Table of contents for Applied stochastic processes / Mario Lefebvre.


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1   Review of Probability Theory . .. ..........          . .....  1
1.1 Elementary probability. . ..... ....         . .... . . 1
1.2  Random  variables  . . . . . .   .          .  . . .... .  8
1.3 Random vectors .........   .    .............           21
1.4  Exercises.... ....   . ..    .              ........ .  34
2   Stochastic Processes      . . .      .   . . . . . .  . . . . . . . . . . 47
2.1 Introduction and definitions  .  . . . . .  .  . . . . .  47
2.2 Stationarity .......                   .    .........  52
2.3 Ergodicity .........     ......        .     ....... 55
2.4 Gaussian and Markovian processes . ................ .58
2.5 Exercises ..  ,....,.,   ,,. ...            ... .....   65
3   Markov Chains ....                               .          73
3.1  Introduction  ....,. .. ..                  ... ...  .  73
3.2 Discrete-time Markov chains . . . . .. . . .  . . . . . . . . . 77
3.2.1  Definitions and  notations . . . . . . . . . . .  ..  . .  .. . .  .  . .  77
3.2.2  Properties  .  . .      .    .  .  . .  . . . . . . . . .  85
3.2.3  Limiting  probabilities ..  .   . . . . . . .  . . . . . . .  . .  94
3.2.4 Absorption problems. . . . . .  . .  . . . . . . . . . . 100
3.2.5  Branching processes  . . .. .....   .  . . . . . . . . . . 104
3.3 Continuous-time Markov chains . . . . . . ...  . . . .. . . . . . . . . 109
3.3.1 Exponential and gamma distributions ........ ...... 109
3.3.2 Continuous-time Markov chains ....... .............  121
3.33 Calculation of the transition f\nction p,,j(t) . ......  124
3.3.4  Particular processes  ..:.. . .                  129
3.3 5 Limiting probabilities and balance equations ......  138
3.4 Exercises . .143
4   Diffusion Processes .    .... .                              173
4.1 The Wiener proc ess .. ....................              173
4.2 Diffusion processes ..  .......... ....... ...           181
4.2.1 Brownian   oton with drift  .............. .      183
4.2.2  Geometric Brownian motion ._. .. .....           185
1.2.3 Integrated Brownian motion ..    ..... ..... __191
4.2.4 Brownian bridge ........... ....... .... 196
4.2.5  T he Ornstein-Uhlenbeck process ......... ...... 199
4.2.6  The Bessel process  ...................... 204
4.3 White noise ...                                    .... 207
4 4 First-passage problems ......................            214
4.5 Exercises . ..... ......      . .   .               ....... 222
5   Poisson Processes           .... 231
5.1 The Poisson process .  ........................          231
5.1.1 The telegraph signal .....................        248
5.2 Nonhomogeneous Poisson processes ............ ... 250
5 .3 Compound Poisson processes.. .      ... ...  ... ...  .. 254
5. 4 Doubly stochastic Poisson processes ....   . .. .. ... ..258
5 5 Filtered Poisson processes ...............       .. .   264
5 6 Renewal processes ...................            ...  . 267
5.6.1 Limit theorems. ...   ..... ... .. ....       ...  278
5.6.2 Regenerative processes ..               .   ..... 284
5.7 Exercises .      . .    . ...... . . . . .. ........ .. .. 289
6   Queueing Theory ..L ..             ....      . ..  . .   . . 315
6 1 Introduction . .. ..  . ......      ..       ...315
6.2  Queues with a single server ...............    ...... 319
6.21 The model AlM/M/1 ...............           .      319
6.2 2 The model AM/M/A1/c ................... ....327
6 3  Queues with many servers .....................   .     . 332
6.3 1 The model M/M/s                               ... 332
),3 2  The model Al/Ml/s/c and loss sysems stems ......336
6 3 3 Networks of queues .                         ... .. 342
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Library of Congress subject headings for this publication: Stochastic processes