## Markov Nets: Probabilistic Models for distributed and concurrent systems (2001)

Venue: | IEEE Transactions on Automatic Control |

Citations: | 30 - 15 self |

### BibTeX

@INPROCEEDINGS{Benveniste01markovnets:,

author = {Albert Benveniste and Eric Fabre and Stefan Haar},

title = {Markov Nets: Probabilistic Models for distributed and concurrent systems},

booktitle = {IEEE Transactions on Automatic Control},

year = {2001}

}

### Years of Citing Articles

### OpenURL

### Abstract

For distributed systems, i.e. large networked complex systems, there is a drastic difference between a local view and knowledge of the system, and its global view. Distributed systems have local state and time, but do not possess global state and time in the usual sense. In this paper, motivated by the monitoring of distributed systems and in particular of telecommunications networks, we develop Markov nets as an extension of Markov chains and hidden Markov models (Hmm) for distributed and concurrent systems. By a concurrent system, we mean a system in which components may evolve independently, with sparse synchronizations. We follow a so-called true concurrency approach, in which neither global state nor global linear time are available. Instead, we use only local states in combination with a partial order model of time. Our basic mathematical tool is that of Petri net unfoldings. Keywords : discrete event systems, stochastic Petri nets, unfoldings. 1 Motivations Distributed network...

### Citations

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Citation Context ...ing work of A.P. Dempster in the late seventies, when the so-called Dempster-Shafer theory of belief functions and belief networks was introduced in statistics and artificial intelligence (see Shafer =-=[28]-=-). Since then, this topic has known significant developments, but to our knowledge no generalization to interacting stochastic processes (i.e., with dynamics and infinite behaviours) has been develope... |

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Citation Context ...ed systems. 2.2 Runs of Petri nets : unfoldings We shall need a representation of all possible runs of a net. Branching Processes in the sense of Engelfriet [7], or Unfoldings as proposed by McMillan =-=[13]-=-, have been further extensively studied by Esparza [8][9] as a compact representation for the concurrent behavior of nets. Occurrence nets. Given two nodes n and n 0 (place or transition) of a Petri n... |

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Citation Context ...type (3) can be regarded as a transition of a Petri net, with its pre-- and post-set of places. Extending this discussion to several concurrent automata, we naturally arrive at considering Petri nets =-=[4]-=-[14] instead of automata and their products. For our purpose, the important facts about nets are the following : ffl States are local. Referring to our example above, the final picture involves only s... |

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Citation Context ...rtunately, such a policy is not local, and does not satisfy our second requirement. Other approaches have been proposed as well. However, as explained in [1], Generalized Stochastic Petri nets (GSPN) =-=[2]-=-[3] and Probabilistic Petri nets (PPN) [9] do not satisfy these requirements either. In this paper, we develop an approach that meets requirements (1,2). 2.2 Background notions on Petri nets and their... |

178 | W.: An improvement of McMillan’s unfolding algorithm
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Citation Context ...l need a representation of all possible runs of a net. Branching Processes in the sense of Engelfriet [7], or Unfoldings as proposed by McMillan [13], have been further extensively studied by Esparza =-=[8]-=-[9] as a compact representation for the concurrent behavior of nets. Occurrence nets. Given two nodes n and n 0 (place or transition) of a Petri net, we say that n causes n 0 , written nsn 0 , if eith... |

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Citation Context ...ve to interleavings. To our knowledge, no previously existing probabilistic model for communicating automata or Petri nets has these features. We have used unfoldings of Nielsen, Plotkin, and Winskel =-=[25]-=-, see also [12] and [13, 14]. We have considered here the case of unfoldings for 1-safe nets, but the reader should remember that the unfolding technique can be extended to general nets, see [17][16].... |

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Citation Context ...tension of stochastic automata to distributed systems. 2.2 Runs of Petri nets : unfoldings We shall need a representation of all possible runs of a net. Branching Processes in the sense of Engelfriet =-=[7]-=-, or Unfoldings as proposed by McMillan [13], have been further extensively studied by Esparza [8][9] as a compact representation for the concurrent behavior of nets. Occurrence nets. Given two nodes ... |

