toll motorway queuing theory

8/12/2019 Toll Motorway Queuing Theory

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Study and Simulation of Queuing Theory inthe Toll Motorway

Houda Mehri* Taoufik Djemel**

*Research Unit in Logistics, Industrial Management and Quality

(LOGIQ) - ISGI-Sfax. BP nr954-3018 Sfax-TUNISIA.

[email protected]** GIAD Laboratory, University of Sfax, FSEG,

Route de lAerodrome, km 4, BP 1088, 3018 Sfax, Tunisia .

[email protected]

ABSTRACT.Adjustment tests in a case study show that arrival times and service times in a

motorway queuing system led to the model M/(, )/c. There are no analytical formula

to calculate performance measures of such a model. In this article, we generalize Pollazeck-Khitchinnes formula to establish approximate formula of these measures and we present a

simulator that calculates performance measures ofA/B/cqueuing system whereA, Bcan be

one of the following distributions: determinist, exponential, Erlang or Gamma. We verify the

theory according to which the type of organization of the queue (an unique queue for all the

servers or multiple queues where each server has his/her own queue) does not influence the

performance measures of the queuing system.

RSUM.Des tests dajustement sur les intervalles de temps entre les arrives et les dures

de service dune tude de cas ont abouti au modle M/(, )/c. Il nexiste pas de formules

analytiques pour calculer les mesures de performance dun tel modle. Nous gnralisons laformule de Pollazeck-Khitchinne pour tablir des formules approximatives de ces mesures et

nous prsentons un simulateur pour calculer les mesures de performance des modles de file

A/B/c o A, Bappartiennent lensemble des distributions {Dterministe, Exponentielle, Er-

lang, Gamma}. Nous vrifions la thorie selon laquelle le type dorganization de la file (une file

unique pour tous les serveurs ou multiple files avec une file devant chaque serveur) ninfluence

pas les mesures de performance du systme dattente routier.

Studia Informatica Universalis.


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Queuing Theory in Toll Motorway 97

KEYWORDS: Queues, statistical analysis, multi agent simulation, exponential distribution,

Gamma distribution.

MOTS-CLS : Files dattente, analyse statistique, simulation multi agent, distribution expo-nentielle, distribution Gama.

1. Introduction

The queueing theory is an operational research technique that modelsa system allowing queue, calculates its performances and determines its

properties in order to help managers in decision making. Results and

theoretical formulations are established for queue models with Poisson

arrivals and exponential service durations(M/M/c)[31], [24]. It is notthe case for all systems with the Poisson arrivals and non exponential

service durationM/G/cwhose analytical survey is very complex.

We suggest in this paper a generalization of Pollazeck-Kitchinne for-

mula to set the performance measures of the model M/G/c. Our nu-meric application of data collected is based on a case study of a toll

motorway. The statistical tests on arrivals and service duration showthat the distribution is respectively Poisson and Gamma. Hence, an

M/(, )/cmodel.

We present a multi-agent simulator that covers any queue whether

A/B/c or A, B belonging to the distribution set {Determinist, Expo-nential, Erlang, Gamma}.This simulator is used to check the hypothe-

sis that the queue type in a toll motorway (an unique queue for all the

servers or multiple queues where each server has his/her own queue)

does not influence the performance measures. To conclude this in-

troduction I will map the overall geography of this article. Section 2

sketches briefly toll motorway queues, section 3 is a case study fol-

lowed in section 4 by a generalization of Pollaczeck-khitchinne formula

to performance measures of models M/G/c. The simulator is presented

in section 5. Section 6 computes and interprets the results, and section

7 concludes the study.


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98 Studia Informatica Universalis.

2. Queues

2.1. Description of Queues

The field of study which would attract the attention of theorists was

Queuing Theory which would develop over the past century by leaps

and bounds. The analytical investigation of stochastic processes has

continued unabated up to the present time.

The literature study on queues contains references to the definition

and historical perspective of the modeling approach. Queueing theoryhas been a well-researched topic for many years and much published in-

formation is available. The literature study will give an overview sample

of Queueing Theory. An important source of information is by Giffin

[13].

Queues are not an unfamiliar phenomenon. To define a queue

requires specification of certain characteristics which describe the

system:

An input process: This may be the arrival of an entity at a servicelocation. The process may involve a degree of uncertainty concerning

the exact arrival times and the number of entities arriving. To describe

such a process the important attributes are the source of the arrivals, the

size of each arrival, the grouping of such an arrival and the inter-arrival

times.

A service mechanism: This may be any kind of service operationwhich processes arriving entities. The major features which must be

specified are the number of servers and the duration of the service.

The queue discipline: It defines the rules of how the arrivalsbehave before service occurs.

The queue capacity: Finite or infinite


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Examples of input and output processes which are as follows:

Situation Input Process Output Process

Bank Entities arriving Tellers serve entities

Toll Plaza Vehicles arriving Toll money is paid

Call Centre Incoming Call Call dealt with

Ferris wheel Tourists arrive Tourists are served

Intermittent Service Channel Entities arrive Entities are served intermittently

Naval Harbour Ships that must unload Unloading of ships

Table 1: Examples of Queuing Systems

The presence of uncertainty makes these systems challenging in respect

of analysis and design. The input rate/arrival rates together with the

output rate/service rate mostly determine whether there are entities in

the queue or not. These factors also determine the length of the queue.

In practice the arrival rate may be measured as the number of arrivals

during a given period. The service rate can be measured in the same

way. This is usually done for a system that has progressed from a tran-

sient state to a steady state. Most of these systems are described by

arrival and service rates. It is however important to also focus on the

transient characteristics of the system.

