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On the pathwise computation of derivatives with respect to the rate of a point process: The phantom RPA method

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The Rare Perturbation Analysis (RPA) method is presented using two approaches: a direct one and an indirect one via a pathwise interpretation of the Likelihood Ratio Method (LRM). These two approaches give a new point of view for the Smoothed Perturbation Analysis (SPA) discussed in Gong [4] and extend the validity of the formulas therein, in particular to the estimation of derivatives of quantities that can be computed over a busy cycle. A heuristic comparison with LRM is given and simulation results are presented to compare the performance of LRM, RPA, and a finite difference RPA in a simple system.

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  1. [1]

    F. Baccelli and P. Brémaud, Virtual customers in sensitivity and light traffic analysis via Campbell's formula for point processes, submitted to AAP (1990).

  2. [2]

    P. Brémaud, Maximal coupling and rare perturbation analysis, in preparation (1990).

  3. [3]

    P. Glynn, Likelihood ratio gradient estimation for stochastic systems, Commun. ACM 33 (1990) 76–84.

  4. [4]

    W.B. Gong, Smoothed perurbation analysis algorithm for aG/G/1 routing problem, in:Proc. 1988 Winter Simulation Conf., eds. M. Abrams, P. Haigh and J. Comfort (1988) pp. 525–531.

  5. [5]

    W.B. Gong and Y.C. Ho, Smoothed perturbation analysis of discrete event dynamical systems, IEEE Trans. Auto. Control AC-10 (1987) 858–866.

  6. [6]

    Y.C. Ho and X.R. Cao, Perturbation analysis and optimization of queueing networks, J. Optim. Theory Appl. 40 (1983) 559–582.

  7. [7]

    P. L'Ecuyer, A unified version of the IPA, SF, and LR gradient estimation techniques, Manag. Sci. 36 (1990) 1364–1383.

  8. [8]

    M.I. Reiman and B. Simon, Open queueing systems in light traffic, Math. Oper. Res. 14 (1989) 26–59.

  9. [9]

    M.I. Reiman and A. Weiss, Sensitivity analysis for simulations via likelihood ratios, Oper. Res. 37 (1989) 830–844.

  10. [10]

    B. Simon, A new estimator of sensitivity measures for simulation based on light traffic theory, ORSA J. Comput. 1 (1989) 172–180.

  11. [11]

    R. Suri, Perturbation analysis: the state of the art and research issues explained via theG/G/1 queue, Proc. IEEE 77 (1989) 114–137.

  12. [12]

    R. Suri and X.R. Cao, The phantom customer and the marked customer methods for optimization of closed queueing networks with blocking and general service times,AGM-Sigmetrics Conf., Minneapolis (1983).

  13. [13]

    F. Vázquez-Abad, Stochastic recursive algorithms for optimal routing in queueing networks, Ph.D. Thesis, Applied Mathematics, Brown University (1989).

  14. [14]

    F. Vázquez-Abad and H. Kushner, A surrogate estimation approach for adaptive routing in communication networks, Brown University Report LCDS/CCS No. 90-2 (May 24, 1990).

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Brémaud, P., Vázquez-Abad, F.J. On the pathwise computation of derivatives with respect to the rate of a point process: The phantom RPA method. Queueing Syst 10, 249–269 (1992). https://doi.org/10.1007/BF01159209

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  • Sensitivity analysis
  • queues
  • point processes
  • perturbation analysis
  • likelihood ratios
  • routing