With the increase in the scale of cloud data centers, more attention is being focused on the issue of energy conservation. In order to achieve greener, more efficient computing in cloud data centers, in this paper, we propose an energy-efficient Virtual Machine (VM) allocation strategy with an asynchronous multi-sleep mode and an adaptive task-migration scheme. The VMs hosted in a virtual cluster are divided into two modules, namely, Module I and Module II. The VMs in Module I are always awake, whereas the VMs in Module II will go to sleep independently, if possible. Accordingly, a queuing model with a partial asynchronous multiple vacations is established to capture the working principle of the proposed strategy. Using the method of a matrix-geometric solution, performance measures in terms of the average response time of tasks and the energy saving rate of the system are mathematically derived. Numerical experiments with analysis and simulation are provided to validate the proposed VM allocation strategy and to estimate the influence of system parameters on performance measures. Finally, a system cost function is constructed to trade off different performance measures, and an intelligent searching algorithm is employed to optimize the number of VMs in Module II and the sleeping parameter in the same time.