This paper considers strictly periodic schedules for Cyclo-Static Dataflow Graphs (CSDFGs) where, for every task, only the start time of the first phase of the first iteration is fixed. This alternative CSDFG scheduling paradigm presents further computational optimization for throughput computations and buffer sizing. It also allows us to consider a wider range of DSP applications and real-time devices, such as time triggered architectures and bufferless software-defined Network-on-chips (NoCs), where the number of periodic executions could be limited. We propose a new framework that defines a necessary and sufficient condition for the existence of such schedules, and present throughput evaluation and buffer sizing use cases.