Input-Output Representation for a Subclass of Timed and Weighted

Event Graphs in Dioids

Bertrand Cottenceau, Laurent Hardouin and Jean-Louis Boimond ∗

July 10, 2012

Abstract

The class of Timed Event Graphs (TEGs) has widely been studied for the last 30 yearsthanks to an algebraic approach known as the theory of Max-Plus linear systems. In par-ticular, the modeling of TEGs via formal power series has led to input-output descriptionsfor which some model matching control problems have been solved. The objective of thiswork is to extend the class of systems for which a similar control synthesis is possible. Tothis end, we define first a subclass of timed Petri nets, called Balanced Timed and WeightedEvent Graphs (B-TWEGs). This class of models contains TEGs and can also describe somedynamic phenomena such as batching and event duplications. Their behavior is describedby rational compositions of four elementary operators γn, δt, µm and βb on a dioid of formalpower series. Then, we show that the series generated by a B-TWEG have a graphical rep-resentation which is three dimensional and have a property of periodicity. In other words,the transfer of a B-TWEG is described by a matrix of formal series that are ultimatelyperiodic.

Keywords: Discrete-Event Systems, Timed and Weighted Event Graphs, Dioid, Formal Power Series, Three

Dimensional Representation.

1 Introduction

Since the beginning of the 80s, it has been known that the class of Timed Event Graphs (TEGs) can bestudied thanks to linear models in some specific algebraic structures called dioids (or idempotent semiring)[18][4][2][8] [13]. Among different representations, a specific approach lies on an operatorial description ofsuch systems. By denoting Σ the semimodule of counter functions1, one can describe the behavior of a TEGby combining two shift operators [4] denoted respectively γ, δ : Σ→ Σ

γ : (γx)(t) = x(t) + 1 δ : (δx)(t) = x(t− 1)

It is known that the input-output behavior of a TEG is then described by a matrix the entries of which areelements of the rational closure2 of the set {ε, e, γ, δ}, i.e. that the transfer matrix of a TEG can be writtenwith a finite composition of these operators. Moreover, due to some fundamental equivalences such as

γn ⊕ γn′ = γmin(n,n′) and δt ⊕ δt′ = δmax(t,t′),

∗Technical Report - LISA EA 4094 - Universite d’Angers, France. Corresponding author B. Cottenceau.Tel. +33 (0)2 41 36 57 31. Fax +33 (0)2 41 36 57 35. E-mail bertrand.cottenceau@univ-angers.fr.

1A counter function x : Z→ Z, t 7→ x(t) gives the cumulative number of occurences of the events labeled x at date t. Such afunction plays the role of signal.

2where ε (resp. e) is the null (resp. neutral) operator.

1

a rational expression has a canonical form which is ultimately periodic [8][15][10] . In other words, we canmanipulate TEG transfer as periodic formal series in two variables γ and δ, with some simplification rules,within a dioid called Max

in Jγ, δK. On the one hand, this fact has made it possible to elaborate software toolsto compute the transfer matrix of any TEG [11] [1] [6]. On the other hand, such an input-output model iswell suited to address some model matching control problems [5] [16] [14] [12]. By analogy with the classicalcontrol theory, controllers can be computed in order to achieve, for the closed-loop system, some prescribedperformances. When applied to manufacturing production, the controllers obtained lead to improve theinternal flaws of products by decreasing internal stocks.

The main objective of this work is to study a class of systems greater than those described by TEGs, butwith similar algebraic tools. We focus here on the class of Timed and Weighted Event Graphs (TWEGs)(see [7][19][17][20]). In addition to synchronisations and delays, TWEGs can also describe other phenomenasuch as batch constitution (several successive input events are necessary to release one output event) andduplication (one input event produces instantaneously several output events). These situations are usual inmanufacturing processes and cannot be accurately modeled with ordinary TEGs. As shown in the worksof Marchetti and Munier [17][20], in comparison to the class of TEGs, the class of TWEGs is slightly morecomplex to analyze, in particular concerning the conditions of liveness.

Our work attempts to be homogeneous with the theory of max-plus linear system as developed by the Max-Plus working group at INRIA (see for instance [4][8][2]) . In particular, the model presented here is mainlyinspired by [3] where an operatorial approach is proposed, but we focus here on the discrete functioning ofTWEGs. According to the conclusions given in [3], the discrete functioning of TWEGs may be arbitrarilyfar from the fluid one. Hence, the class of discrete TWEGs deserves to have a specific study. As in [3],our modeling approach uses the classical shift operators γn and δt, and we introduce two additional onesdenoted βb and µm that represent respectively a batch operation (which is modeled by an integer division3

on a counter variable) and a duplication phenomenon (multiplier operator), ∀x ∈ Σ

βb : (βbx)(t) = bx(t)/bc µm : (µmx)(t) = x(t)×m.

Thanks to these operators, we can deduce that the behavior of a TWEG denoted G can be described by arational expression on the set of elementary operators OM,B = {ε, e, γ, δ, µ2, µ3, ..., µM , β2, ..., βB}, where Mis the maximal multiplier value and B the maximal batch value of G. But, it is not clear if, for the generalcase, there exists a canonical form for a rational expression on OM,B. Equivalently, the formal computationof the behavior of a TWEG does not necessarily lead to a unique expression. In particular, the non unitaryTWEGs seem to be difficult to handle.

We show that a subclass of TWEGs (which includes ordinary TEGs), that we call Balanced Timed andWeighted Event Graphs (B-TWEGs), can be described by rational expressions on OM,B for which a canonicalperiodic form exists. The class of B-TWEGs corresponds to TWEGs such that parallel paths have the samegain4. For these systems, we show that the transfer relation can be expressed, on a dioid of series denotedEJδK, by an ultimately periodic power series in one variable δ, with coefficients in a dioid E of event operators5.The construction of EJδK is done so as to include dioid Max

in Jγ, δK [4][2]. The graphical representation ofseries in EJδK is homogeneous with the ones ofMax

in Jγ, δK and helps us to understand the simplification ruleson operators generated by B-TWEGs. The main feature is that the graphical representation of series inEJδK is three-dimensional : two dimensions to describe event operators in E and a third dimension for timeshift operators.

The paper is organised as follows. In section 2, the subclass of Balanced Timed and Weighted EventGraphs is first defined. Then, the modeling via an operatorial description is presented. Section 3 is devotedto define the dioid of formal series denoted EJδK and to associate a 3D graphical representation. In section4, the result about periodicity of transfer series of B-TWEGs is stated.

