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University Microfilms International 8612388
L ee, H ow oo
A DIFFUSION APPROXIMATION FOR MULTI-SERVER FINITE-CAPACITY BULK QUEUES
The Ohio State University Ph.D. 1986
University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106
Copyright 1986 by Lee, Howoo All Rights Reserved A DIFFUSION APPROXIMATION
FOR
MULTI-SERVER FINITE-CAPACITY BULK QUEUES
Presented in partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Howoo Lee, B.S., M.S.
*****
The Ohio State University
1 986
Reading Committee: Approved by
Dr. Gordon M. Clark
Dr. Walter C. Giffin
Dr. Ramesh C. Srivastava Adviser Department of Industrial and Systems Engineering Copyright by
Howoo Lee
1986 This dissertation is dedicated to my parents.
- ii - ACKNOWLEDGEMENTS
I wish to express my deepest appreciation to my mother and the deceased father. Without their care, enthusiasm for higher education and financial support, my graduate study would have been unsuccessful.
I also wish to express my appreciation to my academic advisor, Dr.
Gordon Clark, for his understanding and patience. His insight and
guidance have greatly influenced me throughout my graduate education.
I wish to acknowledge Dr. Walter Giffin who opened the door for me
to the world of queueing theory. His constructive criticism always brought my poor engineering sense to my attention.
Dr. John Neuhardt deserves my sincere thanks for haveing provided me with the opportunity to work as a research associate and the
financial support when it was most needed. I also wish to thank Dr.
Ramesh Srivastava for serving on my graduate study committee. Dr. Barry
Nelson deserves my appreciation for his help with the variance reduction
techniques.
To my beloved wife, Yoosook, I owe my success. She has encouraged me everytime I needed it and endured those years with patience and
understanding.
My children, Sahngyool and Youngsun, deserve my special thanks. I
always felt like a super man when they rushed to kiss me after a hard
day's work. Sahngyool's most frequent question, "Are you going to
school this evening, too?", was always more difficult to answer than the
- iii - General Exam, questions. He always made me aware of the fact that besides a student, I am a father.
- iv - VITA
January 26, 1954 ...... Born: Wonju, Korea
1976 ...... B.S., Industrial Engineering, Seoul National University, Seoul, Korea
1982 ...... M.S., Industrial and Systems Engineering, The Ohio State University, Columbus, Ohio
1982 - 1983...... Research Associate, The Ohio state University
1983 - 1985...... Teaching Associate, The Ohio State University
FIELDS OF STUDY
Studies in Applied Stochastic Processes
Dr. Gordon M. Clark, Department of Industrial and Systems Engineering, The Ohio State University
Dr. Water C. Giffin, Department of Industrial and Systems Engineering, The Ohio State University
Studies in Mathematical theory of Stochastic Processes
Dr. Ramesh Srivastava, Department of Statistics, The Ohio State University
Studies in Applied Statistics
Dr. John B. Neuhardt, Department of Industrial and Systems Engineering, The Ohio State University
- v - TABLE OF CONTENTS
ACKNOWLEDGEMENTS ...... iii
VITA ...... iv
NOTATIONS ...... viii
LIST OF TAB L E S...... X
LIST OF F I G U R E S ...... xi
Chapter Page
I. INTRODUCTION ...... 1
1.1 Introduction ...... 1 1.2 Motivation...... 2 1.3 Research objectives ...... 5 1.4 Examples of bulk queueing systems...... 6 1.5 Outline and scope of this rese a r c h ...... 9 1.6 Organization of the dissertation...... 11
II. LITERATURE REVIEW...... 12
2.1 Approximation techniques in queueing theory...... 12 2.2 Literature review on diffusion approximation ...... 18 2.2.1 Approximation by solving Kolmogorov equations ...... 20 2.2.2 Diffusion approximation by weak convergence ...... 30 2.2.3 Diffusion approximation by stochastic differential equations...... 32 2.3 Conclusion of literature review...... 33
III. DEVELOPMENT OF DIFFUSION APPROXIMATION ...... 35
3.1 Diffusion process ...... 35 3.1.1 D e f i n i t i o n ...... 35 3.1.2 Forward Kolmogorov Equation (Fokker-Planck equation). . . 37 3.2 Elementary Return Process (ERP)...... 38 3.3 Interpretation in the queueing context ...... 43 3.4 Expressing infinitesimal mean and variance by queueing parameters...... 44 3.5 Solutions of the ERP equations...... 51 3.6 Discretization...... 65
- vi - IV. ANALYSIS OF SYSTEM SIZE DISTRIBUTION ...... 68
4.1 Proof of the legitimacy of the system size distribution. . . 68 4.2 Mean and variance of system s i z e ...... 78 4.3 Derivation of infinite capacity distribution ...... 87 4.4 M/M/1 /k queue as a special case...... 91 4.5 M /M/1 queue as a special c a s e ...... 94
V. ACCURACY EVALUATION ...... 96
5.1 Performance measures ...... 96 5.2 Distributions of interarrival and service times...... 99 5.3 Distributions of arrival and service size...... 103 5.4 Comparison of MX/MX/l/k queue...... 103 5.5 Comparison with simulation estimates for multiple server cases...... 115 5.5.1 Accuracy of simulation estimates ...... 115 5.5.2 Comparison of MX/MY/2/k queues...... 118 5.5.3 Comparison of MX/GY/2/k queues...... 124 5.5.4 Comparison of GI /M /2/k q u e u e s ...... 128 5.5.5 Comparison of GIX/GY/2/k queues ...... 132 5.5.6 Comparison of queues with different variabilities .... 136 5.5.7 Comparison of queues with ultra-heavy traffic ...... 138 5.5.8 Comparison of queues with many s e r v e r s ...... 140 5.6 Bulk queue vs. non-bulk queue...... 141
VI. CONCLUSIONS ...... 146
6.1 Research summary ...... 146 6.2 Suggestions for further study ...... 149 6.2.1 infinitesimal mean and va r i a n c e...... 149 6.2.2 Mean holding t i m e ...... 150 6.2.3 Waiting time...... 151 APPENDIX ...... 152
Little's formula for bulk queues ...... 152 Simulation program coding...... 159 FORTRAN program coding ...... 166
BIBLIOGRAPHY ...... 171
- vii - NOTATIONS
Pr(.) : Probability of an event
X(t) : Diffusion process, or state of the diffusion process at time t f(.) : Density of the diffusion process
Ca : Variability of the interarrival time distribution
Cb : Variability of the service time distribution
Ct(. ) : Infinitesimal variance
$(.) : Infinitesimal mean
A : Arrival rate in groups y : Service rate in groups
X : Arrival size
Y : Service size
P^(i) : Pr ( arrival size = i )
Py(i) : Pr ( service size = i ) m : Number of servers k : System capacity rj : lower boundary i 2 ’• upper boundary
P^ : Pr(jump occurs from boundary ri to boundary rj)
: Pr(jump occurs from boundary r^ into diffusion region) e i : Expected holding time on boundary r^
- viii - j^x) : Density function of the position of the diffusion process after it jumps from boundary r^
6(.) : Dirac delta density function
U (.) : Unit step function
U) : exp (23/ot)
PlNF(n) : Infinite capacity system size distribution obtained by diffusion approximation
PpjN (n) : Finite capacity system size distribution obtained by diffusion approximation
L(.) : Laplace Transform
L~*(.) : Inverson of Laplace Transform
Pj, P 2 : Parameters of the Geometric distributions being used as the distributions of arrival and service sizes
LN : Lognormal Distribution
E 2 : Erlang of order 2
U : Uniform Distribution
- ix - LIST OF TABLES
TABLE PAGE
1. Comparison of mean and variance of system sizeX (M V /M /1/k). . . . 109
2. Comparison of simulation estimates with exact ones ...... 117
3. Comparison of mean and variance of system Size (MX /M Y /2/k). . . . 123
4. Comparison of mean and variance of system sizeX (M Y /G /2/k) . . . 125
X Y 5. Comparison of mean and variance of system Size (GI /M /2/k) . . . 129
6. Comparison of mean and variance of system size (GI X /G Y/2/°°) . . . 134
7. Comparison of mean and variance of system size for various variabilities of interarrival time distribution (Ex/MY/2/°°) . . . 138
8. Comparison of mean and variance of system size for various variabilities of interarrival time distribution (MX/EY/2/°°) .. . 138
9. Comparison of mean and variance in case of ultra-heavy traffic. . 139
10.Comparison of mean and variance in case of many Servers ...... 141
11.Comparison of mean system size of M X /M Y /1/k queue (diffusion approximation vs. M/M/1/k approximation) ...... 143
- x - LIST OF FIGURES
FIGURE PAGE
1. Boundary behavior of E R P ...... 39
2. Shapes of Erlang, Lognormal, and Uniform distributions with same mean and variance, (based on Erlang (4, 2))...... 102
3. Percentage error of mean system size, traffic intensity = 0.5, MX /MY/ l / k ...... 106
4. Percentage error of mean system size, traffic intensity = 0.7, MX /MY/ 1 A ...... 106
5. Percentage error of mean system size, traffic intensity = 0.9, MX /MY/ l / k ...... 107
6. Percentage error of variance of system size, traffic intensity = 0.5, MX /MY/l/k ...... 107
7. Percentage error of variance of system size, traffic intensity = 0.7, MX /MY/l/k ...... 108
8. Percentage difference of mean system size, traffic intensity = 0.5, MX /MY/2/k ...... 120
9. Percentage difference of mean system size, traffic intensity = 0.7, MX /MY/2/k ...... 120
10. Percentage difference of mean system size, traffic intensity = 0.9, MX /MY/2/k ...... 121
11. Percentage difference of variance of system size, traffic intensity = 0.5, MX /MY/2/k ...... 121
12. Percentage difference of variance of system size, traffic intensity =0.7, MX /MY/2/k ...... 122 y v 13. Diffusion approximation vs. M/M/l/k approximation of M /M /1/k queue traffic intensity = 0.9, (Pj*P2 ) = (0.75,0.5) ...... 144
14. Diffusion approximation vs. M/M/l/k approximation of M /MX /1/k Y
queue traffic intensity = 0.5, (Pj«P2 ) = (0.75,0.5) ...... 144
- xi - y y 15. Diffusion approximation vs. M/M/l/k approximation of M/M/l/k
queue traffic intensity = 0.9, (Pj»P2 ) = (0.5,0.5) 145
16. Diffusion approximation vs. M/M/1 A approximation of MX/MY/l/k
queue traffic intensity = 0.5, (Pj»P2 ) = (0.5,0.5) 145
17. Number of customers in a bulk queueing system ...... 154
18. Indexed a r e a s ...... 155
xii - CHAPTER I
INTRODUCTION
1.1 Introduction
Queueing phenomena are commonplace in numerous real world situations. Queues are frequently observed in restaurants, bus stops, ticket offices and manufacturing systems. Queueing theory, as a branch of applied stochastic processes, has developed in an attempt to predict the result of interaction between the fluctuating mechanisms of demand and service and to provide adequate information in analyzing and designing many congestion systems.
Queueing theory originated to solve very practical problems such as Erlang’s work with the telephone congestion problem in the early
20th century. Until the mid-60's, however, the trend of queueing theory was to obtain exact solutions for a variety of queueing models.
Many of the solutions were expressed in the complicated form of transforms, and one needed to invert the transforms to obtain the quantities in which one was really interested. Consequently the practical value of queueing theory had been severely limited by the lack of appropriate approximation techniques which could be easily used to analyze more difficult problems. Attempts, however, were made in mid-
1960's to break away from those standard analytic techniques. Since
1 2 then, approximation in queueing theory has been receiving much attention, and finding approximate solutions have been a challenging branch of queueing theory. A considerable amount of research on approximations to the behaviors of a variety of queueing systems have been done by Kingman [43] [44] [45] and many others.
As the complexity of queueing systems increase, it is obvious that obtaining exact solutions is becoming more and more difficult. Even
though in some simple systems where exact solutions can be achieved, the
form is sometimes so complex that it is hard to directly apply them to practical situations. Approximation techniques can be accepted as a breakthrough in such cases. Emphasizing the need for approximation
techniques, Kleinrock argues [47],
The mathematical structures we have created in attempting to describe these real situations are merely idealized fictions, and one must not become enamoured with them for their own sake if one is really interested in practical answers. We must face the fact that authentical queueing problems seldom satisfy the assumptions made throughout most of the literature available on queueing theory...... Therefore if our mathematical models are so crude, we should be willing to accept much less than an exact solution to the systems of equations they give rise to; rather, we should be happy to accept approximate solutions to those "approximate" mathematical models and hope that such solutions provide information about the behavior of real-world queues. Even more important is the search for "robust" qualitative behavior of queues which provides "rules of thumb" for estimating behavior of complex systems.
1.2 Motivation
Steady progress has been made to solve more complex queueing models
over the last three decades. Nevertheless, not many of the explicit 3 analytical results have been found appropriate for practical
implementation due to their mathematical complexity in handling the solutions which are mostly expressed in transforms. The lack of solutions suitable for ready application is observed in queueing models where customers arrive in groups and/or are served in groups. The queueing models falling into this category are called "bulk queues"^ in the queueing literature. (The utility of the bulk queue models will be
illustrated through an example in section 5.6.)
Needless to say, most of the studies in queueing theory deal with
the system where arrival and service occur singly. But there are many real world situations in which customers arrive in groups and are served
in groups. For example, a family may go to a restaurant and are served as a batch. Considering numerous real world bulk queueing situations,
it is obvious that bulk arrival and service are more realistic assumptions. Moreover, the popular single-unit arrival single-unit service queueing systems are special cases of bulk queueing systems. It
is, therefore, clear that the treatment of customers arriving and served
in goups broadens the range of application of queueing models.
Despite their more realistic and highly applicable character,
explicit solutions available for bulk queues that can be conveniently applied to real world systems are relatively few. One of the reasons is
that bulk queues are, generally speaking, hard to analyze and the mathematics involved are very complicated even for some simplest bulk
queueing models. For example, one needs to evaluate the zeros of a certain polynomial to obtain the steady-state system size distribution 4 of the Markovian single-arrival bulk service queues. (See for example,
Giffin(27:211]. Also readers are urged to see Chaudhry and Templeton
[8] for discussion of bulk queues and related references.)
Most of the studies of bulk queues so far assumed that the size of the waiting area available to customers is unlimited and the number of servers is one. But more realistic treatment of queueing situations involves the finite capacity waiting area and multiple servers. The only exact solution of finite capacity bulk arrival batch service queue exists for MX/MY/l/k queue [2] (i.e., single server finite capacity bulk queueing system with Poisson arrival and negative exponential service).
But the solution is given in transform and one needs to evaluate the transform solution numerically to obtain the performance measures of the queueing system. Undoubtedly, explicit closed form solutions, though approximate, make it much easier to apply the solutions to the practical situations. Moreover, if the solutions are "robust" (in Kleinrock's sense) in estimating the system performance measures, they are more acceptable than the hard-to-invert exact transform solutions. To the best of the author's knowledge, this dissertation is the first attempt to derive the approximate closed form solution of the multiple-server, finite-capacity bulk queues. The purpose of this research is to employ the diffusion approximation method to those complex bulk queues and to evaluate the accuracy of the approximation. 5
1.3 Research Objectives
This dissertation develops a diffusion approximation procedure for the steady-state system size distribution for the bulk-arrival batch- service multiple server queue with finite capacity. We define system size as the sum of the number of customers in the queue and the number being served. In addition, we include numerical comparisons to evaluate the accuracy of the approximation. The queueing systems under study are
y y denoted as GI /G /m/k queue in which GI means general independently and identically distributed (iid) interarrival time distributions, G means general iid service time distributions, X is the arrival size random variable, Y is the service size random variable, m is the number of servers and k is the system capacity.
The queueing systems are described as follows:
1. The arrivals occur in groups of positive random size X, and its
probability distribution is known.
2. Service takes place in groups of positive random size Y, and its
probability distribution is known.
3. The first and second moments of the interarrival and service times
are finite and known. (Exact distribution forms may not be known.)
4. System has m servers and capacity k.
5. Servers are identical and homogeneous.
6. Arriving customers can enter the system until the system is full.
(Break-up of an arriving group is possible when the arrival size is
greater than the available system space.) 6
7. If less customers than the service size are in the queue, all
customers are served. (A server does not wait until the queue size
grows to the service capacity.)
8. The system has no reneging and no balking except for the forced
balking in finite capacity case.
1.4 Examples of bulk queueing systems
Bulk queues exist in many real world situations. Families arriving at a restaurant, an airline terminal where customer baggages
are carried to the baggage pick-up area, a warehouse unloading system,
cars passing through an intersection, elevators serving a floor, and buses serving passengers at a bus stop form examples of bulk queueing
systems. More specific examples appear below.
1) Warehouse unloading system
Consider a warehouse where cargo trucks of variable sizes arrive and wait for being unloaded. The warehouse has two carts for unloading
the packages. The sample data are collected which show the mean and variance of interarrival time of trucks, and the mean and variance of
the time for the unloading cart to be loaded, travel to the warehouse, be unloaded and return to the truck. Since the truck sizes are different, it is reasonable to say that the number of packages arriving
in a truck is a random variable with known probability distribution. 7
Due to the different package sizes, the number of packages that a cart can carry, i.e. the service size, is also a random variable. It is assumed that the packages from two different trucks can be mixed in one cart. The manager of the warehouse is interested in finding following information:
i) Mean number of waiting packages
ii) Mean waiting time until a package is unloaded
iii) The effect of increasing the number of carts on the mean number
of packages and mean waiting time.
2) Subassembly inspection system
Consider an inspection system where baskets of subassemblies arrive and are unloaded for inspection. The number of subassemblies in a basket is a random variable with known probability distribution.. The waiting area of the unloaded subassemblies is limited. Each of two workers inspects two subassemblies at a time. Sample data have been collected which show the means and variances of interarrival time of baskets and time taken to complete an inspection. The manager of the
inspection team wants to know following things.
i) Mean number of waiting subassemblies
ii) Mean delay time of a subassembly
iii) Effect of increasing the number of inspecting workers on the
total number of subassemblies inspected in a day
iv) Effect of the size of the basket carrying subassemblies. 8
3) Fast food store
Consider a problem faced by a businessman who wants to open a small fast food store. A study shows that customers arrive in groups with known probability distribution. The businessman can hire maximum of three servers due to the working space of the store. Customers arrive in groups, form a line, order, and leave the store with the food once their orders are ready. It is assumed that customer orders are taken care of one at a time. Sample data are collected and the means and variances of interarrival and service times are available. The businessman is interested in determining following things.
i) How many servers to hire.
ii) What is the optimal store size ?
4) Rental boat company
Consider a rental boat company on a lakeside. The company owns 5 boats each of which can accomodate three people. Customers arrive in
groups of random size with known probability distribution. The rental
fee is proportional to the length of rental time. A preliminary study
shows that the rental time follows a probability distribution f(t) . The
owner of the company is interested in increasing the number of boats for
the coming summer season. It is expected that three times more customers come to rent a boat during the summer season. The owner of
the company wants to know the effect of increasing the number of boats on the total income.
1.5 Outline and scope of this research
This dissertation has as its objectives developing a diffusion
approximation for a steady-state system size distribution and evaluating
the accuracy of the approximation. The diffusion process with special boundary behaviors called the "Elementary Return Process (ERP)"
developed by Feller (19] will be used. The behavior of this diffusion
process is described by three partial differential equations. (These
equations will be called diffusion equations.) First, the diffusion
equations will be solved after an appropriate interpretation of the
parameters involved. The equations are solved by using the Laplace
Transform technique. The solution of the diffusion equations will be
given in the form of the steady-state density function which governs the
probabilistic rule of the position of the diffusion process. The
continuous density function obtained as the solution of the diffusion
equations will be discretized by replacing the continuous state variable
x of the diffusion process by a discrete variable n which represents the
system size of the finite capacity bulk queueing system. The legitimacy
of this discrete system size distribution (i.e., the probabilities sum
up to one) will be proven under an assumption that service size is
always less than the system capacity. Some analyses of the system size 10 distribution will follow. They include finding the mean and variance of
y y the system size, evaluating some special cases of the GI /G /m/k queues, and obtaining the infinite capacity system size distribution by letting the system capacity k tend to infinity. This infinite capacity system size distribution, interestingly enough, will be seen to be exactly the same as the one obtained by Chiamsiri and Leonard [11]. Hence their study becomes a special case of this research. As far as accuracy is concerned, a complete and comprehensive evaluation of the accuracy is impossible because there do not exist exact solutions of the steady- V Y state system size distribution of GI /G /m/k queues except for the
y y single server Markovian queue (M /M /1/k). This makes it necessary to use simulation estimates in evaluating the accuracy of the approximation. To obtain the reliable simulation estimates, a variance reduction technique called "antithetic variate" technique will be used.
This variance reduction technique turns out to reduce the sample variance by up to 90 percent. To confirm the dependability of the simulation estimates thus obtained, they will be compared with the exact solutions of MX/MY/l/k queue. The comparison shows that the simulation estimates are amazingly accurate. It is also expected that simulation estimates will provide the same accuracy even in the cases where exact solutions do not exist.
This dissertation is not concerend with waiting time. But the celebrated Little's formula could be used to find the mean waiting time once we know the mean system size. A poof will be provided to insure that the Little's formula is valid for bulk queueing systems, too. 11
1.6 Organization of the dissertation
Chapter II is devoted to reviewing literature on diffusion approximation. Chapter III discusses a diffusion process. Especially an emphasis is given on the Elementary Return Process (ERP) which is a diffusion process with special boundary behaviors. The infinitesimal mean and variance are expressed in queueing parameters and then, the differential equations representing the ERP are solved. A discretization scheme is discussed and finally, system size distribution is obtained. Chapter IV presents the proof of the legitimacy of the system size distribution obtained in Chapter III. Also, the mean and variance of the system size are obtained. Some cases such as M/M/1/k, v x v M /M/1 queues will be discussed as special cases of GI /G /m/k queues.
As another special case, the infinite capacity system size distribution is derived by letting the system capacity k tend to infinity.
Chapter V is entirely devoted to evaluating the accuracy of the approximation. Chapter VI discusses conclusions of this research and suggestions for further study. Finally an appendix will include the proof that the Little's formula is valid for bulk queueing systems, a
SIMSCRIPT code used in estimating the system performance measures, and
FORTRAN code used to compute the system size probabilities and its mean and variance. CHAPTER II
LITERATURE REVIEW
In this chapter, literature on approximation techniques in queueing theory and diffusion approximation will be reviewed.
2.1 Approximation techniques in queueing theory
There are a number of approximation techniques in queueing theory.
A list of some of them is provided below.
1) Simulation of queueing models
This approach has become popular in recent years with the advent of the high-speed digital computers. The advantage of simulation is in its wide applicability and flexibility. It provides an alternative method that is applicable when mathematical treatment is impossible, and it allows one to represent a queueing system that is more complicated than mathematical analysis permits. Simulation does not require extensive mathematical development, and its results are more easily understood and communicated among large number of users. These characteristics make it possible to use simulation as an appealing technique in designing 13 various stochastic systems. On the other hand, since a simulation model describes a queueing system by artificially generating arrival
times, service times and many other stochastic variables, the output a
simulation experiment produces is also a stochastic variable in nature,
and will, therefore, require some statistical analysis and careful
interpretation. To obtain reliable results, simulation requires large
sample sizes and large amount of computation. A lot of statistical
analysis and variance reduction techniques have been developed for the
reliable estimation of system performance measures [34][50][71]. These
statistical techniques make it possible to perform simulation
experiments with reduced cost and computation time, but with the same
reliability.
The author's view on using simulation in designing various queueing
systems is that simulation must be the last resort after all
mathematical efforts are exhausted. This is because the sole reliance
on simulation is not economical in the first place. In the second
place, the objective of modeling a queueing system is, in many times, in
gaining insight into the system, and mathematical model can easily
satisfy this objective with less effort. In the third place,
mathematical results of a simple system can be a stepping stone of
predicting the behavior of more complex systems. Although it is very
difficult to handle the entire system mathematically, one can break up
the system into small sub-systems such that mathematical techniques may
be employed. Stating the philosophy of system analysis and design,
Giffin [29] states, 14
Such simplification is justified on several grounds. First, a better understanding of the operation of the simplest system may suggest new approach to more complex problems. Second, the models of the simple system are easy to manipulate and, hence, can yield predictions for many combinations of parameters relatively cheaply. Third, these simple models enable one to highlight a single facet of the problem at a time.
