**5. Linearization of TEGM**

The presence of integer part modeled by operator *μ* induces a nonlinearity in Equation 8 used to represent a TEGM. So, as far as possible, we seek to represent a TEGM with linear equations in order to apply standard results of linear system theory developed in the dioid setting, which leads to transform a TEGM into a TEG (represented without operator *μ*).

#### **5.1. Principle of linearization**

A consistent TEGM has a unique elementary T-invariant in which components are in **N**∗. The used method is based on the use of commutation rules of operators and the impulse inputs (Proposition 1 and 2).

In the next, we suppose that all tokens in a TEGM are "frozen" before time 0 and are available at time 0 which is a classical assumption in Petri Nets theory. Hence, with each counter variable of a TEGM is added a counter variable corresponding to an impulse input *e* (*i.e., e*(*t*) = 0 for *t* < 0 and *e*(*t*)=+∞ for *t* ≥ 0). These initial conditions are weakly compatible. For more details, see [10].

To linearize the expression of counters variables written as Equation 8, one expresses each counter according to an entry impulse. This latter will permit to linearize the mathematical model reflecting the behavior of a TEGM in order to obtain a linear model in (min, +) algebra.

**Figure 8.** Impulse (Point of view of counter).

10 Will-be-set-by-IN-TECH

• *Nq*(*δ*) is the counter *nq*(*t*) associated with the transition *nq*, encoded in Dmin[[*δ*]]. It is equal to the counter *Nq*�(*δ*) shifted by the composition of operators *<sup>μ</sup>Mpq*� , *<sup>δ</sup>τ<sup>p</sup>* , *<sup>γ</sup>mp* and *<sup>μ</sup>M*−<sup>1</sup> *qp* connected

*3.* Let *<sup>N</sup>*(*δ*) such that*,* <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> **<sup>Z</sup>**, *<sup>n</sup>*(*t*) *is a multiple of a, then <sup>μ</sup>a*−<sup>1</sup>*γbN*(*δ*) = *<sup>γ</sup>*�*a*−1*b*�*μa*−<sup>1</sup>*N*(*δ*)*.*

• Point 2: *<sup>μ</sup>a*−<sup>1</sup>*μbN*(*δ*) corresponds to �*a*−1�*b n*(*t*)�� <sup>=</sup> �*a*−1*b n*(*t*)� which leads to

• Point 3: *<sup>μ</sup>a*−<sup>1</sup>*γbN*(*δ*) correspond to �*a*−1(*<sup>b</sup>* <sup>+</sup> *<sup>n</sup>*(*t*))� <sup>=</sup> �*a*−1*b*� <sup>+</sup> *<sup>a</sup>*−1*n*(*t*) since *<sup>n</sup>*(*t*) <sup>∈</sup> **<sup>Z</sup>** <sup>∪</sup>

• Point 4: *<sup>γ</sup>bμaN*(*δ*) corresponds to *<sup>b</sup>* <sup>+</sup> �*a n*(*t*)� <sup>=</sup> �*a*(*a*−1*<sup>b</sup>* <sup>+</sup> *<sup>n</sup>*(*t*))� which leads to

**Example 4.** The TEGM depicted in Figure 4 admits the following representation in Dmin[[*δ*]]:

*ε μ*1/2*γ*4*δ*1*μ*<sup>3</sup> *ε*

⎞

⎛

*N*1

⎞

⎟⎟⎟⎟⎠

*N*2

*N*3

⎜⎜⎜⎜⎝

⎟⎟⎟⎟⎠

*μ*1/3*δ*1*μ*<sup>2</sup> *ε γ*2*δ*1*μ*<sup>2</sup>

*ε μ*1/2*δ*<sup>1</sup> *ε*

The presence of integer part modeled by operator *μ* induces a nonlinearity in Equation 8 used to represent a TEGM. So, as far as possible, we seek to represent a TEGM with linear equations in order to apply standard results of linear system theory developed in the dioid setting, which

A consistent TEGM has a unique elementary T-invariant in which components are in **N**∗. The used method is based on the use of commutation rules of operators and the impulse inputs

In the next, we suppose that all tokens in a TEGM are "frozen" before time 0 and are available at time 0 which is a classical assumption in Petri Nets theory. Hence, with each counter

in series. Let us express some properties of operators *γ*, *δ*, *μ* in dioid Dmin[[*δ*]].

