## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 9, Number 3 (1999), 854-870.

### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 9, Number 3 (1999), 854-870.

**Dates**

First available in Project Euclid: 21 August 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1029962816

**Digital Object Identifier**

doi:10.1214/aoap/1029962816

**Mathematical Reviews number (MathSciNet)**

MR1722285

**Zentralblatt MATH identifier**

0972.90008

**Subjects**

Primary: 90B15: Network models, stochastic

Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 60J27: Continuous-time Markov processes on discrete state spaces

**Keywords**

Jackson network queueing network dynamic routing Markov process superexponential decay

#### Citation

Martin, J. B.; Suhov, Yu. M. Fast Jackson networks. Ann. Appl. Probab. 9 (1999), no. 3, 854--870. doi:10.1214/aoap/1029962816. https://projecteuclid.org/euclid.aoap/1029962816