Structural characterization of decomposition in rate-insensitive stochastic Petri nets

This paper focuses on stochastic Petri nets that have an equilibrium distribution that is a product form over the number of tokens at the places. We formulate a decomposition result for the class of nets that have a product form solution irrespective of the values of the transition rates. These nets where algebraically characterized by Haddad et al.~as $S\Pi^2$ nets. By providing an intuitive interpretation of this algebraical characterization, and associating state machines to sets of $T$-invariants, we obtain a one-to-one correspondence between the marking of the original places and the places of the added state machines. This enables us to show that the subclass of stochastic Petri nets under study can be decomposed into subnets that are identified by sets of its $T$-invariants

A <i>P</i>- and <i>T</i>-invariant characterization of product form and decomposition in stochastic Petri nets

Structural product form and decomposition results for stochastic Petri nets are surveyed,unifed and extended. The contribution is threefold. First, the literature on structural results for product form over the number of tokens at the places is surveyed and rephrased completely in terms of T-invariants. Second, based on the underlying concept of group-local-balance, the product form results for stochastic Petri nets are demarcated and an intuitive explanation is provided of these results based on T-invariants, only. Third, a decomposition result is provided that is completely formulated in terms of both T-invariants and P-invariants

A fluid analysis framework for a Markovian process algebra

Markovian process algebras, such as PEPA and stochastic π-calculus, bring a powerful compositional approach to the performance modelling of complex systems. However, the models generated by process algebras, as with other interleaving formalisms, are susceptible to the state space explosion problem. Models with only a modest number of process algebra terms can easily generate so many states that they are all but intractable to traditional solution techniques. Previous work aimed at addressing this problem has presented a fluid-flow approximation allowing the analysis of systems which would otherwise be inaccessible. To achieve this, systems of ordinary differential equations describing the fluid flow of the stochastic process algebra model are generated informally. In this paper, we show formally that for a large class of models, this fluid-flow analysis can be directly derived from the stochastic process algebra model as an approximation to the mean number of component types within the model. The nature of the fluid approximation is derived and characterised by direct comparison with the Chapman–Kolmogorov equations underlying the Markov model. Furthermore, we compare the fluid approximation with the exact solution using stochastic simulation and we are able to demonstrate that it is a very accurate approximation in many cases. For the first time, we also show how to extend these techniques naturally to generate systems of differential equations approximating higher order moments of model component counts. These are important performance characteristics for estimating, for instance, the variance of the component counts. This is very necessary if we are to understand how precise the fluid-flow calculation is, in a given modelling situation

Fluid aggregations for Markovian process algebra

Quantitative analysis by means of discrete-state stochastic processes is hindered by the well-known phenomenon of state-space explosion, whereby the size of the state space may have an exponential growth with the number of objects in the model. When the stochastic process underlies a Markovian process algebra model, this problem may be alleviated by suitable notions of behavioural equivalence that induce lumping at the underlying continuous-time Markov chain, establishing an exact relation between a potentially much smaller aggregated chain and the original one. However, in the modelling of massively distributed computer systems, even aggregated chains may be still too large for efficient numerical analysis. Recently this problem has been addressed by fluid techniques, where the Markov chain is approximated by a system of ordinary differential equations (ODEs) whose size does not depend on the number of the objects in the model. The technique has been primarily applied in the case of massively replicated sequential processes with small local state space sizes. This thesis devises two different approaches that broaden the scope of applicability of efficient fluid approximations. Fluid lumpability applies in the case where objects are composites of simple objects, and aggregates the potentially massive, naively constructed ODE system into one whose size is independent from the number of composites in the model. Similarly to quasi and near lumpability, we introduce approximate fluid lumpability that covers ODE systems which can be aggregated after a small perturbation in the parameters. The technique of spatial aggregation, instead, applies to models whose objects perform a random walk on a two-dimensional lattice. Specifically, it is shown that the underlying ODE system, whose size is proportional to the number of the regions, converges to a system of partial differential equations of constant size as the number of regions goes to infinity. This allows for an efficient analysis of large-scale mobile models in continuous space like ad hoc networks and multi-agent systems

