A Lagrangian algorithm for computing the optimal service rates in Jackson queueing networks

Jackson queueing-network optimization problems with state-dependent service rates can be decomposed into isolated node problems, each of which is equivalent ot an M/M/1 control model. The optimal service rate for each node is thus monotonically increasing as verified for the exponential models. By using the Lagrangian method, the relationship between the optimal service rates of adjacent states will be explicitly obtained. Moreover, a threshold state, above which the full service rate is optimal, will be specifically found. The existence of the threshold will lead to an efficient numerical algorithm that handles only a finite number of states rather than an infinite state space.