**Proposition 1.** Let *a*, *b* ∈ **N**, we have:

*2. μa*−<sup>1</sup>*μ<sup>b</sup>* = *μ*(*a*−1*b*)*.*

• Point 1 is obvious.

*μ*(*a*−1*b*)*N*(*δ*).

*μaγa*−<sup>1</sup>*bN*(*δ*).

**Proof:**

*1. γaδ<sup>b</sup>* = *δbγa, μaδ<sup>b</sup>* = *δbμ<sup>a</sup>* (commutative properties)*.*

{±∞} is a multiple of *<sup>a</sup>*, which leads to *<sup>γ</sup>*�*a*−1*b*�*μa*−<sup>1</sup>*N*(*δ*).

⎛

⎜⎜⎜⎜⎝

leads to transform a TEGM into a TEG (represented without operator *μ*).

*4. γbμ<sup>a</sup>* = *μaγa*−<sup>1</sup>*b,* or equivalently*, μaγ<sup>b</sup>* = *γabμa.*

⎛

*N*1

⎞

⎟⎟⎟⎟⎠ =

*N*2

*N*3

⎜⎜⎜⎜⎝

**5. Linearization of TEGM**

**5.1. Principle of linearization**

(Proposition 1 and 2).

**Proposition 2.** let *<sup>E</sup>* an impulse input, we have : <sup>∀</sup>*<sup>a</sup>* <sup>∈</sup> **<sup>N</sup>**, *<sup>β</sup>* <sup>∈</sup> *<sup>Q</sup>*+,

$$
\mu\_{\beta} \gamma^{a} \delta^{\tau} E(\delta) = \gamma^{\lfloor \beta a \rfloor} \delta^{\tau} E(\delta). \tag{9}
$$

**Proof:** Thanks to Proposition 1.3, *μβγ<sup>a</sup>δτE*(*δ*) corresponds to �*<sup>β</sup>* <sup>×</sup> (*<sup>a</sup>* <sup>+</sup> *<sup>e</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*))� <sup>=</sup> �*<sup>β</sup>* <sup>×</sup> *<sup>a</sup>*� <sup>+</sup> *<sup>e</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*) since for *<sup>t</sup>* <sup>≥</sup> <sup>0</sup> *<sup>e</sup>*(*t*) �→ <sup>+</sup>∞, hence *<sup>e</sup>*(*t*) is a multiple of *<sup>β</sup>*, which leads to *<sup>γ</sup>*�*βa*�*δτE*(*δ*).

We now give the state model associated to the dynamic of counters of a TEGM. Consider the vector *N* composed of the counter variable. The counter variables corresponding to impulse input *e* added with each transition *ni*:

$$N(\delta) = A \otimes N(\delta) \oplus E(\delta). \tag{10}$$

Knowing that such equation admits the following earliest solution:

$$N(\delta) = A^\* \otimes E(\delta),\tag{11}$$

*<sup>A</sup>*<sup>∗</sup> <sup>=</sup> *<sup>e</sup>* <sup>⊕</sup> *<sup>A</sup>* <sup>⊕</sup> *<sup>A</sup>*<sup>2</sup> ⊕··· .

**Proposition 3.** For initial conditions *weakly compatible*, consistent and conservative TEGM is linearizable without increasing the number of its transitions.

**Proof:** Consider a consistent and conservative TEGM represented by the equation *A*(*δ*) = *A* ⊗ *N*(*δ*) ⊕ *E*(*δ*). Using Equation 11, and then apply the Proposition 2, we obtain a linear equation between transitions of graph (corresponding to a linear TEG). This linearization method may be applied to all transitions of graph, since for any transition, one can involve an impulse input.

