S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....38 task resource systems can be specified coinductively, that is, by means of a behavioural differential equation, and implemented by a finite nd automaton, derived from this differential equation. The relevance of this example is not so much the obtained automaton (which is not new see, e.g. [GM98]) but rather the way in which it illustrates once again our methodology of deriving finite representations from behavioural specifications. A (timed) task resource system is a four tuple (A; R; r; h) consisting of a finite set A of tasks , a finite set R of resources , a function r : A P(R) ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In [Gun98], pages 133--144, 1998.

....of task resource systems can be specified coinductively, that is, by means of a behavioural di#erential equation, and implemented by a finite nd automaton, derived from this di#erential equation. The relevance of this example is not so much the obtained automaton (which is not new see, e.g. [GM98]) but rather the way in which it illustrates our methodology of deriving finite representations from behavioural specifications. A (timed) task resource system is a four tuple (A, R, r, h) consisting of a finite set A of tasks , a finite set R of resources , a function r : A # P(R) assigning to ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) cautomata. In [Gun98], pages 133--144, 1998.

....means at most h Lyapunov exponents to expand. Remark 11 The formulation with the distribution on A 0 may be obtained on each extension results in the same way that we did in (25) in #7.1. 4. 6 Example : stochastic task resource model We consider the following task resource model described in [9]. We recall some notations : # A is a nite set of pieces. # R is a nite set of slots. # R : A P(R) gives the subset of slots covered by a piece. # h : A IR gives the execution time of a task. We assume : A = fa; a 1 ; a 2 g; R = fr 1 ; r 2 g; R(a) fr 1 ; r 2 g; R(a 1 ) fr 1 g; R(a ....

....after some elementary calculations, fl(p 1 ; p 2 ) 1 Gamma 2 1 X n=1 2n Gamma2 n Gamma1 (p 1 p 2 ) n 1 X n=0 2n Gamma1 n [p 1 (p 1 p 2 ) n p 2 (p 1 p 2 ) n ] which is equal to (13) for p 1 p 2 1. In case of generalized task resource models (Tetris type [9]) the formulas of Theorem 5 hold if the memory loss property (A2) is satised, that is if there is a piece of Tetris that occupies all resources or a sequence of pieces of rank one. Here, a guarantees this property. 5 Perturbation representation of the coeOEcients of the expansion In this ....

Gaubert, S. and Mairesse, J. (1998) Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press. INRIA Analytic Expansions of (max; +) Lyapunov Exponents 51

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S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1995.

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S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency, volume 11, pages 133--144. Cambridge University Press, 1997.

....is not always optimal. Key words: Optimal scheduling, timed Petri net, heap of pieces, Tetris game, max, semiring, automaton with multiplicities, Sturmian word. 1 Introduction Heap models have recently been studied as a pertinent model of discrete event systems, see Gaubert Mairesse [18,19] and Brilman Vincent [11,12] They provide a good compromise between modeling power and tractability. As far as modeling is concerned, heap models are naturally associated with trace monoids, see [30] It was proved in [19] that the behavior of a timed onebounded Petri net can be represented ....

.... ( The ALgebraic Approach to Performance Evaluation of Discrete Event Systems ) Preprint submitted to Elsevier Preprint 14 December 2000 surface growth, see [5] The tractability follows essentially from the existence of a representation of the dynamic of a heap model by a (max, automaton, see [12,18]. A heap model is formed by a finite set of slots R and a finite set of pieces A. A piece is a solid block occupying a subset of the slots and having a polyomino shape. Given a ground whose shape is determined by a vector of R and a word w = a 1 Delta Delta Delta a n 2 A , we consider ....

[Article contains additional citation context not shown here]

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency, volume 11, pages 133--144. Cambridge University Press, 1997.

....of the heap on slot i after piling of the pieces of w. Next Theorem was 21 d b a d c a p3 p5 p6 p4 p2 p1 p 6 d p 5 Figure 17: A 1 bounded Petri net and the execution w = abcdabcd in the associated heap monoid proved under various forms: Vincent [26] Brilman Vincent[3] Gaubert Mairesse [11]. We illustrate it below in the case of general pieces (see Heap Automaton II) Theorem 5. The map x : T max is (max, recognizable (see Def. 6) The corresponding (max, Automaton is (T ; Q; 1Q ; M) where M is defined by 1 if piece a occupies the slots i and j, 0 otherwise. In other ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency, volume 11, pages 133--144. Cambridge University Press, 1997.

