An Analytic Approach to a General Class of G/G/s Queueing Systems

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Abstract
We solve the queueing system Ck/Cm/s, where Ck is the class of Coxian probability density functions (pdfs) of order k, which is a subset of the pdfs that have a rational Laplace transform. We formulate the model as a continuous-time, infinite-space Markov chain by generalizing the method of stages. By using a generating function technique, we solve an infinite system of partial difference equations and find closed-form expressions for the system-size, general-time, prearrival, post-departure probability distributions and the usual performance measures. In particular, we prove that the probability of n customers being in the system, when it is saturated is a linear combination of geometric terms. The closed-form expressions involve a solution of a system of nonlinear equations that involves only the Laplace transforms of the interarrival and service time distributions. We conjecture that this result holds for a more general model. Following these theoretical results we propose an exact algorithm for finding the system-size distribution and the system's performance measures. We examine special cases and apply this method for numerically solving the C2/C2/s and Ek/C2/s queueing systems.

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