Modeling and Simulation of Nonstationary Non-Poisson Arrival Processes

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1 Submitted to INFORMS Journal on Computing manuscript (Please, provide the mansucript number!) Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication. Modeling and Simulation of Nonstationary Non-Poisson Arrival Processes Ran Liu SAS Institute, 7 SAS Campus Dr., Cary, North Carolina 27513, ran.liu@sas.com Michael E. Kuhl Department of Industrial and Systems Engineering, Rochester Institute of Technology, 81 Lomb Memorial Drive, Rochester, New York 14623, mekeie@rit.edu Yunan Liu, James R. Wilson Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC {yliu48@ncsu.edu, jwilson@ncsu.edu} We develop and experimentally evaluate CIATA, a combined inversion-and-thinning algorithm for simulating a nonstationary non-poisson process (NNPP) over a finite time horizon, where the target arrival process is specified by a given rate function and its associated mean-value function together with a given variance-to-mean (dispersion) ratio describing the steady-state variability of the process about its mean-value function. CIATA consists of the following steps: (a) constructing a piecewise-constant majorizing rate function that closely approximates the given rate function; (b) constructing the associated piecewise-linear majorizing mean-value function; (c) generating an equilibrium renewal process (ERP) whose interrenewal times have mean equal to one and variance equal to the given dispersion ratio; (d) inverting the majorizing mean-value function at the ERP s renewal epochs to generate the associated majorizing NNPP; and (e) thinning the arrival epochs of the majorizing NNPP to obtain an NNPP having the given rate function and steady-state dispersion ratio. Supporting theorems establish that a CIATA-generated NNPP has the desired mean-value function and steady-state dispersion ratio. Extensive simulation experiments substantiated the effectiveness of CIATA with various rate functions and dispersion ratios. In all cases we found approximate convergence of the dispersion ratio to its steady-state value beyond a relatively short warm-up period. Key words: nonstationary arrival process, non-poisson process, dispersion ratio, time-dependent arrival rate History: Submitted on August 15, Introduction In the development of a high-fidelity stochastic simulation model of a complex system, special attention must often be given to the system s arrival processes. A stream of random arrivals with a constant arrival rate is usually modeled by a homogeneous Poisson process (HPP), which is characterized by interarrival times that are independent and identically distributed (i.i.d.) exponential random variables whose mean is 1

2 2 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) the reciprocal of the arrival rate. Unfortunately many arrival processes of interest have arrival rates that exhibit substantial variation over time. For instance, in many locales, the storm-arrival process depends strongly on the season of the year (Lee et al. 1991); and in many manufacturing systems customer orders have an arrival rate that exhibits daily, weekly, monthly, or seasonal effects as well as long-term trends (Chryssolouris 26). Nonhomogeneous Poisson processes (NHPPs) have been used to model complex time-dependent arrival processes in a broad range of application domains (Lewis and Shedler 1976, 1979; Leemis 1991, 24; Arandia-Perez et al. 214). A noteworthy application involved organ-transplantation policy decisions. The United Network for Organ Sharing (UNOS) used the UNOS Liver Allocation Model (ULAM) to analyze the cadaveric liver-allocation system of the United States (Pritsker et al. 1995; Harper et al. 2). ULAM incorporates NHPP models of the following: (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. Virtually all these arrival streams exhibit 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 an NHPP can accurately represent a given time-dependent arrival rate and the associated meanvalue function describing the expected buildup of arrivals over time, in many simulation studies a critical arrival process exhibits variability about its associated mean-value function that cannot be achieved by an NHPP. For example, Figure 1 depicts the estimated mean-value function, the corresponding estimated variance function, and the estimated variance-to-mean (dispersion) ratio that were observed for arrivals to an endocrinology outpatient clinic of a major teaching hospital in South Korea (Kim et al. 215). Mean and Variance Var[N(t)] E[N(t))] Mean and Variance Var[N(t)] E[N(t))] time of day time of day Figure 1 IDC (Variance to Mean) Var[N(t)]/E[N(t))] time of day IDC (Variance to Mean) Var[N(t)]/E[N(t))] time of day Estimated mean, variance, and variance-to-mean (dispersion) ratio for the number of arrivals to a Korean endocrinology outpatient clinic.

3 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 3 Figure 1 clearly shows that in each morning and afternoon period when the clinic is open, the buildup of patient arrivals since the beginning of the observation period exhibits time-dependent behavior as well as a dispersion ratio that settles down to a value close to.5. On the other hand, similar plots for arrivals to the emergency department of an Israeli hospital over a twenty-four hour period exhibit pronounced timeof-day and day-of-the-week effects as well as a dispersion ratio that was estimated to be about 1.5 after the day-of-the-week effects were properly taken into account (Whitt and Zhang 215). If an NHPP has a rate function that is positive on a nonempty time interval of the form (,a], then the dispersion ratio is exactly equal to one at every nonzero point in a given time horizon [,S]; and therefore an NHPP cannot adequately model the variability of arrivals depicted in Figure 1. Similarly, empirical studies have shown that arrival processes in call centers and healthcare systems can exhibit substantially larger variability than that of a Poisson process (Asaduzzaman and Chaussalet 214; Kim and Whitt 214; Zhang et al. 214). Such inadequacy of the NHPP model of arrivals may invalidate the resulting simulationbased estimates of overall system performance. This article provides an effective approach to modeling and simulating a nonstationary non-poisson process (NNPP) with a constant steady-state dispersion ratio as well as given rate and mean-value functions defined over a given time horizon. The resulting arrival process achieves the desired mean-value function exactly at each point in the time horizon; and although this arrival process may require a warm-up period before approximately achieving the desired steady-state dispersion ratio, in our computational experience this warm-up period is usually relatively short. 1.1 Problem Statement Suppose that {N(t) : t } is a nonstationary arrival process so that N(t) denotes the number of arrivals in the time interval [,t]; and suppose that this process has the given mean-value function E[N(t)] µ(t) t λ(u)du for t, (1) where the associated rate function λ(u) is nonnegative, bounded, and integrable on R +, the nonnegative real numbers. Moreover in each finite time interval [,t], the rate function λ(t) has at most a finite number of discontinuities; and each time interval between the discontinuities of λ(t) has a (finite) partition such that λ(u) is constant or unimodal on each subinterval of the partition. Finally {N(t) : t } is assumed to have a given steady-state dispersion ratio Var[N(t)] C lim, where < C <. (2) t E[N(t)] Since both HPPs and NHPPs have Var[N(t)]/E[N(t)] = 1 for all t > as explained in the previous paragraph, an arrival process with a given time-dependent arrival rate and C 1 must be an NNPP. Given a finite time-horizon length S >, an arrival rate function λ(t), and a steady-state dispersion ratio C >, we seek effective methods for modeling and simulating an NNPP {N(t) : t [,S]} that achieves condition (1) exactly on [, S] and that achieves condition (2) after a relatively brief warm-up period i.e., for sufficiently large values of S and t [,S].

