A simple and efficient computing procedure of the stationary system-length distributions for GI^X / 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 GI(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 (BMAP) 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 BMAP/D/c queue is also derived. Some numerical implementation of the procedure for the GI(X)/D/c and BMAP/D/c queues is performed. Numerical values for the expected system length and waiting time are also obtained.