Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-14T11:12:42.289Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.10.1142/2779Google Scholar
Barlow, R., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Brown, M. (1990). Error bounds for exponential approximations of geometric convolutions. Ann. Prob. 18, 1388–1402.10.1214/aop/1176990750Google Scholar
Cai, J., and Garrido, J. (2002). Asymptotic forms and bounds for tails of convolutions of compound geometric distributions, with applications. In Recent Advances in Statistical Methods (Proc. Statist., Canada, 2001), Imperial College Press, London, pp. 114–131.10.1142/9781860949531_0010Google Scholar
Cai, J., and Kalashnikov, V. (2000). NWU property of a class of random sums. J. Appl. Prob. 37, 283–289.10.1239/jap/1014842286Google Scholar
Cohen, J. (1982). The Single Server Queue, 2nd edn. North Holland, Amsterdam.Google Scholar
Dufresne, F., and Gerber, H. (1991). Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom. 10, 51–59.10.1016/0167-6687(91)90023-QGoogle Scholar
Fagiuoli, E., and Pellerey, F. (1994). Preservation of certain classes of life distributions under Poisson shock models. J. Appl. Prob. 31, 458–465.10.2307/3215038Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Grandell, J. (1991). Aspects of Risk Theory. Springer, New York.10.1007/978-1-4613-9058-9Google Scholar
Miyazawa, M. (1986). Approximation of the queue-length distribution of an M/GI/s queue by the basic equations. J. Appl. Prob. 23, 443–458.10.2307/3214186Google Scholar
Neuts, M. (1986). Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues. Adv. Appl. Prob. 18, 952–990.10.2307/1427258Google Scholar
Prabhu, N. (1997). Foundations of Queueing Theory. Kluwer, Boston, MA.10.1007/978-1-4615-6205-4Google Scholar
Van Hoorn, M. H. (1984). Algorithms and Approximations for Queueing Systems (CWI Tract 8). CWI, Amsterdam.Google Scholar
Willmot, G. E. (2002a). Compound geometric residual lifetime distributions and the deficit at ruin. Insurance Math. Econom. 30, 421–438.10.1016/S0167-6687(02)00122-1Google Scholar
Willmot, G. E. (2002b). On higher-order properties of compound geometric distributions. J. Appl. Prob. 39, 324–340.10.1239/jap/1025131429Google Scholar
Willmot, G. E., and Lin, X. (1996). Bounds on the tails of convolutions of compound distributions. Insurance Math. Econom. 18, 29–33. (Correction: 18, 219.)10.1016/0167-6687(95)00024-0Google Scholar
Willmot, G. E., and Lin, X. S. (2001). Lundberg Approximations for Compound Distributions with Insurance Applications (Lecture Notes Statist. 156). Springer, New York.Google Scholar