Optimal 2-Facility Network Districting in the Presence of Queuing

This paper considers a districting problem for a demand-responsive service system in which queuing is allowed. Customers, located at the nodes of a transportation network, call in a Poisson manner, asking for on-scene service by a mobile service unit. Two such units service the entire network, with unit i(i = 1, 2) responsible for all nodes Nⁱ in its unique “service territory.” In response to a call from within N^ı, unit i, if available, is dispatched immediately to the customer; if the unit is busy with a previous customer, the call is dispatched in a FIFO manner. Each service territory, with its response unit, behaves as an independently operating M/G/1 queuing system. The problem addressed in this paper is determination of the optimal service territories, given fixed home locations for each of the service units, so as to minimize the average response time (queuing delay plus travel time) to a random customer. Exact results are obtained for limiting values of demand rate, and efficient heuristics are presented for arbitrary demand rates.