2.2. Historical Perspective

The ground work for many of the earliest techniques of analysis in

queuing theory was laid byA K Erlang, father of queueing theory, be-tween 1909 and 1929. He is given credit for introducing the Poisson

process to congestion theory, for the method of creating balance state

equations (Chapman-Kolmogorov equations) to mathematically repre-

sent the notion of statistical equilibrium. Pollaczek [26] began studying

non-equilibrium queueing systems by looking at finite intervals. How-

ever, the first truly time dependant solutions were not offered until Bai-

ley [5] using generating functions and Lederman and Reuter [30], using

spectral theory and Champernowne [8], using the combinatorial method

found such solutions. Kendall [21] introduced his method of imbedded

Markov Chains in analyzing non-Markovian queues. The important

technique, known as the supplementary variable technique was intro-


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duced by Cox [10].

Most of the pioneers of Queuing Theory were engineers seeking solu-

tions to practical real world problems. The worth of queuing analysis

was judged on model usefulness in solving problems rather than on the

theoretical elegance of the proofs used to establish their logical consis-

tency.

The focus of the non-research oriented engineer in this expansive

theoretical development was on techniques which demonstrated appli-

cations of the results of the theory. In Operations Research the only field

that has few theoretical models with any useful applications is QueuingTheory. One may speculate why this has occurred. The commonly

mentioned reason is that the equations resulting from many theoretical

investigations are simply too overly complex to apply. The practitioner

then often has to resort to simulation methods for analysis.

In practice common simple queues are scarce. Arrival and service

rates may be constantly shifting over time so it is important to describe

the distributions as functions of time. These systems are contrasted with

steady-state solutions in which the arrival and service patterns are usu-

ally such that the state probability distribution is stationary. Dynamic

systems require robust modeling that can provide useful results eventhough the analysis may violate assumptions used in constructing the

model.

Most of the above discussion relates to what Bhat [6] refers to as be-

havior problems of the system. The focus is to use mathematical models

to seek understanding of a particular process. Other problems are statis-

tical and operational. "Statistical" refers to analysis of empirical data,

estimation of system characteristics and tests of hypotheses regarding

queuing processes. "Operational" refers to design, testing and control

of real life problem.

2.3. Review of Queuing Models and Their Modeling Approaches

Queuing theory is a general theory. It is about the behavior not only

of the queuing process, but also of the generalized symbolic model of

the process. The purpose of a symbolic model is to make predictions


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possible about the behavior of processes under changing conditions or

sometimes about processes that do not yet exist.

Despite the apparent simplicity of having only 3 basic components,

queuing theory has a wide variety of applications and is the starting

point for many practical problems at all levels of complexity. For most

practical problems, one is not so much concerned with properties of a

single given queuing system as with the comparison between various

possible alternatives. In particular, one may wish to predict what effect

various changes in strategy might have on some existing systems. For

example, if there is a large family of possible designs and one wishes tofind the optimal design according to some criteria, one is likely to get

a mixture of mathematical techniques ( including queuing theory) with

various optimization schemes (such as mathematical programming, dy-

namic programming, etc). Moreover, many design problems also con-

tain complex constraints. As a result, the system has to be carefully

designed so as to avoid certain highly undesirable or impossible situa-

tions.

In most queuing problems there is an implied cost associated with

waiting. Various types of service also have their cost, with the one giv-

ing the fastest service being the most expensive. The design problem isto minimize the sum of waiting cost and service cost. In simple situa-

tions, the service facility can be designed to provide any rate of service

(at a price in toll station), but once designed the service is independent

of whether it is fully utilized or not. This is approximately the case

when the service is done by a machine of some sort. The optimal de-

sign is one that minimizes the cost of inadequate capacity during rush

hours and excess capacity at other times.

In most real life situations, there are certain emergency facilities that

can be brought into service when there is a heavy demand. Thus the ser-

vice rate can be considered as time dependent. This is obviously what is

done at banks, grocery stores, etc. where employees are borrowed from

other tasks to serve a rush hour demand.

When it is possible to increase the service rate by the addition of help

from other jobs, the number of parameters associated with any strategy

may become rather large. For example, the cost of adding temporary


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capacity usually has at least two parts. On the one hand, there is a

cost per unit of service such as labour cost, interest on investment, etc.

this may be overtime labour which has a higher rate than the regular

service. On the other hand, there is usually a fixed cost associated with

the introduction of the extra service.

The dynamics of queues has been analysed by using steady-state

mathematics. Such queueing processes are described by using the

Kendall-Lee notation which uses mnemonic characters that specify the

queueing system

A/B/C/D/E/F

A: Specifies the nature of the arrival process.

B: Specifies the nature of the service times.

C: Specifies the number of parallel servers

D: Specifies the queue discipline.

E: Specifies the maximum number of entities in the system.

F: Specifies the size of the population from which entities aredrawn.

This notation is commonly used when deriving expressions for the aver-age system length, number of entities in the queue, the average waiting

time, and many other features.

For queuing models, entity arrivals and service times are summa-

rized in terms of probability distributions normally referred to as arrival

and service time distributions. These distributions may represent sit-

uations where entities arrive and are served individually (e.g. banks,

supermarkets). In other situations, entities may arrive and/or be served

in groups. (e.g. restaurants). The latter case is normally referred to as

a bulk queue. A Poisson stream of entities arriving in groups is served

at a counter in batches of varying size under the general rule for bulkservice in which the server remains idle until the queue size reaches or

exceeds a fixed number whereupon they are served.


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2.4. Steady State Performance

In a steady state, the performance of a system is that any effects

from starting up with no customer present, or with a backlog of cus-

tomers who arrived before the server started working has been reduced

to negligible proportions (i.e. the system has been operating for a long

time sufficient for a large number of customers to pass through it).

For queuing situations such as shops and banks, it may take a large

proportion of the working day before a steady state is achieved. For

mass transport utilities (e.g. MTR, bus, ferry, etc.), it may need onlytwo to three vehicles to bring the system to a steady state.

