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Count the Staying

How full a place is depends on two numbers. Most people only ever watch one of them.

A diner seats ten parties an hour, and each party stays half an hour. At any moment, on average, five parties are at the tables. Now let the same ten parties an hour linger for ninety minutes instead. The door sees exactly the same traffic. The room holds fifteen.

That arithmetic has a name. In 1961 the operations researcher John Little published a proof that it holds far more widely than anyone had needed to assume: the average number of things inside a system equals the rate at which they arrive, multiplied by the average time each one stays. Written short, it is L = λW. L is the number inside; λ, the Greek letter lambda, is arrivals per unit of time; W is the wait, or the stay. It applies to a checkout line, a warehouse, a hospital ward, an email inbox. It does not care in what order things arrive or how unevenly they come. It asks only that, over the long run, the system neither fills without limit nor empties out.

Sourced · Little, "A Proof for the Queuing Formula: L = λW," Operations Research 9(3):383–387 (1961) · the diner is the editor's illustration

CAPTION. Suggested: From the editor's notebook: the diner, worked by hand. Tap to enlarge.

The rule has a consequence that is easy to miss. Walk into the diner at a random moment and look around. The parties you see are not a fair sample of the parties who came in that day. A party that stays three hours is in the room for six times as many moments as one that stays thirty minutes, so it is six times as likely to be there when you look. A snapshot over-counts the people who stay. A count of patients in a hospital on a single day turns up more long stays than the admissions book does, for the same reason.

The second consequence runs the other way, and it concerns where you start the clock. Suppose you watch the diner for one hour and write down only the parties that both arrive and leave while you are there. Every quick lunch makes it into your notebook. A party that sat down before you came in, or will leave after you go, never does. A short watch records arrivals and quick turnovers and misses the long stays and the departures that come late. The longer things stay, the longer you have to watch before the count settles down.

Little's law is the bookkeeping behind both effects. What fills a place is the product of how often things come and how long they remain, and the second number counts exactly as much as the first. People tend to watch the door, because the door is where the events happen. An arrival is something you can see. Staying is a long stretch in which nothing seems to occur, and it is easy to leave out of the count. The term to search is Little's law.

Sourced — J. D. C. Little, "A Proof for the Queuing Formula: L = λW," Operations Research 9(3):383–387 (1961) · J. D. C. Little, "Little's Law as Viewed on Its 50th Anniversary," Operations Research 59(3):536–549 (2011). Diner, hospital and watching-window examples are the editor's illustrations of the formula.