the applicationserver

(iv) Carrying out a transaction on the stored data andreturning results by the database server

(v) Including results in appropriate places on a web pageby the application server after receiving data

(vi) Browsing web pages with results by the client using aweb browser

The characteristic feature of many DWS is a large number ofclients using the Internet services (eg stock trading system)at the same time DWS clients have different response timerequirements In case of the described systems class clients


Workload Workloadmodel

Computer system

Calculated

Performance model

Measured performanceparameters performance parameters

parametersConfiguration


Figure 3 Difference between computer system and performancemodel

are often focused on one event related to the same systemoffer Based on these futures we used a stock trading systemas a benchmark with two-layered architecture In this paperwe consider one class of Internet systems and one class ofInternet clients

33 Modeling of Distributed Web System We have manymodeling methodologies We can indicate some approachesto design like

(i) educated guess(ii) load testing(iii) performance modeling [3]

From many performance engineering models (workloadperformance availability reliability and cost) we have cho-sen and use the performance model (Figure 3) Perfor-mance models provide some recommendations to realize therequired performance level

At this stage of our research it has been decided thatsimulation will be the main mechanism used to do theanalysis of the constructedmodels because our architecture istoo large for analytical solutions (Figure 1) In our simulationswe applied the performance analysis

Typically DWS are composed of layers where each layerconsists of a set of serversmdasha server cluster The layers arededicated to adequate tasks and exchange requests betweeneach other To explain our approach to DWS modeling atypical structure will be modeled and simulated The firstlayer (FE) is responsible for presentation and processingof client requests The nodes of this layer are modeled byProcessor Sharing (PS) (PSmdashrequests are assumed to beserved simultaneously with the server speed being equallydivided among them (with infinitesimally small time slices)mdashit is used for modeling CPUs) queues The next layer (BE)implements system data handling The nodes of this layerare modeled by using the First In First Out (FIFO) queueRequests are sent to the system and then can be processed inboth layersThe successfully processed requests are send backto the client Client is modeled by Infinite Server (IS) queue

Consequently an executable (in a simulation sense) QPNmodel is obtained Tokens generated by the arrival processare transferred in sequence by models of FE layer and by BElayer QPN extend coloured Stochastic Petri Nets by incorpo-rating queues and scheduling strategies into places forming

queueing places This very powerful modeling formalismhas the synchronization capabilities of Petri Nets while alsobeing capable of modeling queueing behaviors The queuemean service time the service time probability distributionfunction and the number of servicing units defined for eachqueueing system in the model are the main parameters ofthe modeled system In the demonstrated model it has beenassumed that queues belonging to the same layer (FE and BE)have identical parameters QPN consists of a set of connectedqueueing places Each queueing place is described by arrivalprocess waiting room service process and additionallydepository We apply several queueing systems most fre-quently used to represent properties of system components

In the Queueing Petri net Modeling Environment soft-ware tool it is possible to construct QPN with queueingsystems having PS and FIFO disciplines As it was mentionedabove the main application of the software tool presented inthe paper is modeling and evaluation of DWS

4 Response Time Analysis

We have many Quality of Service parameters

(i) performance (utilization throughput and responsetime)

(ii) availability(iii) reliability

Monitoring of the above mentioned parameters helps todetermine the system behavior It allows collecting selectedelements of the net state at the moment of an occurrence ofcertain events during the simulation It has been mentionedabove that in each of the model layers response time will bemonitored

In these studies the performance is measured in terms ofmean response time of business transactions Parameters thatdetermine the response time are

(i) workload intensity and hardware and softwareparameters

(ii) residence time and service demand

Response time (3) is equal a sum of residence times where 119894is the number of places

119877 =

119896=1sum

119894

119877

1015840

119896 (3)

Residence time (4) is equal to a sum of queueing time andservice demand

119877

1015840

119896= 119876119896+119863119896 (4)

where queuing time is 119876119896= sum

119896=1119894

119902119896and service demand is

119863119896= sum

119896=1119894

119889119896 Average service time in a particular resource

does not contain the time of waiting for the resource Servicedemand also does not depend on the load


Table 1 Example of mean response time [ms] for 15 requests per second workload

Buy Quote Sell Quote Update Profile Show Quotes Get Home Get Portfolio Show Account100 clients 158329 168806 62192 62595 43929 95894 38257200 clients 197483 208062 91061 91363 83432 117602 78634300 clients 197142 209295 91639 91682 83165 11457 78446

41 Measured Parameters Simulation parameters are basedon a benchmark with realistic workload We present theresults of our earlier experimental analysis in [12] The goalwas to checkmdashamong othersmdashthe service demand parameterfor FE and BE nodes The application servers consideredas a FE layer are responsible for the method execution Allsensitive data is stored in a database system (BE layer) Whena client has to retrieve or update data the application servermakes the corresponding calls to the database system DWSare usually built on middleware platforms such as J2EE Weused the DayTrader [18] performance benchmark which isavailable as an open source application Overall the Day-Trader application is primarily used for performance researchon a wide range of software components and platformsDayTrader is a suite of workloads that allows performanceanalysis of J2EE application server DayTrader is a benchmarkapplication built around the paradigm of an online stocktrading system It drives a trade scenario that allows tomonitor the stock portfolio inquire about stock quotesbuy or sell stock By client business transactions we meanthe stock-broker operations Buy Quote Sell Quote UpdateProfile ShowQuoteGetHomeGet Portfolio ShowAccountand LoginLogout Each business transaction emulates aspecific class of clients

