# An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs
## 4 Stability Analysis of a TEG
The cyclicity is the LCM of the cyclicity of all maximal strongly connected subgraphs of subgraph \(\mathcal{G}^{c}\). Remember that each maximal strongly connected subgraph's cyclicity is equal to the GCD of the arc number of each cycle constituting the subgraph [10]. All maximal strongly connected subgraphs of \(\mathcal{G}^{c}(\mathcal{N})\) have the corresponding connected subgraph of \(\mathcal{N}^{c}\). The number of arcs of each cycle constituting the subgraph in \(\mathcal{G}^{c}(\mathcal{N})\) is the same as the number of tokens in its cycle in \(\mathcal{N}^{c}\). Therefore, the GCD of the number of arcs in \(\mathcal{G}^{c}(\mathcal{N})\) can be interpreted as the GCD of the number of tokens of the original TEG. Although the maximal strongly connected subgraph of the precedence graph has not only the critical cycle but also the connection to the critical cycle, the existence of cycles that exist in only \(\mathcal{G}^{c}(\mathcal{N})\) does not affect the GCD because \(GCD(\mu_{i}+\mu_{j},\mu_{j})=GCD(\mu_{i},\mu_{j})\) by the Euclidean algorithm. A TEG \(\mathcal{N}\)'s cyclicity is the LCM of the cyclities of all maximal strongly connected subgraphs of its critical graph \(\mathcal{N}^{c}\). Theorem 3.1 lets us know that the cycle times of \(\pi_{1}\), \(\pi_{2}\), \(\pi_{3}\) and \(\pi_{4}\) in Figure 1 are \(h_{12}+h_{23}+h_{34}+h_{41}\), \(h_{12}+h_{23}^{\prime}+h_{34}+h_{41}\), \(h_{12}+h_{23}^{\prime}+h_{34}+h_{41}\), \(h_{12}+h_{21}\), and \(h_{34}+h_{43}\), respectively. If \(\pi_{1}\) or \(\pi_{2}\) is a critical cycle, \(\bar{\pi}_{1}\) in Figure 2 is a critical cycle, and its cycle time is \(h_{12}+\max\{h_{23},h_{23}^{\prime}\}+h_{34}+h_{41}\). Cycles \(\pi_{3}\) and \(\pi_{4}\) are transformed into \(\bar{\pi}_{2}\) and \(\pi_{3}\), respectively, and their cycle times are the same even after the transformation, so the critical cycle does not change. Finally, for \(\pi_{4}\) to become a critical cycle, \(\pi_{4}\) must be a critical cycle, and, additionally, \(\pi_{1}\) or \(\pi_{2}\) must be a critical cycle. Table 1 summarizes which cycle in Figure 8 becomes the critical cycle caused by each critical cycle in Figure 1, as well as each cycle's cyclicity. If cycles \(\pi_{1}\) or \(\pi_{2}\) are included in the critical cycle, \(\pi_{1}\) is included in the critical cycle, and its cyclicity is one. When \(\pi_{3}\) or \(\pi_{4}\) is an only critical cycle, the number of tokens in each cycle is three and two, so each cycle's cyclicity is also three and two, respectively. In addition, if \(\pi_{4}\) is a critical cycle with \(\pi_{1}\) or \(\pi_{2}\) together, these cycles are connected, so the cyclicity is also one because of the GCD of the number of tokens, one and three. In this case, \(\pi_{1}\), \(\pi_{3}\) and \(\pi_{4}\) are critical cycles together; they are connected to each other; and their numbers of tokens are one, two, and three, so their GCD, or the cyclicity is one. If \(\pi_{3}\) and \(\pi_{4}\) are critical cycles at the same time, its cyclicity is six because the LCM of token numbers, three and two, and the two cycles are disconnected. In conclusion, cycles \(\pi_{1}\) or \(\pi_{2}\) should be a critical cycle to have a steady schedule in Figure 1.