A monotonic property of the optimal admission control to an M/M/1 queue under periodic observations with average cost criterion

We consider the problem of admission control to an M/M/1 queue under periodic
observations with average cost criterion. The admission controller receives
the system state information every ø :th second and can accordingly adjust the
acceptance probability for customers who arrive before the next state information
update instance. For a period of ø seconds, the cost is a linear function of the
time average of customer populations and the total number of served customers
in that period. The objective is to Ønd a stationary deterministic control policy
that minimizes the long run average cost. The problem is formulated as a discrete
time Markov decision process whose states are fully observable. By taking the
control period ø to 0 or to 1, the model in question generalizes two classical
queueing control problems: the open and the closed loop admission control to an
M/M/1 queue. We show that the optimal policy is to admit customers with a
non-increasing probability with respect to the observed number of customers in
the system. Numerical examples are also given.