On the other side of this forum, there is a raging debate over the potential impact of sequestration on security lines. Unfortunately, lost in the debate is that there is a real possibility, supported by mathematical theory, that lines could get very long.

I do not work for the TSA, I do not like the TSA, I am not here to defend the TSA. I am an expert in queueing theory. Please continue reading only if you would like to understand why an 8% cut in the TSA's checkpoint capacity could result in lines that are an order of magnitude above the length they are today.

Congestion in queues is derived from variability. If people arrived on regular intervals and everyone took exactly the same amount of time to be screened, there would never be a line. However, as in most service systems, the time it takes to complete a security screening varies from person to person.

If we assume a very basic queueing model, the M/M/1 queue, the total waiting time (i.e., time to get through security) is given by 1/(SERV_RATE-ARR_RATE), where SERV_RATE is the service rate (number of people processed per hour) and ARR_RATE is the arrival rate (number of people entering the system per hour). Thus, you can see that the waiting time does not depend linearly on the service rate, meaning an 8% decrease in the service rate does not result in an 8% increase in waiting time. In fact, as the service rate gets closer and closer to the arrival rate, waiting time gets very large.

Example: Consider a small checkpoint where 150 passengers arrive per hour, and there is screening capacity for 165 passengers per hour. Then the expected waiting time is 4 minutes (1/15 hour). Now, assume that the screening capacity is cut by 8% to 151.8 per hour. Then the expected waiting time will be 33.3 minutes (1/1.8 hour), which is over 700% increase. The key point is that depending on the arrival and service rates, an 8% cut could result in a waiting time increase that is much larger than 8%.

Now, using a simple model has some drawbacks. One assumption in the M/M/1 model is that there is only 1 line. In reality, most checkpoints have multiple lines; this could be solved by using a M/M/k model with k lines. However, when all k lines are non-empty almost all the time, as is frequently observed at busy checkpoints, then the M/M/1 model is a very close approximation to the M/M/k model. The M/M/1 model also assumes that service times have a coefficient of variation (cv) of 1. If service times are less variable, it may overestimate the waiting time; if service times are more variable, it may underestimate the waiting time. I don't have access to data on screening times, but my own anecdotal observations point to a highly variable process (think FTer vs. family on the way to Disney). Another assumption is that the arrival rate does not vary with time. In fact, considering time-varying arrivals in this case may exacerbate the problem further. Notice that the above equation for waiting time only makes sense as long as SERV_RATE > ARR_RATE. This is because if the arrival rate exceeds the service rate, the queue will grow longer without bound. High average wait times are bad, but growing queues are even harder to manage. In reality, arrivals to a security checkpoint are likely to increase at peak hours and then decrease. The result is that if cuts to TSA capacity reduces SERV_RATE, we may observe queues that used to be long but manageable may now grow and grow during peak hours, only to subside once the rush is over.

None of what I have said above touches on any of the management actions TSA might take to try to reduce service times (and thus increase SERV_RATE) even as their staffing is cut, for example, by funneling more people through pre-check--these issues cross over too far into the policy debate. I only put this post up because it seems that some FTers are extremely optimistic that lines will not get longer. Travelers who are interested in practical advice should know that as the effects of sequestration are felt, if TSA does not take management action to reduce service times, then it is entirely possible that lines could get very long.

TL;DR:

• Congestion caused by variability will result in exponentially longer waits as the service rate approaches the arrival rate.
• Passengers should be prepared for the very real possibility that TSA lines will get much longer due to sequestration.