If a steady state cannot be assumed, then an analysis of transient

behavior is required. Transient states include not just start up, but tem-

porary periods of over-load such as rush hours. Analysis of transient

behavior is possible with queuing theory, but requires techniques (e.g.

fluid approximation) that are unlikely to be used by non-specialists.

In this study, a queue is made up of customers demanding a service

to one or several servers and a waiting room. The rate of arriving cus-

tomers and the rate of service per time unit are respectively marked

and. A queue model is described according to Kendall-lee nota-tion [31], [24] by A/B/C/Disc/N/P. A and B represent the lawsof arrivals process and the service duration, M stands for exponen-tial, D determinist, (, ) Gamma of parameters and, and G ageneral law (any), C the number of servers, Disc the service disci-pline,N the system capacity and Pthe size of the source population.When the last three optional parameters are omitted, they are considered

FCFS// whereF C F S i.e. First Come, First served.

The performances measures of the queue model are: The average

time of a customer stay within the system (W), the average wait of acustomer (W q), the average service time (W s), the average numberof customers in the system (L), the average number of customers inthe queue (Lq), the average number of customers being served (Ls),the usage rate of servers or traffic intensity =

c.(Where c is the

number of servers), the equilibrium probabilities notedPn (probabilityofn customer in the system) and the wait probability (probabilityof a coming customer waiting before being served). In toll motorways,


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two big schools share the queue organization [25], [11]

Type1 an unique queue for all toll station thereby preventing cus-

tomers to train themselves queues

Type 2 every toll station has its own queue enabling the customer

join the shortest queue.

3. Case Study

In a toll motorway station, the total cost of service contains not onlythe cost of exploitation of server, but also the waiting cost of customer.

The knowledge of average waiting time of a customer by number of

servers in the system allows the administrators to choose the appropri-

ate number of servers which minimizes this total cost of service. The

other factors such as the utilization rate of servers, the average number

of customers in the system or in the queue, the equilibrium state prob-

ability of the number of customers in the system are also useful for a

good management. All these factors are characteristics (measures of

performance) of a queueing system.

During two weeks, we collected data related to cars arrivals and ser-vice durations at Tunis-Msaken motorway. At the toll station of cars

to three available servers were our frameworks of study. Statistical

adjustment tests showed that per second, the arrivals were Poisson of

= 0.015 parameter and service duration Gamma of = 2.42 and= 77.17 parameters. Contrary to multi servers with Poisson arrivalsand exponential service process M/M/c, random distribution modelsare more complex to analyse [22]. There are no analytical formulae

to compute the characteristics of aM/G/c queueing system. The ap-proximation formula of Pollaczek-Khintchine is only valid forM/G/1models. For theM/G/c model [2], [4], Ivo Adan [2] has establishedonly an approximation formula for the average waiting time of a cus-

tomer in the system. Here, we give an approximation of Pollaczek-

Khintchines formula to establish approximation formulae for the other

performance measures of the M/G/c model. The only good theoreticalformulations encountered deal with models not allowing waiting that is

M/G/c/GD/c/ model with loss of customers and M/G/ modelwith an infinite number of servers [9], [22].


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The equilibrium state probabilities of number of customers in the

system are computed with a simulator which we implemented. We were

then able to verify the theory according to which the configuration of

the queue (single line or multiple lines, one for each server) does not

influence the queueing system performance.

3.1. Data Collection

The research goal during this collection was to get time intervals

between cars consecutive arrivals and service length. We take interestto periods where each of the following events occurs: a car arrival, a

beginning of a service in a station, an end of service in a station. Data

collection has been completed thank to a perfected tool that take into

consideration the simultaneous appearance of many events .The main

functionality of this tool is to allow to pick up and save cars arrivals

schedules, beginning and ends of services thank to a simple click or

typing on the keyboard (see figure 1).

Figure 1: Queueing Data Collector

The two initial weeks period retained for the study was from July 17to July 30, 2007. For some independent raisons from our will, July 24

and 25 were brought forward to July 15 and 16.The first day of collec-

tion has allowed us to be in touch with work background and it was a

raison to reduce the period of collection into 2 hours. July 16 night was

punctuated during few minutes of little electricity break which disrupt

collection in that day. For necessity to control passengers behavior dur-


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ing crowded days, July 29 and 30 collections days were brought forward

to August 1 and 2.

3.2. Analysis of Data Collected

3.2.1. Gathered data presentation

Table 2 shows the arrivals effect size during the several days of the

collection period.

17/07/08 18/07/08 21/07/08 23/07/08 01/08/08 02/08/08

Arrivals 405 432 433 419 434 456

Services 401 426 431 403 403 435

Table 2: Arrivals and Service Durations effect Size

An attention has been concentrated on daily periods where we note

crowd. Table 2.2 shows the evolution of arrivals number noted in all

hours during the collection period. We notice that the most important

crowd is observed between 9h30 and 10h30 like the rush hours for mo-

torway. Conversely, the 17h30 to 18h30 period is marked as off periodof the day with 15 arrivals noticed on average.

Range 17/07/08 18/07/08 21/07/08 23/07/08 01/08/08 02/08/08

730-830 42 55 63 34 61 63

830-930 59 64 59 64 63 59

930-1030 49 58 45 75 57 54

1030-1130 55 62 49 47 49 55

1130-1230 23 32 30 26 32 22

1430-1530 68 71 60 63 70 81

1530-1630 58 50 59 58 40 62

1630-1730 38 33 50 47 43 45

1730-1830 15 9 20 7 18 17

Total 407 434 435 421 433 458

Table 3: Arrivals Periodic Evolution during the Collection Period.

Our goal is to determine probabilities distribution that well-describe

time intervals between arrivals and noticed service durations in the sys-

tem under study. Starting from the curve pace of the noticed effect size,


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we apply a chi-square test to fit these data to a known theoretical distri-

bution have the same pace. We illustrate the procedure with data of July

17.