3bxc denotes the greatest integer less than or equal to x.4Thus, we also reduce the problems of liveness.5Event operators are the ones obtained by finite compositions of operators γn, βb and µm

2

2 Balanced Timed and Weighted Event Graphs (B-TWEGs)

2.1 Definitions

Weighted Event Graphs (WEGs) constitute a subclass of generalized Petri Nets given by a set of placesP = {p1, ..., pm} and a set of transitions T = {t1, ..., tn} (see [21] for a survey on Petri nets). An event graphcannot describe concurrency phenomenon, then every place pk ∈ P is defined between an input transitionti and an output transition to. The arcs ti → pk and pk → to are oriented and valued6 by strictly positiveintegers denoted respectively wi(pk) and wo(pk). A transition without input (resp. output) place is called asource or input (resp. sink or output) transition. An initial marking (a set of initial tokens depicted withblack dots) denoted M0(pk) is associated to each place pk ∈ P . A given transition tj is said enabled as soonas each input place pl contains at least wo(pl) tokens. A transition can be fired if it is enabled. At eachfiring of transition, wo(pl) tokens are removed from each input place pl, and wi(pk) tokens are added to eachoutput place pk.

Example 1 For the WEG depicted on fig. 1, t1 (resp. t3) is an input (resp. output) transition. The initialmarking is given by M0(p1) = 1 and M0(p2) = 2. All arcs are assumed to be 1-valued except when mentioned,for instance wi(p1) = 2 and wo(p1) = 3. Transition t3 is enabled when place p1 has 3 tokens and place p2has two tokens. The firing of transition t1 adds 2 tokens in place p1.

Figure 1: Weighted Event Graph

Definition 1 (Gain of a path) The gain of an elementary (oriented) path ti → pk → to is defined asΓ(ti, pk, to) , wi(pk)/wo(pk) ∈ Q. For a general path π passing through places pi, the gain corresponds tothe product of elementary paths, i.e. Γ(π) =

∏pj∈π wi(pj)/wo(pj).

Definition 2 (Neutral and Balanced WEG) A WEG is said neutral if all its circuits have a gain of 1.A WEG is said balanced if ∀ti, tj ∈ T , all the paths from ti to tj have the same gain.

Remark 1 A balanced WEG is necessarily neutral. In [17], a WEG which is neutral and strongly connectedis said unitary. A unitary WEG is necessarily balanced.

For a WEG, a holding time denoted ∆(pk) ∈ N can be associated to each place pk ∈ P . Each token enteringin a place pk has to wait ∆(pk) time units before contributing to enable the output transition. A WEG withholding times is called a Timed and Weighted Event Graph (TWEG). Hereafter, we will mainly considerBalanced Timed and Weighted Event Graphs (B-TWEGs).

Example 2 For the TWEG depicted on fig. 2, holding times are attached to some places: ∆(p1) = 2,∆(p6) = 1,∆(p4) = 1 and ∆(p5) = 2. This is a Balanced TWEG since it is neutral and all the parallel pathsfrom t1 to t4 have the same gain equal to 3/2. For instance, Γ(t1, p1, t2) = 1/2 and Γ(t1, p2, t3) = 3.

Remark 2 (Ordinary TEG) If all the existing arcs are 1-valued, the TWEG is said Ordinary, or simplyTimed Event Graph (TEG). A TEG is obviously a Balanced TWEG.

3

Figure 2: Balanced Timed and Weighted Event Graph

Definition 3 (Earliest Functioning) The earliest functioning of a TWEG consists in firing transitions7 as soon as they are enabled.

2.2 Operatorial representation of B-TWEG

First, we associate a counter function xi : Z → Z ∪ +∞ to each transition ti of a B-TWEG. The set ofcounter functions denoted Σ has a semimodule structure for the internal operation ⊕ = min and for thescalar operation defined by λ.x(t) = x(t) + λ. An operator is a map H : Σ → Σ which is said linear if∀x, y ∈ Σ, a) H(x ⊕ y) = H(x) ⊕H(y) and b) H(λ.x) = λ.H(x). An operator is said additive if only a) issatisfied.

Definition 4 (Dioid O of additive operators [18]) The set of additive operators on Σ, with the opera-tions defined below, is a non commutative complete dioid denoted O : ∀H1,H2 ∈ O

H1 ⊕H2 , ∀x ∈ Σ, (H1 ⊕H2)(x) = min(H1(x),H2(x))

H1 ◦ H2 , ∀x ∈ Σ, (H1 ◦ H2)(x) = H1(H2(x))

The null operator (neutral for ⊕ and absorbing for ◦) is denoted ε : ∀x ∈ Σ, (εx)(t) = +∞ and the unitoperator (neutral for ◦) is denoted e : ∀x ∈ Σ, (ex)(t) = x(t).

For the sequel, we will simply denote by Hx (instead of H(x)) the image of the counter x ∈ Σ by the additiveoperator H ∈ O. And we will also often omit the ◦ symbol for the product of O, H1H2 = H1 ◦ H2.

Definition 5 (Operators for B-TWEGs) The operators found in B-TWEGs are generated from a familyof additive operators defined by : let x ∈ Σ be a counter,

τ ∈ Z, δτ : ∀x, (δτx)(t) = x(t− τ)ν ∈ Z, γν : ∀x, (γνx)(t) = x(t) + νb ∈ N∗, βb : ∀x, (βbx)(t) = bx(t)/bc

m ∈ N∗, µm : ∀x, (µmx)(t) = x(t)×m.

Operators γ, βb and µm (and their ⊕ and ◦ compositions) are considered as event operators (E-operators).Two additive operators h1, h2 ∈ O are equal if ∀x ∈ Σ, h1x = h2x. The unit operator e is then defined ase = γ0 = δ0 = µ1 = β1.

6Graphically, the valuations are depicted directly on the arcs7except source transitions

4

Proposition 1 The next formal equivalences can be stated on operators

γnγn′

= γn+n′δtδt

′= δt+t

′(1)

γn ⊕ γn′

= γmin(n,n′) δt ⊕ δt′

= δmax(t,t′) (2)

γ1δ1 = δ1γ1 µmδ1 = δ1µm βbδ

1 = δ1βb (3)

µmγn = γm×nµm γnβb = βbγ

n×b (4)

Proof: For all counter x ∈ Σ we have (1) : ∀t, (x(t)+n′)+n = x(t)+(n′+n) and x(τ−t−t′) = x(τ−(t+t′)).(2) : ∀t,min(x(t)+n, x(t)+n′) = x(t)+min(n, n′). Since ∀t, x(t) ≥ x(t−1) (x is monotone non-decreasing),then min(x(τ − t), x(τ − t′) = x(τ −max(t, t′)). (3) : immediate (4): m× (x(t) +n) = m×x(t) +m×n and

bx(t)/bc+ n = bx(t)+n×bb c. �

Remark 3 We can note that the equalities (2) are those expressed by the simplification rules in Maxin Jγ, δK.

