Fast Jackson networks
成果类型：
Article
署名作者：
Martin, JB; Suhov, YM
署名单位：
Universite PSL; Ecole Normale Superieure (ENS); University of Cambridge; Russian Academy of Sciences
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
发表日期：
1999
页码：
854-870
关键词：
摘要：
We extend the results of Vvedenskaya, Dobrushin and Karpelevich to Jackson networks. Each node j, 1 less than or equal to j less than or equal to 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),...,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 greater than or equal to 0 may be described by the vector (r) under bar(t) = {r(j)(n, t), 1 less than or equal to j less than or equal to J, n is an element of 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 --> oo. We show that, under a standard nonoverload condition, the limiting invariant distribution (ID! of the process (r) under bar 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.