2) System approximation
A system approximation is a simplification of the system elements of the original queueing system (arrival process, service process,
queue discipline, and system structure) such that the overall
characteristics of the "old system" is maintained by the strong
relationship to the behavior of the "new system". The heavy use of the
exponential distributions for the arrival and service processes can be
understood in this context. Other attempts to express the interarrival
and service time distributions by the phase-type distributions [60] can
be included in this type of approximation, too. This is possible by
matching moments of the actual interarrival and service time
distributions with a phase-type distribution. The advantage of using
the exponential and phase-type distributions is, obviously, in allowing
one to analyze the system behaviors by using the Markovian property or
the memoryless property of the negative exponential distribution, and
thereby to represent the whole system by the Kolmogorov differential
difference equations. Approximately solving Kolmogorov equations, for
example, arising in non-stationary tandem queues [12] is in this
category. Approximating nonstationary M/M/s queue by the Polya 15 distribution [13] is another good example of this type of approximation.
3) Numerical approximation
A numerical approximation is just a numerical manipulation of mathematical expressions. Numerical inversion of Laplace transforms is an example of this approximation. Another example is the Runge-Kutta integration of Kolmogorov equations, A recent work of approximating
GI/G/1 queue by "Laguerre Transforms" [51] is another good example. The use of bounds and inequalities and limit theorems can be classified into this category, too.
The use of bounds and inequalities gives some light on approximately understanding the system behavior by providing the maximum or minimum value that a particular system performance measure can take.
For example, Kingman [45] showed that an upper bound for expected waiting time of GI/G/1 queue can be obtained as
Var(a) + Var(b) E(W) £ ------2 (E(a) - E(b)) where "a" and "b" denote the interarrival and service time random variables. It is interesting to see that the upper bound given above can serve as a good approximation for the expected waiting time itself
[44] in case of heavy traffic (i.e., traffic intensity is close to but strictly less than 1).
Limit and convergence theorems are sometimes established to 16 approximate system behaviors. Typically they describe the behavior of a particular process of interest as one of the system parameters approaches a specific limiting value. For example, Iglehart [35] showed that M/M/m queueing system can be approximated by a special type of diffusion process called the "Ornstein-Uhlenbeck (0-U) process". He obtained the result by using a random process expressed as a function of number of servers, and then let the number of servers tend to infinity after properly scaling and normalizing the random process.
4) Process approximation
A process approximation is to approximate a queueing process by a totally different process. One major difference between the system approximation and the process approximation is that the system approximation is concerned with the simplification of the system itself while the process approximation is interested in representing a queueing process by a different process (deterministic or stochastic) which has the properties similar to the original process. The fluid approximation and the diffusion approximation are examples of process approximation.
The fluid approximation suggested by Newell [59] considers the discrete arrivals and departures as fluid flowing continuously in and out of a water tank. The basic idea of this fluid approximation method is that when the system size is large, it is reasonable to replace the discontinuities of the discrete queueing process by smooth continuous functions. If we define A(t) and D(t) as the number of arrivals and 17 departures in time (o,t], then the number of customers at time t, N(t), is the simple difference between A(t) and D(t) if N(0)=0, i.e., N(t) =
A(t) - D(t). Then the average processes E(A(t)) and E(D(t)) determine the average rates of "flow-in" and "flow-out" by A(t) = d[E(A(t))]/dt and p(t)= d[E(D(t))]/dt, where A(t) is the arrival rate and y(t) is the departure rate (hence the service rate in the case of a saturated system size). Then, average arrival and departure processes are obtained by the integration of arrival and service rates, and thereby, we can find the average number of customers in the system. The main idea of this fluid approximation lies in approximating the arrival and departure processes by deterministic average functions.
A diffusion approximation is to approximate a discrete queueing process by a continuous stochastic process called a diffusion process, which reflects the characteristics of the original queueing process.
The diffusion approximation method has been accepted as a promising technique in approximating many queueing systems. Many researchers have found that this particular approximation method could provide the
"robust" (as Kleinrock put it) estimates of system behavior of a variety of queueing systems. One of the greatest advantages of diffusion approximation is that it allows one to consider more general interarrival and service time distributions. Also the diffusion processes, used in approximating a queueing system, is a continuous stochastic process and is more mathematically tractable than the original discrete queueing processes. The diffusion approximation needs only the first two moments of the interarrival and service times. This 18 is a great advantage in that the exact forms of the distributions are not required. But using the limited amount of information inevitably results in a loss of accuracies and this fact can be considered as a trade-off in employing an approximation technique. The major disadvantage of using diffusion approximation is in its assumption of heavy traffic. The reliance on this heavy traffic assumption brings about the reduced accuracy in light traffic cases. Nevertheless, several efforts have been made to correct this major weakness by some researchers [30] [39] [42].
2.2 Literature review on diffusion approximation
As mentioned earlier, the underlying idea of a diffusion approximation is to approximately represent a discrete queueing process by a continuous stochastic process called a diffusion process. This approximation technique came into being as a result of the heavy traffic approximations initiated by Kingman [43] [44][45]. He studied the behavior of the GI/G/1 queue in heavy traffic. He found that the waiting time is approximately exponentially distributed. Since then the diffusion approximation has attracted much attention for the last twenty years. The reasons why diffusion approximation was applauded by many researchers were that first, generally speaking, it is easier to handle a continuous stochastic process than a discrete process, and second, a considerable amount of work has been already done on the diffusion process by mathematicians [19] [20] [68]. As a result, numerous
interesting facts have been disclosed on first-passage problems, mean
absorption times, boundary behaviors and hitting probabilities of the
diffusion process. Hence diffusion approximations provide available methods for approximating many random processes and quantities
associated with them. For example, first-passage and mean absorption
time can be applied to queueing theory if one is interested in time to
the first overflow in finite capacity queueing systems. Boundary
behaviors can be used to approximate the probability that a queueing
system is full or empty.
The richest works on diffusion process were accomplished by Feller
[19] [20]. He extensively studied one dimensional diffusion processes
with boundaries resulting in the construction of the differential
equations describing the behavior of the diffusion process within and on
the boundaries. He classified the boundaries of the diffusion process
according to their characteristics. They include the Refecting Boundary
(RB), the Instantaneous Return Boundary (IRB), and the Elementary Return
Boundary (ERB). A diffusion process with RB is reflected as soon as it
reaches a boundary and the diffusion starts all over again. A diffusion
process with IRB jumps into the diffusion region as soon as it touches a
boundary where the jump size is a random variable. The difference
between RB and IRB lies in whether the process can jump from the
boundary or not. A diffusion process with ERB is allowed to stay on the
boundary for a random time and then it jumps. This jump size is also a
random variable. Hence it can be said that the IRB is a special case 20 of ERB in that the holding time on the boundaries is zero. This dissertation will employ ERB approach and a detailed description of ERB will be found in section 3.2.
Diffusion approximations in queueing theory can be classified into three categories:
1. Approximation by solving Kolmogorov equations
2. Approximation by using weak convergence theorems
3. Approximation by modeling through stochastic differential
equations
2.2.1 Approximation by solving Kolmogorov equations
Most of the diffusion approximations are based on Kolmogorov equations. It is known that a diffusion process defined on the extended one dimensional space (often called unrestricted diffusion process) follows the equations called the Forward Kolmogorov equation and the
Backward Kolmogorov equation. These second order partial differential equations can be heuristically derived from the usual Chapman-Kolmogorov equations for the continuous Markov Process under the assumptions that the process is time-homogenous, and third and higher infinitesimal moments do not exist [47:66-71] [3:130-135]. One important property of the unrestricted diffusion process is that its overall behavior is characterized by its local characteristics called the infinitesimal mean and variance. The infinitesimal mean and variance are defined as the 21 rates of change of the mean and variance of the displacement of the diffusion process in a small interval of time. (See section 3.1 for mathematical definition of the infinitesimal mean and variance.) But if we impose boundaries on this process, we need additional equations to describe the behaviors of the diffusion process on the boundaries.
The different characteristics of the boundaries mentioned earlier
(RB, IRB, and ERB) differentiates the approaches within the diffusion approximation based on the Kolmogorov equations. One approach employed by Gelenbe [26] and Chiamsiri and Leonard [9] used the ERB. The other approach using RB was employed by many others, e.g., Newell
[55] [56][57] [58], Gaver [22] [25], Halachmi and Franta [30], Heyman
[32][33], Kimura [40], and Kobayashi [48] [49] [64]. The first step in constructing the diffusion approximation based on these Kolmogorov equations is to obtain the infinitesimal mean and variance. This can be achieved by expressing the infinitesimal mean and variance by queueing parameters such as arrival rate, service ' rate, means and variances of interarrival and service times, means and variances of arrival and service sizes (remember that the arrival and service occur in groups of random size), number of servers, and system capacity. The solution of the Kolmogorov equations is given in the form of the density function of the transition of the diffusion process (see section 3.1 for the density function of the transition and its connection with the infinitesimal mean and variance), and this continuous density function is used as the approximation of the distribution of system size. For this purpose, discretization is necessary because the distribution of 22 the system size of a queueing system is discrete.
Since many quantities of the queueing process in which we are interested can not be negative (for example, system size, waiting time, busy period, first overflow time, etc), we need to impose boundaries on the corresponding diffusion process. (Without boundaries, a diffusion process can drift to the negative side, hence result in negative number of customers, negative waiting time, etc.) A boundary is, then, placed at the zero state of the diffusion process. Another boundary can be placed at the system capacity if one is interested in the system size of a finite capacity queueing system.
The final approximate solutions need to be evaluated for their accuracies. This is usually done by comparing the approximate solutions with the exact ones. But in many cases, exact solutions do not exist, and simulation is used for the purpose of comparisons [10] [30] [64].
To summarize the steps of diffusion approximation based on the
Kolmogorov equations:
STEP 1. Find the infinitesimal moments. This can be done by
expressing the infinitesimal moments by queueing parameters.
STEP 2. Solve the equations of the diffusion process with appropriate
boundary behaviors.
STEP 3. Discretize the solution obtained in STEP 2 if necessary.
STEP 4. Evaluate the accuracy of the approximation s 23
The first works on the diffusion approximation based on Kolmogrov equations were achieved by Newell [56] [57] [58]. He investigated, using diffusion approximations, properties of the queue where the time- dependent arrival rate increases and then passes through the saturation condition (arrival rate > service rate). He found that the maximum queue length is approximately normally distributed. The distribution of queue length at the end of rush hour when the queue length returns to equilibrium was also investigated.
While Newell was interested in applying a diffusion approximation to somewhat specific situations, Gaver [22] initiated the use of the diffusion process to approximate a class of queueing systems. He was interested in the waiting time of' M/G/l queue. He suggested that the virtual waiting time process be approximated by a diffusion process with infinitesimal mean and variance given by AE(b)-l and E{a ) where
A is the arrival rate and "a” and "b" are interarrival and service time random variables. He did not specify how he obtained the infinitesimal moments (this can be seen in Kleinrock [47:73]). But it is easy to observe how he obtained the infinitesimal mean of the waiting time process. Since A is the mean number of customers arriving in a unit
I time, and every arriving customer increases the unfinished work by E(b) on the average, AE(b) is the average amount of increased work due to the arrivals in a unit time. But since total amount of unfinished work is reduced by 1 unit in each unit time as the server serves customers,
it is easy to calculate that the average "rate" of change of unfinished work is AE(b) - 1 in each unit time. Of course, the infinitesimal 24 mean and variance were obtained on the assumption that servers are always busy which is possible only in the heavy traffic case, because if servers are not busy (i.e., there is no customer in the system), then the level of unfinished work is zero.
One of his results is that the approximate long-run expected waiting time is exponentially distributed with mean given by
Var(b) + E2 (b) E(W) = ------2 (1 - p) E (a)
where "a" and "b" denote interarrival and service time random variables,
It is interesting to see that if the traffic intensity is close to 1, then A-p , and
Var(b) + E2 (b) Var(b) + (1/p)2 E (W) = ------2 (1 - p) E(a) 2 (1 - p) E(a)
Var(b) + (1/A)2 Var(b) + Var(a)
2 (1 - p) E(a) 2(1 - p) E(a)
The last quantity is the result obtained by Kingman [44] when he studied heavy traffic GI/G/1 queues. Hence Gaver's mean waiting time formula agrees with Kingman's heavy traffic approximation.
Gaver also found that his diffusion approximation seems to
"overestimate" the exact mean waiting time. This observation is not 25 surprising when it is noticed that Kingman's mean waiting time in heavy traffic (hence approximately Gaver's mean waiting time, too) is actually an upper bound of mean waiting time of GI/G/1 queue [45].
A number of diffusion approximations were concentrated on studying network of queues. Initiated by Shedler and Gaver [25], the investigation of network of queues was applied to computer systems.
Shedler and Gaver [25] applied the diffusion approximation obtained by the analysis of cyclic queues to the multiprogramming systems. They compared the approximate estimate of CPU utilizations and the exact solutions, and found that their approximation is fairly accurate.
Kobayashi [48] proposed a diffusion approximation to obtain the steady-state joint distribution of queue length in both single server, infinite capacity open and closed networks of queues. He, for the first time, applied a multi-dimensional diffusion process to come up with the joint distributions. He investigated the interactions among the queueing processes of each station in the form of a variance-covariance matrix and derived the joint queue length distribution as a product form of marginal queue length distributions. His results were applied to a computer network. He [49] also derived the transient solutions of the distribution of queue length in a single-server cyclic queue and provided some examples. He evaluated the accuracies for performance measures of computer systems such as system utilization, throughput, and response time.
Another study applying a diffusion approximation to a network of queues was done by Fischer [21]. He studied a loop service system in 26 which a server serves customers at a station for a fixed amount of time and then moves to the next station, but the total time the server can spend serving all the stations is fixed. He was interested in analyzing the steady-state service load at a station, and used a cost function to find the optimal service time at a station.
Diffusion approximations of tandem queues were acheived by Newell
[55] and Harrison [31]. Newell studied a three-node tandem queue (first node with infinite capacity, and next two nodes' with finite capacities) for general service time distributions. Harrison considered a two- dimensional diffusion process that arises in tandem queues with two single-server facilities with general service time distributions. He showed that the limit distribution is the solution of a first-passage problem for a two-dimensional diffusion process.
While most of the diffusion approximation based on the Kolmogorv equations use the Reflecting Boundary (RB) for the boundary behavior,
Gelenbe [26] used the Elementary Return Process (ERP), i.e., the diffusion process with the Elementary Return Boundary (ERB), for his diffusion approximation of a two server cyclic queueing system.
Recalling the three different boundaries described in the first part of this section, ERB has a property that as soon as the diffusion process hits a boundary, it is absorbed into the boundary for a non-negative random time and then it jumps into the diffusion region (with random jump size), and then diffusion starts all over again from scratch.
Gelenbe [26], erroneously, called this diffusion process Instantaneous
Return Process (IRP). But according to Feller [19] who defined all the 27 boundaries mentioned above, IRP jumps instantly it touches a boundary.
Hence IRP is a special case of ERP in that the expected holding time on the boundary is zero.
Gelenbe [26] investigated a two-server cyclic queueing system which was also studied by Shedler and Gaver [25]. But they used the
reflecting boundary in their study. Gelenbe found, after comparing his results with exact ones, that using the ERP gives estimates which are
less dependent on heavy-traffic approximation. He also observed that using the ERP yields the probability of an empty system automatically while the reflecting boundary method needs some adjustment to provide the same result. His pioneering work turned out to agree with the study by Chiamsiri and Moore [11]. They compared the two different diffusion approximations of the behavior of bulk arrival single server queues: one with Reflecting Boundary and the other with Elementary Return Boundary.
They found that the ERP provides generally superior approximations to the mean and variance of steady-state queue size. The reported superiority seems to come from the facts that first, the ERP can stay on
the boundary, and thereby it can reflect the empty state of a queueing system, and second, the random jump of ERP can reflect the instantaneous
change of the system size due to the arrival of customers.
As far as accuracies of the diffusion approximations are concerned, many studies have shown that diffusion approximation shows improved accuracy in case of heavy traffic, and rather inferior in low traffic cases. This general trend of accuracy was confirmed by an extensive study by Reiser and Kobayashi [64]. They found that the accuracy of the 28 diffusion approximation is quite adequate in many cases. They compared the diffusion approximation for M/G/l and E2 /M/I queues with exact solutions for the mean and variance of system size and queue size distributions. They also considered open and closed networks to evaluate the accuracy for mean queue size and server utilization.
Heyman [32] studied the time-dependent behavior of queue length for the GI/G/1 queue. He [33] also applied the result of Gaver [22] to approximate the density function of the busy period of M/G/l queue. He provided explicit formulae when the service time has constant, Negative
Exponential and Gamma distribution. He found that his approximation gives the correct values of the first two moments.
Kimura et al. [40] presented somewhat different diffusion approximation approach to GI/G/1/k queue. In their study of overflow time, they used the reflecting boundary approach but with a slight modification of the locations of the boundaries which are normally supposed to be placed at the zero state and system capacity. They, however, did not provide any justification of the modified placement of the reflecting boundaries. They [41] also investigated the loss probability (i.e., the probability that an arriving customer can not enter the system) and system size distribution.
Most applications of the diffusion approximation are achieved by assuming that servers are not idle. This assumption is the main culprit that brings about the reduced accuracy in low traffic cases. If one uses the assumption that servers are never idle, then, the resulting infinitesimal moments are expressed in the forms which are independent 29 of the state variable of the diffusion process. For example, if one wants to approximate a GI/G/m queue, then the infinitesimal variance and mean are respectively given by [30:524]
a(x) = X^ Var(a) + m Var(b)
3 (x) = X - m y
where x is the state variable of the diffusion process and "a" and "b" are interarrival and service time random variables. As can be seen, a(x) and g(x) are independent of x. One defect of these infinitesimal moments is that they do not reflect the situations where the number of customers in the system is less than the number of servers, i.e., the situations where some of the servers are idle. The first breakthrough to remedy this problem came from Halactftai and Franta [30] and was improved by Kimura [40]. Halachmi and Franta proposed that m (the number of servers) in the infinitesimal moments be replaced by min(x,m).
This obviously reflects the situation in which number of servers exceeds the number of customers in the system. Their idea was again improved by Kimura [40] in that the continuous state variable x in min(x,m) is replaced by a discrete variable fxl, hence min(x,m) is ultimately replaced by min(fxl,m). Kimura's idea leads to the piecewise continuous functional forms of a(x) and (3(x), which have (m-1) discontinuity points. Kimura incorporated this idea of discretized state variable with the ERP boundary at the origin. Also he applied this to bulk- arrival queues [42]. Yao [75] applied the results of Whitt [74] on 30 approximating point processes by renewal processes and refined the
Kimura's method [76]. Undoubtedly these improved infinitesimal moments would increase the accuracy of the diffusion approximation. In this
sense, their works are considered as great improvements of classical diffusion approximations. But one disadvantage of these modified
infinitesimal moments is that they make it much harder to solve the
Kolmogorov equations.
The first attempt to apply diffusion approximations to bulk queues
Y V was made by Chiamsiri and Leonard [10]. They studied GI /G /I queue
(i.e., single server bulk arrival, batch service queues with general
interarrival and service time distributions) to obtain the distribution of system size. They used the ERP approach and compared their approximate results with exact solutions [2] and simulation estimates.
It will be seen in chapter IV that their solution is just a simple special case of the finite capacity solution produced by this dissertation research.
2.2.2 Diffusion Approximation by Weak Convergence
In many problems of stochastic systems, the behavior of random
quantities can be explained by limit theorems. Limit theorems describe
the behavior of a particular process as one of the system parameters
approaches to a limiting value. This method of approximation is, in
many cases, mathematical treatment of the weak convergence of stochastic 31 processes. This approach is useful if one is interested in the limiting
behavior of some queueing systems.
Some diffusion approximations are obtained by observing the
limiting behavior of a sequence of appropriately normalized and scaled
queueing processes. The limiting arguement involves, in most cases, a
special type of classical convergence theorems, called weak convergence
theorems. These approximations are, in nature, similar to the central
limit theorem for sums of random variables. (Readers are urged to see
Whitt [72] [73] for related references and discussions of weak
convergence theorems in queueing theory.)
Iglehart [35] considered the limiting behavior of multi-server
queues as the number of servers becomes large. If one lets Xn (t) be the
number of customers in the multiserver queueing system with n servers,
Markovian arrival and Markovian service process, then it is well known
that Xn (t) is a birth-death process. Iglehart showed that (t)-np)/✓np
converges to a diffusion process called the "Qrnstein-Uhlenbeck (0-U)
process", where p is the traffic intensity. He applied his results to
the repairman problem where Xn (t), in this case, is the number of
machines undergoing or waiting for repair at time t.
McNeil [54] studied a queueing system with a non-stationary Poisson
input process. He found that the system size process converges to a
non-stationary O-U process.
Kennedy [38] investigated the virtual waiting time process of the
M/G/l queue. He showed that the virtual waiting process, suitably
scaled and normalized, converges to the unsigned Brownian excursion 32 process under some conditions.
Lehoczky and Gaver [23] developed a voice and data communication system where voice operates as a M/M/m/m loss system (i.e., the queueing system where arriving customers are denied if all the servers are busy) and if there is any idle voice channels, data can occupy the channel.
Then data is a Markovian queueing system with a random number of channels. They investigated this problem by stochastic random vectors
(X(t),V(t)) where X(t) is the data system size and V(t) is the voice system size at time t. Then, they proved that finite dimensional distributions of the (X(t),V(t)) process converges to those of the
Wiener process with a reflecting boundary at the origin.
2.2.3 Diffusion approximation by stochastic differential equations
Queueing literature of this type of approach is rare. But stochastic differential equation is one of the modern approaches in dealing with diffusion processes. Differences between ordinary differential equation and stochastic differential equation comes from the fact that stochastic differential equations involve differentiation of a random function (i.e., stochastic process) while ordinary differential equations involve differentiation of deterministic
functions. The random function in most stochastic differential equations is the Wiener process (Brownian motion process) which is a
special type of diffusion process. But since a Wiener process is
composed of infinitely many oscillations in a small interval of time, it 33
is nowhere differentiable [6]. Hence to solve a stochastic differential equation we need a new concept of convergence which is
totally different from the usual concept of convergence of Riemann sums.
(See Arnold (1] for details of stochastic differential equations)
Gaver and Lehoczky [23] applied this type of diffusion
approximation to the repairman problem in which machines break down in different modes. They modeled the problem in the form of a stochastic differential equation and found that their stochastic differential equation describes a non-stationary trivariate 0-U process.
2.3 Conclusion of literature review
In section 2.1, approximation techniques in queueing theory were presented. In section 2.2, literature on diffusion approximation was
reviewed. Most of the diffusion approximations were based on solving
Kolmogorov equations, and only a few of them utilized the weak convergence theorems and stochastic differential equations. Since the diffusion approximation is largely for the heavy traffic systems, it is pointed out in most literature that accuracy of the diffusion approximation decreases as the traffic intensity decreases. In this sense, methods proposed by Halachmi and Franta [29] and Kimura [41][42], and extended by Yao [75][76] were major breakthrough in classical diffusion approximation techniques.
Many researchers dealt with diffusion approximation of queueing network. Major works in this respect are due to Kobayashi [48][49], 34
Gelenbe [26], Shedler and Gaver [25], Newell [53], and Harrison [31].
Kobayashi derived the joint distribution of queue length in both open
and closed networks of queues. His results show that the joint queue
length distribution is the product of marginal queue length
distributions. But his queueing systems were networks with infinite
buffer sizes. The only study of diffusion approximation in the area of
network of queues involving finite capacity was accomplished by Newell
and Harrison. Newell studied a tandem queue with finite buffers on
second and third nodes. He employed multi-dimensional diffusion
approximation to solve the tandem queues, but with limitations on the service rates of each node.
