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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.
Posted by Tyler Cowen on April 10, 2008 at 09:49 AM in Philosophy | Permalink
Comments
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.
Posted by: John Thacker at Apr 10, 2008 9:57:32 AM
Isn't this basically what goes on in traffic? When there's a car wreck, and everyone drives by the wreck and pauses briefly to see the wreck, it essentially creates the problems Zack is describing. The traffic externalities are the worst form of it.
Also, several years ago, Steve Landsburg posted an interesting puzzle - why do we climb stairs but only ride escalators? And I think it was either Margolis or Liebowitz (many people responded to this, but this answer was always interesting) who said that the reason no one climbs the escalator, and instead only passively rides them is because it's a general equilibrium to passively ride. If people were to walk up the escalator, they would pause slightly as they stepped off the escalator at the top or bottom. (Even though I've been riding escalators my entire life, I still mentally think about how I have to time it just right otherwise I'm going to fall on my face when I step off the moving stairwell onto the static path before me). Because they pause slightly, it causes everyone else behind them to pause, and... general equilibrium is everyone is paused and no one climbs the escalator.
Posted by: jason voorhees at Apr 10, 2008 10:08:50 AM
Turn off . Did it work?
Posted by: jason voorhees at Apr 10, 2008 10:10:10 AM
Posted by: Jacqueline at Apr 10, 2008 10:14:11 AM
And don't miss Zack's post on monetary policy.
Posted by: Rich at Apr 10, 2008 10:20:49 AM
What if the people behind you consciously speed up their process by a fraction of a second because they feel delayed by your sneeze? The effect is nullified with just a hanful of passengers. Also, aren't we assuming that the processing of people through the security line is affected (i.e. there is no slack)? That is probably incorrect.
Posted by: HeShootsAndScores at Apr 10, 2008 10:24:24 AM
There's also the possibility that when the line gets too long, they open another checkpoint. That would limit the effect somewhat.
Posted by: Phil at Apr 10, 2008 10:26:33 AM
It annoys the hell out of me when people stop on an escalator, especially if there is a li ne of people trying to go up or down. It significanly reduces the efficiency of the escalator. For this reason, escalators should always be wide enough for people who want to walk to pass.
Posted by: MS at Apr 10, 2008 10:30:07 AM
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!
Posted by: Craig at Apr 10, 2008 10:30:40 AM
I'm distracted from thinking up possible examples by the implications of this for Terminal 5 at Heathrow...
Posted by: Tim at Apr 10, 2008 11:07:42 AM
Craig,
Why isn't the guard's delay simply added on to sneeze delay?
Posted by: josh at Apr 10, 2008 11:23:17 AM
Time value of money is relevant here. Even if it does impact infinite people, with a positive discount rate this will have a finite discounted seconds wasted.
I agree with Phil above that the number of lines is endogenous to the line length, and so even if the line did last forever the delay need not last forever.
Posted by: OneEyedMan at Apr 10, 2008 11:31:32 AM
"the reason no one climbs the escalator, and instead only passively rides them...": eh? Not in London, matey.
Posted by: dearieme at Apr 10, 2008 11:39:52 AM
You have to consider the opportunity cost. The "arbitrarily large quantities of wasted time" would have otherwise been spent waiting in the terminal.
Posted by: Steve R at Apr 10, 2008 12:44:08 PM
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.
Posted by: at Apr 10, 2008 12:54:08 PM
Rather, 32 seconds (not 28 seconds).
Posted by: Steve at Apr 10, 2008 12:55:13 PM
Wouldn't discounting future time lost be accounting for the time value of the money value of time?
Posted by: Steve at Apr 10, 2008 12:57:41 PM
Well, there's an average "delay due to random events" which just decreases the inflow rate from time to time, which in turn decreases average rate of flow for the line. So, if you -don't- sneeze when you're supposed to, all rates and, consequently, all costs go up because you need a marginally larger number of inspectors, assuming each inspector inspects at a constant rate. So, it's OK to sneeze.
Posted by: MattF at Apr 10, 2008 1:22:05 PM
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.
Posted by: ZH at Apr 10, 2008 1:55:04 PM
What if it isn’t a sneeze? What if it is a yawn that sets off a chain of social yawns with exponential growth!!! OH MY GOD!!! The Humanity!!!
And you thought American Airlines and the FCC were the Problem.
Posted by: Chris M. at Apr 10, 2008 2:47:23 PM
FAA, sorry
Posted by: Chris M. at Apr 10, 2008 2:49:17 PM
Take a "master equation" approach to the problem, a la chemical kinetics, and you'll find that if the guard is not "johnny on the spot" with flagging people through, the sneeze gets washed out.
If the person at the head of the line's reaction time is the rate-limiting step, then yes, it is correct that the two second sneeze delay is present for every traveller until the line is clear.
Posted by: Ben Kalafut at Apr 10, 2008 2:49:32 PM
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.
Posted by: Link at Apr 10, 2008 4:07:31 PM
If sneezing isn't working for you, consider that guy who forgets to take off his watch and then leaves his penny collection in his pocket. Or if you think the limiting step isn't queueing, it's getting stuff through the x-ray thingy (which is probably true), consider the woman who takes 18 years shedding her boots.
