One of the key problems facing cloud applications is to reduce their energy consumption, which can increase the working lifetime of a machine, decrease the operation costs of cloud providers, and the environmental impact caused by power consumption. It is very important to design and evaluate an energy-efficient cloud. Recently, two open problems are raised in the literature: 1) what is the optimal solution (the lower bound) for the total energy consumption? and 2) what is the energy-efficiency for a scheduling algorithm? In this paper, we consider two major scheduling policies: 1) always power-on physical machines (PMs) once turning-on and 2) turning-off (hibernating) idle PMs, both with possible virtual machine migrations during evaluation. Focusing on compute-intensive applications on cloud, we propose analytical methods to settle down the two open problems. Our theoretical results are validated by experimental results in different scheduling scenarios and can be applied in cloud computing environments to help energy-efficient design.