Abstract. In this paper we describe PRISM, a tool being developed at the University of Birmingham for the analysis of probabilistic systems. PRISM supports two probabilistic models: continuous-time Markov chains and Markov decision processes. Analysis is performed through model checking such systems against specifications written in the probabilistic temporal logics PCTL and CSL. The tool features three model checking engines: one symbolic, using BDDs (binary decision diagrams) and MTBDDs (multi-terminal BDDs); one based on sparse matrices; and one which combines both symbolic and sparse matrix methods. PRISM has been successfully used to analyse probabilistic termination, performance, dependability and quality of service properties for a range of systems, including randomized distributed algorithms, polling systems, workstation cluster and wireless cell communication. 1

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.

Continuous-time Markov chains (CTMCs) have been widely used to determine system performance and dependability characteristics. Their analysis most often concerns the computation of steady-state and transient-state probabilities. This paper introduces a branching temporal logic for expressing real-time probabilistic properties on CTMCs and presents approximate model checking algorithms for this logic. The logic, an extension of the continuous stochastic logic CSL of Aziz et al., contains a time-bounded until operator to express probabilistic timing properties over paths as well as an operator to express steady-state probabilities. We show that the model checking problem for this logic reduces to a system of linear equations (for unbounded until and the steady-state operator) and a Volterra integral equation system (for time-bounded until). We then show that the problem of model-checking timebounded until properties can be reduced to the problem of computing transient state probabilities for CTMCs. This allows the verification of probabilistic timing properties by efficient techniques for transient analysis for CTMCs such as uniformization. Finally, we show that a variant of lumping equivalence (bisimulation), a well-known notion for aggregating CTMCs, preserves the validity of all formulas in the logic.

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.

. Markov chains are widely used in the context of performance and reliability evaluation of systems of various nature. Model checking of such chains with respect to a given (branching) temporal logic formula has been proposed for both the discrete [17, 6] and the continuous time setting [4, 8]. In this paper, we describe a prototype model checker for discrete and continuous-time Markov chains, the Erlangen--Twente Markov Chain Checker (E MC 2 ), where properties are expressed in appropriate extensions of CTL. We illustrate the general benefits of this approach and discuss the structure of the tool. Furthermore we report on first successful applications of the tool to non-trivial examples, highlighting lessons learned during development and application of E T MC 2 . 1 Introduction Markov chains are widely used as simple yet adequate models in diverse areas, ranging from mathematics and computer science to other disciplines such as operations research, industrial engine...

Obtaining performance models, like Markov chains and queueing networks, for systems of significant complexity and magnitude is a di#cult task that is usually tackled using human intelligence and experience. This holds in particular for performance models of a highly irregular nature. In this paper we argue by means of a non-trivial example --- a plain-old telephone system (POTS) --- that a stochastic extension of process algebra can diminish these problems by permitting an automatic generation of Markov chains. We introduce a stochastic process algebra that separates the advance of time and action occurrences. For the sake of specification convenience we incorporate an elapse operator that allows the modular description of time constraints where delays are described by continuous phase-type distributions. Using this language we provide a formal specification of the POTS and show how a stochastic process of more than 10 7 states is automatically obtained from this system description. ...

Abstract. Considering functional correctness and performance evaluation in a common framework is desirable, both for scientific and economic reasons. In this paper, we describe how the Cadp toolbox, originally designed for verifying the functional correctness of Lotos specifications, can also be used for performance evaluation. We illustrate the proposed approach by the performance study of the Scsi-2 bus arbitration protocol. 1

. Stochastic process algebra have been proven useful because they allow behaviour-oriented performance and reliability modelling. As opposed to traditional performance modelling techniques, the behaviouroriented style supports composition and abstraction in a natural way. However, analysis of stochastic process algebra models is state-oriented, because standard numerical analysis is typically based on the calculation of (transient and steady) state probabilities. This shift of paradigms hampers the acceptance of the process algebraic approach by performance modellers. In this paper, we develop an entirely behaviour-oriented analysis technique for stochastic process algebra. The key contribution is an action-based temporal logic to describe behaviours-of-interest, together with a model checking algorithm to derive the probability with which a stochastic process algebra model exhibits a given behaviour-of-interest. 1 Introduction The analysis of systems with respect to their performance...

Markov chains (and their extensions with rewards) have been widely used to determine performance, dependability and performability characteristics of computer communication systems, such as throughput, delay, mean time to failure, or the probability to accumulate at least a certain amount of reward in a given time.

Dynamic Fault Trees (DFT) extend standard fault trees by allowing the modeling of complex system components’ behaviors and interactions. Being a high level model and easy to use, DFT are experiencing a growing success among reliability engineers. Unfortunately, a number of issues still remains when using DFT. Briefly, these issues are (1) a lack of formality (syntax and semantics), (2) limitations in modular analysis and thus vulnerability to the state-space explosion problem, and (3) lack in modular model-building. We use the input/output interactive Markov chain (I/O-IMC) formalism to analyse DFT. I/O-IMC have a precise semantics and are an extension of continuous-time Markov chains with input and output actions. In this paper, using the I/O-IMC framework, we address and resolve issues (2) and (3) mentioned above. We also show, through some examples, how one can readily extend the DFT modeling capabilities using the I/O-IMC framework.