Semi-Markov Models and Applications

by Janssen, Jacques; Limnios, N.

• ISBN13: 9780792359630
• ISBN10: 0792359631
• Format: Hardcover
• Copyright: 1999-11-01
• Publisher: Springer Verlag

Summary

It is now clear that semi-Markov processes play an increasingly crucia l role in business and industry; this is partially due to the fact tha t, with the powerful mathematical software now existing, the numerical treatment of basic integral equations is easier and so leads to more concrete applications; moreover the development of topics such as non- homogeneous models and statistical estimation make it possible to cons truct more adequate models of real-life problems and to calibrate the basic parameters of the models more accurately from real data. This b ook presents many original models that are or could be truly useful fo r applications in real-life problems, and the editor hopes that it wil l contribute to the stimulation of new interactions between the theore tical development and the applications of semi-Markov models.

Table of Contents

Contributors	xiii
Preface	xix
Part I Extensions of Basic Models
The Solidarity of Markov Renewal Processes	3	(20)
Ronald Pyke
Prologue	3	(4)
Introduction	7	(1)
Preliminaries	8	(4)
Stable laws and solidarity	12	(3)
A basic identity for comparing inter-occurrence sums	15	(4)
Remark	19	(4)
A Generalization of Semi-Markov Processes	23	(10)
Marius Iosifescu
Introduction and summary	23	(1)
Definition and backward equations	24	(1)
The minimal solution	25	(3)
Criteria for non-uniqueness	28	(1)
Asymptotic behavior	29	(1)
The special case of semi-Markov processes	30	(3)
The forward equation	30	(1)
Asymptotic behavior	31	(1)
Open problems	32	(1)
Quasi-stationary Phenomena for Semi-Markov Processes	33	(28)
Mats Gyllenberg
Dmitrii S. Silvestrov
Introduction	33	(4)
Exponential asymptotics for perturbed renewal equation	37	(8)
Asymptotical expansions for moments of hitting times	45	(7)
Nonlinearly perturbed semi-Markov processes	52	(3)
Quasi-stationary distributions for perturbed semi-Markov processes	55	(6)
Semi-Markov Random Walks	61	(16)
Vladimir S. Korolyuk
The semi-Markov random walk	61	(2)
A boundary problem for SMRW	63	(1)
A semi-continuous SMRW	64	(2)
Queueing system M|G|1|Γ	66	(2)
Queueing system G|M|1|Γ	68	(1)
Asymptotic analysis of SMRW in the series scheme	69	(1)
Average scheme. Positive drift, ρ > 0	70	(1)
Diffusion approximation (zero drift, ρ = 0)	71	(1)
Problems of singular perturbation in average and the diffusion approximation schemes	72	(5)
Diffusion Approximation for Processes with Semi-Markov Switches	77	(26)
Vladimir V. Anisimov
Introduction	77	(1)
Switching stochastic processes	78	(7)
Preliminary remarks	78	(2)
Recurrent processes of a semi-Markov type	80	(1)
Recurrent process of a semi-Markov type with additional Markov switches	81	(1)
General case of RPSM	81	(1)
Processes with semi-Markov switches	82	(1)
Switching processes	82	(1)
Examples of Switching processes	83	(1)
PII with semi-Markov switches	83	(1)
Random movements with SMP switches	84	(1)
Dynamical systems in semi-Markov environment	84	(1)
Stochastic differential equations with semi-Markov switches	84	(1)
Switching state-dependent queueing models	85	(1)
Averaging principle and diffusion approximation for RPSM	85	(5)
Processes with semi-Markov switches	90	(4)
Asymptotically mixing environment	91	(2)
Asymptotically consolidated environment	93	(1)
Applications	94	(9)
Random movements	94	(2)
Semi-Markov state-dependent queueing models	96	(1)
Markov models with semi-Markov switches	97	(6)
Approximations for Semi-Markov Single Ion Channel Models	103	(16)
Susan M. Pitts
Introduction	103	(3)
Results	106	(6)
The framework	106	(1)
The functionals	107	(3)
Derivatives	110	(2)
Approximations	112	(7)
Part II Statistical Estimation
Log-likelihood in Stochastic Processes	119	(30)
George G. Roussas
Debasis Bhattacharya
Introduction	120	(3)
Assumptions	123	(3)
Some comments on the assumptions	125	(1)
Asymptotic expansion and asymptotic normality	126	(3)
Some auxiliary results	129	(8)
Proof of theorems	137	(12)
Appendix	140	(9)
Some Asymptotic Results and Exponential Approximation in Semi-Markov Models	149	(18)
George G. Roussas
Debasis Bhattacharya
Introduction, notation, and some auxiliary results	150	(4)
Assumptions and formulation of main results	154	(4)
Some auxiliary results on random times	158	(3)
Assumptions	159	(2)
Proof of main results and some closing comments	161	(6)
Some closing comments	164	(3)
Markov Renewal Processes and Exponential Families	167	(20)
Valeri T. Stefanov
Introduction	167	(2)
Noncurved exponential families associated with Markov renewal processes	169	(1)
Application to limit theory	170	(2)
Clusters of points	172	(3)
Composite events	175	(12)
On Homogeneity of Two Semi-Markov Samples	187	(14)
