1.10. Customers arrive at a facility at random according to a Poisson process of rate A. There is a waiting time cost of c per customer per unit time. The customers gather at the facility and are processed or dispatched in groups at fixed times T, 2T, 3T, .... There is a dispatch cost of K. The process is depicted in the following graph

(a) What is the total dispatch cost during the first cycle from time 0 to time Ti

(b) What is the mean total customer waiting cost during the first cycle?

(c) What is the mean total customer waiting + dispatch cost per unit time during the first cycle?

(d) What value of T minimizes this mean cost per unit time?

Number of Customers Waiting T Time 2T Figure 1.1 The number of customers in a dispatching system as a function of time.