## 2 Preliminaries  
### Basic Concept of Timed Petri Net  
A Petri net is a type of directed graph composed of three types of elements (places, transitions, and directed arcs). The directed arcs connect places to transitions or vice versa [19]. If the arc is an inhibitor, the directed arrow will change to an empty circle. In the real world, not all events occur immediately. Therefore, transitions that incorporate a period of time are considered in a Petri net. This is referred to as a timed Petri net. In this paper, two different types of transitions are presented. One represents immediate firing time and is denoted by a black bar while the other represents deterministic firing time and is denoted by an empty bar. Both types fall under the category of constant firing time. Formally, we have:  
\(\text{TPN}=(P,T,F,W,H,M_{0})\)[19; 20; 21; 27; 28], where  
1. \(P=[p_{1},p_{2},p_{3},\ldots p_{n}]\) which is a finite set of places and can be marked with tokens. It appears as a hollow circle, like \(p_{1},p_{2}\) or \(p_{3}\) of Figure 1.
2. \(T=\{t_{1},t_{2},t_{3},\ldots t_{n}\}\) which is a finite set of transitions and \(T=T_{imm}\cup T_{det}\). \(T_{imm}\) represents immediate firing time and \(T_{det}\) represents deterministic firing time. Hollow transition means a time delay called deterministic (\(T_{det}\)). It appears as an empty bar, like \(t_{1}\) of Figure 1. Solid transition means no time delay called immediate (\(T_{imm}\)). It appears as a solid bar, like \(t_{A11}\) or \(t_{A12}\) in Section 4.1.1 of the \(A_{1}\) timed Petri net model.  
3. The \(F\) is called the set of directed arcs and the characteristic \(F\) is \(F\subseteq(P\times T)\cup(T\times P)\). It is a directional line connecting a place and a transition.
4. \(W(\bullet)\) is marked as the weight in the arc. Furthermore, the weight is a nonnegative integer and the weight number can be omitted if the number is 1.
5. \(H\) is a set of inhibitor arcs and is only from \(p\) to \(t\) that the inhibitor arc connects a place to a transition. The characteristic can be written as \(H\subseteq P\times T\). The symbol of the inhibitor is the empty head circle of the directed arc (Figure 1). In Figure 1a, the arc connected to \(p_{3}\) has a small circle to indicate an inhibitor arc. When \(p_{3}\) has a token, the token in \(p_{1}\) cannot reach \(p_{2}\). In Figure 1b, when \(p_{3}\) has no token, the token in \(p_{1}\) can be enabled and will reach \(p_{2}\).
6. \(M_{0}\) means the initial marking in a Petri net.  
Deadlock [21; 29]: A Petri net (PN) can deadlock if there exists a reachable marking where no transition is enabled. According to the sub-structure formed by the Petri net, two structures may be generated one is a siphon and the other is a trap. The definitions are as follows. A place set \(S\), \(S\
eq\emptyset\) and \(S\subseteq P\). \(S\) is a siphon if \({}^{\bullet}S\subseteq S^{\bullet}\). A place set \(Q\), \(Q\
eq\emptyset\) and \(Q\subseteq P\). \(Q\) is a trap if \({}^{\bullet}Q\subseteq Q^{\bullet}\). Based on these two structural characteristics, two types of deadlocks (empty and full deadlocks) can be generated  
Empty deadlock: In a Petri net system (PNS) with a siphon net, if the PNS produces an empty deadlock, a subnet of the PNS generates fewer input tokens than output tokens.  
Full deadlock: In a Petri net system (PNS) with a trap net, if the PNS produces a full deadlock, a subnet of the PNS generates more input tokens than output tokens.