One of the most important requests [12] Buy Quote(Requests class (type) which has the bigger impact on thebehavior of the system (Table 1)) is used in simulationsExperiments [12] have shown that mean number of requestsper second for a FE layer is about 1400 Respectively themeanmeasured number of requests per second for BE layer isabout 7500 requests per secondWe can also see that the delayin the requests processing is mainly caused by the waitingtime for service in one BE node but the main problem isthe performance of the system response time Our approachpresented in [12] predicts response time for DWS and therelative error is lower than 15[]

42 Queueing Petri Net Models QPN models (Figure 4)are used to predict the system response time We use theQueueing Petri netModeling Environment (QPME) [15] toolQPME is an open-source tool for stochastic modeling andanalysis based on QPN modeling formalism used in manyworks [3 5 6 19] Scheduling strategies service time distribu-tions and number of servers for queues are shown in Table 2Queues are described by Kendall notation (119860119861119898119870119871119873)where 119860 denotes the probability distribution function spec-ifying the interarrival time of tokens (minus means differentrequests interarrival time) 119861 is the probability distributionfunction of a service times (119872 means exponential (Marko-vian) distribution of requests service time) 119898 is the number

Table 2 Queueing systems definitions

Queue place Queueing system DescriptionClients minus1198721119868119878c Clients119865119864 119862119875119880119898

aminus1198721119875119878d FE nodes

119861119864 119868119874119899

bminus1198721119865119868119865119874e BE nodes

a119898 number of FE nodes

b119899 number of BE nodes

cIS service disciplinedPS service disciplineeFIFO service discipline

t1

t2 t3 t4

t5FE BESub-FE Sub-BE

Clients

ThreadsPool

ConnectionsPool

Figure 4 Main model of DWS

of servers (1 means one server) 119870 limits the number ofrequests a queue can hold (if not specified the default isinfin)119871determines the maximum number of requests that can arrivein a queue the size of calling source (if not set 119871 = infin) and119873 is the scheduling strategy (if not specified the default isFIFO)

Model elements are presented in Figure 4 Client thinktime is modeled by IS scheduling strategy (119862119897119894119890119899119905119904 place)Servers of FE layer aremodeled using the PS queuing systems(119865119864 119862119875119880 places) in subnet place (119878119906119887-119865119864 in Figure 5(a))Service in all queueing places is modeled by an exponentialdistribution Service demands in layers are based on experi-mental results [12]

(i) 119889119865119864 119862119875119880

= 0714 (ms)(ii) 119889119861119864 119868119874

= 0133 (ms)

BE servers are modeled by FIFO queue (119861119864 119868119874 place)in subnet place (119878119906119887-119861119864 in Figure 5(b)) Places (119865119864 and119861119864) are used to stop incoming requests when they awaitapplication server threads and database server connectionsrespectively Application server threads and database serverconnections are modeled respectively by 119879ℎ119903119890119886119889119904119875119900119900119897 and


Input-place

Input-transition

Actual population

Output-transition

Output-place

FE_CPU8 FE_CPU9

FE_CPU7FE_CPU6

FE_CPU5

FE_CPU3

FE_CPU1FE_CPU2

FE_CPU4

(a)

Input-place

Output-place

Input-transition Output-transitionActual population

BE_IO1

BE_IO2

BE_IO3

BE_IO4 BE_IO5 BE_IO6

(b)

Figure 5 A subnet place example (a) FE subnet with 9 nodes and (b) BE subnet with 6 nodes

119862119900119899119899119890119888119905119894119900119899119904119875119900119900119897 places The process of requests arrival tothe system is modeled by exponential distribution with the 120582parameter (client think time) corresponding to the numberof clients requests per second The initial marking for placesare

(i) number of clients (number of tokens in119862119897119894119890119899119905119904 place)(ii) application server threads pool (number of tokens in

119879ℎ119903119890119886119889119904119875119900119900119897 place)(iii) database server connections pool (number of tokens

in 119862119900119899119899119890119888119905119894119900119899119904119875119900119900119897 place)

In thesemodels we have three types (A colour specifies a typeof tokens that can be resided in the place) of tokens

(i) Requests(ii) Application server threads(iii) Connections to the database

Based on definition (2) we define the followingmodel (5)of DWS

QPN = (119875 119879 119862 119868119872119876119882) (5)

where

(i) 119875 = 119865119864 119861119864 119879ℎ119903119890119886119889119904119875119900119900119897 119862119900119899119899119890119888119905119894119900119899119904119875119900119900119897(ii) 119879 = 1199051 1199052 1199053 1199054 1199055(iii) 119862 (Table 3)(iv) 119868(119901 119905)(v) 119872(119901) (Table 4)(vi) 119876 = (1198761 1198762 (minus119872infin119868119878

119862119897119894119890119899119905119904 119899119906119897119897 minus1198721

119875119878119878119906119887-119865119864 119899119906119897119897 minus1198721119865119868119865119874

119878119906119887-119861119864 119899119906119897119897 119899119906119897119897)) where