Definition 6 (Kleene star) The Kleene star of an operator in O is defined by : ∀h ∈ O,

h∗ =⊕i∈N

hi = e⊕ h⊕ h2 ⊕ ...

Theorem 1 On a complete dioid D, the implicit equation x = ax⊕ b has x = a∗b as least solution.

Proof: see [2] �

Theorem 2 For all operator h ∈ O, the next equalities are satisfied

h = h(δ−1)∗ = (δ−1)∗h = (γ1)∗h = h(γ1)∗.

Proof: Since a counter function is monotone, ∀x ∈ Σ, γx � x and δ−1x � x. Therefore, ∀x ∈ Σ, ∀h ∈ O,h(γ1)∗x = hx = (γ1)∗hx = h(δ−1)∗x = (δ−1)∗hx �

2.3 Modeling of B-TWEGs

The B-TWEGs are analysed here with the earliest functionning rule (see def 3). We can model a path of aB-TWEG by a product of elementary operators in O, the synchronization of parallel paths by a sum ⊕ andthe circuits by Kleene star of operators. Each elementary path ti → pk → tj of a B-TWEG, where M0(pk)is the initial marking of place pk and τ = ∆(pk) its holding time, can be described by the relation

xj = βw(pk,tj)γM0(pk)µw(ti,pk)δ

τxi, (5)

where xi (resp. xj) is the counter function associated to transition ti (resp. tj).

Example 3 (B-TWEG of Fig. 2) We can link the counter functions xi associated to the transitions tiof the B-TWEG depicted in Fig. 2 as follows. For the earliest functioning, we have

x2(t) = min(bx1(t−2)2 c, x2(t− 2) + 1)x3(t) = min(x1(t)× 3, x3(t− 1) + 2)

Therefore, the counter functions are linked by x2 = β2δ2x1⊕γ1δ2x2 and thanks to Th. 1, x2 = (γ1δ2)∗β2δ

2x1.Similarly, x3 = (γ2δ1)∗µ3x1. Finally, the counter function associated to the output transition is x4 =µ3x2 ⊕ β2γ1δ1x3 = (µ3(γ

1δ2)∗β2δ2 ⊕ β2γ1δ1(γ2δ1)∗µ3)x1. The input-output behavior (or transfer function)

of the B-TWEG is described by the rational expression µ3(γ1δ2)∗β2δ

2 ⊕ β2γ1δ1(γ2δ1)∗µ3 in O.

Theorem 3 (Transfer matrix of a B-TWEG) The behavior of a B-TWEG denoted G = (P, T ) is de-scribed by a matrix the elements of which belong to the rational closure of the set of operators OM,B ={ε, e, γ1, δ1, µ2, ..., µM , β2, ..., βB} where B = maxi,j w(pi, tj) and M = maxi,j w(tj , pi).

Proof: For each place pk we associate an operator µmγnβbδ

t (see (5)). Then, the different graph composi-tions (parallel, cascade, loop) are expressed by operations in {⊕, ◦, ∗}. Since a B-TWEG is a finite graph,the rationality is straightforward. �

5

3 Three dimensional representation of operators

According to (3) in prop. 1, operator δ1 can commute with all simple or composed event operator.For instance, δ1γ1δ2µ3β2δ

1 = γ1µ3β2δ4 = δ4γ1µ3β2. Hence, in every finite word h1h2...hn with hi ∈

{δt, γn, µm, βb}, we can factorize the time-shift operator. The rational expressions on OM,B can be con-sidered as formal power series in one variable δ where coefficients are some E-operators. Moreover, in theparticular case of B-TWEGs, the generated E-operators have a canonical form.

3.1 Bi-dimensional representation of E-operators

3.1.1 Event operators

The set of operators generated by finite sum and composition of operators γn, µm and βb has a dioidstructure.

Definition 7 (Dioid of E-operators E) We denote by E the dioid of operators obtained by finite sumand composition of operators in {ε, e, γn, µm, βb}, with n ∈ Z, and m, b ∈ N∗. The elements of E are calledE-operators hereafter.

Dioid E is a complete subdioid of O (additive operators). Since the ◦ operation is not commutative on E ,checking the equality of two E-operators is not immediate. Nevertheless, the comparison of E-operators ispossible thanks to an associate map called operator function. Since an E-operator h ∈ E induces modificationsonly on the event numbering (no time shift), we can describe its behavior by the means of a counter-to-counter function denoted Fh : Z → Z, ki 7→ ko which maps an input counter value to an output countervalue. For an E-operator, this input-output relation does not depend on time. An E-operator can beconsidered as an instantaneous system. For instance, the γ2 E-operator is described by Fγ2(ki) = ki + 2.This function can be interpreted as follows : for the γ2 E-operator, if ki input events have occurred at date t,then ki+2 output events have occurred at this date. Similarly, E-operator µ2β3γ is described by the functionFµ2β3γ(ki) = b(ki + 1)/3c × 2 (see fig. 3). Function Fh gives an unambiguous representation of E-operatorh. Moreover, we have Fh1⊕h2 = min(Fh1 ,Fh2) and Fh1◦h2 = Fh1 ◦ Fh2 The equality of E-operators can be

0 5

5

I-count

O-c

ount

0

0

5

I-count

O-c

ount

0

Figure 3: Representation of Fµ2β3γ1 and Fγ2β3µ4

checked thanks to the operator function : h1, h2 ∈ E , h1 = h2 ⇐⇒ Fh1 = Fh2 .

3.1.2 Periodic E-operators

The elementary E-operators γn, µm, βb are described by periodic8 operator functions, i.e. the associateoperator function satisfies ∀ki ∈ Z,F(ki + n) = F(ki) + n′. For E-operators γn, µm and βb we obtain

Fγn(0) = n,Fγn(ki + 1) = Fγn(ki) + 1Fµm(0) = 0,Fµm(ki + 1) = Fµm(ki) +m0 ≤ ki < b,Fβb(ki) = 0,Fβb(ki + b) = Fβb(ki) + 1

8More exactly, they are only quasi periodic

6

Operators γn and µm are 1-periodic, and operator βb is b-periodic. The set of periodic E-operators is denotedEper.

Definition 8 (Gain of h ∈ Eper) Let h ∈ Eper be a periodic E-operator s.t. Fh(ki + k) = Fh(ki) + k′. The

gain9 of h is defined as Γ(h) = k′

k . It is the slope of Fh.

