A simple proof of the optimality of the best N-policy in the M/G/l queueing control problem with removable server

Abstract  An M/G/l queueing system with removable server is considered. The following costs are incurred: a holding cost of h per unit time per customer in the system, a cost of r_l(r₂) per unit time when the service mechanism is on (off) and a fixed cost of K₁,(K₂ ,) for turning it on (off).
In this paper we shall give a very simple proof for the well known and intuitively obvious fact that the best N-policy is optimal for the average cost criterion among the class of all policies by first proving that the average cost formula of that policy and its relative cost function satisfy the optimality equation for the average cost criterion. The optimality of the best N-policy is then an immediate consequence.