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UNIVERSITI PUTRA MALAYSIA
A SOLUTION OF SINGLE SERVER AND MULTISERVER GENERAL QUEUEING SYSTEMS USING A LINEAR
ALGEBRAIC APPROACH
MOHAMMAD SALEH MUSTAFA
FSAS 1995 1
A SOLUTION OF SINGLE SERVER AND MULTISERVER
GENERAL QUEUEING SYSTEMS USING A LINEAR
ALGEBRAIC APPROACH
BY
MOHAMMAD SALEH MUSTAFA
Thesis Submitted in Fulfilment of the Requirements for the Degree of Master of Science in the Faculty of
Science and Environmental Studies,
Universiti Pertanian Malaysia
December 1995
UNIVERSITI PUTRA MALAYSIA
A SOLUTION OF SINGLE SERVER AND MULTISERVER GENERAL QUEUEING SYSTEMS USING A LINEAR
ALGEBRAIC APPROACH
MOHAMMAD SALEH MUSTAFA
FSAS 1995 1
Dedicated to My Parents
ACKNOWLEDGMENTS
First, I would like to thank the chairman of my Supervisory Committee, Assoc. Prof. Dr. Abu Talib Othman. His excellent supervIsIon, continuous encouragement, guidance and numerous discussions were instrumental for the completion of the thesis.
My special thanks also go to the members of the Committee, Dr. Abdul Azim Abd. Ghani and Dr. Ramlan Mahniod, for their comments, help and encouragement. I am grateful to Dr. Yazid Mohd. Saman for translating the abstract of this thesis into Bahasa Malaysia. I also wish to extend my thanks Dr. Ali Mamat, head of the Computer Science Department, and all the lecturers and staff in the department.
I am thankful to the very kind man, Mr. Mohammad Noor Azman, the system administrator, for all the belp he offered and is still offering.
To my parents, brothers and sisters, my heartfelt thank for all that no one can give but you.
My gratitude to a very special person without whom, I would not have been able to continue my higher education in this country. My thanks to Belal Mustafa Abu Ata (Abu Al Bara') and his family for all the help, support and encouragement.
I am grateful to the personnel 111 the Graduate School OJTice in UPM for their kindness and cooperation.
The financial support provided by the Public Services Department of Malaysia (JPA), is highly appreciated and gratefully acknowledged.
111
TABLE OF CONTENTS
Page
ACKNOWLEI>GMENTS til
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . VII
ABSTRACT . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
ABSTRAK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XII
CHAPTER
I.
II
INTROI>UCTION
Background of the Pr oble m
Obje cti ve s of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Organiz ation of the The sis . . . . . ... ... . . . .. . . . . .. . . . . . . . . . . . . . . .. . ..... .
SINGLE AND MULTISERVER GENERAL QUEUEING N ETWORKS USING A LINEAR ALGEBRAIC APPROACH . . . . . . .. . . . . . .. . . . . . . . . . . . . .. . .
4
7
8
9
11
Re vie w of Re late d Lite rature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . . . ] ]
Single Se rve r and Multise rve r Ge ne ral Que ue ing Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3
T he Use of t he Ge ne ralize d Expo ne ntial in Describing the General Distribution or! Gcncral Que ue ing Systems ......................................... 16
The L ine ar Alge braic Approach in the Que ue ing T he ory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 16
The Alge braic C ompone nts of a Subsyste m S . . . . 18
Se rvi ce Time Distributi on of S . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
The M/G/) Que ue Ste ady-state Solution . . . . . . . . . . . . . 22
IV
111
IV
The G/M/I Queue Steady-state Solution 25
The G/M/C Queueing System Steady-state Solution . . . . . . . . . . . . . . . . . . . . ....................... ............... 25
M1GE/l AND GE/M/I QUEUEING SYSTEMS AT EQUILIBRIUM USING THE LINEAR ALGEBRAIC APPROACH ................. ................. .
