(An) analysis of general input queues with server vacations

Abstract

In this dissertation, general input queues with server vacations and Markovian service process(MSP) are analyzed. An absorbing Markov chain is used for finding stationary probabilities of the arrival-epoch embedded Markov chains of a class of GI/M/1 queueing systems with generalized vacations. The entries of the so-called fundamental matrix of GI/M/1 queue are explicitly presented. The main significance of this method is that the analysis of GI/M/1 queues with server vacations is reduced to the characterization of the probability vector of the system state as seen by the first arrival during a busy period. This is demonstrated by deriving the arrival epoch queue length distribution of GI/M/1 with N-policy, MEV, and N-policy + MEV. Also, the PGF of the stationary queue length and the LST of the stationary FIFO sojourn time of GI/M/1 with SEV are derived. We make a regeneration cycle that includes one busy period of type 1 (the busy period of the standard GI/M/1 queue) and busy periods of type 2 (the busy period of the GI/M/1/MEV queue). Using the known results of the standard GI/M/1 queue and the GI/M/1/MEV queue, the PGF of the stationary queue length and the LST of the stationary FIFO sojourn time are derived. It is also shown that both the queue length and the sojourn time can be stochastically decomposed into meaningful quantities. This dissertation also presents relationships among the queue length distributions at a random epoch, at an arrival epoch and at a departure epoch for stationary single-server queues with MSP using the elementary balance equation “rate in = rate out” are derived. All these relationships hold for a broad class of infinite- and- finite capacity single-server queues with MSP without any assumption on the arrival process.