**Example 5.** The TEGM depicted in Figure 9 admits the elementary T-invariant *θ<sup>t</sup>* = (3, 2, 1).

These equations are quite (*min*, +) linear. It turns out that the TEG depicted in Figure 10, composed of three elementary circuits: (*n*1, *n*1), (*n*2, *n*2), (*n*3, *n*3), is a possible representation

• **General case**: To evaluate the performance of a TEGM returns to calculate the cycle time

**Definition 9.** [16] The cycle time, *TCm*, of a TEGM is the average time to fire once the T-invariant under the earliest firing rule (i.e., transitions are fired as soon as possible) from

This cycle time is equivalent to the average time between two successive firing of a transition.

*TCm* <sup>=</sup> *<sup>θ</sup><sup>q</sup>*

• *θ<sup>q</sup>* is the component of T-invariant associated with transition *nq*, and *λmq* is the firing rate associated with transition *nq* of TEGM corresponding to the average number of firing of one

• For an industrial system, the cycle time corresponds to the average manufacturing time of a

• **Particular case**: Elements of performance evaluation for TEG. We recall main results characterizing an ordinary TEG modeled in the dioid **Z**min. Knowing that a TEG is a TEGM

**Definition 10.** A matrix *A* is said *irreducible* if for any pair *(i,j)*, there is an integer *m* such that

**Theorem 1.** [5] Let *A* be a square matrix with coefficient in **Z**min. The following assertions

piece, and the firing rate is the average number of pieces produced per unit of time.

with unit weights on the arcs, and their components of T-invariant are all equals 1.

*•* The TEG associated with matrix *A* is strongly connected.

*λmq*

. (12)

Performance Evaluation of Timed Petri Nets in Dioid Algebra 419

of the previous equations.

**Figure 10.** TEG (Linearized TEGM).

the initial marking.

transition per unit time.

(*Am*)*ij* �<sup>=</sup> *<sup>ε</sup>*.

are equivalent:

*•* Matrix *A* is irreducible,

**6. Performance evaluation of TEGM**

It is calculated by the following relation:

and firing rate associated with each transition of a graph.

**Figure 9.** TEGM with impulse inputs added to each transition.

The inputs *e* correspond to the impulse inputs. They have not influence on the evolution of the model. Indeed, ∀*t* ≥ 0, ∀*nq* ∈ T , min(*nq*(*t*),*e*(*t*)) = *nq*(*t*), since *e*(*t*) �→ +∞.

$$
\begin{pmatrix} N\_1 \\ N\_2 \\ N\_3 \end{pmatrix} = \begin{pmatrix} \varepsilon & \mu\_{1/2}\gamma^4 \delta^1 \mu\_3 & \varepsilon \\\\ \mu\_{1/3}\delta^1 \mu\_2 & \varepsilon & \gamma^2 \delta^1 \mu\_2 \\\\ \varepsilon & \mu\_{1/2}\delta^1 & \varepsilon \end{pmatrix} \begin{pmatrix} N\_1 \\ N\_2 \\ N\_3 \end{pmatrix} \oplus \begin{pmatrix} E \\ E \\ E \\ E \end{pmatrix}.
$$

Using Equation 11, *N*(*δ*) = *A*∗*E*(*δ*). The Proposition 2 allows to calculate *A*∗*E*(*δ*):

$$\begin{split} A^\*E(\delta) &= (e \oplus A \oplus A^2 \oplus A^3 \oplus \ldots)E(\delta) \\ &= (E(\delta) \oplus A \, E(\delta) \oplus \underbrace{A \otimes A E(\delta)}\_{A^2 E(\delta)} \oplus \underbrace{A \otimes A^2 E(\delta)}\_{A^3 E(\delta)} \oplus \ldots), \\ A^\*E(\delta) &= \left( \begin{matrix} (\gamma^2 \delta^2)(\gamma^3 \delta^4)^\* \\ \delta^1 (\gamma^1 \delta^2)^\* \\ \delta^4 (\gamma^1 \delta^4)^\* \end{matrix} \right) E(\delta), \end{split}$$

which is the earliest solution of the following equations:

$$
\begin{pmatrix} N\_1(\delta) \\ N\_2(\delta) \\ N\_3(\delta) \end{pmatrix} = \begin{pmatrix} \gamma^3 \delta^4 \\ \gamma^1 \delta^2 \\ \gamma^1 \delta^4 \end{pmatrix} \begin{pmatrix} N\_1(\delta) \\ N\_2(\delta) \\ N\_3(\delta) \end{pmatrix} \oplus \begin{pmatrix} \gamma^2 \delta^2 \\ \delta^1 \\ \delta^4 \end{pmatrix} E(\delta).
$$

