Signature
p_keep = lambda_t / lambda_star
| Inputs | Definition | Unit |
|---|---|---|
lambda_t | Arrival rate lambda(t) at the time of the candidate arrival | arrivals per unit of time |
lambda_star | Constant rate used to generate candidate arrivals, at least as high as lambda(t) at every time in the period | arrivals per unit of time |
p_keep | Probability that the candidate arrival at time t is kept | probability |
|---|
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
Thinning keep decision for a candidate arrival
With the rate at the candidate's time in a cell named RateAtTime and the majorant in MajorantRate, the formula returns TRUE when the candidate is kept. A fresh RAND is used for each candidate.
=RAND()<RateAtTime/MajorantRate
Assumptions
Majorant rate never below the arrival rate
lambda_star is at least lambda(t) everywhere in the period, usually the peak rate. A majorant that is much higher than the rate for most of the period still gives correct arrivals but rejects more candidates.
Each candidate kept or rejected independently
Every candidate is kept or rejected with its own independent uniform draw, compared with lambda(t) divided by lambda_star.
Worked examples
Night-time candidate under a daytime-peak majorant
With the article's Model B rates and a majorant equal to the daytime peak of 3 per hour, a candidate falling overnight, when the rate is 1 per hour, is kept with probability about 0.3333, so night-time arrivals occur at 1 per hour.
lambda_t = 1; lambda_star = 3; p_keep = 0.3333
Daytime candidate kept with certainty
A candidate falling between 08:00 and 20:00, when the rate equals the majorant of 3 per hour, is always kept.
lambda_t = 3; lambda_star = 3; p_keep = 1
Common errors
Majorant set at the average arrival rate
Using the daily average of 2 per hour as the majorant for Model B gives a daytime ratio of 1.5. Every daytime candidate is kept, but candidates arrive at only 2 per hour, so the daytime peak of 3 per hour is understated by a third.
Sources
Thinning method for non-homogeneous Poisson processes
Lewis PAW, Shedler GS. Simulation of nonhomogeneous Poisson processes by thinning. Naval Research Logistics Quarterly. 1979;26(3):403-413. Theorem 1 (delete each candidate with probability 1 minus lambda(x) divided by lambda_star(x); the remaining points form a non-homogeneous Poisson process with rate lambda(x)) and the abstract (no numerical integration of the rate function).
Canonical Identity
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