The waiting time distribution for the random order service M/M/1 queue

The M/M/1 queue is considered in the case in which customers are served in random order. A formula is obtained for the distribution of the waiting time w in the stationary state. The formula is used to show that P9w>t)..t.5/6exp(..t..t1/3) as t.., with the constants .,., and . expressed as functions of the traffic intensity .. The distribution of w for the random order discipline is compared to that of the first in, first out discipline.