4 4 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 1.2 Literature Review We briefly review existing methods for modeling and simulating NHPPs. See Liu (213) for a comprehensive review. Simulation and Estimation of NHPPs. For an NHPP {N(t) : t [,S]} with a finite time horizon [,S], the exact simulation algorithms are generally based on the inverse-transform method or the method of thinning. Introduced by Çinlar (1975), the inverse-transform method consists of two steps: (i) generating an HPP {N (u) : u [, µ(s)]} with arrival rate equal to one and arrival epochs {S n : n = 1,...,N [µ(s)]}; and (ii) inverting the given mean-value function µ(t) at the arrival epochs generated in step (i) to yield {S n = µ 1 (S n) : n = 1,...,N(S)}, the arrival epochs of the target process, where N(S) = N [µ(s)]. However, in general the mean-value function corresponding to an arbitrary rate function can be analytically intractable or difficult to invert numerically, so the inverse-transform method may be either infeasible or computationally inefficient. If the given rate function λ(t) for t [,S] has a known upper bound λ, then it is straightforward to apply the thinning method proposed by Lewis and Shedler (1979). Adapted from the classical acceptance-rejection method for generating random variables, the basic idea of thinning is to generate an HPP { Ñ(t) : t [,S] } with arrival rate λ and arrival epochs { S n : n = 1,...,Ñ(S) } ; then for n = 1,...,Ñ(S), the nth arrival epoch S n of the HPP is independently accepted as an arrival epoch of the target NHPP with probability λ( S n ) / λ. This approach ensures that the resulting NHPP has the instantaneous arrival rate λ(t) at each t [,S]. However, if λ(t) / λ 1 over a substantial range of values for t [,S], then thinning is computationally inefficient because a relatively large number of the arrival epochs { S n } are rejected. For joint use with variance reduction techniques, the inverse-transform method is preferred because it facilitates not only the synchronized sampling of random-number inputs across different simulation experiments, but also the arrangement of a monotonic dependence of the outputs on the random-number inputs. However, this method is limited to special forms of the rate function for which the mean-value function is readily invertible. In particular, it is used mostly with rate functions that are piecewise constant (Harrod and Kelton 26), piecewise linear (Lee et al. 1991), or trigonometric (Chen and Schmeiser 1992). For estimating an NHPP from an observed arrival stream, the relevant statistical methods can be categorized as follows (Kuhl and Wilson 29): (i) nonparametric methods (Leemis 1991; Arkin and Leemis 2; Leemis 24; Henderson 23); (ii) parametric methods (Cox and Lewis 1966; Lee et al. 1991; Kuhl et al. 1997; Kuhl and Wilson 2); and (iii) semiparametric methods (Kuhl and Wilson 21; Kuhl et al. 26, 28). Also see Chen and Schmeiser (213) for methods based on smoothing a piecewise-constant approximation to the rate function of an NHPP. Simulation and Estimation of NNPPs. Gerhardt and Nelson (29) propose an inversion method for simulating an NNPP that allows the user to achieve desired values of both the mean-value function µ(t) and the steady-state dispersion ratio C of the process. The idea is a direct generalization of the inverse-transform

5 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 5 method for simulating an NHPP in which we replace the rate-1 HPP with an equilibrium renewal process (ERP) whose interrenewal times have mean equal to one and variance equal to C. This method is useful when the mean-value function is easily invertible. This idea was originally proposed by Massey and Whitt (1994); see 7 there. Gerhardt and Nelson (29) also propose a method for simulating an NNPP that is based on conventional thinning with known upper bound λ for the rate function λ(t) in which the time-dependent acceptance probability is λ(t) / λ, but the majorizing rate- λ Poisson process is replaced by an ERP with interrenewal times having mean 1 / λ ) 2 and variance C /( λ. The authors show that the resulting thinned point process has the desired mean-value function µ(t) for all t ; however, the thinned process does not in general have the desired steady-state dispersion ratio C. The other disadvantage of this method is that it may be computationally inefficient if λ(t) λ over a substantial range of values for t, resulting in a relatively large number of rejections. Therefore, we are motivated to seek new algorithms for simulating NNPPs that achieve both the given mean-value function and the given steady-state dispersion ratio while being computationally efficient. All the existing methods for estimating NNPPs seem to be based directly on the nonparametric techniques of Henderson (23) for estimating an NHPP with a piecewise constant rate function based on K i.i.d. realizations of the target NHPP, with all the resulting arrival epochs being accumulated in adjacent observation intervals of a common length ζ. For an NHPP, Henderson (23) establishes key asymptotic properties of the associated estimators of the rate and mean-value functions as K, both when ζ is constant and when ζ tends to zero with increasing K. Unfortunately it is unclear whether these asymptotic properties also apply to estimation of NNPPs, because the proofs in Henderson (23) depend critically on key properties of the Poisson distribution. Building on these results, Gerhardt and Nelson (29) also develop a nonparametric estimator of the dispersion ratio C using weighted least-squares regression to fit the relation Var[N(t)] = C µ(t) + ε t for selected values of t in an observation interval of the form [,T E ]. As K, the asymptotic properties of this estimator of C are unclear. In particular, it is unclear how regression through the origin may be biased by a warm-up period preceding convergence to the steady-state dispersion ratio. Improved estimation of NNPPs is the subject of our ongoing work. Organization of this article. In 2 we introduce CIATA, a combined inversion-and-thinning algorithm for simulating an NNPP with a given steady-state dispersion ratio as well as given rate and mean-value functions; and we provide both a formal algorithmic statement of CIATA and the theorems establishing the desired properties of CIATA-generated NNPPs. Depending on the value of C, CIATA uses ERPs with interrenewal times having appropriate phase-type distributions as detailed in 2.3. Specifically when C 1, CIATA uses a two-phase hyperexponential distribution; and when C < 1, CIATA uses a hyper-erlang distribution (i.e., a mixture of two Erlang distributions). In 2.5 we develop SCIATA, a simplified version of CIATA for handling any given positive value of C using an ERP with Weibull interrenewal times. In 3

