Price-directed control of a closed logistics queueing network

Motivated by one of the leading intermodal logistics suppliers in the United States, we consider an internal pricing mechanism for managing a fleet of service units (shipping containers) flowing in a closed queueing network. Nodes represent geographic locations, and arcs represent travel between them. Customer requests for arcs arrive over time, and the problem is to find an accept/reject policy that maximizes the long-run time average reward rate from accepting requests. We formulate the problem as a semi-Markov decision process and give a simple linear program that provides an upper bound on the optimal reward rate. Using Palm calculus, we derive a nonlinear program that approximately captures queueing and stockout effects on the network. Using its optimal Lagrange multipliers, we construct a simple functional approximation to the dynamic programming value function. The resulting policy is computationally efficient and produces superior economic performance as compared with other policies. Furthermore, it provides a methodologically grounded solution to the firm's internal pricing problem.