Closed queueing networks under congestion: Nonbottleneck independence and bottleneck convergence
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We analyze the behavior of closed multiclass 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 nonbottleneck queues converges weakly to the stationary distribution of an ergodic, open product-form queueing network, which is geometric. This open network is obtained by replacing bottleneck queues with per-route Poissonian sources whose rates are uniquely determined by the solution of a strictly concave optimization problem. We strengthen such results by also proving convergence of the first moment of the queue lengths of nonbottleneck stations. 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.