Fewer Monitors and More Efficient Controllability for Deadlock Control in S3 PGR 2 (Systems of Simple Sequential Processes with General Resource Requirements)

Concurrent systems such as distributed operating systems, distributed database systems, flexible manufacturing systems (FMS) etc. employ multiple processors to speed up the execution. However, to fully utilize the resources involved, the competition for resources among processes is strong and often leads to deadlocks where the whole system completely stops. Petri nets have been widely used to model concurrent systems and the associated deadlocks, which arise due to insufficiently marked siphons. A siphon is a structure object or a set of places; once these places become unmarked, they stay so afterwards making their output transitions permanently dead. To avoid deadlocks, monitors and control arcs are added upon problematic siphons, the number of which grows exponentially with the size of the net modeling the FMS. Li and Zhou relieved the problem by adding monitors only to elementary siphons. First, they test the marking linear inequality (MLI). If it fails, then they perform a linear integer programming test which takes exponential time. Thus, the MLI test is only a sufficient (not necessary) one. For systems of simple sequential processes with general resource requirements, there is one additional problem. That is, even though a siphon is not a dependent siphon based on the definition by Li and Zhou, the siphon may become controlled after controlling some elementary siphons. We develop new theory to turn this elementary siphon into a dependent one, thus reducing the number of monitors and simplifying the control hardware. This makes the uncontrolled model less disturbed and hence the controlled system becomes more permissive. Furthermore, we derive the exact controllability (both sufficient and necessary) so that the subsequent integer programming test can be eliminated. As a result, the total time complexity to check controllability of all strongly dependent siphons is reduced from exponential to linear if all n=2 strongly dependent siphons need to be verified against our new MLI test, the number of which is polynomial.