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Further monotonicity properties of renewal processes

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  01 July 2016

Masaaki Kijima
Masaaki Kijima*
Affiliation:
The University of Tsukuba, Tokyo
*
∗Postal address: Graduate School of Systems Management, The University of Tsukuba, Tokyo, 3-29-1 Otsuka, Bunkyo-ku, Tokyo 112, Japan.
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Abstract

In a discrete-time renewal process {Nk, k = 0, 1, ·· ·}, let Zk and Ak be the forward recurrence time and the renewal age, respectively, at time k. In this paper, we prove that if the inter-renewal time distribution is discrete DFR (decreasing failure rate) then both {Ak, k = 0, 1, ·· ·} and {Zk, k = 0, 1, ·· ·} are monotonically non-decreasing in k in hazard rate ordering. Since the results can be transferred to the continuous-time case, and since the hazard rate ordering is stronger than the ordinary stochastic ordering, our results strengthen the corresponding results of Brown (1980). A sufficient condition for {Nk+m – Nk, k = 0, 1, ·· ·} to be non-increasing in k in hazard rate ordering as well as some sufficient conditions for the opposite monotonicity results are given. Finally, Brown's conjecture that DFR is necessary for concavity of the renewal function in the continuous-time case is discussed.

Keywords

RENEWAL AGEFORWARD RECURRENCE TIMEHAZARD RATE ORDERINGSTOCHASTIC ORDERINGDFR INTER-RENEWAL TIMEPF2 DISTRIBUTIONRENEWAL FUNCTION

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 60E05: Distributions
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 3 , September 1992 , pp. 575 - 588
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1] Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. Wiley, New York.Google Scholar
[2] Baxter, L. A. (1988) Some criteria for reliability growth. Microelectron. Reliability 28, 743–750.CrossRefGoogle Scholar
[3] Brown, M. (1980) Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.CrossRefGoogle Scholar
[4] Brown, M. (1981) Further monotonicity properties for specialized renewal processes. Ann. Prob. 9, 891–895.CrossRefGoogle Scholar
[5] Hirayama, T. and Kijima, M. (1992) Single machine scheduling problem when the machine capacity varies stochastically. Operat. Res. 40.CrossRefGoogle Scholar
[6] Karlin, S. (1968) Total Positivity. Stanford University Press, California.Google Scholar
[7] Keilson, J. (1979) Markov Chain Models—Rarity and Exponentiality. Springer-Verlag. New York.CrossRefGoogle Scholar
[8] Keilson, J. and Kester, A. (1978) Unimodality preservation in Markov chains. Stoch. Proc. Appl. 7, 179–190.CrossRefGoogle Scholar
[9] Keilson, J. and Sumita, U. (1982) Uniform stochastic ordering and related inequalities. Canad. J. Statist. 10, 181–198.CrossRefGoogle Scholar
[10] Kijima, M. (1988) Distribution properties of discrete characteristics in M/G/1 and GI/M/1 queues. J. Operat. Res. Soc. Japan 31, 172–189.Google Scholar
[11] Kijima, M. (1989) Uniform monotonicity of Markov processes and its related properties. J. Operat. Res. Soc. Japan 32, 475–490.Google Scholar
[12] Kijima, M. and Nakagawa, T. (1991) A cumulative damage shock model with imperfect preventive maintenance. Naval Res. Logist. 38, 145–156.3.0.CO;2-D>CrossRefGoogle Scholar
[13] Kijima, M. and Sumita, U. (1986) A useful generalization of renewal theory: Counting processes governed by nonnegative Markovian increments. J. Appl. Prob. 23, 71–88.CrossRefGoogle Scholar
[14] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models—An Algorithmic Approach. Johns Hopkins University Press, Baltimore.Google Scholar
[15] Ross, S. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[16] Shanthikumar, J. G. (1988) DFR property of first passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.CrossRefGoogle Scholar
[17] Shanthikumar, J. G., Yamazaki, G. and Sakasegawa, H. (1991) Characterization of optimal order of services in a tandem queue with blocking. Operat. Res. Lett. 10, 17–22.CrossRefGoogle Scholar