(a) 1198761 = 119862119897119894119890119899119905119904 119865119864 119862119875119880119898 119861119864 119868119874

119899

(b) 1198762 =

Table 3 Type (color) to be attached to a token

Placetransition Color DescriptionClients 119865119864 119862119875119880119898 119861119864 119868119874119899 FEBE req Token represents a

clientrsquos requests

ThreadsPool thr Token represents athread

ConnectionsPool con Token represents aconnection

119905119895

req Token represents aclientrsquos requests

Table 4 Initial marking

Place Value DescriptionClients 100 200 300 400 500 Number of clients119865119864 119862119875119880119898 mdash FE nodes119865119864 119868119874119899 mdash BE nodesThreadsPool 30 90 180 270 Number of threadsConnectionsPool 40 120 240 360 Number of connectionsFE mdash Stop incoming requestsBE mdash Stop incoming requests

(vii) 119882 = (11988211198822) where

(a) 1198821 = (b) 1198822 = 119879(c) forall119888 isin 119862(119905

119895) 119908119895(119888) = 1 (all transition firings are

equally likely)

43 Simulation Results Many simulations were performedfor various input parameters (Table 5) Total response timeis a sum of all individual response times of queues and


Table 5 Parameters of simulations (one class of requests corre-sponds with Buy Quote requests)

Elementparameter Namevalue

QPME FE queues 119865119864 119862119875119880119898

a

BE queues 119861119864 119868119874119899

b

Softwarec ThreadsPool place 30d

ConnectionsPool place 40e

Client workload 120582 006f

119862119897119894119890119899119905119904 placeg 100 200 300 400 500Simulation time [s] 300



cInitial marking per noded30 threads for one FE node 90 threads for three FE nodes 180 threads forsix FE nodes and 270 threads for nine nodese40 connections for one BE node 120 connections for three BE nodes 240connections for six BE nodes and 360 connections for nine nodesfClient think time equals 1667 [ms]gClient workload based on 15 30 45 60 requests per second

depositories in a simulation model without the client queueresponse time (client think time)

We investigate the bahaviour of the system during theincrease in workload intensity The number of clients wasincreased in accordance with values (119862119897119894119890119899119905119904 place) from6000 to 30000 requests per second

We used some scenarios in which we have a singlerequests class The results involve the response time of thewhole system Multiple FE (1 3 6 and 9) and BE (1 36 and 9) nodes are the main configuration scenario QPNmodel was used to predict the performance of the systemfor the scenarios (1FE1BE (1 node in FE layer and 1 nodein BE layer) 1FE3BE 1FE6BE 1FE9BE 3FE1BE 3FE3BE3FE6BE 3FE9BE 6FE1BE 6FE3BE 6FE6BE 6FE9BE9FE1BE 9FE3BE 9FE6BE and 9FE9BE) and itwas developedusing QPME

As a result the response time of transactions is improvedfor cases with a higher number of FE and BE nodes Increas-ing number of nodes resulted in simultaneous increase in thenumber of application server threads and connections to thedatabase

Figures 6 and 7 show the mean response time for all testsFigure 6 shows the mean response time from the perspectiveof FE layer and Figure 7 from the perspective of BE layerAs we can see the overall response time decreased while thenumber of nodes was increasing

The response time of one FE node architecture for allcases is the biggest A difference in response time (Figure 6)between FE1 and FE3 is much bigger than between FE3 andFE9 (with different number of nodes in BE layer)We can alsosee the difference between system behavior for the increasingnumber of clients For example for 500 clients we can observebig discrepancy between FE1 and FE3 in all cases

As we can see in Figure 7 an increasing number of nodesdoes not always reduce the response time When more nodesare added the analysis of their impact on other elements ofthe system should be preluded

1003005000

50100150200250300350

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

FE1 FE3 FE6FE9 N

umbe

r of c

lient

s

Mea

n re

spon

se ti

me (

ms)

Number of nodes in FE and BE layer

Response time (60 requests per client)

Figure 6 Mean response time simulation results for differentnumber of nodes in FE (1 3 6 and 9) layer and in BE (1 3 6 and 9)and different numbers of clients

100300500

050

100150200250300350

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9BE1 BE3 BE6

BE9 Num

ber o

f clie

nts

Mea

n re

spon

se ti

me (

ms)

Number of nodes in BE and FE layer


Figure 7 Mean response time simulation results for differentnumber of nodes in BE (1 3 6 and 9) layer and in FE (1 3 6 and 9)and different numbers of clients

In the two Figures 8 and 9 we have presented the sameresults for each part separately We presented tables belowgraphs showing the results to clearer analysis The overallsystem response time increases with the increasing workloadAs we can see in Figure 8mean response time decreases whilethe number of nodes in BE layer increasesThe response timefor the same number of nodes in FE layer is almost the same(Figures 8(a) and 8(b)) We can see that for 6 and 9 nodesin FE layer the mean response time decreases for differentnumber of nodes in BE layer (Figures 8(c) and 8(d))

In the second scenario (Figure 9) the changes of thenumber of nodes in FE layer have a bigger impact on systemresponse time In all cases with the number of clients equalto 200 300 and 400 we can observe linearly decreasingresponse time The response time difference in cases of 100clients may be due to a big number of nodes in BE layercompared to a small number of requests Response timedifference (shortest response time) in cases of 500 clientssignalizes that 3 FE nodes (in this exampled load) is the bestsolution More number of nodes in FE layer is also good butthe benefit is not so obvious

