M/M/1 Bulk Arrival and Bulk Service Queue with Randomly Varying Environment

This paper studies two stochastic bulk arrival and bulk service queueing Models (A) and (B) with k varying environments. 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 in all environments is greater than the maximum of the service sizes in all environments and the infinitesimal generator is partitioned as blocks of k times the maximum of the arrival sizes for analysis and in Model (B) the maximum of the arrival sizes in all environments is less than the maximum of the service sizes in all environments where the generator is partitioned using blocks of k times the maximum of the service sizes. The basic system generator is seen as a block circulant matrix. The stationary queue length probabilities, its expected values, its variances and probabilities of empty levels are derived for the two models using the rate matrix iterated. Numerical examples are presented for illustration. Bulk arrival and bulk service M/PH/1 and PH/M/1 queues become special cases of the models when the PH generator is identified as the generator of the randomly varying environment.