Matrix approach to deadlock avoidance of dispatching in multi-class finite buffer reentrant flow lines

For a very general class of finite-buffer multi-class reentrant flow lines, necessary and sufficient conditions are given for the absence of deadlock in terms of circular wait relations. The result is a multi-class last buffer first serve (LBFS) dispatching policy for finite buffer flow lines. The notion of so-called "critical siphons", as well as the novel notion of "critical traps" are introduced in this paper. Petri net (PN) techniques are used in the analysis. Computationally efficient matrix techniques are given for implementing a multiclass dispatching policy that is guaranteed not only to avoid deadlock, but allows one to obtain efficient utilization of the resources in multi-class reentrant flow lines.