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L IST OF F IGURES 1 2 3 4 5 6 7 8 9 10

A simple TEG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . PN model of an invariant resource sharing problem. . . . . . . . . A split model of an invariant resource sharing problem. . . . . . . A chocolate factory. . . . . . . . . . . . . . . . . . . . . . . . . . . PN model of a chocolate factory. . . . . . . . . . . . . . . . . . . . Machine scheduling in a chocolate factory. . . . . . . . . . . . . . PN a model of an invariant resource sharing problem. . . . . . . . Gantt chart of an invariant resource sharing problem. . . . . . . . . Resources occupancy of an invariant resource sharing problem. . . Resource usage schedule of an invariant resource sharing problem.

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L IST OF TABLES I II III IV

Semantics of figure 2. . . . . . . . . . . . . . . . . . . . . . . Semantics of a railway station PN model. . . . . . . . . . . . Schedule table of an invariant resource sharing problem. . . . Final schedule table of an invariant resource sharing problem.

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A dioid model for invariant resource sharing problems Aurélien Corréïa, Abdeljalil Abbas-Turki, Rachid Bouyekhf and Abdellah El Moudni

Abstract— This paper proposes a model for invariant resource sharing problems in dioid algebra. A strong motivation for investigating the issue is the absence of a general systematic technique which can be used to tackle these problems. (min, +) constraints have been developed to handle resource sharing in Discrete Event Dynamic Systems. In particular, the part that can be modeled by a Timed Event Graph induce (min, +)-linear equations which are constrained by the resource availability. The proposed algebraic model has been proved to describe the actual behavior of the systems dealt with. This paper will show two examples of systems that are modeled and controlled by means of this approach. Index Terms— resource sharing, Petri net, dioid algebra.

EDICS Category: 3-BBND I. I NTRODUCTION ISCRETE Event dynamic Systems (DEDSs) are processes characterized by sequences of events [1]. Whenever two or more users need the same unique resource, a resource sharing problem may occur. Such a problem is usually handled by an operations research technique, where the proposed models are defined through an empirical approach. Thus, each model is only applicable to a particular problem class that depends on the dynamics of the system [2] ,while, for the sake of generality, the approach must include the complete behavior of all the users. The above-mentioned situation has motivated many researchers to develop methods and techniques in order to circumvent the difficulty involved in DEDSs. On the one hand, in [3], [4], algorithms of mutual exclusion have been developed for computer science related issues. However, these are not suitable for analytical modeling. On the other hand, some graphical tools have been proposed to represent the resource sharing problems such as Petri Nets (PNs). In particular, in [5], [6], the authors have avoided the problem of resource sharing by adding a controller to the system. However, such an approach prevents us from gaining a clear insight into the influence of various factors on the behavior of the system as a whole. In [7] and [8], PN-based models which describe parallel and general mutual exclusion phenomena have been proposed, respectively. The main contribution of these papers consists of a simulation of the system. However, although simulation-based approaches are still widely used to evaluate

D

A. Corréïa is with Hitachi Europe SAS [email protected] A. Abbas-Turki, R. Bouyekhf and A. El Moudni are with Université de Technologie de Belfort-Montbéliard (UTBM), 90010 Belfort cedex, France

and compare different scenarios, there is increasing demand for a development of mathematical formulations that produce high-quality solutions to complex problems.The Timed Event Graph (TEG) is a class of the deterministic timed PN that fulfills these two objectives by both providing a useful mathematical specification mechanism and a clear human interface for an analysis and a simulation of the studied systems (see [9, p. 554] and [10, pp. 27 & 125]). Indeed, it admits a linear mathematical modeling in dioid algebra which provides state equations that describe the evolution of the system. However, each place of a TEG has exactly one upstream and one downstream transition. Thus, it can cope with synchronization and parallelism, but it cannot handle concurrency. Several efforts have been made to tackle this modeling constraint. In particular, [11] treated the problem by transforming the PN into a marked graph. However, the authors impose attribution of shared resources to the users following a stringent sequence. Thus, such an approach does not allow us to take into account the many possibilities that the system offers. In [12], the authors proposed a switching (max, +)linear system. They consider a DEDS that can switch between different operation modes. However, when we try to model systems with many shared resources, the number of operating modes can dramatically increase, exponentially even. Hence, we immediately meet computational difficulties, which rapidly grow as the scale of the systems increases. Another notable work in this area is the one presented in [13]. It proposes a combination of TEGs and heap automata that are controlled by an algorithm in dioid algebra based on the Johnson rule. According to the above-mentioned paper, TEGs and Timed State Graphs (TSGs) must be arranged in separate stages. However, as its authors point out, this is a limitation which prevents us from modeling general cases. The present paper endeavors to link dioid algebra to invariant resource sharing problems in DEDSs. More precisely, it proposes a general systematic technique for modeling the entire behavior of Invariant Resource Sharing Systems (IRSSs) in dioid algebra. With such a mathematical model in hand, we gain a fresh perspective on the set of DEDSs that we are working on.

II. R EVIEW OF DIOID ALGEBRA AND TEG This section introduces some terminology and notations of dioid algebra, mostly taken from [1].

