Infinite-server queues with time-varying rates
Summary: 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 \(M(t)/M(t)/\infty\) 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 \(M(t)/M(t)/\infty\) queue. Based on the solutions developed for the \(M(t)/M(t)/\infty\) queue, we provide an alternative and easy to interpret justification for the transition probabilities of the \(M(t)/G(t)/\infty\) queue. Finally we develop expressions to calculate the time dependent version of the mean and variance of the queue length processes.
scientific article; zbMATH DE number 6876009