Advances in Modeling and Simulation of Nonstationary Arrival Processes

We survey various methods for modeling and simulating nonhomogeneous Poisson processes, highlighting their advantages and limitations as approximations to nonstationary arrival processes in simulation applications. We also discuss briefly recent research on inversion and thinning methods for modeling and simulating nonstationary non-Poisson arrival processes, and we propose a combination of these methods that may avoid the disadvantages of both inversion and thinning while retaining the advantages of each method. 1 TIME-DEPENDENT ARRIVAL PROCESSES Time-varying arrival processes are routinely encountered in practical applications of industrial and systems engineering techniques. To analyze or improve system operation in such situations, discrete-event stochastic simulation is often the technique of choice. Consequently, high-fidelity probabilistic input models are frequently needed to perform meaningful simulation experiments. Nonhomogeneous Poisson processes (NHPPs) have been used successfully to model complex time-dependent arrival processes in a broad range of application domains (Lewis and Shedler 1976; Lee, Wilson, and Crawford 1991; and Pritsker et al. 1995). A noteworthy application involved organ-transplantation policy decisions. The United Network for Organ Sharing (UNOS) carried out a large-scale application of NHPPs for modeling and simulating patientand donor-arrival streams in the development and use of the UNOS Liver Allocation Model (ULAM) for analysis of the cadaveric liver-allocation system in the United States (see Harper et al. 2000). ULAM incorporated NHPP models of (a) the streams of liver-transplant patients arriving at 115 transplant centers, and (b) the streams of donated organs arriving at 61 organ procurement organizations in the United States; and virtually all these arrival streams exhibited strong dependencies on the time of day, the day of the week, and the season of the year as well as pronounced geographic effects. Although NHPPs are used to model a large class of nonstationary arrival processes, NHPPs are inappropriate for some applications in manufacturing, telecommunications, marketing, and other areas. In some cases the Poisson postulates are inapplicable (for example, with nearly simultaneous arrivals or pronounced correlation between arrivals in nonoverlapping time intervals); and in other cases, key characteristics of the target arrival process (for example, its variability about the mean-value function) differ substantially from the corresponding characteristics of an NHPP. In this paper we survey various methods for modeling and simulating NHPPs, with emphasis on their advantages and limitations in practice. We also discuss recent research of Gerhardt and Nelson (2009) on methods for modeling and simulating nonstationary non-Poisson arrival processes based on inversion and thinning. We propose a combination of these methods that may avoid the disadvantages of both inversion and thinning while retaining the advantages of each method. This paper is intended as a basis for a larger discussion of not only the characteristics of nonstationary point processes for which current methods are inadequate but also potential methods for addressing these issues. 2 NONHOMOGENEOUS POISSON PROCESSES A nonhomogeneous Poisson process fN.t/ W t 0g is a generalization of a Poisson process in which the instantaneous arrival rate .t/ at time t is a nonnegative integrable function of time. The mean-value function of the NHPP is defined by