The Annals of Applied Probability

Point processes in fast Jackson networks

James B. Martin

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Abstract

We consider a Jackson-type network, each of whose nodes contains N identical channels with a single server. Upon arriving at a node, a task selects m of the channels at random and joins the shortest of the m queues observed.We fix a collection of channels in the network, and analyze how the queue-length processes at these channels vary as $N \to \infty$. If the initial conditions converge suitably, the distribution of these processes converges in local variation distance to a limit under which each channel evolves independently.We discuss the limiting processes which arise, and in particular we investigate the point processes of arrivals and departures at a channel when the networks are in equilibrium, for various values of the system parameters.

Article information

Source
Ann. Appl. Probab., Volume 11, Number 3 (2001), 650-663.

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1015345344

Digital Object Identifier
doi:10.1214/aoap/1015345344

Mathematical Reviews number (MathSciNet)
MR1865019

Zentralblatt MATH identifier
1021.90007

Subjects
Primary: 90B15: Network models, stochastic 60G55: Point processes

Keywords
Queueing network point process Jackson network

Citation

Martin, James B. Point processes in fast Jackson networks. Ann. Appl. Probab. 11 (2001), no. 3, 650--663. doi:10.1214/aoap/1015345344. https://projecteuclid.org/euclid.aoap/1015345344


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References

  • [1] Aalen, O. (1978). Nonparametric inference for a family of countingprocesses. Ann. Statist. 6 701-726.
  • [2] Brown, T. C. and Pollett, P. K. (1982). Some distributional approximations in Markovian queueingnetworks. Adv. Appl. Probab. 14 654-671.
  • [3] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
  • [4] Jackson, J. R. (1957). Networks of waitingtimes. Oper. Res. 5 518-527.
  • [5] Jackson, J. R. (1965). Jobshop-like queueingsystems. Management Sci. 10 131-142.
  • [6] Kabanov, Y. M. and Liptser, R. S. (1983). Convergence of the distributions of multivariate point processes. Z. Wahrsch. Verw. Gebiete 63 475-485.
  • [7] Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, New York.
  • [8] Martin, J. B. and Suhov, Y. M. (1999). Fast Jackson networks. Ann. Appl. Probab. 9 854-870.
  • [9] Vvedenskaya, N. D., Dobrushin, R. L. and Karpelevich, F. I. (1996). Queueingsystem with selection of the shortest of two queues: an asymptotic approach. Problems Inform. Transmission 32 15-27.
  • [10] Williams, D. (1991). Probability with Martingales. Cambridge Univ. Press.