## Algebraic System Analysis of Timed Petri Nets (1997)

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Citations: | 14 - 6 self |

### BibTeX

@INPROCEEDINGS{Cohen97algebraicsystem,

author = {G. Cohen and S. Gaubert and J. P. Quadrat},

title = {Algebraic System Analysis of Timed Petri Nets},

booktitle = {},

year = {1997},

pages = {145--170},

publisher = {Cambridge University Press}

}

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### Abstract

We show that Continuous Timed Petri Nets (CTPN) can be modeled by generalized polynomial recurrent equations in the (min,+) semiring. We establish a correspondence between CTPN and Markov decision processes. We survey the basic system theoretical results available: behavioral (input-output) properties, algebraic representations, asymptotic regime. A particular attention is paid to the subclass of stable systems (with asymptotic linear growth). 1 Introduction The fact that a subclass of Discrete Event Systems equations write linearly in the (min,+) or in the (max,+) semiring is now almost classical [9, 2]. The (min,+) linearity allows the presence of synchronization and saturation features but unfortunately prohibits the modeling of many interesting phenomena such as "birth" and "death" processes (multiplication of tokens) and concurrency. The purpose of this paper is to show that after some simplifications, these additional features can be represented by polynomial recurrences in the ...

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Citation Context ...hat the Scott continuity together with additivity is equivalent to the preservation of arbitrary inf: H(inf i u i ) = inf i Hu i for an arbitrary family. The Scott topology is presented in details in =-=[16]-=-. What we call here Scott continuity is in fact Scott continuity with respect to the algebraic order # of the (min,+) semiring, defined by a # b ## a # b = b ( which is reversed with respect to natura... |

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Citation Context ...ize the subclass of series of A min [[#]] which arise as transfer operators of CTEGM (called transfer series). We recall that given a semiring of formal series K[[#]], the semiring of rational series =-=[4]-=- denoted by K rat [[#]] is the least subsemiring containing polynomials and stable by the operation #, #, #, where a # def = L n#N a n is defined only on series with zero constant coe#cient. An immedi... |

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Citation Context ...me weight). 4.7 Resource Optimization Problems As a by product of the above characterizations of the throughput �, it is possible to address resource optimization problems. The most classical problem =-=[8, 18, 21, 12]-=- relative to TEG consists in optimizing a linear cost function J�m� associated with the initial marking, under the constraint � � �0. Physically, the initial marking represents resources (number of ma... |

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Citation Context ...e systems (with asymptotic linear growth). 1 Introduction The fact that a subclass of Discrete Event Systems equations write linearly in the (min,+) or in the (max,+) semiring is now almost classical =-=[9, 2]. The (min,+) linear-=-ity allows the presence of synchronization and saturation features but unfortunately prohibits the modeling of many interesting phenomena such as "birth" and "death" processes (mul... |

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Citation Context ... ; # # R + and by the powers of ##, where # is a given and fixed value of , will be denoted by R min [##]. It is a particular instance of a classical structure in di#erence algebra: Ore polynomials 7 =-=[26, 19, 13]-=-. 4. The semiring generated by # # , ; # # R, # R +# is isomorphic to the semiring of nondecreasing concave piecewise a#ne maps R # {+#} # R # {+#}, that we denote by A min . A generic element in A mi... |

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Citation Context ... 17 where the latest quantity will be used only when the graph admits potential v. The following periodicity theorem is central. The CTEG case is a consequence of the (max,+)-Perron Frobenius theorem =-=[25, 8, 2, 10]-=-. Another proof has been given by Chretienne [7]. The inequality variant below (4.24) can be found in [12, Ch. IV, Lemma 1.3.8],[14]. The #-discounted case is due to Braker and Resing [5, 6]. Theorem ... |

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Citation Context ...tion of the Isaac Newton Institute, Cambridge University Press, 1997 2 G. Cohen, S. Gaubert and J. P. Quadrat equations given here are much more tractable than the dater equations obtained previously =-=[1]-=-. Similar equations have been introduced by Baccelli et al. [3] in a stochastic context. In 3, we introduce the continuous analogue of Timed Petri Nets. We discuss various natural routing policies, an... |

