Alizadeh, Farid; Eckstein, Jonathan; Noyan, Nilay; Rudolf, Gábor Arrival rate approximation by nonnegative cubic splines. (English) Zbl 1167.90436 Oper. Res. 56, No. 1, 140-156 (2008). 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. Cited in 5 Documents 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 PDFBibTeX XMLCite \textit{F. Alizadeh} et al., Oper. Res. 56, No. 1, 140--156 (2008; Zbl 1167.90436) Full Text: DOI