3.2.2. Arrivals Analysis

We are interested to time intervals between cars consecutive arrivals

(inter-arrivals). We have distributed data over the inter-arrivals into30seconds amplitude classes, starting from 0 to 300 seconds (0to 5 min-utes) and more 1 . Table 4 shows that inter-arrivals less than 30 sec-

onds constitute the distribution mode this means the class that occursthe most frequently in a data set or a probability distribution . (0 30)class constitutes in itself30% to40% of daily effect size and (30to60)represents about20%. These effect sizes decrease to less than10% toeach of more than90 seconds classes (1 minute 30 seconds) howeverother classes of more than150 seconds (2 minutes30 seconds) repre-sent less than 5%. This explains the important decreasing incline ofseveral figure 2. A very light recovery were noticed for more than 300seconds class (5 minutes) for bringing together all inter arrivals higher

than5 minutes,30 seconds amplitude is never considered in this case.

Figure 2: Inter arrivals Effects Sizes Curves

This table shows as well that inter arrivals effects sizes curves have a

decreasing pace. A negative exponential variation might be supposed

1. A class[a, b]will represent higher or equal inter-arrivals to a seconds and strictly lowerto b seconds.


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which leads to orientation for laws of probability with a negative 2 ex-

ponential, like the exponential law or Gamma 3 law.

Class Center(xi) Empirical Effect Size(ni)0-30 15 163

30-60 45 73

60-90 75 68

90-120 105 35

120-150 135 18

150-180 165 18

180-210 195 6

210-240 225 7

240-270 255 4

270-300 285 5

>300 315 8

Total 405

Table 4: Statistic Table of July 17 Service Durations

Inter-arrivals fitting to an exponential lawFrom this table we can conclude: The empirical average:x=

nixiN

=68.185seconds considered an arrival each the1 minute8 seconds. The

empirical variance:S2

=

N

nixi2(

xini)2

N(N1) = 4778.691Considering anarrival each the1 minute8 seconds. This imply that in one second we

have x1 =0.015cars coming to the system per seconds. After fitting,thank to 2 test noticed data to an exponential law of the parameter = 0.015, we get a distance 2

c = 14 .484 . The number of classes

beingr = 10 and as we estimated a parameter (the average ), thefreedom degree is= 10 1 1 = 8.

= 8, 2 table gives as value not to exceed to the threshold = 0.05 the quantity 2 = 15.507. As

2

c < 2

a, with 95% of

confidence degree, we admit the hypothesis according to which July 17

inter-arrivals undergo an exponential law with a parameter = 0.014.shows the curve of noticed effect sizes and theoretical sizes according

to an exponential law.

2. Because it has paces with negative exponential functionf(x) =ex.3. First, we had have proceedat the beginning to a fitting to an Erlang law, k form estimation

provides us only integer values. That is why we go back to a fitting of a Gamma law.


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Figure 3: Theoretical and Noticed Effect Sizes Curve (Exponential

Law) of July 17 Inter arrivals

3.2.3. Service Duration Analysis

Gathered services durations have been also classified into amplitudes

of60 seconds, starting from 0 to 600 seconds (0 to 10 minutes) and

more. According to table4, we obtain the following characteristics: theempirical average: x = 192.044 seconds considering one car servedeach3 minutes12 seconds. The empirical variance2 = 15948.80798.

Class Center (xi) Empirical effect sizeni0-60 30 36

60-120 90 86

120-180 150 101

180-240 210 76

240-300 270 43

300-360 330 24

360-420 390 9420-480 450 6

480-540 510 8

540-600 570 5

> 600 630 7

Total 401

Table 5: Statistic Table of July 17 Service Durations


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Fitting service durations to an Exponential law

In one second we have 1x

= 0.0052served cars.A fitting test with 2 test to an exponential law of the parameter = 1

x = 0.0052 gives a distance2

c = 125.72. Freedom degree of

theoretical2 is = 8 (r = 10 classes), 2 table gives as value notto exceed to the threshold = 0.05 the quantity2

a(8) = 15.507. As

2c

> 2a

; at 95% of the confidence degree, we reject the hypothesis

according to which July 17 service durations undergo an exponentiallaw with a parameter= 0.0052.

Fitting service durations to a Gamma law

Gamma law is characterized by two parameters: and . Its av-eragem= and its variance=2. The parametersancan beestimated by:

=m2

=

192.0442

15948.9 = 2.31 =

m=

15948.9

192.044= 83.04

After fitting test with 2 test to a Gamma law of parameter = 2.31and= 83.04, we obtain a distance2

c = 15.327. The classes number

beingr = 11 and since we estimated two parameters ( and ); sothe freedom degree is = 11 2 1 = 8. For = 8, 2 tablegives as value not to exceed to the threshold = 0.05 the quantity2(8) = 15.507. As

2

c < 2; so at 95% of confidence degree, we

accept the hypothesis according to which July 17 service durations

undergo a Gamma law with parameter = 2.31and= 83.04. Figure4 presents the noticed effect sizes curve compared to theoretical effect

sizes of service durations undergoing a Gamma law.


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Figure 4: Theoretical and Noticed Effect Sizes Curve (Gama Law) of

July 17 Inter arrivals

3.3. Summary of Obtained Results

Tables 2.5 and 2.6 present respectively obtained results about cars

inter-arrivals and service durations for each of the collected days with

threshold = 0.05

Table 2.5 shows that inter-arrivals are adjustable to an exponential

law for the majority of carried out collection days. Fitting service

durations to an exponential law causes negative result because, from thetotality of carried out collection days, none caused a result conclusive

with an approach to an exponential law. However 84% (5 out of6)of carried out collection days produced a satisfactory result for an

approximation of services durations to a gamma law.

