Optimally scheduling N customer arrival times for a single-server system

In this paper, we investigate the problem of scheduling a finite number of customer arrivals for a single-server system. Under the assumptions that the customers arrive at the system by scheduled appointments and they request random amount of times for service, the question is: what are the ‘best’ arrival times so that the total operating cost is minimized? This type of queueing problem has important applications in production systems as well as in service systems. Using phase-type distribution functions and matrix algebraic manipulation, we are able to obtain recursive expressions for the customer flow-time distributions, from which the mean and variance, or higher moments, can be easily computed. A computational procedure is developed that efficiently evaluates the mean flow-times for large number of customers that need to be scheduled. This procedure is integrated with a nonlinear program that determines the optimal customer arrival times.