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Arrival rate approximation by nonnegative cubic splines. (English) Zbl 1167.90436

Summary: 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.

MSC:

90B22 Queues and service in operations research
60K25 Queueing theory (aspects of probability theory)
65D07 Numerical computation using splines
90C30 Nonlinear programming

Software:

Ipopt; NEOS
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