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Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation

Author

Listed:
  • O. J. Boxma

    (Eindhoven University of Technology)

  • E. J. Cahen

    (CWI)

  • D. Koops

    (University of Amsterdam)

  • M. Mandjes

    (University of Amsterdam)

Abstract

We consider a linear stochastic fluid network under Markov modulation, with a focus on the probability that the joint storage level attains a value in a rare set at a given point in time. The main objective is to develop efficient importance sampling algorithms with provable performance guarantees. For linear stochastic fluid networks without modulation, we prove that the number of runs needed (so as to obtain an estimate with a given precision) increases polynomially (whereas the probability under consideration decays essentially exponentially); for networks operating in the slow modulation regime, our algorithm is asymptotically efficient. Our techniques are in the tradition of the rare-event simulation procedures that were developed for the sample-mean of i.i.d. one-dimensional light-tailed random variables, and intensively use the idea of exponential twisting. In passing, we also point out how to set up a recursion to evaluate the (transient and stationary) moments of the joint storage level in Markov-modulated linear stochastic fluid networks.

Suggested Citation

  • O. J. Boxma & E. J. Cahen & D. Koops & M. Mandjes, 2019. "Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 125-153, March.
    • Handle: RePEc:spr:metcap:v:21:y:2019:i:1:d:10.1007_s11009-018-9644-1
    DOI: 10.1007/s11009-018-9644-1
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    References listed on IDEAS

    as
    1. Paul Glasserman & Sandeep Juneja, 2008. "Uniformly Efficient Importance Sampling for the Tail Distribution of Sums of Random Variables," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 36-50, February.
    2. Huang, Gang & Mandjes, Michel & Spreij, Peter, 2016. "Large deviations for Markov-modulated diffusion processes with rapid switching," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1785-1818.
    3. Søren Asmussen & Dominik Kortschak, 2015. "Error Rates and Improved Algorithms for Rare Event Simulation with Heavy Weibull Tails," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 441-461, June.
    4. Blom, Joke & De Turck, Koen & Mandjes, Michel, 2017. "Refined large deviations asymptotics for Markov-modulated infinite-server systems," European Journal of Operational Research, Elsevier, vol. 259(3), pages 1036-1044.
    5. Sezer, Ali Devin, 2009. "Importance sampling for a Markov modulated queuing network," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 491-517, February.
    6. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
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    Cited by:

    1. K. M. D. Chan & M. R. H. Mandjes, 2023. "A Versatile Stochastic Dissemination Model," Methodology and Computing in Applied Probability, Springer, vol. 25(3), pages 1-25, September.

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