Renewal Processes & Repairable Systems (Stand Alone Dup)

133 Pages·2002·0.85 MB·English

by  Suyono

Renewal Processes and Repairable
Systems

Renewal Processes and Repairable
Systems
Proefschrift
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. dr. ir. J. T. Fokkema,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen
op mandaag 3 februari 2003 om 16.00 uur
door
Suyono
Magister Sains Matematika
Universitas Gadjah Mada, Yogyakarta, Indonesi¨e
geboren te Purworejo, Indonesi¨e
Dit proefschrift is goedgekeurd door de promotor:
Prof. dr. F. M. Dekking
Samenstelling promotiecommissie:
Rector Magnificus, voorzitter
Prof. dr. F. M. Dekking, Technische Universiteit Delft, promotor
Prof. dr. R. M. Cooke, Technische Universiteit Delft
Prof. dr. M. Iosifescu, Centre for Mathematical Statistics, Roemeni¨e
Prof. dr. C. L. Scheffer, Technische Universiteit Delft (emeritus)
Prof. dr. R. K. Sembiring, Institut Teknologi Bandung, Indonesi¨e
Prof. dr. H. J. van Zuylen, Technische Universiteit Delft
Dr. J. A. M. van der Weide, Technische Universiteit Delft
The research in this thesis has been carried out under the auspices of the Thomas
Stieltjes Institute for Mathematics, at the University of Technology in Delft.
Published and distributed by : DUP Science
DUP Science is an imprint of
Delft University Press
P.O. Box 98
2600 MG Delft The Netherlands
Telephone: [phone]
Telefax: [phone]
E-mail: [email protected]
ISBN 90-407-xxx-x
Keywords: Poisson point processes, renewal processes, repairable systems
Copyright (cid:176)c 2002 by Suyono
All right reserved. No part of the material protected by this copyright notice
may be reproduced or utilized in any form or by any means, electronic or me-
chanical, including photocopying, recording or by any information storage and
retrieval system, without written permission from the publisher: Delft Univer-
sity Press.
Printed in The Netherlands
To my wife Demonti,
my son Afif,
and my daughters Hana and Nadifa.

Acknowledgments
IamverygratefultomysupervisorDr. J.A.M.vanderWeide,whointroduced
me interesting topics to research, for his creative guidance, his idea, his encour-
agement, and his patience in guiding me doing the research. I also would like
to express my gratitude to my promotor Prof. dr. F. M. Dekking not only for
givingmetheopportunitytocarryoutmyPh.DresearchattheDelftUniversity
of Technology, but also for his constructive comments.
Inaddition,IwishtothankallmembersoftheDepartmentofControl,Risk,
Optimization, Stochastic and Systems (CROSS) for their hospitality and their
assistances, especially to Cindy and Diana for their administrative assistance,
and to Carl for his computer assistance. Also, I wish to thank Durk Jellema
and Rene Tamboer for the arrangement of almost everything I needed during
my research and stay in the Netherlands.
IampleasedtoacknowledgemycolleaguesattheDepartmentofMathemat-
ics, State University of Jakarta. Their encouragement and support have been
invaluable. In this occasion I also would like to thank all of my friends in the
Netherlands for their support and their help.
Thisworkwouldnothavebeenpossiblewithoutthesupportofacooperative
projectbetweenIndonesianandDutchgovernments. InthisconnectionIwould
like to express my appreciation to Prof. dr. R. K. Sembiring and Dr. A. H. P.
van der Burgh who organized a research workshop four years ago in Bandung,
to Dr. O. Simbolon and Dr. B. Karyadi, former project managers of Proyek
Pengembangan Guru Sekolah Menengah (Secondary School Teachers Develop-
ment Project), and to H. P. S. Althuis, director of Center for International
Cooperation in Applied Technology (CICAT).
Finally,Iwouldliketothankmydearestwifeandourbelovedchildren. Saya
menyampaikan banyak terima kasih kepada istriku Dra. Demonti Siswari atas
pengorbananyayangbegitubesaryangdenganreladanikhlasditinggalmenuntut
ilmu di Belanda, atas doanya yang senantiasa dipanjatkan kepadaNya sehingga
saya alhamdulillah banyak mendapatkan jalan keluar ketika dalam kesulitan,
atas motivasinya yang selalu membuat saya kembali bersemangat dikala kehi-
langan semangat, atas kesabarannya yang dengannya alhamdulillah kita banyak
vii
viii
mendapat pertolongan dan kebaikan dari Allah, dan atas jerih payah serta usa-
hanya mengurus dan mendidik anak-anak yang diamanahkan kepada kita se-
hingga insyaAllah menjadi qurata a’yun bagi kita. Juga untuk anak-anakku,
HanaFirdaus,MuhamadAfifAbdurrahim,danNadifaMumtaz,ayahsampaikan
banyak terima kasih atas pengertian dan doa kalian. Pengorbanan kalian sung-
guh sangat besar untuk keberhasilan ayah. Kalian menjalani masa kecil tanpa
memperoleh perhatian dan kasih sayang yang sepantasnya dari ayah. Meskipun
masih kecil kalian sudah harus ikut membantu perjuangan yang ayah dan ibu
lakukan. Semoga ini semua menjadi latihan dan bekal yang berharga bagi kalian
untukmenjalanikehidupandimasayangakandatang. Amiin. Tidaklupauntuk
ayah, ibu, adik, mertua dan saudara-saudaraku semua, saya sampaikan terima
kasihatasdorongan,doa,danbantuannyayangtelahdiberikanpadakudankelu-
argaku khususnya saat saya tinggalkan anak-anak dan istriku untuk menempuh
program Ph.D. selama kurang lebih 4 tahun di Belanda.
Delft,February3,2003 Suyono
Contents
Acknowledgments vii
1 Introduction 1
1.1 Related works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Point processes . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Renewal processes . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Renewal Reward Processes 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Instantaneous reward processes . . . . . . . . . . . . . . . . . . . 14
2.2.1 A civil engineering example . . . . . . . . . . . . . . . . . 20
2.3 Renewal reward processes . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Covariance structure of renewal processes . . . . . . . . . . . . . 33
2.6 System reliability in a stress-strength model . . . . . . . . . . . . 35
2.6.1 Type I models . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6.2 Type II models . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Integrated Renewal Processes 45
3.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . 45
3.2 (N(t)) a Poisson or Cox process. . . . . . . . . . . . . . . . . . . 46
3.3 (N(t)) a renewal process . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 An application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
ix
x CONTENTS
4 Total Downtime of Repairable Systems 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Distribution of total downtime . . . . . . . . . . . . . . . . . . . 66
4.3 System availability . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4 Covariance of total downtime . . . . . . . . . . . . . . . . . . . . 76
4.5 Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.7 Systems consisting of n independent components . . . . . . . . . 88
4.7.1 Exponential failure and repair times . . . . . . . . . . . . 88
4.7.2 Arbitrary failure or repair times . . . . . . . . . . . . . . 93
A The proof of Theorem 2.5.1 95
B Numerical inversions of Laplace transforms 105
B.1 Single Laplace transform . . . . . . . . . . . . . . . . . . . . . . . 105
B.2 Double Laplace transform . . . . . . . . . . . . . . . . . . . . . . 108
Bibliography 113
Samenvatting 117
Summary 119
Ringkasan 121
Curriculum Vitae 123