Anatoliy Swishchuk - Random Motions in Markov and Semi-Markov Random Environments 1

1 Cover

2 Title Page Series Editor Nikolaos Limnios

3 Copyright First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2021 The rights of Anatoliy Pogorui, Anatoliy Swishchuk and Ramón M. Rodríguez-Dagnino to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2020946634 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-547-3

4 Preface

5 Acknowledgments

6 Introduction

7 PART 1: Basic Methods PART 1 Basic Methods 1 Preliminary Concepts 1.1. Introduction to random evolutions 1.2. Abstract potential operators 1.3. Markov processes: operator semigroups 1.4. Semi-Markov processes 1.5. Lumped Markov chains 1.6. Switched processes in Markov and semi-Markov media 2 Homogeneous Random Evolutions (HRE) and their Applications 2.1. Homogeneous random evolutions (HRE) 2.2. Limit theorems for HRE

8 PART 2: Applications to Reliability, Random Motions, and Telegraph Processes 3 Asymptotic Analysis for Distributions of Markov, Semi-Markov and Random Evolutions 3.1. Asymptotic distribution of time to reach a level that is infinitely increasing by a family of semi-Markov processes on the set ℕ; 3.2. Asymptotic inequalities for the distribution of the occupation time of a semi-Markov process in an increasing set of states 3.3. Asymptotic analysis of the occupation time distribution of an embedded semi-Markov process (with increasing states) in a diffusion process 3.4. Asymptotic analysis of a semigroup of operators of the singularly perturbed random evolution in semi-Markov media 3.5. Asymptotic expansion for distribution of random motion in Markov media under the Kac condition 3.6. Asymptotic estimation for application of the telegraph process as an alternative to the diffusion process in the Black–Scholes formula 4 Random Switched Processes with Delay in Reflecting Boundaries 4.1. Stationary distribution of evolutionary switched processes in a Markov environment with delay in reflecting boundaries 4.2. Stationary distribution of switched process in semi-Markov media with delay in reflecting barriers 4.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer: the Markov case 4.4. Application of random evolutions with delaying barriers to modeling control of supply systems with feedback: the semi-Markov switching process 5 One-dimensional Random Motions in Markov and Semi-Markov Media 5.1. One-dimensional semi-Markov evolutions with general Erlang sojourn times 5.2. Distribution of limiting position of fading evolution 5.3. Differential and integral equations for jump random motions 5.4. Estimation of the number of level crossings by the telegraph process

9 References

10 Index

11 Summary of Volume 2

12 End User License Agreement

1 Chapter 4Figure 4.1. A system of two unreliable subsystems, say S 1and S 2, connected in s...Figure 4.2. Efficiency parameter K as a function of reservoir size V for differe...Figure 4.3. Efficiency parameter K as a function of reservoir size V for differe...

1 Chapter 4Table 4.1. Maximum and minimum values of K for different λ/μ ratios. We have fix...Table 4.2. Maximum values of K at V = 5 for different λ/μ ratios and different v...

1 Cover

2 Table of Contents

3 Title page

4 Copyright First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2021 The rights of Anatoliy Pogorui, Anatoliy Swishchuk and Ramón M. Rodríguez-Dagnino to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2020946634 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-547-3

5 Preface

6 Acknowledgments

7 Introduction

8 Begin Reading

9 References

10 Index

11 Summary of Volume 2

12 End User License Agreement

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