c 2007 IEEE 0000–0000/00$00.00

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A. Dioid algebra A set A is said to be ordered if there exists a binary relation  such that the following conditions are satisfied for all x, y and z in A: • reflexive: every element is in relation with itself (x  x); • antisymmetric: if x  y and y  x then x = y; • transitive: if x  y and y  z then x  z. Let (A, A ) be an ordered set, B ⊂ A a non-empty subset of A, and a, b ∈ B. • An element x ∈ A satisfying ∀b ∈ B, b  x is called majorant of set B; • An element y ∈ A satisfying ∀b ∈ B, y  b is called minorant of set B. In particular, if the upper bound (i.e. the least majorant) and/or lower bound (i.e. the greatest minorant) of set {a, b} exist, we denote them by a ∨ b and a ∧ b, respectively. A dioid (D, ⊕, ⊗) is a set D equipped with two operations, ⊕ and ⊗, satisfying the following axioms, for all a, b, c ∈ D [14]: • addition ⊕ is associative (i.e. (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c) = a⊕b⊕c), commutative (i.e. a⊕b = b⊕a) and idempotent (i.e. a ⊕ a = a); • multiplication ⊗ is associative and distributes with respect to addition (i.e. a ⊗ (b ⊕ c) = a ⊗ b ⊕ a ⊗ c); • D has a neutral element, denoted ε: a⊕ε = a and a⊗ε = ε ⊗ a = ε; • D has an identity element, denoted e: a ⊗ e = e ⊗ a = a; In the following, the symbol ⊗ may be omitted and notation (D, ⊕, ⊗) may be abbreviated to D, when their meaning is clear from the context. A dioid D is an ordered set for the relation  defined by ∀a, b ∈ D, a  b ⇔ b = a ⊕ b; a ⊕ b is the least upper bound of set {a, b}. A dioid D is complete if it is closed for infinite sums and if multiplication distributes with respect to infinite sums. In such a dioid, the top element, denoted ⊤, exists and is equal to the sum of all elements in D. A dioid D is commutative if multiplication is commutative. The complete and commutative dioid Zmin , commonly known as (min, +) algebra, is the set Z = Z ∪ {−∞, +∞} containing the following operations [1], [14]: a⊕b

=

min(a, b)

a⊗b

= a+b ,

We commence by recalling some PN terminology and notations. A PN is a four-tuple N = (P, T, F, M0 ) where: • P = {P1 , P2 , . . . , Pm } is a finite set of places (represented by circles), • T = {T1 , T2 , . . . , Tn } is a finite set of transitions (represented by line segments), • F ⊆ (P ×T )∪(T ×P ) is a set of directed arcs F(Pi , Tj ) from places to transitions and F(Ti , Pj ) from transitions to places, • M0 is an initial marking that associates zero or more tokens to each place. Furthermore, n ≥ 0, m ≥ 0, n + m ≥ 1 and P ∩ T = ∅. The state of a PN is defined by the number of tokens in each place and is represented by a vector Mt = [Mt (P1 ), Mt (P2 ), . . . , Mt (Pm )]T , called the marking vector of the PN, where Mt (Pi ) is the number of tokens in place Pi at date t. A transition Tj ∈ T is said to be enabled iff Mt (Pi ) ≥ 1 at t, for all i, j such that F(Pi , Tj ) ∈ F . An enabled transition may fire. When a transition Tj fires, a token is removed from each of its input places and a token is added to each of its output places. A Timed Event Graph (TEG) is a PN in which every place has exactly one input transition and one output transition. It is possible to write the (min, +)- or (max, +)-linear state equations that describe the firings of the transitions of a TEG as long as it follows the Just In Time (JIT) operational rule. To obtain a linear model, we can associate for each transition x either the counter function, which is a mapping Z → Zmin , t 7→ x(t), which gives the number of times x has been fired until date t, or dually the dater function, which is a mapping Z → Zmax , k 7→ x(k) that gives the date of its k th firing. Since in DEDS, states are finite and time is discrete, we assume daters and counters have integer values. We will illustrate this by means of the following example. 2

u1

y

u2 Fig. 1.

[A ⊕ B]ij =aij ⊕ bij = min(aij , bij ) , n M aik ⊗ ckj = min (aik + ckj ), [A ⊗ C]ij = m×n

B. State equations of a TEG

,

for “scalars” a, b ∈ Z, and

k=1

2

k=1,...,n

Consider the TEG presented in figure 1. The behavior of transition y is described as follows: • the associated dater is given by

n×p

for matrices A, B ∈ Z and C ∈ Z . Its neutral element is ε = +∞, its identity element is e = 0 and its top element is ⊤ = −∞. The dual dioid Zmax , also called (max, +) algebra, is obtained by replacing min with max in all previous equations. Its neutral element is ε = −∞, its identity element is e = 0 and its top element is ⊤ = +∞.

A simple TEG.

y(k) = max(u1 (k) + 2, u2 (k − 1))

,

= 2 ⊗ u1 (k) ⊕ u2 (k − 1) in Zmax ; •

the associated counter is given by y(t) = min(u1 (t − 2), u2 (t) + 1)

,

= u1 (t − 2) ⊕ 1 ⊗ u2 (t) in Zmin .

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Now, like the z-transform for series in classical algebra, the γ- and δ-transforms allow us to translate dater and counter functions, respectively, to formal series where: • γ is the backward shift operator in the event domain, e.g. γ x(k − 1) −→ γx(γ), • δ is the backward shift operator in the time domain, e.g. δ x(t − 1) −→ δx(δ). Hence, the behavior of a whole TEG can be described by state equations that are linear in a dioid of formal series in two commutative variables γ, δ with exponents in Z and with Boolean coefficients [15]. This dioid, denoted by Max in Jγ, δK, allows to study both time and event domains. An element L f (n , t )γ ni δ ti s ∈ Max Jγ, δK is defined by s(γ, δ) = i i in i∈Z 2 with f : Z → B a Boolean mapping. When an element of Max in Jγ, δK is used to code information concerning a transition of a TEG, then each of its monomial γ k δ t may be interpreted as “the k th event occurs at least at date t”. For example, the behavior of transition y of figure 1 is described by the following equation in Max in Jγ, δK: y(γ, δ)

= δ 2 u1 (γ, δ) ⊕ γ 1 u2 (γ, δ)

.

Usually, we group the series which describe the firing of the • n source transitions into the input vector U , • p well transitions into the output vector Y , • m other transitions into the state vector X. These give the state equations of TEGs in the form of a matrix:  X(γ, δ) = A ⊗ X(γ, δ) ⊕ B ⊗ U (γ, δ) , Y (γ, δ) = C ⊗ X(γ, δ) . Hence, as in the classical automatic control setting, it is possible to compute a transfer matrix H from these equations: H = CA∗ B, whereL the operator “∗” is called Kleene star and is defined by s∗ = i∈Z si [1]. It represents the input-output relation of the system in a JIT operational rule. More precisely, its entries Hij are causal which means that they represent neither anticipation in event nor in time, i.e. the monomials of the formal series of H are all with positive exponents. It is possible to retrieve counter and dater functions from a series s ∈ Max in Jγ, δK in the following manner: • the counter function is the unique non-decreasing function Cs of Zmin such that M t γ Cs (t)δ ; s= t∈Z