The basic conclusion is that it is difficult to determinewhat would be the behavior of the system after adding morenodes in layers therefore research and performance analysisis necessary


100 200 300 400 500FE1 BE1 54781 12602 197428 268929 339719FE1 BE3 54586 126151 197888 26937 339617FE1 BE6 56792 127864 199597 271809 343988FE1 BE9 60748 131183 203651 276915 346359

0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE1)

(a)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (FE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE6)

(c)

Number of clients


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Response time (FE9)

(d)

Figure 8 Mean response time simulation results for different number of nodes in FE (1 3 6 and 9) layer and in BE (1 3 6 and 9) anddifferent numbers of clients (a) 1 node in FE layer and 1 3 6 and 9 in BE layer (b) 3 nodes in FE layer and 1 3 6 and 9 in BE layer (c) 6nodes in FE layer and 1 3 6 and 9 in BE layer and (d) 9 nodes in FE layer and 1 3 6 and 9 in BE layer

5 Conclusions

We cannot always add new devices to improve performancebecause the initial cost and maintenance will become toolarge Because the overall system capacity is unknown wepropose a combination of benchmarking and modelingsolution

It is still an open issue how to obtain an appropriateDWS Our earlier works propose Performance Engineeringframeworks [10] to evaluate performance during the differentphases of their life cycle The demonstrated research resultsare an attempt to apply QPN formalism to the developmentof a software tool that can support DWS design The result ofthe analysis is the discovery of the DWS structure useful inperformancemodelingThe idea of using QPNwas proposedpreviously by other authors In the presented approach analternative implementation of QPN has been proposed

Earlier [12 13] we set the parameters for the system exper-imentally We verified [12] the influence of the individuallayers on the system performance The paper [13] focuseson the expansion of the model by increasing the numberof elements in layers The current model (5) has only a fewparameters but it is fully functional and can be scaled to

larger systems The modeling approach presented in thispaper differs from previous works because of the following

(i) Realistic workload was not used earlier

(ii) We showed the model of DWS with a greater numberof nodes and different values on arcs

(iii) We applied 50 number of runs for every simulation

(iv) We took into consideration response times of all ele-ments (queues and depositories of QPN) especiallyclients depository

(v) Simplifying the model to a single element in a singlelayer

We develop a framework that helps to identify performancerequirements (response time parameter) The study demon-strates the modeling power and shows how discussed modelscan be used to represent the system bahaviour also inparticular layers (Table 6) Next we shall consider analyzingthe response time characteristics for more classes of clients(a separate token colour) to effectively model the Internetrequests from the clients


100 200 300 400 500BE1 FE1 54781 12602 197428 268929 339719BE1 FE3 4203 110978 182089 210811 232878BE1 FE6 16825 54624 121544 191241 261937BE1 FE9 21661 39059 85621 149563 218018

0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE1)

(a)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE6)

(c)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (BE9)

(d)

Figure 9 Mean response time simulation results for different number of nodes in BE (1 3 6 and 9) layer and in FE (1 3 6 and 9) anddifferent numbers of clients (a) 1 node in BE layer and 1 3 6 and 9 in FE layer (b) 3 nodes in BE layer and 1 3 6 and 9 in FE layer (c) 6nodes in BE layer and 1 3 6 and 9 in FE layer and (d) 9 nodes in BE layer and 1 3 6 and 9 in FE layer

Table 6 Example of mean response time [ms] in system layers(FE6BE9)

100 200 300 400 500FE 13178 37654 99082 168294 23858FE + BE 23534 4802 109423 178639 248921System 24213 48366 109737 178946 249225

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] T Rak and J Werewka ldquoPerformance analysis of interactiveInternet systems for a class of systems with dynamically chang-ing offersrdquo in Advances in Software Engineering Techniquesvol 7054 of Lecture Notes in Computer Science pp 109ndash123Springer Berlin Germany 2012

[2] X Chen C P Ho R Osman P G Harrison and W JKnottenbelt ldquoUnderstanding modelling and improving theperformance of web applications in multicore virtualised envi-ronmentsrdquo in Proceedings of the 5th ACMSPEC InternationalConference on Performance Engineering (ICPE rsquo14) pp 197ndash207March 2014

[3] S Kounev C Rathfelder and B Klatt ldquoModeling of event-based communication in component-based architectures state-of-the-art and future directionsrdquo in Proceedings the 9th Interna-tional Workshop on Formal Engineering Approaches to SoftwareComponents andArchitectures (FESCA 13) vol 295 ofElectronicNotes in Theoretical Computer Science pp 3ndash9 Elsevier May2013

[4] K Jamroz D Pitulej and J Werewka ldquoAdapting enterprisearchitecture at a software development company and the resul-tant benefitsrdquo in Software Architecture vol 8627 of LectureNotesin Computer Science pp 170ndash185 Springer 2014

[5] H Koziolek ldquoPerformance evaluation of component-basedsoftware systems a surveyrdquo Performance Evaluation vol 67 no8 pp 634ndash658 2010