Proposition 2 Let h1, h2 ∈ Eper be two periodic E-operators. We have

h1 ◦ h2 ∈ Eper and Γ(h1 ◦ h2) = Γ(h1).Γ(h2) (6)

if Γ(h1) = Γ(h2) then h1 ⊕ h2 ∈ Eper (7)

if Γ(h1) = Γ(h2) then h1 ∧ h2 ∈ Eper (8)

Proof: The periodic operator functions satisfy Fh1(ki+k1) = Fh1(ki)+k′1 and Fh2(ki+k2) = Fh2(ki)+k′2.Hence, Fh2(ki + k1.k2) = Fh2(ki) + k1.k

′2 and Fh1(Fh2(ki + k1.k2)) = Fh1(Fh2(ki) + k1.k

′2) = Fh1(Fh2(ki)) +

k′1.k′2 = Fh1h2(ki)+k′1.k

′2. Therefore, operator h1h2 is a periodic operator the gain of which is (k′1.k

′2)/(k1.k2).

For the sum of periodic operators with the same gain, we can write both operators with the same periodicity:Fh1(ki+k1.k2) = Fh1(ki)+k

′1.k2 and Fh2(ki+k1.k2) = Fh2(ki)+k

′2.k1 with k′1.k2 = k1.k

′2 (since both operators

have the same gain). Hence, the min of these two operator functions is also periodic. By symmetry, themax (∧) of two periodic E-operators with the same gain is also periodic. �

Remark 4 If Γ(h1) 6= Γ(h2), then h1 ⊕ h2 is not necessarily a periodic operator. Said differently, Eper isnot a subdioid of E.

Proposition 3 The event operators generated by B-TWEGs are periodic.

Proof: Due to the structural definition of the subclass of B-TWEGs (see def. 2), since parallel pathshave the same gain, only E-operators with the same gain are synchronized (with the ⊕ operation). Theperiodicity of E-operators is kept by the compositions of Balanced TWEGs. �

A k-periodic E-operator h ∈ Eper can be handled by the means of a finite representation : a pair (k, k′) ∈ N2

describing the rational gain Γ(h) = k′/k and the values of Fh(i) for one period i ∈ {0, ..., k − 1}. Moreover,a periodic function has a canonical form for which the period is minimal (it is also the case for ultimatelyperiodic formal series in Max

in Jγ, δK see [8]).

Definition 9 (Operator ∇m|b ∈ Eper) We denote by ∇m|b the composed b-periodic E-operator

∇m|b , µmβb.

For periodic operators of gain 1, we also use the notation ∇m , µmβm.

Definition 10 (Canonical form of h ∈ Eper) A periodic E-operator h s.t. Γ(h) = k′/k has a canonicalform which is a finite sum of ∇m|b operators

h =⊕i=N

i=1 γni∇m|bγn′i

where mb = k′

k and N, b are minimal.

As shown in the next example, the canonical form is not necessarily the most concise.

Example 4 To establish the canonical form of γ2β3µ4, we can graphically represent Fγ2β3µ4 which has aperiod equal to 3 and a gain equal to Γ(γ2β3µ4) = 4

3 (see fig. 3). Fγ2β3µ4(0) = 2,Fγ2β3µ4(1) = 3,Fγ2β3µ4(2) =4,Fγ2β3µ4(ki + 3) = Fγ2β3µ4(ki) + 4. Graphically, we can represent Fγ2β3µ4 as a min combination Fγ2β3µ4 =min(Fγ2µ4β3γ2 ,Fγ3µ4β3γ1 ,Fγ4µ4β3). Hence, we have the equality

γ2β3µ4 = γ2∇4|3γ2 ⊕ γ3∇4|3γ

1 ⊕ γ4∇4|3.

Remark 5 We can remark that the periodicity may be reduced by ⊕ combination, for instance we haveγµ2β2γ ⊕ γ2µ2β2 = γ∇2γ ⊕ γ2∇2 = γ.

9A path of a B-TWEG of which the gain is g is described by an E-operator of which the gain is g too.

7

3.1.3 Graphical considerations

The operator function leads to a graphical representation of periodic E-operators. Some features have to bekept in mind.

Partial order on Eper : the comparison of two periodic E-operators s.t. Γ(h1) = Γ(h2) is graphicallyinterpreted as follows

h1 � h2 ⇐⇒ min(Fh1 ,Fh2) = Fh2⇐⇒ epigraph(Fh1) ⊂ epigraph(Fh2)

Graphically, the sum of two E-operators amounts to do the union of their epigraphs10. On fig. 3 the epigraphcorresponds to the gray zone.

Left and right product by γn For h ∈ Eper,

Fγnh ⇐⇒ Fh vertically shifted of n units to the top

Fhγn′ ⇐⇒ Fh horizontally shifted of n’ units to the left .

3.2 Dioid EJδK

The previous subsection shows that E-operators generated by B-TWEGs are periodic and have a canonicalform. Moreover, all the E-operators commute with the time-shift operator δτ (see prop. 1). Hence, foroperators in the rational closure of {γn, µm, βb, δt}, the time-shift δτ can be factorized. Therefore, all theoperators generated by a B-TWEG can be described by the means of formal series in one variable δ denoted⊕

iwiδti , where coefficients wi are taken in Eper.

3.2.1 Three Dimensional representation of operators in B-TWEGs

By analogy with [3], we can describe discrete B-TWEGs as rational combination of periodic E-operatorsand time-shift operators.

Definition 11 (Dioid EJδK) The set of formal power series in one variable δ with exponents in Z andcoefficients in the non commutative complete dioid E, with the simplification rule: ∀s ∈ EJδK,

s = s(δ−1)∗ = (δ−1)∗s, (9)

is a non commutative complete dioid denoted EJδK. A series s ∈ EJδK is written s =⊕

t∈Z s(t)δt with

s(t) ∈ E. For two series s1, s2 ∈ EJδK :

(s1 ⊕ s2)(t) = s1(t)⊕ s2(t)(s1 ⊗ s2)(t) =

⊕τ+τ ′=t s1(τ) ◦ s2(τ ′)

The series of EJδK have a graphical representation which consists in describing for each t ∈ Z the valueof s(t) ∈ E . The convention adopted here is to represent s in a 3D basis, where t is described alongthe z-axis and coefficients s(t) ∈ E are represented by their operator function in the x × y basis (moreexactly, by the epigraph of the operator function). Moreover, according to Th. 2, a series is invariant by aproduct with (δ−1)∗ and by (γ1)∗. Then, each monomial s(t)δt of a series s generates a volume describedby s(t)δt ⊕ s(t)δt−1 ⊕ .... Actually, the 3D representation of a series in EJδK is a volume which looks like astaircase.

Example 5 The simple series (with only one term) γ2β3µ4δ5 ∈ EJδK is depicted on fig. 4. The graphical

representation of Fγ2β3µ4 (see fig. 3) is depicted in a 3D basis at height 5 (value of the time-shit operator).In order to improve the readability of the picture, the 3D representation is truncated to the positive values,i.e. to (x, y, z) ∈ N3.