Days x 2c 2

Results

17/07/08 68.18 0.014 14.48 8 15.507 We acceptH018/07/08 65.76 0.015 3.63 7 14.067 We acceptH021/07/08 69.73 0.014 13.36 8 15.507 We acceptH023/07/08 64.11 0.016 5.156 7 14.067 We acceptH0

01/08/08 67.53 0.015 12.82 9 16.91 We acceptH002/08/08 65.06 0.015 08.194 8 15.507 We acceptH0

Table 6: Result Recapitulation of Inter-arrivals Fitting to an Exponential Law


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Days x 2 2c 2

Results

17/07/08 192.04 15948.80 2.31 83.04 15.327 8 15.507 We accept H0

18/07/08 184.78 13347.843 2.55 76.92 13.671 7 14.067 We accept H0

21/07/08 181.81 15484.87 2.11 90.91 28.71 7 14.067 We reject H0

23/07/08 184.39 15802.30 2.15 89.42 14.056 7 14.067 We accept H0

01/08/08 203.44 14715.18 2.81 76.41 5.273 7 14.067 We accept H0

02/08/08 179.16 11320.38 2.85 63.12 5.038 5 11.070 We accept H0

Table 7: Result Recapitulation of Service Durations Fitting to a Gamma Law

We have no elements to justify the fact that service durations of July23 had not produced satisfactory results neither according to the

exponential law nor according to the Gamma law. This may be due to

some field unforeseen that a study more extended on several months

even several years had allowed us to determine. May be it is important

to point out the anomaly noted in that day; the service was assured

during the first minutes of the morning by only one station, the others

being started after10 to15 minutes.

After a synthesis of obtained results over the analysis of all gathered

data during the collection, the result of arrivals average is 66.71 (= 1

minute6 seconds) and the service durations average is 187.38seconds(= 3minutes7 seconds).

In the final analysis of the carried out study, we come to the model

of the following queue: M/(,/3/FCFS//: Cars come to thesystem following a Poisson distribution according to 53 cars per hour(0.015cars all seconds). The toll area has3 data services with Gammaservice durations of parameter = 2.42 and = 77.17 (in seconds).Service policy is the first arrived the first served. There no limits to

maximum cars number in the motorway as well as in its provenance

source.

For this system, we get the following performance measures:


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Performance Measures Approximation

The trafic intensity 93.5%Probability of having empty system0 0.015

Probability of waiting before servicew 0.81Time of average sojourn in the system W 778

Time of average waitingWq 591.42Time of average serviceWs 186.61

Average number of cars in the systemL 11.76

Average number of cars in the queue Lq 8.86Average number of cars into serviceLs 2.80

Table 8: Performance Measures of the Studied System

3.4. Results and Interpretations

In order to ameliorate results of its systems, motorways managers

often lead to ask several questions like:

1) What effect of stations number increase on systems performance?

2) How much stations are needed in order to get waiting average

time or the average number of waiting passengers lower at a certain

boundary?

3) What effect of the unavailability of one (or n) servers (s) on system

performances?

4) What is the probability that a car wait over a t instant in the queue?

5) What is the probability of more than one car in the queue?

6) How to optimize stations use rate or increase its efficiency if they

are too often unoccupied?

We suggest using queue theory techniques to help these managers in

their decisions. So we need to define the previous performances mea-

sures.

3.5. Used Methodology

The model M/(, )/c is more complex than classical queuingmodels with M/M/c exponential services durations and Poison ar-rivals. So far, no theoretical result on performance measures calcula-

tion for non-exponential queue models has been formally established


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and proven [17]. Only approximations of some performances mea-

sures exist. In literature, several solution approaches were proposed.

A quite detailed of tasks retrospective which were made on queue with

arrivals or non-exponential service durations as well as authors who

introduced it, has been presented by Ivo Adan, W. A van of Waarsen-

burg and J. Wessels in [2]. But none of these results suit to what we

search. Authors think even that multiserver queue systems with arrivals

or non-exponential service durations are a typical example of convivial

or simple models at first sight but which could not be analyzed with sat-

isfaction. This is because it is proving that the behavior of the systemis much more complex and that is a sign of its simple formulation. The

lack of works dealing with queue models with Gamma distributions in

literature, bring us to be oriented rather to a General distribution. An

analysis and measures determination and performance indicator has

been introduced in case of a unique server (M/G/1model) since1950;the problem becomes difficult in the(M/G/c)multi-servers case. Onlytheoretical good approximations of multi-servers models with Poisson

arrivals and General service durations that we meet with, deal with

M/G/c/FCFS/c/ models which do not neglect customers who findthe station service plenty and M/G/ models with unlimited numberof servers which do not fit to our case. Works of Sadowsky J.S and W.

Szpankowskiy [27] about G/G/cmodels could supply us satisfaction;but it focus only on maximum waiting duration and maximum number

of customers in the queue. A certain number of works exists about a

recent model of queue considering rejections and owning established

theoretical results. It is M/GI/c+ GI orGI/GI/c+ GI[32] but itrequire a certain number of conditions that our models do not fit like:

1) Big number of servers(50, 100even1000)

2) A system permanently in overloaded regime where arrivals rate is

upper to services rate( =

c 1.02) so the outflanking of what thecustomers will be is included in rejections rate to bring the system intoa stable regime.

As an alternative solution, we opted to establish from Pollaczek- Khint-

chine formula and inspiring by the approximation of waiting average

duration in an (M/G/c) multiple servers system established by IvoAdan [1], the approximation of performance measures on waiting av-


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erage durations(Wq), of service(Ws)and of sojourn in the system (W)as well as the waiting average number of customers (Lq), in service(Ls)and in the system(L).

We could not establish formula allowing us to find approximate val-

ues for equilibrium probabilities on the number of customers present in

the system (Pn). Therefore, we used simulation to near it.

