Computing the concurrency threshold of sound free-choice workflow nets

Workflow graphs extend classical flow charts with concurrent fork and join nodes. They constitute the core of business processing languages such as BPMN or UML Activity Diagrams. The activities of a workflow graph are executed by humans or machines, generically called resources. If concurrent activities cannot be executed in parallel by lack of resources, the time needed to execute the workflow increases. We study the problem of computing the minimal number of resources necessary to fully exploit the concurrency of a given workflow, and execute it as fast as possible (i.e., as fast as with unlimited resources). We model this problem using free-choice Petri nets, which are known to be equivalent to workflow graphs. We analyze the computational complexity of two versions of the problem: computing the resource and concurrency thresholds. We use the results to design an algorithm to approximate the concurrency threshold, and evaluate it on a benchmark suite of 642 industrial examples. We show that it performs very well in practice: It always provides the exact value, and never takes more than 30 milliseconds for any workflow, even for those with a huge number of reachable markings.