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Deciding the liveness for a subclass of weighted Petri nets based on structurally circular wait

Weighted Petri nets as a kind of formal language are widely used to model and verify discrete event systems related to resource allocation like flexible manufacturing systems. System of Simple Sequential Processes with Multi-Resources (S3PMR, a subclass of weighted Petri nets and an important extension to the well-known System of Simple Sequential Processes with Resources, can model many discrete event systems in which (1) multiple processes may run in parallel and (2) each execution step of each process may use multiple units from multiple resource types. This paper gives a necessary and sufficient condition for the liveness of S3PMR. A new structural concept called Structurally Circular Wait (SCW) is proposed for S3PMR. Blocking Marking (BM) associated with an SCW is defined. It is proven that a marked S3PMR is live if and only if each SCW has no BM. We use an example of multi-processor system-on-chip to show that SCW and BM can precisely characterise the (partial) deadlocks for S3PMR. Simultaneously, two examples are used to show the advantages of SCW in preventing deadlocks of S3PMR. These results are significant for the further research on dealing with the deadlock problem.