Let us express these equations in usual counter setting (dioid **Z**min), we have, ∀*t* ∈ **Z**:

$$\begin{cases} n\_1(t) = 3 \otimes n\_1(t-4) \oplus 2 \otimes e(t-2), \\ n\_2(t) = 1 \otimes n\_2(t-2) \oplus e(t-1), \\ n\_3(t) = 1 \otimes n\_3(t-4) \oplus e(t-4). \end{cases}$$

These equations are quite (*min*, +) linear. It turns out that the TEG depicted in Figure 10, composed of three elementary circuits: (*n*1, *n*1), (*n*2, *n*2), (*n*3, *n*3), is a possible representation of the previous equations.

**Figure 10.** TEG (Linearized TEGM).

12 Will-be-set-by-IN-TECH

**Figure 9.** TEGM with impulse inputs added to each transition.

⎛

⎜⎜⎜⎜⎝

⎛

⎜⎜⎝

which is the earliest solution of the following equations:

*N*1(*δ*) *N*2(*δ*) *N*3(*δ*)

> ⎧ ⎪⎪⎪⎨

> ⎪⎪⎪⎩

⎞

⎟⎟⎠ =

the model. Indeed, ∀*t* ≥ 0, ∀*nq* ∈ T , min(*nq*(*t*),*e*(*t*)) = *nq*(*t*), since *e*(*t*) �→ +∞.

The inputs *e* correspond to the impulse inputs. They have not influence on the evolution of

*ε μ*1/2*γ*4*δ*1*μ*<sup>3</sup> *ε*

*μ*1/3*δ*1*μ*<sup>2</sup> *ε γ*2*δ*1*μ*<sup>2</sup>

*ε μ*1/2*δ*<sup>1</sup> *ε*

⎞

⎟⎟⎠ *E*(*δ*),

*A*2*E*(*δ*)

Using Equation 11, *N*(*δ*) = *A*∗*E*(*δ*). The Proposition 2 allows to calculate *A*∗*E*(*δ*):

= ( *<sup>E</sup>*(*δ*) <sup>⊕</sup> *A E*(*δ*) <sup>⊕</sup> *<sup>A</sup>* <sup>⊗</sup> *AE*(*δ*) � �� �

*<sup>A</sup>*∗*E*(*δ*)=(*<sup>e</sup>* <sup>⊕</sup> *<sup>A</sup>* <sup>⊕</sup> *<sup>A</sup>*<sup>2</sup> <sup>⊕</sup> *<sup>A</sup>*<sup>3</sup> <sup>⊕</sup> ...)*E*(*δ*)

(*γ*2*δ*2)(*γ*3*δ*4)<sup>∗</sup> *δ*1(*γ*1*δ*2)<sup>∗</sup> *δ*4(*γ*1*δ*4)<sup>∗</sup>

⎛

*γ*3*δ*<sup>4</sup> *γ*1*δ*<sup>2</sup> *γ*1*δ*<sup>4</sup>

*n*1(*t*) = 3 ⊗ *n*1(*t* − 4)

*n*2(*t*) = 1 ⊗ *n*2(*t* − 2)

*n*3(*t*) = 1 ⊗ *n*3(*t* − 4)

⎞

⎛

*N*1(*δ*) *N*2(*δ*) *N*3(*δ*)

⎞

⎛

*γ*2*δ*<sup>2</sup> *δ*1 *δ*4

⎞

⎟⎟⎠ *E*(*δ*).

⎜⎜⎝

� <sup>2</sup> ⊗ *<sup>e</sup>*(*<sup>t</sup>* − <sup>2</sup>),

� *<sup>e</sup>*(*<sup>t</sup>* − <sup>1</sup>),

� *<sup>e</sup>*(*<sup>t</sup>* − <sup>4</sup>).

