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

  • Masaaki Kijima (a1) and Ushio Sumita (a1)
Abstract

Let N(t) be a counting process associated with a sequence of non-negative random variables (X j)1 where the distribution of Xn +1 depends only on the value of the partial sum Sn = Σj=1 n Xj . 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.

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Postal address: The Graduate School of Management, The University of Rochester, Rochester, NY 14627, USA.
References
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[1] ÇInlar, E. (1975) Markov renewal theory: A survey. Management Sci. 21, 726752.
[2] Jagerman, D. (1985) Certain Volterra integral equations arising in queueing. Stochastic Models .
[3] Keener, R. W. (1982) Renewal theory for Markov chains on the real line. Ann. Prob. 10, 942954.
[4] Olver, F. W. J. (1974) Introduction to Asymptotics and Special Functions. Academic Press, New York.
[5] Volterra, V. (1959) Theory of Functionals and Integral and Integro-differential Equations. Dover, New York.
[6] Volterra, V. and Peres, J. (1924) Leçons sur la composition et les fonctions permutables. Gauthier-Villars, Paris.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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