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A useful generalization of renewal theory: counting processes governed by non-negative Markovian increments

Published online by Cambridge University Press:  14 July 2016

Masaaki Kijima  and
Ushio Sumita
Masaaki Kijima*
Affiliation:
The University of Rochester
Ushio Sumita*
Affiliation:
The University of Rochester
*
∗Postal address: The Graduate School of Management, The University of Rochester, Rochester, NY 14627, USA.
∗Postal address: The Graduate School of Management, The University of Rochester, Rochester, NY 14627, USA.
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Abstract

Let N(t) be a counting process associated with a sequence of non-negative random variables (Xj)1∞ where the distribution of Xn+1 depends only on the value of the partial sum Sn = Σj=1nXj. In this paper, we study the structure of the function H(t) = E[N(t)], extending the ordinary renewal theory. It is shown under certain conditions that h(t) = (d/dt)H(t) exists and is a unique solution of an extended renewal equation. Furthermore, sufficient conditions are given under which h(t) is constant, monotone decreasing and monotone increasing. Asymptotic behavior of h(t) and H(t) as t → ∞ is also discussed. Several examples are given to illustrate the theoretical results and to demonstrate potential use of the study in applications.

Keywords

A COUNTING PROCESSg-RENEWAL FUNCTIONg-RENEWAL DENSITYg-RENEWAL EQUATIONMONOTONICITYASYMPTOTIC BEHAVIOR
Type
Research Papers
Information
Journal of Applied Probability , Volume 23 , Issue 1 , March 1986 , pp. 71 - 88
Copyright
Copyright © Applied Probability Trust 1986 

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References

[1] ÇInlar, E. (1975) Markov renewal theory: A survey. Management Sci. 21, 726–752.Google Scholar
[2] Jagerman, D. (1985) Certain Volterra integral equations arising in queueing. Stochastic Models .Google Scholar
[3] Keener, R. W. (1982) Renewal theory for Markov chains on the real line. Ann. Prob. 10, 942–954.CrossRefGoogle Scholar
[4] Olver, F. W. J. (1974) Introduction to Asymptotics and Special Functions. Academic Press, New York.Google Scholar
[5] Volterra, V. (1959) Theory of Functionals and Integral and Integro-differential Equations. Dover, New York.Google Scholar
[6] Volterra, V. and Peres, J. (1924) Leçons sur la composition et les fonctions permutables. Gauthier-Villars, Paris.Google Scholar