....linear case boils down to the (max, case by switching to operator GammaT . Such systems appear in many domains of applications, under various forms. For example (without any kind of exhaustivity) ffl Computer science : parallel algorithms, shared memory systems, PERT graphs, see [43] or [23]. ffl Queueing theory : G=G=1=1 queue (see #5) queues in series, queues in series with blocking, fork join networks [3] ffl Operations research and manufacturing : Job shop models, event graphs (a subclass of Petri nets) see [17] 28] and [3] ffl Economy or control theory : dynamic ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1995.

....the convention that A ji (n) Gamma1 if j 62 L(i; n) It is easy to check that such a model, we could call it a task graph with random precedences, verifies Equation (2) A Queuing Network model studied by Baccelli Liu [5] corresponds to this model. The task resource models studied in [14] or [22] also have this kind of structure. 2.2 Cyclic Jackson network We consider a closed Jackson Network. The study of such closed networks can be traced back to Gordon and Newell, 25] In their original model, there is a given number of indistinguishable customers. The routing of the customers ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1995.

....by considering the measure over traces induced by the uniform distribution over words of the same length in the free monoid. In other terms, the probability of a trace is proportional to the number of its representatives in the free monoid. This quantity was introduced in [26] and later studied in [2, 5, 6, 14, 27]. Here we define alternative notions of average parallelism by considering successively the uniform distribution over traces of the same length, the uniform distribution over traces of the same height, and the uniform distribution over Cartier Foata normal forms. We prove in particular that there ....

....P n = n ffi , i.e. P n ftg = n fw : w) tg. The limit below exists: Sigma; D) lim w2 Sigma n h( w) nj Sigmaj : 5) This is proved using Markovian arguments in [26] The existence of can also be proved using sub additive arguments. More precisely, it is shown in [14] that h( is recognized by an automaton with multiplicities over the (max; semiring, which provides a different proof of the existence of . In fact a stronger result holds. Consider a probability space( Omega ; F; P ) Let (x n ) n2N be a sequence of independent random variables ....

[Article contains additional citation context not shown here]

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency, volume 11, pages 133--144. Cambridge University Press, 1998.

....a very intuitive graphical representation of trace monoids. 3. Max, linear representations. A next step was the observation that the height of heaps of pieces is recognized by a (max, automaton, the result holding for general, polyominoshaped, pieces. This was proved in Gaubert and Mairesse [16], and in a different form, by Brilman and Vincent [27] 6] The heap representation theorem for safe timed Petri nets that we give can be seen as a synthesis of these three results. The essential idea is that considering general pieces in heap models enables to model time through the height of a ....

....(given by R(a) during a certain amount of time (u(a; r) Gamma l(a; r) for a resource r 2 R(a) In the simplest case where l(a; r) 0; 8r 2 R(a) the execution of a task begins as soon as all the required resources, used by earlier tasks, become free. For more details along these lines, see [16], 6] Borrowing the terminology of [2] 15] the maps y H and xH are called the dater functions of the heap model. The piling mechanism and the different notations are best understood graphically and on an example, see Fig. 1. Example II.2. Let us consider the following heap model. ffl T = ....

[Article contains additional citation context not shown here]

S. Gaubertand J. Mairesse. Task resourcemodels and(max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1997.

.... Event Systems (DES) a convenient abstraction for many man made systems such as communication networks, digital circuits, or manufacturing systems, can usually be modelled by topical maps and IFS; the extrema of the asymptotic height then correspond to the best and the worst throughput of the DES [Ba, Co, BV, GM1, GM2]. Among topical maps, a special role is played by max plus maps which appear in the modelling of event graphs, 1 bounded Petri nets and Tetris like heap models [Ga2] Topical IFS also appear in other contexts, for example, in various problems of automata and formal language theory [Pin, Sim] ....

