Week 5: First Come, First Served
The rule you have believed in since you were four years old, examined for the first time — including what it costs and what it quietly gets wrong.
22 March 2027
Sometime around 1970, banks in the United States did something that now looks obvious and at the time needed explaining to customers on posters.
They replaced one queue per teller with a single queue feeding every teller.
Customers hated it. It looked longer. It was longer — one line of thirty instead of six lines of five, and the thirty were the same thirty.
Guess the gain before you see it
Four tellers, same customers, same speeds. What does merging the four lines into one actually buy?
Both of the things you might have said, and the first one is far bigger than nearly everyone expects — including the version of me who wrote the first draft of this lecture and put “roughly a third” in it.
It cuts the average visit from nineteen minutes to six. Not by working faster. The tellers are identical, and there are still four of them. By never leaving one of them idle while somebody is waiting.
It destroys the variance. In four lines you might pick the one behind the man depositing a shoebox of coins: the unluckiest tenth of customers wait forty-three minutes. In one line that cannot happen, and the same unluckiest tenth wait thirteen.
I had assumed the historical gain was mostly about fairness and only marginally about speed. That is a common belief, it is wrong by a factor of three, and the branch below is what corrected it.
Where the speed comes from
It is worth being precise about this, because it is not obvious where a free gain could possibly come from when nobody is working faster.
In four separate lines, a teller can be free while three people stand two metres away in the next lane, because nobody switches fast enough to notice. That idle time is pure waste, and it is invisible to everybody in the building — the teller does not look stranded, they look like they are between customers.
A single line makes that state impossible. If anyone is waiting, no window is free. That is the whole mechanism, and the branch below counts it for you.
- W one line
- —
- W four lines
- —
- W₉₀ one line
- —
- W₉₀ four lines
- —
- stranded teller-min an hour
- —
One honest caveat: nobody in the four lines switches lanes. That is the most generous treatment of parallel queues on fairness and the least generous on speed. Real customers jockey, and jockeying recovers part of the gap — never all of it, because you can only switch after you have already lost the time that told you to.
The waste gets worse when the branch is quiet
Turn the load down and watch the pink chip, because it does the opposite of what it should.
At seventy customers an hour the branch strands thirty-seven teller-minutes every hour. Take it down to a comfortable fifty-two and the waits all improve — and the stranding rises to sixty-nine minutes an hour, more than one whole teller permanently standing free beside a queue.
That is not a paradox once you say it out loud. Idle time is what a quiet branch has more of, and every minute of it that coincides with somebody waiting in the wrong lane is a minute nobody can ever get back. A manager looking at this branch sees four busy-looking tellers and a reasonable wait. The rope is eating a salary.
Why the gain shrinks as the system grows
Before you go and pool everything you own, the size effect, because it decides whether this is a good idea or a rounding error.
Think of the branch as needing two separate things. It needs enough tellers to do the work — that quantity is fixed by arithmetic, it is arrivals times service time, and no arrangement can change it. And on top of that it needs some safety capacity: spare teller-time to absorb the moments when several people happen to arrive together.
The useful result is about how much safety you need. A standard rule of thumb in this field sizes a counter as work, plus something proportional to the square root of the work. Which means the safety margin grows much more slowly than the system does.
Put numbers on it. A branch with 4 tellers’ worth of work needs roughly 2 tellers of safety on top — a 50% overhead. A branch with 100 tellers’ worth needs roughly 10 — a 10% overhead. Big pooled systems are intrinsically more efficient than small ones, and the gain is not linear.
Now read that backwards, because that is where it bites.
A bank has 40 tellers across 4 branches. Head office proposes one giant pooled call centre instead. How much is there to gain?
Much less than the branch above suggested, and possibly nothing.
The four-into-one gain was enormous because four is a small number: each separate line was a tiny system with a crushing safety overhead, and pooling them was the difference between four bad systems and one decent one.
Ten tellers is already a reasonably large system. Pooling four of those buys the difference between 10% and 3% safety overhead — real, worth having, and nowhere near the factor of three you get from merging four single-server lines.
Pooling pays most exactly where the pools are smallest. Which means the question is never “should we pool” but “how small are the pieces we currently have” — and the answer for most organisations is that the damage is concentrated in the places where one person is the whole queue.
When pooling is the wrong answer
So why does your supermarket still have one line per till?
Because a pooled queue is not free, and four of its costs are real.
It can make the service itself slower. Pooling usually means generalists. If a specialist teller handles a mortgage in twelve minutes and a generalist takes twenty, you have improved the queueing term and damaged the service term — and week 2’s equation says T appears twice. The gain has to beat that.
