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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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References

[1] Abate, J., Choudhury, G. L. and Whitt, W. (1994) Waiting-time tail probabilities in queues with long-tail service-time distributions. Queueing Systems 16, 311–338.Google Scholar
[2] Abate, J. and Whitt, W. (1994) Transient behavior of the M/G/1 workload process. Operat. Res. 42, 750–764.Google Scholar
[3] Asmussen, S. (1984) Approximations for the probability of ruin within finite time. Scand. Act. J. 31–57; (1985) Scand. Act. J. 57.Google Scholar
[4] Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
[5] Asmussen, S. (1996) Ruin Probabilities. World Scientific, Singapore.Google Scholar
[6] Asmussen, S. and Binswanger, K. (1995) Simulation of ruin probabilities for subexponential claims. Working paper. University of Lund and ETH Zürich.Google Scholar
[7] Asmussen, S. and Klüppelberg, C. (1996) Large deviations results for subexponential distributions, with applications to insurance risk. Stoch. Proc. Appl. (in print).Google Scholar
[8] Asmussen, S. and Schmidt, V. (1995) Ladder height distributions with marks. Stoch. Proc. Appl. 58, 105–119.Google Scholar
[9] Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer, Berlin.Google Scholar
[10] von, Bahr. B. (1975) Asymptotic ruin probabilities when exponential moments do not exist. Scand. Act. J. 6–10.Google Scholar
[11] Barndorff-Nielsen, O. and Schmidli, H. (1995) A saddlepoint aproximation for finite time ruin probabilities. Scand. Act. J. 169–186.Google Scholar
[12] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987) Regular Variation. Cambridge University Press, Cambridge.Google Scholar
[13] Borovkov, A. A. (1976) Stochastic Processes in Queueing Theory. Springer, Berlin.Google Scholar
[14] Choudhury, G. L., Lucantoni, D. and Whitt, W. (1994) Multidimensional transform inversion with applications to the transient M/G/1 queue. Ann. Appl. Prob. 4, 719–740.CrossRefGoogle Scholar
[15] Cohen, J. W. (1973) Some results on regular variation for distributions in queueing and fluctuation theory. J. Appl. Prob. 10, 343–353.CrossRefGoogle Scholar
[16] Cohen, J. W. (1982) The Single Server Queue. 2nd edn. North-Holland, Amsterdam.Google Scholar
[17] De Meyer, A. and Teugels, J. L. (1980) On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1. J. Appl. Prob. 17, 802–813.Google Scholar
[18] Doney, R. A. (1989) On the asymptotic behaviour of first passage times for transient random walk. Prob. Theory Rel. Fields 81, 239–246.Google Scholar
[19] Dufresne, F. and Gerber, H. U. (1988) The surpluses immediately before and at ruin, and the amount of claim causing ruin. Insurance: Math. Econ. 7, 193–199.Google Scholar
[20] Embrechts, P. and Veraverbeke, N. (1982) Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Math. Econ. 1, 55–72.Google Scholar
[21] Feller, W. (1971) An Introduction to Probability Theory and its Applications II. 2nd edn. Wiley, New York.Google Scholar
[22] Fitzsimmons, P. J. (1987) On the excursions of Markov processes in classical duality. Prob. Theory Rel. Fields 75, 159–178.Google Scholar
[23] Lund, R. B., Meyn, S. P. and Tweedie, R. L. (1996) Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Prob. 6, 218–237.Google Scholar
[24] Omey, ?. and Willekens, E. (1986) Second order behaviour of the tail of a subordinated probability distribution. Stoch. Proc. Appl. 21, 339–353.Google Scholar
[25] Pakes, A. (1975) On the tails of waiting-time distributions. J. Appl. Prob. 12, 555–564.Google Scholar
[26] Takács, L. (1967) Combinatorial Methods in the Theory of Stocastic Processes. Krieger, Huntington.Google Scholar
[27] Teugels, J. L. (1974) The class of subexponential distributions. Ann. Prob. 3, 1000–1011.Google Scholar
[28] Teugels, J. L. (1982) Estimation of ruin probabilities. Insurance: Math. Econ. 1, 163–175.Google Scholar
[29] Thorisson, H. (1985) The queue GI/G/1: Finite moments of the cycle variables and uniform rates of convergence. Stoch. Proc. Appl. 19, 85–99.CrossRefGoogle Scholar