S. Gaubert and J. Mairesse, Task resource models and (max; +) automata, in [Gun], pp. 133-144

....the same example as above but with ff = 0; c; 0; 0) T for c 0. The spectral radius is then 8=15 c. In the last part of the paper, we consider the formal power series over IR max : S(ff; f; fi) M k2IN Gamma Phi t;jtj=k fi[f t (ff) Delta x k : It follows from known results ([2, 8, 18, 12]) that this series is recognizable. Here we provide an alternative proof of this result and we give an explicit construction of a triple recognizing S(ff; f; fi) This construction is a priori different from the known one. All the results are presented for bilinear functions. This restriction is ....

....7.1 in [2] the series S(ff; B; fi) is algebraic. Using an adaptation of an original argument by Parikh, see [8, 18] an algebraic series in one indeterminate over a commutative and idempotent semiring is recognizable (the use of Parikh result in the context of (max, algebraic series appears in [12]) Using the notions of simple tree path and simple tree circuit defined above, we obtain an alternative proof of this result. We get an explicit construction of a triple having the required property. Proposition 2. There exists a triple (a; A; b) of dimension O(n2 2n ) which recognizes S(ff; ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency, volume 11, pages 133--144. Cambridge University Press, 1998.

....schedule is not always optimal. Key words: Optimal scheduling, timed Petri net, heap of pieces, Tetris game, max, semiring, automaton with multiplicities, Sturmian word. 1 Introduction Heap models have recently been studied as a pertinent model of discrete event systems, see Gaubert Mairesse [19,20] and Brilman Vincent [12,13] They provide a good compromise between modeling power and tractability. As far as modeling is concerned, heap models are naturally associated with trace monoids, see [31] It was proved in [20] that the behavior of a timed onebounded Petri net can be represented ....

.... ( The ALgebraic Approach to Performance Evaluation of Discrete Event Systems ) Preprint submitted to Elsevier Preprint 26 October 2000 surface growth, see [5] The tractability follows essentially from the existence of a representation of the dynamic of a heap model by a (max, automaton, see [13,19]. A heap model is formed by a finite set of slots R and a finite set of pieces A. A piece is a solid block occupying a subset of the slots and having a polyomino shape. Given a ground whose shape is determined by a vector of R R and a word w = a 1 Delta Delta Delta a n 2 A , we consider ....

[Article contains additional citation context not shown here]

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency, volume 11, pages 133--144. Cambridge University Press, 1997.

....schedule is not always optimal. Key words: Optimal scheduling, timed Petri net, heap of pieces, Tetris game, max, semiring, automaton with multiplicities, Sturmian word. 1 Introduction Heap models have recently been studied as a pertinent model of discrete event systems, see Gaubert Mairesse [19,20] and Brilman Vincent [11,12] They provide a good compromise between modeling power and tractability. As far as modeling is concerned, heap models are naturally associated with trace 1 This work was partially supported by the European Community Framework IV programme through the research ....

.... net can be represented using a heap model (an example appears in Figure 1) We can also mention the use of heap models in the physics of surface growth, see [6] The tractability follows essentially from the existence of a representation of the dynamic of a heap model by a (max, automaton, see [12,19]. A heap model is formed by a finite set of slots and a finite set of pieces A. A piece is a solid block occupying a subset of the slots and having a polyomino shape. With a word w = a 1 Delta Delta Delta a n 2 A , we associate the heap obtained by piling up the pieces a 1 ; a n in ....

[Article contains additional citation context not shown here]

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency, volume 11, pages 133--144. Cambridge University Press, 1997.

....linear case boils down to the (max, case by switching to operator GammaT . Such systems appear in many domains of applications, under various forms. For example (without any kind of exhaustivity) ffl Computer science : parallel algorithms, shared memory systems, PERT graphs, see [43] or [23]. ffl Queueing theory : G=G=1=1 queue (see #5) queues in series, queues in series with blocking, fork join networks [3] RR n Sigma2641 30 F. Baccelli J. Mairesse ffl Operations research and manufacturing : Job shop models, event graphs (a subclass of Petri nets) see [17] 28] and [3] ffl ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1995.

