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Numerical Computation of the Moments of a Probability Distribution from its Transform

Author

Listed:
  • Gagan L. Choudhury

    (AT&T Bell Laboratories, Holmdel, New Jersey)

  • David M. Lucantoni

    (IsoQuantic Technologies, Wayside, New Jersey)

Abstract

We present a simple, fast, and robust algorithm for numerically computing the first N moments (arbitrary N ) of a nonnegative probability distribution from its Laplace-Stieltjes transform (continuous/mixed case) or z -transform (discrete case). The algorithm is based on numerically inverting an adaptively modified moment generating function. It only requires computation of the transform at several complex values of its argument. We also show that the high-order moments may be used in detecting the presence of exponential or geometric tails in distributions, and in case they are present the two parameters characterizing such a tail may be accurately computed. Several numerical examples of interest in the queueing literature illustrate the use of the algorithm. They include commonly used distributions as well as the waiting time, queue length, and busy period in queues with Poisson or more general Markovian arrival processes. Priority queues are also considered.

Suggested Citation

  • Gagan L. Choudhury & David M. Lucantoni, 1996. "Numerical Computation of the Moments of a Probability Distribution from its Transform," Operations Research, INFORMS, vol. 44(2), pages 368-381, April.
    • Handle: RePEc:inm:oropre:v:44:y:1996:i:2:p:368-381
    DOI: 10.1287/opre.44.2.368
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    Citations

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    Cited by:

    1. Dasu, Sriram, 1998. "Class dependent departure process from multiclass phase queues: Exact and approximate analyses," European Journal of Operational Research, Elsevier, vol. 108(2), pages 379-404, July.
    2. Wu-Lin Chen, 2019. "Computing the Moments of Polling Models with Batch Poisson Arrivals by Transform Inversion," INFORMS Journal on Computing, INFORMS, vol. 31(3), pages 515-526, July.
    3. Steve Derkic & James E. Stafford, 2002. "Symbolic Computation of Moments in Priority Queues," INFORMS Journal on Computing, INFORMS, vol. 14(3), pages 261-277, August.
    4. Svetlana Boyarchenko & Sergei Levendorskiu{i}, 2024. "Efficient inverse $Z$-transform and Wiener-Hopf factorization," Papers 2404.19290, arXiv.org, revised May 2024.
    5. Boxma, O. J. & Down, D. G., 1997. "Dynamic server assignment in a two-queue model," European Journal of Operational Research, Elsevier, vol. 103(3), pages 595-609, December.
    6. Tetsuya Takine, 1999. "The Nonpreemptive Priority MAP/G/1 Queue," Operations Research, INFORMS, vol. 47(6), pages 917-927, December.
    7. Brignone, Riccardo & Gonzato, Luca, 2024. "Exact simulation of the Hull and White stochastic volatility model," Journal of Economic Dynamics and Control, Elsevier, vol. 163(C).

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