Gelenbe introduced the Elementary Return boundary into the diffusion approximation. But as will be seen in later chapters, diffusion equations involving Elementary Return boundary are more
complex and thereby make it difficult to combine this boundary with the
improved methods suggested by Halachmi and Franta, and Kimura.
Diffusion approximations of nonstationary queues and bulk queues
are rare. Approximating nonstationary queues involves solving partial differential equations. This is also true for the transient solutions
of stationary queues except for reflecting boundary cases. This is why
it is difficult to find literare on diffusion approximations of
nonstationary queues and transient solutions.
In summary, lack of studies in diffusion approximation is seen in
the areas of networks of queues, waiting time of bulk queues, and nonstationary queues. CHAPTER III
DEVELOPMENT OF DIFFUSION APPROXIMATION
In this chapter, a diffusion approximation to the system size distribution of GI X /G Y /m/k queues is developed. As a preliminary step, a review of the diffusion process will be provided.
3.1 Diffusion process
3.1.1 Definition
It is not easy to define a diffusion process in one statement.
This implies that the diffusion process can be described in a number of different ways, and this fact can be easily verified by taking a look at the different definitions in different texts on mathematical probability and stochastic processes. Wenzell states [70], "The definition of diffusion can be made more precise in different ways, so
that there is no unique standard definition of diffusion. What we have are working definitions or, so to speak, definitions of different
classes of diffusions." But a diffusion process is understood as a
strong Markov process with continuous paths.' Karlin and Taylor [37]
define a diffusion process-as "a continuous time parameter stochastic
process which possesses the strong Markov property and for which the 36 sample paths are (almost always) continuous functions of time".
It is known that every diffusion process satisfies the following condition:
lim [ Pr ( X (t+h)-X(t) > e / X(t)= x ) / h ] = 0 for all e>0 h-K> (3.1) for all x in the state space of the diffusion process. This is a formal
(mathematical) description of the property that the sample paths of the diffusion process X(t) are continuous.
Almost all diffusion processes are characterized by their local characteristics called infinitesimal variance and mean. They are respectively given by
Var [ X (t+h) - X(t) / X(t) = x] a(x) = lim ------h+0 h (3.2)
E [X(t+h) - X(t) / X(t) = x] 3(x) = lim ------h-+0 h (3.3)
where E(.) and Var(.) denote the mean and variance.
The infinitesimal mean and variance are interpreted as "the rates of mean and variance of displacement of a diffusion process in a small interval of time." 37
For a more strict mathematical definition of a diffusion process, readers are urged to see Wentzell [70:197].
3.1.2 Forward Kolmogorov equation (Fokker-Planck equation)
One of the properties of a time-homogenous diffusion process is
that it follows the following second order partial differential equation,
9 92 9 — f(t,y,x) = (1/2) — {a(x) f (t, y, x) } - — {6(x) f(t,y,x)} 9t 9x2 9x (3.4)
This equation is called the Forward Kolmogorov equation (or Fokker-
Planck equation). In the above partial differential equation, f(t,y,x)
is the probability density function of the transition (from state y to
state x in a time interval of length t) of the diffusion process X(t),
i.e.,
f(t,y,x) dx = Pr (x < X(t) < x+dx / X (0) = y) (3.5)
and a(x) and 3(x) are infinitesimal variance and mean defined by (3.2)
and (3.3). This Forward Kolmogorov Equation can be derived from the
usual Chapman-Kolmogorov Equation for continuous Markov Process under
the assumption that a continuous-path Markov Process has time-
homogeneity and the third and higher infinitesimal moments do not
exist. (See Bharucha-Reid [3:130-135] or Kleinrock [47:66-71] for
derivations.) 38
3.2 Elementary Return Process (ERP)
To apply a diffusion process to queueing systems, we need to specify appropriate boundaries, because the system size of a queueing system cannot assume negative numbers (for finite capacity queues, the system size cannot exceed the system capacity, either). Hence the lower boundary of the diffusion process should be zero. Also, if we are interested in finite capacity queues, we need to impose the system capacity (k) as the upper boundary of the diffusion process. The diffusion process to be used in this dissertation for this purpose is the Elementary Return Process (ERP). This process was thoroughly studied by Feller [19] in 1954 and first applied by Gelenbe [26] to queueing theory. This diffusion process is defined within two boundaries rj, and r2 (Figure 1). The boundary behavior of this process states that as soon as the process hits one of the boundaries, say boundary rj at time t, then it is absorbed into the boundary for a non-negative random time T, and then it jumps up either onto the other boundary r2 with probability P12 or into the diffusion region with probability Tj and the process starts all over again from scratch. At this time of jump, the jump size is also a random variable. (Of course we can think of the boundary behavior on boundary r2 with probabilities
P21 and T2 .) Notationally, we have
Pi;) = Pr ( X (t+T) = rj / X (s) = ri, t (3.8) Pij + T i = lf (i * Pr ( r^ < X(t+T) < x ) = T± P± (x) (i=l or 2) (3.9) r2 ------ t+Tt Figure 1. Boundary behavior of ERP The last equation (3.9) is the probability rule according to which a jump occurs from boundary r^ at time t+T. Hence it can be said that P^(x) is the conditional distribution function of the position after a jump from boundary r^ (i=l or 2) under the condition that the jump has occurred into the diffusion region (not onto the other boundary). Feller [19] derived the three partial differential equations to describe the ERP: one for the behavior within the boundaries, one for 40 the behavior on the lower boundary, and one for the behavior on the upper boundary. The behavior within the boundaries is described by the following second order partial differential equation; (3.10) 3 32 3 — f(t,x) = (1/2) — { a(x) f (t, x) } - — { 6 (x) f (t, x) } 3t 3x2 3x 3 T: 3 t2 + — { — P1 (x) Mj (t) } + — { — P2 (x) m2 (t) } 3x e^ 3x e2 This equation corresponds to the Forward Kolmogorov equation (3.4). The additional two terms result from the boundary behaviors of ERP. (Note that the equation (3.4) is for the diffusion process without boundaries.) To define the notation in equation (3.10), f(t,x) = the density function of the position of the diffusion process a(x), B(x) = the infinitesimal variance and mean defined by (3.2) and (3.3) Mi(t) = probability that the diffusion process is on the boundary r^ at time t, i.e., Pr( X(t) = ) e^ = expected holding time on the boundary r^ = probability that the jump occurs into the diffusion region from the boundary r^ as defined by (3.9) 41 P± (x) = conditional probability distribution function of the position of the diffusion process after a jump occurs from the boundary r^ under the condition that the jump has occured into the diffusion region (not onto the other boundary) as defined by (3.9) One thing to note is that in the Forward Kolmogorov eqation (3.4), we were interested in the transition of the process. Hence f(t,y,x) was used in the sense that Pr ( X(t)er / X (0) =y ) = J f (t, y, x) dx r But in equation (3.10), we are interested in the position of the process in the sense that Pr( X(t)er / distribution of X (0) is given ) = / f(t,x) dx r The distribution of X(0) is the initial condition of the system of equations. This initial condition does not affect the steady-state solutions. Since a diffusion process is a Markov Process, the holding time on the boundary r^ is exponentially distributed with mean e^- Hence, we have Pr( holding time on the boundary r^ z ) = 1 - exp(z/e^) The behavior on the lower boundary is described by following first 42 order partial differential equation, (3.11) — Mj (t) ------Mj (t) + M2 (t) dt ej e2 3 + lim { (1/2 ) — [a(x) f(t,x)] - B(x) f(t,x) } . x-*rj 3x and the behavior on the upper boundary is described by the following first order partial differential equation (3.12) a p12 i M, (t) = M. (t) - M, (t) dt ej e 2 3 lim { (1/2 ) — [a(x) f(t,x)] - B(x) f(t,x) } x-^r2 3x In equations, (3.11) and (3.12), p 12 = probability that the jump occurs from the lower to the upper boundary, and P21 = Probability that the jump occurs from the upper to the lower boundary. 43 3 . 3 Interpretation in the Queueing Context In the queueing context, the lower and upper boundaries of the diffusion process become the zero state and system capacity k, respectively. which is the probability that a jump occurs from the lower to the upper boundary can be interpreted as, Pj2 = Pr(system is full with an arrival / system was empty) 00 k - 1 1 Px (i) =1"! Px (i) (3.13) i=k i=l where Px (i) is the probability that the arrival size is i. In the same way, which is the probability that a jump occurs from the upper to the lower boundary can be interpreted as, P2 j = Pr(system is empty after a service completion / system was full) 00 k- 1 1 Py(i) =1-1 Py(i) (3.14) i=k i=l where Py(i) is the probability that the service size is i. Mj(t), and M2 (t) which are the probabilities that the process is on the lower and upper boundary can be interpreted as the probabilities that the queueing system is empty and full at time t. 44 1/ej, which is the reciprocal of the mean holding time on the lower boundary can be interpreted as the arrival rate, and l/^, which is the reciprocal of mean holding time on the upper boundary can be interpreted as the service rate times the number of servers. Hence, we have l/e1 = A (3.15) and l/e2 = m y (3.16) 3.4 Expressing infinitesimal mean and variance by queueing parameters It was stated earlier that the infinitesimal mean and variance characterize a diffusion process. Hence if one wants to use the diffusion process to approximate a queueing system, one needs to express the infinitesimal mean and variance by queueing parameters. Let us define the following variables. X Arrival size (random) with distribution Px(i) Y Service size (random) with distribution Py(i) G(t) Number of batches arriving in (0,t] N(t) Number of customers in the system at time t A(t) Total number of customers arriving in (0,t] 45 D (t) Total number of customers departing in (0,t] H^(t) Number of batches served by i*"*1 server in (0,t] i. u S^t) Number of customers served by i server in (0,t] X Arrival Rate (in batches) y Service Rate (in batches) G(a) Mean of interarrival time Var(a) Variance of interarrival time E(b) Mean of service time Var(b) Variance of service time Then A(t), and D(t), which are the total number of customers arriving and departing in (0 ,t) can be expressed as simple random sums of random variables as follows: A(t) = X: + X2 + ...... + XG(t) (3.17) D (t) = Sj (t) + S2 (t) + ...... + Sm (t) (3.18) where, 46 (3.19) + Y i H. (t) x = size of it ^1 arrival group Y^j = size of j service group served by i server If the system is empty at time zero (this assumption does not affect the steady state behavior of a queueing system), the number of customers in the system at time t is the simple difference between the total number of customers arriving and total number of customers departing in (0,t]. Hence N (t) = A (t) - D (t) (3.20) To find the mean and variance of N(t), we need to find the mean and variance of A(t), and D(t). Since G(t), which is the number of batches arriving in (0 ,t], is a renewal process, we can obtain the following limiting mean and variance (Cox [14]), E (G (t) ) = X t (3.21) Var(G(t)) = X3 Var(a) t (3.22) Now if we assume that the servers are hardly idle, then, H^t), which 47 is the number of groups served by ith server is also a renewal process. Hence, we have (3.23) Var(Hi (t) ) = y3 Var(b) t (3.24) Then, E(N(t) ) {A E(X) - m y E(Y) } t (3.25) ( E(N(t)) represents the change of the system size during the time interval of length t. Hence if E(N(t)) is large as compared with the system capacity, the approximation becomes inaccurate.) Since A(t) and D(t) are random sums of random variables, (see Giffin [28] for the variance of random sum of random variables) Var(A(t)) = E 2 (X) Var(G(t)) + E(G(t)) Var(X), Var (D (t) ) = m [ E 2 (Y) VartH^t)) + EfH^t)) Var(Y) ] and assuming the independence of A(t) and D(t), Var(N(t)) = {e2 (X) A3 Var(a) + A Var(X) (3.26) + m [e2 (Y) y 3 Var(b) + y Var(Y)]} t 48 Finally, we need to derive the infinitesimal mean and variance defined by (3.3) and (3.2). Let E(N(t)) = {A E(X) - m p E(Y)} t = Ij t (3.27) and Var(N(t)) = {e2 (X) A 3 Var(a) + A Var(X) (3.28) + m [e2 (Y) y 3 Var(b) + y Var(Y)]} t = *2 Then, if independent increments of N(t) is assumed, E[N(t+h)-N(t)/N(t)] = E[N(t+h)-N (t)] = (t+h)Ij - tlj = hlj (3.29) and Var[N(t+h)-N(t)/N(t)] = Var(N(t+h)-N(t)] = E[(N(t+h)-N(t))2] - E 2 [N(t+h)-N (t)] = E[N2 (t+h)] - 2 E[N(t)N(t+h)] + E[N2 (t)] - Ij2 h 2 From E [N2 (t) ] = Var(N(t)] + E 2 [N(t)] 49 2 2 = I2 t + Ij t/ Var[N(t+h)-N(t)] = I2 (t+h) + Ij2 (t+h)2 - 2 E [ N (t)N (t+h)] + I2 t + Ij2 t2 - Ij2 h2 = (t+h) I2 + 2 Ij2 t2 + 2 Ij2t h + I2t - 2 E[N(t)N(t+h)] (3.30) To obtain E[N(t)N(t+h)], let us assume that (3.31) N(t) + X with probability Xh + o(h) N(t+h) = < N(t) - Y with probability m y h + o(h) N (t) with probability 1 - Xh - m y h + o(h) where o(h) means a function such that lim f(h)/h = 0, and X h-K) and Y are arrival and service size random variables. This assumption is equivalent to approximating the whole process by using Markovian property. Then, E (N (t) N (t+h) ] E [ (N (t) +X) N (t) ] (Xh+o (h)) + E[ (N(t)-Y)N(t) ] (myh+o(h)) + E[N2 (t )3 (1-Xh-nyh+o(h)) 50 = E [N (t) ] (Ae(X) - myE(Y) ] h + E[N2 (t)] + o (h) = E [N (t) ] Ijh + E [N2 (t) ] + o (h) = Ij2 h t + Var[N(t) ] + E2 [N(t)] + o(h) = lx2 t h + I2 t + Ij2 t 2 + o (h). Finally, from (3.30), Var [N(t+h)-N(t)1 = I2 h + o(h) Hence from the definitions (3.2) and (3.3) (notice that we need to replace the diffusion process X(t) by queueing process N(t)), we can obtain the infinitesimal variance and mean as follows: a(x) = E2 (X) A3 Var(a) + AVar(X) (3.32) + m [e2 (Y) y 3 Var(b) + y Var(Y)] g(x) = A E(X) - m y E(Y) (3.33) Notice that infinitesimal mean and variance can be obtined by dropping "t" from the mean and variance of N(t). This heuristic 51 observation is based on the fact that if the infinitesimal mean and variance are the rates of changes of mean and variance of infinitesimal displacement of the diffusion process, then they can be expressed as B(x) = lim E[N(t)]/t t-K) a(x) = lim Var[N(t)]/t t+ 0 Also, the infinitesimal variance and mean are independent of the current state of the diffusion process, x. Hence they will be used just as a and $ , instead of a(x), and 3 (x). 3.5 Solutions of the ERP equations Let us assume that for all t>0, (t) = Mj (3.34) m2 (t) = m2 (3.35) lim f(t,x) = f(x) (3.36) t-*» i.e., probabilities that the ERP is on the boundaries are time 52 homogeneous and there exists a stationary density function of the position of the diffusion process. Let jj(x) and j2 ^x) be fcbe probability densities according to which the jump occurs from lower and upper boundaries such that (3.37) d , , j.(x) = — {Pr (r. < X(t+T) £ x/jump occurs from r. )j 1 dx (3.38) d , , j,(x) = — iPr (r. < X (t+T) £ x/jump occurs from r 5 )j z dx 1 c where X(t+T) is the position after a jump at time t+T. (See section 3.2 for notations in this regard). Hence it can be seen that jj (x) and j2 After using (3.15), (3.16), (3.32), (3.33), (3.34), (3.35), (3.36), (3.37), and (3.38), the equations (3.10), (3.11) and (3.12) become, in steady state, d 2 d (1/2) a — ,f (x) - 3 — f (x) = - X M. j. (x) -m y M 9 j9 (x) dx dx 1 1 ^ *■ (3.39) 53 lim { (1 /2 ) a — f (x) - 3 f (x) } x-*-k dx (3.41) Now, we need to find j^ (x) and j2 (x). Jump from the lower boundary (i.e. zero) can be interpreted as the arrival of customers into the empty queueing system. Suppose a group of 3 customers arrive. Then as soon as they arrive, the system size instantaneously jumps up to 3. An appropriate probabilistic tool for this purpose is the Dirac Delta density function 6(x-3). (See Giffin [28] for a definition of the Dirac Delta density function.) In the bulk queueing systems of interest in this research, the arrival size is a random variable. Hence Px (i) 6 (x-i) will be used as the distribution of the position after a jump, jj(x). (See Papoulis [63:98] for combined Dirac Delta density functions. Chiamsiri and Leonard [10] also used this.) Now, to find j2 (x), suppose the service of a group of size 3 is finished when the system is full. Then, the system state becomes k-3 and the Dirac Delta density function for this situation will be 6[x-(k-3)]. Hence jj (x) and j2 (x) will be given by k -1 jj(x) = J px ^ (3.42) i=l 54 k -1 j2 (x) = I Py(i) 6 (x-k+i) (3.43) i=l Then final equations for the finite capacity case become (3.44) d 2 d ^ “ 1 (1/2) a — 2f (x) - 3 — f (x) = - X M, £ PY (i) 6 (x-i) dxz dx 1 x k-1 - m y M2 I Py(i) 6 (x-k+i) i=l r d i lim { (1 /2) a — f (x) - 3 f (x) | 1 M, - m p P „ H, x+ 0 dx 1 z it (3.45) r d i lim { (1/2) a — f (x) - 3 fix) = X M, P, 5 - m y M, x+k dx l it t (3.46) To solve these equations, we need boundary conditions lim f(x), and x+ 0 lim f(x). It was mentioned already that once the ERP hits the x+k boundaries, it is absorbed into the boundaries for a non-negative random time. In other words, the boundaries act as absorbing boundaries 55 while the process stays on the boundaries. Hence we have (see Cox and Miller [15:219-220] for a detailed discussion of the boundary conditions for an absorbing boundary), lim f(x) = 0 (3.47) x-K) lim f(x) = 0 (3.48) x+k If we integrate (3.44) once, we have (3.49) d k_1 (1/2) a — f(x) - 8 f (x) = - X Mj I Px (i) U(x-i) dx i=l k -1 - m p M2 I Py (i) U(x-k+i) + C* i=l ♦ where C is an integration constant and U(x-i) is the unit step function defined by, 