### 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 μ_{j}. The network is fed by a family of independent Poisson flows of rates Nλ_{1}, . . . , Nλ_{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 - Σk 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 ≥ 0 may be described by the vector r_{-}(t) = {r _{j}(n, t), 1 ≤ j ≤ J, n ∈ 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 → ∞. 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.

Original language | English (US) |
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Pages (from-to) | 854-870 |

Number of pages | 17 |

Journal | Annals of Applied Probability |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - Aug 1999 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

*Annals of Applied Probability*,

*9*(3), 854-870. https://doi.org/10.1214/aoap/1029962816