How long is this line?

SLOP2710 · Week 3

What you can see from where you stand

Ahead of you — count one switchback, multiply. Say 240.

Speed — pick a fixed point, count crossings for two minutes. Say 80, so 40 a minute.

No access. No data request. Nobody to ask.

240 ÷ 40

= 6 minutes

L = λW

average number in · arrival rate · average time each stays

What it does not care about

  • any arrival pattern
  • any service time distribution
  • any number of servers
  • any discipline whatsoever
L
—
λ
—
L ÷ λ
—
W
—
W90
—
What to watch: Two ways to measure one thing. Counting sweeps a pink plane up the line and everyone it passes turns pink — the vertical cut, and the one you can do standing there. Following paints one traveller pink and runs a clock from joining to clearing the arch — the horizontal cut. Then read L ÷ λ against the separately measured W.

Same area, two cuts

Count across — how many are here right now? Average it. That is L.

Follow one through — how long did this one stay? Average it. That is W.

Both are measuring person-minutes.

The proof, in four lines

  1. Window of length T. N people pass through.
  2. Total time inside = A. So W = A/N.
  3. Same A, integrated over the window: L = A/T.
  4. And λ = N/T.

L/λ = (A/T)÷(N/T) = A/N = W

Now the dangerous part

Where is the box?

Three boxes, one airport

BoxW
The arch0.5 min
The hall60 min
Kerb to gate85 min

All correct. All publishable as “the average wait”.

State the box

before you state the number

What it is for

Estimate a wait from outside.

Size something you have not built.

Catch a lie — 500 people, 20 a minute, and they claim 10 minutes.

Sanity-check a promise — halve W with λ fixed, and L must halve. Which half of these people is leaving?

It is not really about queues

  • Factory: WIP = throughput × cycle time
  • Kanban: lead time = WIP ÷ throughput
  • Warehouse: inventory turns
  • Support desk: 300 open ÷ 60 a day = 5 days
  • Hospital: beds = admissions × length of stay

Three ways to get it wrong

It was filling or emptying. Steady state, or nothing.

Different windows. L at 12:15 and λ across three hours are two systems.

People left without being served. They were in L. Were they in λ?

You can now measure

any queue in the world

So: what did that wait cost?

And who paid it?