In the late 1950s, John Little was a young assistant professor at the Case Institute of Technology in Cleveland, and he had been asked to teach a course on queues. Cars at a toll booth, people in a cafeteria line, instructions waiting inside a computer: anything that arrives, waits, gets served and leaves.
His students had worked through the standard models. One day Little remarked, as his own doctoral adviser, P. M. Morse, did in his textbook, that one formula kept reappearing in all of them. The average number of items in the system, L, equals the rate at which they arrive, λ, times the average time each one spends inside, W. L = λW. It seemed very general. Little sketched a rough argument for why.
After class a student, Sid Hess, came up to him. “How hard would it be to prove it in general?”
“I guess it shouldn’t be too hard,” Little said, on the spur of the moment. Writing about it fifty years later, he added: “Famous last words.”
“Then you should do it!” Hess said.
The question stuck. Little could see that something basic was going on. When he drew the picture, the formula didn’t seem to care how items arrived, how long service took, how many servers there were, or what order people were served in. It seemed to need only a count that went up and down one item at a time, a steady state, and no items lost along the way.
Turning that picture into a proof was another matter. Little had trained in physics and operations research, not pure mathematics. He bought books on measure theory and stochastic processes. In summer his family drove to Nantucket, where his wife’s family had a small house. “Since the beach was our babysitter,” he wrote, he could carve out solid blocks of time to work. He ran into problems the books couldn’t solve, worked his way around them, wrote it all up and sent it to the journal Operations Research. It was accepted, and published in 1961.
Then, by his own account, he decided that “Borel fields and metric transitivity were not going to be my career,” and retired from queuing.
The retirement held until 2004, when a colleague asked him to write a book chapter on the formula, by then widely known as Little’s law. “I don’t know anything about OM and I haven’t looked at L = λW for 40 years,” he told him. He wrote it anyway. By 2011 the law had become standard in operations management textbooks and in the design of computers, used for everything from staffing emergency departments to sizing a hospital’s observation unit.