# An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs

## Abstract

**:**

## 1. Introduction

## 2. Basic Definitions and Notations

#### 2.1. Timed Event Graphs

- ${}^{\u2022}t=\left\{p\right|(p,t)\in F\}$: the set of upstream places in transition t
- ${t}^{\u2022}=\left\{p\right|(t,p)\in F\}$: the set of downstream places in transition t
- ${}^{\u2022}p=\left\{t\right|(t,p)\in F\}$: the set of upstream transitions in place p
- ${p}^{\u2022}=\left\{t\right|(p,t)\in F\}$: the set of downstream transitions in place p

#### 2.2. (max,+) Algebra

**Definition**

**1.**

- 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.

**Definition**

**2.**

## 3. Transformation from a TEG to Its Precedence Graph

**Rule**

**1.**

**Precondition:**- A TEG $\mathcal{N}=(P,T,F,H,{M}_{0})$ if there exists a place ${p}_{io}$ that initially has more than one token (${M}_{0}\left({p}_{io}\right)>1,\exists {p}_{io}\in P$).
**Rule:**- A transformed equivalent TEG ${\mathcal{N}}_{f}=({P}_{f},{T}_{f},{F}_{f},{H}_{f},{M}_{0f})$, such that
- 1.
- ${P}_{f}=P\cup S-\left\{{p}_{io}\right\}$, where $S=\left\{{p}_{io-j}\right|0\le j<{M}_{0}\left({p}_{io}\right)\}$
- 2.
- ${T}_{f}=T\cup {T}^{\prime}-\left\{{t}_{i}\right\}$, where ${T}^{\prime}=\left\{{t}_{i-j}\right|0\le j<{M}_{0}\left({p}_{io}\right)\}$
- 3.
- ${M}_{0}\left(p\right)=1,\forall p\in S$
- 4.
- ${}^{\u2022}{t}_{i-0}{=}^{\u2022}{t}_{i}$
- 5.
- ${}^{\u2022}{t}_{i-j}={p}_{io-(j-1)},\forall j,0<j<{M}_{0}\left({p}_{i}\right)$
- 6.
- ${}^{\u2022}{p}_{io-j}={t}_{i-j},\forall j,0\le j<{M}_{0}\left({p}_{i}\right)$
- 7.
- ${}^{\u2022}{t}_{o}={p}_{io-\{{M}_{0}\left({p}_{io}\right)-1\}}$
- 8.
- $H\left({p}_{io-j}\right)=H\left({p}_{io}\right)$ if $j={M}_{0}\left({p}_{io}\right)-1$, otherwise $H\left({p}_{io-j}\right)=0$

**Rule**

**2.**

**Precondition:**- When a TEG $\mathcal{N}=(P,T,F,H,{M}_{0})$ has Place ${p}_{io}$, which has no initial token, we assume that its upstream and downstream transitions are ${t}_{i}$ and ${t}_{o}$, respectively, and ${t}_{i}$ has only one downstream place ${p}_{io}$ (${t}_{i}^{\u2022}=\left\{{p}_{io}\right\}$)
**Rule:**- A transformed equivalent TEG ${\mathcal{N}}_{f}=({P}_{f},{T}_{f},{F}_{f},{H}_{f},{M}_{0f})$ such that
- 1.
- ${P}_{f}=P\cup S-\left\{{p}_{io}\right\}{-}^{\u2022}{t}_{i}$, where $\forall {p}_{ji}{p}_{io},{p}_{ji}{p}_{io}\in S$
- 2.
- ${T}_{f}=T-\left\{{t}_{i}\right\}$
- 3.
- ${M}_{0}\left({p}_{ji}{p}_{io}\right)={M}_{0}\left({p}_{ji}\right)$
- 4.
- ${}^{\u2022}{p}_{ji}{p}_{io}={\phantom{\rule{3.33333pt}{0ex}}}^{\u2022}{p}_{ji}$
- 5.
- ${p}_{ji}{p}_{io}^{\u2022}={p}_{io}^{\u2022}$
- 6.
- $H\left({p}_{ji}{p}_{io}\right)=H\left({p}_{ji}\right)+H\left({p}_{io}\right)$

