Threshold properties of the M/M/1 queue under T-policy with applications

In this paper, we consider a single server queueing system with a threshold control policy and its application for controlling the energy consumption of a computing server. If the number of customers in the system is less than a threshold, the service rate is set in a low value and it also can be switched to a high value once the number of customers reaches the threshold. We study the monotonicity, convexity or concavity properties of the key performance measures of the system such as the steady-state probability distribution, the expected number of customers in the system and in the queue, the expected sojourn time in the system and the waiting time in the queue, with respect to the threshold. Based on these properties, we further study a real-life problem regarding the energy consumption of a computing server. Numerical results show that the state-dependent service policy is a promising technique to balance the energy consumption and the quality of service of a computing server.