**3. Dynamic behavior of Timed Petri Nets in dioid algebra**

**Definition 5.** An ordinary *Timed Event Graph* (TEG) is a Timed Petri Net such that each place has exactly one upstream transition and one downstream transition. Weights of arcs are all unit.

These graphs are well adapted to model synchronization phenomena occurring in Discrete Event Dynamic Systems. They admit a linear representation on a particular algebraic structure called the *dioid* algebra [1].

**Definition 6.** *A dioid* (D, ⊕, ⊗) is a semiring in which the addition ⊕ is idempotent (∀*a*, *a* ⊕ *a* = *a*). Neutral elements of ⊕ and ⊗ are denoted *ε* and *e* respectively.


*x*∈*A* and if ⊗ distributes at left and at right over infinite sums. The greatest element denoted *<sup>T</sup>* of a complete dioid <sup>D</sup> is equal to *<sup>x</sup> <sup>x</sup>*∈D . The greatest lower bound of every subset *X* of a complete dioid always exists and is denoted *x*.

*x*∈*X*

**Example 1.** The set **Z** ∪ {±∞}, endowed with (*min*) as ⊕ and usual addition as ⊗, is a complete dioid denoted **Z**min and usually called (*min*, +) algebra with neutral elements *ε* = +∞, *e* = 0 and *T* = −∞.

**Example 2.** The set **Z** ∪ {±∞}, endowed with (*max*) as ⊕ and usual addition as ⊗, is a complete dioid denoted **Z**max and usually called (*max*, +) algebra with neutral elements *ε* = −∞, *e* = 0 and *T* = +∞.

**Definition 7.** <sup>A</sup> *signal* is an increasing map from **<sup>Z</sup>** to **<sup>Z</sup>** ∪ {±∞}. Denote *<sup>S</sup>*=(**<sup>Z</sup>** ∪ {±∞})**<sup>Z</sup>** the set of signals.

 

**Figure 3.** Elementary TEGM.

upstream transition.

 

**Assertion 1.** The counter variable associated with the transition *nq* of an elementary TEGM

The inferior integer part is used to preserve the integrity of Equation 7. In general, a transition *nq* may have several upstream transitions {*nq*� ∈ ••*q*} which implies that the associated counter variable is given by the *min* of *transition to transition* equations obtained for each

<sup>2</sup> �,

<sup>2</sup> �.

The mathematical model representing the behavior of this TEGM does not admit a linear representation in (*min*, +) algebra. This nonlinearity is due to the presence of the integer parts generated by the presence of the weights on the arcs. Consequently, it is difficult to use (*min*, +) algebra to tackle, for example, problems of control and the analysis of performances. As alternative, we propose another model based on operators which will be linearized in order

<sup>3</sup> �, 2 + 2*n*3(*t* − 1)),

(under the earliest firing rule) satisfy the following *transition to transition* equation:

*<sup>n</sup>*1(*t*) = � <sup>4</sup>+3*n*2(*t*−1)

*<sup>n</sup>*3(*t*) = � *<sup>n</sup>*<sup>2</sup> (*t*−1)

*<sup>n</sup>*2(*t*) = min(� <sup>2</sup>*n*1(*t*−1)

�*M*−<sup>1</sup>

*nq*(*t*) = min *p*∈•*q*, *q*� ∈• *p*

**Example 3.** Let us consider TEGM depicted in Figure 4.

⎧ ⎪⎪⎪⎨

⎪⎪⎪⎩

**Figure 4.** Timed Event Graph with Multipliers.

to obtain a (*min*, +) linear model.

*qp* (*mp* + *Mpq*�*nq*�(*t* − *τp*))�. (7)

Performance Evaluation of Timed Petri Nets in Dioid Algebra 413

This set is endowed with a kind of module structure, called *min-plus semimodule*, the two associated operations are:


**Definition 8.** An operator Ψ is a mapping defined from **Z** ∪ {±∞} to **Z** ∪ {±∞} is *linear* in (*min*, +) algebra if it preserves the min-plus semimodule structure, *i.e.*, for all signals *x, y* and constant *ρ*,

> Ψ(*x* ⊕ *y*) = Ψ(*x*) ⊕ Ψ(*y*) (additive property), Ψ(*ρ* ⊗ *x*) = *ρ* ⊗ Ψ(*x*) (homogeneity property).