**Lemma**

**1.**

**Proof.**

**Rule**

**3.**

**Precondition:**- When a TEG $\mathcal{N}=(P,T,F,H,{M}_{0})$ has Place ${p}_{io}$, which has no initial token, we assume that its upstream and downstream transitions are ${t}_{i}$ and ${t}_{o}$, respectively, and ${t}_{i}$ has multiple downstream places, unlike Rule 2 (${t}_{i}^{\u2022}-\left\{{p}_{io}\right\}\ne \varnothing $).
**Rule:**- A transformed equivalent TEG ${\mathcal{N}}_{f}=({P}_{f},T,{F}_{f},{H}_{f},{M}_{0f})$ such that
- 1.
- ${P}_{f}=P\cup S-\left\{{p}_{io}\right\}$, where $\forall {p}_{ji}{p}_{io},{p}_{ji}{p}_{io}\in S$
- 2.
- ${t}_{i}^{\u2022}={t}_{i}^{\u2022}-\left\{{p}_{io}\right\}$
- 3.
- ${M}_{0}\left({p}_{ji}{p}_{io}\right)={M}_{0}\left({p}_{ji}\right)$
- 4.
- ${}^{\u2022}{p}_{ji}{p}_{io}={\phantom{\rule{3.33333pt}{0ex}}}^{\u2022}{p}_{ji}$
- 5.
- ${p}_{ji}{p}_{io}^{\u2022}={p}_{io}^{\u2022}$
- 6.
- $H\left({p}_{ji}{p}_{io}\right)=H\left({p}_{ji}\right)+H\left({p}_{io}\right)$.

**Lemma**

**2.**

**Proof.**

**Rule**

**4.**

**Precondition:**- There is an arbitrary subset S of places in which the initial token number, the upstream transition ${t}_{i}$ and the downstream transition ${t}_{o}$ of all places belonging to S are the same.
**Rule:**- All places belonging to the set of places S may be replaced by a single place, specifically a transformed equivalent TEG ${\mathcal{N}}_{f}=({P}_{f},T,{F}_{f},{H}_{f},{M}_{0f})$ such that
- 1
- ${P}_{f}=P\cup \{{p}_{io}^{0}|\cdots |{p}_{io}^{k}\}-S$
- 2
- ${M}_{0}\left({p}_{io}\right)={M}_{0}({p}_{io}^{0}|\cdots |{p}_{io}^{k})$
- 3
- ${}^{\u2022}({p}_{io}^{0}|\cdots |{p}_{io}^{k})={t}_{i}$
- 4
- $({p}_{io}^{0}|\cdots |{p}_{io}^{k}{)}^{\u2022}={t}_{o}$
- 5
- $H({p}_{io}^{0}|\cdots |{p}_{io}^{k})=\underset{\forall {p}_{io}\in S}{max}H\left({p}_{io}\right)$.

**Lemma**

**3.**

**Proof.**

## 4. Stability Analysis of a TEG

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

- 1.
- If and only if there exists a critical cycle $\overline{\pi}$ in $\mathcal{G}\left(\mathcal{N}\right)$, 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 ${\mathcal{G}}^{c}\left(\mathcal{N}\right)$ 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 $\mathcal{G}\left(\mathcal{N}\right)$.

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**The precedence graph for Figure 1.

**Figure 3.**A firing schedule for Figure 1.

**Table 1.**The cyclicity and corresponding critical cycles of each pair of critical cycles of Figure 1.

Critical Cycle (Cyclicity) | ${\mathit{\pi}}_{1}$ | ${\mathit{\pi}}_{2}$ | ${\mathit{\pi}}_{3}$ | ${\mathit{\pi}}_{4}$ |
---|---|---|---|---|

${\pi}_{1}$ | ${\overline{\pi}}_{1}$(1) | ${\overline{\pi}}_{1}$(1) | ${\overline{\pi}}_{1},{\overline{\pi}}_{2}$(1) | ${\overline{\pi}}_{1},{\overline{\pi}}_{3},{\overline{\pi}}_{4}$(1) |

${\pi}_{2}$ | ${\overline{\pi}}_{1}$(1) | ${\overline{\pi}}_{1}$(1) | ${\overline{\pi}}_{1},{\overline{\pi}}_{2}$(1) | ${\overline{\pi}}_{1},{\overline{\pi}}_{3},{\overline{\pi}}_{4}$(1) |

${\pi}_{3}$ | ${\overline{\pi}}_{1},{\overline{\pi}}_{2}$(1) | ${\overline{\pi}}_{1},{\overline{\pi}}_{2}$(1) | ${\overline{\pi}}_{2}$(3) | ${\overline{\pi}}_{2},{\overline{\pi}}_{3}$(6) |

${\pi}_{4}$ | ${\overline{\pi}}_{1},{\overline{\pi}}_{3},{\overline{\pi}}_{4}$(1) | ${\overline{\pi}}_{1},{\overline{\pi}}_{3},{\overline{\pi}}_{4}$(1) | ${\overline{\pi}}_{2},{\overline{\pi}}_{3}$(6) | ${\overline{\pi}}_{3}$(2) |

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Kim, J.-H. An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs. *Appl. Sci.* **2021**, *11*, 4769.
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Kim, Ja-Hee. 2021. "An Asymptotic Cyclicity Analysis of Live Autonomous Timed Event Graphs" *Applied Sciences* 11, no. 11: 4769.
https://doi.org/10.3390/app11114769