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A Markov renewal approach to the asymptotic decay of the tail probabilities in risk and queuing processes. (English) Zbl 1005.60094
In the risk and queueing processes it is interesting to consider them under Markovian environments, when processes are perturbed by continuous Markov chains with finite state space. The asymptotic decay of the ruin probability with MAP claims and stationary workload is particularly important. These asymptotic decays are referred to as the Cramér-Lundberg approximations. The aim of the paper is to get the Cramér-Lundberg approximations and the asymptotics of the related queueing model under a Markovian environment using the Markov renewal structure. The Markov renewal theory for the decay rate problem is introduced, it is applied to risk processes and the MAP/G/1 queue is discussed as a dual of the risk model.

MSC:
60K15 Markov renewal processes, semi-Markov processes
60K25 Queueing theory (aspects of probability theory)
91B30 Risk theory, insurance (MSC2010)
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