Synthesis and Analysis of Product-form Petri Nets

For a large Markovian model, a "product form" is an explicit description of the steady-state behaviour which is otherwise generally untractable. Being first introduced in queueing networks, it has been adapted to Markovian Petri nets. Here we address three relevant issues for product-form Petri nets which were left fully or partially open: (1) we provide a sound and complete set of rules for the synthesis; (2) we characterise the exact complexity of classical problems like reachability; (3) we introduce a new subclass for which the normalising constant (a crucial value for product-form expression) can be efficiently computed.Comment: This is a version including proofs of the conference paper: Haddad, Mairesse and Nguyen. Synthesis and Analysis of Product-form Petri Nets. Accepted at the conference Petri Nets 201

Unbounded Product-Form Petri Nets

Computing steady-state distributions in infinite-state stochastic systems is in general a very difficult task. Product-form Petri nets are those Petri nets for which the steady-state distribution can be described as a natural product corresponding, up to a normalising constant, to an exponentiation of the markings. However, even though some classes of nets are known to have a product-form distribution, computing the normalising constant can be hard. The class of (closed) Pi^3-nets has been proposed in an earlier work, for which it is shown that one can compute the steady-state distribution efficiently. However these nets are bounded. In this paper, we generalise queuing Markovian networks and closed Pi^3-nets to obtain the class of open Pi^3-nets, that generate infinite-state systems. We show interesting properties of these nets: (1) we prove that liveness can be decided in polynomial time, and that reachability in live Pi^3-nets can be decided in polynomial time; (2) we show that we can decide ergodicity of such nets in polynomial time as well; (3) we provide a pseudo-polynomial time algorithm to compute the normalising constant

Towards efficient analysis of Markov automata

One of the most expressive formalisms to model concurrent systems is Markov automata. They serve as a semantics for many higher-level formalisms, such as generalised stochastic Petri nets and dynamic fault trees. Two of the most challenging problems for Markov automata to date are (i) the optimal time-bounded reachability probability and (ii) the optimal long-run average rewards. In this thesis, we aim at designing efficient sound techniques to analyse them. We approach the problem of time-bounded reachability from two different angles. First, we study the properties of the optimal solution and exploit this knowledge to construct an efficient algorithm that approximates the optimal values up to a guaranteed error bound. This algorithm is exhaustive, i. e. it computes values for each state of the Markov automaton. This may be a limitation for very large or even infinite Markov automata. To address this issue we design a second algorithm that approximates the optimal solution by only working with part of the total state-space. For the problem of long-run average rewards there exists a polynomial algorithm based on linear programming. Instead of chasing a better theoretical complexity bound we search for a practical solution based on an iterative approach. We design a value iteration algorithm that in our empirical evaluation turns out to scale several orders of magnitude better than the linear programming based approach.Markov-Automaten bilden einen der ausdrucksstärksten Formalismen um Nebenläufige Systeme zu modellieren. Sie werden benutzt um die Semantik vieler höherer Formalismen wie stochastischer Petri-Netze [Mar95, EHZ10] und Dynamic Fault Trees [DBB90] zu beschreiben. Die zwei herausfordernder Probleme im Bereich der Analyse großer Markov- Automaten sind (i) die zeitbeschränkten Erreichbarkeitwahrscheinlichkeit und (ii) optimale langfristige durchschnittliche Rewards. Diese Arbeit zielt auf das Design effizienter und korrekter Techniken um sie zu untersuchen. Das Problem der zeitbeschränkten Erreichbarkeitswahrscheinlichkeit gehen wir aus zwei verschiedenen Richtungen an: Zum einen studieren wir die Eigenschaften optimaler Lösungen und nutzen dieses Wissen um einen effizienten Approximationsalgorithmus zu bilden, der optimale Werte bis auf eine garantierte Fehlertoleranz berechnet. Dieser Algorithmus basiert darauf, Werte für jeden Zustand des Markov-Automaten zu berechnen. Dies kann die Anwendbarkeit für große oder gar unendliche Automaten einschränken. Um diese Problem zu lösen präsentieren wir einen zweiten Algorithmus, der die optimale Lösung approximiert, und dabei ausschließlich einen Teil des Zustandsraumes betrachtet. Für das Problem der optimalen langfristigen durchschnittlichen Rewards gibt es einen polynomiellen Algorithmus auf Basis linearer Programmierung. Anstelle eine bessere theoretische Komplexität anzustreben, konzentrieren wir uns darauf, eine praktische Lösung auf Basis eines iterativen Ansatzes zu finden. Wie entwickeln einen Werte-iterierenden Algorithmus der in unserer empirischen Evaluation um mehrere Größenordnungen besser als der auf linearer Programmierung basierende Ansatz skaliert