@article{liu_kuhl_liu_wilson_2019, title={Modeling and Simulation of Nonstationary Non-Poisson Arrival Processes}, volume={31}, ISSN={["1526-5528"]}, DOI={10.1287/ijoc.2018.0828}, abstractNote={ We develop CIATA, a combined inversion-and-thinning approach for modeling a nonstationary non-Poisson process (NNPP), where the target arrival process is described by a given rate function and its associated mean-value function together with a given asymptotic variance-to-mean (dispersion) ratio. CIATA is based on the following: (i) a piecewise-constant majorizing rate function that closely approximates the given rate function from above; (ii) the associated piecewise-linear majorizing mean-value function; and (iii) an equilibrium renewal process (ERP) whose noninitial interrenewal times have mean 1 and variance equal to the given dispersion ratio. Transforming the ERP by the inverse of the majorizing mean-value function yields a majorizing NNPP whose arrival epochs are then thinned to deliver an NNPP having the specified properties. CIATA-Ph is a simulation algorithm that implements this approach based on an ERP whose noninitial interrenewal times have a phase-type distribution. Supporting theorems establish that CIATA-Ph can generate an NNPP having the desired mean-value function and asymptotic dispersion ratio. Extensive simulation experiments substantiated the effectiveness of CIATA-Ph with various rate functions and dispersion ratios. In all cases, we found approximate convergence of the dispersion ratio to its asymptotic value beyond a relatively short warm-up period. }, number={2}, journal={INFORMS JOURNAL ON COMPUTING}, author={Liu, Ran and Kuhl, Michael E. and Liu, Yunan and Wilson, James R.}, year={2019}, pages={347–366} }
 @inproceedings{kuhl_damerdji_wilson_1998, title={Least squares estimation of nonhomogeneous Poisson processes}, DOI={10.1109/wsc.1998.745045}, abstractNote={We formulate and evaluate weighted least squares (WLS) and ordinary least squares (OLS) procedures for estimating the parametric mean-value function of a nonhomogeneous Poisson process. We focus the development on processes having an exponential rate function, where the exponent may include a polynomial component or some trigonometric components. Unanticipated problems with the WLS procedure are explained by an analysis of the associated residuals. The OLS procedure is based on a square root transformation of the detrended event (arrival) times - that is, the fitted mean-value function evaluated at the observed event times; and under appropriate conditions, the corresponding residuals are proved to converge weakly to a normal distribution with mean 0 and variance 0.25. The results of a Monte Carlo study indicate the advantages of the OLS procedure with respect to estimation accuracy and computational efficiency.}, booktitle={1998 Winter Simulation Conference: Proceedings: Grand Hotel, Washington, D.C., 13-16 December, 1998}, publisher={Piscataway, New Jersey: IEEE ; New York, New York: Association for Computing Machinery ; San Diego, California: Society for Computer Simulation International}, author={Kuhl, M. E. and Damerdji, H. and Wilson, J. R.}, year={1998}, pages={637–646} }
 @article{kuhl_wilson_johnson_1997, title={Estimating and simulating Poisson processes having trends or multiple periodicities}, volume={29}, ISSN={["0740-817X"]}, DOI={10.1080/07408179708966327}, abstractNote={We develop and evaluate procedures for estimating and simulating nonhomogeneous Poisson processes having an exponential rate function, where the exponent may include a polynomial component or some trigonometric components or both. Maximum likelihood estimates of the unknown continuous parameters of the rate function are obtained numerically, and the degree of the polynomial rate component is determined by a likelihood ratio test. The experimental performance evaluation for this estimation procedure involves applying the procedure to 100 independent replications of nine selected point processes that possess up to four trigonometric rate components together with a polynomial rate component whose degree ranges from zero to three. On each replication of each process, the fitting procedure is applied to estimate the parameters of the process; and then the corresponding estimates of the rate and mean-value functions are computed over the observation interval. Evaluation of the fitting procedure is based on plotted tolerance bands for the rate and mean-value functions together with summary statistics for the maximum and average absolute estimation errors in these functions computed over the observation interval. The experimental results provide substantial evidence of the numerical stability and usefulness of the fitting procedure in simulation applications.}, number={3}, journal={IIE TRANSACTIONS}, author={Kuhl, ME and Wilson, JR and Johnson, MA}, year={1997}, month={Mar}, pages={201–211} }