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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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