Here at the Operations Room, we love queues and few queues have gotten as much attention lately as the lines of drivers waiting to buy gasoline in New Jersey and New York following the hurricane. Stories abounded about wasted time and why allowing price gouging would actually be good. Of course, neither states cleared station owners to jack prices up. Instead, they implemented Nixonian rationing based on license plate numbers. Those with plated ending in even numbers got to buy on even days and those with odd numbers got to buy on odd days.
Now, the Numbers Guy column and blog at the Wall Street Journal is asking whether it worked (Fuel Rationing Is Hard to Gauge, Nov 16, and Does Odd-Even Rationing Work?, Nov 17). Turns out there is no clean answer on this. On the one hand, studies of when similar schemes were used in the early 70s claim to show they were ineffective but even those studies authors think this time may have been different.
“I’m not sure our analysis transfers directly to the Sandy shortage situation,” said Robert Goldfarb, an economist at George Washington University in Washington, D.C. In a 1983 paper, Prof. Goldfarb and co-authors found the odd-even rationing system could lengthen waits, because people who normally spaced out refills by an odd number of days—say, five days—might move up their regular refills to every four days to avoid running out.
William Huss, co-author of a 1981 paper with a similar conclusion, added that while his model “has a solid mathematical foundation, its assumptions regarding driver behavior are hypothetical and logical but not necessarily based on psychological research.”
I have to admit that I love that last quote since it could equally apply to every paper I have written.
So there are a couple of things to note here. First, if one assumes that the need for service was completely exogenous (as would be the case in most standard queuing models), odd/even rationing would work fine. A customer would or would not have a need for service on a given day. If she needed service, then she may or may not be turned away based on an exogenous signal. One would then have fewer arrivals and thus less congestion. One could implement a similar system simply by admitting an arriving customer based on a coin flip. Heads your in, tails your out. For a system like this fly to Mexico. Passengers and their bags may or may not be subject to a detailed search depending on whether a light turns red when they push a button.
There are, of course, problems with a coin flip. Bad luck at one station doesn’t keep the driver from trying a station down the road or even coming back to the same station an hour later to retry. (In the Mexican airport example, no one really wants the detailed inspection so only a five-year old — as my son was — would keep slapping the button until a guard made him stop.) Even if one could stop drivers from retrying within one day, the coin flip method doesn’t provide the insurance of being able to buy tomorrow. There is a one in four chance of being kept from buying two days in a row. License plate rationing solves that.
The problem with buying gas is that the decision to purchase (as opposed to the need to purchase) doesn’t necessarily arrive randomly. Even in normal times, one can apply strategy. You buy now even though you have half a tank because you are near an inexpensive station or because you know you will be using your car a lot in the coming week and it’s just more convenient to stop now.
Once we move out of normal times in which supply is scarce, people are stressed and the government imposes rationing, all bets are off. Whether limiting when people can buy actually reduces congestion depends on how people behave. Thus, one can likely build a model that makes the program look good or bad depending on how one assumes purchasing patterns are affected.
On thing the article mentions is clearly true, however: Small purchases are bad for the system.
The result was that some people were filling up every day, or even more than once a day, with smaller amounts than usual to ensure they had as much fuel as possible in case the situation got worse.
“It wasn’t uncommon to see one-gallon, two-gallon, three-gallon or four-gallon fill-ups,” said Jeff Lenard, spokesman for NACS, a trade group for convenience stores and gas stations. The typical fill-up, he added, is about 10 gallons.
While small gas purchases involve less pumping time than does filling an empty tank, they take as long to pull up, select fuel type, pay and pull out. And the purchasers take up as much space in line. After Sandy, that led to queues that were blocking traffic and complicating the work of emergency responders.
The odd-even system helps break that cycle, by forcing drivers to refill no more than every other day. “It made everybody plan and think a little bit ahead,” said Michael Drewniak, a spokesman for New Jersey Gov. Chris Christie.
The relevant though experiment to think about here is to suppose that we doubled the arrival rate but halved the service time. That is, suppose that the arrival rate goes from 40 per hour to 80 per hour but that we cut the average processing time from one minute to 30 seconds. If that’s what happened as Garden State residence got nervous about gas, then standard queuing models say that we have no problem. The line length is driven by utilization and that hasn’t changed but the wait would actually go down. The problem is that fixed component to the transaction which doesn’t scale. Purchasing half as much then doesn’t translate to half the processing time. (Note that there is similar argument that says you should think twice about getting in the express lane at the grocery store.) Thus, while Governor Christie’s spokesman might have a biased view of events, he hits on the right idea. Odd/even rationing should work if it dissuades people from nuisance, small fill ups.