130 |
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Citation Context ...e (3) can be regarded as a transition of a Petri net, with its pre-- and post-set of places. Extending this discussion to several concurrent automata, we naturally arrive at considering Petri nets [4]=-=[14]-=- instead of automata and their products. For our purpose, the important facts about nets are the following : ffl States are local. Referring to our example above, the final picture involves only state... |

118 | What Is a Petri Net
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Citation Context ...or reasons that are discussed in [1], we restrict our construction to free choice Petri nets, the general case is also discussed in [1]. For completeness, we recall the definition of free choice nets =-=[6]-=-[3] : A (safe) Petri net N = (P; P 0 ; T ; !) is called free choice if jp ffl j ? 1 implies that, for all t 2 p ffl , j ffl tj = 1 holds. The following lemma provides the structure of layers for free ... |

114 | Coordinated decentralized protocols for failure diagnosis of discrete-event systems,” Discrete Event Dynamic Systems
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Citation Context ...ur of finite prefixes of runs. Proof : See Appendix B.7. ⋄ 5 Related work The problem of distributed diagnosis of discrete event systems motivated the present fundamental work, we refer the reader to =-=[11]-=- and the references therein. On the other hand, to our knowledge, net unfoldings are almost unkown in the control community. The only references we are aware of are [21][22], both in the context of su... |

64 | Diagnosis of asynchronous discrete event systems, a net unfolding approach
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Citation Context ...le above, each component has a totally ordered time, but the global system has not. Running example: stating our objectives Fig. 6–left introduces our running example, which is discussed in detail in =-=[7]-=- to illustrate asynchronous fault diagnosis of discrete event systems. It represents the evolution of the failed/safe status of two components interacting via their shared places 3,7. The first compon... |

60 | Stochastic petri nets: An introduction to the theory
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Citation Context ...wed to ignore each other (requirement (2)). A detailed discussion of this topic is found in [1], we give only an outline here. Randomizing Petri nets is performed, for Stochastic Petri nets (SPN) [20]=-=[5]-=-, by applying race policies in which exponential waiting times are allocated to enabled transitions and only the earliest one fires. The use of race policies involves global time and violates our prev... |

53 | Fault detection and diagnosis in distributed systems: an approach by partially stochastic Petri Nets. Journal of Discrete Event Dynamical Systems 8(2):203–231 - Aghasaryan, Fabre, et al. - 1998 |

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39 | S.: An unfolding algorithm for synchronous products of transition systems
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Citation Context ...eed a representation of all possible runs of a net. Branching Processes in the sense of Engelfriet [7], or Unfoldings as proposed by McMillan [13], have been further extensively studied by Esparza [8]=-=[9]-=- as a compact representation for the concurrent behavior of nets. Occurrence nets. Given two nodes n and n 0 (place or transition) of a Petri net, we say that n causes n 0 , written nsn 0 , if either ... |

25 |
Stochastic Petri Nets. Modelling, Stability, Simulation
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Citation Context ...allowed to ignore each other (requirement (2)). A detailed discussion of this topic is found in [1], we give only an outline here. Randomizing Petri nets is performed, for Stochastic Petri nets (SPN) =-=[20]-=-[5], by applying race policies in which exponential waiting times are allocated to enabled transitions and only the earliest one fires. The use of race policies involves global time and violates our p... |

20 |
Randomized non-sequential processes, in
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Citation Context ...lack of space, proofs, bibliographical discussion and some other important aspects are omitted, and the reader is referred for them to the extended version of this paper [1]. We only mention the work =-=[15]-=-, which is closest to ours, but is motivated by the modelling of randomized distributed algorithms, equipped with a coin flip construct. 2 Petri nets and unfoldings, as partial order models of distrib... |

18 | Distributed state reconstruction for discrete event systems - Fabre, Benveniste, et al. - 2000 |

11 | Free - Choice Petri Nets — An Algebraic Approach
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(Show Context)
Citation Context ...reasons that are discussed in [1], we restrict our construction to free choice Petri nets, the general case is also discussed in [1]. For completeness, we recall the definition of free choice nets [6]=-=[3]-=- : A (safe) Petri net N = (P; P 0 ; T ; !) is called free choice if jp ffl j ? 1 implies that, for all t 2 p ffl , j ffl tj = 1 holds. The following lemma provides the structure of layers for free cho... |