6 6 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) we summarize the results of an extensive performance evaluation for CIATA and SCIATA, and we close 3 with a brief comparison of the performance of CIATA with the NNPP-simulation procedures of Gerhardt and Nelson (29). In 4 we prove the main theoretical results supporting both procedures. Conclusions and recommendations for future work are summarized in 5. The Online Supplement to this article contains additional proofs and experimental results. This article is a follow-up to Kuhl and Wilson (29), in which the basic idea for CIATA was first proposed. A preliminary, abridged version of the material on SCIATA will appear in the Proceedings of the 215 Winter Simulation Conference (Liu et al. 215). 2. CIATA: A Combined Inversion-and-Thinning Algorithm for Simulating NNPPs Throughout this article, we make the following assumption about the given rate function λ(t). ASSUMPTION 1. The given rate function λ(t) has an upper bound λ so that λ(t) λ < for t ; and in each finite time interval [,u], λ(t) has at most a finite number of discontinuities. Moreover each time interval between the discontinuities of λ(t) has a (finite) partition such that λ(t) is constant or unimodal on each subinterval of the partition. Finally over the given time horizon [, S], this rate function yields µ(s) >. Assumption 1 ensures that the given arrival process is nontrivial i.e., the time horizon [,S] is sufficiently large so that N(S) > with positive probability. Moreover [,S] can be subdivided into a finite number subintervals on which the given rate function λ(t) is either constant or unimodal (i.e., strictly quasiconvex). In most practical applications, either λ(t) or µ(t) must be estimated by an appropriate statistical method; and in these applications Assumption 1 is usually satisfied. REMARK 1. In the formulation of CIATA and in the proofs of its theoretical properties, without loss of generality we may assume that λ(t) is continuous at each t [,S] for the following reason. If λ(t) has a finite number of discontinuities < t 1 # < < t D # < S in the given time horizon [,S], then with the definitions t # and t D+1, # S, and for i = 1,...,D + 1, we may apply CIATA to each subinterval [t i 1,t # i # ] separately (where we take λ(t i 1) # lim t t # i 1 λ(t) and λ(t i # ) lim t t # i λ(t), the respective right- and left-hand limits within that subinterval) so as to generate the target NNPP over the entire time horizon. 2.1 Overview of CIATA To simulate a given NNPP {N(t) : t [,S]} with < S < that satisfies Equations (1) and (2), we perform the following five steps of CIATA: Step 1. We construct a piecewise-constant majorizing rate function λ Q (t) that closely bounds the given rate function λ(t) for t [,S] based on a partition of [,S] into Q subintervals, where Q is taken sufficiently large to ensure that λ(t) is constant or unimodal in each subinterval. This construction is detailed in 2.2 and specified formally in Algorithm 1 below.

7 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 7 Step 2. We construct the piecewise-linear mean-value function µ Q (t) = [,t]\z λ Q (u)du for t [,S] as detailed in Equation (7) below, where Z denotes the zeros of the given rate function λ(u) for u [,S]. Step 3. We generate an ERP { N (u) : u [, µ Q (S)] } with renewal epochs { S n : n = 1,...,N [ µ Q (S)] } and interrenewal times { X n : n = 1,...,N [ µ Q (S)] } satisfying E[X n ] = 1 and Var[X n ] = C for n 2. This step is detailed in 2.3 below. Step 4. We generate the arrival epochs for the majorizing NNPP { S n : n = 1,...,Ñ Q (S) } where Ñ Q (S) = N [ µ Q (S)] by inverting the majorizing mean-value function at the renewal epochs of the ERP so that we take S n µ 1 Q (Sn) for n = 1,...,Ñ Q (S). Step 5. We apply the method of thinning to the arrival epochs { S n : n = 1,...,Ñ Q (S) } so that for n = 1,...,Ñ Q (S), the nth arrival epoch S n of the majorizing NNPP is independently accepted as an arrival epoch of the target NNPP with probability λ( S n ) / λq ( S n ); and we let {S j : j = 1,...,N Q (S)} denote the resulting sequence of arrival epochs of the CIATA-generated NNPP over the time horizon [,S]. Algorithm 2 in Section 2.4 is a formal statement of CIATA. Theorem 1 below states that E[N Q (t)] = µ(t) for t [,S] so the CIATA-generated NNPP always achieves the given mean-value function (1). Theorem 2 below states that as S so that we can let t, we have lim t lim Q Var[N Q (t)]/e[n Q (t)] = C so that the CIATA-generated NNPP approximately achieves the given steady-state dispersion ratio (2) provided S and Q are both sufficiently large. 2.2 Constructing a Majorizing Rate Function To enable efficient generation of event epochs, we seek to construct a positive piecewise-constant majorizing rate function λ Q (t) that converges uniformly to the rate function λ(t) for t [,S] as Q. If L L l=1 ( ζ l,ζ ) l (3) represents the set of nonoverlapping subintervals of [,S] on which λ(u) is constant, then we assume that the associated endpoints { ζ l,ζ l : l = 1,...,L} are known; and we let { ( ζ l 1 2 ζ l + ζ ) } l : l = 1,...,L denote the corresponding subinterval midpoints so that for each Q 1, we can take { ε/q if t (ζ l λq (t),ζ l ) and λ(ζ } l) = λ(ζ l ) if t (ζ l,ζ l ) and λ(ζ for t L, (4) l) > where ε is an arbitrarily small positive number (in CIATA, we take ε = 1 2 by default). On [,S] \ L we define λ Q (t) separately on each of the subintervals {[ ] } ξ j,ξ j : j = 1,...,M on which λ(t) is nonconstant, M M j=1 [ ] ξ j,ξ j = [,S] \ L. (5) Assumption 1 ensures that we can find a sufficiently large integer Q such that for Q Q and for each subinterval [ξ j,ξ j ] (where j = 1,...,M), the regularly spaced points { z i, j ξ j + i(ξ j ξ j )/Q : i =

