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. We show that this nonlinear program provides a new, alternative derivation of an upper bound due to Harel (1988) for the Erlang loss probability. We also interpret the optimality conditions as optimizing the behaviour of decentralized agents assigned to units. Using optimal solution information in an auxiliary pricing problem, we construct an easily computable functional approximation to the dynamic programming value function. The resulting policy is computationally efficient and produces superior economic performance as compared with other policies and the upper bound. Furthermore, it provides a methodologically grounded solution to the firm's internal pricing problem.
Adelman REF applied a closed queueing network to propose an internal pricing mechanism for managing a fleet of service units, and also used a nonlinear flow model to discuss the price-based policy for the vehicle redistribution.
1 2 Price-Directed Control of a Closed Logistics Queueing Network
Computer Science Mathematics