WAITING FOR A BUS

Stanley L. Sclove, Ph.D.
Statistician
Department of Information and Decision Sciences
University of Illinois at Chicago

Hi, Students,

When we were waiting for the shuttle bus to go to Prof. Chen's talk on the West Side, I was telling you about a related paradoxical result. Here are some details.

A SURPRISING PHENOMENON

Consider arrivals of buses at a given stop. The inter-arrival times have a mean of M minutes. What is the average waiting time of a passenger? The answer usually given is M/2 minutes. But this is so only if buses run at exactly M-minute intervals. The average waiting time depends on the variance of the inter-arrival times. The correct answer is

M/2 + V/(2M) ,

where V is the variance of the inter-arrival time distribution. (If V is infinite, then the average waiting time is also infinite, in spite of the fact that the average inter-arrival time is finite.)

The result depends only upon mild assumptions. Reference: Lajos Takacs, Introduction to the Theory of Queues, Oxford University Press, 1962. (The book may exist in a reprint or later edition.)

Note that this is

M/2 + V/(2M) = (1/2)(M + V/M)
            = (1/2)[(M^2 + V)/M]
            = (1/2)[2nd moment/M] .