•

the dater function is the unique non-decreasing function Ds of Zmax such that M s= γ k δ Ds (k) . k∈Z

surjective, f (x) = y will have no solution for some values of y. If it is not injective, the same equation may have non-unique solutions. One way to always give a unique answer to this problem of mapping inversion is to consider the residuation theory. When it exists, it provides the greatest solution (in accordance with the partial order) to the inequality f (x)  y. Definition 1 (Residuated, residual) Let a mapping f : A → B, with (A, ) and (B, ) ordered sets. f is said to be residuated if for all y ∈ B the least upper bound of the subset {x ∈ A : f (x)  y} exists and lies in this subset. It is then denoted f ♯ (y). Mapping f ♯ is called the residual of f . The mapping Ra : x ⊗ a = y defined over a complete dioid D is residuated. Its residual is the mapping denoted by: _ y = ya = {x ∈ D | x ⊗ a  y} . (1) Ra♯ (y) = a For instance, in Zmin , let a = 2, y = 5. If x ⊗ a = y, then x = ya = 52 = 3. More generally, we have [17]: b a ⊤ a ε a a ε a ⊤

= b − a if a and b are finite, = ⊤

for all a,

= ε

for all a finite,

= ⊤

for all a,

= ε

if a 6= ⊤.

(2)

It should be noticed that ε ⊗ ⊤ = ε (+∞ − ∞ = +∞) whereas εε = ⊤ (+∞ − ∞ = −∞). This shows that the notation a − b is ambiguous for infinite values of a and b. It is basically this fact which raises the difficulty of computing the instantaneous marking of places. To the best of our knowledge, the efforts that have been made up to the present time are limited to computing only the lower and the upper bounds of markings [17]. Recall that u(t) = ε (= +∞) if the associated transition u has not been fired at date t. Hence, we are not able to compute stocks between two transitions u and v unless u(t) and v(t) are finite. Aiming at modeling resource sharing problems in Zmin , we have to overcome this inherent drawback in the following section. This is done both by using a TEG property and by means of a new operator. III. M ODEL OF INVARIANT RESOURCE SHARING PROBLEMS IN DIOIDS

This section both shows how to model invariant resource sharing problems by (min, +) equation systems and simultaneously gives a generic graphical representation of these problems by means of a PN.

C. Residuation in complete dioids Let us recall that ⊗-multiplication of a dioid D is rarely invertible (see [16], [1]). This subsection deals with solving equations of type f (x) = y in complete dioids. If f is not

3

A. Invariant Resource Sharing System First of all, let us introduce the notion of IRSS.

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Definition 2 (Invariant Resource Sharing System) An Invariant Resource Sharing System (IRSS) is a DEDS in which it is possible to identify one set of a finite and a constant, i.e. an invariant number of resources and its users. These users are both holders and suppliers of the resources in the following way: if vi (t) and xi (t) denote the number of times a user i has held and released a resource until t, respectively, then vi (t) − xi (t) ≥ 0, ∀t is the number of resources held by user i at t.

DEDSs containing IRSSs can be found in several applications such as flow-shops and supply chains (the industrial field), railway and motorway systems (the transportation field), and real-time systems (the computer science field). When such a system is not equipped with a rule to attribute resources between users, we meet an invariant resource sharing problem, which is the one of interest. We aim at dealing with this issue on a wide scale, defining the exact solution space. From a theoretical point of view, the problem should be modelled regardless of any particular solution. Since the problem dealt with includes conflicts, we cannot model it with a TEG. Thus, the generic graphical model of IRSSs belongs to a more general PN class, presented in figure 2.

Label vi

4

Meaning input transition: vi (t) ∈ Z is the number of times user i has held a resource until t output transition: xi (t) ∈ Z is the number of times the user i has released a resource until t constant idle time of a resource between two applications (e.g. the reconfiguration of a machine or the repositioning of a tool, ...) number of initial free resources, a.k.a. M0 (Pα ) (mα ≤ r) number of initial resources initially held by user i, a.k.a. M0 (Pi ) (see subsection III-B)

xi τα mα mi

TABLE I S EMANTICS OF FIGURE 2.

Except the work presented in [12] that leads to an exponential number of modes, let us recall that there is no general systematic technique available in the literature for modeling these problems in dioid algebra. The purpose of the next subsections is to fill this theoretical gap. B. Analytic model The PN model of figure 2 is split into two PN parts as it is shown in figure 3: a users PN part, which holds all sub-PNs Gi , and the resource sharing one, denoted by Gα . We recall that we limit our contribution to the discrete case where the invariance of the resource is respected, i.e. there exists τ ∈ N such that xi (t + τ ) = f (vi (t)), ∀t ≥ 0.

vi

vi Gi

Gi

v2

xi

vi

xi

G2 τα mα

v2

mi

x2

v2

Pi τα

xi

m2

mα

P2

Pα

G2 x2

x2 Gα

m1

v1

P1

x1

G1 v1 Fig. 2.

x1

G1 v1

PN model of an invariant resource sharing problem. Fig. 3.

This graphical model represents N users sharing r resources. We assign to each user i ∈ [1, N] a sub-PN Gi that models its behavior while it exploits one or several resources. A sub-PN Gi must have only one input and only one output transition in relation to the shared resource, denoted by vi and xi respectively. This general representation allows us to observe the counters for each transition. Table I gives the details of the model’s semantics.

x1

A split model of an invariant resource sharing problem.

Gα is built around the place Pα , which marking M0 (Pα ) represents idle or unavailable resources. In order to study the behavior of input and output transitions of each subPN Gi under resource sharing conditions, they are duplicated. Without loss of generality, each sub-PN Gi is replaced by a fictive place Pi in Gα , such that its initial marking M0 (Pi ) is the PNnumber of resources held by user i initially (M0 (Pα ) + i=0 M0 (Pi ) = r). This aims at focusing the analysis on the resource sharing regardless of the behavior of its users. In turn, we consider Gα as the elementary model of an IRSS.

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The structural analysis of Gα provides us with the following lemma. Lemma 1 (Constraint) When N users share exclusively r resources, their input and output transition counters vi (t) and xi (t) satisfy the following inequality N X i=1


vi (t) − xi (t)

≤ M0 (Pα ), ∀t

.