It can cost travel. Pooling makes sense when any server can reach any customer instantly. A supermarket single line means a trolley crossing the shop front; a pooled hospital ward means nurses walking further; a warehouse without zones means longer picks. The handoff is not free and is often the whole budget.
It couples things that were independent. One line into four tellers means one stuck transaction can now stall everybody. Four lines contain their own disasters. This is the same argument a systems engineer makes for bulkheads, and it is why pooling is a reliability decision as well as an efficiency one.
It hides the deliberate partitions. The express lane is un-pooling on purpose, and next week is about why that is often right. Skill-based routing in a call centre is partial pooling. A hospital’s resuscitation bay is capacity deliberately withheld from the pool so that it is free when it is needed. Each of those is somebody choosing to pay the pooling gain for something they want more.
Pooling is not a free lunch. It is a good trade that is usually worth making.
Every one of those four costs is a service-side cost, and the pooling gain is a queueing-side gain. You are trading between the two halves of week 2's equation.Why people hate parallel queues anyway
There is one more reason single queues won, and it is not efficiency.
In four lines you can be overtaken. And here is the arithmetic that makes it feel so much worse than it is: with four lanes, the chance that yours is the fastest is one in four. Which means three-quarters of the time, somebody who arrived after you will finish before you — and you will watch them do it.
The single line removes that experience entirely, and removes it completely rather than on average: not one person who arrived after you can leave before you. That is worth more to the people in the line than the thirteen minutes, and it is measurable in complaint volumes long before it is measurable in waits.
Which is week 4’s last beat arriving in a bank: perceived waiting and measured waiting are different quantities, and the single queue happens to improve both. It is the rare case where the fair thing and the fast thing are the same thing, and you should enjoy it, because it does not happen again in this course.
The fairness claim, stated properly
Order of service equals order of arrival, and nothing else is considered.
Transparent, auditable by anyone standing in it, and impossible to game except by arriving earlier.That is a genuine commitment, and for a public system dealing with strangers the properties are not small. It requires no information about the people in it, which means it cannot discriminate on information it does not have.
It has three more properties worth naming, because you will want them in your assignment:
It is verifiable from inside the queue. Anyone standing in it can audit it, in real time, with no access to anything. Almost no other rule in this course has that property — you cannot see somebody’s urgency, their job length or what they paid.
It is strategy-proof, almost. There is exactly one way to do better under FCFS, and it is to arrive earlier. Compare that with every rule in weeks 6 through 8, each of which creates a reason to misrepresent something.
It requires no justification at the point of service. Nobody at the counter has to defend the order to the person in front of them, which is worth more in an angry room than any amount of throughput.
Notice what just happened, though. The single line is faster and it is the one that keeps the promise: in four lines the order you leave in is decided by which rope you guessed at. FCFS is not what four separate queues implement. It is what one queue implements.
What it costs
Three things, and Act II is going to spend a week on each of the first two.
It ignores job length. The person with a thirty-second query waits behind the person with a twenty-minute mortgage restructure, paying twenty minutes for a service worth half a minute to them. That is next week.
It ignores urgency. FCFS applied in an emergency department kills people. Not hyperbole, and it is why no emergency department uses it. Week 7.
It turns a queue into a test of free time. This is the one nobody puts on the poster. “First come” is only fair between people who can equally freely come first. A queue that forms at 6am outside a housing office is allocating by who can afford to lose a night’s sleep and a morning’s wages.
FCFS discriminates on nothing. So how does it manage to land unevenly?
Because access to the rule is not distributed the way the rule assumes.
The rule’s blindness is exactly what stops it noticing. It cannot see that the person at the front took a day off and the person at the back could not, because by design it cannot see anything except arrival order.
A rule that considers nothing does not thereby become fair. It becomes fair conditional on everyone arriving on equal terms, and that condition is almost never checked.
There is a sharper way to put it that will be useful to you all semester. Arrival order is not a neutral fact that the queue discovers. It is an allocation that has already happened, somewhere else, before anybody reached the door — decided by who lives near the branch, who has a car, who knew it opened at nine, who could be free at nine. FCFS then faithfully preserves that allocation and calls the preservation fairness.
Why defaults matter most
FCFS is the default. Defaults are the most powerful design decisions there are, because they get chosen by nobody deciding.
And notice how much work the last two weeks have already done to it. It is not one rule — it is a rule plus a queue geometry, and the geometry turned out to matter more than the rule. Four lines and one line make the same promise and deliver waits that differ by a factor of three.
For the next four weeks we replace it — with job length, with need, with price, and finally with not queueing at all. Each replacement buys something real. Each one charges somebody.
Your job, in the assignment due next week, is to make one of those trades deliberately and then be honest about the bill.
Teaching team
- Adaeze Nkemelu — convenor