1 if x > i (0 if x < i To find C*, evaluate equation (3.49) at x = 0, then C* = (1/2) a f (0) - 8 f (0) dx 56 Using equation (3.45) and replacing P21 by £ Py(i) obtained in i=k (3.14), yields C — X ” id p 1 m2 = A M 1 - m u M 2 I Py(i) (3.50) i=k ♦ By evaluating equation (3.49) at x = k with the value of C given by (3.50), we obtain, (1 /2 ) a — f (k) - 3 f (k) dx k -1 k -1 oo A Mj £ Px (i) - m jj M2 [ Py (i) + A Hj - n y M 2 £ Py (i) i=l i=l i=k k -1 k-1 oo A Mj {l - I Px (i)} - m u M 2 { I Py (i) + I Py (i) ) i=l i=l i=k (from 3.13) — X Mj Pj 2 ~ m |i M2 i which is the same as the right hand side of equation (3.46). Hence C* given by (3.50) satisfies (3.46), too. We, therefore, finally need 57 ♦ to solve the equation (3.49) with C given by (3.50), which becomes d (1/2 ) a — f (x) - 0 f (x) dx k-1 X Mj {l - I Px (i) U(x-i)} i=l k-1 00 - m y M 2 { I Py(i) U(x-k+i) + I Py (i)} i=l i=k (3.51) Still, boundary conditions (3.47) and (3.48) hold. By taking the Laplace transform on (3.51), we have (Notice that L[c] = c/s, and L[U(x-i)] = exp(-is)/s, where L(.) denotes the Laplace Transform. See Giffin [28:220] for Laplace Transform pairs.) (1 /2 ) a {s L[f(x)] - f(0 )} - 0 L[f(x)] k-1 = X Mj { 1 /s - I Px (i) e"ls/s } i=l k-i _ 00 - m y M 2 { I PY (i) e (k l)s/s + £ Py (i)/s } i=l i=k Arranging terms, we have 58 L[f(x)] = k -1 k -1 00 X Mj{l - I Px (i) e-is} - m y M2 { I Py (i) e" (k l)s + I Py (i)} i=l i=l i=k s[ (a/2 )s - 3 ] (3.52) Now, we need to invert L[f(x)] to find the density f(x). Denote L~ *(.) as the inversion of the Lapalce Transform. We will use the following transform pairs and partial fraction in finding L~*(.). L-1[(c exp(-ds))/(s(as/2-b))] = (c/b) U(x-d) [exp(2 b (x-d)/a)- 1] If1 (c/s) = c L-1 [c/(ds+b) ] = (c/d) exp(-bx/d) c/ (s(ds/2-b)) =(-c/b)/s + (c/b)/ (s-(2b/d)] Then we have (3.53) OO f (x) = (1/3) (0)X- 1) [X Mx - m y M2 I Py (i) ] i=k k -1 + (x Mi/3) J LPX (i) U(x-i) (1 - ajx_i)] i= l k-1 + (m y M2/3) I Lpy(i) U(x-k+i) (1 - wx"k+i)] i= l 59 where U) = exp(23/ot) From the definition of the unit step function, we can express the above form equivalently by (3.54) I oo (1/6) (U)X- 1) [) Mj - m p M2 I Py (i) ] i=k if 0 < x < 1 (1/6) ( Wx-1) [) Mj - m p M2 I PY (i)] i=k + (X M j/6) I [Px (i) (1 " a)X_i)] i=l f(x) - < k-1 + (m y M2/6) I (.Py (i) (1 - u>x_k+1)] i=k-n if n < x < n+1 1 < n < k-1 (1/6) (tox— 1) [) Mj - m p M2 I Py (i)] i=k k-1 + (X Mj/6) I [Px (i) (1 ” w )3 i=l k-1 + (m y M 2/6) I [Py(i ) (1 - w1)] i=l if x = k 60 The steady-state density f(x) given by (3.54) involves two unknown quantities, and M 2 which are the probabilities that diffusion process is on the lower and upper boundaries respectively. These can be obtained from one of the boundary conditions and from the fact that sum of probability mass is 1. Hence we have k + / f (x) dx + M2 = 1 , and 0 f(k) = 0 From (3.53) and by letting 0) = exp(23/ot) k 1 00 / f (x) dx = (1/6 ) / (0)x- 1 ) jA Hj - m p H 2 I Py (i) } dx 0 0 i=k + (1/6) / { (0)X-1) (X Mj - m p M 2 I Py (i)) 1 i=k k-i + ^ M1 I »<*-*> " wX_i) i=l k - i + m y M 2 I Py (i) U(x-k+i) (1 - cox_ )} dx i=l cr> g(x-k+i) dx f (x) dx 62 (1/B) (0t/23) (U)k - 1) (X Mj - m y M 2 I P y d ) ) L=k (1/B) k (A Hj - m y M2 I Py (i) ) i=k k-1 + (1/B) I Px (i) (k-i) i=l k-1 (1/B) (a/2B) A Hj I Px (i) (c/ - 1 - 1 i=l k-1 + (1/B) tn y M2 I i Py (i) i=l k-1 - (1/B) m y M2 (a/2B) I Py (i) (w1 1 ) • i = l Hence, k Mj + J f (x) dx + Mj = Mj { 1 + (X a/(2B2 )] (0)k - 1) - X k/B k-1 + (X/B) I Px (i) (k-i) i=l O LO 64 From (3.55) and (3.56), we have Mj = C4 / (CjC^ - C2 C3 ) (3.57) M2 = C3 / (C2 C3 - CjC^) (3.58) where, k-1 Cj = 1 + [X a / (2 $2 )](wk-l) - X k/B + (A/3) I Px (i) (k-i) i = l k-1 - (A a / (2 32) ] I Px (i) (a)k_i - 1) i=l (3.59) 0 0 C2 = 1 + [m y a/ (2 B2)] I Py (i) (l-wk ) i=k 00 k-1 + (myk/B) I Py (i) + (m y/B) I i Py (i) i=k i=l k-1 [m y a / (2 B2 )] I Py (i) (to1 - 1) i=l (3.60) k-1 C3 = (A/B) (0)k- 1) + (A/B) I Px (i) (1 - 0)k_1) i=l (3.61) 65 C4 = (ra m/3 ) (1 - U)k ) I Py(i) i=k k - 1 + (m m/3) I PY (i) (1 - O)1 ) i=l (3.62) and 0) = exp (2 3 /a) As can be seen, Mj and M2 are automatically obtained while solving the ERP equations. This is one of the advantages of using ERP boundaries. Now we can use Mj as the probability that the system is empty and M2 as the probability that system is full. 3.6 Discretization To use the density function f(x) in (3.54) as the distribution of system size, we need to discretize the density function. In the queueing literature, the following four discretization schemes can be found: n+ 1 n n+.5 / f (x) dx , J f (x) dx , / f (x) dx , f(n) n n- 1 n-.5 66 It is hard to say which one is better than others. (Reiser and Kobayashi [64] used the first scheme, and Gelenbe [26] used the second one. The third and fourth schemes were used by Halachmi and Franta [30].) But if the state variable x is replaced by n in (3.54), i.e., if we use f (n) as the system size distribution (the fourth scheme), then the sum of probabilities becomes 1 under a mild assumption that the service size is less than the system capacity, which is a very reasonable assumption. (The proof of this assertion will be given in chapter IV.) Then. P n N (n) = Pr ( n customers in the system with capacity k ) becomes (3.63) C4/(CJC4 - C2C3) if n = 0 (1/B) (U)n- 1) (X - m n M 2 I Py (i)} i=k + (X Mj/B) I Px (i) (1 " U)0"1) i=l PF I N (n) = k- 1 + (m y M2 /B) I Py(i) (1 ~ U)n ' k + i ) i=k-n if 1 < n < k-1 V (C2C3 - c lc4> if n = k 67 where U) = exp(2$/ot), and Cj, C2 , C3 , C4 , Mj and M2 are as obtained in (3.59), (3.60), (3.61), (3.62), (3.57) and (3.58). CHAPTER IV ANALYSIS OF SYSTEM SIZE DISTRIBUTION In this chapter, the system size distribution developed in chapter III will be analyzed. First, the legitimacy of the system size distribution is proved. Then the mean and variance of the system size are derived. The derivation of the system size distribution of a infinite capacity system will follow. Also, some special cases of finite capacity bulk queues will be discussed. 4.1 Proof of legitimacyof the system size distribution In this section, it will be proved that the system size distribution given by (3.63) is a legitimate probability distribution under the assumption that service size is less than the system capacity, i.e., Pr(Y PROOF From the system size distribution (3.63), and for 1 <_ n <_ k-1, 00 PF I N (n) = (1/P) (a>n_1) iX M 1 " n M **2 I P y (i) } i=k 68 69 + (X f^/B) I Px (i) (1 - Un_1) i=l k -1 + (m p M2/$) I Py (i) (1 - 03n"k+1) i=k-n n = Mj { (X/B) I Px (i) (l- Wn_i) + (A/8) (0>n-l) } i=l k- 1 + M 2 { (m y/B) { £ Py (i) (1- a/1-**1 ) } } i=k-n = Mj Aj (n) + M2 A2 (n) k -1 Then, from + £ PpjN (n) + M 2 = 1, we need to show that n=l 1 - Mj - M2 k- 1 k- 1 k- 1 £ Ppijj(n) = Mj £ Aj(n) + M 2 £ A2 (n) , n=l n=l n=l Or rearranging terms, 70 k-1 k-1 **! {l + I Aj(n)) + M2 {i + I A 2 (n)} = 1 (4.1) n=l n=l Now, from (3.59), (3.60), (3.61), and (3.62), k -1 = (my/g) { I Py(i) (1 - w1 )} i=l k- 1 k-1 + (myXa/2g3) {u)k-l - I Px (i) (0)k ~ 1 -1)} { I 1'y(i) ( l-CO1) } i=l i=l k- 1 k- 1 + (mXy/g2) { I Px (i) (k-i) - k} { I Py (i)(l- t I1)} i=l i=l and C2C3 k -1 = (X/B) { I Px (i)(l- o)k_i) + a)k - 1 } i=l 72 (4.1) becomes equivalently k - 1 k -1 C4 {l + I A x (n)} - C3 {l + I A2 (n)} = C ^ - C ^ (4.2) n=l n=l Now, k -1 C4 {l + I A1 (n)} n=l k - 1 n = C4 { 1 + (A/3) I { I Px (i)(l- (J0n_i) + con - 1} } n=l i=l k- 1 = (my/3) I Py (i) (1- U)1) i=l k- 1 n k- 1 + (myA/32 ) I { I Px (i) (1- wn_i)+ U)n-l} { I Py (i) (1- U)1)} and, k -1 C 3 [I + I A 2 (n)} n=l k - 1 k- 1 = C, { 1 + (my/3) I I P„(i) (1- o T k+1) } n=l i=k-n k-1 = (A/B) { I P x (i) (1- wk_1) + iok - 1 } i=l + (myA/32) { I I PY (i) (1- wn_k+i)} { I Px (i) (1- wk"i)+ 0)k- n=l i=k-n L = 1 Then, from (4.2) k - 1 k -1 C4 {1 + I Ajtn)} - C 3 {l + I A 2 (n)} n=l n=l k -1 = (my/3) I Py (i) (1- w1 ) i=l k -1 k- 1 n + (myA/B2){ I Py (i)(1- w 1)} { I I [Px (i) (l-o/1_i)+ U)n- 1 ] } 1 = 1 n=l i=l k_1 - (A/B) { I Px (i) (1- 0)k_1) + 0)k - 1 } i=l - (myA/32) { I Px (i) (l-OJk_i)+wk-l} { I I PY U> n ~ u P ' k+i) i=l n=l i=k-n 74 k-1 = (mn/6 ) { 1 PY (i) (1- W 1)} i=l k- 1 k -1 + (myX/B2 ) { I Px (i) (k-i)-k} { I Py (i) (1 - U)1)} i=l i=l k-1 (X/B) { I Px (i)(l- U)*'1) + 0)k - 1 } i=l k- 1 k - 1 (rayX/B2 ) { I t P y ^ ) } { I Px (i)d" wk_1) + wk -l} i=l i=l Hence, we only need to check the following identity. k- 1 k -1 n { I Py (i) (1- id1) } { I I tPx (i) (l-0)n_i)+ 0)n-l] } i=l n=l i=l - { I Px (i) (l-o)k'i)+wk-l} { I I PY (i) (l-(Jtl-k+i)} i=l n=l i=k-n k -1 k-1 = { I Px (i) (k-i)-k } { I Py (i) (1 - a)1)} i=l i=l II II 11 n 0 a x 3 X 3 x 3 p- x II r~-i i II r—3 i Qi II r—i i n t— j i 3 H- X a II -- I V H- H- X II f—13 II 1 3 II r—iS II E— 3 3 M V U 13 Kj X *0 •d X X H- X X II t"-! I X I p- ►0 X i [(i)( 3 H- X 3 X I II -- 3 I II C'-l I H- H- X II t—3 I 13 3 I X H- 1 3 X H- - H- it r — i i II C--13 0 3 X ) / ( ) II C-~-l I *0 X H- X D 3 II t—1 I X X I I 1 -w) ] 3 X H- 3 II t—I I X l H- ►-> K 13 H- X + X II t - < 3 I •fl 3 X X II t - - 3 I 3 I H- X I H- I X -1 ’- .. o, from Now, = i=k-n n=l - k-1 k-1 1 II i i=k-n k - k-1 k-1 =l n =k-i l i= I I k-1 =l i= I I x() ki - + 1 + k - (k-i) (i) Px ( + > i ( y P 1 - I y() (1- (i) Py k-1 k i=l =l i= -1 I I 0 x() [(l-(ok_i)/(l-(o)] (i) Px x() [(l-OJk_;L) (1-CD) (i) / Px ] ( 1 (ok- )/(l-to) 1 - y(i) Py = ikn =l n=k- k-1 l i= k-1 i=k-n k-1 n=l k-1 I 1 I g (n) g ton-k+i) I = i=k-n n=l - k-1 k-1 i = h() £ (i) h £ = (i) h - k-1 k-1 =l i= I I I Yi CO1 PY(i) y( a/1 Py )(i n=k-i uP'k I -k+i i g g (n) (4.4) 76 77 k - 1 k-1 I i Py (i) I PY (i) t(l- t/l/d-w) . i=l i=l (4.5) But from (4.4) and (4.5), (4.3) becomes k - 1 1 k - 1 b> (l-u)k “ 1 ) ( I Py (i) (l-w1) ] [ 1 I Px (i) (l-t/-1) + ] k - 1 1 k - 1 [ I P x (i) (1—a)k~ 1 ) + w k - 1] [ --- I Py (i) (1 -to1 ) ] i=l 1-to i=l = 0 (4.6) a)(l-U)k _ 1 ) l-0)k From 1 + =------1 —0) 1 -0) it is easy to see that equality (4.6) is satisfied. Proof is finished. 78 4.2 Mean and Variance of system size. In this section, the mean and variance of the system size will be derived under the assumptions that arrival and service sizes are always positive and less than the system capacity, i.e., Pr (X=0) =0 Pr (Y=0) =0 P r (X >_ k) = 0 P r (Y ^ k) = 0 . From (3.59) (3.60) (3.61) (3.62), and by defining following probability generating functions, k -1 Gx (l/u ) ][ o f 1 Px (i) i= l k-1 Gy (0)) l ID1 P y ( i ) i= l we have c1 = l + (Xa/262) (tok-l) - (A/8)E(x) - (Aa/262 ) [u)k Gx (i/w) -l] (4.7) 79 C2 = 1 + (my/g)E(Y) - (raya/232 ) [Gy (tD) -1] (4.8) C3 = (X/3) 0)k [1 - Gx (l/a))] C4 = (ray/3) tl - Gy(0))] (4.10) and, n PF I N (n) = < * V 3 ) { Wn ~ 1 + I PX (i) (1 ‘ a)n_i) } i=l k- 1 + (rayM2 /3) I Py (i) (1 - wn_k+1) i=k-n for 1 < n < k -1 Then k- 1 n E(N) = (XMj/B) I n {ajn - 1 + I Px (i) (1 - w11"1 )} n=l i=l k - 1 k- 1 + (rayM2 /3) I n [ J Py (i) (1 - u)n"k+1) ] n=l i=k-n + k M 2 (4.11) 80 Now, k- 1 I n (a)n- 1 ) n=l 1 - wk _ 1 (k-1 ) 0)k_ 1 k (k-1 ) 0) } - (l-w)2- 1 - 0) (4.12) k-1 n k -1 k - 1 From I g(n) I h(i) = £ h(i) J g(n) , n=l i=l i=l n=i k -1 n k - 1 k - 1 I n I Px (i) = I Px (i) I n n=l i=l i=l n=i k (k-1) - [E(X2) - E(X)] (4.13) and, k -1 n k -1 n I n I px (i> 0)n_i * I pX li) u.'1 n=l i=l n=l i=l k - 1 k- 1 I Px (i) 0)_i I n o)n i=l n=i 81 E (X) - 0)k Gx (l/tD) [k-1 + 1/ (1-0)) ] + 0)/(l-0)) 1 - 0) (4.14) k- 1 k- 1 k- 1 k- 1 From I g(n) ][ h(i)= I h(i) £ g(n), n=l i=k-n i=l n=k-i k -1 k -1 k -1 k -1 I n I Py (i) = I Py (i) I n n=l i=k-n i=l n=k-i (2 k - 1 ) E(Y) - E(Y2 ) 2 (4.15) and k -1 k -1 I n I Py(i) u T k+i n=l i=k-n k -1 k -1 = ofk { I n o)n I Py(i) O)1 } n=l i=k-n k -1 k -1 = 0)-k { I Py(i) O)1 I n u)n } i=l n=k-i 82 k - E(Y) + to/(1 —03) - [k-l+ (1/(1 —to) ) ] Gy (0)) 1 - 0) (4.16) From (4.11), (4.12), (4.13), (4.14), (4.15) and (4.16), E(N) = (XMj/g) { -0.5 E (X2 ) + E (X) [0. 5 - (1/ (1-0)) ) ] + [0)k/(l-0))l Gx (l/0)) [k-l+ (1/ (1-0)) ) ] - 0Jk / (1 -OJ)2 - (k-1 ) 0)k/(l-0)J } + (myM2 /B) { -0.5 E(Y2 ) + E (Y) [k-0.5+(1/(1-0))) ] + [1 /(1 -0)) ] Gy (0)) [k-l + (1 / (1 -0)) ) ] - [1 / (1 -OJ) ] tk+ (0)/ (1 -0)))) } + k M2 (4.17) where M1 = C4 t (CjC4 - C2C3 ) m 2 = c3 / (c2c3 - CjC4 ) I-* VO (81 ’fr) (81 84 k -1 k (2k— 1 ) (k-1 ) I "2 n=l (4.20) k- 1 n k- 1 n I n 2 I to11-1 Px (i) = I n2 ton I PX (i) of1 n=l i=l n=l i=l k-1 k-1 - i I Px (i) w [ n 2 to i=l n=i = Il/d-OJ) ] { E(X2 ) +[2a)/(l-0))] (E(X) - (k— 2) (Ok _ 1 Gy (1 /(jj) + to/(l-a)) - a)k_ 1GY (1 /oj)/(1 -w)) + to/ (l-o)) - to:GX (1/OJ)/ (1-co) - (k-1) 2 tok GX (1/C0) } (4.21) k - 1 n k- 1 k- 1 I n 2 I Px (i) = I Px (i) I n2 n=l i=l i=l n=i k (2k-1) (k-1) -2E(X3 ) + 3E(X2 ) - E(X) 6 (4.22) 85 k-1 k-1 k-1 k-1 I n 2 I Py(i) = I Py (i) I n 2 n=l i=k-n i=l n=k-i (6k2-6k+l)E(Y) + 3 (l-2k)E(Y2 ) + 2E(Y3 ) (4.23) k -1 k - 1 I n 2 I PvY (i) 0)n-k+i n=l i=k-n k -1 k -1 w_k { I PY (i) U)1 I n2 o)n } i=l n=k-i = [1/d-OJ) ] {k2 - 2k E (Y) + E (Y2 ) + [2to/ (1-OJ) ] [ k - E (Y) - (k-2) w-1 Gy(w) + to/ (1—co) - Gy (to) / (tod-O))) ] + to/ (1 —(0 ) - G„ (co) / (1 -0)) - (k-1) 2 Gy (0)) } . (4.24) Hence, 86 E(N2 ) = (XMj/B) { [2 / (l-oo)2] [ 00- (k-1 ) 0)k + 002 (l-00k~2) / (1 -00) ] - oo (l-tok_1) /(l-a» 2 - (k-1 ) 2 ook/(l-oo) - (1/6) [ 2E(X3) - 3 E(X2) + E(X) ] - [ 1 / (1-aj) ] { E (x2) + [2oo/ (1-oj) ] (E(X) - (k—2) a)k_ 1Gx (l/0)) +to/ (1 -co) - (J^Gjjd/urt/d-U))] + to/ (1 -(a)) - 0)kGx ( 1/0))/(1-OJ) - (k-1)2 0)kGx (l/0J) } } + (miiM2/B) { (6 k 2 -6 k+l) E(Y) / 6 + (l-2k) E(Y2 ) / 2 + E(Y3)/3 - 11/(1-tO) ] { k 2 - 2k E(Y) + E (Y2) + (2(0/ (1-OJ) ] [k - E (Y) - (k-2) CO- XGy (to) + 0)/(l-w) - Gy (10)/(00(1-0)))] +00 / (1 -00) - Gy (to) / (l-oo) - (k-1 ) 2 Gy (to) } } + K2 M2 (4.25) Then variance can be obtained from Var(N) = E(N2) - E 2 (N). 87 4.3 Derivation of infinite capacity distribution In this section, the infinite capacity system size distribution will be derived as a limiting case of the finite capacity distribution. By taking limits on Cj, C2 / C-j, and C4 given by (3.59), (3.60), (3.61) and (3.62), lira Cj = 1 - (A a/282 ) (-1 ) - (A k/8 ) k-*» + (A/8) (k - e(x)) + (A a/ 282 ) 1 - (A/B) E(X) 00 lim C2 1 + (m y/B) E(Y) - (m a y/2B2 ) ( I Py (i) w1) k-«o i=l + (ra a y/282) lim C3 (A/B) (-1) + (A/B) 0 k-x» 00 lim C4 (m y/B) (1 - I Py (i) a)1 ) k-*» i=l 88 Then, lira M. = lim [ C. / (C.C, - C,C,) ] Jt-KD -1 1 / lim k-x» = [1 - (A/8) E(X ) ] ~ 1 = 1 - p where p = A E(X)/(m y E(Y)) and lira M, = lim [ Co / (CoC, - 0 ,0 .)]’ = 0 £ k-xo J £ J 1 Q Hence from finite capacity system size distribution (3.63) for n > 1, lim PFIN(n) = k-x» (A/B) (wn - 1) (1 - p) n + (A/8) (1 - P) I Px (i) (1 - 0)n_i) i=l n = (A/B) (1 - p) { wn - 1 + I Px (i) (1 - uf 1' 1 ) } i=l 89 n = - (p/ E(X) ) { 0)n - 1 + I Px (i) (1 - U)n_1) } i=l X X X E (X) { Note that — (1 — p) = (1 ------) 3 X E (X) -m y E (Y) m y E (Y) X m y E (Y) - X E (X) { 1 E(X)-m y E(Y) m y E(Y) — X m y E (Y) E(X) Hence, the infinite capacity system size distribution becomes (4.26) 1 - p if n = 0 PF I N (n) P n ------{a)n - 1 + I Px (i) ( 1 - 0)n_1)} E(X) i=l if n > 1 where, p is the traffic intensity given by X E(X)/(m y E(Y)) and 0) = exp(23/a) 90 This distribution is exactly the same as the one obtained by Chiamsiri and Leonard [10] when they studied the diffusion approximation of infinite capacity bulk queues. Hence their solution can be obtained as a special case. 4.4 M/M/1/k queue as a special case. In this section, diffusion approximate solutions of M/M/1/k queue will be compared with the exact ones. M/M/l/k queue is a special case of GIX/GY/m/k queue, where E(X)=1, Var(X)=0, E(Y)=1, Var(Y)=0, Var(a)=l/A^f Var(b)=l/p2 and m = 1. Hence infinitesimal mean and variance become 0(x) = A - p , and a(x) = A + p . Now let co = exp(20/a). Then, from (3.59), (3.60), (3.61) and (3.62), = l + [(Aa)/(202 )] ujk-1 (orl) - A/6 91 c2 = 1 + y/3 - [(pot)/ (2B2 )] (orl) c3 = (X/B) (o)k - t/"1) c4 = (p/B) (l-w) Then after some manipulation, we have = (l-o)) (p/B) (l-X/B) - (or 1) o)k_1 (X/B) (l+p/B) (or 1) - p2 (or l) o)k_1 (p-1)2 1 - p Ml = c4/(Clc4 - c2c3) = ------—- 1 - p* hT 1 (4.27) and p (1-p) o ^ ' 1 m 2 = - c3/(Clc4 - c2c3) = ------—— i - p* ur 1 (4.28) 92 From system size distribution (3.63), P(n) = Pr(n customers in the system) (X Mj) u)n_1 (u-D/B p o)n_ 1 (1 -w) 1 - p2 w * - 1 (4.29) k -1 E(N) = I n P(n) + k M 2 n=l (after some manipulation) p {l - k p 0) + u) (k p - 1 ) } (1 - p2 W*-1) (l-w) (4.30) In case of heavy traffic (i.e., p - 1), traffic intensity becomes very close to U) . Hence if p is replaced by 0) in (4.27), (4.28), (4.29), and (4.30), then = Pr(empty system) 1 - p (4.31) 93 M2 = Pr(Full system) pk (1 - p) 1 - pk + 1 (4.32) P(n) = Pr(n Customers in the system) (1 - p) pn 1 - pk + 1 (4.33) p - (k+1 ) pk + 1 + k pk + 2 E(n) = (1 - pk+1) (1 - p) (4.34) It can be easily checked that (4.31), (4.32), (4.33), and (4.34) are the exact solutions of M/M/l/k queues. This analysis shows that the approximate system size distribution is expected to be fairly accurate for estimating the performance measures of finite capacity queues. 94 4. 5 /M/1 queue as a_ special case If k -*• oo , then from (3.25), becomes n PINF(n) = ( "P/E(x^ { 0)n - 1 + I Px (i) (1 - U)n_i) } i = l n ^ 1 and a = X E 2 (X) + X Var(X) + y 6 = X E (X) - y Then, from the derived mean (4.17) E (N) = (-p/E (X)) { -0.5 E (X2) + E (X) [0.5 - 1/(1 -o>) ] } P p E(X2 ) + - [ 1 ] 1-0) 2 E (X) Var(X) E(X2 ) p { 1 + ------+ E(X) - 0) [ - 1] } E (X) E(X) 2 (1 - 0)) (4.35) 95 y It is already known that exact mean system size of M /M/1 queue is given by (see, for example, Giffin[27:205]) Var(X) p [1 + ------+ E (X) ] E (X) 2 (1 - p) In heavy traffic case, p 21 U). Then, the difference between exact and diffusion approximate mean system size becomes smaller if E(X ) is close to E(X). In other words, if coefficient of variation of arrival size distribution is getting close to 1 . CHAPTER V ACCURACY EVALUATION In this chapter, the accuracy of the diffusion approximation to the system size distribution will be evaluated. 5.1 Performance Measures As the performance measures, the relative percentage deviation of the mean and variance of system size will be used which are defined by E(Nd ) - E(Nr) x 1 0 0 E(Nr ) and Var(Nd) - Var(Nr) x 1 0 0 Var(Nr ) where E(Nd), Var(Nd) mean and variance of system size from diffusion approximation and E(Nr), Var(Nr) mean and variance of system size from exact (or simulation) results. 