8

Figure 4: 3D representation of γ2β3µ4δ5

Remark 6 (Simplifications in EJδK) The equivalences given in Th. 2 give some rules to simplify seriesin EJδK. Graphically, these equivalences are interpreted as follows. Two operators are equal if they havethe same 3D representation. Said differently, by considering a series s =

⊕s(t)δt of EJδK, if a terml

s(τ)δτ is not visible in the 3D representation of s, then it means that this monomial can be removed froms. For instance, let us consider the series γ2∇2δ

2 ⊕ γ1∇2γ1δ5 ⊕ γ2δ4. The representation of γ2δ4 is not

visible since it is hidden by those of γ1∇2γ1δ5 (see fig. 5). It means that the next simplification holds

γ2∇2δ2 ⊕ γ1∇2γ

1δ5 ⊕ γ2δ4 = γ2∇2δ2 ⊕ γ1∇2γ

1δ5. Finally, the main informations in a series of EJδK arethose coded by the vertices in the 3D representation (the vertices are depicted by blue balls).

Figure 5: Simplifications in EJδK

Due to the specific structure of B-TWEGs, we do not consider the whole set of series of EJδK but only theseries the coefficients of which are periodic E-operators. This subset is denoted EperJδK.

Definition 12 (Balanced series in EperJδK) A series s =⊕s(t)δt ∈ EperJδK is said balanced if all its

coefficients s(t) ∈ Eper have the same gain. The gain of s is denoted Γ(s) and corresponds to the gain of allits coefficients. A balanced series is said conservative if Γ(s) = 1.

3.2.2 Polynomials in EperJδK

The series of EperJδK that can be described by finite sums⊕N

i=1wi)δti are called polynomials. Balanced

polynomials have a canonical form. According to remark 6, it consists in keeping for each monomial wiδti

only information that is not yet contained in monomials wjδtj such that tj > ti. As said in remark 6, it

amounts to keep only visible vertices.

10epigraph(Fh1) , (ki, k) ∈ Z2s.t.k ≥ Fh1(ki).

9

Definition 13 Let us consider a balanced polynomial p =⊕i=T

i=1 wiδti ∈ EperJδK with wi ∈ Eper. The

canonical form of p is such that ∀i, ti < ti+1 and

wi =∧w

{w ⊕⊕j>i

wj =⊕k≥i

wk} (10)

Expression (10) conveys the fact that we only want to keep the essential information, i.e. coefficient wi onlykeeps the information not yet contained in the coefficients wj with j > i.

Example 6 The canonical form is obtained thanks to a backward analysis starting from the monomial withthe greatest exponent. For polynomial p = δ2⊕ (∇3γ

2⊕ γ2∇3)δ4⊕∇3γ

2δ7 depicted on Fig. 6, we obtain thenext simplifications. The monomial ∇3γ

2δ4 is not visible ( since ∇3γ2δ4 � ∇3γ

2δ7), so it can be removedfrom p. Then, the monomial δ2 has a non canonical expanded form δ2 = (γ2∇3 ⊕ γ∇3γ ⊕ ∇3γ

2)δ2. Theonly part of the dynamic of δ2 which is not yet described by γ2∇3δ

4 ⊕ ∇3γ2δ7 is described by the operator

γ∇3γδ2. Finally, we have

p = δ2 ⊕ (∇3γ2 ⊕ γ2∇3)δ

4 ⊕∇3γ2δ7

= γ∇3γδ2 ⊕ γ2∇3δ

4 ⊕∇3γ2δ7

Figure 6: Polynomial γ∇3γδ2 ⊕ γ2∇3δ

4 ⊕∇3γ2δ7

4 B-TWEGs are described by ultimately periodic series of EperJδKIn this section, we show that the behavior of B-TWEGs is described by ultimately periodic and balancedseries of EperJδK. This result has to be compared to the well known result for unweighted Timed EventGraphs (TEGs) : the entries of the transfer matrix of a TEG are ultimately periodic series of Max

in Jγ, δK.For TEG, operations on ultimately periodic series of Max

in Jγ, δK have already been studied in [2] [4][8] [10][15] [6] [11].

Since we consider only B-TWEGs, only balanced series of EperJδK are considered hereafter.

Definition 14 (Ultimately periodic series of EperJδK) A balanced series s ∈ EperJδK is said ultimatelyperiodic if it can be written as s = p⊕q(γνδτ )∗, where p and q are balanced polynomials such that Γ(p) = Γ(q),

p =⊕i=1..n

wiδti q =

⊕j=1..N

WjδTj ,

wi,Wj ∈ Eper.

10

The property of periodicity has a natural graphical interpretation. For the 3D representation of s, therepresentation of q(γνδτ )∗ = q ⊕ qγνδτ ⊕ qγ2νδ2τ ⊕ ... is a periodic staircase. The polynomial q is depictedas a group of steps that is repeated periodically (we have the same steps but shifted by τ units to the topand by ν units toward the decreasing I-count values).

Example 7 Fig 7 gives the graphical description of s = γ2∇3|2δ3 ⊕ γ4∇3|2γ

1δ4 ⊕ [(γ6∇3|2γ1 ⊕ γ7∇3|2)δ

6 ⊕γ7∇3|2γ

1δ7](γ4δ3)∗. From the T-shift value equals to 6, we have the same two-steps repeated each 3 units tothe top but shifted by 4 units toward the decreasing I-count values.

Figure 7: Ultimately periodic series in EperJδK

Even if the product of EJδK is not commutative, an ultimately periodic balanced series of EperJδK has twoperiodic forms.

Proposition 4 (Left/Right periodicity) An ultimately (right) periodic series s = p⊕q(γνδτ )∗ in EperJδKhas also an ultimately left periodic form s = p⊕ (γν

′δτ′)∗q′ where q′ is a balanced polynomial. The left (resp.

right) asymptotic slope is defined as σl(s) = τ ′/ν ′ (resp. σr(s) = τ/ν), and the next equality is satisfiedΓ(s) = σr(s)/σl(s).