A Two-Level Decomposition Scheme for Markovian Process Algebra Models

Die nebenläufige Komposition von Markovketten führt mit steigender Anzahl an involvierten Komponenten schnell zum bekannten Problem der Zustandsraumexplosion, auch bekannt als Largeness Problem. Im Kontext von Markovschen Prozess-Algebren (MPA) ist dieses Problem von besonderer Bedeutung, da kleine und kompakte Modellbeschreibungen in Form von Sprachtermen riesige Markovketten repräsentieren können. Viele altbekannte Gegenstrategien zur Zustandsraumexplosion, wie z.B. Produkt-Form-Lösungen, Lumpability, dünnbesetzte Datenstrukturen, können auf die von der jeweiligen MPA erzeugten Markovkette angewendet werden. Die jüngste Forschung konzentriert sich vornehmlich auf die Klassifikation von syntaktischen Eigenschaften auf der MPA Sprachebene, welche die Anwendbarkeit dieser Strategien garantieren. In der vorliegenden Arbeit schlagen wir einen neuen Ansatz zur Lösung von MPA Modellen vor, der explizit die nebenläufige Struktur des gegebenen Modells ausnutzt. Diese Methode besteht aus zwei Ebenen der Kompositionalität. In der ersten Ebene wird das Modell entlang von globalen Synchronisationspunkten in mehrere Submodelle aufgespalten. Diese Submodelle werden zunächst in Isolation gelöst; anschließend erhält man durch geeignete Kombination der einzelnen Lösungen eine Lösung für das gesamte Modell. Die zweite Ebene der Kompositionalität betrifft die individuellen Submodelle. Unter bestimmten Bedingungen kann jedes Submodell als die parallele Entwicklung mehrerer unabhängiger absorbierender Markovketten beschrieben werden. Diese Unabhängigkeit kann zur Lösung der Submodelle ausgenutzt werden. Als ein Nebenprodukt der Betrachtung der zweiten Ebene der Kompositionalität, präsentieren wir ein neues Resultat über kumulative Ma\ss{}e gemeinsamer absorbierender Markovketten. Falls die marginalen Prozesse unabhängige kontinuierliche Markovketten sind, können die mittlere Zeit bis zur Absorption, sowie die mittlere Verweilzeit in einer transienten Teilmenge des Zustandsraums aus isolierten Lösungen der marginalen Prozesse zusammengesetzt werden. Da bei dieser Methode keine Operationen auf dem gemeinsamen Zustandsraum ausgeführt werden, umgehen wir das Problem der Zustandsraumexplosion. Der Rechenbedarf unserer Methode hängt von Konvergenzeigenschaften der gemeinsamen Markovkette ab, d.h. von der Anzahl an Schritten bis zur Absorption einer in der gemeinsamen kontinuierlichen Markovkette eingebetteten diskreten Markovkette.The concurrent composition of Markov chains quickly leads to the notorious problem of state space explosion, also known as the largeness problem, as the number of involved Markov chains increases. In the context of Markovian Process Algebras (MPAs) this problem is of particular interest, since small and compact model descriptions in form of language terms provided by the MPA may possess huge underlying Markov chains. Of course, many long known counter-strategies to tackle the largeness problem of Markov chains in one way or the other, like, e.g., product-form solutions, lumpability, sparse data structures, nearly-complete decomposability, can also be applied to the Markov chains which are generated by MPA models. Recent research mostly focuses on the classification of syntactical properties on the MPA language term level which ensure the applicability of these strategies. In this work we propose a novel approach to solve MPA models which explicitly exploits the concurrent nature of the given model. The method involves two levels of compositionality. In the first level, the model is decomposed along points of global synchronisation into several sub-models. These sub-models are solved in isolation, and afterwards the individual results are combined to yield a solution of the entire model. The second level of compositionality concerns the individual sub-models. Under certain conditions each sub-model can be described as the parallel evolution of a number of independent absorbing Markov chains. This independence can be exploited to efficiently solve the sub-models. As a side result of the consideration of the second level of compositionality, we derive a novel result on cumulative measures of absorbing joint Markov chains. Provided that the marginal processes are independent continuous time Markov chains (CTMCs), the mean time to absorption and the expected total time in a transient set of the joint Markov chain are computed from the marginal CTMCs in a compositional way. Operations on the state space of the joint Markov chain are never carried out, hence, the problem of state space explosion is avoided. The computational effort of our method rather depends on convergence properties of the joint CTMC, i.e., the number of steps until absorption of a discrete time Markov chain embedded in the joint CTMC