(3)

Mt (Pi ) + Mt (Pα )

= r


vi (t) − xi (t) + M0 (Pi ) + Mt (Pα ) = r

. (4)

Since we have

r−

N X

M0 (Pi ) = M0 (Pα )

then equation 4 can be defined as follows:

i=1

i=1

i=1

but r =

N X

N X


vi (t) − xi (t) + M0 (Pi )

>

r

,

M0 (Pi ) + M0 (Pα ). It follows that:

N X


vi (t) − xi (t) + M0 (Pi ) N X i=1


vi (t) − xi (t)

> M0 (Pα ) +


vi (t) − xi (t) + Mt (Pα ) = M0 (Pα ) .

Hence, since Mt (Pα ) is non-negative for all t, then the conclusion follows. It is important to note that this lemma is a very powerful means to give constraints to the system dealt with. Indeed, it provides us with a relation which implies only counters of transitions involved in Gα and the number of initial free resources. However, from both a theoretical and a practical point of view, it is interesting to verify that the constraint (3) is in agreement with the kind of the behavior of an IRSS. To do this, it suffices to show that: (i) the constraint denies the request for a resource if there is no free resource, (ii) if there is at least one free resource, the constraint must grant the request, (iii) the constraint must not forbid the user to release a resource. Note that, assertion (iii) is obviously satisfied, since the counter of output transitions xi (t) are multiplied by −1 in (3). This means that each firing of such a transition relaxes the constraint. Hence, it remains only to prove (i) and (ii), which is the main concern of the following lemma. Lemma 2 Consider an IRSS. Then under the constraint (3): (a) no transition vi can fire despite the lack of a free resource; (b) a transition vi is allowed to fire if there is at least one free resource.

N X M0 (Pi ) . i=1

> M0 (Pα ) ,

which contradicts (3). Thus, it is not possible to fire vi if there are no free resources. In order to prove (b), it suffices to prove that Mt (Pα ) > 0 implies the strict inequality of (3). Indeed, if we have at least one free resource, it follows that: (5)

Mt (Pα ) > 0 .

,

i=1

N X

Mt (Pi ) =

Hence, we have

where Mt (P) is the instantaneous marking of place P at date t. From [17], we have Mt (Pi ) = vi (t) − xi (t) + M0 (Pi ). Hence,

i=1

N X

i=1

,

i=1

N X

Proof: The proof of (a) is by contradiction. Indeed, if we suppose that all transitions vi can be fired despite the lack of free resource, we can assume the following:

i=1

Proof: From the P-invariant property of Gα , we have: N X

5

With this in mind, the number Mt (Pα ) of free resources at date t is given by: Mt (Pα ) = r −

N X i=1


vi (t) − xi (t) + M0 (Pi )

. (6)

This means that the number of available resources is the result of the total number of resources minus the held resources. From the P-invariant property of Gα , we have: =

r

N X

Mt (Pi ) + Mt (Pα )

.

i=1

This and (6) imply:

Mt (Pα ) = M0 (Pα ) +

N X

M0 (Pi )

i=1

−

N X

M0 (Pi ) −

i=1

= M0 (Pα ) −

N X i=1

N X i=1

From (5), we obtain: N X i=1


vi (t) − xi (t)


vi (t) − xi (t)


vi (t) − xi (t)

.

, (7)

< M0 (Pα ) .

Hence the conclusion follows. The lemma below shows how to combine both the constraint (3) and the transfer relation between xi and vi into one inequation.

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Lemma 3 When N users share exclusively r resources, their input and output transition counters vi (t) and xi (t) satisfy the following inequality N X i=1


vi (t) − xi (t)

≤ M0 (Pα ), ∀t

(8)

.

Proof: We can only act on firings of transitions vi because firings of xi are directly given by those of vi via a transfer relation depending on the behavior of Gi for all i. In addition, there is a transfer relation between transitions vi and xi for all i, depending on the behavior of Gα . This relation is given by N X

vi (t)

≤

N X

xi (t − τα ) + M0 (Pα )

Recall that counter functions are non-decreasing, i.e. ∀i, xi (t) ≥ xi (t − τα ). So N N X X ` ´ ` ´ vi (t) − xi (t) ≤ M0 (Pα ). vi (t) − xi (t − τα ) ≤ M0 (Pα ) ⇒ i=1

i=1

Combining the above results gives the following theorem. Theorem 1 The behavior of an IRSS with N users is described by the following state inequations, for all user i:
xi (t) ≤ vi t − τi (t) + M0 (Pi ) , (9)

where τi (t) is the variable duration, while user i holds a resource (which depends on the internal behavior of Gi ), and each vi must be chosen in such a way that, for all t: N X i=1


vi (t) − xi (t − τα )

⊲

C −→ C

⊲

such that

[C]⊲ ij

( e = [C]ij ⊗ 1

if [C]ij = ε, elsewhere,

for a matrix C. This operator has an interesting property presented in the following lemma. Lemma 4 Let G be a live TEG. Let C(·) be the vector of the counters of all transitions, and M(·) (P) the vector of the instantaneous markings of places P, constituting G. Then we have: =⇒

Mt+∆t (P) = Mt (P)

, ∀t ≥ 0.

In other words, the operator ⊲ preserves markings.

i=1

i=1

for a scalar c and

C(t + ∆t) = C ⊲ (t)

.

6

≤ M0 (Pα )

.

(10)

Proof: Inequalities (9) have been proved to describe the set of possible transition firings in [1]. The constraint (10) has been derived from lemma 3. It has been proved in lemma 2 that the use of this inequation system describes the behavior of an IRSS.

