A Markov renewal approach to the asymptotic decay of the tail probabilities in risk and queuing processes

It is well known that various characteristics in risk and queuing processes can be formulated as Markov renewal functions, which are determined by Markov renewal equations. However, those functions have not been utilized as they are expected. In this article, we show that they are useful for studying asymptotic decay in risk and queuing processes under a Markovian environment, In particular, a matrix version of the Cramer-Lundberg approximation is obtained for the risk process. The corresponding result for the MAP/G/1 queue is presented as well. Emphasis is placed on a straightforward derivation using the Markov renewal structure.