A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications

摘要

The main contribution of this paper is to present a new sufficient condition for the subexponential asymptotics of the stationary distribution of a GI/G/1-type Markov chain with the stochastic phase transition matrix in non-boundary levels, which implies no possibility of jumps from level "infinity" to level zero. For simplicity, we call such Markov chains GI/G/1-type Markov chains without disasters because they are used to analyze semi-Markovian queues without "disasters", which are negative customers who remove all the customers in the system (including themselves) on their arrivals. We first demonstrate the application of our main result to the stationary queue length distribution in the standard BMAP/GI/1 queue. Thereby we present new asymptotic formulas and derive the existing formulas under weaker conditions than those in the literature. We also apply our main result to the stationary queue length distributions in two queues: One is a MAP//1 queue with the -bulk-service rule (i.e., MAP//1 queue); and the other is a MAP//1 retrial queue with constant retrial rate.