Proof: The definition of the operator ⊲, C(t + ∆t) = C ⊲ (t) means that all transitions of G have been fired exactly once during ∆t. Hence, it suffices to prove that for a live TEG, a firing sequence preserves the marking if and only if it fires an equal number of times during each transition. Indeed, on the one hand, it is well known that [9, p. 555] a live TEG is equivalent to a connected marked graph, in which nodes correspond to transitions and arcs to places. On the other hand, it has been proved in [18, p. 561] that for a live connected marked graph, a firing sequence preserves the marking iff it fires every node an equal number of times. It follows immediately from Lemma 4 that the operator ⊲ preserves markings, when it is applied to all counters of a live TEG. Hence, in the following proposition, this interesting property of ⊲ is used to overcome the ambiguity when computing instantaneous markings with infinite counters of a live TEG. Proposition 1 [19] The marking Mt (P) of a place P between two transitions xa and xb in a live TEG at instant t ≥ 0 is given in Zmin by Mt (P)

= M0 (P) ⊗

x⊲ a (t) x⊲ b (t)

,

(11)

with xa (t) 6= ⊤ and xb (t) 6= ⊤. C. (min, +) model Since the description given in theorem 1 is very simple and provides a deep insight in the modeling of a resource sharing problem, it is natural to wonder whether the same description can also be produced in (min, +) algebra. However, remember that subtraction of classical algebra is equivalent to the right division in (min, +) algebra, which is ambiguous in each case where at least one counter has an infinite value (see subsection II-C). Hence, the theorem can not be written in (min, +) algebra without some efforts. To overcome this inherent difficulty, a new operator is introduced. Definition 3 Let ⊲ be an operator defined as follows: ( e if c = ε, ⊲ ⊲ c −→ c = c ⊗ 1 otherwise,

Remark 1 This proposition allows us to compute the number of tokens in every case where xa (t) 6= ⊤ and xb (t) 6= ⊤. Indeed, it is meaningless to try to compute the number of tokens when transitions have been infinitely fired. Now we are able to rewrite constraint (8) in Zmin , as is shown in the following lemma. Lemma 5 (Constraint in Zmin ) When N users share exclusively r ∈ N − {0} resources, their input and output transition counters vi (t) and xi (t) in (min, +) algebra satisfy: N O

vi⊲ (t) ⊲ xi (t − τα ) i=1

 M0 (Pα ),

such that vi⊲ (t), x⊲ i (t − τα ) 6= ⊤.

∀t ≥ 0 ,

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Proof: The proof follows immediately from (1) and (2) and by applying the operator ⊲ to equation (8) which forces the right division to deal with finite values. In summary, since the (min, +) state equations describing the behavior of a live TEG are well known (see section II), then lemma 5 leads to the following theorem.

7

A. Chocolate factory

M1

Packing

M2

P1 P2

Theorem 2 The JIT behavior of a live TEG containing an IRSS with N users, can be described, after a contingent extension of state vectors, by the following state equations in Zmin :  X(t) = AX(t − 1) ⊕ BU (t) ⊕ SV (t) ,       Y (t) = CX(t) , N O vi⊲ (t) with  M0 (Pα ), ∀t ≥ 0,    x⊲  i (t − τα ) i=1   such that vi⊲ (t), x⊲ i (t − τα ) 6= ⊤ ∀i ∈ [1, N],

where vi is the input transition of the critical section of user i and xi is the output one.

P3 Fig. 4.

A chocolate factory.

Let’s consider a chocolate factory that produces three types of chocolate, denoted P1 , P2 and P3 . Each type has two machine-controlled transformation processes M1 and M2 . Finally, a chocolate of each type is arranged in a box. Since this last operation is of no interest here, we suppose it has a negligible process time. The model is depicted in figure 5. M1

The above theorem can easily be transposed to dioid Max in Jγ, δK. This will shorten the dimension of both state and associated matrices (A, B, S, C). Besides, it facilitates readability and further computations (e.g. system control, performance evaluation, . . . ).

Packing

M2

P1 u1

v1

2

v2

3

x1

v3

3

v4

5

x2

v5

1

v6

4

x3

P2

Corollary 1 The JIT behavior of a live TEG containing an IRSS with N users is given on the basis of the following state equations in Max in Jγ, δK:  X = AX ⊕ BU ⊕ SV ,     Y = CX ,   N (t) O C⊲ (∆qi V ) M0 (Pα ), ∀t ≥ 0,   ⊲  C(∆n X) (t − τα ) Zmin   i=1   such thati C⊲ (t), C⊲ (t − τ ) 6= ⊤ ∀i ∈ [1, N], (∆qi V )

(∆n i X)

α

where V is the input transition vector of critical sections T q of users, X is the output one, and ∆qi ∈ Max in Jγ, δK and T n ax n ∆i ∈ Min Jγ, δK are defined for all i by ( e if j = i, [∆i ]j = ε elsewhere. This corollary is needed to use algorithm 1 in example of subsection IV-B. IV. E XAMPLES OF DEDS S CONTAINING IRSS S

This section applies the proposed (min, +) modeling approach to two resource sharing systems. The first example shows an optimal schedule, followed by a check that the (min, +) equation system is not in contradiction with this schedule. The second example shows an algorithm that begins with an unfeasible timetable. Jobs are rescheduled until the new timetable matches the equation system.

u2

y

x4

P3 u3

Fig. 5.

PN model of a chocolate factory.

From the PN of figure 5 and using corollary 1, we can easily obtain an analytic model of the system: 0 1 ε δ3 ε ε ε ε ε γ Bε ε ε δ 5 Bε ε ε γ C @ε ε ε γ A X ⊕ B @ε ε ε ε e e e ε ε ε ε ε ` ´ ε ε ε e X , 0 1 0 ε ε ε ε ε ε e ε 2 Bδ Bε ε ε ε ε ε εC B C B B ε ε ε ε ε εC Bε e V  B > C V ⊕ Bε ε 3 > ε ε δ ε ε ε > B C B > > @ ε ε ε ε ε εA @ε ε > > > 1 > > ε ε ε ε ε ε δ ε > > ⊲ ⊲ ⊲ > > v (t) v (t) v (t) 1 3 5 > > ⊗ ⊗ ∀t, 1 Zmin > > v2 (t)⊲ v4 (t)⊲ v6 (t)⊲ > > ⊲ ⊲ ⊲ > > v2 (t) v4 (t) v6 (t) > : 1 Z ⊗ ⊗ ∀t. min x (t)⊲ x2 (t)⊲ x3 (t)⊲ 1 8 > > > > > > X = > > > > > > > > > > Y = > > > > > > >
> > > > > > > > > > > > > > > > > > X > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > Y > > > > > > > > > > > > > > : 2

