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A Laplace transform representation in a class of renewal queueing and risk processes. (English) Zbl 0942.60086
It is well-known that the probability \(\psi{(x)}\) that the equilibrium waiting time in the G/G/1 queue exceeds \(x\) may be expressed as the tail of a compound geometric distribution, \(\sum (1-\rho){\rho}^n {\overline{H}}^{*n}(x)\), \(x>0\), where \(H(x)\) is a distribution function [see P. Embrechts, C. Kl├╝ppelberg and T. Mikosch, “Modelling extremal events for insurance and finance” (1997; Zbl 0873.62116)]. The author considers queueing processes in which the inter-arrival distributions have rational Laplace-Stieltjes transforms and finds the Laplace transform of \(H(x)\) which can be inverted easily leading to the analytical properties of \(\psi{(x)}\). Special cases are considered.

60K25 Queueing theory (aspects of probability theory)
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
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