....i after piling of the pieces of w. Next Theorem was c d b a d c a p3 p5 p6 p4 p2 p1 b p 6 p 4 a p 1 b p 2 p 3 c d p 5 Figure 17: A 1 bounded Petri net and the execution w = abcdabcd in the associated heap monoid proved under various forms: Vincent [26] Brilman Vincent[3] Gaubert Mairesse [11]. We illustrate it below in the case of general pieces (see Heap Automaton II) Theorem 5. The map x : T R Q max is (max, recognizable (see Def. 6) The corresponding (max, Automaton is (T ; Q; 1Q ; M) where M is defined by M(a) ij = 1 if piece a occupies the slots i and j, 0 ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency, volume 11, pages 133--144. Cambridge University Press, 1997.

....the convention that A ji (n) Gamma1 if j 62 L(i; n) It is easy to check that such a model, we could call it a task graph with random precedences, verifies Equation (2) A Queuing Network model studied by Baccelli Liu [5] corresponds to this model. The task resource models studied in [14] or [22] also have this kind of structure. 2.2 Cyclic Jackson network We consider a closed Jackson Network. The study of such closed networks can be traced back to Gordon and Newell, 25] In their original model, there is a given number of indistinguishable customers. The routing of the customers ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1995.

....IV programme through the research network ALAPEDES ( The ALgebraic Approach to Performance Evaluation of Discrete Event Systems ) 2 JEAN MAIRESSE AND LAURENT VUILLON 1. INTRODUCTION Heap models have recently been studied as a pertinent model of discrete event systems, see Gaubert Mairesse [18, 20] and Brilman Vincent [10, 11] They provide a good compromise between modeling power and tractability. As far as modeling is concerned, heap models are naturally associated with trace monoids, see [30] It was proved in [20] that the behavior of a timed one bounded Petri net can be represented ....

.... net can be represented using a heap model (an example appears in Figure 1) We can also mention the use of heap models in the physics of surface growth, see [5] The tractability follows essentially from the existence of a representation of the dynamic of a heap model by a (max, automaton, see [11, 18]. A heap model is formed by a finite set of slots and a finite set of pieces A. A piece is a solid block occupying a subset of the slots and having a polyomino shape. With a word w = a 1 Delta Delta Delta an 2 A , we associate the heap obtained by piling up the pieces a 1 ; an in ....

[Article contains additional citation context not shown here]

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency, volume 11, pages 133--144. Cambridge University Press, 1997.

....step was the observation that the height of heaps of pieces is recognized by a heap automaton, a special type of (max, automaton, the result holding for general, polyomino shaped, pieces. This MODELING AND ANALYSIS OF TIMED PETRI NETS USING HEAPS OF PIECES 3 was proved in Gaubert and Mairesse [21], and in a different form, by Brilman and Vincent [39, 7] The heap representation theorem for safe timed Petri nets that we give can be seen as a synthesis of these three results. The essential idea is that considering general pieces in heap automata enables to model time through the height of a ....

....(given by R(a) during a certain amount of time (u(a; r) Gamma l(a; r) for a resource r 2 R(a) In the simplest case where l(a; r) 0; 8r 2 R(a) the execution of a task begins as soon as all the required resources, used by earlier tasks, become free. For more details along these lines, see [21, 7]. Borrowing the terminology of [2] the maps y H and xH are called the dater functions of the heap model. The piling mechanism and the different notations are best understood graphically and on an example, see Fig. 1. Example 2.2. Let us consider the following heap model. ffl T = fa; b; c; dg, ....

[Article contains additional citation context not shown here]

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1997.

.... que nous pr#sentons ici a #t# propos#e par Gaubert [10] Un des exemples propos#s, le mod#le de stockage, est une adaptation de celui consid#r# dans [10] Le second exemple, un r#seau de Petri avec choix, est inspir# d un travail en pr#paration [11] Les syst#mes t#ches ressources, voir [6] [12], sont un autre exemple de mod#les se repr#sentant sous forme d automates (max, D# nition 4.1. On appelle automate sur l alphabet A un quadruplet (K; ffi; K e ; K s ) K est l ensemble des #tats, K e ae K l ensemble des #tats d entr#e et K s ae K celui des #tats de sortie. L application ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1995.

