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Closed queueing networks under congestion: non-bottleneck independence and bottleneck convergence

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  • Anselmi, Jonatha
  • D'Auria, Bernardo
  • Walton, Neil

Abstract

We analyze the behavior of closed product-form queueing networks when the number of customers grows to infinity and remains proportionate on each route (or class). First, we focus on the stationary behavior and prove the conjecture that the stationary distribution at non-bottleneck queues converges weakly to the stationary distribution of an ergodic, open product-form queueing network. This open network is obtained by replacing bottleneck queues with per-route Poissonian sources whose rates are determined by the solution of a strictly concave optimization problem. Then, we focus on the transient behavior of the network and use fluid limits to prove that the amount of fluid, or customers, on each route eventually concentrates on the bottleneck queues only, and that the long-term proportions of fluid in each route and in each queue solve the dual of the concave optimization problem that determines the throughputs of the previous open network.

Suggested Citation

  • Anselmi, Jonatha & D'Auria, Bernardo & Walton, Neil, 2012. "Closed queueing networks under congestion: non-bottleneck independence and bottleneck convergence," DES - Working Papers. Statistics and Econometrics. WS ws121711, Universidad Carlos III de Madrid. Departamento de Estadística.
    • Handle: RePEc:cte:wsrepe:ws121711
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    References listed on IDEAS

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    1. B. Pittel, 1979. "Closed Exponential Networks of Queues with Saturation: The Jackson-Type Stationary Distribution and Its Asymptotic Analysis," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 357-378, November.
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    Keywords

    Fluid limit;

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