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On applications of residual lifetimes of compound geometric convolutions

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Gordon E. Willmot  and
Jun Cai
Gordon E. Willmot*
Affiliation:
University of Waterloo
Jun Cai*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
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Abstract

We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.

Keywords

Defective renewal equationNWUNBUNWUENBUE2-NWU2-NBUNWUCNBUCG/G/1 virtual waiting timerenewal risk modelSparre-Anderson modeldeficit at ruinseverity of ruinM/G/c queueruin with diffusionequilibrium distributionintegrated tail distribution

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 802 - 815
Copyright
Copyright © Applied Probability Trust 2004 

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