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Tail probabilities for non-standard risk and queueing processes with subexponential jumps

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

Søren Asmussen ,
Hanspeter Schmidli  and
Volker Schmidt
Søren Asmussen*
Affiliation:
University of Lund
Hanspeter Schmidli*
Affiliation:
Aarhus University
Volker Schmidt*
Affiliation:
University of Ulm
*
∗ Postal address: Department of Mathematical Statistics, University of Lund, Box 118, S-221 00 Lund, Sweden.
∗∗ Postal address: Institute of Mathematics, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark. Email address: schmidli@imf.au.dk
∗∗∗ Postal address: Institute of Stochastics, University of Ulm, D-89069 Ulm, Germany.
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Abstract

A well-known result on the distribution tail of the maximum of a random walk with heavy-tailed increments is extended to more general stochastic processes. Results are given in different settings, involving, for example, stationary increments and regeneration. Several examples and counterexamples illustrate that the conditions of the theorems can easily be verified in practice and are in part necessary. The examples include superimposed renewal processes, Markovian arrival processes, semi-Markov input and Cox processes with piecewise constant intensities.

Keywords

Ruin probabilitystationary waiting time distributionrandom walkergodic inter-occurrence timesintegrated tail distributionregenerative surplus processregular variationsubexponential distribution

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60K25: Queueing theory

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 422 - 447
Copyright
Copyright © Applied Probability Trust 1999 

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References

Ammeter, H. (1948). A generalization of the collective theory of risk in regard to fluctuating basic probabilities. SAJ 48, 171–198.Google Scholar
Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
Asmussen, S. and Højgaard, B. (1996). Ruin probability approximations for Markov-modulated risk processes with heavy tails. Th. Random Proc. 2, 96–107.Google Scholar
Asmussen, S. and Klüppelberg, C. (1996). Large deviations results in the presence of heavy tails, with applications to insurance risk. Stoch. Proc. Appl. 64, 103–125.CrossRefGoogle Scholar
Asmussen, S. and Koole, G. (1993). Marked point processes as limits of Markovian arrival streams. J. Appl. Prob. 30, 365–372.Google Scholar
Asmussen, S., Fløe Henriksen, L. and Klüppelberg, C. (1994). Large claims approximations for risk processes in a Markovian environment. Stoch. Proc. Appl. 54, 29–43.Google Scholar
Asmussen, S., Klüppelberg, C. and Sigman, K. (1999). Sampling at subexponential times with queuing applications. Stoch. Proc. Appl. 79, 265–286. Tail probabilities for M/G/1 queue length probabilities and related random sums. Manuscript.Google Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.Google Scholar
von Bahr, B. (1975). Asymptotic ruin probabilities when exponential moments do not exist. Scand. Act. J. 1975, 6–10.CrossRefGoogle Scholar
Björk, T. and Grandell, J. (1988). Exponential inequalities for ruin probabilities in the Cox case. Scand. Act. J. 1988, 77–111.Google Scholar
Boxma, O. J. (1996). Fluid queues and regular variation. Perf. Eval. 27/28, 699–712.CrossRefGoogle Scholar
Boxma, O. J. (1997). Regular variation in a multi-source fluid queue. Research Report, CWI Amsterdam. In Teletraffic Contributions for the Information Age, eds. Ramaswami, V. and Wirth, P. E. (Proc ITC-15). Elsevier, Amsterdam, pp. 391–402.Google Scholar
Bucklew, J. A. (1990). Large Deviation Techniques in Decision, Simulation and Estimation. Wiley, New York.Google Scholar
Chistyakov, V. P. (1964). A theorem on sums of independent, positive random variables and its applications to branching processes. Theory Prob. Appl. 9, 640–648.Google Scholar
Choudhury, G. L. and Whitt, W. (1997). Long-tail buffer-content distributions in broadband networks. Perf. Eval. 30, 177–190.Google Scholar
Cline, D. B. H. (1986). Convolution tails, product tails and domains of attraction. Prob. Theory Rel. Fields 72, 529–557.Google Scholar
Cline, D. B. H. and Samorodnitsky, G. (1994). Subexponentiality of the product of independent random variables. Stoch. Proc. Appl. 49, 75–98.Google Scholar
Cohen, J. W. (1973). Some results on regular variation for distributions in queueing and fluctuation theory. J. Appl. Prob. 10, 343–353.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Proc. Appl. 13, 263–278.Google Scholar
Embrechts, P. and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1, 55–72.Google Scholar
Embrechts, P., Grandell, J. and Schmidli, H. (1993). Finite-time Lundberg inequalities in the Cox case. SAJ 93, 17–41.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Extremal Events in Finance and Insurance. Springer, Heidelberg.Google Scholar
Franken, P., König, D., Arndt, U. and Schmidt, V. (1982). Queues and Point Processes. John Wiley, Chichester.Google Scholar
Glynn, P. W. and Whitt, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. Studies in Applied Probability, eds Galambos, J. and Gani, J. (J. Appl. Prob. 31A). Applied Probability Trust, Sheffield, UK, pp. 131–156.Google Scholar
Grandell, J. (1997). Mixed Poisson Processes. Chapman & Hall, London.CrossRefGoogle Scholar
Heath, D., Resnick, S. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in on/off processes and associated fluid models. Math. Operat. Res. 23, 145–165.CrossRefGoogle Scholar
Heath, D., Resnick, S. and Samorodnitsky, G. (1997). Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Prob. 7, 1021–1057.CrossRefGoogle Scholar
Jelenković, P. R. and Lazar, A. A. (1996). Multiple time scales and subexponential asymptotic behaviour of a network multiplexer. In Stochastic Networks: Stability and Rare Events, eds Glasserman, P., Sigman, K. and Yao, D. D.. Springer, New York, pp. 215–235.Google Scholar
Jelenković, P. R. and Lazar, A. A. (1999). Asymptotic results for multiplexing on-off sources with subexponential on periods. Adv. Appl. Prob. 31, 394–421.Google Scholar
Klüppelberg, C., (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
Miyazawa, M. and Schmidt, V. (1993). On ladder height distributions of general risk processes. Ann. Appl. Prob. 3, 763–776.Google Scholar
Neuts, M. F. (1977). A versatile Markovian point process. J. Appl. Prob. 16, 764–779.CrossRefGoogle Scholar
Pakes, A. G. (1975). On the tails of waiting-time distributions. J. Appl. Prob. 12, 555–564.Google Scholar
Rolski, T., Schlegel, S. and Schmidt, V. (1999). Asymptotics of Palm-stationary buffer content distributions in fluid flow queues. Adv. Appl. Prob. 31, 235–254.Google Scholar
Schmidli, H. (1999). Compound sums and subexponentiality. To appear in Bernoulli 6.Google Scholar
Schonmann, R. H. (1989). Exponential convergence under mixing. Prob. Theory Rel. Fields 81, 235–238.Google Scholar
Thorin, O. and Wikstad, N. (1977). Calculation of ruin probabilities when the claim distribution is lognormal. ASTIN Bull. 9, 231–246.Google Scholar