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| Àâòîðèçàöèÿ |
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| Ïîèñê ïî óêàçàòåëÿì |
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| Kao E. — Introduction to Stochastic Processes |
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| Ïðåäìåòíûé óêàçàòåëü |
 queue 91
 queue 67—68
 queue 65—67
 queue, departure process 68—69 82—83
 queue, waiting time, distribution of 25—26
 queue, time dependent probability 90
Absorbing state 174 241
Accessibility 172
Alternating renewal process see "Renewal process"
Aperioidic state and chain 171
Arbitrage 402
Arrival theorem see "Closed queueing network"
Asymptotic-rate theorem 27 42 100
Balance equation see also "Detailed balance equation"
Balance equation for continuous-time Markov chain 253—254
Balance equation for discrete-time Markov chain 177—179
Bellman's functional equation 409
Beta of a security 407
Birth and death process 242—243 254—255 260—263
Birth rates 242
Black — Scholes differential equation 402—403
Blackwell's renewal theorem 128
Branching process 40—41
Brownian motion 374—375
Brownian motion, arithmetic 397 399
Brownian motion, geometric 398—401
Brownian motion, integrated 417
Brownian motion, reflected at origin 419
Brownian motion, standard 394
Brownian motion, with two barriers 384—385
Call option, European 403—404
Canonical form of Markov chain 173 185—186
Capital asset pricing 407—409
Cash management problem 190—192 214 282—284
Censored discrete-time Markov chain 200—201
Central limit theorem 128 401
Central server model 306
Cesaro limit see "Time average"
Chapman — Kolmogorov equation of continuous-time Markov chain 240
Chapman — Kolmogorov equation of diffusion process 385—386
Chapman — Kolmogorov equation of discrete-time Markov chain 165—166
Characteristic function 28
Closed queueing network 293—294 308
Closed queueing network with blocking 296 314—315
Closed queueing network, arrival theorem for 295
Closed queueing network, Buzen's algorithm 294—295 320
Closed queueing network, normalizing constant 294
Closed queueing network, product form solution 294
Closed queueing network, waiting time at server 295
Coefficient of variation 73 305
Communication 168
Compering exponential 19—21 242 350
Compound random variable 10
Compound random variable, generating function 10
Compound random variable, mean and variance 10—11
Contingent claim 402
Continuous distribution, random variable,  with balanced means 25 305
Continuous distribution, random variable, bivariate normal 408
Continuous distribution, random variable, Erlang 21
Continuous distribution, random variable, exponential 19
Continuous distribution, random variable, gamma 21 123 126
Continuous distribution, random variable, generalized Erlang 267
Continuous distribution, random variable, hyperexponential 268
Continuous distribution, random variable, mixture of generalized Erlang 269
Continuous distribution, random variable, mixture of two Erlang 305
Continuous distribution, random variable, multivariate normal 395
Continuous distribution, random variable, normal 281 386 400 403
Continuous distribution, random variable, Weibull 205—206
Continuous-time Markov chain 239
Continuous-time Markov chain as Markov renewal process 331
Continuous-time Markov chain of QBD type 309—310
Continuous-time Markov chain two-dimensional 306
Continuous-time Markov chain, absorbing chains 256—258
Continuous-time Markov jump process 276
Convolution 18 335
Counting process 5
Coupon collection problem 199—200
Covariance 395 417
Current life, or age, or backward recurrence time 107
Current life, or age, or backward recurrence time, distribution of 114—115
Data base management 89 219 381—382
Death rates 242
Defective random variable 130 170
Delayed renewal process see "Renewal process"
Denumerable Markov chain 162
Derivative security 402
Detailed balance equation of reversible Markov chain 209
Diffusion equations for Brownian motion, backward equation 376—377
Diffusion equations for Brownian motion, forward equation 377—378 418
Diffusion process 385
Diffusion process with reflecting barrier 386
Diffusion process, generalized 391—393
Diffusion process, limiting density 392
Directly Riemann integrable 102 120
Discounted reward 203 277—280
Discrete distribution, random variable, binomial 8
Discrete distribution, random variable, generalized negative binomial 198—199