Proof: Let Γ(s) = k′/k be the gain of s. The coefficients of polynomial q =⊕wjδ

tj in their canonical

form are given by wj =⊕

i γnij∇mj |bjγ

n′ij with k′/k = mj/bj . Let us remark that thanks to (4), ∇mj |bjγbj =

µmjβbjγbj = µmjγ

1βbj = γmjµmjβbj = γmj∇mj |bj . More generally, ∇mj |bjγαbj = γαmj∇mj |bj . Therefore, if

we take B = lcm(bj) and M = B.k′/k, then ∀i, j, γnij∇mj |bjγn′ijγB = γMγnij∇mj |bjγ

n′ij , and consequently

∀i, wiγB = γMwi. Since we can develop (γνδτ )∗ = (e⊕ γνδτ ⊕ ...⊕ γ(B−1)νδ(B−1)τ )(γBνδBτ )∗, then

q(γνδτ )∗ = q(e⊕ ...⊕ γ(B−1)νδ(B−1)τ )(γBνδBτ )∗

= q(γBνδBτ )∗(e⊕ ...⊕ γ(B−1)νδ(B−1)τ )= (γMνδBτ )∗q(e⊕ ...⊕ γ(B−1)νδ(B−1)τ )= (γMνδBτ )∗q′

Finally, σr(s) = τ/ν and σl(s) = (Bτ)/(Mν) and σr(s)/σl(s) = Γ(s) = k′/k. �As for series inMax

in Jγ, δK, the ultimately periodic series of EperJδK have different expressions. But, we canprovide two canonical forms (left/right) where the periodicity is minimal.

Definition 15 (Canonical forms) An ultimately periodic and balanced series of EperJδK has a left and aright canonical form for which the degree of p (resp. p′) is minimal and the value of ν (resp. ν ′) is minimal.

Example 8 Let us consider the periodic series given by its canonical left form s = γ1β3µ2δ1⊕(γ4δ3)∗(γ3β3µ2δ

3⊕γ5µ2β3γ

1δ4) which is depicted in fig. 8. The gain of s is Γ(s) = 2/3. The canonical right form of this seriesis s = γ1β3µ2δ

1 ⊕ (γ3β3µ2δ3 ⊕ γ5µ2β3γ1δ4)(γ6δ3)∗. We check that Γ(s) = σr(s)/σl(s) = [3/6]/[3/4] = 4/6.

11

Figure 8: Ultimately periodic series s = γ1β3µ2δ1 ⊕ (γ4δ3)∗(γ3β3µ2δ

3 ⊕ γ5µ2β3γ1δ4)

The next lemma is given in Gaubert’s PhD thesis [8].

Lemma 1 Let m1 = γn1δt1(γν1δτ1)∗ and m2 = γn2δt2(γν2δτ2)∗ be two simple periodic series in Maxin Jγ, δK

such that τ1/ν1 > τ2/ν2. There exists an integer K such that

γn2δt2γKν2δKτ2(γν2δτ2)∗ � m1.

The next lemma is an extension of lemma 1 to some simple series in EperJδK.

Lemma 2 Let m1 = w1(γν1δτ1)∗ and m2 = w2(γ

ν2δτ2)∗ be two simple series in EperJδK where w1, w2 ∈ Eperare periodic E-operators such that Γ(w1) = Γ(w2). If series m1 and m2 are such that τ1/ν1 > τ2/ν2, thereexists a positive integer K such that

w2γKν2δKτ2(γν2δτ2)∗ � m1.

Proof: If w1 and w2 are periodic E-operators and have the same gain, then we can find an integer N s.t.w2γ

N � w1. Graphically, we can shift w2 to the right to become less than w1. Therefore, w2γN (γν1δτ1)∗ �

w1(γν1δτ1)∗. By applying Lemma 1, γN (γν1δτ1)∗ is asymptotically greater than m2. Therefore m1 is also

asymptotically greater than m2. �

Proposition 5 Let us consider two ultimately periodic and balanced series of EperJδK s1 = p1 ⊕ q1(γν1δτ1)∗

and s2 = p2 ⊕ q2(γν2δτ2)∗. If Γ(s1) = Γ(s2) then s1 ⊕ s2 is an ultimately periodic and balanced series ofEperJδK such that

σr(s1 ⊕ s2) = max(σr(s1), σr(s2))

Proof: This proof is similar to the one for ultimately periodic series of Maxin Jγ, δK (see [8]). Let us remark

first that if Γ(s1) = Γ(s2), then s1 ⊕ s2 is balanced and such that Γ(s1 ⊕ s2) = Γ(s1) = Γ(s2). Two caseshave to be considered. If σr(s1) = σr(s2), then we can expand each series with a non canonical right forms1 = P1 ⊕Q1(γ

lcm(ν1,ν2)δlcm(τ1,τ2))∗ and s2 = P2 ⊕Q2(γlcm(ν1,ν2)δlcm(τ1,τ2))∗ with the same periodicity. The

sum is then ultimately periodic. If σr(s1) > σr(s2), lemma 2 shows that for an integer K large enough, seriess1 dominates s2 i.e. s1 � q2(γKν2δKτ2)(γν2δτ2)∗. Expressed differently, s1⊕s2 = s1⊕p2⊕q2⊕q2γν2δτ2⊕ ...⊕q2γ

(K−1)ν2δ(K−1)τ2 , which is an ultimately periodic series. In each case, σr(s1 ⊕ s2) = max(σr(s1), σr(s2))and also σl(s1 ⊕ s2) = max(σl(s1), σl(s2)). �

Example 9 Let us consider series s1 = γ3µ2β2(γ2δ3)∗ and s2 = µ2β2γ

1δ1(γ2δ2)∗. The sum s1 ⊕ s2 isdepicted in fig. 9. Series s1 dominates s2 asymptotically since σr(s1) = 3/2 is greater than σr(s2) =2/2. If we develop series s1, we obtain s1 = γ3µ2β2 ⊕ γ3µ2β2γ

2δ3 ⊕ γ3µ2β2γ4δ6 ⊕ ... ⊕ γ3µ2β2γ

10δ15 ⊕γ3µ2β2γ

12δ18 ⊕ ... = γ3µ2β2 ⊕ γ5µ2β2δ3 ⊕ γ7µ2β2δ6 ⊕ ... ⊕ (γ2δ3)∗γ13µ2β2δ15. We graphically see that the

terms in (γ2δ3)∗γ13µ2β2δ15 are definitely greater than the terms in s2. Finally, s1 ⊕ s2 = µ2β2γ

1δ1 ⊕µ2β2γ

3δ3 ⊕ ...⊕ (γ2δ3)∗γ13µ2β2δ15.

