D. N. Shanbhag, C. Radhakrishna Rao0444500138, 9780444500137
Stochastic Processes: Modelling and Simulation.
It is concerned mainly with the theme of reviewing and in some cases, unifying with new ideas the different lines of research and developments in stochastic processes of applied flavour. This volume consists of 23 chapters addressing various topics in stochastic processes. These include, among others, those on manufacturing systems, random graphs, reliability, epidemic modelling, self-similar processes, empirical processes, time series models, extreme value theory, applications of Markov chains, modelling with Monte carlo techniques, and stochastic processes in subjects such as engineering, telecommunications, biology, astronomy and chemistry. (A complete list of the topics addressed in the volume is available from the “Contents” of the volume.)
An attempt is made to cover in this volume, as in the case of its predec
Table of contents :
Front cover……Page 1
Series……Page 2
Title page……Page 3
Date-line……Page 4
Preface……Page 5
Table of contents……Page 7
Contributors……Page 15
1. Introduction……Page 19
2. Controlled piecewise deterministic processes……Page 21
3. Continuous flow model for production control……Page 33
4. Preventive maintenance and production control model……Page 37
5. Maintenance model without considering the machine aging……Page 50
6. Robust controller for a class of production and maintenance……Page 54
7. Conclusion……Page 64
References……Page 65
1. Introduction……Page 69
2. Overview of the Erdos-Renyi model……Page 75
3. Applications of the Erdos-Renyi model……Page 81
4. Geometric random graphs……Page 83
5. Random cluster models……Page 87
6. Random randomly coloured graphs……Page 90
7. Other models of random graphs……Page 96
References……Page 105
1. Introduction……Page 111
2. Locally self-similar processes……Page 113
3. Generalized fractional Brownian motion……Page 115
4. Estimating the scaling function……Page 120
5. Implementation of the estimation procedure……Page 122
6. Simulations……Page 124
7. Applications……Page 135
Appendix……Page 142
References……Page 151
1. Variation in biology……Page 155
2. Stochastic chemistry……Page 156
3. Exponentially distributed waiting times……Page 159
4. The biological cell……Page 160
5. A glib mathematical abstraction……Page 161
6. The spatial pattern of replication origins……Page 162
7. The time to separation of a long DNA molecule……Page 164
8. The proportion of origins initiated……Page 168
9. Mean eye lengths and eye-to-eye distances……Page 169
10. What is happening inside the eyes?……Page 171
12. The expectations of $N_t$ and $P_t$: renewal equations……Page 173
13. The expectation of $D_t$: the quasi-renewal equation……Page 175
14. Relationship between fragment length and primer-site spacing……Page 177
16. The competing theory……Page 178
17. Notations for the competing theory……Page 179
18. Analysis of the lagging strand for Model B……Page 180
19. Analysis of the leading strand for Model B……Page 181
20. Concluding remarks……Page 182
References……Page 183
1. Introduction……Page 185
2. The empirical integrated lack-of-memory process……Page 188
3. Connection with certain test statistics and empirical processes……Page 190
4. Asymptotic behaviour of the process……Page 199
5. Statement of results; examples and comparisons……Page 208
6. Integral statistics. Applications to testing……Page 214
7. Asymptotic efficiency in the continuous case……Page 225
References……Page 241
1. Early encounters: random numbers and the theory of runs……Page 245
2. Sequences of events with repetitions……Page 248
3. Strings and string overlaps……Page 250
4. Further classical and martingale methods……Page 253
5. Markov chain techniques……Page 256
References……Page 258
1. Introduction……Page 261
2. Traffic models……Page 262
3. Network performance using traffic models……Page 268
4. LAN (multiaccess communication) models……Page 286
5. Other topics and models……Page 290
Acknowledgements……Page 298
References……Page 299
2. Chain binomial models……Page 303
3. The simple and general stochastic epidemic models……Page 306
4. Spatial models……Page 319
5. Stochastic models for control of epidemics……Page 326
6. Specific applications……Page 330
7. Stochastic processes in parameter estimation and hypothesis testing……Page 339
8. Summary and conclusions……Page 346
References……Page 348
1. Introduction……Page 355
2. The asymptotic variance of empirical estimators for Markov chains……Page 359
3. Efficient estimation for Markov chain models……Page 363
4. Improving empirical estimators by conditioning……Page 366
5. Asymptotic variance of empirical estimators for Gibbs samplers……Page 369
6. Asymptotic variance bounds for Gibbs samplers……Page 371
7. Improving empirical estimators for random fields with local interactions……Page 377
8. Exploiting symmetries of random fields……Page 381
References……Page 384
1. Introduction……Page 389
2. Fractal geometry……Page 391
3. Fractals and stochastic processes……Page 406
4. Dynamic fractal models……Page 409
5. Further applications and conclusion……Page 421
References……Page 422
1. Introduction……Page 425
2. Numerical inversion of Laplace transforms……Page 428
