an:01682654
Zbl 0990.60004
Limnios, N.; Opri??an, G.
Semi-Markov processes and reliability
EN
Statistics for Industry and Technology. Basel: Birkh??user. xii, 222 p. (2001).
00422169
2001
b
60-02 60K15 60K20
Markov renewal process; semi-Markov process; functionals; limit theorems; reliability; availability; maintainability; performability
This book is in two parts: the first presents semi-Markov process theory from scratch; the second discusses applications of these processes to problems in reliability. The authors claim that the text is ``within reach of those who have a first course in probability theory'', but it assumes rather more than this: it is really a handbook for researchers.
In the first part, after a brief roll call of stochastic processes, emphasizing renewal theory, the formal machinery of general state-space Markov renewal processes is set up. There is no motivation at this stage. In a similar vein, semi-Markov processes are then developed, and there is a discussion of the Markov renewal equation, process functionals and jump time asymptotics. For the countable state-space case some more specialized topics are covered, such as limit theorems for recurrence time processes, statistical estimation of semi-Markov kernels and renewal functions, and phase-type kernels.
The second part, which refers to the first rather sparingly, explains how semi-Markov processes can be applied in studying various aspects of the dependability of systems: reliability, availability, maintainability, performability, etc. The emphasis is on finite state-space models; in particular, six methods are outlined for calculating system transition probabilities. The generalities are illustrated by a selection from the literature of specific system models. The book closes with a brief account of Monte Carlo methodology.
There are numerous minor inconsistencies and slips, by far the most amusing of which is the phrase ``finitely maNew York states' that appears in the bibliography.
J.Preater (Keele)