Due to the ubiquity of batch data processing in cloud computing, the related problem of scheduling malleable batch tasks and its extensions have received significant attention recently. In this paper, we consider a fundamental model where a set of n tasks is to be processed on C identical machines and each task is specified by a value, a workload, a deadline and a parallelism bound. Within the parallelism bound, the number of machines assigned to a task can vary over time without affecting its workload. For this model, we obtain two core results: a sufficient and necessary condition such that a set of tasks can be finished by their deadlines on C machines, and an algorithm to produce such a schedule. These core results provide a conceptual tool and an optimal scheduling algorithm that enable proposing new algorithmic analysis and design and improving existing algorithms under various objectives.