In this paper we compare ruin functions for two risk processes with respect to stochastic ordering, stop-loss ordering and ordering of adjustment coefficients. The risk processes are as follows: in the Markov-modulated environment and the associated averaged compound Poisson model. In the latter case the arrival rate is obtained by averaging over time the arrival rate in the Markov modulated model and the distribution of the claim size is obtained by averaging the ones over consecutive claim sizes.
A general concept is considered of expanding the expectation of a wide class of functionals of marked point processes, which expresses this expectation by a sum of integrals over higher-order factorial moment measures of the underlying point process. The idea of factorial moment expansion is applied in order t o derive approximation formulas for stationary characteristics of multi-server queues with Markov-modulated arrival process and with the first-come-first-served queueing discipline. Besides real-valued queueing characteristics like waiting time and total work load, we also give approximations for the Kiefer-Wolfowitz work-load vector. A boundedness condition on the service time distributions is given which ensures t h a t the components of the expected stationary work-load vector are analytic functions of the arrival intensity in a neighborhood of zero. If the service times have phase-type distributions, the factorial moment expansion provides a useful computational technique for approximations of moments of the stationary work-load vector.Some numerical examples are given which show how the algorithm works in light traffic.
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