The Annals of Applied Probability

Fast Jackson networks

J. B. Martin and Yu. M. Suhov

Full-text: Open access

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


Export citation

References

  • [1] Ethier, S. N. and Kurtz, T. G. (1985). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [2] Jackson, J. R. (1957). Networks of waiting times. Oper. Res. 5 518-527.
  • [3] Jackson, J. R. (1965). Jobshop-like queueing sy stems. Management Sci. 10 131-142.
  • [4] Karpelevich, F. I., Pechersky, E. A. and Suhov, Y. M. (1996). Dobrushin's approach to queueing network theory. J. Appl. Math. Stochastic Anal. 9 373-397.
  • [5] Kelly, F. P. (1991). Loss networks. Ann. Appl. Probab. 1 319-378.
  • [6] Mitzenmacher, M. (1996). The power of two choices in randomized load balancing. Ph.D. dissertation, Univ. California, Berkeley.
  • [7] Turner, S. R. E. (1998). The effect of increasing routing choice on resource pooling. Probab. Engrg. Inform. Sci. 12 109-124.
  • [8] Vvedenskay a, N. D., Dobrushin, R. L. and Karpelevich, F. I. (1996). Queueing sy stem with selection of the shortest of two queues: an asy mptotic approach. Problems Inform. Transmission 32 15-27.
  • [9] Vvedenskay a, N. D. and Suhov, Y. M. (1997). Dobrushin's mean-field approximation for a queue with dy namic routing. Markov Processes and Related Fields 3 493-526.