Optimal Dispatching of a Finite Capacity Shuttle under Delayed Information

We consider the discrete time problem of optimally dispatching a single, nite capacity shuttle in a two terminal system under incomplete state observations. Passengers arrive to each terminal and must be moved to the other terminal. The durations of successive shuttle trips follow a general distribution, but have a lower bound D 1 . There is an information delay of I time units between the two terminals. That is, if at one terminal, the shuttle controller observes the history of the arrival process at that terminal with no delay, while observations of the history of the arrival process at the other terminal are delayed by I units of time, where I D 1 . We impose convex holding costs, for waiting customers and a linear dispatching cost for each shuttle trip. Subject to the above information pattern, the shuttle controller must decide at each instant whether to dispatch the shuttle or to wait at least an additional unit of time. We prove that it is optimal to dispatch the shuttle from a given terminal if and only if the number of customers at that terminal exceeds a threshold which depends on the probability distribution of the number of customers in the other terminal. We show that the threshold is a monotone nonincreasing function of the most recent delayed observation. Finally, under linear holding costs, we derive one necessary and several su cient conditions for dispatching. These conditions reduce the computational e ort required to determine an optimal threshold policy. The signi cant role of transportation networks in today's society motivates the need to develop further insight into the fundamental issues of control and optimization problems associated with these networks. We see a need to address the issue of network control in the absence of complete state observations. An understanding of the e ects of partial information on optimal control policies will be very useful for e ectively designing and controlling networks in which incomplete (imperfect) information is a realistic consideration. As a rst modest step in this direction we focus on the e ect that delayed observations have on an optimal shuttle dispatching policy in a simple two-node transportation network. In discrete time, we examine a two terminal network with a single, nite capacity shuttle providing transportation between the terminals. Passengers arrive at either terminal and must be transported to the other terminal, whereupon they exit the system. At a given terminal at any time t, the controller's (shuttle dispatcher's) decisions are based on the following information: (i) the history of the arrival process to that terminal through time t and (ii) the history of the arrival process of the other terminal through time t I. By imposing a holding cost per passenger per unit time held at either terminal, we provide an incentive for prompt service. On the other hand, a dispatching cost is incurred by each shuttle trip, thus discouraging frequent dispatching. The objective is to characterize a shuttle dispatching policy which is a function of the abovementioned information and minimizes an expected discounted cost due to passenger waiting and the dispatching of the shuttle. Results for this type of network have implications for many existing systems, mass transit and shipment of goods being obvious examples. The queueing network considered here captures fundamental features of transportation networks because: (i) it models service occurring in batches of up to Q customers and (ii) it incorporates a switching cost which re ects the cost of initiating service. The transportation problem posed above was introduced by Ignall and Kolesar, (1972) and (1974), where various dispatching schemes were investigated for the case where the