To study a TEG in (*min*, +) algebra, considered state variable is a *counter* , denoted *xq*(*t*). This latter denotes the cumulated number of firings of transition *xq* up to time *t* (*t* ∈ **Z**). To illustrate the evolution of a counter associated with the transition *xq* of a TEG, we consider the following elementary graph:

**Figure 2.** Elementary TEG

$$\chi\_q(t) = \min\_{p \in \Upsilon^q q, q' \in \Upsilon^p} (m\_p + \chi\_{q'}(t - \tau\_p)). \tag{5}$$

Note that this equation is nonlinear in usual algebra. This nonlinearity is due to the presence of the (*min*) which models the synchronization phenomena <sup>1</sup> in the transition *xq*. However, it is linear equation in (*min*, +) algebra:

$$\mathfrak{x}\_q(t) = \bigoplus\_{p \in \bullet q, \, q' \in \bullet p} (m\_p \otimes \mathfrak{x}\_{q'}(t - \tau\_p)). \tag{6}$$

In the case where weight of an arc is greater than one, TEG becomes weighted. This type of model is called Timed Event Graph with Multipliers, denoted TEGM.

The *earliest* functioning rule of a TEGM is defined as follows. A transition *nq* fires as soon as all its upstream places {*p* ∈ •*q*} contain enough tokens (*Mqp*) having spent at least *τ<sup>p</sup>* units of time in place *p*. When transition *nq*� fires, it produces *Mpq*� tokens in each downstream place *p* ∈ *q*�•.

<sup>1</sup> Synchronization phenomena occurs when multiple arcs converge to the same transition.

**Figure 3.** Elementary TEGM.

6 Will-be-set-by-IN-TECH

**Definition 7.** <sup>A</sup> *signal* is an increasing map from **<sup>Z</sup>** to **<sup>Z</sup>** ∪ {±∞}. Denote *<sup>S</sup>*=(**<sup>Z</sup>** ∪ {±∞})**<sup>Z</sup>**

This set is endowed with a kind of module structure, called *min-plus semimodule*, the two

• pointwise minimum of time functions to add signals: ∀*t* ∈ **Z**,(*x* ⊕ *y*)(*t*) = *x*(*t*) ⊕ *y*(*t*) =

• addition of a constant to play the role of external product of a signal by a scalar: ∀*t* ∈

**Definition 8.** An operator Ψ is a mapping defined from **Z** ∪ {±∞} to **Z** ∪ {±∞} is *linear* in (*min*, +) algebra if it preserves the min-plus semimodule structure, *i.e.*, for all signals *x, y* and

> Ψ(*x* ⊕ *y*) = Ψ(*x*) ⊕ Ψ(*y*) (additive property), Ψ(*ρ* ⊗ *x*) = *ρ* ⊗ Ψ(*x*) (homogeneity property).

To study a TEG in (*min*, +) algebra, considered state variable is a *counter* , denoted *xq*(*t*). This latter denotes the cumulated number of firings of transition *xq* up to time *t* (*t* ∈ **Z**). To illustrate the evolution of a counter associated with the transition *xq* of a TEG, we consider the

 

Note that this equation is nonlinear in usual algebra. This nonlinearity is due to the presence of the (*min*) which models the synchronization phenomena <sup>1</sup> in the transition *xq*. However, it

In the case where weight of an arc is greater than one, TEG becomes weighted. This type of

The *earliest* functioning rule of a TEGM is defined as follows. A transition *nq* fires as soon as all its upstream places {*p* ∈ •*q*} contain enough tokens (*Mqp*) having spent at least *τ<sup>p</sup>* units of time in place *p*. When transition *nq*� fires, it produces *Mpq*� tokens in each downstream place

(*mp* + *xq*�(*t* − *τp*)). (5)

(*mp* ⊗ *xq*�(*t* − *τp*)). (6)

the set of signals.

associated operations are:

following elementary graph:

**Figure 2.** Elementary TEG

*p* ∈ *q*�•.

is linear equation in (*min*, +) algebra:

**Z**, ∀*ρ* ∈ **Z** ∪ {±∞},(*ρ*.*x*)(*t*) = *ρ* ⊗ *x*(*t*) = *ρ* + *x*(*t*).

*xq*(*t*) = min *p*∈•*q*, *q*� ∈• *p*

*xq*(*t*) =

model is called Timed Event Graph with Multipliers, denoted TEGM.

