This is rare: I think my earlier post about the sneeze guy was wrong, or at least didn't contain the full picture. What's even rarer, is that it took an email from Tyler Cowen to prod me to reverse myself.
Although Mike in the comments of my last post had talked about the sneeze working its effects backwards through the line--much like a traffic accident can cause a delay to ripple backwards through the highway--I don't think that's the way to analyze this situation. Or at least, that's not what the sneeze guy had in mind.
Suppose the security line originally has 10 people in it. After 6 seconds one person walks through the metal detector, and a new person arrives at the end of the line. So the line is a constant 10 people long, and each person has to wait a total of one minute to get through security.
Now some guy at the front has a sneezing fit. He ends up taking 12 total seconds to go through; i.e. he "wastes" 6 seconds. This means that the line now grows to 11 people in length, and it stays like that.
So because of that guy's 6-second delay, every single person who ever joins this line has to wait for an extra person to go through, i.e. is delayed an additional 6 seconds. If a million people go through in the next year, that's 6 million people-seconds wasted.
Now relax the assumption that the line is fixed in length, but still insist that it never hits zero. Even so, the initial 6-second delay causes the line to always have one more person in it, than it otherwise would have--so long as we assume that people entering the line don't take its length into account.
I think the above analysis is correct, insofar as it goes. Since that's all the sneeze guy was claiming, I think we have to admit he was right.
There are still weird things with this. I think we've established above that if one million people go through that line, they each lose 6 seconds. But now suppose that it's the same 10 people who keep going around and around the security gate. I.e. once you go through the metal detector, you go around to the back of the line.
To get rid of the infinities, suppose for some reason that 10 people need to go through the metal detector a total of 1,000 times each.
Now if we had just said there were an initial pool of 10,000 people who had to go through the metal detector once, the analysis above shows the sneeze costs a total of 60,000 person-seconds in lost time.
But what in our new scenario? Does it cost the same amount?
That seems weird. Just imagine you're watching the group of ten people from the ceiling. They form a ring that is rotating, where every 6 seconds the people all walk a little bit around the ring, with one of them going through the gate. Do you mean to tell me that if this rotation pauses for six additional seconds, that each person loses 6,000 seconds of time? No way.
And yet, this seems to be equivalent to the original scenario, with 10,000 strangers forming the line. Surely the amount of time wasted shouldn't depend on the identity of the person in line. If the guy sneezing at the front causes the 8,000th person going through to lose 6 seconds, why does it matter who the 8,000th person is?
I'm pretty sure I know the solution to this riddle, but it's just beyond the reach of perfect obviousness in my mind. So I'm hoping someone can spell it out clearly and I can say, "Yep, you got it!"