Sequential Mutual Exclusion

Our study of Petri net synthesis for manufacturing systems begins with the classification of places in a Petri net model. The modeling approach in which places are used to model operation processes and the availability of resources, and in which transitions are used to model the start and/or end of operations makes such a distinction among places possible. Section 2.3 has divided places into three classes: operation places, fixed resource places, and variable resource places, called A, B, and C, respectively. Such a division depends upon a modeler’s choice in many cases. A-places represent the operation processes and are not initially marked. B-places model the availability of fixed resources like robots and machines. C-places model the availability of variable resources such as fixtures or pallets, whose number needs to be specified. The number of the initial tokens in C-places can influence the qualitative properties of some systems. These systems contain resources which are shared by several sequential processes. If too many parts are allowed to enter such a system in a given period, the system may become deadlocked or blocked. Conceptualizing such phenomena, we define sequential mutual exclusion (SME) in the context of Petri net theory and find the underlying relationship between a net with such a structure and the net’s initial marking over C-places. Also, to capture characteristics of a system with the resource shared by independent processes, we have defined and discussed the parallel mutual exclusion (PME) concept in Chapter 3. Each SME structure can be visualized as the sequential composition of several PMEs. The token capacity between two consecutive PMEs is introduced and, roughly speaking, is the number of tokens which can be provided by the first PME without consumption by the second. Each token capacity is calculated in a reduced net which eliminates the shared resource. The results are used to determine the number of initial tokens in each C-place. This number can be interpreted as the number of jobs which can be dispatched to the system in the same interval without blocking the system.