Periodic structure and liveness of Petri net rings

For k⩾2, let R^k be the ring consisting of k identical copies C₀, C₁, ..., C_k-1 of a component C, where C is given as a Petri net. Assume that all components of R^k except possibly C₀ have an identical initial marking. We consider the problem of testing whether all rings R^k , k⩾2, are live, for the case when the rings are either state machines or marked graphs. We present various sufficient conditions under which all rings are live, and discuss the complexity issue of testing the liveness of marked graph rings. Such conditions can greatly simplify the analysis of large scale process rings given as a Petri net