Two lanes

One has a queue

SLOP2710 · Week 2

Both drive-throughs

  • serve 60 cars an hour
  • have one window, one person
  • average 55 seconds an order
  • run at 92% utilisation

One never has a queue. The other has eleven cars in it.

Nothing a manager measures

can tell these two apart

The only difference

Lane A — cars arrive on the minute, every order takes 55 seconds.

Lane B — cars arrive whenever. Some are ordering for the office.

ρ
—
L clockwork
—
W clockwork
—
L real
—
W real
—
What to watch: Drag irregularity down to clockwork and the right-hand queue disappears with the load unchanged. Drag it back up and the queue rebuilds out of nothing but timing. The window flashes pink each time a car arrives to find it occupied — nought per cent in one lane, about eighty-five in the other.

Queues are made of variability,

not of load.

What actually makes a queue

A gap — work arrives while the server is still busy with something that ran long.

Bad luck accumulates. Good luck does not.

Measuring raggedness

c = standard deviation ÷ mean, on the gaps between arrivals

  • c = 0 — a metronome
  • c = 1 — memoryless. People arriving without coordinating
  • c > 1 — bursty. A lecture ending

The last few per cent

ρρ/(1−ρ)
50%1
80%4
90%9
95%19
99%99

95% → 99%

is four points on a dashboard

and five times the wait

One equation

W ≈ V × U × T

V — variability · U — ρ/(1−ρ) · T — service time

Three systems, one equation

Motorways — flow breaks down and the bottleneck then discharges slower than before it jammed. Ramp metering is a V-and-U device.

Hospital beds — the 85% occupancy target is this table, with a bed as the server.

Servers — p99 latency takes off long before the mean. Autoscaling at 70% CPU is buying the gap.

Two sources — and one you make yourself

Arrivals — mostly cannot remove. Can reshape (week 9).

Service — often can reduce. Short menus, split stations, separate desks.

Batching — variability you manufacture, to save a setup cost, and bill to the queue.

Hold on to this aircraft

Boarding through one door.

Every bag stowed blocks everyone behind it, for an unpredictable interval, in a corridor nobody can overtake in.

V enormous. U pinned at 1. T set by the slowest person in the aisle.

Next: measuring a queue

you have no access to

Before the lab

Find a queue with an obvious batch in it.

Write down what setup cost the batch is saving, and who pays for it.

One paragraph. Bring it.