4. Approximation of the Performance Measures ofM/G/cModels

According to the average value formula [2] of Pollaczek-

khintchinne, a new arriving customer should wait for a customer be-

ing served and all customers queuing behind before being served. The

average wait is calculated as follows:

Wq =E(R) + LqE(B) (1)

Ris the residual service time of a customer being served, B the ser-vice time and=

the probability of finding a customer being served.

In accordance with Ivo Adan [2], we suppose that the time needed to

empty the queue with c servers isc times shorter than with one server.Then, we get:

Wq =1

c(wE(R) + LqE(B)) (2)

Here the probability of a customer waiting before being served is noted

w (The valuew of theM/M/cmodel can be used as an approxima-tion ofw of theM/G/cmodel). Ris the residual service time and Bthe average service time. According to the theorem of Little Lq =Wq,sinceE(B) = 1

, we get:

Wq

wE(R)

c(1 ) with=

c (3)

The average distribution of residual time [24], [31] is:

E(R) =1

2(c2B+ 1)E(B)

Where c2Bis the coefficient of variation of service durations (quotient ofthe standard deviation from the average).


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Then we get:

Wq = 1

c

w2(1 )

(c2B+ 1). (4)

Hence

Ws=E(B) = 1

(5)

W=Ws+ Wq = 1

+

1

c

w2(1 )

(c2B+ 1) (6)

Lq =Wq =

1

c

w2(1 ) (c

2B+ 1) (7)

Ls= Ws= =

(8)

(Not to confuse to the use rate c

).

L= Ls+ Lq =

+

1

c

w2(1 )

(c2B+ 1) (9)

Since we cannot elaborate formulae to work out approximate values for

the equilibrium state probabilities of the number of cars present in the

system(P n), we simulated them.Derivation ofw:To Ivo Adan [1] the value ofw ofM/M/c model can be used as anapproximation ofw of theM/G/c model. InM/M/c model,w isthe probability for a customer to wait before profiting from the service,

with c servers we havew =Pc+Pc+1+Pc+2+. . . . but for the M/M/cmodel:, with , this imply that Pc+n =

nPc, with =

, this imply that

w = Pc1

Case of a unique queueThe statistic analysis of the set of global data allows us to conclude that

the cars arrivals are Poisson with a rate = 0.015so the average 1

=66.71 seconds and that service durations are Gamma (with parameter = 2.42 and = 77.17) of an average 1

= 187.38 seconds and of

variance2 = 14461.96. The number of servers isc = 3.Table 8 gives us the obtained values of performance indicators. The last

column shows the results produced by the simulator.

The following figure illustrates equilibrium probabilities to get n cars


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Performance Measures Approximation Simulation

93.7% 93.5%0 0.015 0.014w 0.87 0.80W 786.107 744.073

Wq 608.727 557.463Ws 187.38 186.61

L 11.941 11.576Lq 9.13 8.770Ls 2.81 2.805

Table 9: Performance Measures of the Studied System

Figure 5: Equilibrium Probabilities to getnCars in the System

in the systemPn according ton.Case of multiple queuesTheoretical models, which include all aspects of this approach, do not

exist. Results obtained at the end of several simulations of different

durations but under the same initial conditions as the results for a unique

queue.

5. Simulation

Multi-agent simulation aims at creating an artificial world in which

interact agents evolving in an environment[19].


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5.1. Functionality

The simulator programmed in Java is multi-agent. It implements

both types of organization of toll motorway queues A/B/c modelswhereA, B belong to the set of distributions {Determinist, Exponen-tial, Erlang, Gamma}. The simulator input parameters are: the type of

organization of the queue of the simulated model, the simulation dura-

tion, the distributions of services and arrivals durations and the number

of tollbooths. The simulator releases the model performance measures.

Figure 6 gives an idea about the simulator.

Figure 6: Glimpse of the Simulator

5.2. Description

The agent is represented by an independent thread, the know-how by

a method, the state by the set of values of its features and its behavior by

the Run method of the thread. The agents are endowed with two facul-ties: Perceive and Act. The agent should first perceive ones surround-

ings before acting. The system has three types of agents: customers,

tollbooth staff and the head agent who supervises and coordinates the

simulation process.

The choice of time units and working out the dependant values is

important. The time unit is governed by the service and arrival rates.


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These rates are stated in relation to the same time unit. If for example

the arrivals rate is stated according to the number of arriving customers

per hour, then the service rate should be stated in terms of the number

of served customers per hour. The simulation duration should also be

stated in the same time unit.

In type2, when a customer chooses a queue,s/he remains there un-til being served. The simulator uses the reverse transformation method

[9] to generate the random variables. To validate the simulator, we com-

pared the output to the known theoretical results.

5.3. Performance Measures Calculation

Calculation ofL, LqandLsWe use two tables of integers Lq[ ] andLs[ ]. A simulation step cor-responds to 100ms (1/10second). At theiiem step we assign to Lq[i](respectively Ls[i]) the number of customers in the queue (Qsize) (re-spectively the number of customers being served S size). At the end of

the simulation we get:

Lq =T

pi=1 Lq[i]Tp

; Ls =T

pi=1 Ls[i]Tp

; L= Lq+ Ls. (10)

Tp is the total number of simulation steps. IfT is the simulation dura-tion thenTp= 10 T.Calculation ofW,W qandW sWe use also two integers tablesWq[ ]and Ws[ ]corresponding respec-tively to the wait and service time of customers in the system. Each

customer agent has two variables Date (creation date Dc and servicestarting dateDd) and a service duration variableDs. Upon creation ofthe customer, its creation date is updated to the current date and the

computer generates the time before creating the next customer accord-

ing to the distribution of time intervals among arrivals. The created

customer will be served if a tollbooth is free. Otherwise; it is attributed

a positionPos =Qsize+ 1in the queue (the shortest in the case of sev-eral queues) that is decremented as far as a tollbooth is freed (its own

queue in the case of multiple queues). When the iiem customer occupiesposition1 and a tollbooth is freed (its tollbooth in the case of multiple


25/35


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For the Exponential distribution with average, the simulator gen-erates an uniform random numberr and returns the 1

In(r)value.

