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Trueconcurrency probabilistic models Branching cells and distributed probabilities for event structures
, 2006
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Concurrency, σalgebras, and probabilistic fairness
"... We extend previous constructions of probabilities for a prime event structure E by allowing arbitrary confusion. Our study builds on results related to fairness in event structures that are of interest per se. Executions of E are captured by the set Ω of maximal configurations. We show that the inf ..."
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We extend previous constructions of probabilities for a prime event structure E by allowing arbitrary confusion. Our study builds on results related to fairness in event structures that are of interest per se. Executions of E are captured by the set Ω of maximal configurations. We show that the information collected by observing only fair executions of E is confined in some σalgebra F0, contained in the Borel σalgebra F of Ω. Equality F0 = F holds when confusion is finite (formally, for the class of locally finite event structures), but inclusion F0 ⊆ F is strict in general. We show the existence of an increasing chain F0 ⊆ F1 ⊆ F2 ⊆... of subσalgebras of F that capture the information collected when observing executions of increasing unfairness. We show that, if the event structure unfolds a 1safe net, then unfairness remains quantitatively bounded, that is, the above chain reaches F in finitely many steps. The construction of probabilities typically relies on a Kolmogorov extension argument. Such arguments can achieve the construction of probabilities on the σalgebra F0 only, while one is interested in probabilities defined on the entire Borel σalgebra F. We prove that, when the event structure unfolds a 1safe net, then unfair executions all belong to some set of F0 of zero probability. Whence F0 = F modulo 0 always holds, whereas F0 ̸ = F in general. This yields a new construction of Markovian probabilistic nets, carrying a natural interpretation that “unfair executions possess zero probability”.
Diagnosis with Petri Net Unfoldings
 C. SEATZU ET AL. (EDS.): CONTROL OF DISCRETEEVENT SYSTEMS, LNCIS 433, PP. 301–318
, 2013
"... Large systems or softwares are generally obtained by designing independent modules or functions, and by assembling them through appropriate interfaces to obtain more elaborate functions and modules. The latter can in turn be assembled, up to forming huge systems providing sophisticated services. Con ..."
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Large systems or softwares are generally obtained by designing independent modules or functions, and by assembling them through appropriate interfaces to obtain more elaborate functions and modules. The latter can in turn be assembled, up to forming huge systems providing sophisticated services. Consider for instance the various components of a computer, telecommunication networks, plane ticket reservation softwares for a company, etc. Such systems are not only modular in their design, but often multithreaded, in the sense that many events may occur in parallel. From a discrete event system perspective, such modular or distributed systems can be modeled in a similar manner, by first designing component models and then assembling them through an adequate composition operation. A first approach to this design principle has been presented in Chapter 5 (see Section 5.5): composition can be defined as the synchronous product of automata. The transitions of each component carry labels, and the product proceeds by synchronizing transitions with identical labels, while all the other transitions remain private. This construction is recalled in Fig. 15.1 on the simple case of three tiny automata. The size of the resulting system is rather surprising, given the simplicity of the three components! And this deserves a detailed study. One first notices the classical state space explosion phenomenon: the number of states in the global system is the product of the number of states of their components (here 2×2×3 = 12). So, the number of states augments exponentially fast with the
Markov Concurrent Processes
, 2010
"... We introduce a model for probabilistic systems with concurrency. The system is distributed over two local sites. Global trajectories of the system are composed of local trajectories glued along synchronizing points. Global trajectories are thus given as partial orders of events, and not as paths. As ..."
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We introduce a model for probabilistic systems with concurrency. The system is distributed over two local sites. Global trajectories of the system are composed of local trajectories glued along synchronizing points. Global trajectories are thus given as partial orders of events, and not as paths. As a consequence, time appears as a dynamic partial order, contrasting with the universal chain of integers we are used to. It is surprising to see how natural it is to adapt mathematical techniques for processes to this new conception of time. The probabilistic model has two features: first, it is Markov, in a sense convenient for concurrent systems; and second, the local components have maximal independence, beside their synchronization constraints. We construct such systems and characterize them by finitely many real parameters, that are the analogous to the transition matrix for discrete Markov chains. This construction appears as a generalization of the “synchronization of Markov chains ” developed in an earlier collaboration.