Abstract. We consider the timed automata model of [3], which allows the analysis of real-time systems expressed in terms of quantitative timing constraints. Traditional approaches to real-time system description express the model purely in terms of nondeterminism; however, we may wish to express the likelihood of the system making certain transitions. In this paper, we present a model for real-time systems augmented with discrete probability distributions. Furthermore, using the algorithm of [5] with fairness, we develop a model checking method for such models against temporal logic properties which can refer both to timing properties and probabilities, such as, "with probability 0.6 or greater, the clock x remains below 5 until clock y exceeds 2".

We consider the timed automata model of [3], which allows the analysis of real-time systems expressed in terms of quantitative timing constraints. Traditional approaches to real-time system description express the model purely in terms of nondeterminism; however, we may wish to express the likelihood of the system making certain transitions. In this paper, we present a model for real-time systems augmented with discrete probability distributions. Furthermore, using the algorithm of [5] with fairness, we develop a model checking method for such models against temporal logic properties which can refer both to timing properties and probabilities, such as, "with probability 0.6 or greater, the clock x remains below 5 until clock y exceeds 2".

Probabilistic timed automata, a variant of timed automata extended with discrete probability distributions, is a modelling formalism suitable for describing formally both nondeterministic and probabilistic aspects of real-time systems, and is amenable to model checking against probabilistic timed temporal logic properties. However, the previously developed verification algorithms either suffer from high complexity, give only approximate results, or are restricted to a limited class of properties. In the case of classical (non-probabilistic) timed automata it has been shown that for a large class of real-time verification problems correctness can be established using an integral model of time (digital clocks) as opposed to a dense model of time. Based on these results we address the question of under what conditions digital clocks are sufficient for the performance analysis of probabilistic timed automata and show that this reduction is possible for an important class of systems and properties including probabilistic reachability and expected reachability. We demonstrate the utility of this approach by applying the method to the performance analysis of three probabilistic real-time protocols: the dynamic configuration protocol for IPv4 link-local addresses, the IEEE 802.11 wireless local area network protocol and the IEEE 1394 FireWire root contention protocol. concerns the ZeroConf dynamic configuration protocol for IPv4 link-local addresses [18] (preliminary results concerning this case study can be found in [37]). The second extends the results of [40], and investigates the performance of the contention resolution protocol of the IEEE 802.11 wireless local area network standard [33]. In the third case study we consider the root contention protocol of the IEEE 1394 FireWire standard [32], previously studied using probabilistic timed automata in [41,20]. In the latter two publications, the analysis was with respect to probabilistic reachability properties, whereas in this paper we study expected reachability properties.

Markov chains are a well-known stochastic process that provide a balance between being able to adequately model the system '

Probabilistic timed automata are timed automata extended with discrete probability distributions, and can be used to model timed randomised protocols or faulttolerant systems. We present symbolic model-checking algorithms for probabilistic timed automata to verify both qualitative temporal logic properties, corresponding to satisfaction with probability 0 or 1, and quantitative properties, corresponding to satisfaction with arbitrary probability. The algorithms operate on zones, which represent sets of valuations of the probabilistic timed automaton's clocks. Our method considers only those system behaviours which guarantee the divergence of time with probability 1. The paper presents a symbolic framework for the verification of probabilistic timed automata against the probabilistic, timed temporal logic PTCTL. We also report on a prototype implementation of the algorithms using Difference Bound Matrices, and present the results of its application to the CSMA/CD and FireWire root contention protocol case studies.

Abstract. We consider parity games on infinite graphs where configurations are represented by control-states and integer vectors. This framework subsumes two classic game problems: parity games on vector addition systems with states (VASS) and multidimensional energy parity games. We show that the multidimensional energy parity game problem is inter-reducible with a subclass of singlesided parity games on VASS where just one player can modify the integer counters and the opponent can only change control-states. Our main result is that the minimal elements of the upward-closed winning set of these single-sided parity games on VASS are computable. This implies that the Pareto frontier of the minimal initial credit needed to win multidimensional energy parity games is also computable, solving an open question from the literature. Moreover, our main result implies the decidability of weak simulation preorder/equivalence between finite-state systems and VASS, and the decidability of model checking VASS with a large fragment of the modal µ-calculus.

Probabilistic timed automata (PTAs) are a formalism for modelling systems whose behaviour incorporates both probabilistic and real-time characteristics. Applications include wireless communication protocols, automotive network protocols and randomised security protocols. This paper gives an introduction to PTAs and describes techniques for analysing a wide range of quantitative properties, such as "the maximum probability of the airbag failing to deploy within 0.02 seconds", "the maximum expected time for the protocol to terminate" or "the minimum expected energy consumption required to complete all tasks". We present a temporal logic for specifying such properties and then give a survey of available modelchecking techniques for formulae specified in this logic. We then describe two case studies in which PTAs are used for modelling and analysis: a probabilistic non-repudiation protocol and a task-graph scheduling problem. IntroductionAutomated verification techniques, such as model checking, provide powerful methods for rigorously analysing the correctness of systems. Increasingly, this analysis must also take into account quantitative aspects of the systems being verified, including both real-time characteristics and probabilistic behaviour. Embedded systems, for example, whether in communication and multimedia devices or in automotive and avionic control systems, often operate under timing constraints. The need for automated formal verification techniques in this domain is clear, as evidenced by the take-up of timed automata verification tools such as UPPAAL [13]. On the other hand, many real-life systems also exhibit stochastic behaviour,

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