8 8 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!),1,...,q } constitute a partition of [ξ j,ξ j ] and λ(t) is unimodal on each subsubinterval [z i 1, j,z i, j ] for i = 1,...,Q. The majorizing rate function λ Q (t) will therefore be based on a partition of the time horizon [,S] into a total of Q = L + MQ subintervals, where Q Q. REMARK 2. In the definition (3) of L, we arbitrarily chose to take its constituent subintervals as open rather than closed or half open, half closed (i.e., including one endpoint and excluding the other endpoint). This choice was made merely for convenience in the complementary definition (5) of M ; as demonstrated in the following development, it has no bearing on the validity of the fundamental properties we establish for the majorizing rate function, the majorizing mean-value function, or the NNPPs generated by CIATA. REMARK 3. The following development, in particular the proof of Theorem 1, depends on the boundedness and measurability of the function λ(t) / λq (t) for all t [,S]. The definition of λ Q (t) ensures that λq (t) > for t [,S] so that the required properties of λ(t) / λq (t) are guaranteed. A formal statement of the scheme to compute λ Q (t) is given in Algorithm 1. To construct λ Q (t) on [z i 1, j,z i, j ] for fixed i and j, we use the golden section search procedure to find the maximum value λ i, j of the given rate function λ(t) on that subsubinterval. Let φ ( 1+ 5 )/ denote the golden ratio. We let a z i 1, j, b z i, j, x 1 a + (2 φ)(b a), and x 2 a + (φ 1)(b a) so that we have a < x 1 < x 2 < b before performing the first iteration of the golden section search. Each iteration of the procedure begins by evaluating λ(x 1 ) and λ(x 2 ) in the current interval of uncertainty [a,b]. If λ(x 1 ) < λ(x 2 ), then the desired maximum is in [x 1,b]; and we make the assignment λ i, j λ(x 2 ) in case the termination condition is satisfied on the current iteration, followed by the assignments a x 1, x 1 x 2, x 2 a + (φ 1)(b a) in case the termination condition is not satisfied so that another iteration must be performed. Otherwise if λ(x 1 ) λ(x 2 ), then the desired maximum is in [a,x 2 ]; and we make the assignment λ i, j λ(x 1 ) in case the termination condition is satisfied on the current iteration, followed by the assignments b x 2, x 2 x 1, and x 1 a + (2 φ)(b a) in case the termination condition is not satisfied so that another iteration must be performed. After the interval of uncertainty [a,b] has been updated, the last step of the current iteration is to check for the termination condition (b a) < δ, where δ > is the user-specified maximum length of the final interval of uncertainty (in CIATA, δ = 1 4 by default). If the termination condition is satisfied, then λ i, j is delivered as the (approximate) maximum value of λ(t) on [z i 1, j z i, j ]; otherwise another iteration of the procedure is performed. For i = 1,...,Q, the golden section search is successively performed on each ] subsubinterval [z i 1, j,z i, j ] of the jth subinterval [ξ j,ξ j M. After the procedure of the preceding paragraph has been performed on the subinterval [ξ j,ξ j ] for j = 1,...,M, the majorizing rate function is defined on M by { Q i=1 M j=1 λi, ji (zi 1, λq (t) = j,z i, j ](t) for t M \ {ξ j : j = 1,...,M}, if t = ξ j for some j {1,...,M}, λ 1, j where I (zi 1, j,z i, j ](t) is the indicator function for (z i 1, j,z i, j ] so that we have I (zi 1, j,z i, j ](t) 1 if z i 1, j < t z i, j and otherwise I (zi 1, j,z i, j ](t). (6)

9 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 9 Algorithm 1 Constructing the piecewise-constant majorizing rate function 1: Initialization: Set δ 1 4, ε 1 2, and Q 2; then assign Q Q. 2: for j = 1,...,M do 3: Partition [ ξ j,ξ ] j into Q equal-length subintervals {[z i 1, j,z i, j ] : i = 1,...,Q }: Set z i, j ξ j + i(ξ j ξ j )/Q for i =,1,...,Q. 4: for i = 1,...,Q do 5: Set a z i 1, j, b z i, j, x 1 a + (2 φ)(b a), x 2 a + (φ 1)(b a). 6: while (b a) δ do 7: if λ(x 1 ) < λ(x 2 ) then 8: Set λ i, j λ(x 2), a x 1, x 1 x 2, x 2 a + (φ 1)(b a); 9: else 1: Set λ i, j λ(x 1), b x 2, x 2 x 1, x 1 a + (2 φ)(b a). 11: end if 12: end while 13: end for 14: end for 15: Set Q L + MQ, and define λ Q ( ) on L and M by (4) and (6), respectively. as Using the majorizing rate function defined by (4) and (6), we obtain the majorizing mean-value function µ Q (t) t λq (u)du for t S. (7) The next proposition shows that the majorizing mean-value function µ Q (t) converges uniformly to the target µ(t) from above as the majorizing interval diminishes. The proof of this property is given in 4. PROPOSITION 1. (Asymptotic accuracy of the majorizing mean-value function) The majorizing meanvalue function given in Equation (7) is asymptotically accurate as Q increases, in particular, µ Q (t) µ(t) uniformly on [,S] as Q. (8) REMARK 4. In many applications the given rate function λ(t) is smooth (i.e., differentiable) and dλ(t)/dt has a finite number of zeros in the time horizon [,S] so that L = /, L =, M = [,S], M = 1, and Q = Q. All the test problems in 3 have this property. In such situations we have found that taking Q between 1 and 3 is usually sufficient to guarantee that λ Q (t) adequately approximates λ(t) from above for t [,S]. In general finding a satisfactory value of Q may require some trial and error coupled with visual inspection of a graph of λ Q (t) superimposed on λ(t) for t [,S]. As we will see in , the execution time of CIATA is relatively insensitive to Q for 1 Q 3.