[6] P Meier S Kounev and H Koziolek ldquoAutomated trans-formation of component-based software architecture modelsto queueing petri netsrdquo in Proceedings of the 19th AnnualIEEEACM International Symposium on Modeling Analysisand Simulation of Computer and Telecommunication Systems(MASCOTSrsquo 11) pp 339ndash348 Singapore July 2011

[7] C Rathfelder D Evans and S Kounev ldquoPredictive modellingof peer-to-peer event-driven communication in component-based systemsrdquo inComputer Performance Engineering vol 6342of Lecture Notes in Computer Science pp 219ndash235 SpringerBerlin Germany 2010

[8] S Samolej and T Szmuc ldquoHTCPNs-based modelling andevaluation of dynamic computer cluster reconfigurationrdquo in


Advances in Software Engineering Techniques vol 7054 ofLecture Notes in Computer Science pp 97ndash108 Springer BerlinGermany 2012

[9] K Zatwarnicki ldquoOperation of cluster-based web system guar-anteeing web page response timerdquo in Computational CollectiveIntelligence Technologies and Applications vol 8083 of LectureNotes in Computer Science pp 477ndash486 Springer BerlinGermany 2013

[10] T Rak and S Samolej ldquoDistributed internet systems modelingusing TCPNsrdquo in Proceedings of the International Multiconfer-ence on Computer Science and Information Technology (IMCSITrsquo08) pp 559ndash566 October 2008

[11] F Bause ldquoQueueing petri vetsmdasha formalism for the combinedqualitative and quantitative analysis of systemsrdquo in Proceedingsof the 5th InternationalWorkshop on Petri Nets and PerformanceModels pp 14ndash23 IEEE Toulouse France October 1993

[12] T Rak ldquoPerformance analysis of distributed Internet systemmodels using QPN simulationrdquo in Proceedings of the FederatedConference on Computer Science and Information Systems pp769ndash774 September 2014

[13] T Rak ldquoPerformance analysis of cluster-basedweb systemusingthe QPNmodelsrdquo in Information Sciences and Systems 2014 pp239ndash247 Springer Cham Switzerland 2014

[14] S Kounev Performance Engineering of Distributed Component-Base Systems Benchmarking Modeling and Performance Predic-tion Shaker 2006

[15] S Kounev S Spinner and P Meier ldquoQPME 20mdasha tool forstochastic modeling and analysis using Queueing Petri netsrdquoin From Active Data Management to Event-Based Systems andMore vol 6462 of Lecture Notes in Computer Science pp 293ndash311 Springer Berlin Germany 2010

[16] R Osman D Coulden and W J Knottenbelt ldquoPerformancemodelling of concurrency control schemes for relationaldatabasesrdquo inAnalytical and StochasticModeling Techniques andApplications vol 7984 of Lecture Notes in Computer Science pp337ndash351 Springer Berlin Germany 2013

[17] R Nou S Kounev F Julia and J Torres ldquoAutonomic QoS con-trol in enterprise Grid environments using online simulationrdquoJournal of Systems and Software vol 82 no 3 pp 486ndash5022009

[18] httpsgeronimoapacheorgGMOxDOC22daytrader-a-more-complex-applicationhtml

[19] D Coulden R Osman and W J Knottenbelt ldquoPerformancemodelling of database contention using queueing Petri netsrdquo inProceedings of the 4th ACMSPEC International Conference onPerformance Engineering (ICPE rsquo13) pp 331ndash334 ACM April2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Internetsystem

Experiments with Experiments with

Mathematicalmodel

Analiticalsolution Simulation

the real system a formal model ofthe system

Figure 1 Performance modeling possibilities

The remaining work is organized as follows Section 2describes QPN Section 3 presents DWS architecture anddescribes the modeling approach Section 4 presents perfor-mance analysis results The final section contains concludingremarks

2 Queueing Petri Nets

In our solution we propose a popular formal methodmdashQueueing Petri Net [11] This method is based on QueueingNets and Petri Nets Queuing Theory deals with modelingand optimizing different types of service units QueueingNet usually consists of a set of connected queuing systemsThe various queue systems represent computer componentsQueueing Nets (QN) are very popular for the quantitativeanalysis [14] QNs have a queue scheduling discipline andare suitable for modeling competition of equipment Toanalyze any queue system it is necessary to determine arrivalprocess service distribution service discipline and waitingroom (scheduling strategies) Petri Nets (PN) are used tospecify and analyze the concurrence in systems The systemdynamics is described by the rules of tokens flow The netscheme can be subjected to a formal analysis in order tocarry out a qualitative analysis based on determining itslogical validity PN have tokens representing the tasks andare suitable for modeling software PN are referred to asthe connection between the engineering description and thetheoretical approach Petri Nets are well-known models usedto describe and analyze service units PN cannot be used fora quantitative analysis due to lack of time aspectsThe studiesfocus on incoming load measuring for example measureof the response time or presentation of an overall modelingplanQNmdashquantitative analysismdashhas a queue and schedulingdiscipline and are suitable formodeling competition of equip-ment PNmdashqualitative analysismdashhave tokens representingthe tasks and are suitable for modeling software

Queueing Petri Net (QPN) formalism is a very popularformal method of functional and performance modeling(performance analysis) These nets provide sufficient power