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Citation Context ...m [25, 8, 2, 10]. Another proof has been given by Chretienne [7]. The inequality variant below (4.24) can be found in [12, Ch. IV, Lemma 1.3.8],[14]. The #-discounted case is due to Braker and Resing =-=[5, 6]-=-. Theorem 4.16. Consider a strongly connected CTEG. There exists N # 0 and c # 1 (cyclicity) such that, for all initial condition v, t # N # Z(t + c) = #c + Z(t) , (4.21) where # = min C |C| # |C| # (... |

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Citation Context ... P. Quadrat 4.7 Resource Optimization Problems As a by product of the above characterizations of the throughput #, it is possible to address resource optimization problems. The most classical problem =-=[8, 18, 21, 12]-=- relative to TEG consists in optimizing a linear cost function J(m) associated with the initial marking, under the constraint # # # 0 . Physically, the initial marking represents resources (number of ... |

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Citation Context ... P. Quadrat 4.7 Resource Optimization Problems As a by product of the above characterizations of the throughput #, it is possible to address resource optimization problems. The most classical problem =-=[8, 18, 21, 12]-=- relative to TEG consists in optimizing a linear cost function J(m) associated with the initial marking, under the constraint # # # 0 . Physically, the initial marking represents resources (number of ... |

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Citation Context ...##]] is rational i# it is a merge of ultimatelys#-periodic series. The CTEG case (i.e. # = 1) is proved in [9, 2] for the subclass of monotone 15 series h n+1 # h n . It was already noticed by Moller =-=[22]-=- in the non monotone case. It is essentially known to the tropical community [20]. The #-generalization was announced in [13]. The proof will appear in a paper in preparation [15]. No such simple char... |

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Citation Context ... ; # # R + and by the powers of ##, where # is a given and fixed value of , will be denoted by R min [##]. It is a particular instance of a classical structure in di#erence algebra: Ore polynomials 7 =-=[26, 19, 13]-=-. 4. The semiring generated by # # , ; # # R, # R +# is isomorphic to the semiring of nondecreasing concave piecewise a#ne maps R # {+#} # R # {+#}, that we denote by A min . A generic element in A mi... |

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Citation Context ...(i.e. # = 1) is proved in [9, 2] for the subclass of monotone 15 series h n+1 # h n . It was already noticed by Moller [22] in the non monotone case. It is essentially known to the tropical community =-=[20]-=-. The #-generalization was announced in [13]. The proof will appear in a paper in preparation [15]. No such simple characterization seems to exist for A rat min [[#]]: the coe#cient h # of # # in h is... |

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Citation Context ...aph admits potential v. The following periodicity theorem is central. The CTEG case is a consequence of the (max,+)-Perron Frobenius theorem [25, 8, 2, 10]. Another proof has been given by Chretienne =-=[7]-=-. The inequality variant below (4.24) can be found in [12, Ch. IV, Lemma 1.3.8],[14]. The #-discounted case is due to Braker and Resing [5, 6]. Theorem 4.16. Consider a strongly connected CTEG. There ... |

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Citation Context ... is a consequence of the (max,+)-Perron Frobenius theorem [25, 8, 2, 10]. Another proof has been given by Chretienne [7]. The inequality variant below (4.24) can be found in [12, Ch. IV, Lemma 1.3.8],=-=[14]-=-. The #-discounted case is due to Braker and Resing [5, 6]. Theorem 4.16. Consider a strongly connected CTEG. There exists N # 0 and c # 1 (cyclicity) such that, for all initial condition v, t # N # Z... |

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Citation Context ... continuous version of TPN (in which the number of tokens are real numbers instead of integers). Such continuous models occur naturally when fluids rather than tokens flow in networks (see [2, 1.2.7],=-=[24]-=- for an elementary example). They also arise as approximation of (discrete) Petri Nets since they provide an upper bound for the real behavior. A continuous TPN (CTPN) is defined as a TPN, but the mar... |

8 |
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Citation Context ... P. Quadrat 4.7 Resource Optimization Problems As a by product of the above characterizations of the throughput #, it is possible to address resource optimization problems. The most classical problem =-=[8, 18, 21, 12]-=- relative to TEG consists in optimizing a linear cost function J(m) associated with the initial marking, under the constraint # # # 0 . Physically, the initial marking represents resources (number of ... |

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Citation Context ...a change of units (they are linearized by a non linear change of variable in the (min,+) semiring). The importance and tractability of the (non continuous) version of these systems, called expansible =-=[23]-=- was first recognized by Munier. 3. #-discounted TEG are the TEG-analogue of uniformly discounted deterministic optimization problems. They represent systems with constant birth (or death) rate #. 4. ... |

