Analyzing the timing properties of asynchronous systems is essential for characterizing their performance and power. Previous work on timing showed that such systems under and-causality and fixed delay exhibit periodicity properties. We give a different graph-based rigorous proof of the exact timing behavior of more general classes of such systems, and conclude their exact periodicity property, where each of the signal transition will occur with the same period after finite occurrences. We established our results under weaker assumption about system connectivity/topology, and this paper provides the theoretical foundation, for the exact periodicity property to be applied and exploited in circuits containing a combination of synchronous and asynchronous components. We provide simulation-based results for several typical asynchronous circuit topologies to quantify this time period in practical circuits. We also provide an extension of our analysis and methods to the case of bounded delay systems. A key result that is a consequence of our analysis is that asynchronous circuits can be integrated with synchronous logic via a metastability-free interface, thereby eliminating the high-overhead synchronizers when an asynchronous circuit is fully surrounded by synchronous logic.