⎟⎟⎠ ⊕

⎜⎜⎝

⎟⎟⎠

Let us express these equations in usual counter setting (dioid **Z**min), we have, ∀*t* ∈ **Z**:

⎜⎜⎝

⎞

⎛

*N*1

⎞

⎛

*E*

⎞

⎟⎟⎟⎟⎠ .

⊕ ...).

*E*

*E*

⎜⎜⎜⎜⎝

⎟⎟⎟⎟⎠ ⊕

*N*2

*N*3

<sup>⊕</sup> *<sup>A</sup>* <sup>⊗</sup> *<sup>A</sup>*2*E*(*δ*) � �� � *A*3*E*(*δ*)

⎜⎜⎜⎜⎝

⎟⎟⎟⎟⎠

⎛

*N*1

⎞

⎟⎟⎟⎟⎠ =

*N*2

*N*3

*A*∗ *E*(*δ*) =

⎛

⎜⎜⎝

⎜⎜⎜⎜⎝

### **6. Performance evaluation of TEGM**

• **General case**: To evaluate the performance of a TEGM returns to calculate the cycle time and firing rate associated with each transition of a graph.

**Definition 9.** [16] The cycle time, *TCm*, of a TEGM is the average time to fire once the T-invariant under the earliest firing rule (i.e., transitions are fired as soon as possible) from the initial marking.

This cycle time is equivalent to the average time between two successive firing of a transition. It is calculated by the following relation:

$$T\mathbb{C}\_{m} = \frac{\theta\_{q}}{\lambda\_{m\_{\theta}}}.\tag{12}$$

• *θ<sup>q</sup>* is the component of T-invariant associated with transition *nq*, and *λmq* is the firing rate associated with transition *nq* of TEGM corresponding to the average number of firing of one transition per unit time.

• For an industrial system, the cycle time corresponds to the average manufacturing time of a piece, and the firing rate is the average number of pieces produced per unit of time.

• **Particular case**: Elements of performance evaluation for TEG. We recall main results characterizing an ordinary TEG modeled in the dioid **Z**min. Knowing that a TEG is a TEGM with unit weights on the arcs, and their components of T-invariant are all equals 1.

**Definition 10.** A matrix *A* is said *irreducible* if for any pair *(i,j)*, there is an integer *m* such that (*Am*)*ij* �<sup>=</sup> *<sup>ε</sup>*.

**Theorem 1.** [5] Let *A* be a square matrix with coefficient in **Z**min. The following assertions are equivalent:


One calls *eigenvalue* and *eigenvector* of a matrix *A* with coefficients in **Z**min, the scalar *λ* and the vector *υ* such as:

$$A \otimes \upsilon = \lambda \otimes \upsilon.$$

**Theorem 2.** [5] Let *A* be a square matrix with coefficients in **Z**min. If *A* is irreducible, or equivalently, if the associated TEG is strongly connected, then there is a single eigenvalue denoted *λ*. The eigenvalue can be calculated in the following way:

$$\lambda = \bigoplus\_{j=1}^{n} (\bigoplus\_{i=1}^{n} (A^j)\_{ii})^{\frac{1}{7}}.\tag{13}$$

About the firing rate associated with each transition of the graph, using the relation (12):

, *<sup>λ</sup>m*<sup>2</sup> <sup>=</sup> <sup>1</sup>

2

Confirmation of these results can be deducted directly to the following marking graph of the

Performance evaluation of TEGM is the subject of this chapter. These graphs, in contrast to ordinary TEG, do not admit a linear representation in (*min*, +) algebra. This nonlinearity is due to the presence of weights on the arcs. For that, a modeling of these graphs in an algebraic structure, based on operators, is used. The obtained model is linearized, by using of pulse inputs associated with all transitions of graphs, in order to obtain representation in linear (*min*+) algebra, and apply some results basic spectral theory, usually used to evaluate the performance of ordinary TEG. The work presented in this chapter paves the way for other development related to evaluation of performance of these models. In particular, the calculation of cycle time for any timed event graph with multipliers is, to our knowledge, an