10 1 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 2.3 Generating an ERP with the Desired Dispersion Ratio To obtain an ERP with the desired dispersion ratio, we use phase-type distributions to generate interrenewal times. Specifically, we use the hyperexponential distribution for the case C 1 and the hyper-erlang distribution for the case < C < 1 as recommended by Gerhardt and Nelson (29) High Dispersion Ratio (C 1). Hyperexponential distributions have larger coefficients of variation than the exponential distribution. For the 2-phase hyperexponential (H 2 ) distribution, with probability p (,1) the upper exponential phase with parameter µ 1 is sampled; and with probability (1 p) the lower exponential phase with parameter µ 2 is sampled. The probability density function (p.d.f.) and cumulative distribution function (c.d.f.) of the H 2 distribution are respectively given by {, if x <, f H2 (x; p, µ 1, µ 2 ) = pµ 1 e µ 1x + (1 p)µ 2 e µ 2x, if x, and {, if x <, F H2 (x; p, µ 1, µ 2 ) = 1 pe µ 1x (1 p)e µ 2x, if x. (9) The first two noncentral moments of a hyperexponential random variable X with c.d.f. (9) are respectively given by E[X] = p/µ 1 +(1 p)/µ 2 and E[X 2 ] = 2p/µ (1 p)/µ 2 2. The squared coefficient of variation (SCV) of X is CV 2 [X] = Var(X)/E 2 [X] = [2p/µ (1 p)/µ 2 2 ]/[p/µ 1 + (1 p)/µ 2 ] 2 1. The following proposition characterizes the desired ERP based on an H 2 distribution. The proof of this result is given in S1 of the Online Supplement. PROPOSITION 2. (Parameters of the ERP with H 2 interrenewal distribution for C 1) If C 1 and p = 1 +C ± C 2 1, 2(1 +C) then with the interrenewal times X 1 G e (x) F H2 [x; 1/2,2p,2(1 p)] and {X i : i = 2,3,...} i.i.d. G(x) F H2 [x; p,2p,2(1 p)], we have E[X 2 ] = 1, Var[X 2 ] = C so that the associated ERP {N (t) : t } has the steady-state dispersion ratio C. REMARK 5. If C = 1, then we have p = 1 2 and µ 1 = µ 2 = 1 so that in Proposition 2, the resulting ERP is a rate-1 Poisson process. REMARK 6. In the case that C 1, we use the two-phase balanced-means hyperexponential distribution for {X i : i = 2,3,...} so that we have µ 1 = 2p and µ 2 = 2(1 p). With this setup, the two solutions to the equations E[X 2 ] = 1, Var[X 2 ] = C yield the same interrenewal distribution. Moreover in our computational experience this setup leads to relatively short warm-up periods beyond which the associated NNPP approximately achieves its steady-state dispersion ratio.

11 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) Low Dispersion Ratio (C < 1). The hyper-erlang distribution gives smaller coefficients of variation than the exponential distribution. For the hyper-erlang distribution, with probability [, 1) we sample the upper Erlang distribution with shape parameter k 1 (k is an integer with k 2) and scale parameter β; and with probability (1 p), we sample the lower Erlang distribution with shape parameter k and scale parameter β. The c.d.f. of an Erlang distribution with shape parameter k 1 and scale parameter β > is F Er (x; k,β) = x uk 1 exp( u/β)/[γ(k)β k ]du for x and F Er (x; k,β) = for x <, where Γ(k) is the gamma function. The first two noncentral moments of this distribution are respectively given by E[X] = kβ and E[X 2 ] = k(k +1)β 2. Let x denote the ceiling of x. The following proposition, characterizes the desired ERP based on a hyper-erlang distribution. The proof of this result is given in S1 of the Online Supplement. PROPOSITION 3. (Parameters of the ERP with hyper-erlang interrenewal distribution for < C < 1) If < C < 1 and we take then with the interrenewal times k = 1/C, p = kc k(1 +C) k 2 C, and β = 1/(k p), 1 +C X 1 G e (x) x[1 F Er (x; k 1,β)] + F Er (x; k,β) and {X i : i = 2,3,...} i.i.d. G(x) pf Er (x; k 1,β) + (1 p)f Er (x; k,β), we have E[X 2 ] = 1, Var[X 2 ] = C so that the associated ERP {N (t) : t } has the steady-state dispersion ratio C. REMARK 7. In the case that C < 1, for noninitial interrenewal times {Xi : i = 2,3,...} we use the mixture c.d.f. of the form pf Er (x; k 1,β)+(1 p)f Er (x; k,β) for two reasons. With this setup and with the choice of the integer k 2 such that k = 1/C, there is a unique feasible solution (β, p) to the equations E[X 2 ] = 1, Var[X 2 ] = C. Moreover in our computational experience this setup leads to relatively short warm-up periods beyond which the associated NNPP approximately achieves its steady-state dispersion ratio. 2.4 Using CIATA to Generate NNPPs A formal statement of CIATA is given in Algorithm 2 below. The following theorem establishes that CIATA delivers an arrival process with the desired mean-value function. We give its proof in 4. THEOREM 1. If the arrival process {N Q (t) : t S} is generated by CIATA as given in Algorithm 2, then t E[N Q (t)] = µ(t) = λ(u)du for t S.

