Fast Jackson networks

Abstract

We extend the results of Vvedenskaya, Dobrushin and Karpelevich to Jackson networks. Each node $j, 1 \leq j \leq J$ of the network consists of N identical channels, each with an infinite buffer and a single server with service rate $\mu_j$. The network is fed by a family of independent Poisson flows of rates $N\lambda_1,\dots, N\lambda_J$ arriving at the corresponding nodes. After being served at node j, a task jumps to node k with probability $p_{jk}$ and leaves the network with probability $p_j^* = 1 - \Sigma_k p_{jk}$. Upon arrival at any node, a task selects m of the N channels there at random and joins the one with the shortest queue. The state of the network at time $t \geq 0$ may be described by the vector $\underline{\mathbf{r}}(t) = {r_j(n, t), 1 \leq j \leq J, n \epsilon \mathbb{Z}_+}$, where $r_j(n, t)$ is the proportion of channels at node j with queue length at least n at time t. We analyze the limit $N \rightarrow \infty$. We show that, under a standard nonoverload condition, the limiting invariant distribution (ID) of the process $\underline{\mathbf{r}}$ is concentrated at a single point, and the process itself asymptotically approaches a single trajectory. This trajectory is identified with the solution to a countably infinite system of ODE's, whose fixed point corresponds to the limiting ID. Under the limiting ID, the tail of the distribution of queue-lengths decays superexponentially, rather than exponentially as in the case of standard Jackson networks--hence the term "fast networks" in the title of the paper.