<sup>1</sup> Synchronization phenomena occurs when multiple arcs converge to the same transition.

*p*∈•*q*, *q*� ∈• *p*

min(*x*(*t*), *y*(*t*));

constant *ρ*,

**Assertion 1.** The counter variable associated with the transition *nq* of an elementary TEGM (under the earliest firing rule) satisfy the following *transition to transition* equation:

$$n\_q(t) = \min\_{p \in \Upsilon q, q' \in \Upsilon p} \left| M\_{qp}^{-1} (m\_p + M\_{pq'} n\_{q'} (t - \tau\_p)) \right|. \tag{7}$$

The inferior integer part is used to preserve the integrity of Equation 7. In general, a transition *nq* may have several upstream transitions {*nq*� ∈ ••*q*} which implies that the associated counter variable is given by the *min* of *transition to transition* equations obtained for each upstream transition.

**Example 3.** Let us consider TEGM depicted in Figure 4.

$$\begin{cases} n\_1(t) &= \lfloor \frac{4+3n\_2(t-1)}{2} \rfloor, \\ n\_2(t) &= \min(\lfloor \frac{2n\_1(t-1)}{3} \rfloor, 2+2n\_3(t-1)), \\ n\_3(t) &= \lfloor \frac{n\_2(t-1)}{2} \rfloor. \end{cases}$$

**Figure 4.** Timed Event Graph with Multipliers.

The mathematical model representing the behavior of this TEGM does not admit a linear representation in (*min*, +) algebra. This nonlinearity is due to the presence of the integer parts generated by the presence of the weights on the arcs. Consequently, it is difficult to use (*min*, +) algebra to tackle, for example, problems of control and the analysis of performances. As alternative, we propose another model based on operators which will be linearized in order to obtain a (*min*, +) linear model.

### **4. Operatorial representation of TEGM**

We now introduce three operators, defined from **Z** ∪ {±∞} to **Z** ∪ {±∞}, which are used for the modeling of TEGM.

• **Operator** *<sup>μ</sup><sup>r</sup>* to represent a scaling of factor *<sup>r</sup>* (*<sup>r</sup>* <sup>∈</sup> **<sup>Q</sup>**+). It is defined as follows:

(*μ<sup>r</sup>* <sup>⊗</sup> *<sup>δ</sup>τ*)*n*�

(*t*)=(*γν*×*<sup>r</sup>* <sup>⊗</sup> *<sup>μ</sup>r*)*n*�

(*t*) = �*r* × *ν* + *r* × *n*�

 

Denote by <sup>D</sup>min the (noncommutative) dioid of finite sums of operators {*μr*, *<sup>γ</sup>ν*} endowed with pointwise *min* (⊕) and composition (⊗) operations, with neutral elements equal to

1≤*i*≤*k*

*i*∈**Z**

Elements of Dmin[[*δ*]] allow modeling the transfer between two transitions of a TEGM. A

**Assertion 2.** The counter variables of an elementary TEGM satisfies the following equation

*kτ i*=1 *μr<sup>τ</sup> i γν<sup>τ</sup>*

*<sup>ε</sup>* <sup>=</sup> *<sup>μ</sup>*+∞*γ*+<sup>∞</sup> and *<sup>e</sup>* <sup>=</sup> *<sup>μ</sup>*1*γ*<sup>0</sup> respectively. Thus, an element in <sup>D</sup>min is a map *<sup>p</sup>* <sup>=</sup> *<sup>k</sup>*

The set of these formal power series endowed with the two following operations:

*f*(*i*) ⊗ *h*(*τ* − *i*) = inf

is a dioid denoted <sup>D</sup>min[[*δ*]], with neutral elements *<sup>ε</sup>* <sup>=</sup> *<sup>μ</sup>*+∞*γ*+∞*δ*−<sup>∞</sup> and *<sup>e</sup>* <sup>=</sup> *<sup>μ</sup>*1*γ*0*δ*0.

to the fact that it is also equal to *n* ⊗ *e* (by definition of neutral element *e* of Dmin).

