Matrix approach to deadlock-free dispatching in multi-class finite buffer flowlines

For finite-buffer manufacturing systems, the major stability issue is ‘deadlock’, rather than ‘bounded-buffer-length stability.’ The paper introduces the concept of ‘system deadlock,’ defined rigorously in Petri net terms, and system operation with uninterrupted part-flow is characterised in terms of the absence of this condition. For a large class of finite-buffer multiclass re-entrant flowline systems an analysis of ‘circular waits’ yields necessary and sufficient conditions for the occurrence of ‘system deadlock’. This allows the formulation of a maximally permissive one-step-look-ahead deadlock-avoidance control policy for dispatching jobs, while maximizing the percent utilisation of resources. The result is a generalized kanban dispatching strategy, which is more general than the standard multi-class last buffer first serve (LBFS) dispatching strategies for finite buffer flowlines that typically under-utilize the resources. The problem of computational complexity associated with Petri net applications is overcome by using certain sub-matrices of the PN incidence matrix. Computationally efficient matrix techniques are given for implementing the deadlock-free dispatching policy.