For an Erlang distribution with kform and average, the simulatorgenerates k random numbers r1, r2 . . . rk and returns the amount

5 of

distributed exponential random numbersk with the average 1

, that is to

say, the value 1

In(r1, r2, . . . , rk).

For a Gamma distribution, we will use the Erlang distribution that

is one of its special case 6. we used as well the acceptance and rejec-

tion method algorithm [6] which consists on identifying an quantity

(which corresponds to the contribution of the decimal part of noted )using the generation of a variable (, 1) (with0 < < 1). Then weapply an "-addition" with an Erlang random variable corresponding tothe entire part of. So the generated value is: 1

( ln(U1Ur2. . . U []))

That follows a distribution(, )

5.5. The Use of Simulator

Our simulator contains several functionalities distributed over four

panels namely:

Data PanelIt gives the user controls allowing entering simulation data mainly: Re-

spectively three unrolling lists for the approach choice, the arrivals dis-

tribution, and that of services as well the parameters that characterize it.

Textszones for stations number and the simulation duration. A box to

tick for the infinite mode 7

Progress Check-up PanelIt displays dynamically during the simulation: The number of arrived

cars, the number of served cars, the number of busy server, the number

of free servers and the queue size.

Chart PanelIt gives panels showing graphics and end of simulation results. The

displayed graphics show: the queue size, the average duration of wait-

5. We saw before that an Erlang random variable with ak form is in fact the amount ofkexponential random variables.

6. When is an integer then the Gamma distribution reduces to an Erlang distribution.7. In infinite mode, the simulation has not a limited duration; it runs until the user stops it.


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ing cars, the number of busy servers, the average duration service. The

presented results at the end of the simulation are the following: The av-

erage number of cars in the systemL, the average number of cars in theserviceLs, the average number of waiting carsLq, the average durationof sojourn of a car in the systemW, the average duration that one carpasses in the service Ws, the average time that one car passes to thequeueWq and the equilibrium stationary probabilities Pn.The Control PanelIt gives a certain number of orders allowing controlling the simulation

progress like: Run to start the simulation, Pause to interrupt shortlya simulation in progress and see current results, Resumeto take againafter an interruption,Reinitialiseto take again data and graphics withinsight of a new simulation,Stopto stop the simulation, it allows also tochoose the mode of graphics display (curves or areas).

5.6. Simulator Validation

The results of our simulator are valid thanks to some comparisons

with known theoretical results. For exponential queues, the simulation

results were compared to obtained results analytically. For queues ofone server with Poisson arrivals and non exponential services dura-

tions (Erlang, Determinist or Gamma), results concerning performance

measures were compared to that of the Pollaczek-Khintchine formula.

For the other cases, the analytical results are not available.

6. Results and Interpretation

For type 1, we used the approximations made in section 3 andsimulated the equilibrium probabilities. For type 2, there are no

theoretical models that take in charge all the aspects of this approach.The results were simulated under the same initial conditions.

Initial dataThe statistical analysis of the set of collected data in our study led

to a model M/(, )/3 and enabled us conclude that car arrivalsare Poisson with the rate = 0.015 per second and that the service


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durations are Gamma with the average 1

= 187.38 seconds and the

variance2 = 14461.96. Thus = 2.42and= 77.17.ResultsTable4 shows the values of performances indicators from both cases.

Performance Measures A single queue Multiple queues

93.7% 93.5%0 0.015 0.015w 0.87 0.81W 796.107 778

Wq 608.727 591.42Ws 187.38 186.61

L 11.941 11.76Lq 9.13 8.86

Ls 2.81 2.80

Table 10: Comparative Performances Measures of Both Approaches

Figure 7 shows the equilibrium probabilities varying according to n for

both types of queue organization.

Figure 7: Equilibrium Probabilities for Both Types of Queue Organiza-tion

We notice that the intensity of service is high because tollbooth staff

are busy more than 93% of their working hours which ensures a fullusage of the motorway resources. Practically there are always three

cars being served (exactly 2.81). There are on average12 cars in the


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system with a bit more than9 cars on average queuing. A car remainson average 13 minutes 16 seconds in the at the tollbooth 10 minutes8 seconds of which queuing. A new coming car80% chance to waitbefore being served. Notice that both approaches give practically

the same results. We can then conclude that there are no significant

differences between both queuing schemes if not from an organization

(order, comfort and spaces). In fact the single queue model has certain

advantages:

(i) for the tollbooth, the existence of zero or an unique queue ensuresorder and comfort of passengers and making the most of space;

(ii) for the passengers, the numbering system does not require a

physical presence in the queue. So, they can sit and wait or keep

oneself busy while waiting.

But if we vary the number of stations?

If it is a factor that motorway managers can easily act on, so it is the

number of available stations for the service.

Table5 shows us performance measures for configuration to2,4then5stations. With two stations, an overloaded system will explode. In this

case, the performance measures do not exist because we cannot obtain

exploited stations over a100%use rate and cars arrivals rate higher thanthe maximum capacity of service.

