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Uniform renewal theory with applications to expansions of random geometric sums

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  01 July 2016

J. Blanchet  and
P. Glynn
J. Blanchet*
Affiliation:
Harvard University
P. Glynn*
Affiliation:
Stanford University
*
∗ Postal address: Statistics Department, Harvard University, 1 Oxford Street, Cambridge, MA 02138, USA. Email address: blanchet@fas.harvard.edu
∗∗ Postal address: Management Science and Engineering, Stanford University, 380 Panama Way, Stanford, CA 94305, USA.
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Abstract

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Consider a sequence X = (Xn: n ≥ 1) of independent and identically distributed random variables, and an independent geometrically distributed random variable M with parameter p. The random variable SM = X1 + ∙ ∙ ∙ + XM is called a geometric sum. In this paper we obtain asymptotic expansions for the distribution of SM as p ↘ 0. If EX1 > 0, the asymptotic expansion is developed in powers of p and it provides higher-order correction terms to Renyi's theorem, which states that P(pSM > x) ≈ exp(-x/EX1). Conversely, if EX1 = 0 then the expansion is given in powers of √p. We apply the results to obtain corrected diffusion approximations for the M/G/1 queue. These expansions follow in a unified way as a consequence of new uniform renewal theory results that are also developed in this paper.

Keywords

Renewal theoryrandom geometric sumcorrected diffusion approximation

MSC classification

Primary: 60K05: Renewal theory 60K25: Queueing theory
Secondary: 90B22: Queues and service 90B20: Traffic problems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 4 , December 2007 , pp. 1070 - 1097
Copyright
Copyright © Applied Probability Trust 2007 

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