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Multiscale diffusion approximations for stochastic networks in heavy traffic

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  • Budhiraja, Amarjit
  • Liu, Xin

Abstract

Stochastic networks with time varying arrival and service rates and routing structure are studied. Time variations are governed by, in addition to the state of the system, two independent finite state Markov processes X and Y. The transition times of X are significantly smaller than typical inter-arrival and processing times whereas the reverse is true for the Markov process Y. By introducing a suitable scaling parameter one can model such a system using a hierarchy of time scales. Diffusion approximations for such multiscale systems are established under a suitable heavy traffic condition. In particular, it is shown that, under certain conditions, properly normalized buffer content processes converge weakly to a reflected diffusion. The drift and diffusion coefficients of this limit model are functions of the state process, the invariant distribution of X, and a finite state Markov process which is independent of the driving Brownian motion.

Suggested Citation

  • Budhiraja, Amarjit & Liu, Xin, 2011. "Multiscale diffusion approximations for stochastic networks in heavy traffic," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 630-656, March.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:3:p:630-656
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    References listed on IDEAS

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    1. Martin I. Reiman, 1984. "Open Queueing Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 441-458, August.
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    Cited by:

    1. Amarjit Budhiraja & Xin Liu, 2012. "Stability of Constrained Markov-Modulated Diffusions," Mathematics of Operations Research, INFORMS, vol. 37(4), pages 626-653, November.
    2. Li, Ming, 2020. "Multi-fractional generalized Cauchy process and its application to teletraffic," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 550(C).

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