The M/GEI1 Queueing System Solution Using the
29
Linear Algebraic Approach . . . . . . . . . . . . . . . . . . . . . ............ ... 29
The M I H2 1 1 Queueing System Using the
Linear Algebraic Approach . . . . . . . . . . . . . . . . 30
Changing of Parameters .. .... ... ........ .... 35
The Generalized Exponential as the Limit of the Hyperexponential . . . . . . . . . . . .. . . . ...... . . . .. . . . . .. 36
The GE/M/I Queue Using the Linear Algebraic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Numerical Examples . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . .. . .. . . . . . . . . . . . . . . . . . . . 45
AN EXPLICIT SOLUTION OF THE GE/M/C QUEUEING SYSTEM AT EQUILIBRIUM USING THE LINEAR ALGEBRAIC
APPROACH . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The GE/M/C Queueing System Solution Using the Linear Algebraic Approach .................................. .
The H 2 I M I C Queueing System Using the ,
Linear Algebraic Approach ........................... .
Constructing the Solution of the Generalized Exponential Using its Algebraic Components
Abbreviation to MIMIC Queueing System . .. .. . . . . . . . . .
Abbreviation to GE/MI1 Queueing System . .................... .
Numerical Examples ...................................... .................... .
v
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63
71
74
75
v CONCLUSION AN[) FUTURE RESEARCH
Summary ....
Suggestion for Future Research ., .. ........ .
BIBLIOGRAPHY
APPENDIX
A The Generalized Exponential (GE) Distribution Model
B Definitions Related to the Algebraic Components of an M/G/C Queueing System
VITA .............................. .
vi
81
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Figure
2
3
4
5
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7
8
9
1 0
1 1
12
LIST OF FIGURES
A Des cript io n of a Subs yst em S Us ing t he L inear Alg ebraic Co mpo nents Fo rmalis m
An M/G/1 Clos ed L o o p Queueing Syst em. Des cript io n Us ing t he Linear Alg ebraic Co mpo nents . . . . . . . . . . . . . . . . . . . . . .
An M/G/C Clos ed Lo o p Queueing Syst em Des cript io n Us ing t he Linear Alg ebraic Co mpo nents . . . . . . . . . . . . . . . . . . . . . .
An M/H 2 11 Queueing Syst em . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
The Queue Lengt h Pro babilit ies of an M/G E/l and G E/MI 1 Queueing Syst ems w it h C� = C; = 1 . 0 and p:::: OJ ............... ... ........................................... . ........... .
The Queue Lengt h Pro babilit ies of an M/G E/ 1 and GE/MI l Queueing Syst ems wit h C; = C; == 1 .0 and p == 0.1, 0 .3 , 0 .5 , 0 .7 and 0 .9 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .
T he Queue L engt h P ro babil it ies o f an M/GE/l
Queueing Syst em w it h C; = 2 .5 and p = 0. 1 , 0. 3, 0 .5, 0 . 7 and 0 .9 . .. . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . , . . .. . . .
The Queue Lengt h P ro babilit ies of a G E/M/I Queueing Syst em w it h C: = 2 .5 and p == 0.1,0.3,0 .5,0.7 and 0.9 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Queue Lengt h Pro babilit ies o f an M/G EI1 Queueing Syst em with C; = 1 . 0,1 .5,2 . 0,2 .5,3. 0 and p = 0.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Queue Length Probabilities of a GE/MII Q ue uei ng S yst em wit h C� = 1. 0,1 .5,2 . 0,2 . 5,3 . 0 and p = 0.4 . . . .
The Queue Lengt h Pro babi lit ies of an M/G EI l and GEIMIl Queueing Syst ems w it h C; = C; = 1.5 and p = 0. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
T he Queue L engt h Pro babilit ies o f an M/GE/l and GE/MJl Queueing Syst ems w it h C; = C� :::: 3 . 0 and p = 0 . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .
vii
Page
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13
1 4
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1 9
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The Queue Length Probabi li ties of an M/GEII and GE/MI1 Queueing Systems w ith C � = C ; = 1.75 and p=0. 2 . . . . ... . . . . . . . . . . .. . .... . .. . .. . ...... . .. . . .... . . . . . ... . . . ... . . . . . .
