Augmented Marked Graphs and the Analysis of Shared Resource Systems

Augmented marked graphs were first introduced in 1997 (Chu & Xie, 1997). They are not well known as compared to other sub-classes of Petri nets such as free-choice nets (Desel & Esparza, 1995), and the properties of augmented marked graphs are not studied extensively. However, augmented marked graphs possess a structure which is desirable for modelling shared resources, and for this reason, they are often used in modelling shared resource systems, such as manufacturing systems (Chu & Xie, 1997; Zhou & Venkatesh, 1999; Jeng et al., 2000; Jeng et al., 2002; Huang et al., 2003; Cheung & Chow, 2005). There are a few published works on augmented marked graphs. Based on siphons and mathematical programming, Chu and Xie proposed a necessary and sufficient condition of live and reversible augmented marked graphs, which checks the existence of potential deadlocks (Chu & Xie, 1997). However, this involves the flow of tokens during execution and cannot be checked simply by looking into the structure. Chu and Xie also proposed another siphon-based characterisation for live and reversible augmented marked graphs but it provides a sufficient condition only. On the other hand, the boundedness and conservativeness of augmented marked graphs were not investigated. In the literature, apart from (Chu & Xie, 1997), the studies of augmented marked graphs mainly focus on their property-preserving synthesis or composition. Jeng et al. proposed a synthesis of process nets (covering augmented marked graphs) for manufacturing system design, where the condition of liveness and reversibility are based on siphons and the firability of transitions (Jeng et al., 2000; Jeng et al., 2002). Huang et al. also investigated the composition of augmented marked graphs so that properties such as liveness, boundedness and reversibility can be preserved (Huang et al., 2003). In our earlier works on augmented marked graphs, we proposed characterisations for their liveness, boundedness, reversibility and conservativeness and applied the characterisations to the analysis and design of manufacturing systems, object-oriented systems and shared resource systems (Cheung, 2004; Cheung, 2005; Cheung & Chow, 2005a; Cheung & Chow, 2005b; Cheung & Chow, 2005c; Cheung, 2006; Cheung & Chow, 2006; Cheung et al., 2006; Cheung, 2007). This paper consolidates our previous works with a special focus on the properties of augmented marked graphs. First, we provide a number of characterisations for live and reversible augmented marked graphs, based on siphons and cycles. In particular, a property called R-inclusion property is