Scalable Performance Analysis of Massively Parallel Stochastic Systems

The accurate performance analysis of large-scale computer and communication systems is directly inhibited by an exponential growth in the state-space of the underlying Markovian performance model. This is particularly true when considering massively-parallel architectures such as cloud or grid computing infrastructures. Nevertheless, an ability to extract quantitative performance measures such as passage-time distributions from performance models of these systems is critical for providers of these services. Indeed, without such an ability, they remain unable to offer realistic end-to-end service level agreements (SLAs) which they can have any confidence of honouring. Additionally, this must be possible in a short enough period of time to allow many different parameter combinations in a complex system to be tested. If we can achieve this rapid performance analysis goal, it will enable service providers and engineers to determine the cost-optimal behaviour which satisfies the SLAs. In this thesis, we develop a scalable performance analysis framework for the grouped PEPA stochastic process algebra. Our approach is based on the approximation of key model quantities such as means and variances by tractable systems of ordinary differential equations (ODEs). Crucially, the size of these systems of ODEs is independent of the number of interacting entities within the model, making these analysis techniques extremely scalable. The reliability of our approach is directly supported by convergence results and, in some cases, explicit error bounds. We focus on extracting passage-time measures from performance models since these are very commonly the language in which a service level agreement is phrased. We design scalable analysis techniques which can handle passages defined both in terms of entire component populations as well as individual or tagged members of a large population. A precise and straightforward specification of a passage-time service level agreement is as important to the performance engineering process as its evaluation. This is especially true of large and complex models of industrial-scale systems. To address this, we introduce the unified stochastic probe framework. Unified stochastic probes are used to generate a model augmentation which exposes explicitly the SLA measure of interest to the analysis toolkit. In this thesis, we deploy these probes to define many detailed and derived performance measures that can be automatically and directly analysed using rapid ODE techniques. In this way, we tackle applicable problems at many levels of the performance engineering process: from specification and model representation to efficient and scalable analysis

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

Modeling SMS driven conversion of ceramide to sphingomyelin reveals the existence of a positive feedback mechanism

In questa tesi presentiamo un modello matematico minimo per la conversione di un ceramide in sfingomielina catalizzata dall'enzima sfingomielina sintasi 1 (SMS1) basato sulle leggi della cinetica chimica. Viene dimostrato, utilizzando tecniche di sampling per la stima parametrica e metodi di analisi matematica, che questo modello non è in grado di riprodurre qualitativamente delle misure sperimentali sulla composizioni dei lipidi in seguito ad alterazione dell'attivita enzimatica di SMS1. Concludiamo quindi che è necessario considerare un meccanismo di feedback positivo fra i prodotti e i reagenti della reazione, che esiste effettivamente in vivo tramite la proteina chinasi D e la proteina di trasporto di ceramide CERT. Di conseguenza, proponiamo un secondo modello modificato in modo da comprendere questo meccanismo di feedback, che risulta essere in grado di spiegare i risultati sperimentali // Here we present a minimal mathematical model for the Sphingomyelin synthase 1 (SMS1) driven conversion of ceramide to sphingomyelin based on chemical reaction kinetics. We demonstrate, via sampling-based parameter estimation and mathematical analysis, that this model is not able to qualitatively reproduce experimental measurements on lipid compositions after altering SMS1 activities. We conclude that a positive feedback mechanism is required from the products to the reactants of the reaction, which in fact exists in vivo via protein kinase D and the ceramide transfer protein CERT. Accordingly, a modified model that comprises this feedback mechanism was able to reproduce experimental findingsope

Studying the effects of adding spatiality to a process algebra model

We use NetLogo to create simulations of two models of disease transmission originally expressed in WSCCS. This allows us to introduce spatiality into the models and explore the consequences of having different contact structures among the agents. In previous work, mean field equations were derived from the WSCCS models, giving a description of the aggregate behaviour of the overall population of agents. These results turned out to differ from results obtained by another team using cellular automata models, which differ from process algebra by being inherently spatial. By using NetLogo we are able to explore whether spatiality, and resulting differences in the contact structures in the two kinds of models, are the reason for this different results. Our tentative conclusions, based at this point on informal observations of simulation results, are that space does indeed make a big difference. If space is ignored and individuals are allowed to mix randomly, then the simulations yield results that closely match the mean field equations, and consequently also match the associated global transmission terms (explained below). At the opposite extreme, if individuals can only contact their immediate neighbours, the simulation results are very different from the mean field equations (and also do not match the global transmission terms). These results are not surprising, and are consistent with other cellular automata-based approaches. We found that it was easy and convenient to implement and simulate the WSCCS models within NetLogo, and we recommend this approach to anyone wishing to explore the effects of introducing spatiality into a process algebra model

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