M-timed Petri nets and Markov chains in modelling of computer systems

It is shown that the behavior of enhanced free-choice bounded M-timed Petri nets, i.e., Petri nets with two classes of transitions, immediate and timed ones, and with exponentially distributed firing times of timed transitions, can be represented by state-transition graphs which are finite continuous-time homogeneous Markov chains. Moreover, for each finite continuous-time homogeneous Markov chain there exists an enhanced free-choice bounded M-timed Petri net with the state-transition graph isomorphic to this chain. These two classes of models are thus equivalent. A straight-forward application of (enhanced) M-timed Petri nets is modelling and performance evaluation of Markovian queueing systems, and in particular closed-network models of computer systems. Simple models of interactive systems are used as an illustration of modelling. as an illustration of modelling

Quantitative evaluation of Pandora Temporal Fault Trees via Petri Nets

© 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Using classical combinatorial fault trees, analysts are able to assess the effects of combinations of failures on system behaviour but are unable to capture sequence dependent dynamic behaviour. Pandora introduces temporal gates and temporal laws to fault trees to allow sequence-dependent dynamic analysis of events. Pandora can be easily integrated in model-based design and analysis techniques; however, the combinatorial quantification techniques used to solve classical fault trees cannot be applied to temporal fault trees. Temporal fault trees capture state and therefore require a state space solution for quantification of probability. In this paper, we identify Petri Nets as a possible framework for quantifying temporal trees. We describe how Pandora fault trees can be mapped to Petri Nets for dynamic dependability analysis and demonstrate the process on a fault tolerant fuel distribution system model

Process algebra for performance evaluation

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. The main emphasis of this paper is the treatment of operational semantics, notions of equivalence, and (sound and complete) axiomatisations of these equivalences for different types of Markovian process algebras, where delays are governed by exponential distributions. Starting from a simple actionless algebra for describing time-homogeneous continuous-time Markov chains, we consider the integration of actions and random delays both as a single entity (like in known Markovian process algebras like TIPP, PEPA and EMPA) and as separate entities (like in the timed process algebras timed CSP and TCCS). In total we consider four related calculi and investigate their relationship to existing Markovian process algebras. We also briefly indicate how one can profit from the separation of time and actions when incorporating more general, non-Markovian distributions

CSL model checking of Deterministic and Stochastic Petri Nets

Deterministic and Stochastic Petri Nets (DSPNs) are a widely used high-level formalism for modeling discrete-event systems where events may occur either without consuming time, after a deterministic time, or after an exponentially distributed time. The underlying process dened by DSPNs, under certain restrictions, corresponds to a class of Markov Regenerative Stochastic Processes (MRGP). In this paper, we investigate the use of CSL (Continuous Stochastic Logic) to express probabilistic properties, such a time-bounded until and time-bounded next, at the DSPN level. The verication of such properties requires the solution of the steady-state and transient probabilities of the underlying MRGP. We also address a number of semantic issues regarding the application of CSL on MRGP and provide numerical model checking algorithms for this logic. A prototype model checker, based on SPNica, is also described

Analysis of Timed and Long-Run Objectives for Markov Automata

Markov automata (MAs) extend labelled transition systems with random delays and probabilistic branching. Action-labelled transitions are instantaneous and yield a distribution over states, whereas timed transitions impose a random delay governed by an exponential distribution. MAs are thus a nondeterministic variation of continuous-time Markov chains. MAs are compositional and are used to provide a semantics for engineering frameworks such as (dynamic) fault trees, (generalised) stochastic Petri nets, and the Architecture Analysis & Design Language (AADL). This paper considers the quantitative analysis of MAs. We consider three objectives: expected time, long-run average, and timed (interval) reachability. Expected time objectives focus on determining the minimal (or maximal) expected time to reach a set of states. Long-run objectives determine the fraction of time to be in a set of states when considering an infinite time horizon. Timed reachability objectives are about computing the probability to reach a set of states within a given time interval. This paper presents the foundations and details of the algorithms and their correctness proofs. We report on several case studies conducted using a prototypical tool implementation of the algorithms, driven by the MAPA modelling language for efficiently generating MAs.Comment: arXiv admin note: substantial text overlap with arXiv:1305.705

Construction and Verification of Performance and Reliability Models

Over the last two decades formal methods have been extended towards performance and reliability evaluation. This paper tries to provide a rather intuitive explanation of the basic concepts and features in this area. Instead of striving for mathematical rigour, the intention is to give an illustrative introduction to the basics of stochastic models, to stochastic modelling using process algebra, and to model checking as a technique to analyse stochastic models

A model checker for performance and dependability properties

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 [8] and the continuous time setting [1], [3]. In this short paper, we describe the prototype model checker $E \vdash M C^2$ for discrete and continuous-time Markov chains, where properties are expressed in appropriate extensions of CTL.We illustrate the general benefits of this approach and discuss the structure of the tool