Batch Arrival Multiserver Queue with State-Dependent Setup for Energy-Saving Data Center

Abstract

Queues with setup time are extensively studied because they have application in performance evaluation of power-saving data centers. In data centers, there are a huge number of servers which consume a large amount of energy. In the current technology, an idle server still consumes about 60% of the energy when it is busy. Thus, a simple way to save energy is to turn off idle servers. However, when there are some waiting jobs, we have to turn on the OFF servers in order to reduce the waiting time. A server needs some setup time to be active during which it consumes energy but cannot process jobs. Therefore, there exists a trade-off between power consumption and delay performance. Gandhi et al. (Eval Rev 38:48–50, 2010; Perform Eval 67:1123–1138, 2010) analyze this trade-off using an M/M/c queue with staggered setup (one server in setup at a time). In this paper, using an alternative approach, we obtain generating functions for the joint stationary distribution of the number of active servers and that of jobs in the system for a more general model with batch arrivals and state-dependent setup time. We further obtain moments for the joint queue length. Numerical results reveal that under the same traffic intensity, the mean power consumption decreases with the mean batch size. One of the main theoretical contributions is a new conditional decomposition formula showing that the number of waiting customers under the condition that all servers are busy can be decomposed to the sum of two independent random variables with clear physical interpretation.