....a very intuitive graphical representation of trace monoids. 3. Max, linear representations. A next step was the observation that the height of heaps of pieces is recognized by a (max, automaton, the result holding for general, polyominoshaped, pieces. This was proved in Gaubert and Mairesse [16], and in a different form, by Brilman and Vincent [27] 6] The heap representation theorem for safe timed Petri nets that we give can be seen as a synthesis of these three results. The essential idea is that considering general pieces in heap models enables to model time through the height of a ....

....(given by R(a) during a certain amount of time (u(a; r) Gamma l(a; r) for a resource r 2 R(a) In the simplest case where l(a; r) 0; 8r 2 R(a) the execution of a task begins as soon as all the required resources, used by earlier tasks, become free. For more details along these lines, see [16], 6] Borrowing the terminology of [2] 15] the maps y H and xH are called the dater functions of the heap model. The piling mechanism and the different notations are best understood graphically and on an example, see Fig. 1. Example II.2. Let us consider the following heap model. ffl T = ....

[Article contains additional citation context not shown here]

S. Gaubertand J. Mairesse. Task resourcemodels and(max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1997.

....linear case boils down to the (max, case by switching to operator GammaT . Such systems appear in many domains of applications, under various forms. For example (without any kind of exhaustivity) ffl Computer science : parallel algorithms, shared memory systems, PERT graphs, see [43] or [23]. ffl Queueing theory : G=G=1=1 queue (see #5) queues in series, queues in series with blocking, fork join networks [3] ffl Operations research and manufacturing : Job shop models, event graphs (a subclass of Petri nets) see [17] 28] and [3] ffl Economy or control theory : dynamic ....

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1995.

....of trace monoids. 3. Max, linear representations. A next step was the observation that the height of heaps of pieces is recognized by a heap automaton, a special type of (max, automaton, the result holding for general, polyomino shaped, pieces. This was proved in Gaubert and Mairesse [22], and in a different form, by Brilman and Vincent [41] 7] The heap representation theorem for safe timed Petri nets that we give can be seen as a synthesis of these three results. The essential idea is that considering general pieces in heap automata enables to model time through the height of ....

....(given by R(a) during a certain amount of time (u(a; r) Gamma l(a; r) for a resource r 2 R(a) In the simplest case where l(a; r) 0; 8r 2 R(a) the execution of a task begins as soon as all the required resources, used by earlier tasks, become free. For more details along these lines, see [22], 7] 1 It corresponds for example to the mechanism of the Tetris game. 2 We recall the following standard notations. Given a finite set (alphabet) T , we denote by T n the set of words of length n on T . We denote by T = n2N T n the free monoid on T , that is, the set of finite ....

[Article contains additional citation context not shown here]

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. Appears in [25].

....by trace languages and trace monoids. The isomorphism between trace monoids and heap monoids was noticed in Viennot [23] The fact that the height of heaps of pieces is recognized by a (max, automaton, the result holding for general, polyomino shaped, pieces was proved in Gaubert and Mairesse [14], and in a different form, by Brilman and Vincent [24] 6] Heap representations are particularly well adapted to algebraic computations. As a typical illustration, we derive an heapbased performance evaluation method for safe jobshops. The assignment of the jobs on the machines is fixed but not ....

....(given by R(a) during a certain amount of time (u(a; r) Gamma l(a; r) for a resource r 2 R(a) In the simplest case where l(a; r) 0; 8r 2 R(a) the execution of a task begins as soon as all the required resources, used by earlier tasks, become free. For more details along these lines, see [14], 6] Borrowing the terminology of [2] 13] the maps y H and xH are called the dater functions of the heap model. The piling mechanism and the different notations are best understood graphically and on an example, see Fig. 1. Example II.2. Let us consider the following heap model. ffl T = ....

[Article contains additional citation context not shown here]

S. Gaubert and J. Mairesse. Task resource models and (max,+) automata. In J. Gunawardena, editor, Idempotency. Cambridge University Press, 1997.

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