One-Eyed makes a good point re time discounting, but it's still going to add up to something huge.
Posted by: Dr. Zeuss at Apr 10, 2008 5:29:43 PM
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.
Posted by: Nick M. at Apr 10, 2008 5:48:12 PM
In Europe, people stand on the right, the left is open to run by it in scalators
Posted by: karl at Apr 10, 2008 9:10:06 PM
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.
Posted by: xan at Apr 10, 2008 10:22:26 PM
Being a frequent sneezer AND understanding queues from operating systems and queuing theory, I'm an expert on this.
The answer is, because it's a queue (really a series) that Zack's condition CAN happen, but it's really, really, really unlikely. thanks to queue magic.
The usual case is there's already at least SOME queue of people there, though shortish. In that case, your sneeze doesn't matter, because the people behind you hafta wait in line anyway. If you sneeze right when you're going forward, it only matters to the guy right behind you.
But if the queue's empty, it STILL probably doesn't matter, because not enough people are there in line to form a line; it'll just make you 2 seconds later, because nobody's behind you.
The ONLY case where it matters is where the crowd is just gluey enough that the entire's crowd's moving in sync. The math's against such synchony happening much or involving much of a crowd (I'll go through it only if anybody cares), and that theory's backed by the fact that I've only seen in happen once or twice for that kind of individual delaying action, and only fiveish people were affected.
Any questions, class? It'll be on your quiz on Monday.
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Posted by: fdf at Apr 10, 2008 10:57:42 PM
In DC, natives usually observe protocol and stand on the right of Metro escalators so you can climb on the left, but flatland tourists are oblivious. So the hypothesis is that you need a public esclator culture, preferably connected with hurried commuting, to get this result. I see Landsburg has mostly lived outside real cities, and while Chicago counts as a real city, I seem to remember the El having stairs.
Don't get me started on people who clog up on moving walkways. Anyway, if you want a real example of one person delaying many, a clot of more than four people on an escalator turns it from climb/ride to ride/ride.
Posted by: Colin Danby at Apr 11, 2008 4:11:52 AM
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?
Posted by: Tom at Apr 11, 2008 8:05:35 AM
I think people ride the escalator so they do not have to walk up stairs. If the benefit of not exerting the energy of walking exceeds the value of the lost time from not walking up the escalator, then it is completely rational to simply ride it. Besides, why don’t we run up the escalator? It would save even more time.
As for me, riding the escalator reminds me of being a kid.
This is my first post to marginal revolution. It is very exciting.
Posted by: Eric from PA at Apr 11, 2008 10:45:45 AM
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.
Posted by: greatzamfir at Apr 11, 2008 10:49:22 AM
1. A sneeze is much less than 2 seconds.
2. You are already moving much slower than your potential, the time is easily made up.
Read The Goal, by Goldratt and Cox.
It provides very simple easy to understand basics of optimization and queuing principles.
Posted by: aaron at Apr 11, 2008 11:10:07 AM
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.
Posted by: TomC at Apr 11, 2008 1:26:04 PM
this is silly. the security queue is not the only and permanent bottleneck. People who get through this queue (the queue to have your papers examined) get buffered in another queue to be scanned through the x-ray, then buffered AGAIN before boarding the plane.
And all of these queues are adjustable by various means, so there is no permanent change in any of them.
The sneeze time costs ONLY those people who miss a plane due to the 2-second delay before something adjusts back to equilibrium (like the security worker being 0.1% faster for an hour), which is likely to be zero.
Posted by: Dagon at Apr 11, 2008 4:38:01 PM
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?
Posted by: Alistair Morley at Apr 11, 2008 9:20:03 PM
The escalator is a great example of the opposite case from the TSA line, because it does create entire crowds of synchronous travelers, where any pause means a pause in many people behind you. That's why you do need escalator protocol. Another case where the sneeze (for me, turning around after remembering I'd forgotten my jacket) does matter in travel is when your boarding group has been announced and you get a clogged group all going the same way.
sneezing will increase the need for an extra desk.
A good point, but only infinitesimally, because it's only going to matter when sneezing on the agent (hmm...way too tempting ;-)). Except when the agents are dealing with your case, it doesn't matter, because otherwise it has no effect on service time / time the agents take to deal with you. Otherwise you're just another pod in the pipeline.
Tom, all kinds of traffic tend to self-organize itself to travel in waves, like sound does. That gives regions of alternate higher density (more cars) and lower density (fewer cars). It's alot harder to reshape traffic to travel smoothly than it looks, though. So far, the best we have (a good start, actually) is traffic timing lights entering expressways. That'll just cut down on hopeless jams, though - it does little to smooth pressure waves.
Alistair, see my case analysis, above.
Posted by: Jon Kay at Apr 11, 2008 11:37:29 PM
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.
Posted by: Shane M at Apr 13, 2008 5:57:54 PM
I am on escalators and moving walkways on a regular basis. I live in the DC area and travel for work a fair amount. Look at the people that walk the escalator and walkway and those that ride. Three main types ride. They have a lot of baggage or other heavy things they are carrying. They have younger children. They are overweight. Walk the escalator or the walkway just for the extra exercise. Stand right, walk left. Lead or get out of the way.
Posted by: techreseller at Apr 14, 2008 3:35:40 PM