12 12 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) Algorithm 2 Using CIATA to Generate an NNPP 1: Construct the majorizing rate function λ Q (t) given by (4) and (6) using Algorithm 1. 2: if C 1 then 3: Define the c.d.f. s G e (x) and G(x) as in Proposition 2, 4: else 5: Define the c.d.f. s G e (x) and G(x) as in Proposition 3. 6: end if 7: Set n 1, j, S, S, and S. Generate X n G e and set S n S n 1 + X n. Set S n µ 1 Q (S n) and X n S n S n 1. 8: while S n S do 9: Generate U n Uniform[,1]. 1: if U n λ( S n ) / λq ( S n ) then 11: Set j j + 1, S j S n. Set the interarrival time X j S j S j 1. 12: end if 13: Set n n + 1. Generate X n G. Set S n S n 1 + X n, S n µ 1 Q (S n), and X n S n S n 1. 14: end while For t [,S], let C Q (t) Var[N Q (t)]/e[n Q (t)] = Var[N Q (t)]/µ(t) denote the dispersion ratio for the CIATA-generated arrival count N Q (t). Theorem 2 below establishes conditions under which C Q (t) converges to C as S, Q, and t all increase. In terms of the noncentral moments of X 2, θ l E [( X 2 ) l ] for l = 1,2,... Note that θ 1 = 1 and θ 2 = C +1 in Propositions 2 and 3. We assume that θ 3 < so that θ l < for l = 1,2; and we define the constant θ 5θ 2 2 2θ 3 θ 2 4θ1 4 3θ1 3 θ1 2 = 5C2 + 6C θ 3, (1) that appears in the statement of Theorem 2. Section 4 contains the proof of this theorem. THEOREM 2. If θ 3 <, then for large S and large t [,S], we have lim C Q(t) = C + [θ + o(1)]/µ(t), (11) Q where in general the notation o{h[µ(t)]} denotes a function g[µ(t)] such that g[µ(t)]/h[µ(t)] if µ lim t µ(t) =. (12) If (12) holds, then lim lim lim C Q(t) = C. (13) t S Q

13 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 13 REMARK 8. The condition (12) ensures that C Q (t) C for sufficiently large values of S, Q, and t. A simple sufficient condition for (12) is to require that λ(t) λ > for all t for some positive lower bound λ on the arrival rate. In the next section we provide some experimental evidence supporting the steady-state result (13). 2.5 SCIATA: A Simplified Version of CIATA with Weibull Interrenewal Times In this section we formulate SCIATA, a simplified version of CIATA with a Weibull distribution for the noninitial interrenewal times {X i : i = 2,3,...}. SCIATA has the following advantages: (i) it provides a simple, unified approach for achieving any dispersion ratio C > ; and (ii) it provides faster generation of all interrenewal times. Especially when.5 C 1.5, we have found that SCIATA achieves the desired value of C after a relatively short warm-up period. The Weibull p.d.f. and c.d.f. are respectively given by f We (x; α,β) = (α/β)(x/β) α 1 exp [ (x/β) α] and F We (x;α,β) = 1 exp [ (x/β) α] for x, where the required value of the shape parameter α is the unique positive root of the equation and the associated scale parameter is given by α { 2Γ(2/α) (1/α)[Γ(1/α)] 2} [Γ(1/α)] 2 = C ; (14) β = α/γ(1/α). (15) The required distribution of X 1 is also easily derived. In terms of the auxiliary gamma p.d.f. with shape parameter η and scale parameter 1, namely f Ga (x;η) = [1/Γ(η)]x η 1 e x for x, and the associated gamma c.d.f., F Ga (x;η) = x f Ga(u;η)du for x, the time to the first renewal is sampled according to the relation Equation (16) is equivalent to the relation X 1 G e (x) F Ga [ (x/β) α ;1/α ] for x. (16) Generate Y F Ga (y;1/α) and deliver X 1 β ( Y 1/α). All subsequent interrenewal times are generated from the relation {X i : i = 2,3,...} i.i.d. G(x) F We (x;α,β) for x, (17) where α and β are given by Equations (14) and (15), respectively. The following proposition, whose proof is given in S1 of the Online Supplement, characterizes the desired ERP based on a Weibull distribution.

14 14 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) PROPOSITION 4. (Parameters of the ERP with Weibull interrenewal distribution) If C >, then Equation (14) has a unique positive root; and if we take α and β as specified by Equations (14) and (15), respectively, then with the initial and subsequent interrenewal times defined by Equations (16) and (17), respectively, we have E[X 2 ] = 1, Var[X 2 ] = C, and the associated ERP {N (t) : t } has the steady-state dispersion ratio C. Because the formal statement of SCIATA is a minor modification of Algorithm 2, it is omitted. 3. Experimental Performance Evaluation In this section we discuss a comprehensive performance evaluation of CIATA and a more limited performance evaluation of SCIATA. First we describe the experimental setup in 3.1, and then in 3.2 we detail our performance-estimation methods. In , we summarize all the simulation results for CIATA and SCIATA; and finally in 3.6 we compare the performance of CIATA with that of the NNPP-simulation procedures of Gerhardt and Nelson (29). 3.1 Experimental Setup For t S, we choose rate functions of the type exponential-polynomial-trigonometric with multiple periodicities (EPTMP), which means they have the form λ(t) = exp{h(t;m, p,θ)} with h(t;m, p,θ) = m i= α i t i + p j=1 γ j sin(ω j t + φ j ), (18) where Θ = [α,α 1,...,α m,γ 1,...,γ p,ω 1,...,ω p,φ 1,...,φ p,] is the vector of continuous parameters of the designated rate function (Kuhl et al. 1997). The first m + 1 terms in Equation (18) define a degree-m polynomial representing a possible long-term evolutionary trend in the arrival rate over time. The next p terms in Equation (18) define the trigonometric functions representing possible periodic effects exhibited by the arrival process. The use of an exponential rate function is a convenient means of ensuring that the instantaneous arrival rate is always positive. In a specific application, we can assign the appropriate degree m for the polynomial rate component as well as the appropriate oscillation amplitude (γ j ), oscillation frequency (ω j ), and phase delay (φ j ) for each of the cyclic rate components (if there are any). For more details on estimation and simulation of EPTMP-type rate functions, see Kuhl et al. (1997). Table 1 displays the parameters for the five test cases of EPTMP-type rate functions that are used in our performance evaluation of CIATA. All five test cases have a time horizon of S = 8 time units and contain two nested cyclic effects. In cases 1 3, the first effect has a period of 1 time unit (so that ω 1 = 2π) and the second effect has a period of.5 time units (so that ω 2 = 4π); on the other hand in cases 4 and 5, the first effect has a period of 16 time units (so that ω 1 = π/8) and the second effect has a period of 8 time

