Fast Jackson networks

Abstract

We extend the results of Vvedenskaya, Dobrushin and Karpelevich to Jackson networks. Each node j, 1 ≤ j ≤ J of the network consists of N identical channels, each with an infinite buffer and a single server with service rate μj. The network is fed by a family of independent Poisson flows of rates Nλ1, . . . , NλJ arriving at the corresponding nodes. After being served at node j, a task jumps to node k with probability pjk and leaves the network with probability p*j = 1 - Σk pjk. 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 ≥ 0 may be described by the vector r-(t) = {r j(n, t), 1 ≤ j ≤ J, n ∈ ℤ+}, where rj(n, t) is the proportion of channels at node j with queue length at least n at time t. We analyze the limit N → ∞. We show that, under a standard nonoverload condition, the limiting invariant distribution (ID) of the process 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.