(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*)� = (*δ<sup>τ</sup>* <sup>⊗</sup> *<sup>μ</sup>r*)*n*�

*<sup>q</sup>*(*t*) = *μrn*�

(*t*)=(*δ<sup>τ</sup>* <sup>⊗</sup> *<sup>μ</sup>r*)*n*�

(*t*).

**Property 3.** Operator *μ<sup>r</sup>* satisfies the following rules when composed with operators *δ<sup>τ</sup>*

*<sup>q</sup>*�(*t*) = �*r* × *n*�

(*t*),

(*t*), for *<sup>ν</sup>* <sup>∈</sup> *<sup>r</sup>*−<sup>1</sup> <sup>×</sup> **<sup>N</sup>**.

(�*ri*(*ν<sup>i</sup>* + *n*(*t*))�).

(*f*(*i*) + *h*(*τ* − *i*)),

*τ*∈**Z**

(*t*)� = *r* × *ν* + �*r* × *n*�

*<sup>q</sup>*�(*t*)�,

Performance Evaluation of Timed Petri Nets in Dioid Algebra 415

(*t*)� = (*γν*×*<sup>r</sup>* <sup>⊗</sup>

*<sup>i</sup>*=<sup>1</sup> *μri γνi*

*<sup>i</sup>* . We define the power series

*n*(*τ*) *δτ*, simply due

*h*(*τ*)*δτ*.

*τ*∈**Z**

*<sup>μ</sup>M*−<sup>1</sup> *qp <sup>γ</sup>mp <sup>δ</sup>τpμMpq*� *Nq*�(*δ*). (8)

<sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> **<sup>Z</sup>**, <sup>∀</sup> *nq*� <sup>∈</sup> **<sup>Z</sup>Z**, *<sup>n</sup>*�

with *<sup>r</sup>* <sup>∈</sup> **<sup>Q</sup>**+(*<sup>r</sup>* is equal to a ratio of elements in **<sup>N</sup>**).

(*μ<sup>r</sup>* <sup>⊗</sup> *<sup>γ</sup>ν*)*n*�

(*t*) = �*r* × *n*�

*b* )

defined from *S* to *S* such that ∀ *t* ∈ **Z**, *p* (*n*(*t*)) = min

Let a map *<sup>h</sup>* : **<sup>Z</sup>** → Dmin, *<sup>τ</sup>* �→ *<sup>h</sup>*(*τ*) in which *<sup>h</sup>*(*τ*) =

*F*(*δ*) ⊕ *H*(*δ*): (*f* ⊕ *h*)(*τ*) = *f*(*τ*) ⊕ *h*(*τ*) = min(*f*(*τ*), *h*(*τ*)),

*i*∈**Z**

*<sup>H</sup>*(*δ*) in the indeterminate *<sup>δ</sup>* with coefficients in Dmin by: *<sup>H</sup>*(*δ*) =

formal series of Dmin[[*δ*]] can also represent a signal *<sup>n</sup>* as *<sup>N</sup>*(*δ*) =

*p*∈•*q*, *q*� ∈• *p*

*Nq*(*δ*) =

• ∀*<sup>ν</sup>* <sup>∈</sup> *<sup>r</sup>*−<sup>1</sup> <sup>×</sup> **<sup>N</sup>**, (*μ<sup>r</sup>* <sup>⊗</sup> *<sup>γ</sup>ν*)*n*�

(*t*), since *ν* × *r* ∈ **N**.

*and γ<sup>ν</sup> :*

Indeed, we have: • (*μ<sup>r</sup>* <sup>⊗</sup> *<sup>δ</sup>τ*)*n*�

**Figure 7.** Operator *μr*(*r* = *<sup>a</sup>*

*<sup>F</sup>*(*δ*) ⊗ *<sup>H</sup>*(*δ*) : (*<sup>f</sup>* ⊗ *<sup>h</sup>*)(*τ*) =

in dioid Dmin[[*δ*]]:

*μr*)*n*�

• **Operator** *<sup>γ</sup><sup>ν</sup>* to represent a shift of *<sup>ν</sup>* units in counting (*<sup>ν</sup>* <sup>∈</sup> **<sup>Z</sup>** ∪ {±∞}). It is defined as follows:

$$\forall t \in \mathbb{Z}, \quad \forall \ n\_{q'} \in \overline{\mathbb{Z}}^{\mathbb{Z}}, \qquad n\_q(t) = \gamma^\nu n\_{q'}(t) = n\_{q'}(t) + \nu.$$

