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

This paper studies two stochastic bulk arrival and bulk service PH/PH/1 queue Models (A) and (B) with randomly varying k* distinct environments. The arrival and service distributions are (αi , Ti) and (βi , Si) in the environment i for 1 ≤ i ≤ k* respectively. Whenever the environment changes from i to j the arrival PH and service PH distributions change from the i version to the j version with the exception of the first remaining arrival time and first remaining service time have stationary PH distributions of the j version which is known as equilibrium PH distribution for 1 ≤ i, j ≤ k* and on completion of the same the arrival and service distributions become initial versions (αj , Tj ) and (βj , Sj ). The queue system has infinite storing capacity and the state space is identified as five dimensional one to apply Neuts’ matrix methods. The arrivals and the services occur whenever absorptions occur in the corresponding PH distributions. The sizes of the arrivals and the services are finite valued discrete random variables with distinct distributions with respect to environments and with respect to PH phases from which the absorptions occur. 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 mostly as blocks of the sum of the products of PH arrival and PH service phases in the various environments 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 mostly using blocks of the same sum-product of phases times the maximum of the service sizes. Block circulant matrix structure is noticed in the basic system generators. The stationary queue length probabilities, its expected values, its variances and probabilities of empty levels are derived for the two models using matrix methods. Numerical examples are presented for illustration.