# An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs
## 4 Stability Analysis of a TEG
The right TEG has an additional new cycle connecting them in the following way: decrease the cycle in the sequence of \(t_{b}\), the new place \(p_{bi}p_{io}\), \(t_{o}\), the path removing \(p_{io}\) from the cycle including \(p_{ai}\) and \(p_{io}\) (\(t_{o}\rightarrow\cdots\rightarrow t_{a}\to p_{ai}\to t_{i}\)), and the path removing \(p_{bi}\) from the cycle including \(p_{bi}\) and \(p_{ia}\) (\(p_{ia}\rightarrow\cdots\to t_{b}\)). Rule 4 does not change transitions. It merges places whose upstream transition, downstream transition, and token number are identical. Given that these merged places have the same upstream transition and downstream transition, the reachability does not change. That is, if and only if there was a cycle that a merged place \(p_{io}\) could make with \(t_{o}\rightarrow\cdots\to t_{i}\), all other merged places also make a cycle use the same path. We can also compose a cycle using the new place with the same path. Thus, if the original TEG has \(n\times m\) cycles because there are \(m\) different paths from \(t_{o}\) to \(t_{i}\) and \(n\) places are merged by Rule 4, the number of cycles decreases to \(m\) after the rule is applied. The cycle time of transformed TEG becomes the maximum cycle time of \(n\) merged cycles. The number of tokens in the corresponding cycle does not change. We can now comprehend the meaning of the proof of Lemma 5 using Figure 8. The first step merges places \(p_{23}\) and \(p_{23}^{\prime}\) by applying Rule 4, so the cycles \(\pi_{1}\) and \(\pi_{2}\) are combined into one cycle. The cycle time of the combined cycle is the same as the maximum of cycles \(\pi_{1}\) and \(\pi_{2}\), and its token numbers are unchanged. Other cycles \(\pi_{3}\) and \(\pi_{4}\) remain, so the number of cycles decreases to three. The next step decomposes the corresponding transitions from \(t_{2}\) and \(t_{4}\) by Rule 1. Consequently, cycles \(\pi_{3}\) and \(\pi_{4}\) increase the number of transitions and places, but the transformed TEG keeps the cycles and cycle times. The third step removes the places \(p_{12}\) and \(p_{34}\) using Rule 2. Cycle \(\pi_{3}\) becomes \(\bar{\pi}_{2}\), each of whose places has exactly one token. Cycle \(\pi_{2}\) still keeps the number of tokens and the sum of the holding times of places that make up the cycle, so the cycle times of \(\pi_{3}\) and \(\pi_{2}\) are the same. Similarly, \(\pi_{4}\) becomes \(\bar{\pi}_{3}\), and their cycle times are equal. In the final step, applying Rule 3 causes the place \(p_{41}p_{12}\) to disappear. As a result, we can remove all zero-tokened places in the combined cycle from the first step, and we call it \(\bar{\pi}_{1}\). The cycle time of \(\pi_{1}\) is still the maximum of \(\pi_{1}\) and \(\pi_{2}\), and the number of tokens is maintained as one. Additionally, this step creates a new cycle \(\bar{\pi}_{4}\) by connecting \(\bar{\pi}_{1}\) and \(\bar{\pi}_{3}\). The newly created \(\bar{\pi}_{4}\) has the cycle time of the sum of the maximum cycle time of cycles \(\pi_{1}\) and \(\pi_{2}\) plus \(\pi_{4}\). Its number of tokens is three, which is the sum of the number of tokens in the connected cycle \(\pi_{1}\) and \(\pi_{3}\). As such, Figure 8 satisfies Lemma 5. From the proof of Lemma 5, we can also derive Collarories 1-3. **Corollary 1**. : _If a cycle \(\tilde{\pi}\) of a precedence graph \(\mathcal{G}(\mathcal{N})\) corresponds to one cycle \(\pi\) of the original TEG \(\mathcal{N}\), its length is the same as the number of tokens of \(\pi\). By comparison, if a cycle \(\pi\) is the concatenation of more than two cycles \(\pi_{i},\cdots\pi_{j}\), its length equals the sum of the number of tokens in \(\pi_{i}\cdots\pi_{j}\)._  
**Corollary 2**. : _If a cycle \(\tilde{\pi}\) of a precedence graph \(\mathcal{G}(\mathcal{N})\) corresponds to one cycle \(\pi\) of the original TEG \(\mathcal{N}\), the sum of the holding times of the places it comprises is the same as the sum of the holding times of the places that make up \(\pi\).