**Property 1.** Operator *γ<sup>ν</sup>* satisfies the following rules:

$$(\gamma^{\nu} \oplus \gamma^{\nu'}) n\_q'(t) = \gamma^{\min(\nu, \nu')} n\_q'(t).$$

$$(\gamma^{\nu} \otimes \gamma^{\nu'}) n\_q'(t) = \gamma^{\nu+\nu'} n\_q'(t).$$

Indeed, we have


**Figure 5.** Operator *γ<sup>ν</sup>*

• **Operator** *<sup>δ</sup><sup>τ</sup>* to represent a shift of *<sup>τ</sup>* units in dating (*<sup>τ</sup>* <sup>∈</sup> **<sup>Z</sup>** ∪ {±∞}). It is defined as follows:

$$
\forall t \in \mathbb{Z}, \quad \forall \ n\_{q'} \in \overline{\mathbb{Z}}^{\mathbb{Z}}, \qquad n\_q(t) = \delta^\mathsf{T} n\_{q'}(t) = n\_{q'}(t-\tau).
$$

**Property 2.** Operator *δ<sup>τ</sup>* satisfies the following rules:

$$(\delta^{\tau} \oplus \delta^{\tau'}) n\_q'(t) = \delta^{\max\left(\tau, \tau'\right)} n\_q'(t).$$

$$(\delta^{\tau} \otimes \delta^{\tau'}) n\_q'(t) = \delta^{\tau + \tau'} n\_q'(t).$$

• Knowing that the signal *nq*(*t*) is non decreasing, we have :

$$(\delta^{\tau} \oplus \delta^{\tau'})n\_q'(t) = \min(n\_q'(t-\tau), n\_q'(t-\tau')) = n\_q'(t-\max(\tau, \tau')) = \delta^{\max(\tau, \tau')}n\_q'(t).$$

$$(\delta^{\tau} \otimes \delta^{\tau'})n\_q'(t) = \delta^{\tau}n\_q'(t-\tau') = n\_q'(t-\tau'-\tau) = \delta^{\tau'+\tau}n\_q'(t).$$

**Figure 6.** Operator *δ<sup>τ</sup>*

• **Operator** *<sup>μ</sup><sup>r</sup>* to represent a scaling of factor *<sup>r</sup>* (*<sup>r</sup>* <sup>∈</sup> **<sup>Q</sup>**+). It is defined as follows:

$$\forall t \in \mathbb{Z}, \ \forall \ n\_{q'} \in \overline{\mathbb{Z}}^{\mathbb{Z}}, \qquad n'\_{q}(t) = \mu\_{r} n'\_{q'}(t) = \lfloor r \times n'\_{q'}(t) \rfloor\_{\mathcal{I}}$$

with *<sup>r</sup>* <sup>∈</sup> **<sup>Q</sup>**+(*<sup>r</sup>* is equal to a ratio of elements in **<sup>N</sup>**).

**Property 3.** Operator *μ<sup>r</sup>* satisfies the following rules when composed with operators *δ<sup>τ</sup> and γ<sup>ν</sup> :*

$$(\mu\_r \otimes \delta^\tau) n'(t) = (\delta^\tau \otimes \mu\_r) n'(t)\_\nu$$

$$(\mu\_r \otimes \gamma^\nu)n'(t) = (\gamma^{\nu \times r} \otimes \mu\_r)n'(t), \text{for } \; \nu \in r^{-1} \times \mathbb{N}.$$

Indeed, we have:

8 Will-be-set-by-IN-TECH

We now introduce three operators, defined from **Z** ∪ {±∞} to **Z** ∪ {±∞}, which are used for

• **Operator** *<sup>γ</sup><sup>ν</sup>* to represent a shift of *<sup>ν</sup>* units in counting (*<sup>ν</sup>* <sup>∈</sup> **<sup>Z</sup>** ∪ {±∞}). It is defined as

<sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> **<sup>Z</sup>**, <sup>∀</sup> *nq*� <sup>∈</sup> **<sup>Z</sup>Z**, *nq*(*t*) = *<sup>γ</sup><sup>ν</sup>nq*�(*t*) = *nq*�(*t*) + *<sup>ν</sup>*.