96 97 Hence, if the percentage deviation is greater(less) than zero, diffusion approximation overestimates (underestimates) the true or simulation mean and variance of system size. It can be seen from (3.63), (3.32), (3.33), (3.57) and (3.58) that the system size distribution is expressed in terms of E(X), E(Y), Var(X), Var(Y), E(a), E(b), Var(a), Var(b), m and k, where E (X) , Var(X) = Mean and variance of the arrival size E (Y) , Var(Y) = II (1 service size E (a), Var(a) = II II interarrival E (b), Var(b) = II II service time m = Number of servers k = System capacity P = Traffic intensity a = Infinitesimal variance 3 = Infinitesimal mean Also, it is easy to observe that $/a , A/a and y/a are involved in the approximate solutions. Then, 8 A E (X) - m y E (Y) a E2(X) A3 Var(a) + A Var(X) + m E2 (Y) y3 Var(b) + m y Var(Y) By dividing both numerator and denominator by my , and denoting Ca = A2 Var(a), = y2 Var(b), 98 0 a P E(Y) - E(Y) [p E(Y)/E(X)] Ca + [p E(Y) /E(X)]var(X) + E 2 (Y) Cfa + Var(Y) X X 1 3 X E(X) - m y E(Y) E(X) [1 - (1/p)] y y i 3 X E(X) - m y E(Y) m E(Y) (p - 1) where X E (X) p = ------m y E(Y) Hence, once p, m, k, and the first two moments of X and Y are specified, approximate solutions depend only on Cfl, C^ and the distributions of arrival and service size. Note that Ca {= X 2 Var(a)) and (= y2 Var(b)) are variabilities (squared coefficients of variation) of the interarrival and service 2 2 time distributions defined by var(a)/E (a) and Var(b)/E (b). (Because 99 X = 1/E(a), and p = 1/E(b)). Also note that the variability of the Exponential distribution is equal to 1 and that of the Erlang distribution is less than 1. 5.2 Distributions of interarrival and service time The diffusion approximation of interest in this dissertation does not need the exact forms of interarrival and service time distributions. What is needed is the first two moments of the interarrival and service times. To be more specific, if the first two moments are the same, the diffusion approximation produces the same system size distribution, thereby the same mean and variance. Using this limited amount of information will inevitably result in reduced accuracy of the approximation. The purpose of this accuracy evaluation is to check the degree of deviation of the diffusion approximation from the exact solutions. The only exact solution for finite capacity bulk-arrival, bulk- y y service queues exist for M /M /1/k queues [2]. For other finite capacity cases, exact solutions are not available. Hence simulation estimates will be used. To estimate the mean and variance of a bulk queueing system by simulation, it is needed to specify the exact forms of the interarrival and service time distributions. For these interarrival and serivce time distributions, the Erlang, Lognormal, and Uniform distributions will be used. Even though the diffusion approximation depends on the first two moments, (not on the exact forms 100 of interarrival and service times) it is expected that exact distributional forms affect the system size distribution and the performance measures. We will compare the results for those three different distributions with same mean and variance to assess the magnitude of this effect. The Erlang distribution can assume a variety of density patterns by varying its parameters. The density function of the Erlang of order n and with parameter y (will be denoted by Erlang(n,y) ) is given by g(x) = (yn/r (n)) xn_1 e""Yx x > 0 (5.1) Its mean and variance are given by n/y, and n /y respectively. If X is the arrival rate, then 1/X is the mean of the interarrival time distribution. By letting 1/X equal to n/y, we have y = n X (5.2) Then, the variance of the Erlang distribution of order n is expressed by n/y^ = n/(n^ X^) = l/(n X^). (5.3) The density function of the Lognormal distribution is given by 1 g(x) = exp[-(ln x - £)^/2O^] x > 0 (5.4) x ✓2ti a l Its mean and variance are given by Mean = exp(£ + 0.5 0^) 101 Variance = exp(2£ + 2o^) - exp(2fj + o2 ) To make these mean and variance the same as those of the Erlang distribution, n/y = exp(£ + 0.5 o 2 ) and n /y 2 = exp(2^ + 2a2 ) - exp(2^ + a2 ) . Then, we have a2 = In (1/n + 1) (5.5) £ = In (1/X) - 0.5 In (1/n + 1) (5.6) Hence, we expressed the mean and variance of the Lognormal distribution by the arrival rate and the order of the Erlang distribution. The Uniform distribution is given by g(x) = l/(d-c) c < x < d (5.7) Its mean and variance are given by Mean = (d+c)/2 Variance = (d-c)^/12 Once again, by equating these mean and variance with those of the Erlang 102 distribution, we have c = (n - /3 n ) /y = (n - ^/3n) / (n X) (5.8) d = (n + /3 n ) /y = (n + /3n) / (n X) (5.9) It is observed from (5.8) that the order of the Erlang distribution (n) should be at leat 3 to obtain a positive Uniform random variate. As can be seen from (5.2), (5.3), (5.5), (5.6), (5.8) and (5.9), if the arrival rate and the order of the Erlang distribution are given, the parameters of the Lognormal and Uniform distributions are automatically determined to have the same mean and variance with the Erlang distribution. The differences of shapes between the three different densities with the same means and variances can be seen on Figure 2. «D O Erlang(4,2) Lognormal Uniform m to ogo .00 1.00 2.00 3.00 11.00 5.00 6.00 Figure 2. Shapes of Erlang, Lognormal and Uniform distributions with same mean and variance, (based on Erlang(4,2)) 103 5.3 Distributions of arrival and service size As the distributions of the arrival and service sizes, the Geometric distribution will be used. The Geometric distribution was already used in their study of bulk queues by Bagchi and Templeton [2], and Chiamsiri and Leonard [10] as the distribution of the arrival and service size. The Geometric distribution is given by q(x) = p (l-p)x_1 x = 1,2,.... 2 Its mean and variance are 1/p and (l-p)/p . Hence if p=l, the variance becomes zero, and it means single-unit arrival or service size. One thing to note here is that since the Geometric distribution is defined on the positive integers, the derived mean and variance given by (4.17) and (4.25) can not be used because they were derived under the assumption that the arrival and service sizes are less than the system capacity. The mean and variance, therefore, in the accuracy evaluation of this chapter will be calculated directly from the system size distribution given by (3.63). 5.4 Comparison of MX/MY/1/k queue In this case, exact solutions are available from Bagchi and 104 Templeton [2]. Their solutions are given in transforms and exact quantities can be obtained after numerous iterations. They also provided FORTRAN source code to obtain the mean system size, even though some errors had to be fixed. (Some parameters were mixed up and interchanged.) Also, it was needed to slightly modify their source code to obtain the variance of system size. The mean and variance of the system size will be compared for three different traffic intensities (.5, .7, and .9), different combinations of parameters of the Geometric distributions (p^ for the arrival size, and P 2 for the service size), and various system capacities. Based on the Table 1 and Figures 3, 4, and 5, the evaluation of mean system size can be summarized as follows: a. The percentage error in the mean decreases as the difference between mean arrival and service size becomes smaller. b. If the mean arrival size is smaller than the mean service size, the diffusion approximation seems to underestimate the true mean system size as indicated by a negative percentage error in the mean. c. If the mean service size is smaller than the mean arrival size, diffusion approximation seems to overestimate the true mean system size as indicated by a positive percentage error in the mean. d. For a fixed mean service size, the percentage error in the mean decreases as the difference between mean arrival and service size decreases. 105 e. For a fixed mean arrival size, the percentage error in the mean decreses as the difference between arrival and service size decreases. f. In lighter traffic cases, the ratio of convergence of mean percentage error to the system size is higher. g. As system capacity increases to infinity, maximum percentage error in the mean is within 5 % and higher traffic gives smaller percentage error in the mean. Based on Table 1 and Figures 6, 7, the evaluation of variance of system size is summarized as follows. 1) In low and moderate traffic intensity cases a. Percentage error in the variance shows consistent behavior depending on the combination of arrival and service size. b. If arrival size and service size are close, the percentage error in the variance monotonically decreases. c. If mean arrival size is less than mean service size, then variance is overestimated in the beginning and then gradually unde res t ima ted. 2) in heavy traffic cases a. Variance is overestimated for any combination of arrival and service size. b. As the mean arrival and service sizes increase, the percentage error in the variance increases. 106 (.75, 1.0) (0.5, .75) 20 (0.5, 1.0) (.75, .75) (0.5, 0.5) (1.0, .75) (.75, 0.5) -20 c Figure 3. Percentage error of mean system size, traffic intensity = 0.5, MX/MY/l/k 10 20 10 Figure 4. Percentage error of mean system size, traffic intensity = 0.7, MX/M*/l/k 10 Figure 5. Percentage error of mean system size, traffic intensity = 0.9, MX/MY/l/k 20 H -20 Figure 6. Percentage error of variance of system size, traffic intensity = 0.5, MX/M*/l/k 108 20 H AO 20 Figure 7. Percentage error of variance of system size, traffic intensity = 0.7, MX/My/l/k 109 v v Table 1. Comparison of mean and variance of system size (M /M /1/k) Pj = parameter of the Geometric distribution of arrival size p2 = parameter of the Geometric distribution of service size p Pi P2 k exact diffusion % errors mean var. mean var. mean var. 0.9 1.0 1.0 20 6.14 27.53 6.42 29.18 4.63 5.97 15 5.32 18.39 5.36 18.54 0.79 0.82 10 3.97 9.36 3.97 9.36 0.02 0.04 7 2.95 5.07 2.95 5.07 0.01 0.04 5 2.19 2.86 2.19 2.86 0.01 0.04 3 1.37 1.24 1.37 1.24 0.00 0.04 2 0.93 0.66 0.93 0.66 0.00 0.03 20 7.09 31.86 7.04 33.14 - 0.69 4.01 15 5.92 20.16 5.69 20.51 - 3.84 1.71 10 4.31 10.03 4.07 10. 24 - 5.53 2.63 7 3.19 5.41 2.97 5.65 - 6.90 4.56 5 2.37 3.05 2.18 3.27 - 8.32 7.35 3 1.50 1.32 1.34 1.49 -10.97 13.47 2 0.93 0.66 0.93 0.66 0.00 0.03 25 9.45 52.05 9.08 55.45 - 3.95 6.53 20 8.27 36.58 7.60 38.09 - 8.06 4.14 15 6.67 22.27 5.95 23.45 -10.72 5.29 10 4.78 10.93 4.13 12.00 -13.69 9.84 7 3.54 5.87 2.96 6.81 -16.41 16.01 5 2.65 3.29 2.16 4.05 -18.61 23 .07 4 2.18 2.25 1.76 2.86 -19.22 27.46 3 1.70 1.38 1.39 1.81 -18.24 31 .08 30 11.17 75.37 10.99 79. 18 - 1.67 5.06 25 9.96 56.26 9.55 58.34 - 4.18 3.69 20 8.43 38.38 7.95 40.04 - 5.68 4.31 10 4.70 11.42 4.29 12.78 - 8.78 11.93 7 3.43 6.69 3.07 7.32 -10.55 18.58 5 2.53 3.48 2.23 4.37 -11.99 25.58 4 2.07 2.39 1.82 3.09 -12.25 29.36 3 1.59 1.48 1.41 1.94 -11.16 31.29 110 (Table 1 continued) k exact diffusion % errors mean var. mean var. mean var. 0.9 .75 .75 25 8.76 50.23 8.93 51.97 1.90 3.46 20 7.58 35.11 7.61 35.95 0.33 2.39 15 6.07 21.42 6.07 22.12 0.01 3.28 10 4.30 10.56 4.29 11.18 0.00 5.96 7 3.11 5.70 3.11 6.24 - 0.03 9.57 5 2.27 3.22 2.27 3.67 - 0.35 14.06 4 1.84 2.22 1.82 2.61 - 1.00 17.44 3 1. 39 1.39 1.36 1.68 - 2.39 21.29 0.9 .75 1.0 25 7.88 45.37 8. 34 47.22 5.80 4.07 20 6.96 32.62 7.25 33.23 4.04 1.85 15 5.66 20. 19 5.91 20.64 4. 34 2.20 10 4.04 9.98 4.27 10.39 5.91 4.06 7 2.92 5.37 3.14 5.71 7.48 6.41 5 2. 12 3.02 2.30 3.29 8.90 9.19 4 1.70 2.08 1.86 2.31 9.63 11.08 3 1.27 1.29 1.40 1.46 10.01 12.97 0.9 0.5 0.5 40 14.54 133.87 14.79 139.38 1.73 4.12 30 11.87 82.92 11.89 86.23 0.14 3.99 20 8.48 40.77 8.48 43.70 0.00 7. 17 15 6.57 24.61 6.57 27.25 - 0.01 10.74 10 4.52 12.16 4.51 14.32 - 0.18 17.70 7 3. 22 6.60 3.19 8.26 - 0.93 25. 12 5 2.32 3.73 2.27 4.88 - 2.21 30.89 3 1.41 1.58 1.37 2.06 - 3.10 30.82 0.9 0.5 .75 35 12.09 99.01 12.71 102.34 5. 16 3.36 30 10.90 77.12 11.43 79.60 4.78 3.20 25 9.49 56.51 9.98 58.78 5. 19 4.01 20 7.89 38.26 8.37 40.42 6.15 5.66 15 6. 11 23.04 6.58 25.00 7.56 8.54 10 4.18 11.30 4.58 12.90 9.64 14.22 7 2.94 6.08 3.26 7.32 11.11 20.46 5 2.08 3.41 2.32 4. 31 11.36 26.26 4 1.65 2.34 1.82 3.02 10.36 29. 18 3 1. 22 1.44 1.31 1 .88 7.66 30.41 Ill (Table 1 continued) p Pl p2 k exact diffusion % errors mean var. mean var. mean var. 0.9 0.5 1.0 30 10.28 73.22 11.03 75.34 7.40 2.89 25 9.01 54.04 9.74 56.00 8.13 3.63 20 7.52 36.70 8.25 38.65 9.74 5.30 15 5.84 22.08 6.56 23.88 12.23 8. 15 10 3.98 10.76 4.62 11.23 16.14 13.68 7 2.77 5.74 3.32 6.87 19.54 19.82 5 1.94 3.19 2.37 4.01 21.93 25.61 4 1.52 2.17 1 .87 2.79 22.61 28. 56 3 1.11 1.33 1.35 1.72 21.92 30. 12 0.7 1.0 1.0 00 2.33 7.78 2.35 7.94 0.90 2.03 15 2.28 6.92 2.30 7.03 0.79 1.63 10 2.11 5.29 2.12 5.35 0.59 1.13 8 1.95 4. 23 1.96 4.27 0.47 0.92 7 1.84 3.62 1.85 3.65 0.40 0.82 6 1.71 2 .99 1.71 3.01 0.33 0.73 5 1.53 2.34 1.54 2.35 0.26 0.65 4 1.32 1.71 1.33 1.72 0.19 0. 57 3 1.07 1.13 1.07 1.13 0. 11 0. 50 2 0.77 0.63 0.77 0.63 0.05 0.38 00 3.11 12.78 3.13 15.02 0.48 17.50 15 2.93 9.87 2.87 10.82 - 2.02 9.61 10 2.60 6.76 2.48 7.19 - 4.59 6.32 7 2.19 4.32 2.04 4.56 - 6.85 5.56 5 1.78 2.68 1.62 2.85 - 8.79 6.57 4 1.52 1 .92 1.37 2 .07 -10.06 7.98 3 1.22 1.25 1.08 1.37 -11.62 10.39 0.7 1.0 0.5 CO 4.67 26.44 4.68 35.93 0. 19 35.88 20 4.34 19.25 4.09 22.52 - 5.72 17.02 15 3.99 14.64 3.63 16.48 - 9.24 12.54 7 2.69 5.23 2.22 5.94 -17.47 13.67 5 2.14 3.11 1.71 3.68 -20.04 18.32 4 1.82 2.18 1.44 2.66 -20.80 21.78 3 1.46 1.38 1.67 1.73 -19.96 25.23 112 (Table 1 continued) ? Pi P2 k exact diffusion % errors mean var. mean var. mean var. 0.7 .75 0.5 o» 5.44 36.67 5.45 46.95 0. 17 21.43 20 4.78 23.56 4.57 26.00 - 4.44 10. 36 15 4.26 16.93 3.98 18.58 - 6.51 9.73 10 3.41 9.67 3.09 10.84 - 9.19 12.07 7 2.67 5.62 2.37 6.57 -11.27 16.82 5 2.07 3.30 1 .81 4.05 -12.78 22.50 4 1.73 2.31 1.50 2.91 -13.07 25.85 3 1.36 1.45 1.20 1 .86 -11.97 27.97 0.7 .75 .75 00 3.89 21.59 3.90 21.39 0. 31 3.71 20 3.67 16.82 3.68 17.52 0.26 4.17 15 3.37 13.10 3.40 13.75 0. 19 4.97 10 2.83 8.06 2.83 8.47 0.12 7.25 7 2.26 4.85 2.26 5.36 0.06 10.53 5 1.76 2.91 1.76 3.33 - 0.21 14.54 4 1.47 2.05 1.46 2.41 - 0.80 17.35 3 1.31 1.15 1. 12 1.57 - 2.18 20.33 0.7 .75 1.0 00 3.11 14.86 3.13 13.57 0. 15 - 8.69 15 2.86 10.65 2.94 10.48 2 .87 - 1 . 6 0 10 2.46 6.97 2.59 7.21 5.41 3.42 7 2.00 4.32 2.16 4.65 7.77 7.69 5 1.57 2.62 1.72 2.92 9.74 11.42 4 1.31 1.86 1.45 1.11 10.65 13.51 3 1.02 1.19 1.13 1.37 11.05 15.25 0.7 0.5 0.5 00 7.00 70.00 7.01 72.96 0. 1 4.23 20 5.36 29.71 5.37 31.33 0.04 8.82 15 4.55 19.99 4.56 22.41 0.03 12.10 10 3.43 10.80 3.42 12.80 - 0.11 18.59 7 2.57 6.10 2.55 7.64 - 0.76 25.21 5 1.92 3.53 1.88 4.59 - 2.04 30.13 4 1.57 2.44 1.52 3.21 - 2.84 31 .38 3 1.20 1.52 1.16 1 .97 - 3.19 29.68 .75 00 5.43 45.93 5.45 41.68 0.49 - 9.25 20 4.54 24.79 4.76 25.79 4.94 4.01 15 3.93 17.26 4.21 18.82 7. 22 9.04 10 3.00 9.54 3.31 11.11 10.26 16.55 7 2.25 5.42 2.52 6.68 12.19 23.30 5 1.66 3. 12 1.87 4.02 12.55 28.69 4 1.34 2.16 1.50 2.82 11.47 30.91 3 1.01 1.34 1 .09 1.75 8.47 31 .09 113 (Table 1 continued) P Pi Po k exact diffusion % errors mean var. mean var. mean var. 0.7 0.5 co 4.67 35.78 4.68 29.41 0.24 -17.81 20 4.05 21.85 4.33 21.60 6.72 - 1.12 15 3.56 15.60 3.94 16.55 10.75 6.12 10 2.75 8.76 3.21 10.16 16.68 15.98 7 2.06 4.98 2.50 6. 18 21.21 23.94 5 1.51 2.86 1.88 3.71 24.00 30.05 4 1.21 1.96 1.51 2.60 24.63 32.65 3 0.90 1.20 1.11 1.61 23.61 33.37 0.5 1.0 00 1.00 2.00 1.03 2.14 2.80 7.00 9 0.99 1 .90 1.02 2.02 2.54 6.05 7 0.97 1.75 0.99 1.84 2.25 5.17 6 0.94 1.61 0.96 1.68 2.01 4.58 5 0.90 1.42 0.92 1.48 1.69 3.91 4 0.84 1.17 0.85 1.20 1.30 3.20 3 0.73 0.86 0.74 0.88 0.85 2.45 2 0.77 0.63 0.77 0.63 0.05 0.38 0.5 1.0 CO 1.33 3.11 1.35 4.14 1.43 32.90 9 1.30 2.76 1.28 3. 35 - 1.57 21.35 7 1.25 2.41 1.20 2.81 - 3.81 16.59 6 1.20 2.15 1.14 2.46 - 5.29 14.26 5 1.13 1.83 1.05 2.05 - 7.03 12.26 4 1.03 1.44 0.94 1.60 - 9.04 10.83 3 0.89 1.03 0.79 1.13 -11.32 10.32 2 0.68 0.61 0.60 0.67 -12.30 10.71 0.5 1.0 00 2.00 6.00 2.01 10. 13 0.6 68.85 15 1 .98 5.67 1.91 8.40 - 3.36 48.03 10 1.90 4.75 1.72 6.26 - 9.40 31.82 8 1.80 4.01 1.57 5.05 -13.06 25.79 7 1.73 3.54 1.47 4.37 -15.14 23.38 6 1.63 3.02 1.35 3.67 -17.31 21.61 5 1.51 2.44 1.21 2.95 -19.43 20.69 4 1.34 1.84 1.06 2.22 -21.06 20.79 3 1.13 1.24 0.89 1.51 -21.02 21.66 0.5 .75 00 2.33 9.33 2.34 13.02 0.47 39.54 15 2.26 8.06 2.19 10.22 - 3.14 26.79 10 2.07 6.08 1.93 7.38 - 7.07 21.36 8 1.91 4.87 1.74 5.87 - 9.19 20.57 7 1.80 4.19 1.62 5.06 -10.33 20.80 6 1.67 3.47 1.48 4.21 -11.51 21.58 5 1.50 2.73 1.31 3.36 -12.58 22.95 4 1.30 2.00 1. 13 2.50 -13.20 24.75 3 1.06 1.32 0.93 1.66 -12.38 25.89 114 (Table 1 continued) f Pi k exact diffusion % errors mean var. mean var. mean var. 0.5 .75 00 1.67 5.56 1.68 6.14 0.96 10.51 10 1.56 4. 35 1.58 4.86 0.73 11.60 8 1.47 3.66 1.48 4.13 0.60 12.80 7 1.41 3.22 1.41 3.67 0.51 13.77 6 1.31 2.73 1.32 3.15 0.39 15.09 5 1.20 2.20 1.20 2. 57 0.17 16.79 4 1.05 1.65 1.04 1.96 - 0.39 18.80 3 0.86 1.11 0.84 1.34 - 1 .79 20.54 0.5 .75 1.0 00 1.33 4.00 1.36 3.70 1.73 - 7.50 10 1.28 3.41 1.33 3.38 3.63 - 0.70 8 1.22 2.96 1.29 3 .06 5.23 3.42 7 1.18 2.65 1.25 2.81 6.34 6.02 6 1.11 2.28 1.19 2.49 7.66 8.96 5 1.02 1.87 1.11 2.10 9.14 12. 11 4 0.89 1.42 0.99 1.64 10.57 15.19 3 0.73 0.96 0.81 1.13 11.40 17.36 0.5 0.5 00 3.00 18.00 3.01 20. 14 0.30 11.88 15 2.68 12.09 2.68 14.04 0.19 16.09 10 2.26 7.87 2.26 9.55 0.05 21.39 7 1.83 4.91 1.82 6.23 - 0.52 26.82 6 1.64 3.94 1.63 5.07 - 1.03 28.81 5 1.43 3 .00 1.41 3.92 - 1 .78 30.45 4 1.20 2.13 1.17 2.79 - 2.68 31.00 3 0.94 1.35 0.91 1.74 - 3.21 28.86 0.5 0.5 CO 2.33 12.44 2.35 11.48 0.56 - 7.76 15 2. 16 9.44 2.26 9.90 4.31 4.91 10 1.87 6.45 2.03 7.47 8.43 15.85 7 1.53 4.11 1.70 5. 13 11.50 24.64 5 1.19 2.53 1.34 3.30 12.57 30.46 4 0.99 1.80 1.11 2.38 11 .74 32.27 3 0.76 1.14 0.83 1.49 8.74 31.49 00 2.00 10.00 2.02 8.15 0.85 -18.50 15 1.89 8.03 1.99 7.63 5.30 - 4 . 