# An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs
## 2 Basic Definitions and Notations
### (max,+) Algebra
The Kleene star is defined as \(e\oplus A\oplus A^{2}\oplus\cdots\) where \(e\) denotes the _tropical identity matrix_ of the same size as Matrix \(A\)[14,17]. The general standard autonomous equation form is the following:  
\[\vec{x}^{r}=\bar{A}\otimes\vec{x}^{r-1}, \tag{3}\]  
where \(\vec{x}^{r}=\{\vec{x}^{r},\cdots\vec{x}^{r+1-M_{0}}\}\) and \(M_{0}\) is the maximum initial markings of TEG \(\mathcal{N}\): \(\tilde{A}=\left( \begin{array}{cccc}\tilde{A}(1)&\cdots&\tilde{A}(M_{0})\\ e&\epsilon&\epsilon\\ e&e&\epsilon\end{array} \right)\). Generally, a standard autonomous equation's size increases in proportion to a maximum number of initial tokens greater than its original size. Therefore, it is necessary to simplify the model's dynamic behavior without changing it for calculation convenience. It is known that the two TEGs are said to be _equivalent_ if there is a bijective mapping that lets two corresponding transitions of two TEGs fire simultaneously from the initial TEG set to the subset of the transformed TEG [10]. To put it another way, any observation of the dynamic behavior of the corresponding transitions of the two equivalent TEGs is identical. This article extends the concept of equivalency by not distinguishing the initial TEG from the derived TEG. **Definition 1**. : _Two TEGs \(\mathcal{N}_{1}=(P_{1},T_{1},F_{1},H_{1},M_{01})\) and \(\mathcal{N}_{2}=(P_{2},T_{2},F_{2},H_{2},M_{02})\) are said to be equivalent when the following conditions are satisfied:_  
* _the set of the transaction of TEG_ \(\mathcal{N}\) _is a subset of the transaction of the other TEG_ \(\mathcal{N}_{2}\) _(_\(T_{1}\subseteq T_{2}\)_)._
* _there exists a bijective mapping that can fire the corresponding transactions of these two TEGs simultaneously._  
There are many ways to reduce the size of matrix \(\bar{A}\) such that the reduced TEG is equivalent to the original [10]. Using the first way, if the number of initial tokens of any place of \(t^{\star}\) of transition \(t\) is zero, we remove the column and the row for the transition. The other way depends on whether all elements of the column are \(\epsilon\). For example, the size of \(\bar{A}\) is originally \(12\times 12\) because Figure 1 has four transitions and, maximally, three tokens. Next, we can remove the corresponding columns and rows of \(t_{1}\) and \(t_{3}\) of \(\bar{A}_{1}\), \(\bar{A}_{2}\), and \(\bar{A}_{3}\) because the transitions have no downstream place that initially has a token. Finally, we eliminate the columns whose values are all \(\epsilon\). In conclusion, only \((t_{2}^{\prime},t_{4}^{\prime},t_{2}^{\prime-1},t_{4}^{\prime-1},t_{2}^{\prime -2})^{\prime}\) survives, so Equation (2) is transformed into a standard autonomous equation as follows:  
\[\bar{x}^{\prime}=\bar{A}\otimes\bar{x}^{\prime-1} \tag{4}\]  
\[\bar{A}=\left( \begin{array}{cccccc}\max\{h_{23},h_{23}^{\prime}\}+h_{34}+h_{ 41}+h_{12}&\epsilon&\epsilon&h_{43}+h_{34}+h_{41}+h_{12}&h_{12}+h_{21}\\ \max\{h_{23},h_{23}^{\prime}\}+h_{34}&\epsilon&\epsilon&h_{43}+h_{34}&\epsilon \\ e&e&\epsilon&\epsilon&\epsilon\\ \epsilon&e&\epsilon&\epsilon&\epsilon\\ \epsilon&e&\epsilon&\epsilon&\epsilon\\ \epsilon&e&\epsilon&e&\epsilon\\ \end{array} \right).\]  
If a standard autonomous equation is visualized in TEG form again, it becomes a special form of a TEG with exactly one token in every place, so it is not necessary to represent places and tokens in the TEG. For simplicity, we use a _precedence graph_, which maps the columns and rows of the matrix to the graph's nodes and maps the elements of the matrix to the weight of arcs in the graph. The precedence graph of TEG \(\mathcal{N}\) is defined as \(\mathcal{G}(\mathcal{N})=(\mathcal{V},\mathcal{A},\mathcal{W})\). In \(\mathcal{G}(\mathcal{N})\), \(\mathcal{V}\) is the surviving transition after conversion, \(\mathcal{A}\) is the set of places in which one token connects the transitions, and \(\mathcal{W}\) is the set of holding times.