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On the Rate of Convergence of the Ross Approximation to the Renewal Function

Published online by Cambridge University Press:  27 July 2009

John E. Angus  and
Xiao Hong
John E. Angus
Affiliation:
Department of Mathematics, The Claremont Graduate School, Claremont, California 91711
Xiao Hong
Affiliation:
Department of Mathematics, The Claremont Graduate School, Claremont, California 91711
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Abstract

Consider a renewal process [N(t), t>0]. For fixed t > 0 and each n ≥ 1, let yn,1, …, Yn,n be independent exponentials each having mean t/n, independent of the renewal process. Ross [2] developed a recursion for the sequence of approximations mn = EN(Yn,1 + … + Yn,n) that converges to m(t)if the renewal function m(·) = EN(·) is continuous at t > 0. In this note, we derive an upper bound on the rate of convergence of this sequence under mild conditions on m near t. Tightness of this bound is discussed in terms of regularity conditions on m.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 2 , April 1996 , pp. 207 - 211
Copyright
Copyright © Cambridge University Press 1996

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References

1.Feller, W. (1971). An introduction to probability theory and its applications. Vol. II, 2nd ed.New York: John Wiley.Google Scholar
2.Ross, S.M. (1987). Approximations in renewal theory. Probability in the Engineering and Informational Sciences 1: 163–173.CrossRefGoogle Scholar