Signature
L = lambda * W
| Inputs | Definition | Unit |
|---|---|---|
lambda | Average number of entities arriving per unit of time | entities per unit of time, for example per day |
W | Average time an entity spends in the system, waiting plus service | time, in the unit used for lambda |
L | Long-run time-average number of entities in the system, waiting or in service | entities |
|---|
Function
Event time sampling function
Maps a uniform random number and a fitted time-to-event distribution to a sampled time to the next event for one simulated patient. The sampled time is the value at which the survival function equals the random number, so repeated draws reproduce the fitted distribution. The simulation clock then advances to the earliest scheduled event.
Try this function
Implementations
Excel
Little's law comparison in one cell
Excel returns the relative gap between the simulated time-average number in the system and the value implied by Little's law, using named cells.
=ABS(SimulatedL-ArrivalRate*SimulatedW)/SimulatedL
Assumptions
Stable system over the observation window
Arrivals, capacity and service rules do not shift during the observation period, and the system is not in a warm-up phase. The law does not apply mechanically to a transient system.
Consistent definitions of the system
L, lambda and W refer to the same boundary, for example waiting and service together, and use the same time unit.
Worked examples
Diagnostic service with 18 arrivals per day
In the article's illustrative diagnostic service, 18 patients arrive per day and the simulation estimates an average time in the system of 0.30 days, so about 18 × 0.30 = 5.4 patients should be in the system on average.
lambda = 18; W = 0.30; L = 5.4
Common errors
Mixing time units
Combining an arrival rate per day with a time in the system in hours gives 18 × 7.2 = 129.6 instead of 5.4, which a check with consistent units would flag.
Including the warm-up period
Averaging over a simulation that starts empty includes an artificial low-congestion period, so L and W can both be understated and the check can pass on a misleading run.
Sources
Original proof of Little's law
Little JDC. A proof for the queuing formula: L = λW. Operations Research. 1961;9(3):383-387.
Simulation output analysis and queueing measures
Law AM. Simulation Modeling and Analysis. 5th ed. New York: McGraw-Hill Education; 2015.
Canonical Identity
Stable URI · Machine-readable · Resolvable · CC BY 4.0