3. The ubiquity of Markov chains……Page 431
4. Finite Markov chains……Page 432
5. Infinite Markov chains……Page 436
6. The BMAP/G/1 queue……Page 441
7. The quasi birth-and-death process……Page 443
References……Page 446
1. Introduction……Page 449
2. The Markov Chain imbedding technique……Page 451
3. Success runs and pattern distributions……Page 453
4. Markov Chain imbeddable variables of binomial type……Page 461
5. Waiting time distributions associated with $MVB$’s……Page 464
6. The number of runs and patterns as members of the $MVB$ family……Page 466
7. Multivariate $MVB$ distributions……Page 472
8. Multivariate success runs distributions……Page 475
9. Alternative methods for exact distribution evaluation……Page 484
References……Page 488
1. Introduction……Page 491
2. Image labeling……Page 493
3. Markov random fields and Gibbs distributions……Page 498
4. Useful MRF models……Page 507
5. The MAP-MRF framework……Page 517
References……Page 525
1. Introduction……Page 533
2. Semi-Markov kernel……Page 534
3. Markov renewal processes (MRP)……Page 536
4. Semi-Markov processes with an arbitrary state space……Page 543
5. Markov renewal equation……Page 546
6. The countable case……Page 550
7. Classification of states……Page 555
8. Asymptotic behavior……Page 558
9. Some recent approaches to semi-Markov processes……Page 560
10. Reliability modeling and estimation……Page 569
References……Page 572
1. Introduction……Page 575
2. Characterization/identifiability via output processes……Page 576
3. Characterization/identifiability via infinite divisibility property……Page 582
4. Strong unimodality and other relevant properties……Page 586
References……Page 588
1. Introduction……Page 591
2. Markov chains……Page 593
3. The $DARMA$ models……Page 594
4. Models based on thinning……Page 596
5. Regression models……Page 612
6. State space and Bayesian models……Page 615
References……Page 620
1. Introduction……Page 625
2. Limit laws in univariate extremes and characterizations……Page 626
3. Rates of convergence……Page 631
4. Generalized extreme value (GEV) distribution……Page 636
5. Generalized Pareto (GP) distribution……Page 638
6. Joint distribution of the $r$-largest order statistics……Page 641
7. A point process characterization……Page 642
8. Extremes of stochastic processes……Page 643
9. Limit laws for multivariate extremes……Page 653
10. Characterizations of the domain of attraction……Page 655
11. Characterizations of multivariate extreme value distributions……Page 667
12. Rates of convergence……Page 670
13. Parametric families for bivariate extreme value distributions……Page 672
14. Parametric families for multivariate extreme value distributions……Page 681
15. Extremes of multivariate stochastic processes……Page 695
Acknowledgements……Page 697
References……Page 698
0. Introduction……Page 711
1. History, surnames, and sex……Page 713
2. Genetics and evolution……Page 721
3. Epidemic modelling……Page 746
4. Ecology and conservation modelling……Page 756
References……Page 780
1. Introduction……Page 793
2. Modified versions of some basic results on damage models……Page 794
3. Characterizations based on modified Rao-Rubin conditions……Page 800
4. Characterization via conditional expectations……Page 806
References……Page 811
1. Introduction: what’s the point?……Page 813
2. Unique features of astronomical point processes……Page 815
3. Naive point process theory……Page 816
4. The mystery of Gamma Ray bursts……Page 819
5. Other examples of astronomical point processes……Page 833
6. Conclusion……Page 840
References……Page 841
1. Introduction……Page 845
2. Linear time series models and cumulant spectra……Page 851
3. Volterra expansion and bilinear models……Page 852
4. Higher-order moments and identification……Page 855
5. Estimation of higher-order cumulants and the bilinear models……Page 856
6. Multivariate nonlinear time series and higher-order cumulants of random vectors……Page 861
7. Spurious regression and cointegration, nonlinearity……Page 865
8. Time dependent nonlinear models……Page 868
9. Long range dependence……Page 870
10. Stationary bilinear process in continuous time……Page 879
Acknowledgement……Page 884
References……Page 885
1. Introduction……Page 889
2. State-space model……Page 892
3. Nonlinear and non-Gaussian state-space modeling……Page 905
4. Monte Carlo studies……Page 924
5. Summary and concluding remarks……Page 935
Appendix A. Linear and normal system……Page 937
Appendix B. Sampling methods……Page 938
Appendix C. Recursive versus non-recursive algorithms……Page 941
References……Page 944
1. Introduction……Page 949
2. Aggregated Markov chains……Page 952
3. Theoretical bursts……Page 957
4. Empirical bursts……Page 963
5. A five-state ligand-activated ion channel model……Page 967
6. A linear sequential model with drug blockade……Page 969
7. A model showing biphasic drug effects……Page 973
8. A model for a supergated double-barrelled chloride channel……Page 976
9. Some comments on statistical inference……Page 981
Note added in proof……Page 982
References……Page 983
Subject Index……Page 987
Contents of Previous Volumes……Page 997
Back cover……Page 1019
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