Preface ......................................................... x
1 Preliminaries .............................................. 1
1.1 Basics of Probability ...................................... 1
1.1.1 Introduction ........................................ 1
1.1.2 Conditional Probability ............................. 2
1.2 Discrete Random Variables and Distributions ................ 4
1.3 Discrete Moments ........................................... 8
1.4 Continuous Random Variables, Density, and Cumulative
Distribution Functions .................................... 13
1.5 Continuous Random Vector .................................. 17
1.6 Functions of Random Variables ............................. 19
1.7 Continuous Moments ........................................ 23
1.8 Difference Equations ...................................... 25
1.8.1 Introduction ....................................... 25
1.8.2 Basic Definitions and Properties ................... 25
1.9 Methods of Solving Linear Difference Equations with
Constant Coefficients ..................................... 27
1.9.1 Characteristic Equation Method ..................... 27
1.9.2 Recursive Method ................................... 29
1.9.3 Generating Function Method ......................... 30
1.9.4 Laplace Transform Method ........................... 32
Exercises ................................................. 36
2 Stochastic Processes ...................................... 39
2.1 Introduction and Basic Definitions ........................ 39
2.2 Markov Chain .............................................. 43
2.2.1 Classification of States ........................... 53
2.3 Markov Process ............................................ 58
2.3.1 Markov Process with Discrete Space State ........... 58
2.4 Random Walk ............................................... 61
2.5 Up-and-Down Biased Coin Design as a Random Walk ........... 69
Exercises ................................................. 75
3 Birth and Death Processes ................................. 77
3.1 Overviews of the Birth and Death Processes ................ 77
3.2 Finite В-D Process ........................................ 86
3.3 Pure Birth Process (Poisson Process) ...................... 94
3.4 Pure Death Process (Poisson Death Process) ................ 96
Exercises ................................................. 97
4 Standard Queues .......................................... 101
4.1 Introduction of Queues (General Birth and Death
Process) ................................................. 101
4.1.1 Mechanism, Characteristics, and Types of Queues ... 103
4.2 Remarks on Non-Markovian Queues .......................... 108
4.2.1 Takacs's Waiting Time Paradox ..................... 108
4.2.2 Virtual Waiting Time and Takács's Integro-
Differential Equation ............................. 109
4.2.3 The Unfinished Work ............................... 113
4.3 Stationary М/М/1 Queueing Process ........................ 116
4.4 A Parallel M/M/C/K with Baking and Reneging .............. 119
4.5 Stationary M/M/1/K Queueing Process ...................... 120
4.6 Busy Period of an M/M/1/K Queue .......................... 122
4.7 Stationary M/M/l and M/M/1/K: Queueing Processes with
Feedback ................................................. 124
4.7.1 Stationary Distribution of the Sojourn Time of
a Task ............................................ 126
4.7.2 Distribution of the Total Time of Service by
a Task ............................................ 128
4.7.3 Stationary Distribution of the Feedback Queue
Size .............................................. 129
4.7.4 Stationary Distribution of ςn (Sojourn Time of
the nth task) .................................... 130
4.8 Queues with Bulk Arrivals and Batch Service .............. 131
4.9 A Priority Queue with Balking and Reneging ............... 133
4.10 Discrete Time M/M/l Queueing Process, Combinatorics
Method (Lattice Paths) ................................... 137
4.10.1 The Basic Ballot Problem .......................... 138
4.10.2 Ballot Problem (based on Takács 1997) ............. 140
4.10.3 Transient Solution of the M/M/l by Lattice Path
Method ............................................ 149
4.11 Stationary M/M/C Queueing Process ........................ 153
4.11.1 A Stationary Multiserver Queue .................... 154
5 Queues With Delay ........................................ 159
5.1 Introduction ............................................. 159
5.2 A Queuing System with Delayed Service .................... 163
5.3 An M/G/1 Queue with Server Breakdown and with Multiple
Working Vacation ......................................... 172
5.3.1 Mathematical Formulation of the Model ............. 173
5.3.2 Steady-State Mean Number of Tasks in the System ... 173
5.3.3 A Special Case .................................... 183
5.4 A Bulk Queuing System Under N-Policy with Bilevel
Service Delay Discipline and Start-Up Time ............... 185
5.4.1 Analysis of the Model ............................. 186
5.5 Interrelationship between N-Policy M/G/l/K and
F-Policy G/M/1/K Queues with Start-up Time ............... 188
5.5.1 N-policy M/M/1 queueing system with exponential
startup time ...................................... 189
5.5.2 F-Policy G/E/l/K Queuing System with Exponential
Start-up Time ..................................... 195
5.6 A Transient M/M/1 Queue Under (M, N)-Policy, Lattice
Path Method .............................................. 199
5.6.1 Solution in Discrete Time ......................... 200
5.6.2 Solution in Continuous Time ....................... 206
5.7 Stationary М/М/1 Queuing Process with Delayed Feedback ... 208
5.7.1 Distribution of the Queue Length .................. 209
5.7.2 Mean Queue Length and Waiting Time ................ 213
5.8 Single-Server Queue with Unreliable Server and
Breakdowns with an Optional Second Service ............... 222
5.9 A Bulk Arrival Retrial Queue with Unreliable Server ...... 229
5.9.1 The Model ......................................... 231
5.9.2 Model Analysis .................................... 233
5.9.3 Steady-State System Analysis ...................... 237
5.9.4 Performance Measures .............................. 244
5.9.5 Numerical Illustration ............................ 248
5.10 Multiserver Queue with Retrial Feedback Queuing System
with Two Orbits .......................................... 253
5.11 Steady-State Stability Condition of a Retrial Queuing
System with Two Orbits, Reneging, and Feedback ........... 258
5.11.1 Necessary Stability Condition for the Steady-
State System ...................................... 259
5.12 Batch Arrival Queue with General Service in Two
Fluctuating Modes and Reneging During Vacation and
Breakdowns ............................................... 263
5.12.1 The Model ......................................... 263
5.12.2 Analysis .......................................... 265
Exercises ................................................ 266
6 Networks of Queues with Delay ............................ 267
6.1 Introduction to Networks of Queues ....................... 267
6.2 Historical Notes on Networks of Queues ................... 270
6.3 Jackson's Network of Queues .............................. 272
6.3.1 Jackson's Model ................................... 273
6.4 Robustness of Networks of Queues ......................... 298
6.5 A MAP Single-Server Queueing System with Delayed
Feedback as a Network of Queues .......................... 302
6.5.1 Description of the Model .......................... 304
6.5.2 Servicestation .................................... 307
6.5.3 Stepwise Explicit Joint Distribution of the
Number of Tasks in the System: General Case When
Batch Sizes Vary Between a Minimum к and
a Maximum K ....................................... 319
6.6 Unreliable Networks of Queueing System Models ............ 336
6.6.1 Unreliable Network Model of Goodman and Massey .... 337
6.6.2 Unreliable Network of Queues Model of Mylosz and
Daduna ............................................ 340
6.6.3 Unreliable Network of Queues Model of Gautam
Choudhury, Jau-Chuan Ke, and Lotfi Tadj:
A Queueing System with Two Network Phases of
Services, Unreliable Server, Repair Time Delay
under N-Policy .................................... 348
6.7 Assessment of Reliability of a Network of Queues ......... 363
6.8 Effect of Network Service Breakdown ...................... 365
6.8.1 The Model (CoginfoCom System) ..................... 366
6.8.2 Analysis .......................................... 368
6.8.3 Numerical Example ................................. 370
Exercises ................................................ 374
References .................................................... 377
Index ......................................................... 391
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