Batch Arrival Multiserver Queue with Setup Time

Abstract

Queues with setup time are extensively studied because they have application in performance evaluation of power-saving data centers. In a data center, 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 its peak processing a job. Thus, the only way to save energy is to turn off servers which are not processing a job. However, when there are some waiting jobs, we have to turn on the OFF servers. A server needs some setup time to be active during which it consumes energy but cannot process a job. Therefore, there exists a trade-off between power consumption and delay performance. Gandhi et al. \cite{Gandhi10a,Gandhi10} 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 queue size. Numerical results reveal that keeping the same traffic intensity, the mean power consumption decreases with the mean batch size for the case of fixed batch size. One of the main theoretical contribution 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 where the first is the same quantity in the corresponding model without setup time while the second is the number of waiting customers before an arbitrary customer.