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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.

References

[1] Anscombe, F. J. (1952) Large-sample theory of sequential estimation. Proc. Camb. Phil. Soc. 48, 600–607.CrossRefGoogle Scholar
[2] Block, H. W., Borges, W. S. and Savits, T. H. (1985) Age-dependent minimal repair. J. Appl. Prob. 22, 370–385.CrossRefGoogle Scholar
[3] Brown, M. (1981) Further monotonicity properties for specialized renewal processes. Ann. Prob. 9, 891–895.CrossRefGoogle Scholar
[4] Harris, T. E. (1977) A correlation inequality for Markov processes in partially ordered state spaces. Ann. Prob. 5, 451–454.CrossRefGoogle Scholar
[5] Jacobs, P. A. (1986) First passage times for combinations of random loads. SIAM J. Appl. Math. 46, 643–656.Google Scholar
[6] Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[7] Shaked, M. and Shanthikumar, J. G. (1987) Temporal stochastic convexity and concavity. Stoch. Proc. Appl. 27, 1–20.CrossRefGoogle Scholar
[8] Shaked, M. and Shanthikumar, J. G. (1988) Stochastic convexity and its applications. Adv. Appl. Prob. 20, 427–446.CrossRefGoogle Scholar
[9] Shanthikumar, J. G. and Sumita, U. (1983) General shock models associated with correlated renewal sequences. J. Appl. Prob. 20, 600–614.CrossRefGoogle Scholar
[10] Shanthikumar, J. G. and Sumita, U. (1984) Distribution properties of the system failure time in a general shock model. Adv. Appl. Prob. 16, 363–377.CrossRefGoogle Scholar
[11] Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
[12] Sumita, U. and Shanthikumar, J. G. (1985) A class of correlated cumulative shock models. Adv. Appl. Prob. 17, 347–366.CrossRefGoogle Scholar