Semi-Markov Processes and Reliability
At first there was the Markov property. The theory of stochastic processes, which can be considered as an exten sion of probability theory, allows the modeling of the evolution of systems through the time. It cannot be properly understood just as pure ma
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Series Editor N Balakrishnan McMaster University Department of Mathematics and Statistics 1280 Main Street West Hamilton, Ontario L8S 4K1 Canada
Editorial Advisory Board Max Engelhardt EG&G Idaho, Inc. Idaho Falls, ID 83415 Harry F. Martz Group A-I MS F600 Los Alamos National Laboratory Los Alamos, NM 87545 Gary C. McDonald NAO Research & Development Center 30500 Mound Road Box 9055 Warren,MI48090-9055 Peter R. Nelson Department of Mathematical Sciences Clemson University Martin Hall Box 341907 Clemson, SC 29634-1907 Kazuyuki Suzuki Communication & Systems Engineering Department University of Electro Communications 1-5-1 Chofugaoka Chofu-shi Tokyo 182 Japan
Semi-Markov Processes and Reliability
N. Limnios G. Oprişan
With 16 Figures
Springer-Science+Business Media, LLC
N. Limnios Division Mathematiques Appliquees Universite de Technologie de Compiegne Compiegne Cedex 60205 France
G. Oprişan University of "Politehnica" of Bucharest 313 Spaiul Independentei Sector 6R 77206 Bucharest Romania
Library of Congress Cataloging-in-Publication Data Limnios, N. (Nikolaos) Semi-Markov processes and reliability / N. Limnios, G. Oprişan. p. cm. - (Statistics for industry and technology) Includes bibliographicaI references and index. ISBN 978-1-4612-6640-2 ISBN 978-1-4612-0161-8 (eBook) DOI 10.1007/978-1-4612-0161-8 1. Markov processes. 2. Reliability (Engineering)-StatisticaI methods. 1. Oprisan, Gheorghe. II. Title. III. Series. QA274.7 .L56 2000 519.2'.33- T} belong to both a-algebras Fs and FT.
< T}, {S = T},
6. Suppose that the filtration is continuous on the right and that Tn is a sequence of stopping times; then the random variable lim infn-too Tn and lim sUPn-too Tn are stopping times. Moreover, suppose that (Tn) is nonincreasing and denote by T its limit. Then we have FT = nnFTn' Given a stochastic process, we often use the notion of the hitting time of a subset of states. More precisely, let (X(t), t E 1R.+) be a stochastic process with values (E,E) and let A E E. We consider the function TA : 0 -+ 1R.+: TA(W)
= inf{s > 0: X(s,w)
E
A}
= +00).
(inf0
If this function is a random variable and if moreover it is a stopping time with respect to the family F t = a(X(s), s ::; t), then it is called the hitting (or enter) time of the set A. Sometimes, we have to distinguish TA(W) = inf{s;::: 0: X(s,w) E A}. Let (X(n), n E N) be a discrete-time stochastic process with values in a finite or countable set E. For j E E, let Tj(W)
the random variable we have {Tj
where Fn
1.3
= n} =
Tj
= inf{n > 0: X(n,w) = j};
is the hitting time of the state j since for all n E N*,
{X{k) of j, 1 ::; j ::; n - 1, X{n)
= j} E F n ,
= a(X{k), 0::; k ::; n).
Important Families of Stochastic Processes
1.3.1
Second-Order Stochastic Processes
Let L2 = L2(S1, F, 1P) be the space of all (complex) random variables Z such that IZI 2< 00. The random variable Z is called a second-order random variable. In fact, L2(S1, F, 1P) is a Hilbert space (or unitary) Hilbert space of random variables considered on the probability space (0, F,
