Introduction to Matrix-Analytic Methods in Queues 2

4Busy Period in Queues

The analysis of the busy period (BP) in queues is more complicated (see Bhat (2015); Brill (2017); Gross and Harris (1998); Kleinrock (1975)) and very involved in terms of the tools needed as compared to the analysis of other items like queue lengths and waiting times. This is not due to the choice but rather to the inherent nature of the study of the BP. Even for the classical M/M/1 queue, the probability distribution of the BP is known in terms of the modified Bessel functions. In queueing literature, several approaches such as complex analysis, combinatorics, lattice path and matrix- analytic methods (MAM) have been applied to study some selected queueing models by several authors. The study of BP got revived since the introduction of MAM by Neuts (1981, 1989) in the context of M/G/1 and GI/M/1 paradigms, which were presented in Volume 1. A summary of such approaches as well as some interesting results on the BP in the context of GI/G/c and MAP/G/c queues via simulation can be seen in Chakravarthy (2019a).

Recall that the BP is defined as the duration of the time interval that begins with an arrival of a customer to an empty system and ends with the system becoming empty again at the departure of a customer in the case of a single-server system. In the case of multi-server systems, there are two ways to define them. One is partial and the other is full. In this book, we define a partial busy period to be the duration of the time interval that begins ...