An introduction to queueing for people who schedule other people's time.
Operations 20190 minutesSample course deck
By the end of this session
You will be able to…
✓Explain why a queue grows even when average capacity exceeds average demand.
✓Calculate utilisation and read what it does to waiting time.
✓Tell variability in arrivals apart from variability in service.
✓Choose between adding capacity and reducing variation, and say why.
01
Where the waiting comes from
Concept
A queue is a store of time.
Work arrives unevenly and is served unevenly. When the two do not line up, the gap has to be held somewhere, and the only place to hold it is in front of the server. That holding place is the queue.
Worked example — press → to step through
A clinic with one room
1Patients arrive at 7 an hour on average.
2One room, average appointment 7.5 minutes.
3Capacity is 8 an hour, so on paper there is room to spare. ρ = 7 / 8 = 0.875
4Arrivals are not evenly spaced and appointments are not all 7.5 minutes.
5Waiting rises with ρ / (1 − ρ), which at 0.875 is 7.
6Seven times the service time is spent waiting — about 52 minutes.
The one to remember
Waiting time, in one line
Wq ≈ ( ρ / (1 − ρ) ) × ( Ca² + Cs² ) / 2 × ts
ρ is utilisation, Ca and Cs are how variable arrivals and service are, ts is the average service time. Only the first term explodes; the second is the one you can usually change.
02
What to do about it
Two levers
Add capacity, or take out variation
Add capacity
Works immediately and predictably.
Costs money every month, for ever.
At high utilisation, small additions buy a lot.
Does nothing about a bad process.
Reduce variation
Slower to take effect and harder to measure.
Usually a one-off cost.
Helps most when utilisation is already high.
Makes the capacity you have behave better.
Common mistake
Planning to 100% utilisation
"We have capacity for 8 and demand of 8, so we are fully used and nothing waits."
At ρ = 1 the waiting term has no finite value. A system with any variation at all and no slack does not settle — the queue grows for as long as the day lasts. Plan to about 0.80 and keep the rest for the variation you did not remove.
Quick check — press a, b or c
Utilisation goes from 0.80 to 0.90. Waiting roughly…
ρ/(1−ρ) moves from 4 to 9, so the waiting term more than doubles. The last tenth of utilisation is the expensive one.
Summary
Five lines to take away
A queue is where unevenness is stored.
Average capacity above average demand does not stop a queue forming.
Waiting rises with ρ/(1−ρ), so the last tenth costs the most.
Variation in arrivals and in service are two separate problems.
Plan to roughly 0.80 and spend the slack deliberately.
Before next session
Measure your own queue
Pick one service point you are responsible for. Log arrival times and service times for two hours. Work out ρ, estimate Ca and Cs from your log, and bring the number you get for Wq alongside the waiting you actually observed.
Reading: one chapter on variability, one paper on pooled versus dedicated servers. Both are on the course page.