Optimal Dispatching of a Finite Capacity Shuttle

We consider the problem of determining the optimal operating policy of a two terminal shuttle with fixed capacity Q \le \infty . The passengers arrive at each terminal according to Poisson processes and are transported by a single carrier operating between the terminals. The interterminal travel time is a positive random variable with finite expectation. Under a fairly general cost structure, we show that the policy which minimizes the expected total discounted cost over infinite time horizon has the following form: Suppose the carrier is at one of the terminals with x passengers waiting there and y passengers waiting at the other terminal. Then the optimal policy is to dispatch the carrier if and only if x \ge G(y), where G(y) is a monotone decreasing control function. Furthermore, G(y) is always less than or equal to the carrier capacity Q. This control function can be approximated by the linear function G(y) = K - \betay.