Optimal control of service in tandem queues

Customers arrive in a Poisson stream into a network consisting of two
M/M/1
service stations in tandem. The service rate
u \in [0, a]
at station 1 is to be selected as a function of the state (
x_{1}, x_{2}
) where x
i
is the number of customers at station
i
so as to minimize the expected total discounted or average cost corresponding to the instantaneous cost
c_{1}x_{1} + c_{2}x_{2}
. The optimal policy is of the form
u=a
or
u=0
according as
x_{1} < S(x_{2}) or x_{1} \geq S(X_{2})
and
S
is 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.