9 9 10 1.65 5.65 1.86 6.23 12.28 10.34 7 1.36 3.65 1.61 4.49 18.85 23.09 5 1.06 2.25 1.30 2.96 23.18 31.59 4 0.87 1.59 v 1.06 2.15 24.40 34.69 3 0.66 1.00 0.82 1.35 23.72 35.18 115 5.5 Comparison with simulation estimates for multiple server cases 5.5.1 Accuracy of Simulation Estimates In cases of multiple server finite capacity bulk queues, exact solutions are not available . Hence simulation estimates will be used. To increase the dependability of the simulation estimates, a variance reduction technique called "antithetic variate technique" is employed. (See Kleijnen [46] for more detailed discussions of variance reduction techniques.) To eliminate the initial experimental bias, transient period was taken care of by collecting data long time after the system started. In the actual simulation runs, data collected for the first 5,000 time units are deleted and the random number seeds at this moment are stored. Then, the data are collected for the next 10,000 time units. Next, using the random number seeds already stored, antithetic variates of interarrival and service times are generated for the 10,000 time units again. In this way, we have two different streams of data sets. Then those two data sets are sent to the "batch means procedure" which is specially designed by Dr. Gordon M. Clark of the department of the Industrial and Systems Engineering at the Ohio State University. This batch means procedure is a subroutine designed to compute the batch means. This antithetic variate technique turned out to reduce the sample 116 variance by up to 90 %. In simulating a stochastic system, it is very important to take a good care of the transient period. A transient period is the part of a random process before the process is stabilized. when estimating the steady-state value of a simulated system performance measures, the starting conditions can be a major source of error. For example, when one simulates a queueing system one usually starts with an empty system. This artificially introduced initial condition can affect the length of the transient period. In this case, we say that "the simulation output is contaminated by an initialization bias" [65]. Schruben [65] provides an initialization bias test based on the asymptotic convergence of a standardized sequence of samples to a Brownian Bridge process. Even though the Shruben's test does not assure that the system reaches a steady-state, the test can be used to support the arguement that the initialization bias is adequately controlled. To test the accuracy of the simulation, the simulation estimates Y V for the M /M /1/k queue will be compared with those of exact results. As can be seen in the table 2, the simulation gives very exact estimates. The maximum error is around 1 % for the mean system size, and around 2.5 % for the variance of system size. Hence it can be said that the simulation provides very accurate estimates of the mean and variance of system size of bulk queueing systems. 117 X Y Table 2. Comparison of simulation estimates with exact ones (M /M /1/k) Pj = parameter of the Geometric distribution of arrival size P 2 = parameter of the Geometric distribution of service size k simu. exact % error mean var. mean var. mean var. 40 14.59 134.66 14.54 133.87 0. 31 0.59 30 11 .87 81.23 11.87 82.92 -0.01 -2 .04 20 8.44 40.56 8.48 40.56 -0. 36 -0.53 15 6.62 24.76 6.57 24.61 0.80 0.63 10 4.53 12.29 4.52 12.16 0.37 1.01 7 3.22 6.65 3.22 6.60 0. 13 0.74 5 2.33 3.74 2.32 3.73 0.40 0.21 3 1.41 1.59 1.41 1.58 0.19 0.58 0.5 .5 .5 15 2.65 12.08 2.68 12.09 -1.06 -0. 12 10 2.27 8.00 2.26 7.87 0.40 1.68 7 1.83 4.93 1.83 4.91 0.31 0.41 6 1.66 4.04 1.64 3.94 0.94 2.53 5 1.44 3.02 1.43 3.00 0.41 0.48 4 1.20 2.15 1.20 2. 13 0.42 1.15 3 0.94 1.36 0.94 1.35 0.47 0.60 0.9 .75 0.5 25 10.01 56.96 9.96 56.26 0.48 1.23 20 8.44 39.14 8.43 38.38 0. 15 1.97 10 4.68 11.50 4.70 11.42 -0.53 0.74 7 3.42 6.20 3.43 6. 17 -0.23 0.46 5 2.53 3.53 2.53 3.48 -0.12 1.46 4 2.07 2.40 2.07 2.39 -0.0.2 0.36 3 1.59 1.49 1.59 1.48 -0.19 0.72 0.5 .75 .5 15 2.25 8. 19 2.26 8.06 -0.44 1.58 10 2.06 6.26 2.07 6.08 -0.48 2.90 8 1.91 4.99 1.91 4.87 -0.31 2.40 7 1.79 4. 24 1.80 4. 19 -0.50 1.32 6 1.67 3.55 1.67 3.47 -0. 12 2.50 5 1.51 2.77 1.50 2.73 0.40 1.39 4 1.31 2.04 1.30 2.00 0.69 1.77 3 1.07 1. 32 1 .06 1. 32 0.27 0.19 118 k simui. exact % error mean var. mean var. mean var. 30 11.01 75.46 10.91 77.12 1.00 -2.16 20 9.58 56.27 9.49 56.51 0.98 -0.43 20 7.92 37.40 7.89 38.26 0.41 -3.04 15 6. 13 22.98 6. 11 23 .04 0.31 -0.26 10 4.22 11.34 4.18 11.30 0.98 0.38 7 2.96 6. 13 2.94 6.08 0.78 0.78 5 2.10 3.45 2.08 3.41 0.77 0.95 4 1.66 2. 35 1.65 2. 34 0.55 0.47 3 1.23 1.45 1.22 1.44 1.07 0.58 0.5 .5 .75 15 2.14 9.53 2. 16 9.44 -0.93 0.96 10 1.87 6.61 1.87 6.45 0.27 2.58 7 1.53 4.18 1.53 4. 11 0.39 1.53 5 1.20 2.55 1.19 2.53 0.76 0.45 4 0.99 1.81 0.99 1 .80 0.30 0.78 3 0.77 1.14 0.76 1.14 0.52 0.76 5-5.2 comparison of MX/MY/2/k queues. Diffusion approximation will be compared with simulation estimates for three different traffic intensities (.5, .7, and 0.9) and three different parameter combinations ( (p^, P 2 > = (.5, .5), (.5, 1.), and (1., .5)). Based on Table 3 and Figures 8, 9, and 10, the evaluation of mean system size can be summarized as follows: a. The percentage difference in the mean decreases as the difference between mean arrival and service size becomes smaller. b. If mean arrival size is smaller than the mean service size, diffusion approximation seems to underestimate the true mean. 119 c. if mean service size is smaller than mean arrival size, diffusion approximation seems to overestimate in the beginning and then gradually underestimate the simulation mean. d. As system capacity increases to infinity, percentage difference in the mean gradually decreases. e. The overall behavior of mean error is very similar to the case where comparison of MX/MY/l/k queues were made. Based on Table 3 and Figures 11, and 12, evaluation of variance of system size can be summarized as follows: a. percentage difference in the variance shows consistent behaviour depending upon combination of arrival size and service size. b. if mean arrival size is smaller than mean service size, percentage difference in the variance first decreases and then increases, but diffusion approximation constantly overestimates the simulation variance. c. if mean arrival size is larger than mean service size, diffusion approximation overestimates the variance first and then underestimates it. d. if mean arrival size is equal to mean service size, then variance consistently decreases but diffusion approximation seems to overestimate the true variance. e. Based upon the observations above, the percentage difference in the variance behaves similarly as the percentage error in the variance of MX/MY/l/k queues. Figure 8. Percentage difference of mean system size, traffic intensity-«= 0.5, M /M /2/k (0.5,1.0) 20 (0.5,0.5) 1.0,0.5) Figure 9. Percentage difference of mean system size, traffic intensity = 0.7, MX/HY/2/k 121 10 k 30 (0.5,0.5) -10 Figure 10. Percentage difference of mean system size, traffic intensity = 0.9, MX/MY/2/k 0,0.5) (0.5,0.5) 20 (0.5,1 -30 • Figure 11. Percentage difference of variance of system size, traffic intensity =0.5, MX/MY/2/k 122 30 20 10 (0.5,0.5) (0.5,1.0) -10 Figure 12. Percentage difference of variance of system size, traffic intensity = 0.7, MX/M*/2/k 123 X Y Table 3. Comparison of mean and variance of system size (M /M /2/k) Pj = parameter of the Geometric distribution of arrival size P 2 = parameter of the Geometric distribution of service size diffusion % difference p Pi P2 k simu. mean var. mean var. mean var. 0.9 1.0 0.5 30 10.98 70.07 10.39 74.96 - 5.40 6.98 20 8.35 35.95 7.60 38.10 - 8.97 5.96 10 4.83 10.95 4.13 12.00 -14.48 9.59 5 2.66 3.26 2.15 4.05 -18.98 23 .93 0. 5 1.0 0.5 10 2.12 6.31 2.01 10.13 - 5.00 60.51 7 1.82 3.62 1.47 4.37 -19.46 20.69 5 1.59 2.47 1 .21 2.95 -23.44 19.29 3 1.19 1.25 1.39 1.51 -16.52 21. 33 0.7 1.0 0.5 15 4.08 14.71 3.63 16.48 -11.22 12.03 10 3.41 9.08 2.86 9.77 -15.97 7.56 5 2.19 3.15 1 .71 3.68 -21.96 16.81 3 1.49 1.37 1.17 1.73 -21.95 26.22 0.9 0.5 1.0 30 10.66 73.13 11.03 75.34 3.50 3.02 20 7.78 35.96 8.25 38.65 6.03 7.48 10 4.60 10.56 4.62 12.23 2.66 15.72 5 2 .09 3.18 2.37 4.01 11.05 25.95 0. 5 0.5 1.0 10 1.93 6. 10 1.86 6.23 - 4.09 2.18 7 1.62 3 .99 1.61 4.49 - 0.33 12.64 5 1.28 2.45 1.30 2.96 1.83 20.70 3 0.83 1.13 0.82 1.35 - 1.59 19.29 0.7 0.5 1.0 15 3.85 15.85 3.94 16.55 2.46 4.42 10 3.01 8.84 3.21 10.16 6.56 14.90 5 1.70 2.93 1.88 3.71 10. 12 26.78 3 1.04 1.27 1.11 1.61 6.49 26.77 0.9 0.5 0.5 35 13.50 108.98 13.40 111.67 - 0.70 2.46 20 8.60 40.21 8.48 43.70 - 1.50 8.67 10 4.61 12.10 4.51 14.32 - 2.23 18.34 5 2.40 3.67 2.27 4.88 - 5.23 32.85 0.5 0.5 0.5 10 2.47 8. 20 2.26 9.55 - 8.34 16.48 7 2.02 5. 11 1 .82 6.23 - 9.80 21.94 5 1.59 3.11 1.41 3.92 -11.63 25.80 3 1.06 1.41 0.91 1.74 -14.74 23.68 124 p Pl p2 k simu. diffusion % difference mean var. mean var. mean var. 0.7 0.5 0.5 15 4.73 19.95 4.56 22.41 - 3.75 12.33 10 3.58 10.82 3.42 12.80 - 4.27 18.37 5 2.02 3.52 1.88 4.59 - 7.09 30.53 3 1.29 1.52 1.16 1.97 -10.10 28.71 5.5.3 Comparison of MX/GY/2/k queues It was mentioned earlier that the diffusion approximation needs the first and second moments of the interarrival and service time distributions. Also even if the true distributions are different, same diffusion approximate solutions are obtained as long as their variabilities (coefficients of variation) are the same. This may not be true for the true system size distribution and performance measures. To see the distributional effects, three different service time distributions will be used (Erlang of order 4, Lognormal and Uniform distributions). But parameters of the Lognormal and the Uniform distributions will be selected such that they have the same variabilities as the Erlang distribution. Based on Table 4, evaluation of mean and variance of system size can be summarized as follows: a. if mean arrival size is less than mean service size, diffusion approximation underestimates the simulation estimates of mean system size. b. if mean arrival size is equal to mean service size, percentage difference in the mean is getting smaller c. no regular pattern can be found about the percentage difference in the variance 125 Table 4 Comparison of mean and variance of system size (MX/GY/2A) P j = parameter of the Geometric distribution of arrival size p2 = parameter of the Geometric distribution of serivce size E = Erlang of order 4 as the service time distribution L = Lognormal as the service time distribution U = Uniform as the service time distribution f Pi k simu. diffusion % difference mean var. mean var. mean var. 0.9 1.0 30 E 9. 59 62. 16 8.68 58.23 - 9.45 - 6.32 L 9.76 64.50 -11.02 - 9.71 U 9.57 61.88 - 9.31 - 5.89 20 E 7.66 31.64 6.79 31.30 -11.38 - 1 .05 L 7.57 30.93 -10.40 1 .22 U 7.57 31.95 -10.36 - 2.01 10 E 4.66 9.42 3.91 9.91 -16.01 5.19 L 4.69 9.36 -16.52 5.87 U 4.70 9. 45 -16.68 4.87 5 E 2.63 2 .89 2.08 3.39 -21.10 17.61 L 2.64 2.87 -21.21 18.28 U 2.64 2.91 -21.31 16.70 0.5 10 E 1 .73 3. 17 1.23 3.17 -28.72 - 0. 15 L 1.73 3.13 -28.58 1.28 U 1.72 3.11 -28.09 1.72 7 E 1.65 2.73 1.15 2.60 -30.53 - 4.93 L 1.67 2.72 -30.07 - 4.21 U 1.66 2.76 -30.84 - 5.74 5 E 1.51 2.05 1.02 1.99 -32.48 - 3.16 L 1.52 2.05 -33.00 - 3.22 U 1.52 2.09 -32.92 - 4.99 3 E 1.19 1.15 0.80 1.20 -32.69 4.96 L 1.20 1.15 -32.91 4.99 U 1.19 1.16 -32.42 3.86 126 (Table 4 continued) k simu. diffusion % difference p Pi p2 mean var. mean var. mean var. 0.7 1.0 0.5 10 E 2.93 2.72 2. 37 2.68 -19.01 - 0.60 L 3.03 2.68 -21.52 - 0.05 U 3.02 6.64 -21.46 - 0.60 7 E 2.57 4.26 1.96 4.37 -23.80 2.59 L 2.58 4. 29 -24.22 1.76 U 2.58 4.29 -24.18 1.80 5 E 2. 12 2.67 1.56 2.85 -26.36 6.65 L 2.10 2.62 -25.63 8.92 U 2. 12 2.68 -26.11 6.47 3 E 1.48 1.26 1.10 1.47 -26.06 17.16 L 1.48 1.25 -25.92 17.97 U 1.48 1.27 -26.00 16.16 0.9 0.5 1.0 30 E 10. 18 68.80 10. 30 67.89 1.13 - 1 .32 L 10.18 67.28 1.13 - 0.92 U 10. 20 68.61 0.92 - 1.04 20 E 7.64 33.68 7.92 35.60 3.76 5.69 L 7.64 34.31 3.71 3.76 U 7.66 33.26 3.44 7.02 10 E 5.88 9.00 4.56 11.29 -22.36 25.53 L 4.17 9.65 9.38 17.07 U 4. 17 9.67 9.46 16.86 5 E 2.60 2.84 2.37 3.73 - 8.72 31.27 L 2.10 2.89 12.99 13.53 U 2.09 2.91 13.08 27.93 0.5 0.5 1.0 10 E 1.96 5.79 1.59 4.61 -18.75 -20.33 L 1.95 5.74 -18.34 -14.70 U 1.96 5.77 -18.49 -20.09 7 E 1 .64 3 .82 1.45 3.61 -11.83 - 5.54 L 1.64 3.77 -11.81 - 4.18 U 1 .64 3.77 -11.63 - 4.19 5 E 1.32 2.40 1.22 2.51 - 7.60 6.26 L 1.32 2.39 - 7.78 4.78 U 1.32 2.40 - 7.70 4.75 3 E 0.88 1.13 0.79 1.21 - 9.30 10.59 L 0.87 1. 12 - 9.14 8.41 U 0.87 1.13 - 9.02 7.29 127 (Table 4 continued) simu. diffusion % difference p Pi P2 k mean var. mean var. mean var. 0.7 0.5 1.0 10 G 2.98 8. 14 2.98 8.70 0.01 6.92 L 2.97 8.03 0.19 8.36 U 2.98 8.09 0. 15 7.53 7 E 2.30 4.69 2.40 5.48 4.45 16.90 L 2.29 4.67 4.59 17.26 U 2.28 4.64 5.29 18.22 5 E 1.72 2.74 1.83 3.37 6.35 22.95 L 1.73 2.74 5.85 22.76 U 1.72 2.75 6.35 22.30 3 E 1 .08 1.22 1.10 1.50 1.88 22.57 L 1.08 1.23 1.62 22.12 U 1 .07 1.23 2.54 21.85 0.9 0.5 0.5 35 E 12.75 96.56 12.45 98.76 - 2.36 2.28 L 12 .80 97.95 - 2.76 0.84 U 12.74 96.44 - 2.30 2.41 20 E 8.37 36.71 8.18 39.10 - 2.33 6.50 L 8.39 36.89 - 2.51 5.99 U 8.40 37.10 - 2.62 5. 39 10 E 4.56 10.82 4.44 12.83 - 2.53 18.64 L 4.56 10.74 - 1.93 19.49 U 4.57 10.90 - 2.83 17.70 5 E 2.39 3.41 2.25 4.56 - 5.70 34.53 L 2.38 3.39 - 5.43 34.31 U 2.38 3.43 - 5.44 32.85 0.5 0.5 0.5 10 E 2.44 7.40 1.97 7. 17 -19.33 - 3.01 L 2.44 7.32 -19.31 - 2.04 U . 2.45 7.48 -19.37 - 4.06 7 E 2.03 4.78 1.66 5.08 -17.94 6.34 L 2.03 4.71 -18.15 7.82 U 2.02 4.81 -17.65 5.74 5 E 1.62 2.97 1.32 3.40 -18.51 14.40 L 1.62 2.96 -18.37 14.90 U 1.62 2.97 -18.17 14.48 3 E 1.11 1.40 0.87 1.63 -21.31 16.84 L 1.11 1.40 -21.56 16.77 U 1.11 1.40 -21.28 37.42 128 (Table 4 continued) p Pj P2 k simu. diffusion % difference mean var. mean var. mean var. 0.7 0.5 0.5 15 E 4.49 17.52 4. 10 18.01 - 8.90 2.79 L 4.52 17.38 - 9.41 3.59 U 4.50 17. 31 - 8.88 4.04 10 E 3.49 9.63 3.22 10.85 - 7.84 12.70 L 3.51 9.64 - 8.25 12.60 U 3.50 9.74 - 7.94 11.52 5 E 2.02 3.29 1.83 4. 18 - 9.54 26.92 L 2.03 3.28 - 9.70 27.44 U 2.02 3. 31 - 9.33 26.20 3 E 1.31 1.47 1.14 1.89 -13.56 28.35 L 1.32 1.47 -13.90 28.44 U 1.31 1.48 -13.47 27.38 5.5.4 Comparison of GIX/MY/2/k queue In this case, the Erlang distribution of order 4, the Lognormal and Uniform distributions with same mean and variances will be used. Based on Table 5, evaluation can be summarized as follows: 1. if mean arrival size is smaller than mean service size, diffusion approximation seems to underestimate the true mean. 2. if mean arrival size is greater than mean service size, diffusion approximation seems to underestimate the true mean in low traffic but increasingly overestimate it in heavy traffic. 3. diffusion approximation seems to overestimate the variance for all cases. 129 Table 5. Comparison of mean and variance of system size. (GI X /M Y/2/k) Pj = parameter of the Geometric distribution of arrival size P 2 = parameter of the Geometric distribution of serivce size E = Erlang of order 4 as interarrival time distribution L = Lognormal as interarrival time distribution U = Uniform as interarrival time distribution p Pj P 2 k simu. diffusion % difference mean var. mean var. mean var. 30 E 10.56 64. 55 9.78 68.75 - 7.36 6.50 L 10.45 64.98 - 6.35 5.79 U 10.68 63.33 - 7.68 8.56 20 E 8.25 34.39 7.32 35.43 -11.24 3.02 L 8.23 34.83 -11.04 1.72 U 8.27 33.94 -11.48 4.38 10 E 4.91 10. 32 4.05 11.15 -17.41 8.00 L 4.89 10.32 -17.09 8.05 U 4.90 10.28 -17.28 8.51 5 E 2.81 2.98 2.13 3.79 -24.24 27.48 L 2.80 2.98 -23.98 27.28 U 2.80 2.98 -24.09 27.26 10 E 1.91 3.59 1.62 5.55 -15.17 54.87 L 1.87 3.44 -13.69 61.44 U 1.90 3.61 -15.06 53.73 7 E 1.79 2.89 1.40 4.00 -21.57 38.43 L 1.78 2.84 -20.97 40.83 U 1.80 2.86 -21.80 39.54 5 E 1.64 2.09 1. 18 2.76 -28.10 31.92 L 1.63 2.09 -27.80 31.83 U 1.64 2.09 -28.23 31.73 3 E 1.32 1.11 0.88 1.46 -32.77 31. 12 L 1.31 1.12 -32.93 30.59 U 1.31 1.12 -32.93 29. 56 130 simu. diffusion % difference p Pi P2 k mean var. mean var. mean var. 0.7 1.0 0.5 15 E 3.67 10.79 3. 37 14.44 - 8.07 33.82 L 3.66 11.06 - 8.03 30.58 U 3.66 10.75 - 7.97 34.29 10 E 3.24 7.48 2.73 8.85 -15.66 18.22 L 3.22 7.42 -15.01 19.23 U 3.24 7.39 -25.42 17.73 5 E 2.25 2.76 1.67 3.44 -25.56 24.72 L 2.23 2.75 -25.02 25.11 U 2.24 2.75 -25.40 24.88 3 E 1.60 1.22 1.15 1.66 -28.18 35.73 L 1.60 1.22 -28.16 35.79 U 1.60 1.23 -28.15 35.33 0.9 0.5 1.0 30 E 9. 15 58. 10" 9.44 59.76 3.14 2.84 L 9. 13 56.73 3.38 5.33 U 9. 10 56.62 3.75 5.54 20 E 7.29 31.60 7.52 32.53 3.26 2.97 L 7.25 31.07 3.81 4.72 U 7.16 30.03 5.16 8.33 10 E 4.19 9. 36 4.48 10.43 6.92 11.38 L 4.17 9.31 7.43 12.04 U 4.15 9.22 8.11 13.06 5 E 2.24 2 .84 2.37 3.45 5.44 21.59 L 2.24 2.83 5.58 21.93 U 2.22 2.81 6.71 22.70 0.5 0.5 1.0 10 E 1.79 4.45 1.59 4.61 -11.03 3.50 L 1.77 4.36 - 9.76 5.71 U 1.78 4.54 -10.66 1.54 7 E 1.60 3.23 1.45 3.61 - 9.30 11.79 L 1.60 3.23 - 9.39 11.71 U 1.57 3.26 - 8.92 10.86 5 E 1.36 2. 19 1.22 2. 51 -10.60 14.61 L 1.35 2.22 -10.24 13.26 U 1.34 2.21 - 9.57 13.45 3 E 0.96 1.11 0.79 1.21 -17.39 9.42 L 0.96 1.09 -17.44 10.91 U 0.95 1.11 -16.22 9.17 131 p p 1 p2 k simu. diffusion % difference mean var. mean var. mean . var. 0.7 0.5 1.0 10 E 2.78 7. 18 2.86 7.99 2.63 11.41 L 2.79 7.15 2.35 11.86 U 2.78 6.91 2.83 15.76 7 E 2.27 4.31 2.34 5.15 2.81 19.47 L 2.28 4.32 2.40 9.04 U 2.26 4.28 3.66 20.10 5 E 1.79 2.61 1.81 3.20 0.76 22.37 L 1.79 2.60 0.98 22.79 U 1.78 2.57 1.65 24.18 3 E 1.18 1.20 1.09 1.45 - 7. 12 20.52 L 1.18 1.19 - 7.22 21.01 U 1.16 1.19 - 5 .97 20.95 0.9 0.5 0.5 35 E 12.70 96. 39 12.57 100.27 - 1.07 4.04 L 12.65 95.25 - 0.63 5.27 U 12.75 94.92 - 1.40 5.95 20 E 8.51 37.22 8.21 39.61 - 3.48 6.42 L 8.46 37.39 - 2.85 5.93 U 8.46 36.45 - 2.95 8.67 10 E 4.78 11.02 4.45 13.00 - 6.93 17.96 L 4.75 10.97 - 6.34 18.54 U 4.76 10.97 - 6.53 18.48 5 E 2.59 3.34 2.25 4.60 -13.04 37.01 L 2.60 3.35 -13.35 37.04 U 2.59 3.35 -12.91 37.21 0.5 0.5 0.5 10 E 2.36 6.48 2.14 8.46 - 9.67 30.48 L 2.36 6.44 - 9.66 31.38 U 2.36 6.45 - 9.39 30.48 7 E 2.04 4.37 1.75 5.72 -14.30 30.94 L 2.05 4.33 -14.55 32.16 U 2 .05 4.38 -14.56 30.47 5 E 1.71 2.84 1.37 3.69 -11.61 30.20 L 1.71 2.80 -19.66 31.79 U 1.70 2.81 -19.15 31.22 3 E 1.21 1.36 0.89 1.69 -26.13 24.33 L 1.21 1.35 -26.27 14.58 U 1.19 3.36 -25.37 24.97 132 p Pl p2 k simu diffusion % difference mean var. mean var. mean var. 0.7 0.5 0.5 15 E 4.43 16.60 4.26 19.48 - 3.80 17.39 L 4.41 16.35 - 3.39 23.48 U 4.40 16. 