Transient Analysis of Two-Dimensional State Non-Markovian Queueing System with Multiple Working Vacations

This paper analyses the transient behaviour of a first-come, first-served, two-dimensional state, non-Markovian queueing system with multiple working vacations such that the server works with different service rate rather than completely stop service during a vacation period. This queue approximates a multi-queue system in which each queue operates either at a fast service rate or a nominal service rate, and the fast service mode cyclically moves from queue to queue with an exhaustive schedule. For such a queue, we provide a recursive method, using the supplementary variable technique as the remaining service time during busy period, to obtain the Laplace transform of the probabilities of exact number of arrivals and departures by a given time t, number of units arrived by time t and number of units departed by time t. It is assumed that the service time during working vacation period and working vacation times are both exponentially distributed. The emphasis in this paper is theoretical, but numerical assessment of operational consequences is also given and presented graphically for exponential service time during the busy period. Finally, some particular cases are derived.