Theory and Methodology
Modeling and simulating Poisson processes having trends or nontrigonometric cyclic effects

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Abstract

We formulate a nonparametric technique for estimating the (cumulative) mean-value function of a nonhomogeneous Poisson process having a long-term trend or some cyclic effect(s) that may lack familiar trigonometric characteristics such as symmetry over the corresponding cycle(s). This multiresolution procedure begins at the lowest level of resolution by estimating any long-term trend in the target counting process; then at progressively higher levels of resolution, the procedure yields estimates of the cyclic behavior associated with progressively smaller cycle lengths. We also formulate an efficient algorithm for generating realizations of such counting processes.

Introduction

In many simulation studies, we encounter arrival processes having a long-term trend or multiply periodic behavior. A prominent recent example is found in a large-scale simulation model of the organ procurement and transplantation network of the United States that was developed for the United Network for Organ Sharing (UNOS) (Pritsker, 1998). The UNOS Liver Allocation Model (ULAM) is currently being used by UNOS to evaluate alternative liver-allocation policies for the United States. In analyzing ULAM's arrival streams of liver-transplant donors and patients, we found some arrival rates to exhibit significant growth over time as well as daily, weekly, or annual effects – that is, cyclic patterns of behavior with periods of 1 day, 1 week, or 1 year, respectively (Pritsker et al., 1995). In building ULAM, Pritsker et al. (1995) implemented the modeling, estimation, and simulation procedures introduced by Kuhl et al. (1997b) for representing a nonhomogeneous Poisson process (NHPP) having an exponential rate function, where the exponent may include a polynomial component or multiple trigonometric components. Because some of the observed cyclic effects deviated substantially from the symmetric behavior exhibited by trigonometric functions with fundamental frequencies of 1 cycle per day, 1 cycle per week, or 1 cycle per year, often we had to augment the fitted rate function with several harmonics of these fundamental frequencies in order to obtain accurate models of the observed periodic behavior.

In this paper, we present a method for estimating the mean-value function of an NHPP that possesses a long-term trend or several periodic effects, where the periodic effects may exhibit lack of symmetry or other nontrigonometric characteristics. In particular, this multiresolution procedure focuses on processes having several nested cyclic effects so that each larger cycle (period) includes an integral number of smaller cycles; moreover, although successive cycles of a given length may have different expected numbers of arrivals because of a long-term trend, all cycles of the same length have in common the key property that the same cumulative percentage of each cycle's expected number of arrivals is achieved at the same relative position within the cycle. The benefits of this nonparametric approach include the ability to model more accurately nontrigonometric periodic rate components without having to: (a) specify a parametric model; (b) include a large number of trigonometric rate components (as is often required in some parametric approaches); or (c) sample from a lengthy historical record (as in trace-driven simulation).

The rest of this paper is organized as follows. In Section 2, we explain the basic assumptions about the target NHPP, we formulate the corresponding mean-value function, and we give an overview of the proposed multiresolution technique for estimating the mean-value function from a realization of the target process. A simple numerical example illustrating the steps of the multiresolution procedure is presented in Section 3. Basic large-sample properties of the multiresolution procedure are established in Section 4. In Section 5 we present an application of the multiresolution procedure to modeling the arrival stream of liver-transplant patients at an organ transplant center. In Section 6 we present an efficient procedure for simulating NHPPs having the proposed multiresolution-type mean-value function. Conclusions and recommendations for future work are summarized in Section 7. Although this paper is based on Kuhl (1997), in Kuhl et al. (1997a) we presented a brief description of our estimation and simulation procedures along with an illustrative example but without any analysis or justification of the properties of these procedures.

Section snippets

The multiresolution procedure for estimating NHPPs

We consider counting processes that represent the buildup of events (arrivals) over time. For such processes we are able to observe each arrival time exactly, and in general the arrival intensity (rate) changes over time. Under certain assumptions a nonstationary arrival process can be represented as an NHPP {N(t):t⩾0}, where N(t) is the number of arrivals in the time interval (0,t] for all t⩾0, such that the instantaneous arrival rate at time t, λ(t), is a nonnegative integrable function of

An example illustrating the multiresolution procedure

To illustrate the multiresolution procedure, we applied the procedure to the arrival process introduced in the previous section. Recall that in this example, we took the overall time horizon to be 35 days; and the arrival rate exhibited a long-term trend as well as weekly and daily cyclic effects. We constructed the underlying NHPP so as to satisfy Assumption 1, Assumption 2 and hence Eq. (4), where we took μ(S)=740 andR0(s)=0.0258s+0.0000647s2,0⩽s<b0=35,R1(s)=0.2042s+0.04426s2−0.01772s3

Theoretical basis for the multiresolution procedure

The multiresolution procedure exploits basic properties of Poisson processes as well as the periodic structure of NHPPs that satisfy Assumption 1, Assumption 2. From Eq. (4), we see that the theoretical mean-value function can be written in the formμ(t)=μ(S)R0(s0,t)+∑ℓ=1pR(sℓ,t)∏i=0ℓ−1(Ri(si,t+bi+1)−Ri(si,t)),where si,t=(ji+1,t−1)bi+1−(ji,t−1)bi for i=p−1,…,1,0 and sp,t=t−(jp,t−1)bp. We can easily verify that (15) is the proper form of the mean-value function by noticing that μ(S)R0(s0,t) is

An application to the arrival process of liver transplant patients

In the previously mentioned simulation study sponsored by the UNOS, Pritsker et al. (1995) modeled the streams of liver-transplant patients arriving at transplant centers in the United States using NHPPs with exponential rate functions, where the exponent might include a polynomial component or some trigonometric components (Kuhl et al., 1997b). In this section, we discuss the application of the multiresolution procedure to the arrival stream of liver patients at Transplant Center 11 during the

Simulating NHPPs having a multiresolution-type mean-value function

We use the method of inversion (Bratley et al., 1987) to generate realizations of the fitted NHPP. The following discussion is in terms of the theoretical NHPP. To generate variates from an estimated NHPP, we replace the theoretical values with their corresponding estimates from the preceding discussion. For an NHPP having rate function λ(t),t∈[0,S], the cumulative distribution function of the next event time τi conditioned on the observed value τi−1=ti−1 of the last event time is given byFτi

Conclusions and recommendations

The multiresolution procedure developed in this paper provides a nonparametric estimate of the mean-value function of an NHPP having a long-term trend or cyclic effects that may exhibit nontrigonometric characteristics. Through this procedure we are able to model asymmetric periodic behavior without having to store all of the observed data. We have also developed an algorithm for simulating realizations of the fitted process using the method of inversion.

To make the multiresolution procedure

Acknowledgements

The authors thank the referees for several constructive suggestions that substantially improved the readability of this paper.

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