12

Figure 9: The sum of s1 = γ3µ2β2(γ2δ3)∗ and s2 = µ2β2γ

1δ1(γ2δ2)∗

Proposition 6 Let s1 and s2 be two ultimately periodic and balanced series of EperJδK. Then, s1 ⊗ s2 is anultimately periodic and balanced series s.t. Γ(s1 ⊗ s2) = Γ(s1).Γ(s2). The left and the right slope of s1⊗ s2are given by

σr(s1 ⊗ s2) = max(σr(s2),Γ(s2)× σr(s1))σl(s1 ⊗ s2) = max(σl(s1), σl(s2)/Γ(s1))

Proof: We can write s1 and s2 with their right and left forms :

s1 ⊗ s2 = (p1 ⊕ q1(γν1δτ1)∗)

⊗(p2 ⊕ (γν′2δτ′2)∗q′2)

= p1p2 ⊕ p1(γν′2δτ′2)∗q′2

⊕q1(γν1δτ1)∗p2⊕q1(γν1δτ1)∗(γν

′2δτ′2)∗q′2

Series p1(γν′2δτ

′2)∗q′2 and q1(γ

ν1δτ1)∗p2 are finite sums of periodic series, due to prop. 5, the result is periodic.The last term (γν1δτ1)∗(γν

′2δτ′2)∗ is also an ultimately periodic series in Max

in Jγ, δK (see [8]), and therefore inEJδK too. �

Finally, the computation of the transfer relation of a B-TWEG also needs the computation of the Kleenestar of some conservative (Γ(s) = 1) ultimately periodic and balanced series s ∈ EperJδK. Indeed, the Kleenestar operation describes the behavior of the circuits of the B-TWEG graph that are necessarily neutral. Thisproperty is the more technical to check and needs several intermediate results.

Lemma 3 Let p =⊕I

i=1(⊕Ji

j=1 γnj∇miγ

n′j )δti be a conservative balanced polynomial of EperJδK. Polynomialp can be written with a non canonical form as

p =K⊕k=1

γnk∇Mγn′kδtk

where M = lcm(mi).

Proof: It suffices to remark that we can develop ∇m according to a non canonical form

∇m =

j=n−1⊕j=0

γj×m∇n×mγ(n−1−j)×m.

Then, each coefficient wi of p can be developped as a sum of γn∇Mγn′

operators with M = lcm(mi). �

13

Example 10 Let p = γ2∇3γ2δ2 ⊕ γ3∇2γ

1δ3. We can write p as a sum of γn∇lcm(2,3)γn′ operators. On

the one hand, γ2∇3γ2δ2 = γ2(∇6γ

3 ⊕ γ3∇6)γ2δ2. On the other hand, γ3∇2γ

1δ3 = γ3(∇6γ4 ⊕ γ2∇6γ

2 ⊕γ4∇6)γ

1δ3. Finally, p = γ2∇6γ5δ2 ⊕ γ5∇6γ

2δ2 ⊕ γ3∇6γ5δ3 ⊕ γ5∇6γ

3δ3 ⊕ γ7∇6γ1δ3.

Lemma 4 The next equality is satisfied

∇MγK∇M = γbKcM∇M = ∇MγbKcM

where bKcM = bK/Mc ×M is the greatest integer in MZ less than K.

Proof: For all counter function x ∈ Σ, we have (∇Mx)(t) = bx(t)cM . Then, (∇MγK∇Mx)(t) = bbx(t)cM +KcM = bbx(t)cM + bK/Mc ×M + (K mod M)cM = bbx(t)cMcM + bKcM = bx(t)cM + bKcM . �

Thanks to Lemma 3, one can consider that a polynomial p can always be written as

p =i=N⊕i=1

wiδti with wi = γni∇Mγn

′i ,

i.e. a form where the E-operators wi are constituted with the same ∇M operator (see Example 10). Then,the Kleene star of p can be seen as an infinite sum given by

p∗ = e⊕ p⊕ p2 ⊕ ...= e⊕

⊕Ni1=1wi1δ

ti1

⊕[⊕N

i1=1

⊕Ni2=1wi1δ

ti1wi2δti2 ]

⊕...⊕[⊕N

i1=1 ...⊕N

iK=1

⊗j=Kj=1 wijδ

tij]

⊕...

It is an infinite sum of operators such as⊗j=K

j=1 wijδtij that are obtained by K products of operators wiδ

ti

(i ∈ {1, .., N}), with K a finite integer.First, we can focus on the product of K operators wijδ

tij , where each index ij is taken in {1, .., N},⊗j=Kj=1 wijδ

tij = wi1wi2 ...wiKδ(ti1+ti2+...+tiK )

= γni1∇Mγn′i1 ...γniK∇Mγ

n′iK δ(ti1+...+tiK )

By applying Lemma 4, we can simplify as follows⊗j=Kj=1 wijδ

tij = γκδτ ⊗ γni1∇Mγn′iK

withκ = bn′i1 + ni2cM + ...+ bn′iK−1

+ niKcM=

∑j=K−1j=1 bn′ij + nij+1cM .

τ =∑j=K

j=1 tij

Finally, we can express such an operator as⊗j=Kj=1 wijδ

tij =⊗j=K

j=1 γnij∇Mγn′ij δtij

=[⊗K−1

j=1 γbn′ij+nij+1

cMδtij

]γni1∇Mγ

n′iK δtiK(11)

From (11), it comes that the operators⊗j=K

j=1 wijδtij such that i1 = I and iK = J are all written

γκδτγnI∇Mγn′J δtJ . Therefore, we can give a rational expression for this sum.

14

Lemma 5 Let K be a finite integer, then we have⊕ij∈{1,..,N}

{⊗j=K

j=1 wijδtij |i1 = I, iK = J}

= (ϕK−1)IJγnI∇Mγn

′J δtJ

with ϕ a square matrix of Maxin Jγ, δKN×N defined by

ϕab = γbn′a+nbcM δta

Proof: According to that definition of ϕ, the (K − 1)-th exponent of matrix ϕ satisfies

(ϕK−1)IJ =⊕ij∈{1,..,N}

{⊗K−1

j=1 γbn′ij+nij+1

cMδtij |i1 = I, iK = J}

By extending the expression obtained in (11) for one operator⊗j=K

j=1 wijδtij , the result is straightforward.

�

Lemma 6 The infinite sum⊕+∞

K=1[⊗K

i=1wiδti ] such that ∀K, i1 = I and iK = J is described by a rational

expression ⊕K≥1

[ ⊕ij∈{1,..,N}

{⊗j=K

j=1 wijδtij |i1 = I, iK = J}

]= (ϕ∗)IJγ

nI∇Mγn′J δtJ

Proof: By using Lemma 6 for operators⊗j=K

j=1 wijδtij where K is an integer greater or equal than 1,we

obtain ⊕K≥1

[⊕ij∈{1,..,N}{

⊗j=Kj=1 wijδ

tij |i1 = I, iK = J}]

=⊕

K≥1

[(ϕK−1)IJγ

nI∇Mγn′J δtJ

]= (ϕ0 ⊕ ϕ1 ⊕ ...)IJγnI∇Mγn

′J δtJ

= (ϕ∗)IJγnI∇Mγn

′J δtJ .

