Abstract
Recently multiserver queues with setup times have been extensively studied because they have applications in power-saving data centers. A challenging model is the M/M/c/Setup queue where a server is turned off when it is idle and is turned on if there are some waiting jobs. Recently, Gandhi et al. (in: Proceedings of the ACM SIGMETRICS, pp. 153---166, ACM, 2013; Queueing Syst. 77(2):177---209, 2014) obtain the generating function of the number of jobs in the system, as well as the Laplace transform of the response time using the recursive renewal reward approach and the distributional Little's law (Keilson and Servi in Oper Res Lett 7(5):223--- 227, 1988). In this paper, we derive exact solutions for the joint stationary queue length distribution of the same model using two alternative methodologies: generating function approach and matrix analytic method. The generating function approach yields exact closed form expressions for the joint stationary queue length distribution and the conditional decomposition formula. On the other hand, the matrix analytic approach leads to an exact recursive algorithm to calculate the joint stationary distribution and performance measures so as to provide some application insights.
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