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

TL;DR

This paper investigates the asymptotic behavior of large closed queueing networks, proving convergence of non-bottleneck queues to open network distributions and showing fluid concentration on bottlenecks.

Contribution

It establishes the weak convergence of non-bottleneck queues to an open network model and characterizes the long-term fluid distribution focusing on bottleneck queues.

Findings

Non-bottleneck queues converge to an open network distribution.

Fluid concentrates on bottleneck queues over time.

Long-term proportions solve the dual optimization problem.

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.