Place Queueingplace

Subnetplace

Queue Depository

Input OutputUser

subnet

Actualpopulation

Figure 2 Elements of QPN

to express modeling and analyzing of complex online sys-tems The choice of QPN was caused by a possibility ofobtaining different character information The main idea ofQPN is to add queueing and timing aspects to the net placesQPN have the advantages of QN (eg evaluation of thesystemperformance and the network efficiency) and PN (eglogical assessment of the system correctness) QN consistsof a collection of service stations and clients The servicestations (queues) represent system resources while the clientsrepresent users or transactions A service station is composedof one or more servers and a waiting area (Figure 2) Tokensenter the queueing place through the firing of input transi-tions as in other PN When a request arrives at a servicestation it is immediately serviced if a free server is availableOtherwise the request has to wait in the waiting area Dif-ferent scheduling strategies can be used to serve the requestswaiting in the waiting area Places (QPN) are of two typesordinary and queued A queueing place (resource or state)is composed of a queue (service station) and a depositoryfor tokens that completed their service at a queue (Figure 2)After being served by the service station (coloured) tokensare placed onto a depository Input transitions are fired andthen tokens are inserted into a queueing place accordingto the queuersquos scheduling strategy Queueing places canhave variable scheduling strategies and service distributions(timedqueueing places) Tokens in the queue are not availablefor output transitionswhile tokens in the depository are avail-able to all output transitions of the queued place Immediatequeueing places impose a scheduling discipline on arrivingtokens without a delay [11]The last place on Figure 2 is calleda subnet place Hierarchical QPN (HQPN) has a dedicatedinput and output place which are ordinary places of a PNTokens being inserted into a subnet place after a transition fir-ing are added to the input place of the corresponding HQPNsubnet The semantics of the output place of a subnet place issimilar to the semantics of the depository of a queueing placeEvery subnet contains actual population place used to keeptrack of the total number of tokens fired into the subnet place

QPN is a tuple (2) where CPN is Coloured Petri Net (1)[11 15]

CPN = (119875 119879 119862 119868119872) (1)


where








QPN = (CPN 119876119882) (2)

where

(i) 119876 = (1198761 1198762 (1199021 119902|119875|)) where



119894denotes the



(119901119894) equals 119899119906119897119897)

(ii) 119882 = (11988211198822 (1199081 119908|119879|)) where


119895isin [119862(119905

119895) 997891rarr




119895isin 1198822























Computer system

Calculated

Performance model























119877 =

119896=1sum

119894

119877

1015840

119896 (3)


119877

1015840

119896= 119876119896+119863119896 (4)


119896=1119894


119863119896= sum

119896=1119894












119861119864 119868119874119899

bminus1198721119865119868119865119874e BE nodes




t1

t2 t3 t4


Clients

ThreadsPool

ConnectionsPool




(i) 119889119865119864 119862119875119880

= 0714 (ms)(ii) 119889119861119864 119868119874

= 0133 (ms)



Input-place

Input-transition

Actual population

Output-transition

Output-place

FE_CPU8 FE_CPU9

FE_CPU7FE_CPU6

FE_CPU5

FE_CPU3

FE_CPU1FE_CPU2

FE_CPU4

(a)

Input-place

Output-place


BE_IO1

BE_IO2

BE_IO3


(b)





in 119862119900119899119899119890119888119905119894119900119899119904119875119900119900119897 place)




QPN = (119875 119879 119862 119868119872119876119882) (5)

where


119862119897119894119890119899119905119904 119899119906119897119897 minus1198721

119875119878119878119906119887-119865119864 119899119906119897119897 minus1198721119865119868119865119874

119878119906119887-119861119864 119899119906119897119897 119899119906119897119897)) where

(a) 1198761 = 119862119897119894119890119899119905119904 119865119864 119862119875119880119898 119861119864 119868119874

119899

(b) 1198762 =






119905119895




(vii) 119882 = (11988211198822) where

(a) 1198821 = (b) 1198822 = 119879(c) forall119888 isin 119862(119905


equally likely)





QPME FE queues 119865119864 119862119875119880119898

a

BE queues 119861119864 119868119874119899

b















1003005000

50100150200250300350

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

FE1 FE3 FE6FE9 N

umbe

r of c

lient

s

Mea

n re

spon

se ti

me (

ms)




100300500

050

100150200250300350

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9BE1 BE3 BE6

BE9 Num

ber o

f clie

nts

Mea

n re

spon

se ti

me (

ms)









0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE1)

(a)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (FE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE6)

(c)

Number of clients


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Response time (FE9)

(d)


5 Conclusions













0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE1)

(a)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE6)

(c)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (BE9)

(d)



100 200 300 400 500FE 13178 37654 99082 168294 23858FE + BE 23534 4802 109423 178639 248921System 24213 48366 109737 178946 249225



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where








QPN = (CPN 119876119882) (2)

where

(i) 119876 = (1198761 1198762 (1199021 119902|119875|)) where



119894denotes the



(119901119894) equals 119899119906119897119897)

(ii) 119882 = (11988211198822 (1199081 119908|119879|)) where


119895isin [119862(119905

119895) 997891rarr




119895isin 1198822























Computer system

Calculated

Performance model























119877 =

119896=1sum

119894

119877

1015840

119896 (3)


