Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-20T02:37:23.844Z Has data issue: false hasContentIssue false
  • English
  • Français

A laplace transform representation in a class of renewal queueing and risk processes

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Gordon E. Willmot
Gordon E. Willmot*
Affiliation:
University of Waterloo
*
∗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

For a class of renewal queueing processes characterized by a rational Laplace–Stieltjes transform of the arrival inter-occurrence time distribution, the Laplace–Stieltjes transform of the equilibrium (actual) waiting time distribution is re-expressed in a manner which facilitates explicit inversion under certain conditions. The results are of interest in other contexts as well, as for example in insurance ruin theory. Various analytic properties of these quantities are then obtained as a result.

Keywords

Compound geometricfailure rateresidual lifetimesubexponentialregular variationErlang distributionCoxian distributionLagrange interpolation

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 570 - 584
Copyright
Copyright © Applied Probability Trust 1999 

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. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
Asmussen, S. (1989). Aspects of matrix Wiener-Hopf factorisation in applied probability. Math. Scientist 14, 101–116.Google Scholar
Asmussen, S. (1992). Phase-type representations in random walk and queueing problems. Ann. Prob. 20, 772–789.Google Scholar
Bondesson, L. (1983). On preservation of classes of life distributions and reliability operations: some complementary results. Naval Res. Logistics Quart. 30, 443–447.Google Scholar
Cohen, J. (1982). The Single Server Queue (rev. edn). North-Holland, Amsterdam.Google Scholar
De Smit, J. (1995). Explicit Weiner-Hopf factorizations for the analysis of multidimensional queues. In Advances in Queueing: Theory, Methods, and Open Problems, ed. Dshalalow, J. H. CRC Press, Boca Raton, FL, pp. 293–309.Google Scholar
Dickson, D., and Hipp, C. (1999). Ruin probabilities for Erlang (2) risk processes. Insurance Math. Econom. 22, 251–262.Google Scholar
Embrechts, P. (1983). A property of the generalized inverse Gaussian distribution with some applications. J. Appl. Prob. 30, 537–544.Google Scholar
Embrechts, P., Maejima, M., and Teugels, J. (1985). Asymptotic behaviour of compound distributions. Astin Bull. 15, 45–48.Google Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
Fagiuoli, E., and Pellerey, F. (1993). New partial orderings and applications. Naval Res. Logistics. 40, 829–842.Google Scholar
Fagiuoli, E., and Pellerey, F. (1994). Preservation of certain classes of life distributions under Poisson shock models. J. Appl. Prob. 31, 458–465.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications,Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Gerber, H. (1973). Martingales in risk theory. Vereinigung Schweizerischer Versicherungs Mathematiker Mitteilungen, 205–216.Google Scholar
Grandell, J. (1997). Mixed Poisson Processes. Chapman and Hall, London.Google Scholar
Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory and Rel. Fields 82, 259–269.CrossRefGoogle Scholar
Lin, X. (1996). Tail of compound distributions and excess time. J. Appl. Prob. 33, 184–195.Google Scholar
Lin, X., and Willmot, G. (1997). Analysis of a defective renewal equation arising in ruin theory. Research Report 97-10, Institute of Insurance and Pension Research, University of Waterloo.Google Scholar
Neuts, M. (1989). Structured Stochastic Matrices of M/G/1 Type and their Applications. Marcel Dekker, New York.Google Scholar
Ross, S. (1974). Bounds on the delay distribution in GI/G/1 queues. J. Appl. Prob. 11, 417–421.CrossRefGoogle Scholar
Ross, S. (1996). Stochastic Processes, 2nd edn. John Wiley, New York.Google Scholar
Shanthikumar, J. (1988). DFR property of first passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.CrossRefGoogle Scholar
Tijms, H. (1986). Stochastic Modelling and Analysis: A Computational Approach. John Wiley, Chichester.Google Scholar
Tijms, H. (1994). Stochastic Models: An Algorithmic Approach. John Wiley, Chichester.Google Scholar
Willmot, G., and Lin, X. (1997). Simplified bounds on the tails of compound distributions. J. Appl. Prob. 34, 127–133.Google Scholar