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 ∈ [0, a]at station 1 is to be selected as a function of the state (x₁, x₂) where xiis the number of customers at stationiso as to minimize the expected total discounted or average cost corresponding to the instantaneous costc₁x₁ + c₂x₂. The optimal policy is of the formu=aoru=0according asx₁ < S(x₂) or x₁ ≥ S(X₂)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.