[1] Baccelli, F., Cohen, G., Olsder, G.-J. & Quadrat, J.-P. [1992]. *Synchronization and Linearity:*

[2] Chao, D., Zhou, M. & Wang, D. [1993]. Multiple weighted marked graphs, *IFAC 12th*

[3] Cohen, G., Gaubert, S. & Quadrat, J.-P. [1998]. Timed-event graphs with multipliers and homogenous min-plus systems, *IEEE Transaction on Automatic Control*

[4] David, R. & Alla, H. [1992]. *Du Grafcet au réseaux de Petri*, Editions Hermès, Paris. [5] Gaubert, S. [1995]. Resource optimization and (min,+) spectral theory, *IEEE Transaction*

 

, *<sup>λ</sup>m*<sup>3</sup> <sup>=</sup> <sup>1</sup>

4 .

 

Performance Evaluation of Timed Petri Nets in Dioid Algebra 421

 

*<sup>λ</sup>m*<sup>1</sup> <sup>=</sup> <sup>3</sup> 4

> 

• *kni*/*t*: after t time units, the transition *ni* is firing *k* time.

Samir Hamaci, Karim Labadi and A.Moumen Darcherif *EPMI, 13 Boulvard de l'Hautil, 95092, Cergy-Pontoise, France*

*An Algebra for Discrete Event Systems*, Wiley and Sons.

*Triennial World Congress*, Sydney, Australie, pp. 371–374.

 

**Figure 11.** Marking graph of the initial TEGM.

 

 

initial TEGM.

**7. Conclusion**

open problem to date.

**Author details**

**8. References**

Vol.43(No.9): 1296–1302.

*on Automatic Control* 40(11): 1931–1934.

*λ* corresponds to the firing rate which is identical for each transition. This eigenvalue *λ* can be directly deduced from the TEG by:

$$\lambda = \min\_{\mathcal{c}\in\mathcal{C}} \frac{M(\mathcal{c})}{T(\mathcal{c})},\tag{14}$$


In the case of Ordinary TEG strongly connected, The inverse of eigenvalue *λ* is equivalent to cycle time, denoted TC.

$$T\mathcal{C} = \frac{1}{\lambda'} \tag{15}$$

**Example 6.** The TEG depicted in Figure 10, which is not strongly connected, is composed of three circuits : (*n*1, *n*1), (*n*2, *n*2) and (*n*3, *n*3). Each circuit admits a T-invariant composed of one component equals 1.

Using the Definition 9 and Equation 15, one deduce that each circuit, which is an elementary TEG strongly connected, admits the following cycle time:


The cycle time of TEGM depicted in Figure 4, corresponds to the time required to fire each transition a number of times equal its corresponding elementary T-invariant component. Hence

$$T\mathbb{C}\_1 = 3 \times \frac{4}{3} \prime \qquad T\mathbb{C}\_2 = 2 \times \frac{2}{1} \prime \qquad T\mathbb{C}\_3 = 1 \times \frac{4}{1} \prime$$

Note that the cycle time is identical for all transitions of the graph which is equal to 4 time units. This means that each transition is asymptotically fired once every four time units.

About the firing rate associated with each transition of the graph, using the relation (12):

$$
\lambda\_{m\_1} = \frac{3}{4}, \qquad \lambda\_{m\_2} = \frac{1}{2}, \qquad \lambda\_{m\_3} = \frac{1}{4}.
$$

Confirmation of these results can be deducted directly to the following marking graph of the initial TEGM.