11 - 3.03 20.96 10 E 3.55 9.44 3.29 11.51 - 7.17 21.88 L 3.55 9.42 - 7.13 22.14 U 3.54 9.29 - 3.04 23.92 5 E 2.17 3.23 1.85 4.32 -15.02 34.00 L 2.19 3.23 -15.49 33.78 U 2.17 3.21 -14.96 34.78 3 E 1.44 1.43 1.14 1.92 -20.71 34.37 L 1.45 1.42 -20.97 34.65 U 1.43 1.4 2 _ -20.15 34.66 5.5.5 Comparison of GIX/GY/2/m queues In this section, mean and variance of system size in case of non- V V Exponential interarrival and service time distributions (E2 /E^ /2/m LN1X/LN2Y/2/m , and LNjX/UY/2/m ) will be compared for various traffic intensities. (LN and U denote Lognormal and Uniform distributions.) The Lognormal and Uniform distributions will have the same mean and variance with the Erlang distribution. It can be seen from the table 6 that difference between mean system sizes of E2X/E4Y/2/m , LN^x/LN2Y/2/m and LNjX/UY/2/m queues are minor. Also, the Uniform distribution is markedly different from the Erlang and the Lognormal distributions under the condition that the first two 133 moments are the same (Figure 2). This fact suggests that diffusion approximation yields robust estimation of mean system size regardless of exact distributional forms of interarrival and service time distributions, but further testing is needed to verify this conclusion. The evaluation can be summarized as follows: a. diffusion approximation underestimates the mean system size. b. as traffic intensity increases, percentage difference in the mean decreases and in most heavy traffic cases, percentage difference is around 5 %. c. most of the time, diffusion approximation underestimates the variance. d. as traffic intensity increases, percentage difference in the variance decreases. 134 y y Table 6. Comparison of mean and variance of system size (GI /G /2/») Pj = parameter of the Geometric distribution of arrival size P 2 = parameter of the Geometric distribution of service size E = E2X/E4Y/2/(B queue L = LN1X/LN2Y/2/b> queue U = LN1X/UY/2/« queue simu. diffusion % difference Pi P2 ? mean var. mean var. mean var. 0.5 0.5 0.3 E 1.50 4.28 0.92 3.21 -39.04 -24.97 L 1.48 3.91 -38.03 -17.98 U 1.47 3.91 -37.71 -17.82 0.4 E 2 .00 9. 23 1. 38 5.11 -30.92 -17.96 L 1.96 5.78 -29.49 -17.96 U 1 .95 5.79 -28.95 -13.28 0. 5 E 2.65 9.61 2.02 8. 15 -23.95 -15.18 L 2.58 8.80 -21.83 - 7.98 U 2.52 8.23 -20.08 - 1.08 0.6 E 3.49 13 .96 2.94 13.75 -15.79 - 1.01 L 3.43 12.55 -14.20 1.46 U 3.42 13.51 -14.02 1.77 0.7 E 5. 11 29.58 4.45 26.16 -13.08 -11.55 L 4.83 26.98 - 7.94 - 3.02 U 4.85 26.80 - 8.34 - 2.37 0.8 E 8.18 75.96 7.41 63.31 - 9.42 -16.66 L 7.72 72.53 - 4.05 -12.72 U 7.70 66.33 - 3.80 -16.66 0.75 .75 0.3 E 0.87 1.28 0.49 0.82 -43.51 -36.07 L 0.86 1.16 -43 .04 -29.39 U 0.85 1.17 -42.38 -29.73 0.4 E 1.14 1 .78 0.73 1.25 -36.59 -29.73 L 1.11 1.60 -34.80 -27.76 U 1.11 1.60 -34.39 -21.79 0. 5 E 1.47 2.57 1.04 1.93 -29.42 -24.61 L 1.43 2.30 -27.40 -16.07 U 1.42 2.29 -27.18 -15.62 0.6 E 1.93 4.04 1.48 3.18 -23.33 -21.12 L 1.86 3.58 -20.21 -10.94 U 1 .86 3.65 -20.27 -12.78 135 Pi P2 ? simu. diffusion % difference mean var mean var. mean var. 0.7 E 2.67 7.31 2.20 5.97 -17.60 -18.35 L 2.54 6.67 -13.37 -10.52 U 2.58 6.33 -14.44 -18.35 0.8 E 4.15 19.22 3.62 14.32 -12.85 -25.50 L 3.91 15.25 - 7.36 - 6. 10 U 3 .93 15.21 - 7.97 - 6.21 0.75 .5 0.3 E 1.09 1.72 0.66 1.61 -39.74 - 6.81 L 1 .07 1.58 -38.74 1.49 U 1.07 1.61 -38.50 0.01 0.4 E 1.44 2.61 0.99 2.61 -31.29 16.62 L 1.38 2.24 -28.42 0. 16 U 1.38 2.28 -28.54 14.77 0. 5 E 1.89 3.97 1.44 4.25 -23.82 7.01 L 1 .83 3.75 -21.64 13.35 U 1.83 3.67 -27.60 15.72 0.6 E 2.56 7.09 2.09 7.27 -18.06 2.64 L 2.48 6.64 -15.44 9.53 U 2.50 6.17 -16.13 17.81 0.7 E 3.69 16.10 3.17 13.96 -15.65 -13.30 L 3.52 11.80 -10.17 18.29 U 3.50 12.08 - 9.55 15.59 0.8 E 5.82 38.43 5.28 33.70 - 9.26 12.31 L 5.59 32.21 - 5.52 4.64 U 5.55 30.88 - 9.26 4.63 0.5 .75 0.3 E 1.26 3.52 0.75 2.13 -40.47 -39.57 L 1.25 3.17 -40.00 -32.90 U 1.24 3.25 -40.47 -34.53 0.4 E 1.68 4.86 1.12 3.23 -33.46 -33.54 L 1.66 4.53 -32.68 -28.70 U 1.63 4.40 -31.46 -26.45 0.5 E 2.19 6.96 1.61 4.93 -26.62 -29.10 L 2.13 6.22 -24.61 -20.68 U 2.12 6.22 -24.18 -20.68 E 2.92 10.80 2.32 8.01 -20.46 -25.83 L 2.81 9.38 -17.40 -14.59 U 2.85 10. 10 -18.44 -20.67 136 simu. diffusion % difference mean var. mean var. mean var. 0.7 E 4.05 19.22 3.48 14.83 -14.18 -22.83 L 3.95 17.59 -12.04 -15.69 U 3.96 17.89 -12.07 -14.23 0.8 E 6.35 45. 13 5.75 35.47 - 9.53 -21.39 L 5.99 34.23 - 4.10 3.63 U 6.05 36.62 - 4.99 - 3.13 5.5.6 Comparisons of queues with different variabilities. In this section, queues with different variabilities will be compared. As the interarrival and service time distributions, the Erlang distributions of different orders will be used. Remember that the variability of a distribution is defined as variance/mean . Hence the variability of the Erlang distribution of order n becomes 1/n. Both diffusion approximation and simulation estimates show that as the variabilities decrease, the mean and variance of system size decreases, but the decreasing rate of variance is higher than that of the mean. This can be expected from the fact that zero variability means deterministic interarrival or service time. 137 Table 7 Comparison of mean and variance of system size for various variabilities of interarrival time distribution.(EX/MY/ 2 / ® ) Pj = parameter of the Geometric distribution of arrival size P 2 = parameter of the Geometric distribution of service size Ca = variability of the Erlang distribution C, simu. diffusion % difference P Pi a mean var. mean var. mean var. 0.7 1.0 0.5 1/2 4.36 18.68 4.27 29.57 - 2.02 58.28 1/3 4.01 14.80 4.13 27.59 3.17 86.48 1/4 3.96 13.00 4.07 26.63 2.62 104.81 1/5 3 .94 12.99 4.25 26.05 2.10 100.60 0.7 0.5 1.0 1/2 4.06 22.37 3.06 18.94 - 4.80 -15.33 1/3 3.64 16.04 3.59 16.01 - 1.19 - 0.19 1/4 3.58 16.34 3.46 14.64 - 3.50 -10.40 1/5 3.49 14.20 3.38 13.85 - 3.20 - 2.46 0.5 1/2 6.03 42.43 6. 19 51.41 2.59 21.16 1/3 5.86 39.42 5.92 50.12 0.96 27.14 1/4 5.55 32.78 5.78 47.57 4.30 45.14 1/5 5.45 33.95 5.70 46.08 4.64 35.74 138 Table 8. Comparison of mean and variance of system size for various variabilities of interarrival time distribution.(M /E /2/« ) Pj = parameter of the Geometric distribution of arrival size p2 = parameter of the Geometric distribution of service size Cjj = variability of the Erlang distribution diffusion % difference { Pi p2 °b simu mean var. mean var. mean var. 0.7 1.0 0.5 1/2 3.90 15.24 3.51 19.41 - 9.87 27.31 1/3 3.66 12 .78 3.13 15.02 -14.48 17.54 1/4 3.53 12.72 2.93 13.04 -20.14 2.51 1/5 3.36 11.11 2 .82 11 .92 -16.26 7.26 0.7 0.5 1.0 1/2 4.67 30.57 4. 10 21.68 -12.87 -29.09 1/3 4.55 29.66 3 .90 19.38 -14.27 -34.65 1/4 4.55 30.18 3.81 18.29 -16.34 -39.39 1/5 4.53 29.47 3.75 17.65 -17.32 -40. 11 0.7 0.5 0.5 1/2 6.48 51.28 5.84 48.65 - 9.81 - 5.12 1/3 6.23 49.96 5.45 41.68 -12.46 -16.58 1/4 6. 20 54.36 5. 26 38.40 -15.11 -29.37 1/5 5.98 45.48 5.14 36.50 -14.00 -19.74 5.5.7 Comparisons of queues with ultra-heavy traffic. In this section, queues with ultra-heavy traffic (traffic intensity Y V > 0.9) will be compared for M /M /2/k queues. It is very hard to obtain simulation estimates in ultra-heavy traffic cases because it takes much longer time for a system to reach a steady-state. Hence infinity capacity cases will be excluded from the comparison. Table 9 shows that as the system capacity increases, percentage difference in the mean decreases, and in large capacity cases, is within 5 %. In most cases, percentage difference in the variance seems to show the same behavior. 139 Table 9. Comparison of mean and variance in case of ultra-heavy traffic Geometric distribution of size Pi = parameter of P2 = parameter of Geometric distribution of size f Pi P2 k simu. dif on % diffe rence mean v< mean var. mean var. 0.93 0.5 1.0 40 25.42 713 61 26.57 736.50 4.53 3.21 25 14.66 128 33 15.22 132.29 3.83 2 .99 10 4.31 10 77 4.82 12.32 11.80 14.39 5 2. 13 3 17 2.44 4.02 14.92 26.69 0.93 1.0 0.5 40 15.30 125 42 14.51 131.99 - 5.12 5.24 25 10.81 55 96 9.98 58.35 - 7.75 4.26 10 5.03 11 15 4.32 12.17 -14.20 9. 12 5 2.74 3 27 2.22 4.08 -19.19 24.50 0.93 0.5 0.5 50 19..44 215 82 19.49 217.54 0.26 0.79 40 16.,34 147 51 16.35 141.40 - 0.02 4.32 20 8..94 44 44 9.04 41.28 - 1.06 7.66 10 4..66 14 41 4.75 12.02 - 2.02 19.83 0.95 0.5 1.0 50 19..77 207 56 18.78 205.25 5.30 1.13 40 16.,69 139 87 15.83 133.99 4.37 4.38 20 9..33 40 49 8.71 38.42 7.05 5.40 10 4..94 12 36 4.43 10.80 11.46 14.40 0.95 1.0 0.5 50 19.01 207 26 19.66 194.79 - 3.33 6.40 40 15.91 139 35 16.83 130.30 - 5.45 6.95 20 8.62 40 15 9.59 38.27 -10.11 4.91 10 4.44 12 26 5.20 11.14 -14.61 9.97 0.95 0.5 0.5 100 35. 55 745 95 35.62 763.92 0.20 2.41 70 28. 13 405 89 27.60 413.14 - 1.89 1.79 40 17. 45 146 53 17.39 151.44 - 0.36 3.35 10 4. 84 12 15 4.75 14.45 - 1.77 18.93 140 pj P2 k sirau. diffusion % difference mean var. mean var. mean var. 0.97 0.5 1.0 100 29.86 690.11 37.69 771.92 26.22 26.22 70 23.61 380.27 28.97 408.19 22.66 7.34 40 15.83 133.99 18.19 145.03 14.95 8.23 10 4.43 10.80 5.07 12.37 14.40 14.54 0.97 1.0 0.5 100 37.66 639.27 36.91 773.03 - 1.97 20.92 70 29.59 368.00 28.14 407.89 ■4.92 10.84 40 18.44 137.45 17.36 144.58 ■ 5.83 5.19 10 5.31 11 .04 4.56 12.32 ■14.16 11.55 0.97 0.5 0.5 150 55.39 1591.20 55.95 1739.95 1.02 9.22 110 43.52 876.35 44.43 999.28 2.09 14.03 70 30.04 402.91 ..30.53 433.78 1.63 7.66 30 14.14 87.97 ”*14.08 91.55 - 0.41 4.07 5.5.8 Ooiparlson of queues with many servers In this section, finite capacity queues with six servers are y v compared for M /M /6/k queues. As can be seen on Table 10, the percentage difference is, in most cases, within 10 % for p = 0.7. If the mean service size is greater than the mean arrival size, the percentage difference in the mean decreases as the system capacity increases. It is easy to see that percentage difference in the variance decreases as the system capacity increases. 141 Table 10. Comparison of mean and variance in case of many servers Pj = parameter of the Geometric distribution of arrival size P 2 = parameter of the Geometric distribution of service size diffusion % difference f Pi P2 k simu mean var. mean var. mean var. 0.7 1.0 0.5 15 4.12 14.87 3.63 16.48 -12.04 10.81 12 3.43 11.22 3.21 12.46 -13.12 10.15 9 3.24 7.67 2.67 8.45 -17.59 10. 19 6 2.49 4.09 1 .97 4.77 -20.79 16.73 0.7 0.5 1.0 15 4.38 17.12 3.94 16.55 - 9.91 - 3.31 12 3.84 12.43 3.56 12.82 - 7.30 3.18 9 3. 14 7.93 3.00 8.82 - 4.52 11.16 6 2.28 4.15 2.21 4.91 - 3.14 18.41 0.7 0.5 0.5 15 4.88 20.31 4.56 22.41 - 6.71 10.35 12 4.20 14.36 3 .92 16.54 - 6.59 15.15 9 3.41 9. 24 3. 15 11.02 - 7.65 19.23 6 2.47 4.84 2.22 6.07 - 9.80 25.54 5.6Bulk queue vs. non-bulk queue In this section, the utility of the bulk queue model is illustrated through an example. If bulk-queue solutions are not available, one alternative way is to approximate the bulk queueing system by a non-bulk queue model. Suppose we have a single server finite capacity bulk queueing system MX/MY/l/k with arrival rate A. and service rate fj.. Assume 142 that the arrival and service size follow the Geometric distributions with parameter pj and P 2 respectively. Hence the mean arrival and service size become I/P2 and 1/P2* Now we are going to approximate the Mx/MY/ 1 A queue by M/M/1/k queue. Then, the arrival and service rate of M/M/l/k queue will be \ E(X) and E(Y) respectively. y v Under this scheme, the mean system size of exact M /M /1/k solution, diffusion approximation and approximation by M/M/l/k queue will be compared. As can be seen in Table 11, approximation of MX/MY/l/k queue by M/M/l/k queue results in severe deviation from the true mean system size. The maximum percentage error in the mean is as high as 63 %, and the error is expected to increase as the system capacity increases. This single example is enough to show us the utility of the bulk queueing models. 143 Table 11. Comparison of mean system size of MX/MY/l/k queue (diffusion approximation vs. approximation by M/M/l/k queue) p ^ : parameter of the Geometric distribution of arrival size P2 : parameter of the Geometric distribution of service size % errors k exact diffu. M/M/l/k diff. M/M/1 /\ ? 0.9 40 14.54 14.79 8.45 1.73 -41 .9 30 11.87 11.89 7.70 0. 14 -34.6 20 8.48 8.48 6.42 0.00 -24.2 15 6. 57 6.57 5. 36 - 0.01 -18.4 10 4.52 4.51 3.97 - 0.18 -12.1 7 3.22 3.19 2.95 - 0.93 - 8.2 5 2.33 .. 2.74 2.20 - 2.21 - 5.6 3 1.41 1.37 1.37 - 3.10 - 2.8 0.5 15 2.68 2.68 1.00 0.19 -62.7 10 2.26 2.26 0.99 0.05 -56.0 7 1 .83 1.82 0.97 - 0.52 -30.4 6 1.64 1.62 0.95 - 1.03 -42.5 5 1.43 1.41 0.91 - 1.78 -36.9 4 1.20 1.17 0.84 - 2.68 -29.9 3 0.94 0.91 0.73 - 3.21 -21.7 0.9 30 11.17 10.99 7.70 - 1.67 -31.1 20 8.43 7.95 6.42 - 5.68 -23 .8 10 4.70 4.29 3.97 - 8.78 -15.6 7 3.43 3.07 2.95 -10.55 -13 .9 5 2.53 2.23 2.20 -11.99 -13.2 4 2.07 1 .82 1.79 -12.25 -13.4 3 1.59 1.41 1.37 -11.16 -13.9 15 2.26 2.19 1 .00 - 3.14 -55.8 10 2.07 1.93 0.99 - 7.07 -51.9 8 1.91 1.74 0.98 - 9.19 -48.6 7 1.80 1.62 0.97 -10.33 -46.2 6 1.67 1.48 0.95 -11.51 -43.4 5 1.50 1.31 0.91 -12.58 -39.7 4 1.30 1.13 0.84 -13.20 -35.5 3 1.06 0.93 0.73 -12.38 -30.9 0 true M /M /1/k d iffu sio n approx rt/M/1/k approx. Figure 13. Diffusion approximation vsvs. M/M/1 A approximation of MX/My/l/k queue, traffic intensity = 0.9, (p1»P2 ) = (0.75,0.5) E(N) true MVMVI/k d iffu sion approx M/M/1/k approx. 5 10 15 Figure 14. Diffusion approximation vs. M/M/l/k approximation of MX/MY/l/k queue, traffic intensity ■ 0.5, (P!.P2) *= (0.75,0.5) 145 E(N) 10 true M /M /1 /k 5 diffu sion approx M/M/1/k approx. 10 20 30 Figure 15. Diffusion approximation vs. M/M/l/k approximation of MX/My/l/k queue, traffic intensity = 0.9, E(N ) (Pi»P2^ * (0.5,0.5) 3 true M /M /I /k d iffu sion approx M/M/1/k approx. 2 1 Figure 16. Diffusion approximation vs. M/M/l/k approximation of M /MY/l/k queue, traffic intensity = 0.5, (Pj*P2) “ (0.5,0.5) CHAPTER VI CONCLUSIONS 6.1. Research Summary A diffusion approximation of multiserver, finite capacity bulk queues was developed and its accuracy was evaluated. The diffusion process employed was the Elementary Return Process (ERP) developed by Feller [19]. Advantages of using ERP are that it can more accurately represent the actual queueing phenomena, and probabilities that the system is full and empty can be automatically obtained while solving the diffusion equations. The infinitesimal mean and variance were developed by using random sums of random variables and applying the limiting mean and variance of a renewal process. One of the drawbacks of the infinitesimal mean and variance used in this research is that they are independent of the state variable of the diffusion process (i.e., a and $ are independent of x. See (3.32) and (3.33).) and thereby induce some inherent inaccuracies of the approximation. The boundary behaviors of the ERP were interpreted and the diffusion equations were solved by using Laplace Transform Technique. The final solutions are expressed in terms of all the queueing parameters available. The distribution of system size obtained is 1 46 147 complex, but that is inevitable in consideration of the complexities of the queueing systems under study. It was proved that the approximate distribution of system size was legitimate in the sense that sum of the probabilities is one under the assumption that service size is less than the system capacity. The mean and variance of the system size were derived from the system size distribution and resulted in complex formulae. Accuracy of the diffusion approximation was evaluated for a variety of bulk queues. Due to the rare availability of exact solutions of finite capacity bulk queues, simulation was used to estimate the mean and variance of system size and the accuracy of the simulation estimates were evaluated. Based on the accuracy evaluation of chapter V, diffusion approximation developed in this dissertation seems to be fairly accurate in following cases: 1. heavy traffic 2. large system capacity 3. if the arrival size distribution and the service size distribution is the same (possibly if the coefficient of variation of arrival size distribution and that of service size distribution are close to each other.) 4. if the coefficient of variation of service time distribution is getting close to that of interarrival time distribution. The fact that the diffusion approximation is accurate in heavy traffic cases has been pointed out by many researchers and it is 148 confirmed again in this dissertation research. In heavy traffic cases (traffic intensity greater than 0.8), the percentage error of mean system size is well within 10 % if the system capacity is relatively large (approximately greater than 13). The diffusion approximation of this research shows decreased accuracy in following cases : 1. low traffic 2. small system capacity 3. if the difference between mean arrival size and mean service size is large. 