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Computing the Stationary Distribution of Queueing Systems with Random Resource Requirements via Fast Fourier Transform

Author

Listed:
  • Valeriy A. Naumov

    (Service Innovation Research Institute, 8 A Annankatu, 00120 Helsinki, Finland)

  • Yuliya V. Gaidamaka

    (Applied Informatics and Probability Department, Peoples’ Friendship University of Russia (RUDN University), Miklukho-Maklaya St. 6, Moscow 117198, Russian
    Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilov St. 44-2, Moscow 119333, Russian)

  • Konstantin E. Samouylov

    (Applied Informatics and Probability Department, Peoples’ Friendship University of Russia (RUDN University), Miklukho-Maklaya St. 6, Moscow 117198, Russian
    Institute of Informatics Problems, Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Vavilov St. 44-2, Moscow 119333, Russian)

Abstract

Queueing systems with random resource requirements, in which an arriving customer, in addition to a server, demands a random amount of resources from a shared resource pool, have proved useful to analyze wireless communication networks. The stationary distributions of such queuing systems are expressed in terms of truncated convolution powers of the cumulative distribution function of the resource requirements. Discretization of the cumulative distribution function and the application of the fast Fourier transform are a traditional way of calculating convolutions. We suggest finding truncated convolution powers of the cumulative distribution functions by calculating the convolution powers of the truncated cumulative distribution functions via fast Fourier transform. This radically decreases computational complexity. We introduce the concept of resource load and investigate the accuracy of the proposed method at low and high resource loads. It is shown that the proposed method makes it possible to quickly and accurately calculate truncated convolution powers required for the analysis of queuing systems with random resource requirements.

Suggested Citation

  • Valeriy A. Naumov & Yuliya V. Gaidamaka & Konstantin E. Samouylov, 2020. "Computing the Stationary Distribution of Queueing Systems with Random Resource Requirements via Fast Fourier Transform," Mathematics, MDPI, vol. 8(5), pages 1-9, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:5:p:772-:d:356862
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    References listed on IDEAS

    as
    1. Grübel, Rudolf & Hermesmeier, Renate, 2000. "Computation of Compound Distributions II: Discretization Errors and Richardson Extrapolation," ASTIN Bulletin, Cambridge University Press, vol. 30(2), pages 309-331, November.
    2. Grübel, Rudolf & Hermesmeier, Renate, 1999. "Computation of Compound Distributions I: Aliasing Errors and Exponential Tilting," ASTIN Bulletin, Cambridge University Press, vol. 29(2), pages 197-214, November.
    3. Ruckdeschel, Peter & Kohl, Matthias, 2014. "General Purpose Convolution Algorithm in S4 Classes by Means of FFT," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 59(i04).
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