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Citation Context ... 17 where the latest quantity will be used only when the graph admits potential v. The following periodicity theorem is central. The CTEG case is a consequence of the (max,+)-Perron Frobenius theorem =-=[25, 8, 2, 10]-=-. Another proof has been given by Chretienne [7]. The inequality variant below (4.24) can be found in [12, Ch. IV, Lemma 1.3.8],[14]. The #-discounted case is due to Braker and Resing [5, 6]. Theorem ... |

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Citation Context ... 17 where the latest quantity will be used only when the graph admits potential v. The following periodicity theorem is central. The CTEG case is a consequence of the (max,+)-Perron Frobenius theorem =-=[25, 8, 2, 10]-=-. Another proof has been given by Chretienne [7]. The inequality variant below (4.24) can be found in [12, Ch. IV, Lemma 1.3.8],[14]. The #-discounted case is due to Braker and Resing [5, 6]. Theorem ... |

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Citation Context ...oughput # is characterized as the unique value for which a finite vector v is solution of v = min p (# p - ## p + P p v) . (5.15) Indeed, the asymptotic behavior of Z(t) is known in much more details =-=[27]-=-. Note that the e#ective computation of # from (5.15) proceeds from standard algorithms (Policy Improvement [28], Linear Programming). Proof. This is an adaptation of standard stochastic control resul... |

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Citation Context ...+ P p v) . (5.15) Indeed, the asymptotic behavior of Z(t) is known in much more details [27]. Note that the e#ective computation of # from (5.15) proceeds from standard algorithms (Policy Improvement =-=[28]-=-, Linear Programming). Proof. This is an adaptation of standard stochastic control results [28, Chap. 33, Th. 4.1]. The growth rate # is independent of the initial point q for the subclass of communic... |

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Citation Context ...m [25, 8, 2, 10]. Another proof has been given by Chretienne [7]. The inequality variant below (4.24) can be found in [12, Ch. IV, Lemma 1.3.8],[14]. The #-discounted case is due to Braker and Resing =-=[5, 6]-=-. Theorem 4.16. Consider a strongly connected CTEG. There exists N # 0 and c # 1 (cyclicity) such that, for all initial condition v, t # N # Z(t + c) = #c + Z(t) , (4.21) where # = min C |C| # |C| # (... |

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Citation Context ... to SISO systems (the MIMO case is not more di#cult, although the notation is more intricate). We introduce the family of indeterminatessu # , # # N. The series s given by (5.10) is a Volterra series =-=[11]-=- if for all # , the series is a polynomial in the indeterminate u # (equivalently, if the indeterminate u # appears in (5.10) with a finite number of exponents). The evaluation su of the Volterra seri... |

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Citation Context ...dy noticed by Moller [22] in the non monotone case. It is essentially known to the tropical community [20]. The #-generalization was announced in [13]. The proof will appear in a paper in preparation =-=[15]-=-. No such simple characterization seems to exist for A rat min [[#]]: the coe#cient h # of # # in h is an element of A min , but its complexity 16 grows in general as # ##. 4.6 Asymptotic Behavior of ... |

1 |
Quelques exemples de series formelles utilisees en algebre non commutative. Rapport de recherche 90-2, Universite de Paris 7
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Citation Context ... ; # # R + and by the powers of ##, where # is a given and fixed value of , will be denoted by R min [##]. It is a particular instance of a classical structure in di#erence algebra: Ore polynomials 7 =-=[26, 19, 13]-=-. 4. The semiring generated by # # , ; # # R, # R +# is isomorphic to the semiring of nondecreasing concave piecewise a#ne maps R # {+#} # R # {+#}, that we denote by A min . A generic element in A mi... |

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Citation Context ... [25, 8, 2, 10]. Another proof has been given by Chretienne [7]. The inequality variant below (4.24) can be found in [12, Ch. IV, Lemma 1.3.8],[14]. The ff-discounted case is due to Braker and Resing =-=[5, 6]. Th-=-eorem 4.16. Consider a strongly connected CTEG. There exists Ns0 and cs1 (cyclicity) such that, for all initial condition v, tsN ) Z(t + c) = c + Z(t) ; (4.21) wheres= min C jCjsjCj �� (4.22) (the... |