The highly dynamic nature of the cloud environment leads to a time-varying resource utilization and the cloud provider can potentially accommodate secondary jobs with the remaining resource. To better implement the idea of resource reutilization in the cloud environment, the problem of secondary job scheduling with deadlines under time-varying resource capacity is considered in this paper. A transformation is proposed to reduce the offline problem with time-varying processor capacity to that with constant capacity. For online scheduling of underloaded system, it is shown that the earliest deadline first (EDF) scheduling algorithm achieves competitive ratio 1. For the overloaded system, an online scheduling algorithm V-Dover is proposed with asymptotically optimal competitive ratio when a certain admissibility condition holds. It is further shown that, in the absence of the admissibility condition, no online scheduling algorithm exists with a positive competitive ratio. Simulation results are presented to illustrate the performance advantage of the proposed V-Dover algorithm.