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Arrival Rate Approximation by Nonnegative Cubic Splines

Author

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  • Farid Alizadeh

    (Department of Management Science and Information Systems, Business School and RUTCOR, Rutgers University, Piscataway, New Jersey 08854)

  • Jonathan Eckstein

    (Department of Management Science and Information Systems, Business School and RUTCOR, Rutgers University, Piscataway, New Jersey 08854)

  • Nilay Noyan

    (Faculty of Engineering and Natural Sciences, Sabanci University, Orhanli, Tuzla, 34956 Istanbul, Turkey)

  • Gábor Rudolf

    (RUTCOR, Rutgers University, Piscataway, New Jersey 08854)

Abstract

We describe an optimization method to approximate the arrival-rate function of a nonhomogeneous Poisson process based on observed arrival data. We estimate the function by cubic splines, using an optimization model based on the maximum-likelihood principle. A critical feature of the model is that the splines are constrained to be nonnegative everywhere. We enforce these constraints by using a characterization of nonnegative polynomials by positive semidefinite matrices. We also describe versions of our model that allow for periodic arrival-rate functions and input data of limited time precision. We formulate the estimation problem as a convex nonlinear program, and solve it with standard nonlinear optimization packages. We present numerical results using both an actual record of e-mail arrivals over a period of 60 weeks, and artificially generated data sets. We also present a cross-validation procedure for determining an appropriate number of spline knots to model a set of arrival observations.

Suggested Citation

  • Farid Alizadeh & Jonathan Eckstein & Nilay Noyan & Gábor Rudolf, 2008. "Arrival Rate Approximation by Nonnegative Cubic Splines," Operations Research, INFORMS, vol. 56(1), pages 140-156, February.
    • Handle: RePEc:inm:oropre:v:56:y:2008:i:1:p:140-156
    DOI: 10.1287/opre.1070.0443
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    References listed on IDEAS

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    1. NESTEROV, Yu., 2000. "Squared functional systems and optimization problems," LIDAM Reprints CORE 1472, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Kuhl, Michael E. & Wilson, James R., 2001. "Modeling and simulating Poisson processes having trends or nontrigonometric cyclic effects," European Journal of Operational Research, Elsevier, vol. 133(3), pages 566-582, September.
    3. Michael E. Kuhl & Sachin G. Sumant & James R. Wilson, 2006. "An Automated Multiresolution Procedure for Modeling Complex Arrival Processes," INFORMS Journal on Computing, INFORMS, vol. 18(1), pages 3-18, February.
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    Cited by:

    1. Farid Alizadeh & David Papp, 2013. "Estimating arrival rate of nonhomogeneous Poisson processes with semidefinite programming," Annals of Operations Research, Springer, vol. 208(1), pages 291-308, September.
    2. Lakshman S. Thakur & Mikhail A. Bragin, 2021. "Data Interpolation by Near-Optimal Splines with Free Knots Using Linear Programming," Mathematics, MDPI, vol. 9(10), pages 1-12, May.

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