This paper studies a generalized busy-time scheduling model on heterogeneous machines. The input to the model includes a set of jobs and a set of machine types. Each job has a size and a time interval during which it should be processed. Each job is to be placed on a machine for execution. Different types of machines have distinct capacities and cost rates. The total size of the jobs running on a machine must always be kept within the machine's capacity, giving rise to placement restrictions for jobs of various sizes among the machine types. Each machine used is charged according to the time duration in which it is busy, i.e., it is processing jobs. The objective is to schedule the jobs onto machines to minimize the total cost of all the machines used. We develop an $O(1)$-approximation algorithm in the offline setting and an $O(μ)$-competitive algorithm in the online setting (where $μ$ is the max/min job length ratio), both of which are asymptotically optimal.