Max-Min Optimality of Service Rate Control in Closed Queueing Networks

In this technical note, we discuss the optimality properties of service rate control in closed Jackson networks. We prove that when the cost function is linear to a particular service rate, the system performance is monotonic w.r.t. (with respect to) that service rate and the optimal value of that service rate can be either maximum or minimum (we call it Max-Min optimality); When the second-order derivative of the cost function w.r.t. a particular service rate is always positive (negative), which makes the cost function strictly convex (concave), the optimal value of such service rate for the performance maximization (minimization) problem can be either maximum or minimum. To the best of our knowledge, this is the most general result for the optimality of service rates in closed Jackson networks and all the previous works only involve the first conclusion. Moreover, our result is also valid for both the state-dependent and load-dependent service rates, under both the time-average and customer-average performance criteria.