A modern introduction to the Poisson process, with general point processes and random measures, and applications to stochastic geometry.

"Simulation is a technique of creating representations or models of real world systems or processes and conducting experiments to predict behavior of actual systems. Input modeling is a critical aspect of simulation modeling. Stochastic input models are used to model various aspects of the system under uncertainty including process times and interarrival times. This research focuses on input models for nonstationary arrival processes that can be represented as nonhomogeneous Poisson processes (NHPPs). In particular, a smooth flexible model for the mean-value function (or integrated rate function) of a general NHPP is estimated. To represent the mean-value function, the method utilizes a specially formulated polynomial that is constrained in least-squares estimation to be nondecreasing so the corresponding rate function is nonnegative and continuously differentiable. The degree of the polynomial is determined by applying a modified likelihood ratio test to a set of transformed arrival times resulting from a variance stabilizing transformation of the observed data. Given the degree of polynomial, final estimates of the polynomial coefficients are obtained from original arrival times using least-squares estimation. The method is extended to fit an NHPP model to multiple observed realizations of a process. In addition, the method is adapted to a multiresolution procedure that effectively models NHPPs with long term trend and cyclic behavior given multiple process realizations. An experimental performance evaluation is conducted to determine the capabilities and limitations of the NHPP fitting procedure for single and multiple realizations of test processes. The method is implemented in a Java-based programming environment along with a web interface that allows user to upload observed data, fit an NHPP, and generate realizations of the fitted NHPP for use in simulation experiments."--Abstract.