In this paper we introduce PRISM, a probabilistic model checker, and describe the ecient symbolic techniques we have developed during its implementation. PRISM is a tool for analysing probabilistic systems. It supports three models: discrete-time Markov chains, continuous-time Markov chains and Markov decision processes. Analysis is performed through model checking speci cations in the probabilistic temporal logics PCTL and CSL. Motivated by the success of model checkers such as SMV, which use BDDs (binary decision diagrams), we have developed an implementation of PCTL and CSL model checking based on MTBDDs (multi-terminal BDDs) and BDDs. Existing work in this direction has been hindered by the generally poor performance of MTBDD-based numerical computation, which is often substantially slower than explicit methods using sparse matrices. We present a novel hybrid technique which combines aspects of symbolic and explicit approaches to overcome these performance problems. For typical examples, we achieve orders of magnitude speed-up compared to MTBDDs and are able to almost match the speed of sparse matrices whilst maintaining considerable space savings.

Abstract. This paper examines the influence of the Raf Kinase Inhibitor Protein (RKIP) on the Extracellular signal Regulated Kinase (ERK) signalling pathway [5] through modelling in a Markovian process algebra, PEPA [11]. Two models of the system are presented, a reagentcentric view and a pathway-centric view. The models capture functionality at the level of subpathway, rather than at a molecular level. Each model affords a different perspective of the pathway and analysis. We demonstrate the two models to be formally equivalent using the timingaware bisimulation defined over PEPA models and discuss the biological significance. 1

Abstract. This paper presents an abstract machine for a variant of the stochastic pi-calculus, in order to correctly model the stochastic simulation of biological processes. The abstract machine is proved sound and complete with respect to the calculus, and then used as the basis for implementing a stochastic simulator. The correctness of the machine helps ensure that the simulator is correctly implemented, giving greater confidence in the simulation results. A graphical representation for the pi-calculus is also presented, as a potential front-end to the simulator. 1

We introduce a stochastic process algebra, PEPA, as a high-level modelling paradigm for con-tinuous time Markov chains (CTMC). Process algebras are mathematical theories which model concurrent systems by their algebra and provide apparatus for reasoning about the structure and behaviour of the model. Recent extensions of these algebras, associating random variables with actions, make the models also amenable to Markovian analysis. A compositional structure is inherent in the PEPA language. As well as the clear advantages that this offers for model construction, we demonstrate how this compositionality may be exploited to reduce the state space of the CTMC. This leads to an exact aggregation based on lumpability. Moreover this technique, taking advantage of symmetries within the system, may be formally defined in terms of the PEPA description of the model. An equivalence relation, strong equivalence, developed as a process algebra bisimulation relation, is used to partition the derivation graph. 1

Stochastic Process Algebras have been proposed as compositional specification formalisms for performance models. In this paper, we describe a tool which aims at realising all beneficial aspects of compositional performance modelling, the TIPPtool. It incorporates methods for compositional specification as well as solution, based on state-of-the-art-techniques, and wrapped in a user-friendly graphical front end. Apart from highlighting the general benefits of the tool, we also discuss some lessons learned during development and application of the TIPPtool. A non-trivial model of a real life communication system serves as a case study to illustrate benefits and limitations.

This paper surveys the theoretical developments in the field of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems --- like large-scale computers, client-server architectures, networks --- can accurately be described using such stochastic specification formalisms.

Performance Evaluation Process Algebra (PEPA) is a formal language for performance modelling based on process algebra. It has previously been shown that using the process algebra apparatus compact performance models can be derived which retain the essential behavioural characteristics of the modelled system. However no efficient algorithm for this derivation was given. In this paper we present an efficient algorithm which recognises and takes advantage of symmetries within the model and avoids unnecessary computation. The algorithm is illustrated by a multiprocessor example. Keywords: Performance modelling, model aggregation, performance evaluation tools, stochastic process algebras. 1 Introduction In recent years several Markovian process algebras (MPAs) have been presented in the literature. These include PEPA [1], MTIPP [2], and EMPA [3]. As with classical process algebras, these formalisms allow models of systems to be constructed which are amenable to functional or behavioural an...

In each of the current stochastic process algebras all non-competitive interactions between components or agents are modelled using a single combinator, variously called the parallel, synchronisation or cooperation operator. This paper aims to compare the definitions of this combinator which have been used; in particular, looking at the different ways in which rates are associated with the actions which result from such interactions. The implications of the chosen definitions, from a modelling point of view, will be described. When we consider concrete systems rather than abstract representations many different types of interactions between systems are exhibited. Some of these possible interactions are presented in the latter half of the paper and we analyse the extent to which these can be captured using the combinators available in the SPA languages. To conclude some observations about current modelling practice are made together with suggestions of potential extensions to the set of combinators. 1

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