# Waiting in airports

I was visiting the University of Utah this past week.  I gave talks on the Kinetic Theory of Coupled Oscillators and on Deriving Moment Equations for Neural Networks. On my way to the airport I wondered what would be the optimal arrival time so that you spend the least amount of time waiting in the airport balanced by the cost of missing a flight.  If you make some basic assumptions, it’s not too hard to derive a condition for the optimum.  Let’s say the only thing we’re concerned about is minimizing wasted time.  Then what we would want to do is to balance the average time waiting in airports with the average time lost to make up for  a missed flight.

Let $t_a$ be the time between arrival at the airport and boarding the plane and $\sigma$ be the standard deviation in this time due to traffic, the check in line, going through security, etc.  The average amount of time spent waiting in the airport is thus the expectation value of $t_a$, $\bar t_a$.  Suppose we let C be the time wasted if you miss a flight.  Then the expected time wasted for missing a flight is  CP, where P is the probability of missing a flight.  So, optimality would be given by $\bar t_a = C P$.  Now the probability for missing a flight will be a function of the waiting time.  Assuming a normal distribution gives $P= .5{\rm erfc}(\bar t_a/\sqrt{2}\sigma)$, where erfc is the complementary error function.  Hence, if your expected waiting time is zero then you would miss half of your flights.  The optimal arrival time is then given by the condition $\bar t_a= .5C{\rm erfc}(\bar t_a/\sqrt{2}\sigma)$.

So let’s say the standard deviation is an hour and a missed flight costs about 5 hours, then solving numerically (on Mathematica) gives $\bar t_a = 0.9$.  So the optimal time to arrive at the airport is a little less than an hour before you board.   The optimal time is not very sensitive to the cost of missing the flight.  Making it 20 hours only increases the optimal arrival time to an hour and a half.   Reducing the standard deviation to half an hour reduces the optimal time to 36 minutes.

By this calculation it would seem that by arriving about an hour before departure, which is what I usually do, is close to optimal.  However, there is a flaw in this calculation because I can only recall missing one flight in my life and by optimality I should be missing about one in five flights (given that I arrive at the airport an hour before my flight and my estimated cost per missed flight is 5 hours).  What this implies is that the transit time to the gate distribution is much narrower than a normal so that while the uncertainty in transit time from my house to the gate seems to be about half an hour to an hour, it almost never takes much longer.  However, having a narrower distribution means that  the optimal waiting time won’t change very much because the probability of missing a plane increases very quickly as you shorten the waiting time (i.e.  the difference between arriving 45 minutes before departure versus an hour could mean missing many more flights). So an hour before the flight is still pretty close to optimal.  Having said all this, I actually don’t mind showing up at the airport a little earlier than necessary since it gives me a chance to read.