Larisa Afanasyeva
Peter Radchenko
Introduction	187	(1)
The model	188	(2)
Calculation of DU	190	(4)
Proof of the theorem	194	(4)
The test	198	(3)
Product-Type Estimator of Convolutions	201	(6)
Ilya Gertsbakh
I. Spungin
The product-type estimator of a convolution	201	(1)
Generating an estimate of Bk (T)	202	(1)
Crude Monte Carlo vs product-type estimator	203	(1)
Comparison of the product-type estimator with Brown's estimator	204	(1)
Convolutions in simulating the time to absorption in a semi-Markov process	204	(3)
Failure Rate Estimation of Semi-Markov Systems	207	(12)
Brahim Ouhbi
Nikolaos Limnios
Introduction	207	(1)
Preliminaries	208	(2)
Reliability and failure rate functions of a semi-Markov system	210	(1)
Estimator of the failure rate of a semi-Markov system	211	(6)
Numerical application	217	(2)
Estimation for Semi-Markov Manpower Models in a Stochastic Environment	219	(10)
Sally McClean
Erin Montgomery
Introduction	219	(2)
The stochastic environment	221	(1)
Estimation	221	(4)
Estimation for the compromise matrix	221	(1)
Estimation for the S-NHMS	222	(1)
Estimation for the S-NHSMS	223	(2)
Using the model	225	(1)
Conclusion	226	(3)
Semi-Markov Models for Lifetime Data Analysis	229	(12)
Rafael Perez-Ocon
Juan Eloy
Ruiz-Castro
M. Luz Gamiz-Perez
Introduction	230	(1)
The semi-Markov model	231	(2)
Transition probability functions	231	(1)
Maximum-likelihood function	232	(1)
Some data on survival times	233	(1)
Some data on failure times	234	(7)
Part III Non-Homogeneous Models
Continuous Time Non Homogeneous Semi-Markov Systems	241	(12)
Aleka A. Papadopoulou
Panagiotis C.G. Vassiliou
Introduction	241	(1)
The model	242	(3)
Limiting behavior	245	(2)
Illustration: a special case	247	(6)
The Perturbed Non-Homogeneous Semi-Markov System	253	(16)
Panagiotis C.G.
Vassiliou Helena Tsakiridou
Introduction	253	(1)
The perturbed non homogeneous semi-Markov system	254	(5)
The expected interval transition probabilities	259	(4)
The expected population structure	263	(6)
Part IV Queueing Systems Theory
Semi-Markov Queues with Heavy Tails	269	(16)
Soren Asmussen
Introduction	269	(2)
Sub-exponential distributions. Random walk results	271	(1)
Markov--modulated M/G/1 queues and related models	272	(6)
Structured Markov chains of the M/G/1 type	278	(7)
MR Modelling of Poisson Traffic at Intersections Having Separate Turn Lanes	285	(28)
Rudy Gideon
Ronald Pyke
Introduction	286	(2)
The 4-state MRP on S1 and S2	288	(6)
An r-car right turn lane and ergodicity	294	(5)
Computations for an r-car right turn lane	299	(3)
Special cases of r = 0, 1 and +∞	302	(6)
The general case	308	(5)
Part V Financial Models
Stochastic Stability and Optimal Control in Insurance Mathematics	313	(12)
Anatoly Swishchuk
Introduction	313	(1)
Semi-Markov risk processes	314	(1)
Semi-Markov risk process as discontinuous semi-Markov random evolution	315	(1)
Analogue of Dynkin's formula for semi-Markov random evolutions	316	(1)
Boundary value problem for semi-Markov random evolutions	316	(1)
Stochastic stability of semi-Markov risk processes	317	(3)
Stochastic optimal control of semi-Markov risk processes	320	(5)
Option Pricing with Semi-Markov Volatility	325	(12)
Jacques Janssen
Raimondo Manca
Ernesto Volpe
Introduction	325	(1)
The JMC semi-Markov model in stochastic finance	326	(3)
The JMC semi-Markov model (1995, 1998)	327	(1)
The explicit expression of S(t)	328	(1)
Call option pricing	329	(2)
Stationary option pricing formula	331	(1)
Conclusions	332	(5)
Part VI Controlled Processes & Maintenance
Applications of Semi-Markov Processes in Reliability and Maintenance	337	(12)
Mohamed Abdel-Hameed
Introduction and summary	337	(1)
Basic results	338	(7)
Optimal replacement under the expected total discounted cost criterion	345	(2)
Optimal replacement under the long run average cost criterion.	347	(2)
Controlled Queueing Systems with Recovery Functions	349	(18)
Tadashi Dohi Shunji
Osaki Naoto Kaio
Introduction	350	(1)
Model description	351	(3)
Retry model	354	(3)
Repair model	357	(2)
Numerical examples and some remarks	359	(8)
Part VII Chromatography & Fluid Mechanics
Continuous Semi-Markov Models for Chromatography	367	(24)
Boris P. Harlamov
Introduction	367	(1)
Continuous semi-Markov process	368	(5)
Semi-Markov model of chromatography	373	(18)
Chromatography	373	(2)
Model of liquid column chromatography	375	(3)
Some monotone semi-Markov processes	378	(3)
Processes of transfer with terminal absorption	381	(3)
Process of transfer with diffusion	384	(7)
The Stress Tensor of the Closed Semi-Markov System. Energy and Entropy	391	(10)
George M. Tsaklidis
Introduction	391	(1)
The closed continuous-time semi-Markov model	392	(2)
The continuum viewpoint	394	(1)
The stress tensor of the closed continuous time HSMS	395	(2)
The energy equation	397	(1)
The entropy of the system	397	(4)
Index	401