The Queue Length Probabi l ities of an M/GEI I and GE/MIl Queueing Systems with C � = C � = 1.75 and p = 0.8 . . . .. . . . . . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An H2 /M/C Queueing System . . .. . . . . ... . .... .. . . . . . . .. . . . ........ .
The H2/M/C Queueing System with A{n) Defined in [4 .2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Queue Length Probabil it ies i n a GE/M/C Queueing System with Coefficient of Variation C ; = 4, P = 0.9 and C = I , 2, 3 and 20 . ,', .. , . . " . . . . . , . . . . . . .
The Queue Length Probabil i ties in a GE/M/C Queueing System with Coefficient of Variation C � = 4, P = 0.9 and C = 4, 5, 6 and 10 . . """ . . " . . """ . . " . , , . . . . .
The Departure o r Arrival P robabi l it ies for a GE/M/C Queueing System with p:= 0,9 , C � = 4 and C = 4, 5, 6 and 1 0 , . . " . , , . , , . . . , , . . , . . , , . . . , . . , , . . , , , . , . . . , , . , , . . . . . , . " , , . , . , . . . . .
The Departure or Arrival Probabi l ities for a GE/M/C Queueing System with p = 0.9 , C � = 4 and C = I, 2, 3 and 20 , . . , . . .. . . . , . . , . . . .. , , . . . . , , . , , . . . , . . , . . . , , . . , , . . . . . . . . . . .. . . . , . , . . .
The Queue Length Probabi l ities for a GE/M/C Queueing System with p = 0.8, C ; =4 and C = 1 , 2, 3
and 20 , . . ,', ., . ' . . . . . . , . , ., . . . . " . . . . , . " ' , " ' . " ' ," ,., . . . , . . , . . . . . . , . . . . , . , . . . " .
The Queue Length Probabilities for a GE/M/C Queueing System with p:= 0.4, C � := 4 and C = 1, 2, 3 and 20 " " " " " " " " " " " " " " " , . . . . . . , . . . . " . , . , . " . . , . ,' , . . . ,. " . " . . . , . . . ,
The Queue Length Probabi lities for a GE/M/C Queueing System with p = 0.2, C� = 4 and C = I, 2, 3 and 20 . . " ' , " " ' , . , . . , . , . . , . . " . " . . . , . . . , ', . , . . . , " ,' , . . . . . ,' , . . , . . . , . " " . ,, ., , .
The Queue Length Probabilit ies for a MIMIC Queueing System with p = 0.9 and C:= 1, 2, 3 and 20 . . . . " . . " . . .
An M I H2 I C Queueing System "" . . . . . . . . . . . . . . . . . " . " . . . . . ..
viii
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26 A < p, B > Server in an M I H 2 I C �ueuelIlg
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
ix
Abstract of thesis submiued to the Senate of Universiti Pertanian Malaysia in fulfilment of the requirements for the I degree of Master of Science .
A SOLUTION OF SINGLE SERVER AND MULTISERVER GENERAL QUEUEING SYSTEMS USING A LINEAR
ALGEBRAIC APPROACH
by
MOHAMMAD SALEH MUSTAFA
December, 1995
Chairman Assoc. Prof Dr. Abu Talib Othman
Faculty Science ancl Environmental S tudies
Performance evaluation of computer systems and communication
networks is very important due to the essential need of finding such
systems which can meet all the objectives, for which these resources are
buil t, at an acceptable cost. Many methods have been applied to describe
the performance of these resources and to find possible ways of
improving performance . Each of these methods has its own characterist ics
and procedure in dealing with the performance concept . These procedure
describe these systems such that a previously specified technique can be
used to evaluate usefu l performance measurements which wi l l help
decision making .
