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Decentralized control of service rates in a closed Jackson network

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. Assume that both costs are time-homogeneous, and that the operating cost is a linear function of the service rate. Allow, however, both costs to be arbitrary functions of the number of jobs at the node. The objective is to minimize the time-average expected total cost. We show 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. The model also allows additional constraints (on average delay or throughput, for instance) to be imposed at each node. In this case, the optimal threshold control is adjusted by adding a set of randomized points, the number of which equals the number of additional constraints.