$$
\begin{array}{c}
\begin{array}{c}
\text{P}\_{\text{P}\_{1}} \\
\text{P}\_{\text{P}\_{2}} \\
\text{P}\_{4} \\
\text{P}\_{3}
\end{array}
\xrightarrow{\{2\text{ n}\_{1}\}/0}
\begin{array}{c}
\text{(2\text{ n}\_{1})/0} \\
\text{(2\text{ n}\_{2})} \\
\text{(2\text{ n}\_{3})}
\end{array}
\xrightarrow{\{1\text{ n}\_{2}\}/1}
\begin{array}{c}
\text{(1\text{ n}\_{2})/1} \\
\text{(1\text{ n}\_{1})}
\end{array}
\xrightarrow{\{1\text{ n}\_{1}\}/1}
\begin{array}{c}
\text{(1\text{ n}\_{2})} \\
\text{(1\text{ n}\_{3})}
\end{array}
\xrightarrow{\{1\text{ n}\_{2}\}/1}
\begin{array}{c}
\text{(1\text{ n}\_{2})} \\
\text{(2\text{ n}\_{1}\text{ n}\_{3})}
\end{array}
\xrightarrow{\{1\text{ n}\_{2}\}/1}
\end{array}
$$

**Figure 11.** Marking graph of the initial TEGM.

• *kni*/*t*: after t time units, the transition *ni* is firing *k* time.

#### **7. Conclusion**

14 Will-be-set-by-IN-TECH

One calls *eigenvalue* and *eigenvector* of a matrix *A* with coefficients in **Z**min, the scalar *λ* and

*A* ⊗ *υ* = *λ* ⊗ *υ*. **Theorem 2.** [5] Let *A* be a square matrix with coefficients in **Z**min. If *A* is irreducible, or equivalently, if the associated TEG is strongly connected, then there is a single eigenvalue

*λ* corresponds to the firing rate which is identical for each transition. This eigenvalue *λ* can

In the case of Ordinary TEG strongly connected, The inverse of eigenvalue *λ* is equivalent to

*TC* <sup>=</sup> <sup>1</sup>

**Example 6.** The TEG depicted in Figure 10, which is not strongly connected, is composed of three circuits : (*n*1, *n*1), (*n*2, *n*2) and (*n*3, *n*3). Each circuit admits a T-invariant composed of one

Using the Definition 9 and Equation 15, one deduce that each circuit, which is an elementary

The cycle time of TEGM depicted in Figure 4, corresponds to the time required to fire each transition a number of times equal its corresponding elementary T-invariant component.

Note that the cycle time is identical for all transitions of the graph which is equal to 4 time units. This means that each transition is asymptotically fired once every four time units.

2 1

, *TC*<sup>3</sup> = 1 ×

4 1 .

, *TC*<sup>2</sup> = 2 ×

*M*(*c*)

*λ* = min *c*∈ *C*

*<sup>j</sup>* . (13)

*<sup>T</sup>*(*c*) , (14)

*<sup>λ</sup>*, (15)

denoted *λ*. The eigenvalue can be calculated in the following way:

*<sup>λ</sup>* <sup>=</sup> *<sup>n</sup> j*=1 ( *n i*=1 (*A<sup>j</sup>* )*ii*) 1

the vector *υ* such as:

be directly deduced from the TEG by:

cycle time, denoted TC.

component equals 1.

• Circuit (*n*1, *n*1), *TC* = <sup>4</sup>

• Circuit (*n*2, *n*2), *TC* = <sup>2</sup>

• Circuit (*n*3, *n*3), *TC* = <sup>4</sup>

Hence

• C is the set of elementary circuits of the TEG. • T(c) is the sum of holding times in circuit *c*. • M(c) is the number of tokens in circuit *c*.

TEG strongly connected, admits the following cycle time:

3 .

1 .

1 .

*TC*<sup>1</sup> = 3 ×

4 3 Performance evaluation of TEGM is the subject of this chapter. These graphs, in contrast to ordinary TEG, do not admit a linear representation in (*min*, +) algebra. This nonlinearity is due to the presence of weights on the arcs. For that, a modeling of these graphs in an algebraic structure, based on operators, is used. The obtained model is linearized, by using of pulse inputs associated with all transitions of graphs, in order to obtain representation in linear (*min*+) algebra, and apply some results basic spectral theory, usually used to evaluate the performance of ordinary TEG. The work presented in this chapter paves the way for other development related to evaluation of performance of these models. In particular, the calculation of cycle time for any timed event graph with multipliers is, to our knowledge, an open problem to date.
