Abstract
In this paper martingales methods are applied for analyzing limit non-stationary behavior of the queue length processes in closed Jackson queueing networks with a single class consisting of a large number of customers, a single infinite server queue, and a fixed number of single server queues with large state independent service rates. It is assumed that one of the single server nodes forms a bottleneck. For the non-bottleneck nodes we show that the queue length distribution at timet converges in generalized sense to the stationary distribution of the M/M/1 queue whose parameters explicitly depend ont. For the bottleneck node a diffusion approximation with reflection is proved in the moderate usage regime while fluid and Gaussian diffusion approximations are established for the heavy usage regime.
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Kogan, Y., Liptser, R.S. Limit non-stationary behavior of large closed queueing networks with bottlenecks. Queueing Syst 14, 33–55 (1993). https://doi.org/10.1007/BF01153525
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DOI: https://doi.org/10.1007/BF01153525