x
The queueing network model is the most widely used computer
systems and communication networks description method. Traditional
techniques used in solving the above method use unrelated mathematical
techniques . The linear algebraic approach has been chosen as the
mathematical approach in this thesis . This approach transforms the
queuemg theory problems from those of integral equations into those of
algebraic equations over a finite dimensional vector space. The solutions
obtained using this approach can now make use of the high-speed
parallel processors since the components of these solutions are in matrix-
vector format . In this research, the linear algebraic approach is used to
derive explicit solutions for single server and multi server general queueing
systems . The research approach uses the generalized exponential to
represent the G-type distr ibution because of the tractable facilit ies that it
offers e .g ., i t facilitates the derivation of product form solutions . The
performance measurements derived are the queue length probabilit ies, mean
; queue length, throughput, mean time a customer spends in the system
and departure and arrival probabil i ties . The der ived solutions can be
comprehended, gIve a clear insight about the behaviour of the systems
and are easy to evaluate for any gIven system parameters. Numer ical
examples showing the clTccts of changing the parall1eters or the solutions
obtained on the performance of the systems are studied.
XI
Absttak tesis yang dikemukakan kepada Senat Universiti Pertanian Malaysia bagi memenuhi keperluan untuk Ijazah Master Sains.
PENYELESAIAN SISTEM G1LlRAN AM
SATU PELAYAN DAN MULTIPELAYAN MENGGUNAKAN PENDEKATAN AL.JABAR LINEAR
Oleh
MOHAMMAD SALEH MUSTAFA
Disember, 1 995
Pengerusi: Prof Madya. Dr. Abu Talib bin Othman
Fakulti: Sains dan Pengajian Alam Sekitar
Penilaian pres tasi unluk sislcl11 komputcr dan rangkaian kOlllll nikasi
adalah sangat mustahak. Ini d isebabkan oleh keperlllan ulama untllk
mencari satll sistem dengan segala penggunaan sumber yang mahal
di laksanakan. Tambahan pula, sel11ua objektif untuk pembinaan sumber ini
dapat dicapai dengan kos yang munasabah. Terdapat banyak kaedah untuk
menerangkan prestasi sllmber ini dan I11cl1cari jalan lIntlik I11cl1Ipcrbaikinyn
Setiap kaedah mempunyai ciri-ciri dan prosedurnya dalam mengendal i
konsep prestasi ini. Prosedur ini menerangkan sistclll berdasarkan tcknik
yang terdahulu untuk menilai pengukur prestasi yang akan membantu
dalam menetapkan keputusan.
xii
Selalllnya, teknik tradisi dalam penyelesaian model rangkaian giliran
(scpcrli dalam kacdah pCl1crangal1 dalalll sislcl11 kOl11pulcr dan l"angkaian
komunikasi) juga ter l ibat dalam teknik Matematik yang t iada kai tannya.
Pendekatan atjabar l inear dipil ih sebagai pendekatan Matemat ik dalam tesis
in i. Ia menterjemah masalah dalam teori gil iran dari persamaan integral ke
dalam persamaan algebra untuk ruang vektor berdimensi terhad. Hasil dari
penggunaan pendekatan ini boleh digunakan dalam pemproses selari yang
laju kerana komponen hasil berformat matrik-vektor. Di dalam
penyelidikan In! pendekatan aljabar l inear digunakan untllk mcncart
penyelesaian tersurat untuk sistem gil iran satu pelayan dan multi pelayan.
Penyelidikan in i menggllnakan kaedah eksponen yang am untuk mewakil i
taburan jenis-G kerana kemlldahan yang ada berbentuk hasi l darab
Pengukuran prestasi yang diperolehi terdir i dari probabi l iti pal�ang gi l iran,
purata pa�ang giliran truput, purata masa pelanggan mcnllnggu dalam
sistem dan probabil it i ketibaan dan keluar . Hasil yang dicapai boleh
difahami, member ikan pandangan yang jelas berkenaan sistem dan mudah
lIntuk dinilai dengan memberikan sebarang parameter . Contoh nllmerikal
yang menu�lIkkan kesan perubahan parameter hasi l yang diperolehi dar i
prestasi sistem juga c l ikaji.
xiii
CHAPTER I
INTRODUCTION
Computer communication network can be viewed as a facility that
makes possible communication between computers and other devices.