0

B B B B B B B B =B B B B B B B @

ε ε ε ε ε ε ε ε ε ε ε

ε Bε Bε B Be B Bε B ⊕B Bε Bε B Bε B Bε @ε

ε ε ε ε e ε ε ε ε ε ε

0

ε 0

e B ε =@ ε ε Zmin

ε ε ε ε ε ε ε ε ε ε ε

ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε e ε ε ε ε

ε ε ε ε

3 O i=1

2

ε ε δ ε ε ε ε ε ε ε ε γδ ε ε ε δ 2 δ 4 γδ 2 ε ε ε ε ε ε ε ε ε ε ε ε ε ε ε 0 1 ε ε ε εC Bε ε Bε ε εC B C Be ε C εC B Be ε εC B C Bε ε εC U ⊕ B C Bε e εC B C Bε ε εC B C εC Bε ε @ε ε A ε ε

e

ε ε ε ε ε ε ε ε ε e ε ε C⊲ (t) q (∆ V )

ε ε ε ε

i

C⊲ (∆n X) (t − 3)

ε

ε ε ε ε

,

ε ε ε ε ε ε ε ε ε ε ε ε ε ε δ4 ε ε δ2 ε δ3 ε ε 1 ε εC εC C εC C εC C εC CV εC C εC C εC εA

ε e ε ε ε ε ε ε ε ε ε

ε e ε ε ε ε ε ε ε ε ε

1

ε ε δ3 ε ε ε ε ε ε ε ε

C C C C C C C C CX C C C C C C A

,

Algorithm 1

ε ε ε ε

ε e ε ε

ε ε e ε

1 ε ε C X ε A ε

Inputs DEDS containing an IRSS state and evolution equations, X = f (X, U, V ) and Y = g(X) respectively. An input trajectory U.

,

(13)

∀t ≥ 0,

Outputs

i

T

The objective is to plan train arrivals in a tight schedule, while respecting track availability. Table III shows both the deadline for a container delivered by train of type 2 and the latest possible train departure times.

20 25 -

2) Control: We propose a control algorithm that consists of two phases: phase 1 gives the best (i.e. latest) dates regardless of the resource sharing part by means of an openloop control [21]. Phase 2 identifies trains that can not access simultaneously according to track limitation on the basis of the proposed equation system. In in order to choose which train to advance, we use the Early Due Date (EDD) rule. This second phase will be processed until U , V and X match the equation system. The EDD or Jackson rule [22] proposes to order train arrival dates U by ascending order of desired departure dates Y with respect to resource availability. Thus, at least one train is definitively scheduled at each iteration of phase 2. Since this train is withdrawn from the set of trains considered for the next round, the algorithm is of complexity O(n2 ), where n is the initial number of trains. Indeed, in the worst P1 case scenario, i.e. where all due dates are similar, we have i=n−1 i operations.

e

q nT n where ∆qi ∈ Max ∈ Max are in Jγ, δK and ∆i in Jγ, δK defined for all i by ( e if j = i, [∆i ]j = ε elsewhere.

Train of type 1

9

Container from a train of type 2 22 -

Train of type 2

Train of type 3

19 -

11 16 21

TABLE III S CHEDULE TABLE OF AN INVARIANT RESOURCE SHARING PROBLEM .

On the basis of this table, we can now write the objective in the form of firing trajectories Z.   δ 20 ⊕ γδ 25 ⊕ γ 2 δ ∗   δ 22 ⊕ γδ ∗  . Z =    δ 19 ⊕ γδ ∗ δ 11 ⊕ γδ 16 ⊕ γ 2 δ 21 ⊕ γ 3 δ ∗

Subsequently, we will have to compute U (arrival at station) and V (access permission) that give the greatest Y (departure from station) such that Y  Z with respect to (13).

Optimal control and output trajectories V and Y respectively. Phase 1 1. Initialize V , U and X: 1.1. Merge V and U in U ′ : X

= AX ⊕ B ′ U ′ = AX ⊕ (B | S)

  U V

;

ˆ′ = 1.2. Compute ideal (unreachable) input trajectories U Z ; CA∗ B ′ ˆ ′ into U ˆ and Vˆ to compute X ˆ = A∗ (B U ˆ⊕ 1.3. Split U ˆ S V ); ˆ , X to X, ˆ V to Vˆ . Let wi the number 1.4. Initialize U to U of trains of type i to
be scheduled: initialize t to max D[X]i (wi − 1) . i∈[1,N]

Phase 2

2. While equation (13) is matched at t, t takes value t − 1. If t = 0, algorithm ends. 3. While equation (13) is not satisfied, do: 3.1. Compute the number g of tasks to be advanced at this step:

g

=

N O i=1

C⊲ ([V ]i ) (t) C⊲ ([X]i ) (t − τα ) r

;

3.2. Build the set L of tasksl which, so far, are planned to finish at date t: L = l ∈ N | xl (t) > xl (t − 1) ;

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3.3. Build the set L∗ ∈ L of tasks to be advanced at this step: L∗

=

8 > > >
> > :

` ´ D [X] ˆ i C [X]i (t) M ` ´ D [V ]j C[V ]j (t + τα )

^

i∈L

j∈L−{i}

9 > > > = > > > ;

If there are more than g tasks in this set, pick g arbitrarily. If there are less than g tasks in this set, once again select tasks in the same way, but from L − L∗ . Once L∗ has g elements, order this set by decreasing values of Dl , l ∈ L∗ ; 3.4. For all l in L∗ , do: 3.4.1. Consider the greatest X such that X  Q with Q defined by [Q]i = 8 > > < > > :

„ L [X]i ∧ δ j6=i

D[V ] j

`

C[V ] (t+τα ) j τα

´

⊕ γ 1C [X]i (t) δ ∗

[X]i

«

if i ∈ L∗ , elsewhere;

3.4.2. Compute the greatest U and V that induce such an X: 3.4.2.1. Merge again U and V in U ′ : X ⇐⇒ A B U ′ ∗

′

′

 Q ,  Q, ∀U ′ ∗

;

′

3.4.2.2. Compute U = QA B . Then extract V and U and compute the induced X: X = A∗ (BU ⊕ SV ); 3.5. Go back to step 2. initializes state

