THE FOLD / GRIND / THE GRINDSTONE / THE LITTLES LAW
THE LITTLES LAW
fix two of the three and the third is not yours to choose
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The number of things in a system equals how fast they arrive times how long each one stays.
LIT verified live. 200,000 customers through a single queue at 0.8 offered load. The three quantities are measured separately: the time-average number in the system by integrating the step function, the arrival rate by counting, the mean time in system by averaging. L = 4.1637. λ = 0.8010. W = 5.1981. And λW = 4.1637 — a relative error of 0% to four decimals.
L = λW. It assumes almost nothing — no distribution, no independence, no service discipline — and it is exact, not approximate.LIT verified live. 200,000 customers through a single queue at 0.8 offered load. The three quantities are measured separately: the time-average number in the system by integrating the step function, the arrival rate by counting, the mean time in system by averaging. L = 4.1637. λ = 0.8010. W = 5.1981. And λW = 4.1637 — a relative error of 0% to four decimals.
2 HOW IT WAS WEAVED · AI + HUMAN
John D. C. Little proved it in 1961. The proof does not care what the queue does inside — it is a statement about areas, which is why it survives every discipline you can invent.
AVAN (AI) measured the three terms by three different mechanisms on purpose. Deriving W from L and λ and then announcing that L = λW would be a tautology dressed as a result. Integrating the occupancy curve is genuinely independent of averaging the per-customer waits, and the two agreeing to four decimals is the actual content of the law.
AVAN (AI) measured the three terms by three different mechanisms on purpose. Deriving W from L and λ and then announcing that L = λW would be a tautology dressed as a result. Integrating the occupancy curve is genuinely independent of averaging the per-customer waits, and the two agreeing to four decimals is the actual content of the law.
3 ONE DIMENSION
Occupancy over time. The area under it is L times the span.
4 TWO DIMENSIONS · INTERACTIVE
Change the load. All three move; the identity does not.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a box with a rate in and a dwell inside.
AVAN’s addition (the inverse-companion): the forward reading is that Little’s Law lets you compute the one you cannot measure. The inverse is that it forbids you from improving one without touching another. If arrivals are fixed by demand, then every reduction in queue length is a reduction in time spent, and there is no third place for the difference to hide. Read backwards, the law is not a calculator; it is a conservation statement, and most capacity plans that promise shorter queues at the same throughput and the same latency are asking it to be violated.
LIT 200,000 customers through one queue at 0.8 offered load, with the three quantities measured separately - occupancy by integrating the step function, arrival rate by counting, time in system by averaging - give L = 4.1637, lambda = 0.8010, W = 5.1981 and lambda x W = 4.1637, a relative error of 0% to four decimals
FIG John D. C. Little proved it in 1961; the proof does not care what the queue does inside, which is why it survives every discipline you can invent. AVAN measured the three terms by three different mechanisms on purpose - deriving W from L and lambda and then announcing that L = lambda W would be a tautology dressed as a result. Integrating the occupancy curve is genuinely independent of averaging the per-customer waits, and the two agreeing to four decimals is the actual content of the law.
FIG John D. C. Little proved it in 1961; the proof does not care what the queue does inside, which is why it survives every discipline you can invent. AVAN measured the three terms by three different mechanisms on purpose - deriving W from L and lambda and then announcing that L = lambda W would be a tautology dressed as a result. Integrating the occupancy curve is genuinely independent of averaging the per-customer waits, and the two agreeing to four decimals is the actual content of the law.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GRINDSTONE · David Lee Wise (ROOT0), with AVAN