According to Hammond and O'Reilly (1986), it came into being when
intelligent devices such as the devices of a data communication system
including control and graphic terminals, rotation and applications seIvers,
multiplexing and switching devices, modems, videos, facsimile, scanners
and printers were moved from the main frame to remote locations.
Increased use of sophisticated computer and communication
systems which are expensIve resources, makes it imperative that they are
used efficiently. To achieve this ann, performance evaluation of such
systems must be considered at all stages of the system life cycle (design,
development, configuration and tuning).
Computer and communication systems performance methods arc
dividcd into two main areas: performance measures and performance
modeling methods (Ferrari, 1978). The measurement method is applied by
using a direct measure to the performance of the system using hardware
dedicated and/or software dedicated tools. This method is intractable
since the system needs to be operational. As a result, the performance of
the system during design and development stages can not be analyzed by
2
usmg this method. Moreover, this method is very complex because it
involves considerable human and machine cost. I Ience, model ing
techniques are preferred.
According to Buchholz ( 1 992), performance analysis of comp lex
computer systems is often accomplished by means of models which can
be mapped on Markov processes. Evaluation and prediction of the
stochastic behaviour of work flow in the system are undertaken using
these models. As a result, usefu l system performance measurements can
be obtained to aid decision making. In specifying these models,
description techniques such as queueing networks (QN) (Lavenberg,
1983), generalized Petri nets (PN) (Ajmone-Marsan el al., 1984) or
networks of stochastic automata (Plateau, 1985) are used.
A computer system can be viewed as a network of queues. It is
generally characterized as a set of software and hardware resources
(limited number) and a set of multiple jobs or tasks using these
resources, in which a queueing situation wil l occur. From this point of
view, a computer system can be thought of as a queueing system 1 11
which computer resources represent the servIce centres and the jobs
represent the customers. The results developed frol11 this queuell1g system
such as queue lengths, queue delay, etc., are then used to predict the
performance of the real system.
According to Lazowska et al. (1984), the importance of queueing
network models as modeling tools lies in their ability to achieve a
favourable balance between accuracy and efficiency m most applications.
f'or this reason, queueing network models arc used 1 11 dincrelll
application areas such as telecommunication systems, Iransportation
systems, banking systems, production systems, computer systems, health
care systems and in many service systems where a queueing situation
occurs. Even simple queueing network models are able to yield fa irly
accurate predictions of the behaviour of complex systems. Thi s has
helped in stimulating queueing network research .
The application of probabi l ity theory to these problems can be
traced to 1917 when a Danish engineer (Erlang, 1917) analyzed the
behaviour of simple queues to assist him in designing a telephone
exchange (King, 1990). Jackson (1957) was responsible for a key paper
entitled "Network of Waiting Lines". A review of related l i terature
covenng a sixty-year period unti l 1967 can be found in Hhat (1969).
The use of queueing nctwork models to evaluatc the performance of
computer systems and computer communication networks began In the
seventies (Kleinrock, 1975; Allen, 1978) . Extensive research was carried
out on this fiel d in the seventies and eighties. Disney and Konig (1985)
reviewed more than 300 related articles whi le Prabhu (1987) compiled a
bibliography of books and survey papers on queueing systems which
included the surveys by Lavcnberg (1983) and Kleinrock (1988). As a final comment, performance evaluation cannot be studied In
isolation as i t cannot be separated from queueing systems and networks.
It is generally agreed that a research in performance evaluation research
IS realy research on queueing theory. Moreover, this theory has had an
important effcct on communication systcms and nctworks. Indeed, queuclllg
theory was first applied in communication systems, whcrc it is still in llse
today (Pekergin, 1990).
Background of the Problem
A queUCIl1g network can be modeled using three mall1 techniques.
