Semi Markov Models and theory and applications.pdf

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SEMI-MARKOV MODELS Theory and Applications

SEMI-MARKOV MODELS Theory and Applications Edited by Jacques Janssen Free University of Brussels Brussels, Belgium Springer Science+Business Media, LLC

Library of Congress Cataloging in Publication Data International Symposium on Semi-Markov Processes and Their Applications (1984: Brussels, Belgium) Semi-Markov models. "Proceedings of an International Symposium on Semi-Markov Processes and Their Applications, held June 4-7, 1984, in Brussels, Belgium"—T.p. verso. Bibliography: p. 1. Markov, processes—Congresses. 2. Renewal theory—Congresses. I. Janssen, Jac . II. Title. ques, 1939- QA274.7.I58 1984 ISBN 978-1-4899-0576-5 519.2'33 86-12389 ISBN 978-1-4899-0576-5 DOI 10.1007/978-1-4899-0574-1 ISBN 978-1-4899-0574-1 (eBook) Proceedings of an International Symposium on Semi-Markov Processes and Their Applications, held June 4-7, 1984, in Brussels, Belgium © 1986 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1986 Softcover reprint of the hardcover 1st edition 1986 All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher

PREFACE This book is the result of the International Symposium on Semi Markov Processes and their Applications held on June 4-7, 1984 at the Universite Libre de Bruxelles with the help of the FNRS (Fonds National de la Recherche Scientifique, Belgium), the Ministere de l'Education Nationale (Belgium) and the Bernoulli Society for Mathe matical Statistics and Probability. This international meeting was planned to make a state of the art for the area of semi-Markov theory and its applications, to bring together researchers in this field and to create a platform for open and thorough discussion. Main themes of the Symposium are the first ten sections of this book. The last section presented here gives an exhaustive biblio graphy on semi-Markov processes for the last ten years. Papers selected for this book are all invited papers and in addition some contributed papers retained after strong refereeing. Markov additive processes and regenerative systems Semi-Markov decision processes Sections are I. II. III. Algorithmic and computer-oriented approach Semi-Markov models in economy and insurance IV. V. Semi-Markov processes and reliability theory VI. Simulation and statistics for semi-Markov processes VII. Semi-Markov processes and queueing theory VIII. Branching IX. Applications in medicine X. Applications in other fields v

PREFACE XI. A second bibliography on semi-Markov processes It is interesting to quote that sections IV to X represent a good sample of the main applications of semi-Markov processes i.e. Economy, Insurance, Reliability, Simulation, Queueing, Branching, Medicine (including survival data), Social Sciences, Language Model ling, Seismic Risk, Analysis, Biology and Computer Science. This strong interaction between recent theoretical results presented in the first three sections and applications given above was clearly pointed out by the participants. I should like to thank all of them for their contribution and their enthusiasm. In particular, I should like to mention the strong support I found by A.A. BOROVKOV (Institue of Mathematics, Novosibirsk), E.~INLAR (Civil Engineering Department, Princeton University), D.R.COX (Department of Mathematics, Imperial College, London), V.S.KOROLYUK (Institute of Mathematics, Ukrainian Academy of Sciences, Kiev), M.NEUTS (Department of Mathematical Sciences, University of Delaware, Newark) and J.TEUGELS (Department of Mathematics, Katho lieke Universiteit te Leuven). This last one provides me with in valuable support. Finally, I should mention the CADEPS (Centre d'Analyse des Donnees et Processus Stochastiques, Ecole de Commerce Solvay, Brussels) and the CEME (Centre d'Economie Mathematique et d'Econo metrie, Brussels) which give me effective support by taking in hands the local organization. Here too, I have to retain particularly the strong help of Mr ABIKHALIL (CADEPS) for the final preparation of this volume. J. Janssen

CONTENTS SECTION I. MARKOV ADDITIVE PROCESSES AND REGENERATIVE SYSTEMS Some limit theorems for Markov additive processes . • • • • •• P.Ney and E.Nummelin Stationary regenerative processes H.Kaspi and B.Maisonneuve Asymptotic analysis of some non-homogeneous semi-Markov processes R.V.Benevento SECTION II. SEMI-l~OV DECISION PROCESSES Markov and semi-Markov decision models and optimal stopping M.Schal Markov decision drift processes F.A.Van der Duyn Schouten The functional equations of undiscounted denumerable state rfurkov renewal programming • . • • • • • E.Mann SECTION III. ALGORITHMIC AND COMPUTER-ORIENTED APPROACH Recursive moment formulas for regenerative simulation • • .• P.W.Glynn and D.L.lglehart 3 13 23 39 63 79 99 Computation of the state probabilities in a class of semi- regenerative queueing systems H.Schellhaas • • 111

viii CONTENTS The superposition of two PH-renewal processes • • • • . . . • • 131 M.Neuts and G.Latouche SECTION IV. SEMI-MARKOV MODELS IN ECONOMY AND INSURANCE Pension accumulation as a semi-Markov reward process, with applications to pension reform • • • • • . • • . • 181 Y.Balcer and I.Sahin The structure of a firm's optimal non-decreasing wage policy when recruitment is a wage dependent Poisson process N.H.Schager SECTION V. SEMI-MARKOV PROCESSES AND RELIABILITY THEORY Markov renewal processes in reliability analysis V.S.Korolyuk • 201 . 217 Deterioration processes • • • • • • • • • • • • . • • • • . • • 231 M.Abdel-Hameed Stochastic processes with an embedded point process and their application to system reliability analysis • . •• 253 P.Franken and A.Streller SECTION VI. SIMULATION AND STATISTICS FOR SEMI-MARKOV PROCESSES Semi-Markov models for manpower planning S.McClean . • • • . • • • • . . 283 Statistical analysis of semi-Markov processes based on the theory of counting processes • • • • • • • • • • • 301 N.Keiding SECTION VII. SEMI-MARKOV PROCESSES AND QUEUEING THEORY Approximation of stochastic models V.V.Kalashnikov • 319 The method of renovating events and its applications in queueing theory . • . • • • • • • . • • • • • 337 S.G.Foss On non-time homogeneity • • • . • • • • . • • • • • • . • . . • 351 H.Thorisson

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