Optimal Operation of an M/M/2 Queue with Removable Servers

Abstract

The form of the optimal policy is investigated in an average cost, infinite horizon M/M/2 problem where the number of servers working can be adjusted at arrival or service completion epochs. The costs considered are linear holding costs, linear servers' wages and set-up (shut-down) charges per server turned on (off). It is shown that an optimal policy has a hysteresis form characterized by four parameters, R1, R2, S0, and S1, denoting numbers of customers in the system when the number of working servers should be adjusted upward to 1, 2, and downward to 0, 1, respectively. Allowing for the possibilities that S0 = −1 (or S1 = −1) denoting the fact that the number of working servers are never adjusted downward to 0 (or 1), the following relationships between R1, R2, S0, and S1 are shown: (i) Ri ≥ i, i = 1, 2, (ii) R2 ≥ R1, (iii) −1 ≤ S0 ≤ S1, (iv) R2 > S1, and (v) S0 ≤ 0. Such a policy has the property that it never reacts to the arrival of a customer by immediately reducing the number of working servers and never reacts to a service completion by immediately increasing the number of working servers.

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