After 6 iterations, we obtain the following vectors:   δ 16 ⊕ γδ 25 ⊕ γ 2 δ ∗   δ 14 ⊕ γδ ∗   4 . δ ⊕ γδ 7 ⊕ γ 2 δ 21 ⊕ γ 3 δ ∗     δ 12 ⊕ γδ 21 ⊕ γ 2 δ ∗     δ 10 ⊕ γδ 19 ⊕ γ 2 δ ∗     δ 14 ⊕ γδ 23 ⊕ γ 2 δ ∗   X=   δ 7 ⊕ γδ ∗   11 ∗   δ ⊕ γδ   13 ∗   δ ⊕ γδ   14 ∗   δ ⊕ γδ    e ⊕ γδ 3 ⊕ γ 2 δ 17 ⊕ γ 3 δ ∗  δ 2 ⊕ γδ 5 ⊕ γ 2 δ 19 ⊕ γ 3 δ ∗   δ 10 ⊕ γδ 19 ⊕ γ 2 δ ∗  δ 7 ⊕ γδ ∗ V = e ⊕ γδ 3 ⊕ γ 2 δ 17 ⊕ γ 3 δ ∗   δ 12 ⊕ γδ 21 ⊕ γ 2 δ ∗  δ 10 ⊕ γδ 19 ⊕ γ 2 δ ∗   U =   δ 7 ⊕ γδ ∗ e ⊕ γδ 3 ⊕ γ 2 δ 17 ⊕ γ 3 δ ∗   δ 16 ⊕ γδ 25 ⊕ γ 2 δ ∗   δ 13 ⊕ γδ ∗  Y = 14 ∗   δ ⊕ γδ 4 7 2 21 3 ∗ δ ⊕ γδ ⊕ γ δ ⊕ γ δ

values of state

,

,

,

.

From V and X one can read the schedule of trains which is presented in table IV. Train of type 1 Arrival Departure 10 16 19 25 -

Train of type 2 Arrival Departure 7 14 -

Train of type 3 Arrival Departure 0 4 3 7 17 21

TABLE IV F INAL SCHEDULE TABLE OF AN INVARIANT RESOURCE SHARING PROBLEM .

,

It is easy to build a Gantt chart on the basis of the final values of X and V . See figure 8. Infinite resouces (phase 1)

# n ai Tr pe Ty

3) Numerical results: Step 1 of algorithm variables as follows:   δ 20 ⊕ γδ 25 ⊕ γ 2 δ ∗   δ 19 ⊕ γδ ∗  11  δ ⊕ γδ 16 ⊕ γ 2 δ 21 ⊕ γ 3 δ ∗      δ 16 ⊕ γδ 21 ⊕ γ 2 δ ∗   14 19 2 ∗   δ ⊕ γδ ⊕ γ δ   18 23 2 ∗   δ ⊕ γδ ⊕ γ δ ˆ   X= 12 ∗  δ ⊕ γδ   16 ∗   δ ⊕ γδ   18 ∗   δ ⊕ γδ   19 ∗   δ ⊕ γδ  7   δ ⊕ γδ 12 ⊕ γ 2 δ 17 ⊕ γ 3 δ ∗  δ 9 ⊕ γδ 14 ⊕ γ 2 δ 19 ⊕ γ 3 δ ∗   δ 14 ⊕ γδ 19 ⊕ γ 2 δ ∗  δ 12 ⊕ γδ ∗ Vˆ =  7 12 δ ⊕ γδ ⊕ γ 2 δ 17 ⊕ γ 3 δ ∗   δ 16 ⊕ γδ 21 ⊕ γ 2 δ ∗   δ 14 ⊕ γδ 19 ⊕ γ 2 δ ∗ ˆ =  U 12 ∗   δ ⊕ γδ 7 12 2 17 3 ∗ δ ⊕ γδ ⊕ γ δ ⊕ γ δ   δ 20 ⊕ γδ 25 ⊕ γ 2 δ ∗   δ 18 ⊕ γδ ∗  Yˆ =  19 ∗   δ ⊕ γδ 11 16 2 21 3 ∗ δ ⊕ γδ ⊕ γ δ ⊕ γ δ

10

3 2

Final

1 2 1 1

,

1

2 3 0

,

.

Fig. 8.

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

t

Gantt chart of an invariant resource sharing problem.

We would like to draw the attention of the reader to the fact that corollary 1 is used directly in the algorithm at steps 2 and 3. As a result, the three assertions of subsection III-B are satisfied. Indeed, we can see in figure 8 that equation (13) prohibits a train from being on a track that is not available (assertion (i)). Figure 9 shows tracks that are used all the time,

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which means that equation (13) allows each track to be used as soon as it is available (assertion (ii)). Finally, we can see in figure 10 that when a train leaves the station, it releases a track (assertion (iii)). Headway time

Type 1 train

Type 2 train

Type 3 train

Track 1 Track 2 0

Fig. 9.

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

t

Resources occupancy of an invariant resource sharing problem.

Track1 busy unavailable idle 0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

t

Track 2 busy unavailable idle t

Fig. 10. Resource usage schedule of an invariant resource sharing problem.

V. C ONCLUSION This paper has proposed a general model of IRSSs using dioid algebra. The model introduces a set of constraints to express the limitations of the behavior of the system. In fact, it graphically extracts those parts of the system which correspond to the shared resources in the PN. Once this extraction has taken place and if the resulting PN is a TEG, it is possible to describe the entire behavior of the system by means of a fully analytic model. As a consequence, the model allows us to write equations that describe the behavior of many systems that could not be represented by dioids so far. Thus, we broadened the scope of this powerful algebra for the analysis of DEDSs. There are several issues that deserve further investigation. One is to use our model to define optimal control methods for the systems that are studied. Indeed, it may be possible to obtain a generic control law based on the residuation theory, at least for, for example, subsection IV-B and similar problems. Another issue to work on is how to extend the model to deal with other structural conflict types encountered in general PNs.

