Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-11T04:06:10.440Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic tail behaviour of Poisson mixtures by applications

Published online by Cambridge University Press:  01 July 2016

Gordon E. Willmot
Gordon E. Willmot*
Affiliation:
University of Waterloo
*
∗Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ont., Canada, N2L 3G1.
Get access Rights & Permissions [Opens in a new window]

Abstract

This expository paper deals with the right tail behaviour of a class of Poisson mixtures. An Abelian-type result is obtained using basic theory of regular variation. Applications to compound distributions in insurance risk theory and queue length distributions under various queue disciplines in the case of Poisson arrivals are then discussed.

Keywords

REGULAR VARIATIONCOMPOUND DISTRIBUTIONSINSURANCE RISK THEORYM/G/1 QUEUE LENGTHFEEDBACK QUEUESQUEUES WITH VACATIONSQUEUES WITH WARM-UP TIMES
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 1 , March 1990 , pp. 147 - 159
Copyright
Copyright © Applied Probability Trust 1990 

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

Albrecht, P. (1984) Laplace transforms, Mellin transforms, and mixed Poisson processes. Scand. Actuarial J., 58–64.CrossRefGoogle Scholar
Arnold, B. (1983) Pareto Distributions. International Co-operative Publishing House, Fairland, Maryland.Google Scholar
Bender, E. (1974) Asymptotic methods of enumeration. SIAM REV. 16, 485–515; Corrigendum 18 (1976), 292.CrossRefGoogle Scholar
Bingham, N., Goldie, C. and Teugels, J. (1987) Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Borovkov, A. (1976) Stochastic Processes in Queueing Theory. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Borovkov, A. (1984) Asymptotic Methods in Queueing Theory. Wiley, New York.Google Scholar
Bühlmann, H. (1970) Mathematical Methods in Risk Theory. Springer-Verlag, New York.Google Scholar
Chover, J., Ney, P. and Wainger, S. (1973) Functions of probability measures. J. Analyse Math. 26, 255–302.CrossRefGoogle Scholar
Disney, R., König, D. and Schmidt, V. (1984). Stationary queue-length and waiting-time distributions in single-server feedback queues. Adv. Appl. Prob. 16, 437–446.CrossRefGoogle Scholar
Doshi, B. (1986) Queueing systems with vacations—a survey. Queueing Systems 1, 29–66.CrossRefGoogle Scholar
Embrechts, P. (1983) A property of the generalized inverse Gaussian distribution with some applications. J. Appl. Prob. 20, 537–544.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. (1982) On convolution tails. Stoch. Proc. Appl. 13, 263–278.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.CrossRefGoogle Scholar
Embrechts, P. and Hawkes, J. (1982) A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory. J. Austral. Math. Soc. A 32, 412–422.CrossRefGoogle Scholar
Embrechts, P., Maejima, M. and Teugels, J. (1985) Asymptotic behaviour of compound distributions. ASTIN Bull. 15, 45–48.CrossRefGoogle Scholar
Evgrafov, M. (1961) Asymptotic Estimates and Entire Functions. Gordon and Breach, New York.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Kleinrock, L. (1975) Queueing Systems–Volume 1, Theory. Wiley, New York.Google Scholar
Meir, A. and Moon, J. (1987) Some asymptotic results useful in enumeration problems. Aequationes Math. 33, 260–268.CrossRefGoogle Scholar
Neuts, M. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore.Google Scholar
Neuts, M. (1986) Generalizations of the Pollaczek–Khinchin integral equation in the theory of queues. Adv. Appl. Prob. 18, 952–990.CrossRefGoogle Scholar
Neuts, M. and Ramalhoto, M. (1984) A service model in which the server is required to search for customers. J. Appl. Prob. 21, 157–166.CrossRefGoogle Scholar
Sichel, H. (1971) On a family of discrete distributions particularly suited to represent long tailed frequency data. Proc. Third Symp. Mathematical Statistics, ed. Laubscher, N.. CSIR, Pretoria.Google Scholar
Stam, A. (1973) Regular variation of the tail of a subordinated probability distribution. Adv. Appl. Prob. 5, 308–327.CrossRefGoogle Scholar
Stein, G., Zucchini, W. and Juritz, J. (1987) Parameter estimation for the Sichel distribution and its multivariate extension. J. Amer. Statist. Assoc. 82, 938–944.CrossRefGoogle Scholar
Teugels, J. (1977) On the rate of convergence of the maximum of a compound Poisson process. Bull. Soc. Math. Belg. 29, 205–216.Google Scholar
Teugels, J. and Willmot, G. (1987) Approximations for stop-loss premiums. Insurance: Math. Econ. 6, 195–202.Google Scholar
Tijms, H. (1986) Stochastic Modelling and Analysis: A Computational Approach. Wiley, Chichester.Google Scholar
Welch, P. (1964) On a generalized M/G/1 queueing process in which the first customer of each busy period receives exceptional service. Operat. Res. 12, 736–752.CrossRefGoogle Scholar