119877

1015840

119896= 119876119896+119863119896 (4)


119896=1119894


119863119896= sum

119896=1119894












119861119864 119868119874119899

bminus1198721119865119868119865119874e BE nodes




t1

t2 t3 t4


Clients

ThreadsPool

ConnectionsPool




(i) 119889119865119864 119862119875119880

= 0714 (ms)(ii) 119889119861119864 119868119874

= 0133 (ms)



Input-place

Input-transition

Actual population

Output-transition

Output-place

FE_CPU8 FE_CPU9

FE_CPU7FE_CPU6

FE_CPU5

FE_CPU3

FE_CPU1FE_CPU2

FE_CPU4

(a)

Input-place

Output-place


BE_IO1

BE_IO2

BE_IO3


(b)





in 119862119900119899119899119890119888119905119894119900119899119904119875119900119900119897 place)




QPN = (119875 119879 119862 119868119872119876119882) (5)

where


119862119897119894119890119899119905119904 119899119906119897119897 minus1198721

119875119878119878119906119887-119865119864 119899119906119897119897 minus1198721119865119868119865119874

119878119906119887-119861119864 119899119906119897119897 119899119906119897119897)) where

(a) 1198761 = 119862119897119894119890119899119905119904 119865119864 119862119875119880119898 119861119864 119868119874

119899

(b) 1198762 =






119905119895




(vii) 119882 = (11988211198822) where

(a) 1198821 = (b) 1198822 = 119879(c) forall119888 isin 119862(119905


equally likely)





QPME FE queues 119865119864 119862119875119880119898

a

BE queues 119861119864 119868119874119899

b















1003005000

50100150200250300350

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

FE1 FE3 FE6FE9 N

umbe

r of c

lient

s

Mea

n re

spon

se ti

me (

ms)




100300500

050

100150200250300350

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9BE1 BE3 BE6

BE9 Num

ber o

f clie

nts

Mea

n re

spon

se ti

me (

ms)









0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE1)

(a)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (FE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE6)

(c)

Number of clients


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Response time (FE9)

(d)


5 Conclusions













0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE1)

(a)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE6)

(c)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (BE9)

(d)



100 200 300 400 500FE 13178 37654 99082 168294 23858FE + BE 23534 4802 109423 178639 248921System 24213 48366 109737 178946 249225



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Computer system

Calculated

Performance model























119877 =

119896=1sum

119894

119877

1015840

119896 (3)


119877

1015840

119896= 119876119896+119863119896 (4)


119896=1119894


119863119896= sum

119896=1119894












119861119864 119868119874119899

bminus1198721119865119868119865119874e BE nodes




t1

t2 t3 t4


Clients

ThreadsPool

ConnectionsPool




(i) 119889119865119864 119862119875119880

= 0714 (ms)(ii) 119889119861119864 119868119874

= 0133 (ms)



Input-place

Input-transition

Actual population

Output-transition

Output-place

FE_CPU8 FE_CPU9

FE_CPU7FE_CPU6

FE_CPU5

FE_CPU3

FE_CPU1FE_CPU2

FE_CPU4

(a)

Input-place

Output-place


BE_IO1

BE_IO2

BE_IO3


(b)





in 119862119900119899119899119890119888119905119894119900119899119904119875119900119900119897 place)




QPN = (119875 119879 119862 119868119872119876119882) (5)

where


119862119897119894119890119899119905119904 119899119906119897119897 minus1198721

119875119878119878119906119887-119865119864 119899119906119897119897 minus1198721119865119868119865119874

119878119906119887-119861119864 119899119906119897119897 119899119906119897119897)) where

(a) 1198761 = 119862119897119894119890119899119905119904 119865119864 119862119875119880119898 119861119864 119868119874

119899

(b) 1198762 =






119905119895




(vii) 119882 = (11988211198822) where

(a) 1198821 = (b) 1198822 = 119879(c) forall119888 isin 119862(119905


equally likely)





QPME FE queues 119865119864 119862119875119880119898

a

BE queues 119861119864 119868119874119899

b















1003005000

50100150200250300350

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

FE1 FE3 FE6FE9 N

umbe

r of c

lient

s

Mea

n re

spon

se ti

me (

ms)




100300500

050

100150200250300350

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9BE1 BE3 BE6

BE9 Num

ber o

f clie

nts

Mea

n re

spon

se ti

me (

ms)









0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE1)

(a)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (FE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE6)

(c)

Number of clients


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Response time (FE9)

(d)


5 Conclusions













0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE1)

(a)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE6)

(c)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (BE9)

(d)



100 200 300 400 500FE 13178 37654 99082 168294 23858FE + BE 23534 4802 109423 178639 248921System 24213 48366 109737 178946 249225



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119861119864 119868119874119899

bminus1198721119865119868119865119874e BE nodes




t1

t2 t3 t4


Clients

ThreadsPool

ConnectionsPool




(i) 119889119865119864 119862119875119880

= 0714 (ms)(ii) 119889119861119864 119868119874

= 0133 (ms)



Input-place

Input-transition

Actual population

Output-transition

Output-place

FE_CPU8 FE_CPU9

FE_CPU7FE_CPU6

FE_CPU5

FE_CPU3

FE_CPU1FE_CPU2

FE_CPU4

(a)