Discrete distribution, random variable, geometric 9 11 130 171
Discrete distribution, random variable, negative binomial 9
Discrete distribution, random variable, Poisson 8
Discrete renewal process 144—146
Discrete renewal process, delayed 171
Discrete renewal process, excess life 145—146
Discrete renewal process, limiting excess life 211—212
Discrete-time Markov chain 162
Discrete-time Markov chain of GI/M/1 type 223
Discrete-time Markov chain of QBD type 221
Discrete-time Markov chain, absorbing chain 173—174 188—190
Discrete-time Markov chain, classification of state 167—169
Discrete-time Markov chain, decomposition of state space 172—173 229—230
Discrete-time Markov chain, embedded in a continuous-time Markov chain 242 274
Discrete-time Markov chain, embedded in a Markov renewal process 337 343
Discrete-time Markov chain, equivalence class 168
Discrete-time Markov chain, n-step transition probability 165
Discrete-time process 2
Drift of a Brownian motion 376
Elementary renewal theorem 101 128
Equilibrium renewal process see "Stationary renewal process"
Ergodic discrete-time Markov chain 175—176
Erlang loss formula 288
Excess life, or residual life, or forward recurrence time 107
Excess life, or residual life, or forward recurrence time under semi-regenerative process paradigm 349
Excess life, or residual life, or forward recurrence time, distribution of 109
Excess life, or residual life, or forward recurrence time, limiting distribution of 110
Excess life, or residual life, or forward recurrence time, limiting mean excess life 110
Excess life, or residual life, or forward recurrence time, time average of expected excess life 127
Filtered nonhomogeneous Poisson process 91
Filtered Poisson process 76—80
Filtered Poisson process, mean and variance for 79
Filtered Poisson process, probability generation function for 79
Filtered Poisson process, response function of 77
Final value property of generating function 6 35 41
Final value property of Laplace transform 18 35 41
First passage time of Brownian motion 379—380 395 419
First passage time of discrete-time Markov chain 168—169 175 190
Flexible manufacturing system 296—297 306—308 365—366
Fork Join queue 308
Forward contract 401—403
Function of bounded variation 31
Fundamental matrix of continuous-time Markov chain 257
Fundamental matrix of discrete-time Markov chain 189
Gain rate 203 277
Gamma function 21
Gauss kernel 418
Gaussian process 395
Generating function 6 294
Generating function, finding moments 8
Generating function, numerical inversion 16—17 44—45
Geometric transform see "Generating function"
GI/G/1 queue, diffusion approximation 386—389
| GI/G/c queue, diffusion approximation 393—394
GI/M/1 queue under Markov renewal paradigm 329
GI/M/1 queue, limiting probabilities 181—183
GI/M/1 queue, queue length distribution 359—362
GI/M/c queue queue 86—87
Gordon — Newell network see "Closed queueing network"
Hamilton — Jacobi — Bellman equation 411—412 414—416
Holding time distribution 323
Holding time distribution, mean 337
Independent increments 4 49 51 400
Index bond, Fisher's model of 405—407
Index set 2
Infinitesimal generator 242
Instantaneous state 241
Insurance problem 21—24
Integrate intensity function 56
Intensity function 56
Inventory models 88
Inventory models, (S-1,S) policy 90
Inventory models, continuous review (s,S) policy 120
Inventory models, continuous review, unit demand 147—148
Inventory models, discrete-time with batch demands 346—347
Inventory models, periodic review (s,S) policy 113 149 164 167 204
Inventory models, two-echelon repairable item 312—313
Irreducible Markov chain 168
Ito's lemma, process 398—399 404—405
Jackson network see "Open Jackson network"
Jensen's inequality 407
Kelly's lemma 289
Key renewal theorem 102
Kolmogorov criteria see "Reversible Markov chain"
Kolmogorov differential equations, backward 245—246
Kolmogorov differential equations, forward 246 265
Kolmogorov differential equations, Laplace transform 246—247 265
Lack of anticipation assumption (LAA) 85
Laplace transform 17
Laplace transform of renewal function 99—100
Laplace transform, finding moments 18
Laplace transform, numerical inversion 25 26 45—46
Laplace — Stieltjes transform 32—34 331
Laws of total probability 12 19 103 331 351
Left limit 28
Leibnitz's rule 35—36
Length biased sampling 108 157
Limiting state probability of continuous-time Markov chain 252—254
Limiting state probability of discrete-time Markov chain 175—178 233
