Infinite-server queues with time-varying rates. (1st January 2013)

• Record Type: Journal Article
• Authors: Zhang, Guichang; Srinivasan, Raj
• Is Part Of: International journal of mathematics in operational research. Volume 5:Number 1(2013)
• Page Start: 91
• Page End: 109
• Publication Date: 2013-01-01
• Subjects: Kolmogorov-type equation -- multi-server queues -- martingale -- time-varying rates; Operations research -- Mathematical models -- Periodicals; Operations research -- Mathematics -- Periodicals; Decision making -- Mathematical models -- Periodicals; 658.4033
• Languages: English
• ISSNs: 1757-5850

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.