Mean Waiting Time as a Measure of Effectiveness

In the application of probabilistic models to waiting time problems it is sometimes necessary to compare two or more different service patterns to determine which pattern is most effective in decreasing waiting times. In many cases the mean waiting time of the customer is the criterion used to measure the effectiveness of the service because it is the easiest property of the waiting time distribution to calculate. It is the purpose of this note to give an example of a simple waiting time problem in which it is intuitively clear that the mean waiting time gives a misleading indication of the effectiveness of the service and that some other measure should be considered. Suppose there are N independent bus routes passing a particular bus stop A which by different paths arrive at a particular destination point B. Let T,,(') (i = 1, 2, . . . N; n1, 2, . . .) be the instant at which the nth arrival of a bus on the ith route occurs at point A. We will assume that On(i) = T1+1'i'Tn"), (n = 1 2 . . .) are independent identically distributed positive random variables such that