Two types of customer behavior are considered: (1) if a customer is acquired for service before he has waited a time τ 0 , he remains in the queue until served irrespective of whether or not his total waiting time exceeds τ 0 . Only those customers who wait for a time τ 0 without being acquired for service become “lost” customers, and (2) a customer whose total waiting time is τ 0 becomes a lost customer irrespective of whether he is acquired for service or not.

A waiting-line problem in which the arrival and service rates are of Poisson type is considered subject to the additional conditions that (a) each customer is limited to a fixed time during which he is available for service and after which he automatically leaves the system and (b) customers are selected at random from among those in the system rather than on a first-come-first-served basis. The equilibrium distribution of the length of the queue and the expected rate at which customers escape service are obtained. The expected escape rate is plotted as a function of the limited time of customer availability for selected arrival and service rates and compared with the result of Monte Carlo trials of the model.