With a4 stations configuration, the servers are busy during70% of itsavailability time; the service duration remains unchanged while passen-

gers go fewer than a minute in the queue. We have permanently on

average4 cars in the system, less than one car is in the queue. For apassage to5 stations, these last stations pass practically the half of its

empty time. We get on average, few than three cars in the service andan almost empty queue. The waiting cars stay in the queue on average

11seconds.Figure8 shows the equilibrium probabilities according to that we had a3,4, or5 servers configuration. We consider that curves increase at thebeginning then collapse later. Yet, we notice a pic on 4 and5 stationscurves with a number of cars included between3 and 5. In3 stations,


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P M c= 2 c= 3 c= 4 c= 5 1.4 93.7 0.70 0.56

0 - 0.049 0.057 0.015w - 0.87 0.40 0.18W - 796.107 (13 min,16 sec) 232.244 (3 min,52 sec) 198.41 (3 min,18 sec)

Wq - 608.727 (10 min,8 sec) 44.864 (0 min,44 sec) 11.03 (0 min,11 sec)Ws - 187.38 (3 min,7 sec) 187.38 (3 min,7 sec) 187.38 (3 min,7 sec)

L - 11.941 3.48 2.97Lq - 9.13 0.67 0.16Ls - 2.81 2.81 2.81

Table 11: System Performance Measure for2,3,4and 5Servers

it is possible to see at a given time 30 cars and more, present in thesystem as it will be less likely to see more than 11 cars for a4 stationsconfiguration and more than9 cars for a5 stations configuration. Thisexplains the fact that the curves of4 and 5 stations configurations col-lapse quicker than the3 stations one.

Figure 8: Equilibrium Probabilities for 3, 4 and 5 Servers

And if the service durations are exponential?

Despite the fact that the data analysis has shown us that the services

durations do not follow an exponential distribution, we noticed the sys-

tem behavior if it has been considered as exponential, to gauge the im-

pact that such consideration had on results. Table6 contains the ob-tained performance measures values. We can note that the average du-

ration of service and the average number of cars in service are still the


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same as well as the stations use rate. Yet, we can remark a notable dif-

ference in comparison to results of non-exponential service durations.

Mainly on waiting average duration (more than4 minutes) and the av-erage number of cars waiting (around 3 cars). These differences affectlogically the average duration of sojourn of a car in the system and the

average number of cars in the system. In the absence of theoretical

formulations for the(M/G/c) models, these results confirm us in ourdecision not to choose as approximation by the M/M/cmodels.

0 w W Wq Ws L Lq Ls0.93 0.015 0.82 1061.56 874.18 187.38 15.92 13.11 2.81

(17 min,41sec)(14min,34sec)(3min,7sec)

Table 12: Performance Measures when Services Durations are Considered Exponentials

Performance measures vary considerably when passing from3to4or5stations. Cost knowledge related resources exploitations to the func-

tioning of a station would allow determining optimal configuration.

Given the fact that for 4 or 5 stations, the performance measures arepractically in the same order, the crossing from 4 to 5 stations would

not be profitable in the motorway. When we increase the number ofstations to4, the number of waiting customers and the average durationof waiting collapse very strongly, consequently in one side, it assures

certainly the comfort and the satisfaction of customers, in other side,

with traffic decrease, the servers become underused. To decide if it is

necessary to increase or not the number of stations, we have to take into

account cost related to a station functioning and to non-comfort and to

non-satisfaction of customers.

7. Conclusion

The queue theory is a technique of optional research that allows mod-

eling a system assuming a waiting phenomenon, calculating its perfor-

mance measures and determining its characteristics to help managers in

their decisions. theoretical results and formulations are well-established

for multi-servers queues models with Poisson arrivals andM/M/cex-ponential services durations. But not to all systems such as those with


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Poisson arrivals andM/G/c non-exponential service durations whoseanalytical study is complex.

We used in this study the Pollaczek-Kitchinne formula to suggest

formulas that enable approximations of the performances models of the

M/G/c queue models. We set a multi-agent simulator that measuresthe performances ofA/B/c models where A, Bare distributions of thetype Determinist, Exponential, Erlang or Gamma. The numeric appli-

cations with data from a case study at Tunis-Msaken motorway were

used to check the theory according to which in a tollbooths system the

type of organization of the queue (a single queue for all toll booths ormultiple queues one per tollbooth ) does not influence the performance

measures of the system.

In conclusion, the time that customers spend in a service queue has

an important impact on their evaluations of the providers overall qual-

ity. By causing anger and uncertainty, the perceived duration of a wait

appears to be more influential than the quality of service delivered. Ser-

vice managers should do all they can to manage customers responses

to queues, such as offer explanation and apology. However, to the cus-

tomer faced with high waiting costs, perception management techniques

in isolation from actions that actually reduce the queue are akin to re-arranging deck furniture on a sinking ship. Waiting time should be re-

duced in reality as well as in customer perception.

Acknowledgements

I am heartily thankful to each person whose encouragement, guidance and support

from the initial to the final level enabled me to develop an understanding of the paper.

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Houda MEHRIis doctoral student in operational research at the National School of Engineers

(ENIM). Its research field is the study of chaotic queue within the framework of congestion and

large scale integer programming problems, especially arising from transportation and logistics

applications. She is is more specifically interested in the transition behaviour of a dynamic

system using the mathematical devices of the theory of CHAOS. Holding a Masters degree in

Operational Research and Production Control (2006) from the University of Sfax in Tunisia(FSEGS). She was also an assistant teacher of the Linear Programming course. She took part

in the drafting of some cases and teaches the course of Statistical and numerical systems. She

has also taken part in several national and international scientific conferences and seminars on

various topics of scientific research.

Taoufik DJEMEL Professor at the University of Tunis (El Manar), holder of a doctorate (Ph.D)

from "Graduate School of Business, University of Wisconsin At Madison", former director of

the department of Management at the Faculty of Economic Sciences and Management of Tunis,

Teaching several courses in several Tunisian and foreign university institutions for several lev-

els. He carried out and published several studies and research pieces. He also wrote a book

of statistics in 3 volumes. Currently, He is supervising a group of assistant-teachers preparing

their doctoral theses. He was several times rapporteur of thesis and viva member or president.

Previously, He was the first director of the Department of the Quantitative Methods and Data

Processing at the Faculty of Economic Sciences and Management of Sfax. He was named pres-

ident or member of several national juries for the recruitment of teachers in management and

quantitative methods on several levels.

toll motorway queuing theory

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