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Convergence rates for M/G/1 queues and ruin problems with heavy tails

Part of: Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

Søren Asmussen  and
Jozef L. Teugels
Søren Asmussen*
Affiliation:
University of Lund
Jozef L. Teugels*
Affiliation:
Universiteit Leuven
*
∗Postal address: Department of Mathematical Statistics, University of Lund, Box 118, 221 00 Lund, Sweden.
∗∗Postal address: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium.
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Abstract

The time-dependent virtual waiting time in a M/G/1 queue converges to a proper limit when the traffic intensity is less than one. In this paper we give precise rates on the speed of this convergence when the service time distribution has a heavy regularly varying tail.

The result also applies to the classical ruin problem. We obtain the exact rate of convergence for the ruin probability after time t for the case where claims arrive according to a Poisson process and claim sizes are heavy tailed.

Our result supplements similar theorems on exponential convergence rates for relaxation times in queueing theory and ruin probabilities in risk theory.

Keywords

BUSY PERIODEQUILIBRIUM DISTRIBUTIONREGULAR VARIATIONRELAXATION TIMERUIN PROBABILITYSUBEXPONENTIAL DISTRIBUTION

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 1181 - 1190
Copyright
Copyright © Applied Probability Trust 1996 

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