Limiting state probability of Markov renewal process 338
Limiting state probability, solution by state reduction 201—203
Little's formula 140—141 293
Little-oh function 20 22—23 34—35 65—66
M/G/1 queue under Markov renewal paradigm 325—327
M/G/1 queue with group arrivals 366
M/G/1 queue, control with an N policy 138—140 366
M/G/1 queue, Laplace — Stieltjes transform of sojourn time 223
M/G/1 queue, limiting probabilities 180—181
M/G/1 queue, Pollaczek — Khintchine formula 218
M/G/1 queue, queue length distribution 136—138 350—351 357—359
M/G/1 queue, special service after an idle period 366
M/G/c queue 86
M/M/1 queue 263
M/M/3 queue, departure process of 327—328
M/M/s queue 244 255—256 263—264 285—286
M/M/s/s queue 287—288
M/PH/1 queue 224 310
Marked Poisson process see "Poisson process"
Markov chain subordinated to a Poisson process 273 278
Markov jump process with reward 277—282
Markov process 385
Markov renewal equation 336
Markov renewal equation, limiting solution 339
Markov renewal function 331—336
Markov renewal process 323
Markov renewal process, embedded in semi-Markov process 339—340
Markov renewal process, embedded in semi-regenerative process 348—351
Markov renewal type of equation 352
Markov reward processes, continuous-time 277—281
Markov reward processes, discrete-time 203
Markovian property 162
Matrix exponential 247—248 251
Maximum likelihood estimator 62
Mean holding time 337
Mean value function 57
Memory less property 21 241
Merton's Portfolio selection problem 398 415—417
Method of embedding 164
Moment generating function 27
Moment generating function of normal distribution 28
Multiprogramming system 390—391
Nonhomogeneous Poisson process 56—58
Nonhomogeneous Poisson process, arrival time condition on arrival count 58—59
Nonhomogeneous Poisson process, filtered see "Filtered nonhomogeneous Poisson process"
Nonhomogeneous Poisson process, transform arrival time to that of Poisson 60—61
Nonstationary Poisson process see "Nonhomogeneous Poisson process"
Null recurrent state 169 175
Occupancy-time statistics of continuous-time Markov chain 257—258
Occupancy-time statistics of discrete-time Markov chain 171 189
Open Jackson network 290—293
Open Jackson network, product form solution 291
Ornstein — Uhlenbeck process 417
Palm theorem 56
Palm theorem, with nonhomogeneous Poisson arrivals 91
Partial fraction expansion 14
Periodic discrete-time Markov chain 176
Periodic discrete-time Markov chain, canonical form of 185—188 232—233
Periodicity of discrete-time Markov chain 171
Perpetual warrant 382—383
PH/M/1 queue 224
PH/M/c queue 311—312
Phase-type distribution, approximation by 270—271
Phase-type distribution, continuous-time 264—266 361—362 387
Phase-type distribution, convolution 266—267
Phase-type distribution, discrete-time 197—198
Phase-type distribution, limiting excess-life distribution 311
Phase-type distribution, moments 266
Phase-type distribution, order of 265
Phase-type renewal process 271—273
Pointwise limit 35
Poisson arrival see time averages (PASTA) 83—87
Poisson arrival see time averages (PASTA), applied to GI/M/c queue 86—87
Poisson arrival see time averages (PASTA), applied to M/G/1 queue 137 359
Poisson process 48
Poisson process as a birth and death process 244
Poisson process, arrival time condition on arrival count 52—53
Poisson process, compound 72—76
Poisson process, decomposition 54
Poisson process, generating arrival count by simulation 52
Poisson process, generating arrival times by simulation 52
Poisson process, interarrival time of 51
Poisson process, logarithmic Poisson 74—75
Poisson process, marked 80
Poisson process, stuttering Poisson 75—76
Poisson process, two-dimensional 80—83
Positive recurrence 169 175
Probability generating function 7
Probability generating function, finding moments 8
Product form solution, nonexistence of 296
Product warranty problems 112 148
Pure birth process 304
Quality control 183—185
Quasi birth and death (QBD) process 222
Queueing network with blocking 315
R matrix of QBD process 272
Random sum see "Compound random variable"
Random walk 221
Random walk with a reflecting barrier 179
Random walk, approximate a Brownian motion 374—375
Random walk, Markov chain formulation 220—221
Random walk, simple 41 104—105 150—151
rcll 29
Recurrence time of a renewal process see "Total life"
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