The Möbius framework is an environment for supporting multiple modeling formalisms and solution techniques. Models expressed in formalisms that are compatible with the framework are translated into equivalent models using Mobius framework components. This translation preserves the structure of the models, allowing e#cient solutions. The framework is implemented in the tool by a well-defined abstract functional interface. Models and solution techniques interact with one another through the use of the standard interface, allowing them to interact with Mobius framework components, not formalism components. This permits novel combinations of modeling techniques, and will be a catalyst for new research in modeling techniques. This paper describes our approach, focusing on the "atomic model." We describe the formal description of the Mobius components as well as their implementations in our software tool.

The development of techniques for quantitative, model-based evaluation of computer system dependability has a long and rich history. A wide array of model-based evaluation techniques are now available, ranging from combinatorial methods, which are useful for quick, rough-cut analyses, to state-based methods, such as Markov reward models, and detailed, discreteevent simulation. The use of quantitative techniques for security evaluation is much less common, and has typically taken the form of formal analysis of small parts of an overall design, or experimental red team-based approaches. Alone, neither of these approaches is fully satisfactory, and we argue that there is much to be gained through the development of a sound model-based methodology for quantifying the security one can expect from a particular design. In this work, we survey existing model-based techniques for evaluating system dependability, and summarize how they are now being extended to evaluate system security. We find that many techniques from dependability evaluation can be applied in the security domain, but that significant challenges remain, largely due to fundamental differences between the accidental nature of the faults commonly assumed in dependability evaluation, and the intentional, human nature of cyber attacks.

The PEPA language [1] is supported by a suite of modelling tools which assist in the solution and analysis of PEPA models. The design and development of these tools have been influenced by a variety of factors, including the wishes of other users of the tools to use the language for purposes which were not anticipated by the tool designers. In consequence, the suite of PEPA tools has adapted to attempt to serve the needs of these users while continuing to support the language designers themselves. In this paper we report on our use of the PEPA tools and give some advice gained from our experiences. 1 Introduction PEPA (Performance Evaluation Process Algebra) extends classical process algebra with the capacity to assign rates to the activities which are described in an abstract model of a system. Taken together, the information about the rates of performance of activities and the definition of the outcome of performing an activity specify a stochastic process and thus PEPA is said to b...

In this paper we describe a formalism which uses the stochastic process algebra PEPA as the inscription language for labelled stochastic Petri nets. Viewed in another way, the net is used to provide a structure for linking related PEPA systems. The combined modelling language naturally represents such applications as mobile code systems where the PEPA terms are used to model the program code which moves between network hosts (the places in the net). We describe the implementation of a tool to support this modelling formalism and apply this to model a hierarchical cellular network. 1

Many formalisms and solution methods exist for performance and dependability modeling. However, different formalisms have different advantages and strengths, and no one formalism is universally used. The Möbius

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. Stochastic process algebras such as PEPA provide ample support for the component-based construction of models. Tools compute the numerical solution of these models; however, the stochastic process algebra methodology has lacked support for the specification and calculation of complex performance measures. In this paper we present a stochastic modal logic which can aid the construction of a reward structure over the model. We discuss its relationship to the underlying theory of PEPA. We also present a performance specification language which supports high level reasoning about PEPA models, and allows queries about their equilibrium behaviour. The meaning of the specification language has its foundations in the stochastic modal logic. We describe the implementation of the logic within the PEPA Workbench and a case study is presented to illustrate the approach. 1 Introduction It has long been recognised that whilst Markovian models of simple computer systems can be constru...

A key advantage of the Möbius modeling environment is the ease with which one can incorporate new modeling formalisms, model composition and connection methods, and model solution methods. We present a new state-level abstract functional interface (AFI) for Möbius that allows numerical solution methods to communicate with Möbius state-level models via the abstraction of a labeled transition system. This abstraction and its corresponding implementation yield an important separation of concerns: It is possible to treat separately the problem of representing large labeled transition systems, like generator matrices of continuous-time Markov chains, and the problem of analyzing these systems. For example, any numerical solver (e.g., Jacobi, SOR, or uniformization) that accesses a model through the Möbius state-level AFI can operate on a variety of state-space representations, including "on-the-fly," disk-based, sparse-matrix, Kronecker, and matrix-diagram representations, without requiring that the solver implementation be changed to match the statespace representation. This abstraction thus avoids redundant implementations of solvers and state-generation techniques, eases research cooperation, and simplifies comparisons of approaches as well as benchmarking. In addition to providing a formal definition of the Möbius state-level AFI, we illustrate its use on two state-space representations (a sparse matrix and a Kronecker representation) and on several numerical s...

Stochastic Process Algebras have been introduced to enable compositional performance analysis of parallel and distributed systems. As with other high level modelling formalisms, state space explosion is a frequently observed problem, especially if the system consists of many cooperating components. However, if the components are identical replicas of each others, the state space can be reduced by means of equivalence preserving aggregation. This paper introduces symmetric parallel composition, an operator to specify sets of identical replicas cooperating in parallel. Its operational semantics is consistent with usual parallel composition whereas the state space explosion problem is drastically reduced. We illustrate this beneficial effect, and provide an interpretation of symmetric parallel composition in terms of Petri Nets. INTRODUCTION Continuous time Markov chains (CTMC), which are widely used as performance models in many diverse areas, are usually generated from high level descr...

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.