One queue is shared. The other isn't. 🗄️🧲
Lockers need more places to serve kiddos at once than pouches do. It isn't that a lock is slower than a magnet — give both the same seconds per kiddo and the gap is still there. Here's where it comes from.
Same kids, two ways of organizing them.
One compartment each, so one place to be served. The unit three feet away sits idle and it does them no good — look at the empty second line next to the crowded third.
Any kid unlocks at any dock, so whoever is next takes whichever frees up first. No dock sits idle while somebody waits.
Operations researchers call the first dedicated capacity and the second pooled capacity. Pooling wins — at bank tellers, at airport security, and in any line a school forms at dismissal.
The bell rings and every kiddo is already standing at the lockers or the docks. Nobody is still walking from class. So nothing ever sits idle for want of a customer — the only question is how fast the line moves.
We model them as an exponential distribution with your number as the average. That's the standard choice for how long a task takes when most are quick and a few run much longer, and it's what lets the rest of this page come out as clean formulas rather than a computer simulation.
What it means in practice, if you enter 30 seconds: most kiddos are done well inside that, but about a third take longer than 30 seconds, roughly one in seven takes over a minute, and one in twenty takes more than a minute and a half. That long tail is not a rounding error — it is the reason lockers and pouches come out differently at all.
While there's still a line, every dock is busy. Docks finish at different moments, and each time one frees up the next kiddo steps in. With c docks going, somebody finishes every s/c seconds on average. Getting through everyone except the last c takes:
Then the last few. Now the final c kiddos are each at a dock with nobody behind them, and you're waiting on whichever of them is slowest. That works out to:
Here is where that chain of fractions comes from. The last c kiddos are all being served at the same moment, so you are not waiting for one whole retrieval — you are waiting for the first of c of them to end, which happens after about s/c seconds. Once that kiddo leaves, c − 1 are still going, so the next one to finish takes s/(c − 1). Fewer and fewer docks are in use at once, so each finish takes longer than the last, until one kiddo is left alone at a dock and takes a full s.
600 kiddos, 30 seconds each, 30 docks: 9.5 minutes with every dock in use, then 2 minutes waiting on the last few. About 11.5 minutes.
A locker unit takes one kiddo at a time, which is what most of them are wide enough for. So c units means c kiddos served at once, and we can compare units to docks straight across. Split the school across those units and each one has n = m/c kids to work through, one after another, which takes it n × s on average.
But the units don't all finish together, because some kiddos are quick and some are slow. One unit gets several jammed locks in a row and falls behind; another gets an easy group and finishes early. How far apart the units finish works out to about s × √n.
And here's the part that costs you: everyone isn't done until the slowest unit is done. The unit that finished early can't help, because those kids' phones are locked in a different cabinet. So the question isn't how long a unit takes on average — it's how long the worst of them takes.
That has an exact answer, and it's simpler than it sounds. Every unit has to be finished for the school to be finished, and the units don't affect each other, so:
And a unit is done by t only if all n of its retrievals fit inside t. We know exactly how likely that is, because we said each retrieval is exponential and we know how many there are — n of them added up. So the whole curve is known, and the average clearing time comes straight off it. The calculator works that out for your numbers.
600 kiddos, 30 seconds each, 30 units: 10 minutes if every unit finished at the same moment, plus about 5 minutes waiting on the slowest one. About 15.1 minutes.
Same 600 kiddos, same 30 seconds each — 30 locker units against 30 docks:
| Steady part | Waiting on the slow ones | Total | |
|---|---|---|---|
| 🧲 Pouches | 9.5 min | 2.0 min | 11.5 min |
| 🗄️ Lockers | 10.0 min | 5.1 min | 15.1 min |
Look at the middle column, because that's the entire difference. Both methods move through the bulk of the school at about the same rate. What separates them is what happens around the slow kiddos.
With pouches, a slow kiddo holds up one dock and nobody else — the next kid in line simply goes to a different dock. With lockers, a slow kiddo holds up every kid assigned to that same unit, and they can't go anywhere else, because their phone is in that cabinet and no other. So all of them wait.
That's what you're buying when you buy a pooled line. Not speed — protection from the slow ones.
Everything above is about when the last kiddo leaves, because that's what has to fit inside your window. The calculator also shows the average wait, which is a different thing: what a typical kiddo stands there for. Both have exact formulas too.
With pouches there's one line, so a kiddo waits for everyone ahead of them to clear a dock. With every dock in use somebody finishes every s/c seconds, and averaging over the whole school gives:
With lockers a kiddo only waits on the kiddos ahead of them at their own unit — the rest of the school is irrelevant to them. There are n kiddos at a unit, so on average:
Here's the part worth knowing before you compare the two numbers. If you size each method to hit the same window, lockers show the shorter average wait. 600 kiddos and a 10-minute window: 36 docks gives an average wait of 3.7 minutes, and the 55 locker units you'd need gives 2.5 minutes.
That is not lockers being better. It's you having bought 55 of something instead of 36. Put the same number of each in the building and they're nearly identical — at 30 places apiece it's 4.5 minutes for pouches against 4.8 for lockers. The average wait is not where these two methods differ. The end of the line is.
Every number in the calculator rests on these. Each one is stated plainly first, then in the terms a colleague would want.
Nobody is still walking from class, so no dock or unit ever stands empty for want of a kiddo. If your students take three minutes to reach the lockers, add three minutes to every time on the calculator — and it costs both methods about the same.
Formally: all m arrivals occur at t = 0, so utilization is 1 until the queue empties.
Retrieval times vary, and your number is the average, not a guarantee.
Formally: retrieval times are independent draws from an exponential distribution with mean s. Independence is doing real work here — if your slow kiddos cluster (a whole class arriving together after gym), both methods do worse than shown, lockers more so.
Compartments are handed out so that no unit is responsible for many more students than any other. If one unit is, lockers do worse than shown.
Formally: each of the c units serves exactly n = m/c students. Arbitrary assignment instead makes the busiest unit hold about m/c + √(2(m/c) ln c), and the school waits on that one.
Which is what most units are wide enough for. If two kiddos really can open compartments side by side, you need fewer units than the calculator says — set that on the calculator and it will account for it.
Formally: units and serving positions are the same count, so c units gives c servers.
One line feeds all the docks, and whoever is next takes whichever comes free. Tie kiddos to a particular dock and the whole advantage disappears — pouches then behave exactly like lockers.
Formally: a single queue with c servers, not c queues with one server each.
Neither number on the calculator comes from a simulation, and neither is a rule of thumb. Both fall out of the assumptions above.
Formally: the pouch time is (m − c)s/c + sHc in closed form. The locker time is the mean of the largest of c Erlang(n) variables, which is a single integral we evaluate numerically. Against a simulation of 600 kiddos across 30 units it lands within a tenth of a percent.
Nobody has published how long it actually takes a kiddo to fetch a phone from a locker, so the calculator starts from a guess and lets you replace it. If you time yours, please send it — that single number would improve everything here.
None of this covers dollars, or how well either method actually keeps phones out of hands during the day. Those are real questions and this is not the tool for them.
This calculator was built by Samantha Keppler — former high school teacher, PhD engineer, and expert in education operations. Get in touch with feedback or your cell phone ban implementation stories. She'd love to hear from you!
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