In a modern data center (DC), a large majority of costs arise from energy consumption. The most popular technique used to mitigate this issue is virtualization and more precisely virtual machine (VM) consolidation. Although consolidation may increase server usage by about 5-10%, it is difficult to actually witness server loads greater than 50%. By analyzing the traces from our cloud provider partner, confirmed by previous research work, we have identified that some VMs have sporadic moments of data computation followed by large periods of idleness. These VMs often hinder the consolidation system which cannot further increase the energy efficiency of the DC. In this paper we propose a novel DC power management system called Drowsy-DC, which is able to identify the aforementioned VMs which have matching patterns of idleness. These VMs can thus be colocated on the same server so that their idle periods are exploited to put the server to a low power mode (suspend to RAM) until some data computation is required. While introducing a negligible overhead, our system is able to significantly improve any VM consolidation system; evaluations showed improvements up to 81% and more when compared to OpenStack Neat.