The setting for the problem discussed here is a service facility which is to be "time-shared" by two customers. A precise notion of a processing schedule, which prescribes the times at which the facility is available to each customer, is introduced. Associated with each schedule is the expected total waiting time of the two customers. The schedules which minimize this time are called optimum schedules and are determined here. A number of examples and extensions are given which indicate, the scope of the methods used. © 1968 Academic Press, Inc.