10 |
On the existence of liveness-enforcing supervisory policies of discreteevent systems modeled by n-safe Petri nets
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Citation Context ...rk, we refer the reader to [11] and the references therein. On the other hand, to our knowledge, net unfoldings are almost unkown in the control community. The only references we are aware of are [21]=-=[22]-=-, both in the context of supervisory control of discrete event systems modelled by Petri nets. Having random variables indexed by sets that are not linearly ordered has already been considered in the ... |

8 |
Liveness verification of discrete-event systems modeled by n-safe Petri nets
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Citation Context ...l work, we refer the reader to [11] and the references therein. On the other hand, to our knowledge, net unfoldings are almost unkown in the control community. The only references we are aware of are =-=[21]-=-[22], both in the context of supervisory control of discrete event systems modelled by Petri nets. Having random variables indexed by sets that are not linearly ordered has already been considered in ... |

8 |
Networks and Distributed Computation: Concepts, Tools and Algorithms
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Citation Context ...stantaneous (see Fig. 1). Hence, events from different nodes are only partially ordered. Similarly, building a global state requires gathering a consistent set of local states, which is not easy (see =-=[26]-=- for this topic). To summarize, networked systems possess local state and totally ordered local time. Global time, however, is only partially ordered, and it is preferred not to consider the global st... |

7 |
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Citation Context ...ts, but to our knowledge no generalization to interacting stochastic processes (i.e., with dynamics and infinite behaviours) has been developed. Benveniste, Lévy and Fabre have proposed more recently =-=[6]-=- a theory of interacting finite systems of random variables, but again no generalization to stochastic processes with dynamics was available. In [1] Fabre et al. have considered for the first time a n... |

6 | Branching Processes of general S/T-Systems - Haar - 1998 |

6 | Occurrence Net Logics - Haar |

5 |
Probabilistic Unfoldings and Partial Order Fairness in Petri Nets
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(Show Context)
Citation Context ...hout restrictions to arbitrary Petri nets. The cost is, of course, the use of a different semantics, with an additional effort necessary to include the scheduling of clusters; see the application, in =-=[18]-=-, of this to partial order fairness. Both branching process and cluster approaches have their respective merits and drawbacks, and should be seen as complementing one another. 6 Discussion and perspec... |

5 | Probabilistic cluster unfoldings
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Citation Context ... safe Petri nets. On the other hand, he focuses on the comparison of sequential versus concurrent semantics, and fairness. Finally, motivated by the findings of the present work, S. Haar has proposed =-=[19]-=- an approach to probabilizing partial order behaviors, in which the unfolding semantics is modified. In fact, rather than the token/place-oriented view of the branching process semantics, his approach... |

4 |
Fairness, Randomisierung und Konspiration in verteilten Algorithmen
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(Show Context)
Citation Context ...immediately vL : ωB ′ ⊙vL PπL ( vL | ωB from formula (10) since Pπ L is a probability. This finishes the construction of Pπ . Some comments are in order about the present construction, versus that of =-=[30]-=-. The construction of [30] applies to free choice conflicts only, whereas ours is more general. It uses no Kolmogorov’s extension argument, but rather performs a direct construction of the probability... |

3 |
P.A.Meyer: Probabilités et Potentiel, Chap
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Citation Context ...2) expresses that , where B ranges over the set of choice-conformal branching processes of P, the family Pπ B forms a projective family of probability distributions. By Kolmogorov’s extension theorem =-=[24]-=- (see Appendix B.3 for a detailed argument), there exists a unique probability Pπ over the projective limit of the choice-conformal B’s, this projective limit identifies with ΩP. This construction def... |

1 |
Free choice Petri nets—an algebraic approach
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(Show Context)
Citation Context ... next. Free choice nets are popular, they are regarded as an interesting intermediate class between automata and general Petri nets. For completeness, we recall the definition of free choice nets [10]=-=[4]-=-: Definition 5 (free choice Petri net) A Petri net P = (P, P0, T, →) is called free choice if |p • | > 1 implies that, for all t ∈ p • , | • t| = 1 holds. Lemma 4 Free choice nets are choice-compact—b... |