*<sup>q</sup>*(*t*) = *<sup>γ</sup>min*(*ν*,*ν*�

*<sup>q</sup>*(*t*) = *<sup>γ</sup>ν*+*ν*�

) = *n*�

*<sup>q</sup>*(*t*) + *ν*� + *ν* = *γν*+*ν*�

) *n*� *<sup>q</sup>*(*t*).

*<sup>q</sup>*(*t*) + *min*(*ν*, *ν*�

*n*� *<sup>q</sup>*(*t*). ) = *γmin*(*ν*,*ν*�

)) = *δmax*(*τ*,*τ*�

<sup>+</sup>*τn*� *<sup>q</sup>*(*t*). ) *n*� *<sup>q</sup>*(*t*).

)*n*� *<sup>q</sup>*(*t*).

*n*� *<sup>q</sup>*(*t*).

)*n*�

)*n*�

*<sup>q</sup>*(*t*) + *ν*�

• **Operator** *<sup>δ</sup><sup>τ</sup>* to represent a shift of *<sup>τ</sup>* units in dating (*<sup>τ</sup>* <sup>∈</sup> **<sup>Z</sup>** ∪ {±∞}). It is defined as

)*n*�

)*n*�

*<sup>q</sup>*(*t* − *τ*�

*<sup>q</sup>*(*t* − *τ*�

 

<sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> **<sup>Z</sup>**, <sup>∀</sup> *nq*� <sup>∈</sup> **<sup>Z</sup>Z**, *nq*(*t*) = *<sup>δ</sup><sup>τ</sup>nq*�(*t*) = *nq*�(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*).

*<sup>q</sup>*(*t*) = *<sup>δ</sup>max*(*τ*,*τ*�

)) = *n*�

) = *n*�

*<sup>q</sup>*(*t*) = *<sup>δ</sup>τ*+*τ*�

) *n*� *<sup>q</sup>*(*t*).

*<sup>q</sup>*(*t* − *max*(*τ*, *τ*�

*<sup>q</sup>*(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*� <sup>−</sup> *<sup>τ</sup>*) = *<sup>δ</sup>τ*�

*n*� *<sup>q</sup>*(*t*).

) = *n*�

(*δ<sup>τ</sup>* <sup>⊕</sup> *<sup>δ</sup>τ*�

(*δ<sup>τ</sup>* <sup>⊗</sup> *<sup>δ</sup>τ*�

*<sup>q</sup>*(*t* − *τ*), *n*�

*<sup>q</sup>*(*t*) = *δτn*�

• Knowing that the signal *nq*(*t*) is non decreasing, we have :

**4. Operatorial representation of TEGM**

**Property 1.** Operator *γ<sup>ν</sup>* satisfies the following rules:

(*γ<sup>ν</sup>* <sup>⊕</sup> *<sup>γ</sup>ν*�

(*γ<sup>ν</sup>* <sup>⊗</sup> *<sup>γ</sup>ν*�

*<sup>q</sup>*(*t*) + *ν*, *n*�

*<sup>q</sup>*(*t*) + *ν*�

**Property 2.** Operator *δ<sup>τ</sup>* satisfies the following rules:

*<sup>q</sup>*(*t*) = *min*(*n*�

)*n*�

(*δ<sup>τ</sup>* <sup>⊗</sup> *<sup>δ</sup>τ*�

the modeling of TEGM.

Indeed, we have • (*γ<sup>ν</sup>* <sup>⊕</sup> *<sup>γ</sup>ν*�

• (*γ<sup>ν</sup>* <sup>⊗</sup> *<sup>γ</sup>ν*�

**Figure 5.** Operator *γ<sup>ν</sup>*

(*δ<sup>τ</sup>* <sup>⊕</sup> *<sup>δ</sup>τ*�

**Figure 6.** Operator *δ<sup>τ</sup>*

)*n*�

follows:

)*n*�

)*n*�

*<sup>q</sup>*(*t*) = *min*(*n*�

*<sup>q</sup>*(*t*) = *γν*(*n*�

follows:


