Cost Sharing for Capacity Transfer in Cooperating Queueing Systems

We study the problem where independent operators of queueing systems cooperate to generate a win–win solution through capacity transfer among each other. We consider two types of costs: the congestion cost in the queueing system and the capacity transfer cost, and two types of queueing systems: M/M/1 and M/M/s. Service rates are considered to be capacities in M/M/1 and are assumed to be continuous, while numbers of servers are capacities in M/M/s. For the capacity transfer problem in M/M/1, we formulate it as a convex optimization problem and identify a cost†sharing scheme which belongs to the core of the corresponding cooperative game. The special case with no transfer cost is also discussed. For the capacity transfer problem in M/M/s, we formulate it as a nonlinear integer optimization problem, which we refer to as the server transfer problem. We first develop a marginal analysis algorithm to solve this problem when the unit transfer costs are equal among agents, and then propose a cost†sharing rule which is shown to be in the core of the corresponding game. For the more general case with unequal unit transfer costs, we first show that the core of the corresponding game is non†empty. Then, we propose a greedy heuristic to find approximate solutions and design cost allocations rules for the corresponding game. Finally, we conduct numerical studies to evaluate the performance of the proposed greedy heuristic and the proposed cost allocation rules, and examine the value of capacity transfer.