4. if the number of servers is large. In some low traffic or small capacity systems, the percentage error of mean system size grows up to 30 %. The reason why small system capacity shows low accuracy is that in a small capacity system, the probabilities that the system is empty or full are large and thereby approximation by a continuous stochastic process is inappropriate. The increasing number of servers turns out to decrease the accuracy of the approximation. The reason is that the infinitesimal variance and mean given by (3.32) and (3.33) do not represent the system where some of the servers are idle. But if the traffic intensity is extremly high in large capacity systems, then the diffusion approximation developed in this research is expected to be accurate even in many server cases. As far as the contribution of this dissertation research is 149 concerned, to the best of the author's knowledge, this dessertation is the first attempt to derive the closed form solution (not in transform) of the system size distribution of the multiple server, finite capacity bulk queues, even though it is approximate. Although, in some cases, the approximation is inaccurate (this is true for any approximation technique), it can be reasonably applied to analyzing and designing many real world bulk queueing systems, such as the examples given in chapter I. A final suggestion to the possible users is that the diffusion approximation developed in this research is dependable if the system capacity is greater than 15 and ..traffic intensity is greater than 0.7. But even in low traffic cases, if the arrival and service size distributions are close to each other, the accuracy turns out to be very accurate. Although the accuracy in many server cases is lower than the cases with small number of servers, if the system traffic is heavy, then it turns out to be reliable even in many server cases (up to approximately 6 servers). 6.2 Suggestions for Further Study 6.2.1 Infinitesinal sean and variance. It was pointed out that the infinitesimal moments developed in this research are independent of state variable of diffusion process. This indicates that even if the system size is less than the potential mean service capacity of the system (m y E(Y)), the decreasing rate of system 150 size is still m y E(Y), which brings about inaccuracy in the approximation. One remedy of this situation is to use the infinitesimal moments suggested by Halachmi and Franta[30] and Kimura [39] [40]. Halachmi and Franta studied a diffusion approximation for GI/G/m queue. The infinitesimal variance and mean of GI/G/m queue can be obtained by letting Var(X)=Var(Y)=0 and E(X)=E(Y)=1 in (3.32) and (3.33). Then the infinitesimal variance and mean become a(x) = X3 Var(a) + m y3 Var(b) B (x) = X - m y But Halachmi and Franta used min(x,m) instead of m . Kimura improved the idea of Halachmi and Franta and used min(rxl,m) where Fxl is the largest integer smaller than x. Yao [75][76] improved the solution procedures of Kimura and applied the results of Whitt [74] on approximating point processes by renewal processes. The advantage of these improved infinitesimal mean and variance is that they can represent the situations where some of the servers are idle. Further research needs to be concentrated on this problem. Another possibility is a mathematical treatment imbedding a discrete process into the diffusion process. This approach was suggested by Walsh [69] and Chacon [7]. 6.2.2 Mean holding time In section 3.3, when we interpreted the mean holding times (e^ 151 and &2 on diffusion equations (3.10), (3.11) and (3.12)), we used 1/A and l/'my . But this is true only when the queueing process is Markovian. Hence if the queueing system is not Markovian, this interpretation may contribute to the error of the approximation. An alternative way would be to apply the concept of mean residual time of a renewal process. But the service process of a queueing system is not a renewal process in a strict sense, this idea also is expected to induce some errors too. 6.2.3 waiting ti»e This research is not concerned with waiting time. But according to the well-known Little's formula, L = A W, if we know the arrival rate and the mean system size, we can calculate the mean waiting time. Little [52] proved the formula under the assumptions that the queueing process is strictly stationary and the arrival process is metrically transitive (Doob [17:457]). The formula is remarkably free of specific assumption about arrival and service time distributions, independence of interarrival time, number of servers, queue discipline, etc. Jewell [36], Eilon [18], Maxwell [53] and Stidham [66] [67] also provided different proofs. But all these proofs were based on the non-bulk queues. The proof that the Little's formula is valid for bulk queues will be provided in Appendix. Appendix 1. PROOF OF LITTLE'S FORMULA FOR BULK QUEUES 2. SIMULATION PROGRAM CODING OF FINITE CAPACITY BULK QUEUE 3. FORTRAN PROGRAM CODING FOR FINDING SYSTEM SIZE DISTRIBUTION, MEAN AND VARIANCE 1 52 153 LITTLE*S FORMULA FOR BULK QUEUES Suppose groups of customers arrive into a bulk queueing system at time tj, t2 and the customers in the group are randomly ordered. Let a ^ a2 »... be the arrival time of the randomly ordered customers. K 4* Vi Hence it is possible that a.^ = ai+1 if the i customer and (i+1) customer belong to the same arritf&l group. Assume that the first arrival occurs when the system is empty at time 0, and the queueing process continues until time T such that J=[max j/ tj< T] is a predetermined value and N(T)=0 where N(t) is the number of customers in the system at time t. This means that arrival is the last arrival into the queueing system and the queueing system is observed until the customers in the group are served and leave the system. Let be the waiting time of the ith customer and d^ be the departure time of the i1**1 customer. This situation is illustrated in Figure 17. 1 54 N (t) 7 6 5 4 3 2 1 t2 d i t3 fc4 d6 fc5 d7 ‘10 a l a4 d2 a5 a8 a10 d8 a o. a2 a6 a9 a ll d9 a3 a7 al 2 Figure 17. Number of customers in a bulk queueing system Proposition 1. If n is the total number of arriving customers in time [OfT]f then T n J N(t) = I W± 0 i=l 1 55 PROOF N (t) 7 6 5 12 4 11 3 12 2 10 12 1 11 ft *-* t2 d l *3 t4 d3 d4 d6 fc5 l10 a l a4 d2 a5 a8 d5 a10 a a a2 a6 a9 a ll a3 a7 a l 2 Figure 18. Indexed areas Let A^ be the area under the undex i in Figure 18. Then, Aj = dj - = Wj A2 = d 2 - a 2 = w2 A3 = (d3-a5)+(a5-d1)+(d1-a3) = W3 W A11 " *dl l ”d8*+ *d8- a l 1 * 11 ‘12 ^d12_d10^+ ^dl10 n"^R)+ 8 ^R8 -ai “ 12 ' = ^12 1 56 T J N(t) dt is the total area under the curve. 0 Hence T 11 11 J N(t) dt = I Pi± = I Wi . 0 i=l i=l The generalization is obvious: T n J N(t) dt = I 0 i=l Theorem 1. (Little [52:385]) Let u i=ti+l-ti- means that u^^ is the time between (i+l)th and ifcl1 group arrival. If we assume that the stochastic process {ui} is strictly stationary and metrically transitive with mean 1/A , then as J -*• oo, tj w with probability 1. (See Doob [17:457] for metric transitivity.) 1 57 Proposition 2 If n is the total number of customers arriving in time [0,T] and X^'s are iid random variables denoting the group size of i*"*1 arrival with E(Xi)<<» . Then n/T -► XE (X1) with probability 1, where X is the group arrival rate defined by X = lim J/T PROOF Since n is the total number of arriving customers, we have n = Xj + X2 + + Xj Then, from Theorem 1 and the strong law of large numbers (See Billingsly (5:250, theorem 22.5), we have Pr [ lim | n/J - E(Xj)| = 0 ) = 1 J-KO But from X = lim J/T , J-KX3 n/T = (n/J) (J/T) -*■ X E(X1) with probability 1, 158 Proposition 3 T Let L = (1/T) J N(t) dt 0 n and W* = (1/n) J i=l Then, L ■+• X E(Xj) W with probability 1, where W is the mean waiting time. PROOF From proposition 1, T n T L = J N(t) dt = I = n W* . 0 i = l 4s Obviously, W -► W with probability 1, and from proposition 2, L -*■ A E(Xj) W with probability 1 SIMULATION PROGRAM CODING // JOB , // TIME=(5,00), REGION=6000K /*JOBPARM LINES=5000,DISKIO=20000,V=SLACK //PROCLIB DD DSN=TSA.SIMSCRIT.CNTL, DISP=SHR //SIM EXEC GMCSIM //PRE.SYSIN DD * PREAMBLE LAST COLUMN IS 72 '' SANDS MODULES INCLUDE BATCH.MEANS NORMALLY MODE IS INTEGER TEMPORARY ENTITIES EVERY CUSTOMER HAS A T.ARR AND MAY BELONG TO THE QUEUE THE SYSTEM OWNS THE QUEUE EVENT NOTICES INCLUDE ARRIVAL, SHRUBEN.TEST, END.TRANS, VAR.EVENT AND BATCHEVENT EVERY SERVICE HAS A PERSON DEFINE T.TRANS, T.STOP, LAMBDA, MU, T.ARR, T.ZERO, SIG, PI, P2, NNCUST AS REAL VARIABLES DEFINE NIDLE, K, M, A, B, C, D, N.OBS, ALT, ARRAY, NCUST, NNNCUST AS INTEGER VARIABLES DEFINE N.OBSl,N.OBS2,INIT.STATE, IJ,IIJJ,NUM.DATA AS INTEGER VARIABLES DEFINE MDFD AS A FORTRAN ROUTINE DEFINE Y AS A 1-DIM VARIABLE DEFINE X AS A 2-DIM VARIABLE DEFINE T AS A 2-DIM REAL VARIABLE DEFINE STATE AS A 2-DIM VARIABLE DEFINE Z AS A 1-DIM REAL VARIABLE DEFINE TT AS A 1-DIM REAL VARIABLE ACCUMULATE VAR.CUST AS THE VARIANCE OF NNNCUST END MAIN LET A=1 LET B=2 LET C=3 LET D=4 READ M, K, LAMBDA, MU, T.TRANS, T.STOP, PI, P2 LET NIDLE=M LET ARRAY=({T.STOP-T.TRANS)*LAMBDA+M*(T.STOP-T.TRANS)*MU)*1.2 RESERVE Y (*) AS ARRAY 1 60 RESERVE T (*,*) AND X(*,*) AS 2 BY ARRAY PRINT 1 LINE WITH M, K, LAMBDA, MU, T. TRANS, T. STOP, PI ,P2 THUS M**K***LAMBDA**. ***T>TR*****. * Pi**. **P2**. ** SKIP 1 LINE PRINT 1 LINE WITH ARRAY THUS ARRAY SIZE = ***** FOR ALT=1 TO 2, DO IF ALT=2, SKIP 3 LINES PRINT 1 LINE THUS $$$$ ANTITHETIC RUN $$$$ ELSE SKIP 2 LINES PRINT 1 LINE THUS $$$$ NON-ANTITHETIC RUNS $$$$ ALWAYS LET SEED.V(A)=2735473 LET SEED.V(B)=7379643 LET SEED.V (0=237634 LET SEED.V(D)=3673 SCHEDULE AN ARRIVAL NOW SCHEDULE AN END.TRANS IN T.TRANS UNITS SCHEDULE AN SHRUBEN.TEST IN T.STOP UNITS START SIMULATION CALL INITIALIZE LOOP LET TIME.V=0. LET BM.NUM.EST=1 LET BM.M=200 LET BM.TPVS=1 LET BM.TPDEL=.1 CALL BM.INIT CALL DATA.BATCH STOP END ROUTINE DATA.BATCH LET DATA.POINT=(N.0BS1+N. CBS2)*1. 2 RESERVE Z(*) AS DATA.POINT RESERVE T T (*) AS DATA.POINT LET 11=1 LET JJ=1 FOR 1=1 TO DATA.POINT UNTIL II>N.OBSl OR JJ>N.0BS2, DO IF T(l,II)=T(2,JJ) LET T T (I)=T (1,11) - T.TRANS LET Z(I)=(X(1,II)+X(2,JJ))/2 LET 11=11+1 LET JJ=JJ+1 ELSE IF T(l, IIXT(2, JJ) LET T T {I)= T (1,11) - T.TRANS IF JJ=1 LET Z(I)=(X(1,II)+INIT.STATE)/2 ELSE LET Z(I) = (X(1, II)+X(2, JJ-1) )/2 ALWAYS LET 11=11+1 ELSE LET TT(I)=T(2,JJ) - T.TRANS IF 11=1 LET Z (I)=(INIT.STATE+X(2,JJ))/2 ELSE LET Z(I)=(X(1,II-1)+X(2,JJ))/2 ALWAYS LET JJ=JJ+1 ALWAYS ALWAYS LOOP LET NUM.DATA=I-1 FOR JJI=1 TO N.OBS1 LET T (1,JJI)=T(1,JJI)-T.TRANS SKIP 1 LINE PRINT 1 LINE WITH NUM.DATA THUS NUM.DATA=******* LET NNCUST=INIT.STATE LET IJ=1 SCHEDULE AN BATCHEVENT AT TT(1) START SIMULATION LET TIME.V=0. LET NNNCUST=INIT.STATE LET IIJJ=1 SCHEDULE A VAR.EVENT AT T(l,l) START SIMULATION RETURN END EVENT VAR.EVENT LET NNNCUST=X(1,IIJJ) IF IIJJ+1>N.OBS1, CALL OUTPUT2 RETURN ALWAYS SCHEDULE A VAR.EVENT AT T(1,IIJJ+1) LET IIJJ=IIJJ+1 RETURN END EVENT BATCHEVENT LET NNCUST=Z(IJ) LET BM.OBS(l)=NNCUST IF IJ+1>NUM.DATA CALL OUTPUT RETURN ALWAYS 162 SCHEDULE A BATCHEVENT AT TT(IJ+1) LET IJ=IJ+1 RETURN END ROUTINE OUTPUT PRINT 1 LINE WITH TIME.V THUS RESULTS AT *******.** CALL BM.OUT(1) FOR EVERY BATCHEVENT IN EV.S(I.BATCHEVENT), DO REMOVE THIS BATCHEVENT FROM EV.S(I.BATCHEVENT) DESTROY THIS BATCHEVENT LOOP RETURN END ROUTINE OUTPUT2 SKIP 2 LINES PRINT 1 LINE WITH VAR.CUST THUS VARIANCE OF SYSTEM SIZE = *****.***** SKIP 1 LINE PRINT 1 LINE WITH M, K,LAMBDA, MU,T.TRANS,T.STOP,PI,P2 THUS ** **** **^ * **.**** ++***_* *****_* +_**** STO/ END EVENT ARRIVAL CALL GEO.RANDOM(PI,C) YIELDING GEORAN LET N=GEORAN IF NCUST+N > K, LET N=K-NCUST ELSE LET HOWOO=100. ALWAYS FOR 1=1 TO N, DO CREATE A CUSTOMER FILE CUSTOMER IN QUEUE LOOP LET NCUST=NCUST+N LET N .QBS=N.OBS+1 LET X(ALT/ N. OBS)=NCUST LET T(ALT,N.OBS)=TIME.V IF NIDLE>0 LET NIDLE=NIDLE-1 SCHEDULE A SERVICE IN EXPONENTIAL.F(1./MU,B) UNITS ALWAYS SCHEDULE AN ARRIVAL IN EXPONENTIAL.F(1./LAMBDA, A) UNITS RETURN END EVENT SERVICE CALL GEO.RANDOM(P2,D) YIELDING GEORAN LET N.SERVE=MIN.F(N.QUEUE,GEORAN) FOR 1=1 TO N.SERVE, DO REMOVE FIRST CUSTOMER FROM QUEUE DESTROY CUSTOMER LOOP LET NCUST=NCUST-N.SERVE LET N .QBS=N.OBS+1 LET X (ALT, N. CBS) =NCUST LET T(ALT, N.OBS) =TIME.V IF QUEUE IS EMPTY LET NIDLE=NIDLE+1 RETURN ELSE SCHEDULE A SERVICE IN EXPONENTIAL.F(1/MU,B) UNITS RETURN END EVENT SHRUBEN.TEST DEFINE PROBAB AS A 1-DIM REAL VARIABLE DEFINE F,H.F, H.L AS REAL VARIABLES IF ALT=2, LET N.OBS2=N.OBS ELSE LET N.QBSl=N.OBS ALWAYS FOR 1=1 TO N.CBS, LET Y (I)=X(ALT, I) LET NNN=N.GBS/2 PRINT 1 LINE WITH NNN THUS NNN=*J*‘*3*t)*c CALL H GIVEN Y (*), NNN YIELDING H.F LET NNN=N. CBS-NNN FOR 1=1 TO NNN, LET Y(I)=Y(I+NNN) CALL H GIVEN Y(*), NNN YIELDING H.L RESERVE PROBAB(*) AS 1 IF H.L=0. LET PROBAB(1)=1.0 ELSE LET F=H.F/H.L LET N1=3 LET N2=3 CALL MDFD(F, N1,N2,PROBAB(*),IER) ALWAYS LET S IG=1. -PROBAB (1) SKIP 3 LINES PRINT 3 LINES WITH H.F, H.L, SIG THUS NUMERATOR OF F-STAT IS *********. *** DENOMINATOR OF F-STAT IS *********.*** SIGNIFICANCE LEVEL OF TEST IS **.*** SKIP 2 LINES IF SIG>0.05, PRINT 1 LINE THUS PASS THE SHRUBEN'S TEST ELSE PRINT 1 LINE THUS FAIL THE SHRUBEN TEST ALWAYS CALL INITIALIZE RETURN END EVENT END.TRANS PRINT 1 LINE WITH NCUST, TIME.V THUS NCUST=***** TIME.V=***** LET N.OBS=0 LET INIT.STATE=NCUST IF ALT=2 LET A=-A LET B=-B LET C=-C LET D=-D ALWAYS RETURN END ROUTINE INITIALIZE LET TIME.V=0. LET NIDLE=M LET NCUST=0 LET N.OBS=0 FOR EVERY ARRIVAL IN EV.S(I.ARRIVAL), DO REMOVE THIS ARRIVAL FROM E V .S (I.ARRIVAL) DESTROY THIS ARRIVAL LOOP FOR EVERY SERVICE IN EV.S(I.SERVICE), DO REMOVE THIS SERVICE FROM E V .S (I.SERVICE) DESTROY THIS SERVICE LOOP FOR EVERY CUSTOMER IN THE QUEUE, DO REMOVE THIS CUSTOMER FROM THE QUEUE DESTROY THIS CUSTOMER LOOP RETURN END ROUTINE GEO.RANDOM(PARA, STREAM) YIELDING GEORAN DEFINE GEORAN AND STREAM AS INTEGER VARIABLES DEFINE U,PX AND PARA AS REAL VARIABLES LET U=RANDOM.F(STREAM) LET 1=0 LET PX=0. FOR J=1 TO 100, UNTIL PX>U, DO 1 6 5 ADD 1 TO I LET PX=PX+PARA*(l.-PARA)**(1-1) LOOP LET GEORAN=J-1 RETURN END ROUTINE H GIVEN Y, NNN YIELDING H.F DEFINE Y AS A 1-DIM VARIABLE DEFINE T .HAT, AVE, H.F AS REAL VARIABLES DEFINE I, NNN, K.HAT AS INTEGER VARIABLES FOR 1=1 TO NNN, DO ADD Y (I) TO SUM LET Y(I)=SUM LOOP LET AVE=SUM/NNN FOR 1=1 TO NNN, DO LET Y (I)=AVE*I-Y(I) COMPUTE K.HAT AS THE MAX(I) OF Y(I) LOOP LET T.HAT=K.HAT/NNN IF T.HAT=1. LET H.F=0. ELSE LET H.F=Y(K.HAT)**2/NNN/(3*T. HAT*(1.-T.HAT)) ALWAYS RETURN END /* //LKED.SYSLIB DD // DD // DD DSN=SYS1.IMSL.SINGLE, DISP=SHR // DD DSN=SYS1.VFORTLIB,DISP=SHR // DD DSN=SYSl.FORTLIB, DISP=SHR //GO.SYSIN DD * 2 900 3.6 3.0 5000. 15000 .9999 .5 /* // 166 FORTRAN PROGRAM CODING C FOLLOWING FORTRAN PROGRAM IS TO FIND THE DISTRIBUTION C OF SYSTEM SIZE, ITS MEAN AND VARIANCE OF C FINITE CAPACITY BULK QUEUES. ALSO THE ROUTINES TO C FIND THE EXACT MEAN AND VARIANCE OF MX/MY/1/K QUEUE C IS INCLUDED (SEE BAGCHI & TEMPLETON [2]). A‘ SLIGHT C MODIFICATION WAS NEEDED TO FIND THE VARIANCE. C C C DIMENSION PR (68) ,F(120) , G (120, 68) ,D(120),A(120) ,E(120) DIMENSION SQEN(120), V (12 0) REAL Ml,M2,MOVERB, LOVERB,LAMBDA, MU CALL ASSIGNd, 'OUTPUT',6) WRITE(5,10) 10 FORMAT(IX,'TYPE IN RHO, M, K, Pi, P2, CA, CB'/) READ (5, *) RHO, M, K,Pi ,P2,CAj.CB EX=1./P1 EY=1./P2 VARX=(1 .-PI) /P1 **2 VARY=(1.-P2)/P2**2 BOVERA=(RHO*EY-EY)/ (RHO*EY*EX*CA+RHO*EY/EX*VARX+EY**2*CB+VARY) LOVERB=l./(EX*(1.-(1./RHO))) MOVERB=l./(M*EY*(RHO-1.))*M RHOHAT=EXP(2 *BOVERA) WRITE(5,123)RHOHAT 123 FORMAT(IX, 'RHOHAT = \F13.6/) C C C FOLLOWING IS TO CALCULATE Cl C C CALL PONLY(1,K-1,Cl 1,Pi) CALL PXI(1,K-1, C12,Pi) CALL PPOWER(1, K-1,Cl 3,P 1,1./RHOHAT) Cl=l.+.5*LOVERB/BOVERA*(RHOHAT**K-1.) 1 -LOVERB*K+(LOVERB*K+.5*L0VERB/B0VERA)*Cl1 2 -LOVERB*Cl2- . 5*LOVERB/BOVERA*RHOHAT**K*C13 C C WRITE(5,139)C11,C12,C13,C1 139 FORMAT(IX, 'Cl 1= ',F13.6/ 1 2X,'C12= ’,F13.6/ 2 2X, 'C13= ',F13.6/ 3 /// 2X,'Cl = ' ,F13.6) C FOLLOWING IS TO CALCULATE C2 C C 1 67 CALL P0NLY(1,K-1, C21,P2) CALL PPOWER(l,K-l,C22,P2,RHOHAT) CALL PXI(1,K-1, C23,P2) C2=l.+.5*MOVERB/BOVERA*(1.-RHOHAT**K*(1.-C21)-C22) 1 +K*MOVERB* (1 .-C21 )+MOVERB*C23 C WRITE (5,8 36) C2 3 , C2 836 FORMAT(IX,'C23 = ',Fl3.6///2X,'C2 = ',F13.6/////) C C FOLLOWING IS TO CALCULATE C3 C C C3=LOVERB*(RHOHAT**K-1.)+LOVERB*(Cl 1-RHOHAT**K*Cl3) WRITE(5,174)C3 174 FORMAT(IX,'C3=',Fl3.6////////) C C C FOLLOWING IS TO CALCULATE C4 C C C4=MOVERB*(1.-RHOHAT**K*(1.-C21)-C22) C WRITE(5,435)C4 435 FORMAT(IX, 'C4 = ',Fl3.6///////) C M1=C4/(C1*C4-C2*C3) M2=C3/(C2*C3-C1*C4) C WRITE(5,843)Ml,M2 843 FORMAT(IX,'Ml = ',F13.6,10X,'M2 = ',F13.6///////) C DO 200 N=1,K-l CALL PONLY(1,N, CC1,PI) CALL PPOWER(1,N,CC2,PI, 1./RHOHAT) CALL PONLY(K-N, K, CC3,P2) CALL PPOWER(K-N,K,CC4,P2,RHOHAT) C C PR(N)=(RHOHAT**N-l.)*(LOVERB*Ml-MOVERB*M2*(1.-C21)) 1 +L0VERB*M1*(CCl-RHOHAT**N*CC2)+MOVERB*M2 2 *(CC3-RHOHAT* *(N-K)*CC4) 200 CONTINUE C C C PR(K)=M2 PSUM=Ml DO 300 N=1, K PSUM=PSUM+PR(N) 300 CONTINUE WRITE (5,100DPSUM 1001 FORMAT(/////IX, 'PSUM = \F13.6) C C C EN=0. ENSQ=0. DO 400 N=1, K EN=EN+N*PR(N) ENSQ=ENSQ+N**2*PR(N) 400 CONTINUE VARN=ENSQ-EN**2 C C C WRITE(5,500)RHO, M,K,PI,P2,CA, CB, EN,VARN FORMAT(1 HI, IX, 'RHO = ',F13.6/ 1 2X, 'M = ’,12/ 2 2X, 'K = ',14/ 3 2X, 'PI = '-,£13.6/ 4 2X, 'P2 = ',F13.6/ 5 2X, 'CA = ',Fl3.6/ 6 2X, 'CB = ',F13.6/ 7 2X, 'EN = ',Fl3.6/ 8 2X, 'VARN = ',F13.6) C C C WRITE(1,500)RHO,M,K,PI,P2,CA,CB, EN,VARN C C IF(M.EQ.1) GO TO 3377 GO TO 3388 C C C C C 3377 HOWOO=100. K=K+1 LAMBDA=RH0*P1/P2 MU=1. AL=LAMBDA AMU=MU P=1.-P1 Q=1. -P2 ALPMU=AL*£H-AMU SLMU=AL+AMU DO 2910 1=1,K 169 G(l,I) =0. 2910 CONTINUE C C INITIALIZE THE SYSTEM C G (1,1)=1.0 F (1)=AL/SLMU F (2)=AL*AMU*(1 --Q)/SLMU**2 DO 2911 1=3, K F (I)=F(I-1)*ALPMU/SLMU 2911 CONTINUE DO 2918 N=1,115 D (1) =0. D (2) =1.0-P DO 2912 1=3,K D(I)=D(I-1)*P 2912 CONTINUE SA=0. Kl=K-l DO 2914 1=1,K1 A(I)=0. DO 2915 J=1,I A (I)= A (I)+ G (N,J )* D (I-J+1) 2915 CONTINUE SA=SA+A(I) 2914 CONTINUE A(K) =1.-SA S=0. KI=K-1 DO 2916 1=1,KI J=I-1 G (N+1, K-J) =0. DO 2917 L=1,1 IL=L-1 G(N+1,K-J)=G(N+1,K-J)+F(L)*A(K-J+IL) 2917 CONTINUE S=S+G(N+1,K-J) 2916 CONTINUE G(NU,1)=1.-S E(N) =0. SQEN(N)=0. DO 2919 1=1,K-l AI=I E(N)=E(N)+AI*G (N, 1+1) SQEN(N)=SQEN(N)+AI* *2 *G (N, 1+1) V(N)=SQEN(N)-E(N)**2 2919 CONTINUE 2918 CONTINUE C C 170 C ERRORM=(EN-E(115))/E(115)*100. ERRORV= (VARN-V (115) ) /V(115) *100. WRITE(5,3999) E (115), ERRORM,V(115), ERRORV 3999 FORMAT(//IX, ’EXACT MX/MY/M/K MEAN SYSTEM SIZE = ', F13.6/ 1 IX, 'MEAN PERCENT ERROR = F13.6 2 //IX, 'VARIANCE OF SYSTEM SIZE= ' ,F13.6/ 3 IX,'VARIANCE PERCENT ERROR =',F13.6) 3388 HOWOO=100. CALL CLOSE(1) STOP END C C SUBROUTINE PXI(ISTART,IEND,VALUE, PARA) VALUE=0. DO 10 I=ISTART,IEND VALUE=VALUE+I*PARA*(1.-PARA)**(I-1) 10 CONTINUE RETURN END C C C SUBROUTINE PPOWER(ISTART,IEND,VALUE, PARA, X) VALUE=0. DO 10 I=ISTART,IEND VALUE=VALUE+PARA*(1.-PARA)**(I-1)*X**I 10 CONTINUE END C C C SUBROUTINE PONLY(ISTART,IEND,VALUE, PARA) VALUE=0. 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