Optimal control of service in tandem queues

Customers arrive in a Poisson stream into a network consisting of twoM/M/1service stations in tandem. The service rateu in [0, a]at station 1 is to be selected as a function of the state (x_{1}, x_{2}) where xiis the number of customers at stationiso as to minimize the expected total discounted or average cost corresponding to the instantaneous costc_{1}x_{1} + c_{2}x_{2}. The optimal policy is of the formu=aoru=0according asx_{1} < S(x_{2}) or x_{1} geq S(X_{2})andSis a switching function. For the case of discounted cost, the optimal process can be nonergodic, but it is ergodic for the case of average cost.