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[6] J. O. Moody and P. J. Antsaklis, “Petri net supervisors for des with uncontrollable and unobservable transitions,” in IEEE Transactions on Automatic Control, vol. 45, no. 3, Mar. 2000, pp. 462–476. [7] A. A. Desrochers and R. Y. Al-Jaar, Applications of Petri nets in manufacturing systems: modeling, control, and performance analysis. Press of the Institute of Electrical and Electronics Engineers (IEEE), 1995. [8] A. Giua, F. Di Cesare, and M. Silva, “Generalized mutual exclusion contraints on nets with uncontrollable transitions,” in IEEE international conference on Systems, Man and Cybernetics (SMC), vol. 2, Chicago, IL, USA, Oct. 1992, pp. 974–979. [9] T. Murata, “Petri nets: properties, analysis and applications,” Proceedings of the Institute of Electrical and Electronics Engineers (IEEE), vol. 77, no. 4, pp. 541–580, Apr. 1989. [10] R. David and H. Alla, Du GraFCET aux réseaux de Petri. Hermès, 1992. [11] B. Trouillet and A. Benasser, “Cyclic scheduling problems with assemblies: an approach based to the search of an initial marking in a marked graph,” in IEEE international conference on Systems, Man and Cybernetics (SMC), vol. 3, Hammamet, Tunisia, Oct. 2002. [12] T. J. J. van den Boom and B. De Schutter, “Modelling and control of discrete event systems using switching max-plus-linear systems,” Control engineering practice, vol. 14, no. 10, pp. 1199–1211, 2006. [13] M. Al Saba, J.-L. Boimond, and S. Lahaye, “On just-in-time control of flexible manufacturing systems via dioid algebra,” in IFAC symposium on INformation COntrol in Manufacturing (INCOM), vol. 2, Saint-Étienne, France, 2006, pp. 137–142. [14] R. Cunninghame-Green, “Minimax algebra,” in Lecture notes in economics and mathematical systems. Springer-Verlag, 1979, vol. 166. [15] G. Cohen, P. Moller, J.-P. Quadrat, and M. Viot, “Algebraic tools for the performance evaluation of discrete event systems,” Proceedings of the Institute of Electrical and Electronics Engineers (IEEE), vol. 77, no. 1, pp. 39–58, 1989. [16] T. S. Blyth and M. F. Janowitz, Residuation theory. Pergamon Press, 1972. [17] G. Cohen, S. Gaubert, R. Nikoukhah, and J.-P. Quadrat, “Second order theory of min-linear systems and its application to discrete event systems,” in IEEE international Conference on Decision and Control (CDC), Brighton, England, Dec. 1991. [18] T. Murata, “Circuit theoretic analysis and synthesis of marked graphs,” IEEE Transactions on Circuit Systems, vol. CAS-24, no. 7, pp. 400–405, July 1977. [19] A. Corréïa, A. Abbas Turki, R. Bouyekhf, and A. El Moudni, Advanced technologies: research-development-application. pIV pro literatur Verlag Robert Mayer-Scholz, Sept. 2006, ch. Controlling a model of flowshop based on dioids, pp. 215–228. [20] S. M. Johnson, “Optimal two- and three-stage production schedules with setup times included,” Naval Research Logistic Quaterly, vol. 1, pp. 61– 68, 1954. [21] B. Cottenceau, “Contribution à la commande de systèmes à événements discrets: synthèse de correcteurs pour les graphes d’événements temporisés dans les dioïdes,” Ph.D. dissertation, Institut des Sciences et Techniques pour l’Ingénieur d’Angers (ISTIA), Université d’Angers, France, 1999. [22] J. R. Jackson, “Scheduling a production line to minimize maximum tardiness,” University of California, Los Angeles, Research report 43, July 1955.

R EFERENCES [1] F. Baccelli, G. Cohen, G.-J. Olsder, and J.-P. Quadrat, Synchronization and linearity, an algebra for discrete event systems. Wiley, 1992. [Online]. Available: http://www-rocq.inria.fr/metalau/cohen/SED/book-online.html [2] R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, “Optimization and approximation in deterministic sequencing and scheduling: a survey,” Annals of discrete mathematics, vol. 5, pp. 287–326, 1979. [3] L. Lamport, “Time, clocks, and the ordering of events in a distributed system,” Association for Computing Machinery (ACM), vol. 7, no. 21, pp. 558–565, July 1978. [4] M. Naimi, M. Trehel, and A. Arnold, “A log(n) distributed mutual exclusion algorithm based on path reversal,” Journal of parallel and distributed computing, vol. 34, pp. 1–13, 1996. [5] J. O. Moody, P. J. Antsaklis, and M. D. Lemmon, “Automated design of a petri net feedback controller for a robotic assembly cell,” in IEEE international conference on Emerging Technologies and Factory Automation (ETFA), vol. 2, Paris, France, Oct. 1995, pp. 117–128.

Aurélien Corréïa is a Reseach Engineer at Hitachi Europe SAS in the field of navigation systems. He received a Master degree in control systems and computer science from the University of Angers, France, in 2004, and a Ph.D degree in control systems from UTBM and University of Besançon, France, in 2007. He has developed a model for IRSSs in dioid algebra, formerly devoted to urban traffic issues. His research interests include dioid algebra, Petri nets, urban traffic and operations research.

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Abdeljalil Abbas-Turki is Lecturer at UTBM where he teaches transportation engineering and computer science. At the end of 2000, he joined the laboratoire Systèmes et Transports of UTBM where he obtained a PhD degree three years later. Besides research and teaching activities, he participates in several national and international projects related to the field of transport. Through these various projects, he had the opportunity to apply system-engineering techniques, face realistic transport problems and gain multidisciplinary experiences. Momentarily, he is involved in scheduling and resource allocation research for transportation systems.

Rachid Bouyekhf received a Ph.D degree in automatic control in 1998 from Franche-Comté University, Besançon, France. He is currently an Associate Professor at UTBM, where he is in charge of courses in automatic control and information theory. His research interests include hybrid systems, H∞ control, transportation systems, full cell systems and discrete event systems.

12

Abdellah El Moudni received a Ph.D degree in automatic control in 1985 from Lille University, Lille, France. He is currently a Professor at UTBM, where he is in charge of courses in automatic control. His research interests include transportation systems, nonlinear control theory, dynamic systems, control of discrete event systems and singular perturbation methods in discrete time.