Modeling and Simulation of Nonstationary Non-Poisson Processes

We develop and experimentally evaluate various methods for modeling and simulating nonstationary non-Poisson point processes (NNPPs) with the following key properties: (i) a given mean-value function over a finite time horizon of interest; (ii) a given "rate" function whose integral yields the given mean-value function over that time horizon; and (iii) a given finite positive asymptotic value for the ratio of the process variance to its mean-value function as the time horizon for the process increases. Recent research has focused on inversion and thinning methods for modeling and simulating NNPPs. Unfortunately the inversion method is limited to mean-value functions with an inverse that is analytically or numerically tractable; and the current thinning methods are not guaranteed to yield the desired dispersion (variance-to-mean) ratio for the given process even approximately. In this dissertation we propose CIAT, a combined inversion-and-thinning method for simulating NNPPs that avoids the disadvantages of the methods of inversion and thinning while retaining the advantages of each method.;First we construct a strictly positive step function that majorizes and closely approximates the given rate function and that also has an easily invertible majorizing mean-value function over the given time horizon. Next we generate a stationary renewal process whose interrenewal times have mean one and the desired variance-to-mean ratio; and finally the resulting sequence of renewal epochs is transformed via the inverse of the majorizing mean-value function to generate the associated "majorizing" point process. To complete CIAT we apply the thinning method to the arrival epochs of the majorizing NNPP to obtain the desired sequence of arrival epochs from an NNPP with the given rate function and the desired asymptotic dispersion ratio as follows. Each arrival epoch in the majorizing NNPP is independently selected for inclusion in the desired NNPP with probability equal to the ratio of the following : (i) the given rate function evaluated at the current arrival epoch in the majorizing NNPP; and (ii) the majorizing rate function evaluated at the same arrival epoch. CIAT is specifically designed so that at every arrival epoch in the majorizing NNPP, the ratio of (i) to (ii) is close to one, guaranteeing that a high percentage of the generated arrival epochs survive the thinning step of CIAT.;We establish the following key properties of a CIAT-generated NNPP: · The resulting point process has the desired mean-value function over the given time horizon. · If the desired mean-value function grows without bound as the time horizon increases, then the resulting point process asymptotically achieves the desired dispersion ratio, provided that the step rate function for the majorizing process is based on a sufficiently fine subdivision of the time horizon into subintervals over which the step function is constant.;In an experimental performance evaluation involving a wide range of test processes, we examine the quality of CIAT-generated NNPPs and the efficiency of CIAT by evaluating the following: (i) the closeness between the estimated mean-value function of the CIAT-generated NNPP and the corresponding desired mean-value function, and the closeness between the estimated asymptotic dispersion ratio and the desired asymptotic dispersion ratio; (ii) the convergence rate of CIAT, expressed by the warm-up time, after which the estimated confidence intervals for the mean-value function and dispersion ratio of the CIAT-generated NNPP include their desired respective values; and (iii) the computational cost for CIAT to generate 50 realizations of selected NNPPs over the given time horizon.;We also develop CIATL, a simplified version of CIAT using lognormal distributions for the interrenewal times in the underlying stationary renewal process. In the selected test processes, the performance of CIATL was similar to that of CIAT.