Queues, part 3. Part 1 asked which lane to join, and part 2 asked whether to have an express lane. This one is about self-checkout.
Picture a self-checkout bank has six kiosks and one attendant. When it’s slow, the obvious fix is more kiosks. In a simulation, that’s almost always the wrong lever.
The setup
One snake queue feeds a bank of kiosks. Kiosks are about 1.5× slower per item than a staffed till, because you’re the one scanning. With probability p, a customer needs the attendant (age check, unexpected item in the bagging area, a barcode that won’t scan). The attendant takes about 40 seconds per fix.
The detail that matters: while you wait for the attendant, you keep your kiosk. You can’t step aside. The kiosk sits there, occupied and doing nothing.
The bank holds up, then collapses
Compare six kiosks plus one attendant against three staffed tills (mean wait about 75 s):
| p (needs help) | Mean wait, 6 kiosks + 1 attendant |
|---|---|
| ≤ 0.5 | ~69 s, beats the tills |
| 0.6 | 218 s |
| 0.7 | ~6 800 s |
Up to 2.8 of the 6 kiosks were parked, each holding a customer who was waiting for help. The attendant’s queue doesn’t show up as a queue. It shows up as kiosks taken out of service.
Which lever works
At p = 0.7 the system is so close to breaking that runs vary a lot. In this run the baseline wait was about 9 000 s:
- +1 kiosk: 1 450 s
- Attendant 25% faster: 103 s
- A second attendant: 31 s
Making the attendant faster beats adding a kiosk by about 14×. Once p goes past about 0.5, five kiosks with two attendants beats seven kiosks with one.
When the attendant has to walk
Real attendants walk between kiosks, so a bigger bank makes each fix slower. Model a fix as 30 + 2.5·K seconds for K kiosks. Then, from p ≈ 0.5 up, there’s a best size (seven kiosks at 0.5, six at 0.6). Past it, adding kiosks makes the wait longer.
The rule
Effective kiosk service time is S + p·(attendant wait + fix time). The attendant’s queue leaks into the kiosk bank as lost capacity. So cut p (better scanners, fewer age-restricted surprises) or make the attendant faster. Hardware is the wrong thing to buy.