15 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 15 units (so that ω 2 = π/4). Cases 1 and 4 do not contain a general trend over time. Cases 2, 3, and 5 contain general trends that are represented by polynomials of degrees 1, 2, and 2, respectively. In the performance evaluation of CIATA for cases 1 3, the steady-state dispersion ratio is assigned the values C =.2,.8, 1.5, and 1.. Cases 4 and 5 demonstrate the performance of CIATA in test processes for which the rate function varies relatively slowly over the time horizon. In cases 4 and 5 the steady-state dispersion ratio is assigned the values C =.2 and 1.5. The performance evaluation of SCIATA is limited to case 1 with C =.5 and C = 1.5. A more comprehensive performance evaluation of SCIATA is the subject of ongoing work. Table 1 Parameters of the NNPPs with EPTMP-Type Rate Functions That Are Used in the Experimental Performance Evaluation of CIATA and SCIATA Test Parameter Case α α 1 α 2 γ 1 φ 1 ω 1 γ 2 φ 2 ω To evaluate the performance of CIATA and SCIATA, we carry out a metaexperiment for each procedure consisting of R = 1 independent basic experiments; and in each basic experiment, we execute K = 2 independent replications of the relevant procedure to generate 2 realizations of an NNPP {N Q (t) : t [,S]} for each of the relevant test cases in Table 1 with each of the associated values of the dispersion ratio. This experimental setup is designed to yield valid point and CI estimators of the mean-value function µ(t) = E[N Q (t)] and the dispersion-ratio function C Q (t) Var[N Q (t)]/e[n Q (t)] for selected values of t (, S], enabling quantitative and visual assessment of the extent to which the simulation-generated NNPP satisfies the desired conditions (1) and (2). In the next section we detail the statistical methods used in the experimental performance evaluation of CIATA and SCIATA. 3.2 Performance Estimation Methods Based on 1 replications of each basic experiment, we compute point and 95% CI estimators of µ(t i ) and C Q (t i ) at the times {t i : t i = iζ, for i = 1,...,T}, where the spacing ζ between successive observations of the two statistics µ(t i ) and C Q (t i ) is chosen so that T = S/ζ is a positive integer. For all the experiments in this section, we set ζ =.2 so that T = 4. The following methods are used to compute the point and CI estimators required for the experimental performance evaluation. On the kth realization of a simulation-generated NNPP in the rth basic experiment, we let N Q (t;r,k) denote the number of accepted arrivals up to time t [,S], where r = 1,...,R, and k =

16 16 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 1,...,K. In the rth basic experiment, the estimators of the mean-value and variance functions are the sample statistics N Q (t;r) = 1 K Var[N Q (t;r,1)] = 1 K 1 K k=1 K k=1 N Q (t;r,k) The estimated dispersion-ratio function at time t is [ NQ (t;r,k) N Q (t;r) ] 2 for t (,S] and r = 1,...,R. Ĉ Q (t;r) = Var[N Q (t;r,1)] N Q (t;r) for t (,S] and r = 1,...,R. Based on the entire metaexperiment, the overall estimator of the desired mean-value function µ(t) is µ Q (t) = 1 RK R K r=1 k=1 N Q (t;r,k) for t (,S]; and a pooled estimator of Var[N Q (t)] is Var[N Q (t)] = 1 R R r=1 Var[N Q (t;r,1)] for t (,S]. The overall point estimator of C Q (t) is Ĉ Q (t) = 1 R and the overall estimator of Var[Ĉ Q (t;1)] is R r=1 Ĉ Q (t;r) for t (,S]; (19) Var[Ĉ Q (t;1)] = 1 R 1 R r=1 [ĈQ (t;r) Ĉ Q (t) ] 2 for t (,S]. Therefore an approximate 1(1 θ)% CI estimator for C Q (t) is Ĉ Q (t) ± z 1 θ/2 Var[Ĉ Q (t;1)]/r for t (,S], where z 1 θ/2 is the 1 θ/2 quantile of the standard normal distribution. Similarly, an approximate 1(1 θ)% CI estimator for µ(t) is µ Q (t) ± z 1 θ/2 Var[N Q (t)]/r for t (,S]. To provide additional criteria for evaluating the performance of CIATA and SCIATA, we examine the relative errors of the point estimators µ Q (t) and Ĉ Q (t) (i.e., their respective percentage deviations from the associated target values µ(t) and C), which we call closeness measures; and we study the sensitivity of these closeness measures to the dispersion ratio C and to the number of majorizing intervals Q. We define