�

Proposition 7 Let us consider a conservative polynomial p =⊕i=N

i=1 wiδti with ∀i ∈ {1, .., N}, wi =

γni∇Mγn′i. The Kleene star of p is a conservative and ultimately periodic series in EperJδK given by

p∗ = e⊕i=N⊕i=1

j=N⊕j=1

(ϕ∗)ijγni∇Mγn

′jδtj

. (12)

Proof: Thanks to Lemma 6, the Kleene star is obtained by enumerating all the N2 cases where the leftand the right factors are one of the N terms wiδ

ti with i ∈ {1, .., N}. The Kleene star of matrix ϕ, since itis a matrix of Max

in Jγ, δK (without ∇M operator), is known to be composed of periodic series of Maxin Jγ, δK.

Then, the Kleene star p∗ can be obtained by the sum of N2 ultimately periodic series of EperJδK. Thanks toproposition 5, the result is periodic too. �

Proposition 8 Let s = p⊕ q(γνδτ )∗ be a conservative balanced (Γ(s) = 1) and ultimately periodic series inEperJδK. Then s∗ is also a conservative balanced and ultimately periodic series in EperJδK.

15

Proof: Thanks to lemma 4, we can write p =⊕

i γni∇Mγn

′iδti and q =

⊕j γ

Nj∇MγN′jδTj , i.e. as decom-

posed on the basis of the same ∇M operator. If we take r = γMνδMτ then pr = rp and qr = rq, i.e. themonomial r can commute with polynomials p and q. By setting r = γMνδMτ , then we have

(γνδτ )∗ = (e⊕ γνδτ ⊕ γ2νδ2τ ⊕ ...⊕ γ(M−1)νβ(M−1)τ )r∗

and consequently we can rewrite the periodic conservative series as s = p⊕q(e⊕γνδτ⊕...⊕γ(M−1)νδ(M−1)τ )r∗ =p ⊕ q′r∗. Moreover r also commutes with q′, i.e. q′r = rq′. Therefore, since (a ⊕ b)∗ = a∗(ba∗)∗, thens∗ = (p⊕ q′r∗)∗ = p∗(q′r∗p∗)∗. Moreover, since rp = pr, then r∗p∗ = (r ⊕ p)∗. So, s∗ = p∗(q′(r ⊕ p)∗)∗. In anon commutative dioid, one also have (ab∗)∗ = e⊕a(a⊕b)∗. Thus, we obtain that s∗ = p∗(e⊕q′(q′⊕r⊕p)∗).Since q′ ⊕ r ⊕ p is a conservative polynomial, then (q′ ⊕ r ⊕ p)∗ is a periodic series (see prop. 7). Finally,since the product of periodic series is periodic too (see Proposition 6), then the Kleene star of a conservativebalanced and ultimately periodic series is a balanced and ultimately periodic series. �

Proposition 9 (The transfer matrix of a B-TWEG) The transfer matrix of a B-TWEG is composedof balanced and ultimately periodic series of EperJδK.

Proof: Due to the structure of a B-TWEG, the rational expressions generated are such that the sum ofsame gain balanced periodic series are done, product of balanced periodic series are done and Kleene starof conservative balanced periodic series are done. �

5 Applications

5.1 Example of fig. 2

First, we state the transfer relation of fig. 2 in its canonical form. We obtained its transfer relation inexample 3, x4 = µ3(γ

1δ2)∗β2δ2 ⊕ β2γ1δ1(γ2δ1)∗µ3x1, say x4 = H1x1. The gain of series H1 is clearly the

gain of all paths from t1 to t4, Γ(H1) = 3/2.Series H1 is depicted in its 3D representation in Fig. 10. The left and the right canonical forms of H1 are

given below (where coefficients are also described in their canonical form in Eper)

H1 = p⊕ q(γ2δ3)∗ = p⊕ (γ1δ1)∗q′

withp = ∇3|2δ

2 ⊕ γ2∇3|2γ1δ3 ⊕ γ3∇3|2δ

4

⊕γ4∇3|2γ1δ5 ⊕ (γ5∇3|2γ

1 ⊕ γ6∇3|2)δ6

q = [(γ6∇3|2γ1 ⊕ γ8∇3|2)δ

7

⊕(γ7∇3|2γ1 ⊕ γ9∇3|2)δ

8]

q′ = [(γ6∇3|2γ1 ⊕ γ8∇3|2)δ

7]

The left and the right slopes are given by σr(H1) = 3/2 and σl(H1) = 1/1.

5.2 Kleene star calculus

A second example of transfer is given here. The B-TWEG depicted on fig. 11 has a circuit which needs thecomputation of a Kleene star in EperJδK.

The modeling of this B-TWEG leads to the next equations

x1 = u⊕ β3γ5µ2x3x2 = β2µ3x1 ⊕ γ2δ2x2x3 = δ3x2y = x3

16

Figure 10: Transfer series of the B-TWEG of Fig. 2

Figure 11: Balanced Timed and Weighted Event Graph

By applying Th. 1 to eliminate implicit relations, the B-TWEG transfer is described by the next rationalexpression

y = H2u = δ3(γ2δ2)∗β2µ3[β3γ5µ2δ

3(γ2δ2)∗β2µ3]∗u

We first compute the canonical form of β3γ5µ2δ

3(γ2δ2)∗β2µ3 = (γ4δ6)∗(γ1δ3 ⊕ γ3∇2δ5 ⊕ γ4δ7) Then ,

(β3γ5µ2δ

3(γ2δ2)∗β2µ3)∗ = ((γ4δ6)∗(γ1δ3 ⊕ γ3∇2δ

5 ⊕ γ4δ7))∗ = e⊕ (γ1δ3 ⊕ γ3∇2δ5 ⊕ γ4δ7)(γ1δ3 ⊕ γ3∇2δ

5 ⊕γ4δ7 ⊕ γ4δ6)∗ = (γ1δ3 ⊕ γ3∇2δ

5 ⊕ γ4δ7)∗.Finally, the result in its canonical (left/right) forms is

H2 = (γ3δ6)∗[(∇3|2γ1 ⊕ γ1∇3|2)δ

3

⊕(γ1∇3|2γ1 ⊕ γ3∇3|2)δ

6]

= (∇3|2γ1 ⊕ γ1∇3|2)δ

3(γ1δ3)∗

This series is depicted on fig. 12.

Figure 12: Transfer series of B-TWEG in fig. 11

17

6 Conclusion

This work presents a modeling approach for the class of Balanced Timed and Weighted Event Graphs (B-TWEGs) in a dioid of additive operators. Four elementary operators denoted γn, δt, µm and βb are necessaryto describe the dynamical phenomena modeled by a B-TWEG. The input-output behavior of B-TWEGscan be embedded into some rational formal series in a dioid denoted EJδK. Each formal series has a naturalthree dimensional graphical representation which has an ultimate periodicity property. This work providesa natural extension of the max-plus theory for Timed Event Graphs to a class of weighted TEGs.

References

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