Infinite-server queues with time-varying rates

Guichang Zhang and Raj Srinivasan

International Journal of Mathematics in Operational Research, 2013, vol. 5, issue 1, 91-109

Abstract: In this paper, we study infinite server queues with time-varying rates. Under the assumption of non homogenous Poisson process for the arrivals and service processes, in addition to deriving the standard Kolmogorov forward equation, we introduce and develop a generalised backward equation for the M(t)/M(t)/∞ queue using martingales. Explicit solution for these two equations are provided. We also introduce two more Kolmogorov type equations namely, the quasi Kolmogorov Backward Equation and the quasi Generalised Forward Equation for the M(t)/M(t)/∞ queue. Based on the solutions developed for the M(t)/M(t)/∞ queue, we provide an alternative and easy to interpret justification for the transition probabilities of the M(t)/G(t)/∞ queue. Finally we develop expressions to calculate the time dependent version of the mean and variance of the queue length processes.

Keywords: Kolmogorov-type equation; multi-server queues; martingales; time-varying rates; infinite server queues; generalised backward equation; queue length. (search for similar items in EconPapers)
Date: 2013
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