PH/PH/1 Bulk Arrival and Bulk Service Queue

This paper studies two stochastic bulk arrivals and bulk services PH/PH/1 queue Models (A) and (B) with k_(1 ) and k_2 as the number of phases of PH arrival and PH service distributions respectively. The system has infinite storing capacity and the arrival and service sizes are finite valued random variables. Matrix partitioning method is used to study the models. In Model (A) the maximum of the arrival sizes is greater than the maximum of the service sizes and the infinitesimal generator is partitioned as blocks of k_1 k_2 times the maximum of the arrival sizes for analysis. In Model (B) the maximum of the arrival sizes is less than the maximum of the service sizes. The generator is partitioned using blocks of k_1 k_2 times the maximum of the service sizes. Block circulant matrix structure is noticed in the basic system generator. The stationary queue length probabilities, its expected values, its variances and probabilities of empty levels are derived for the two models using matrix geometric methods. Numerical examples are presented for illustration.