17 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) 17 the average percentage discrepancy (APD) T ( ) and maximum percentage discrepancy (MPD) T( ) as two measures of closeness: T ( µ Q ) = 1 T T (Ĉ Q ) = 1 T T i=1 T i=1 { D i ( µ Q ), T( µ Q ) = max 1 i T Di ( µ Q ) }, { D i (Ĉ Q ), T(Ĉ Q ) = max 1 i T Di (Ĉ Q ) }, where the percentage discrepancy D i ( ) is the percentage difference of the estimators µ Q (t i ) and Ĉ Q (t i ) from their respective target values µ(t i ) and C at each observation epoch t i : D i ( µ Q ) = µ Q (t i ) µ(t i ) µ(t i ) 1% and D i(ĉ Q ) = Ĉ Q (t i ) C C 1% Finally, we compute the approximate 95% CI estimator for the expected value of T (Ĉ Q ), { S (Ĉ Q ) T (Ĉ Q ) ± z 1 θ/2, where S (Ĉ Q ) = T 1 T 1 T i=1 for i = 1,...,T. [ Di (Ĉ Q ) T (Ĉ Q ) ] } 1/2 2, and θ =.5. For the APD estimator T ( µ Q ), the standard-deviation estimator S ( µ Q ), and the CI estimator of E[ T ( µ Q )] are defined similarly. All our algorithms have been implemented in MATLAB, and all the experiments in this article were carried out on a 2.66 GHz Intel 2 Quad processor with 3.25 GB RAM, running Windows XP Professional. 3.3 Performance of CIATA Here we report results for CIATA in cases 1, 3, and 4, with the target dispersion ratios C =.2 and 1.5. In the Online Supplement, we summarize the results for CIATA in cases 2 and 5 and with other dispersion ratios (i.e., C =.8 and 1). For case 1 and t [,8], we plot the following in Figure 2: (i) the given rate function λ(t) (dashed curve) and the corresponding majorizing step rate function λ Q (t) (solid step function); (ii) the 95% CI estimators of the mean-value function µ(t); and (iii) the 95% CI estimators for the dispersion-ratio function C Q (t). Here Q = 16 so the majorizing step size is S/Q =.5. Figures 3 and 4 depict the CIATA-generated results for cases 3 and 4, respectively, based on the same layout used in Figure 2. Also see Tables S8 S11 and Tables S24 S27 in the Online Supplement for the values of the CI estimators for µ(t) and C Q (t) depicted in Figures 2 through 4. The third parts of Figures 2 through 4 exemplify the warm-up periods required for an CIATA-generated arrival process to achieve approximate convergence to the desired steady-state dispersion ratio. Except for extreme situations such C =.2 or C = 1, we found that the time-dependent dispersion ratio usually achieved approximate convergence to C within a relatively short warm-up period; consequently we concluded that CIATA could deliver reasonably fast convergence to a given nonextreme dispersion ratio. See Table S5 in the Online Supplement for a comparison of the warm-up times across different dispersion ratios. (2)

18 Variance-tp-mean ratio Cumulative Mean Arrivals Liu et al.: Simulating Nonstationary Non-Poisson Processes 18 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) Arrival rate Target arrival rate l(t) Majorizing arrival rate l Q (t) Time Figure Time t Time t a) Case 1, C=1.5 b) Case 1, C=.2 Performance of CIATA for Case 1 with Dispersion Ratio C = 1.5,.2: (i) (Top Panel) the Majorizing Rate Function λ Q (t); (ii) (Middle Panel) 95% CIs for the Mean-Value Function µ(t) with C = 1.5 (Left) and C =.2 (Right); and (iii) (Bottom Panel) 95% CIs for C Q (t) with C = 1.5 (Left) and C =.2 (Right). Table 2 provides the closeness measures T ( µ Q ), T( µ Q ), T (Ĉ Q ), and T(Ĉ Q ) and the approximate 95% CIs for E[ T ( µ Q )] and E[ T (Ĉ Q )] for the cases 1, 3, and 4 with C =.2 and 1.5. For the situations in which C = 1.5, Table 2 shows that the APD and MPD for the mean-value function of the CIATA-generated NNPP are less than 1% and 3%, respectively; and the APD and MPD for the dispersion ratio of the CIATAgenerated NNPP are less than 2% and 8%, respectively. For the more extreme situations in which C =.2, we see that the APD and MPD for the mean-value function of the CIATA-generated NNPP are less than 1% and 3%, respectively; and the APD and MPD for the dispersion ratio of the CIATA-generated NNPP are less than 4% and 21%, respectively. We judged these results to provide good evidence of the effectiveness and efficiency of the CIATA-based estimators of the given rate and mean-value functions as well as the given dispersion ratio. See Table S6 in the Online Supplement for case-1 results with C = 2,15,1, and.8 and for a detailed discussion of these results. We remark that the adequacy of a CIATA-generated NNPP can also be impacted by Q, the number of subintervals that are used to generate the majorizing rate function λ Q (t). See Table S7 in the Online Sup-

19 Cumulative Mean Arrivals Liu et al.: Simulating Nonstationary Non-Poisson Processes Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) Arrival rate Variance-tp-mean ratio Target arrival rate l(t) Majorizing arrival rate l Q (t) Time Figure Time t Time t a) Case 3, C=1.5 b) Case 3, C=.2 CIATA Performance for Case 3 with Dispersion Ratio C = 1.5,.2: (i) (Top Panel) the Majorizing Rate Function λ Q (t); (ii) (Middle Panel) 95% CIs for the Mean-Value Function µ(t) with C = 1.5 (Left) and C =.2 (Right); and (iii) (Bottom Panel) 95% CIs for C Q (t) with C = 1.5 (Left) and C =.2 (Right). plement for case-1 results of the four discrepancy measures and the 95% CIs for E[ T ( µ Q )] and E[ T (Ĉ Q )] with different values of Q.

20 Variance-to-mean ratio Cumulative Mean Arrivals Liu et al.: Simulating Nonstationary Non-Poisson Processes 2 Article submitted to INFORMS Journal on Computing; manuscript no. (Please, provide the mansucript number!) Time t Time t a) Case 4, C=1.5 b) Case 4, C=.2 Figure 4 CIATA Performance for Case 4 with Dispersion Ratio C = 1.5,.2: (i) (Top Panel) the Majorizing Rate Function λ Q (t); (ii) (Middle Panel) 95% CIs for the Mean-Value Function µ(t) with C = 1.5 (Left) and C =.2 (Right); and (iii) (Bottom Panel) 95% CIs for C Q (t) with C = 1.5 (Left) and C =.2 (Right). Table 2 CIATA-based Closeness Measures for Cases 1, 3, and 4. Case 1 C T ( µ Q ) T( µ Q ) T (Ĉ Q ) T(Ĉ Q ).2.918% ±.197% 2.538% 3.53% ±.266% 6.15% % ±.176% 2.463% 1.91% ±.35% 5.233% Case 3 C T ( µ Q ) T( µ Q ) T (Ĉ Q ) T(Ĉ Q ).2.56% ±.9%.22% 1.34% ±.22% 5.% % ±.38%.56% 1.196% ±.196% 3.89% Case 4 C T ( µ Q ) T( µ Q ) T (Ĉ Q ) T(Ĉ Q ).2.37% ±.25%.7% 3.297% ±.9% 2.93% % ±.53% 1.21%.913% ±.293% 7.97%