Using Factorization in Analyzing D-bmap/g/1 Queues

The discrete-time batch Markovian arrival process (D-BMAP) was first defined in [2]. The D-BMAP can represent a variety of arrival processes which include, as special cases, the Bernoulli arrival process, the Markov-modulated Bernoulli process (MMBP), the discrete-time Markovian arrival process (D-MAP), and their superpositions. It is the discrete-time version of the versatile Markovian point process introduced by Neuts [28], the N-process of Ramaswami [31], and the batch Markovian arrival process of Lucantoni [25, 26]. The objective of this paper is to demonstrate how one can apply the factorization property to the derivation of the queue-length distributions of the D-BMAP/G/1 queues with complex operational behavior during the idle period. To demonstrate how this new approach works, we are going to analyze the D-BMAP/G/1 queueing system under a double threshold policy and a setup time, which becomes the basic model for many production systems. The approach in this paper is simpler than the conventional matrix analytic method (MAM) and the supplementary variable technique. The MAM was pioneered by Neuts [29]. It starts with the analysis of the imbedded Markov renewal process at departure epochs. This method is cumbersome, especially in a system with a high degree of behavioral complexities during the idle period, in that it involves manipulating the vast amount of matrices without knowing the practical meaning of the resulting matrices. Works based on MAM are many. Blondia and Casals [2] modeled a digital video communication system by D-BMAP. Hashida et al. [7] analyzed the system with switched batch Bernoulli process (SBBP) with and without priorities. Ishizaki et al. [10] analyzed the SBBP/G/1 system in which the staying time of the underlying