**Figure 7.** Operator *μr*(*r* = *<sup>a</sup> b* )

Denote by <sup>D</sup>min the (noncommutative) dioid of finite sums of operators {*μr*, *<sup>γ</sup>ν*} endowed with pointwise *min* (⊕) and composition (⊗) operations, with neutral elements equal to *<sup>ε</sup>* <sup>=</sup> *<sup>μ</sup>*+∞*γ*+<sup>∞</sup> and *<sup>e</sup>* <sup>=</sup> *<sup>μ</sup>*1*γ*<sup>0</sup> respectively. Thus, an element in <sup>D</sup>min is a map *<sup>p</sup>* <sup>=</sup> *<sup>k</sup> <sup>i</sup>*=<sup>1</sup> *μri γνi* defined from *S* to *S* such that ∀ *t* ∈ **Z**, *p* (*n*(*t*)) = min 1≤*i*≤*k* (�*ri*(*ν<sup>i</sup>* + *n*(*t*))�).

Let a map *<sup>h</sup>* : **<sup>Z</sup>** → Dmin, *<sup>τ</sup>* �→ *<sup>h</sup>*(*τ*) in which *<sup>h</sup>*(*τ*) = *kτ i*=1 *μr<sup>τ</sup> i γν<sup>τ</sup> <sup>i</sup>* . We define the power series *<sup>H</sup>*(*δ*) in the indeterminate *<sup>δ</sup>* with coefficients in Dmin by: *<sup>H</sup>*(*δ*) = *τ*∈**Z** *h*(*τ*)*δτ*.

The set of these formal power series endowed with the two following operations: *F*(*δ*) ⊕ *H*(*δ*): (*f* ⊕ *h*)(*τ*) = *f*(*τ*) ⊕ *h*(*τ*) = min(*f*(*τ*), *h*(*τ*)), *<sup>F</sup>*(*δ*) ⊗ *<sup>H</sup>*(*δ*) : (*<sup>f</sup>* ⊗ *<sup>h</sup>*)(*τ*) = *i*∈**Z** *f*(*i*) ⊗ *h*(*τ* − *i*) = inf *i*∈**Z** (*f*(*i*) + *h*(*τ* − *i*)), is a dioid denoted <sup>D</sup>min[[*δ*]], with neutral elements *<sup>ε</sup>* <sup>=</sup> *<sup>μ</sup>*+∞*γ*+∞*δ*−<sup>∞</sup> and *<sup>e</sup>* <sup>=</sup> *<sup>μ</sup>*1*γ*0*δ*0.

Elements of Dmin[[*δ*]] allow modeling the transfer between two transitions of a TEGM. A formal series of Dmin[[*δ*]] can also represent a signal *<sup>n</sup>* as *<sup>N</sup>*(*δ*) = *τ*∈**Z** *n*(*τ*) *δτ*, simply due to the fact that it is also equal to *n* ⊗ *e* (by definition of neutral element *e* of Dmin).

**Assertion 2.** The counter variables of an elementary TEGM satisfies the following equation in dioid Dmin[[*δ*]]:

$$N\_q(\boldsymbol{\delta}) = \bigoplus\_{p \in \bullet q, q' \in \bullet p} \mu\_{M\_{qp}^{-1}} \gamma^{m\_p} \delta^{\tau\_p} \mu\_{M\_{pq'}} N\_{q'}(\boldsymbol{\delta}).\tag{8}$$

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

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.

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.

**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

**Proposition 3.** For initial conditions *weakly compatible*, consistent and conservative TEGM is

**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

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

**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>*+,

Knowing that such equation admits the following earliest solution:

linearizable without increasing the number of its transitions.

Performance Evaluation of Timed Petri Nets in Dioid Algebra 417

*μβγ<sup>a</sup>δτE*(*δ*) = *<sup>γ</sup>*�*βa*�*δτE*(*δ*). (9)

*N*(*δ*) = *A* ⊗ *N*(*δ*) ⊕ *E*(*δ*). (10)

*N*(*δ*) = *A*<sup>∗</sup> ⊗ *E*(*δ*), (11)

For more details, see [10].

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

input *e* added with each transition *ni*:

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

input.

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

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

$$
\mathcal{Z}.\ \ \mu\_{a^{-1}}\mu\_b = \mu\_{(a^{-1}b)}\cdot 
$$

*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*)*.*

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

### **Proof:**


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

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