Decentralized control of service rates in a closed Jackson network

The authors consider a closed Jackson network with M nodes. The service rate at each node is controllable in a decentralized manner, i.e., it will be controlled based on local information extracted from that node only. For each node, there is a holding cost and an operating cost. It is assumed that both costs are time-homogeneous, and that the operating cost is a linear function of the service rate. However, both costs are arbitrary functions of the number of jobs at the node. The objective is to minimize the time-average expected total cost. It is shown that there exists an optimal control characterized by a set of thresholds (one for each node), such that it is optimal for each node to serve at zero rates if the number of jobs there is below or at the threshold, and serve at maximum allowed rates when the number of jobs exceeds the threshold. This structure of the optimal control is identified by using the product-form solution of the Jackson network and the duality theory of linear programming