Input-place

Output-place


BE_IO1

BE_IO2

BE_IO3


(b)





in 119862119900119899119899119890119888119905119894119900119899119904119875119900119900119897 place)




QPN = (119875 119879 119862 119868119872119876119882) (5)

where


119862119897119894119890119899119905119904 119899119906119897119897 minus1198721

119875119878119878119906119887-119865119864 119899119906119897119897 minus1198721119865119868119865119874

119878119906119887-119861119864 119899119906119897119897 119899119906119897119897)) where

(a) 1198761 = 119862119897119894119890119899119905119904 119865119864 119862119875119880119898 119861119864 119868119874

119899

(b) 1198762 =






119905119895




(vii) 119882 = (11988211198822) where

(a) 1198821 = (b) 1198822 = 119879(c) forall119888 isin 119862(119905


equally likely)





QPME FE queues 119865119864 119862119875119880119898

a

BE queues 119861119864 119868119874119899

b















1003005000

50100150200250300350

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

FE1 FE3 FE6FE9 N

umbe

r of c

lient

s

Mea

n re

spon

se ti

me (

ms)




100300500

050

100150200250300350

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9BE1 BE3 BE6

BE9 Num

ber o

f clie

nts

Mea

n re

spon

se ti

me (

ms)









0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE1)

(a)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (FE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE6)

(c)

Number of clients


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Response time (FE9)

(d)


5 Conclusions













0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE1)

(a)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE6)

(c)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (BE9)

(d)



100 200 300 400 500FE 13178 37654 99082 168294 23858FE + BE 23534 4802 109423 178639 248921System 24213 48366 109737 178946 249225



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Input-place

Input-transition

Actual population

Output-transition

Output-place

FE_CPU8 FE_CPU9

FE_CPU7FE_CPU6

FE_CPU5

FE_CPU3

FE_CPU1FE_CPU2

FE_CPU4

(a)

Input-place

Output-place


BE_IO1

BE_IO2

BE_IO3


(b)





in 119862119900119899119899119890119888119905119894119900119899119904119875119900119900119897 place)




QPN = (119875 119879 119862 119868119872119876119882) (5)

where


119862119897119894119890119899119905119904 119899119906119897119897 minus1198721

119875119878119878119906119887-119865119864 119899119906119897119897 minus1198721119865119868119865119874

119878119906119887-119861119864 119899119906119897119897 119899119906119897119897)) where

(a) 1198761 = 119862119897119894119890119899119905119904 119865119864 119862119875119880119898 119861119864 119868119874

119899

(b) 1198762 =






119905119895




(vii) 119882 = (11988211198822) where

(a) 1198821 = (b) 1198822 = 119879(c) forall119888 isin 119862(119905


equally likely)





QPME FE queues 119865119864 119862119875119880119898

a

BE queues 119861119864 119868119874119899

b















1003005000

50100150200250300350

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

FE1 FE3 FE6FE9 N

umbe

r of c

lient

s

Mea

n re

spon

se ti

me (

ms)




100300500

050

100150200250300350

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9BE1 BE3 BE6

BE9 Num

ber o

f clie

nts

Mea

n re

spon

se ti

me (

ms)









0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE1)

(a)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (FE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE6)

(c)

Number of clients


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Response time (FE9)

(d)


5 Conclusions













0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE1)

(a)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE6)

(c)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (BE9)

(d)



100 200 300 400 500FE 13178 37654 99082 168294 23858FE + BE 23534 4802 109423 178639 248921System 24213 48366 109737 178946 249225



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












QPME FE queues 119865119864 119862119875119880119898

a

BE queues 119861119864 119868119874119899

b















1003005000

50100150200250300350

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

BE1

BE3

BE6

BE9

FE1 FE3 FE6FE9 N

umbe

r of c

lient

s

Mea

n re

spon

se ti

me (

ms)




100300500

050

100150200250300350

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9

FE1

FE3

FE6

FE9BE1 BE3 BE6

BE9 Num

ber o

f clie

nts

Mea

n re

spon

se ti

me (

ms)









0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE1)

(a)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (FE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE6)

(c)

Number of clients


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Response time (FE9)

(d)


5 Conclusions













0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE1)

(a)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE6)

(c)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (BE9)

(d)



100 200 300 400 500FE 13178 37654 99082 168294 23858FE + BE 23534 4802 109423 178639 248921System 24213 48366 109737 178946 249225



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE1)

(a)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (FE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (FE6)

(c)

Number of clients


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Response time (FE9)

(d)


5 Conclusions













0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE1)

(a)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE6)

(c)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (BE9)

(d)



100 200 300 400 500FE 13178 37654 99082 168294 23858FE + BE 23534 4802 109423 178639 248921System 24213 48366 109737 178946 249225



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE1)

(a)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE3)

(b)


0

100

200

300

400

Mea

n re

spon

se ti

me (

ms)

Number of clients

Response time (BE6)

(c)

Mea

n re

spon

se ti

me (

ms)


0

100

200

300

400

Number of clients

Response time (BE9)

(d)



100 200 300 400 500FE 13178 37654 99082 168294 23858FE + BE 23534 4802 109423 178639 248921System 24213 48366 109737 178946 249225



References





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Response Time Analysis of Distributed Web...

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