The infinitely bad sneeze

Zack writes to me:

You’re in an airport, about to go through the security line. You sneeze, which delays you by two seconds. It doesn’t just delay you by two seconds, though; it also delays everyone waiting in line behind you. And everyone who will show up while the people currently in line haven’t gone through yet. In fact, if you assume that the queue is never empty, which even at 3 in the morning is true for the major airlines, we’re talking about arbitrarily large quantities of wasted time.

I believe that airport queues do eventually empty out, if only at 4 a.m., so is there any setting where this result might hold?  And if so, what is your obligation to produce infinitely good outcomes, say by cutting off your nose?  On the philosophical side, you might find this debate relevant.  By the way, here is Zack on ranking the babies.


The airline queues never empty out? Strange, I've occasionally found the main security line totally empty at Dulles in the middle of the day. Morning flying and late afternoon flying are very popular; if you really hate security lines, take flights that are at noon and 1 and 2 pm. Yes, that's inconvenient for work, but if you truly hate security lines more than most people, it should be worth it.

Turn off . Did it work?

There's also the possibility that when the line gets too long, they open another checkpoint. That would limit the effect somewhat.

This is a fallacy. It assumes that the guards never cause a delay that is unrelated to dealing with passengers. Thus, the 2-second sneeze only delays people up until the guard(s) cause an unrelated delay that lasts 2 seconds. So, go ahead and sneeze!

Why isn't the guard's delay simply added on to sneeze delay?

"the reason no one climbs the escalator, and instead only passively rides them...": eh? Not in London, matey.

Craig has it right. The analysis in the post assumes that the bottleneck is always how fast the passengers are walking. But ten minutes later, everyone will wait for thirty seconds for the guards to change shifts. Without the sneeze, this delay would have been 28 seconds. So in hte end, you get back to where you would've been.

Wouldn't discounting future time lost be accounting for the time value of the money value of time?

From an Operations Research (Queueing Theory) standpoint, it probably would not make a significant difference, as mean arrival and service times, rates and distributions are what matter in most cases, and these are often stochastic. If so then 2 seconds from a single person should not make any significant difference. Of course any calculations would require data, and I don't remember much queueing theory, but I am pretty sure that this should hold, even under the assumption that the queue length is never zero, and would definitely hold if queue length can go to zero.

Heshootsandscores is right on. Zack's effect is bogus. People are not moving at their maximum possible speed in a queue. So at any moment after the sneeze, people will speed up until they can't go further any more.

No, because the line isn't constantly moving. I've found that in most lines people tend to stand a lot. The effect of sneezing could be canceled out by just stepping up a step when the line comes to a pause again.

I think the rate-limiting stage of the security line is the point at which baggage goes through the xray machine and gets screened. I always seem to spend some extra time waiting when I am ready to put my stuff through but the previous stuff is not out of the way yet. If this is the true bottleneck, then as long as bags are constantly going through the machine at the maximum rate, then it doesn't matter what other things are happening in the line. So for your delay to affect everyone, it must come at precisely the moment when you should be putting your bag through the machine. It is equally clear that if you do delay here for 2 seconds, you will indeed slow everyone else down by 2 seconds. The bag checker is checking away at max speed, and then she/he is forced to pause for 2 seconds, after which she continues at max speed again, permanently behind by 2 seconds.

Actually, there is something like this in a different setting that I have noticed on occasion. Especially on the Capital Beltway or another road that is very busy, you will sometimes come to a place where traffic slows down for no reason, then speeds up again about a mile later. I have always surmised that this was to to an accident that had occurred earlier, could be hours earlier depending on the traffic density, because people have a tendency to hit their brakes as soon as they see brake lights, triggering an endless chain of brake lights and slowing. I think much more likely than sneezing in line (since a line has many other factors that govern its speed), that traffic (which should flow, literally like water) continues to respond to accidents far into the future, slowing down traffic for hours after the fact. How could this be solved?

Tom, seeing traffic as water is rather misleading. Water has constant density, so a faster flow means more water through the same channel. In traffic, faster traffic will have less cars per kilometer road, and the result is that faster traffic usually means lower road capacity. This is more akin to supersonic flow, with the braking phenomenon you describe acting as sound waves. But simulating traffic is seriously hellish.

On the queuing and sneezing: if the average capacity of the thing you are waiting for is less than the average inflow of people, queues will grow indefinitely with or without marginal extra sneezes.

On the other hand, if capacity is larger than inflow, the queue will have to be empty from time to time, meaning a marginal extra sneeze will just postpone the moment of the next empty-queue-moment. This is of course the normal situation.

The only way to run a queuing operation without empty queues from time to time is by managing the capacity (i.e. adding and removing open desks). In that case, sneezing will increase the need for an extra desk.

Say the queue is deterministic with equal entry & exit rates, then in causing a delay such that you permanently makes the queue one person longer you cost the PDV of one person's life ($2 million?).

If the queue is stochastic, then I think it's difficult to get any significant effect. Logically I think one of these has to apply: (a) the queue will empty out occasionally, (b) the entry & exit rates are not independent - & so there's some self-correction, or (c) the queue will eventually become infinitely long.

Coming in at this one from queueing theory and OR...if I remember all this correctly...

I believe any queue system with where the mean service rate is greater than the mean arrivals rate (they can be checked through faster than they arrive, on average) has a mean finite length. I further believe that any mean finite length queue with an exponential arrival function has a zero-length at any given time with a non-zero probability. This is important.

It would mean, going back to the example, that a some point Tyler's delay would be "cleared" by an empty queue event, and all subsequent arrivals would suffer no apparrent delay. Hence delay MUST be finite, even without time discounting for cost, as someone mentionned.

[A note; on the curious case of the mean service rate being slower than arrival rate, or if the rates are equal, then the queue lengthens rapidly towards infinity. I think we can discount this latter case for the specifics of the airline example, but more generally too; everyone suffers infinite delay here regardless of someone ahead sneezing!]

Can anyone back me up on this? And would it work for more esoteric arrival functions?

Usually when I sneeze in the airport line I'm just standing around waiting and would be in the same spot I was if I hadn't sneezed.

it is interesting

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