Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds
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We study a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traffic limits for the distributions of the joint workload, waiting-time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traffic limits. The marginals of these limiting distributions are shown to be exponential with rates that can be computed by matrix-analytic methods. Moreover, we show how to numerically compute the joint distributions, by viewing the limit processes as multi-dimensional semi-martingale reflected Brownian motions in the non-negative orthant.
KeywordsFunctional central limit theorem Layered queueing networks Machine-repair model Semi-martingale reflected Brownian motion
Mathematics Subject Classification60K25 68M20 90B22
The authors are indebted to Sem Borst and Onno Boxma for providing valuable comments on earlier drafts of this paper. Furthermore, the authors wish to thank an anonymous referee for providing constructive criticism and for making several suggestions that led to an improved exposition of the contents in this paper. Funded in the framework of the STAR-project ‘Multilayered queueing systems’ by the Netherlands Organization for Scientific Research (NWO). The research of Maria Vlasiou is also partly supported by an NWO individual Grant through Project 632.003.002. The research of Bert Zwart is partly supported by an NWO VIDI grant.
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