| 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. |