## 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. Klüppelberg 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.

### MSC:

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

### Keywords:

failure rate; Erlang distribution; Lagrange interpolation

Zbl 0873.62116
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