These are simulation, approximation, and analytical techniques. Continuolls
and discrete-event queueing model simulations (Mitrani, 1982; Sauer and
MacNair, 1983) are effective techniques for evaluating systcm pcr/tlrlnancc.
However, simulation techniques are costly and have two main constraints.
The first is that the levcl of details in the model and computation is timc
consumIl1g while the second is the difficulty of creating the simulation
programme and verifying its accuracy.
Approximate methods arc orten uscd for solving qucucll1g nctwork s
Il1 cases where thc exact solution IS not known or computationally
intractable. These techniques are applied only under certain considerations
which cannot be satisfied for a wide range of systems. For instance,
diffusion methods, which arc used to approximate the discrcte now of
customcrs by continuous flow, are only accurate under heavy traffic
conditions. On the other hand, the maxImum entropy method needs certain
known results which have to be obtained independently by some other
technique (Harrison and Patel, 1993).
descriptions of the system under consideration. These techniques arc
preferred to than the other techniques because they (i) provide
quantitative predictions and insights into the structure and behaviour of
the real system, (ii) provide cost effective methods, (iii) are applicable in
all stages of the system life-cycle, and (iv) a considerable amount of cost
savings can be obtained as a result of early recovery and elimination of
distressed designs from performance viewpoint in these stages
Some of the computer systems and communication networks
aspects can be modeled as product form queueing networks (Kelly, 1 979;
Walranc1, 1988; Van D�jk, \993) which is an important class of
stochastic analytic modeling technique. These tractable models can be
solved efficiently with a variety of computatior1al algori thms such as mean
value analysis (MY A) (Reiser and Lavenberg, 1980), recursion by chain
(Conway and Georagallas, 1(86), and convolution (BuZCI1, 1 973)
However, not all of computer and communication network aspects
can be modeled as product form networks. In addition, to that, there is
another general and important class of stochastic models whicb can be
modeled by the presence of imbedded Markov chains that are two
dimensional generalizations of elementary M/G/I and (J/M/I queues.
These are the matrix-geometric techniques. These techniques enable the
consttuction of computationally efficient algorithms which arc applicable
for a large number of general queueing networks and lead to closed-form
solutions (Nellts, 1981; Neuts, 1989; Nelson, 19(1).
(i
Throughout this thesis, the linear algebraic approach is used. In
this approach queueing theory problems are transformed frol11 those of
integral equations into those of algebra ic equations over a finite
dimensional vector space (Lipsky, [985a). This is done by representing
each nonexponential service time (or interarrival lime) distribution by a
collection of exponential servers, with the constraint that the col lection
can only be accessed by one customer at a time (Carrol el ar, 1982)
In using the traditional techniques to solve the queueing system, problems
arise from the fact that these techniques usually involve uSlIlg unrelated
mathematical techniques in which integrals play a very imp0l1ant rule .
Solutions to problems 111 simple systems can be understood using such
techniques. However, ""hen problems anse III more complex systems,
such techniques become very complicated, ditlicult to analyze and in
many cases, it can be quite dinicu lt to evaluate the solution rot a given
system parameters.
The linear algebraic approach is used to derive the solution of
single server general queueing systems from the type M/G/I and (J/M/I
and the solution of multi server general queueing networks using the
generalized exponential (GE) pdf (sec Appendix A) to represellt the (i-
type interarrival or service time distribution with known first two
moments. The generalized exponential (GE) distribution which is defined
as (E1-Affendi and Kouvatsos, 1983; Kouvatsos, 1986a; 1988),
t � 0 r 1. 1]
7
where �l i s the mean servIce rate, C is the coefficient of variation and
U o ( t ) IS the unit impulse function. This has been chosen as the best
choice to represent the G-type distribution. According to Kouvatsos
( 1 988), the interest in the GE distribution as a universal model to
approximate G-type d istributions (when only the first two moments are
known) has been motivated by some of its robust and versati le properties
which make it particularly useful for the analysis of multiserver queues
and general queuing networks. Moreover, it faci l itates the derivation of
product form approximations for isolated queues and networks
(Kouvatsos, 1 986a; I 986b).
