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An age-dependent counting process generated from a renewal process

Published online by Cambridge University Press:  01 July 2016

Ushio Sumita  and
J. George Shanthikumar
Ushio Sumita*
Affiliation:
University of Rochester
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
*
∗Postal address: W. E. Simon Graduate School of Business Administration, University of Rochester, Rochester, NY 14627, USA.
∗∗Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
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Abstract

Let {N(t)} be a renewal process having the associated age process {X(t)}. Of interest is the counting process {M(t)} characterized by a non-homogeneous Poisson process with age-dependent intensity function λ (X(t)). The trivariate process {Y(t) = [M(t), N(t), X(t)]} is analyzed obtaining its Laplace transform generating function explicitly. Based on this result, asymptotic behavior of {S(t) = cM(t) + dN(t)} as t → ∞ is discussed. Furthermore, a sufficient condition is given under which {M(t), –N(t), X(t)} is stochastically monotone and associated. This condition also assures increasing stochastic convexity of {M(t)}. The usefulness of these results is demonstrated through an application to the age-dependent minimal repair problem.

Keywords

AGE-DEPENDENT NON-HOMOGENEOUS POISSON PROCESSASYMPTOTICSSTOCHASTIC MONOTONICITYCONVEXITYASSOCIATIONAGE-DEPENDENT MINIMAL REPAIR
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 4 , December 1988 , pp. 739 - 755
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research partially supported by NSF Grant ECS-8600992 and by the IBM Program of Support for Education in the Management of Information Systems.

Research partially supported by NSF Grant ECS-8601210.

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