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.

TSA should eliminate its unnecessary practices of harassment to compensate for fewer unnecessary employees.

If TSA continues gate harassment, use of nudeoscopes over WTMD, ID and BP scrutiny, Shoe Carnival, concern with items not related to airport/aircraft security, sexual assault, etc, then Congress should eliminate 100% of its funding, not just 8.

Well, we know that TSA already does this somewhat.. when it gets really backed up they start passing people through the metal detectors, which are a LOT faster.

Unclear the extent to which they're willing to increase this practice... at busy times it may be that the only way to keep lines from growing is to put all passengers through the WTMD.

^ As usual right on. A BIG + 1.

While I agree with Spiff, I think one of the very few benefits of the sequester could be the TSA adapting. Reduce gate groping and stop using NOS as primary in order to speed lines.

A naive, simplistic question: suppose the arrival rate exceeds the service rate, thus leading to an apparent negative situation. How does queueing theory deal with this? Throw hands up in the air? Shrug shoulders? Would this be classic gridlock?

If the arrival rate exceeds the service rate (in steady state) then the average waiting time is infinity (because the line gets longer and longer). Now, in reality this won't happen. In the airport case, arrival rates are actually time-varying. So the line will grow, and then eventually the arrivals will slow down because it reaches an off-peak hour, and the lines will shrink.

A better answer to your question is that if arrival rates exceed service rates in the long run, people will become discouraged from arriving (i.e., people will change their behavior and travel less). One way to model this is economic or game-theoretic: individuals are heterogeneous, and each individual has a value for the service of air travel (if I need to go to a business meeting, my value is probably greater than the value of the guy going to vegas for the weekend). I am willing to pay a certain cost (waiting in line) to obtain the valuable service. As the length of the line increases, the cost I have to pay increases (have to get to the airport earlier, which takes valuable time out of my day). Thus, eventually the cost exceeds the benefit and I decide not to go through the hassle of air travel. Because individuals have different values, the low-value travelers will reduce their travel first and line lengths will reach an equilibrium that is tolerable for high-value travelers. This is actually not far-fetched at all-- a lot of people believe that post-9/11 security is somewhat responsible for the very slow rebound in air travel as a lot of people decided it was just as easy to drive.

In fact, from a mathematical standpoint, one does not even need to assume that individuals are heterogeneous to get this discouragement effect; even among homogenous individuals, if the service slows down, they will choose to receive the service less often, which slows the arrival rate and eventually puts the service system in equilibrium. This bit of analysis is perhaps a bit too technical to describe qualitatively.

Thanks very much for your amplification. Seems to involve rate processes with a dash of behavioral economics ... season to taste <gr>

The problem is that the TSA is not interested in science or efficiency. They are about maintaining their bloated kingdom. If they were about science and efficincy they would never have rolled out the BDO program. They would have a mechanical system in place to move empty bins from airside to landside without the need for a highly paid human to move them. They would know that gate ID checks and VIPR teams at train stations will do nothing to make anyone safer.

Any business that was forced to cut 8% of their budget would not do ANYTHING that would impact the customer in any significant way. Travel gets cut, housekeeping reduced, new purchases deferred, etc. Hours reductions and layoffs come in if the less painful cuts can't meet the requirement. Those left get stuck working harder to meet customer needs.

In the government case, they jump straight to "you will have to wait longer Mr. Forced Customer" and don't seem to care about any of the massive wastes already in the system.

Just wanted to say I took a semester on this topic as part of a computer science degree. Fascinating stuff but I still remember how it befuddled me and that was a long time ago in a galaxy far, far away.

So I can appreciate those who are experts in this field!

In most airports I travel through, they DO NOT do this. The lines are allowed to back up with no use of the metal detectors. In some airports there is a directive that - regardless of backups - the metal detectors are not to be used (except for children/families and other specified exceptions).
THere is all the incentive in the world right now for them NOT to increase the practice. From the agency's perspective, making the lines grow will increase justification for more staff. From the Administration's perspective, longer lines will serve the political ends of forcing Congress and trying to change control of congress to all D in 2014. From the paranoia side, the powers that control agency risk avoidance will think that terry-wrists will see such a step as a vulnerability and "test" the system, therefore the guard should be raised not lowered.

I would assume that Pre-Check will be one of the first areas to close.

Thank you for the very clear explanation! One benefit of this post - an MBA candidate I know has finally been able to show to the rest of the family the usefulness of his ops research course!Until now we've been getting lots of hand-waving...

As an engineer, I thank you for taking the time to provide this analysis, which makes sense mathematically but which I've never seen before. I appreciate that you've chosen to separate this analysis discussion from the debate on "the other side."

And if we were talking about lines at a grocery store or gas station or movie theater, where the owners and employees have an interest in customer service and repeat business, your analysis would be completely relevant.

But as tev9999 and GHF have said, TSA is not interested in efficiency or deploying resources to minimize the effect on passengers, and in fact is willing to manipulate the process to exacerbate the problem. Exhibit A is Janet Napolitano all over the media rubbing her hands with glee at the longer lines three days into sequestration. Subtext is not "we're going to try our best to keep things moving smoothly", it's quite clearly [Wicked Witch of the West]"give us all the money we want or you'll all suffer."[/WWOTW]

Get ready for a fun ride. The sequester is just the first of a trifecta of potential for government induced pain. The current continuing resolution funding government ends March 27. The debt limit has been suspended until May 18th by legislation. Had it not been for the legislation allowing it, it would have hit in late February or early March. In other words, right now.

If "Every nation gets the government it deserves.", y'all have been very, VERY bad.

(Oh, don't take offense. Australia has apparently been naughty, too.)