The solutions derived for the systems mentioned with the (GE)
distribution representing the G-type using the l inear algebraic techniques
can be understood, give a clear insight about t he system and are easily
evaluated for any given system parameters.
Objectives of the Study
This study has been undertaken work i n this thesis had been done
to achieve four main objectives. The first is to prove the power and
ability of the li near algebraic approach in the qucueing thcory by two
means: ( i ) comparing different techniques used to represent the theory to
show the potential and l imitations of each, and (ii) representing the l inear
algebraic approach and its mechanism to derive the solutions. The second
objective is to derive the solution of single server general queueing
networks the M/GII and G/MII with the generalized exponential 'GE'
representing the G-type pdf. Deriving an expl icit solution for the
multi server general queueing systems from the type G/M/C with the (GE)
representing the G-type pdf is the third objec�ive of this thesis The Iinal
objective i s to provide a clear and proven method for on-going research
on this topic by suggesting a way to explore an explicit solution fl.)! the
M/GIC queueing system with the 'GE' representing the G-type pdf. The
approach is d iscussed briefly in the section on further research on this
queuetng system.
The Approach
The research methodology starts with formulation of the algebraic
components describing the general queuel11g system uSl11g the
hyperexponcntia[ (H 2 ) pdf to represent the G-type. Then the solution of
the system using these algebraic components i s constructed. There are
infini tely many two-stage models representing the G-type with the same
mean value and squared coefficient of variation . The prevIOus point
shows the need to control the manner i n which such representations are
constructed. This necessitates the changing of parameters (EI-Affendi and
Kouvatsos, 1 983), which leaves us with the mean, squared coellicicnt or
variation and a new parameter k. Subsequantly, the two-stage
convergence to the (GE) pdf i s followed through by taking the l imit
when the parameter k approaches infinity. The resultant solution i s
simplified and rearranged to achieve the expl ici t solution. As a result, the
I)
system can be studied under various conditions to gIVe an instant view
of the behaviour of the real system easi ly, without encountering
evaluat ion difficulties.
Organization of the Thesis
The thes is cons ists of five chapters, organized m a manner that
gives a clear indication of the focus of the research, pavmg the way for
further research. The outline of the chapters fol lows:
Chapter I: This chapter presents an overview of the importance of
queue1l1g theory in the performance evaluation of computer systems and
communication networks. The potential and l imitations of the vanous
techniques used to evaluate a computer system or a communicat ion
network are discussed . This chapter alsohighlights the power of the l inear
algebraic approach as the approach used throughout this study . This
chapter takes a brief look at the methodology used and the objectives of
the thes is and concludes with an outline of the thesis.
Chapter II: The basis of the l inear algebraic approach of the queuemg
theory is discussed in this chapter by a review of the approach stages
until the current stage. Literature which relate to the analysis and
modeling of the s ingle server general queueing networks and multiserver
general queueing network are reviewed and documented. Final ly, the
linear algebraic approach is discussed briefly showing the construction of
the algebraic components of a given system and the mechanism to derive
the solution from these components .
\0
Chapter HI: The solution of single server general queueing networks
M/G/l and G/M/l with the (GE) pdf describing the G-type is derived In
this chapter uSing the linear algebraic approach. The performance
measurements of the systems such as queue length probabilities,
throughput, etc. are gIven. The systems are compared under d ifferent
conditions which may reflect the behaviour of real l ife systems.
Chapter IV: This chapter d iscusses the queue length probabi l it ies
departure and arrival probabil it ies of multi server general queueing systems
of the type GE/M/C i .e . , for load dependent multi server general queueing
systems when the (GE) pdf represents the G-type. The system is
analyzed uSing this approach under different conditions to give an insight
i nto real systems performance.
Chapter V: This chapter concludes the research with a summary of the
main ideas. Further work is encouraged on . the exploration of the
solution of M/GE/C queueing system using the l inear algebraic approach
