Subexponential tail equivalence of the queue length distributions of BMAP/GI/1 queues with and without retrials

Abstract: The main contribution of this paper is to prove the subexponential tail equivalence of the stationary queue length distributions in the BMAP/GI/1 queues with and without retrials. We first present a stochastic-decomposition-like result of the stationary queue length in the BMAP/GI/1 retrial queue, which is an extension of the stochastic decomposition of the stationary queue length in the M${}^X$/GI/1 retrial queue. The stochastic-decomposition-like result shows that the stationary queue length distribution in the BMAP/GI/1 retrial queue is decomposed into two parts: the stationary conditional queue length distribution given that the server is idle; and a certain matrix sequence associated with the stationary queue length distribution in the corresponding standard BMAP/GI/1 queue (without retrials). Using the stochastic-decomposition-like result and matrix analytic methods, we prove the subexponential tail equivalence of the stationary queue length distributions in the BMAP/GI/1 queues with and without retrials. This tail equivalence result does not necessarily require that the size of an arriving batch is light-tailed, unlike Yamamuro's result for the M${}^X$/GI/1 retrial queue (Queueing Syst. 70:187--205, 2012). As a by-product, the key lemma to the roof of the main theorem presents a subexponential asymptotic formula for the stationary distribution of a level-dependent M/G/1-type Markov chain, which is the first reported result on the subexponential asymptotics of level-dependent block-structured Markov chains.