A simple and efficient computing procedure of the stationary system-length distributions for GIX/D/c and BMAP/D/c queues

This article gives closed-form analytic expressions as well as a computational analysis of the stationary system-length distribution for the renewal-input, bulk-arrival, and multi-server continuous-time queueing model. The service times are equal to the constant D for any customer. The queueing model may be denoted as G I X / D / c queue. Using the steady-state equations, the system-length probability generating function is derived. Subsequently, by inverting this probability generating function the stationary system-length distribution is obtained using the roots of a characteristic equation. Next, a similar analysis for the corresponding multi-server queueing model with batch Markovian arrival process ( B M A P ) is carried out using the roots of a characteristic equation associated with the vector generating function of the system-length distribution. The distribution function of the stationary actual waiting-time for the first customer of an arrival batch in a B M A P / D / c queue is also derived. Some numerical implementation of the procedure for the G I X / D / c and B M A P / D / c queues is performed. Numerical values for the expected system length and waiting time are also obtained.