Concave Renewal Functions do not Imply DFR Interrenewal Times

Yu, Yaming

doi:10.1239/jap/1308662647

Journal of Applied Probability

2011

DOI: 10.1239/jap/1308662647

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Concave Renewal Functions do not Imply DFR Interrenewal Times

Yaming Yu¹

Abstract: Brown (1980), (1981) proved that the renewal function is concave if the interrenewal distribution is DFR (decreasing failure rate), and conjectured the converse. This note settles Brown's conjecture with a class of counterexamples. We also give a short proof of Shanthikumar's (1988) result that the DFR property is closed under geometric compounding.

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Cited by 4 publications

References 19 publications

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Inventory models with nonlinear shortage costs and stochastic lead times; applications of shape properties of randomly stopped counting processes

Badía

Sangüesa

2015

Naval Research Logistics

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Abstract:In this article, we study generalizations of some of the inventory models with nonlinear costs considered by Rosling in (Oper. Res. 50 (2002) 797-809). In particular, we extend the study of both the periodic review and the compound renewal demand processes from a constant lead time to a random lead time. We find that the quasiconvexity properties of the cost function (and therefore the existence of optimal (s, S) policies), holds true when the lead time has suitable log-concavity properties. The results are derived by structural properties of renewal delayed processes stopped at an independent random time and by the study of log-concavity properties of compound distributions.

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Inventory models with nonlinear shortage costs and stochastic lead times; applications of shape properties of randomly stopped counting processes

Badía

Sangüesa

2015

Naval Research Logistics

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Monotonicity properties for solutions of renewal equations

Dermitzakis

Politis

2022

Statistics & Probability Letters

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Log-Concavity of Compound Distributions With Applications in Operational and Actuarial Models

Badía

Sangüesa

Federgruen

2019

Prob. Eng. Inf. Sci.

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We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s,S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters.

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Concave Renewal Functions do not Imply DFR Interrenewal Times

Cited by 4 publications

References 19 publications

Inventory models with nonlinear shortage costs and stochastic lead times; applications of shape properties of randomly stopped counting processes

Inventory models with nonlinear shortage costs and stochastic lead times; applications of shape properties of randomly stopped counting processes

Monotonicity properties for solutions of renewal equations

Log-Concavity of Compound Distributions With Applications in Operational and Actuarial Models

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