# An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs
## 4 Stability Analysis of a TEG
If \(\pi\) is derived from the concatenation of multiple cycles \(\pi_{i},\cdots\pi_{j}\), the sum of the holding times of all places constituting \(\tilde{\pi}\) equals the holding times of the places included in \(\pi_{i},\cdots\pi_{j}\). However, if there are places merged by Rule 4 among \(\pi_{i},\cdots\pi_{j}\), only the maximum value is added without adding their individual values._  
**Corollary 3**. : _Rules 1-4 do not change the reachability between transitions._  
Proof. : Lemma 5 shows that the reachability is unchanged except in the case of Rule 3. Rule 3 eliminates the place directly connecting \(t_{o}\) from \(t_{j}\), so we suspect that \(t_{o}\) may no longer reach \(t_{i}\). However, given that every transition in a live autonomous TEG is included in any cycle, \(t_{i}\) has a path to \({}^{\bullet\bullet}t_{i}\) and \({}^{\bullet\bullet}t_{i}\) can connect to \(t_{o}\) via new places. Therefore, there is a path, \(t_{i}\rightarrow\cdots\rightarrow\ ^{\bullet\bullet}t_{i}\rightarrow\text{new place} \to t_{o}\). In conclusion, Rule 3 also does not change the reachability. It is known that the stationary behavior is \(d\)-cyclic if the transitions of a strongly connected TEG fire under the earliest starting strategy [3]. From the lemmas and corollaries proved thus far, we can find the cyclicity in a live autonomous TEG as Theorem 1. **Theorem 1**. : _When a live autonomous TEG \(\mathcal{N}\) is strongly connected, the cyclicity of a TEG \(\mathcal{N}\) is the LCM of the cyclicities of all maximal strongly connected subgraphs of \(\mathcal{N}^{c}\), which is a critical graph consisting only of critical cycles. The cyclicity of each maximal strongly connected subgraph of \(\mathcal{N}^{c}\) is the GCD of the number of tokens of all its cycles. In other words, if the cyclicity of a TEG \(\mathcal{N}\) is \(d\), it has an asymptotic \(d\)-cyclic schedule, and, if it is an asymptotic one-cyclic schedule, this system is said to be stable or to have stability._  
Proof. : Definition 2 said that the one-cyclic schedule is stable, so this proof focuses on whether the TEG's cyclicity calculation is correct. Additionally, it is known that the asymptotic cyclicity of a strongly connected precedence graph \(\mathcal{G}(\mathcal{N})\) can be calculated by the LCM of the cyclicities of all maximal strongly connected subgraphs of its critical graph \(\mathcal{G}^{c}(\mathcal{N})\)[10]. The cyclicity of each strongly connected subgraph of \(\mathcal{G}^{c}(\mathcal{N})\) is the GCD of the number of arcs of all its cycles. Therefore, we prove this theorem in the following order:  
1. If and only if there exists a critical cycle \(\tilde{\pi}\) in \(\mathcal{G}(\mathcal{N})\), there exists a corresponding critical cycle and/or a connection of the corresponding critical cycles in \(\mathcal{N}\). 2. \(\pi_{i}\) and \(\pi_{j}\) belong to the same component of \(\mathcal{N}^{c}\) if and only if the corresponding cycle of \(\pi_{i}\) and \(\pi_{j}\) and their connection in \(\mathcal{G}^{c}(\mathcal{N})\) also belong to the same component. 3. The GCD of the number of tokens of all its cycles in each maximal strongly connected subgraph of \(\mathcal{N}^{c}\) is the same as the GCD of the number of arcs of all its cycles in each maximal strongly connected subgraph of \(\mathcal{G}(\mathcal{N})\). 1. First, we prove that \(\mathcal{G}(\mathcal{N})\) has a critical cycle \(\tilde{\pi}\) if and only if its corresponding cycle is critical and the cycle including it is critical in \(\mathcal{N}\). A critical cycle is a cycle that has the maximum cycle time, calculated by the sum of the holding times of the places divided by the number of tokens in the cycle. The number of tokens and the sum of holding times of each cycle remain after the transformation, according to Corollaries 1 and 2 with the exception of Rule 4. Even Rule 4 does not change which cycle is critical because the cycle time after the transformation depends on larger cycle times among merged cycles by applying Rule 4.