Signature
m = lambda_1 * t_1 + lambda_2 * t_2
| Inputs | Definition | Unit |
|---|---|---|
lambda_1 | Arrival rate during the first period, for example daytime | arrivals per unit of time |
t_1 | Length of the first period, in the time unit of lambda_1 | time, for example hours |
lambda_2 | Arrival rate during the second period, for example overnight | arrivals per unit of time |
t_2 | Length of the second period, in the time unit of lambda_2 | time, for example hours |
m | Expected number of arrivals over the window, the area under the arrival rate curve | arrivals |
|---|
Function
Arrival process count, gap and arrival-time function
Maps an arrival rate, constant or varying with time, to the number of patients or other entities that arrive in a window, the gaps between successive arrivals and the clock time of each arrival. The three descriptions carry the same information: S_n is the time of the nth arrival, X_i the gap before arrival i and N(t) the number of arrivals up to time t, in the notation of the Arrival Process article. The records cover Poisson counts, time-varying rates, sampling of gaps and arrival times in a discrete event simulation, and the single-server queueing result that links the arrival rate to waiting.
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Implementations
Excel
Expected arrivals over any number of rate periods
With the rate of each period in a range named PeriodRates and the matching period lengths in PeriodLengths, SUMPRODUCT returns the area under a piecewise-constant rate curve for any number of periods.
=SUMPRODUCT(PeriodRates,PeriodLengths)
Assumptions
Rate constant within each arrival period
The arrival rate is flat within each period and changes only at the period boundaries. Where the rate drifts within a period, the window is cut into more periods or the rate curve is integrated directly.
Independent arrivals in separate intervals
Counts in non-overlapping intervals are independent, the defining property of the non-homogeneous Poisson process. The gaps between arrivals are then neither independent nor identically distributed, so the process is not a renewal process.
Worked examples
Daytime peak with the same daily demand
In the article's Model B, 3 patients an hour arrive for 12 daytime hours and 1 an hour for 12 night-time hours, so 3 × 12 + 1 × 12 = 48 arrivals are expected, the same daily mean as a constant 2 an hour.
lambda_1 = 3; t_1 = 12; lambda_2 = 1; t_2 = 12; m = 48
Constant rate written as two equal arrival periods
Model A, with 2 patients an hour in both halves of the day, also gives 48 expected arrivals. The two models differ in how the arrivals are spread, not in the daily mean.
lambda_1 = 2; t_1 = 12; lambda_2 = 2; t_2 = 12; m = 48
Common errors
One gap distribution fitted to pooled busy and quiet periods
Because the gaps of a time-varying Poisson process are neither independent nor identically distributed, a single exponential or other distribution fitted to gaps pooled across the peak and the night describes neither period. Rates are estimated for each period from arrivals divided by observed time.
Pooling the same hour across weeks with different rates
Combining counts for a fixed hour and weekday over several weeks whose arrival rates differ makes the pooled counts more variable than a Poisson count, so a test can reject the Poisson model for the wrong reason. Kim and Whitt list this overdispersion, data rounding and subintervals over which the rate varies too much as causes of false rejection.
Sources
Non-homogeneous Poisson process with time-varying rate
Gallager RG. Discrete Stochastic Processes, course text for MIT 6.262, chapter 2 (Poisson processes). MIT OpenCourseWare; 2011. Section 2.4 on the non-homogeneous Poisson process with time-varying arrival rate lambda(t), equations 2.31 to 2.33 and Theorem 2.4.1 on the Poisson count with mean equal to the integral of the rate.
Gaps of a non-homogeneous Poisson process are not identically distributed
Lewis PAW, Shedler GS. Simulation of nonhomogeneous Poisson processes by thinning. Naval Research Logistics Quarterly. 1979;26(3):403-413. Section 1: the number of points in any interval is Poisson, and the intervals between points are neither independent nor identically distributed.
Data problems when testing hospital arrivals against a Poisson model
Kim SH, Whitt W. Are call center and hospital arrivals well modeled by nonhomogeneous Poisson processes? Manufacturing and Service Operations Management. 2014;16(3):464-480. Abstract: hospital emergency department arrivals are consistent with a non-homogeneous Poisson process only when rounding, subinterval choice and overdispersion from pooling weeks are handled.
Canonical Identity
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