Abstract-In the past, logics of several kinds have been proposed for reasoning about discrete-time or continuous-time Markov chains. Most of these logics rely on either state labels (atomic propositions) or on transition labels (actions). However, in several applications it is useful to reason about both state properties and action sequences. For this purpose, we introduce the logic asCSL which provides a powerful means to characterize execution paths of Markov chains with actions and state labels. asCSL can be regarded as an extension of the purely state-based logic CSL (continuous stochastic logic). In asCSL, path properties are characterized by regular expressions over actions and state formulas. Thus, the truth value of path formulas depends not only on the available actions in a given time interval, but also on the validity of certain state formulas in intermediate states. We compare the expressive power of CSL and asCSL and show that even the state-based fragment of asCSL is strictly more expressive than CSL if time intervals starting at zero are employed. Using an automaton-based technique, an asCSL formula and a Markov chain with actions and state labels are combined into a product Markov chain. For time intervals starting at zero, we establish a reduction of the model checking problem for asCSL to CSL model checking on this product Markov chain. The usefulness of our approach is illustrated with an elaborate model of a scalable cellular communication system, for which several properties are formalized by means of asCSL formulas and checked using the new procedure.

Model checking has been introduced as an automated tech-

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Abstract. In this paper algorithms for model checking CSL (continuous stochastic logic) against infinite-state continuous-time Markov chains of so-called quasi birth-death type are developed. In doing so we extend the applicability of CSL model checking beyond the recently proposed case for finite-state continuous-time Markov chains, to an important class of infinite-state Markov chains. We present syntax and semantics for CSL and develop efficient model checking algorithms for the steady-state operator and the time-bounded next and until operator. For the former, we rely on the so-called matrix-geometric solution of the steady-state probabilities of the infinite-state Markov chain. For the time-bounded until operator we develop a new algorithm for the transient analysis of infinite-state Markov chains, thereby exploiting the quasi birth-death structure. A case study shows the feasibility of our approach.

The usage of mobile devices like cell phones, navigation systems, or laptop computers, is limited by the lifetime of the included batteries. This lifetime depends naturally on the rate at which energy is consumed, however, it also depends on the usage pattern of the battery. Continuous drawing of a high current results in an excessive drop of residual capacity. However, during intervals with no or very small currents, batteries do recover to a certain extend. We model this complex behaviour with an inhomogeneous Markov reward model, following the approach of the so-called Kinetic battery Model (KiBaM). The state-dependent reward rates thereby correspond to the power consumption of the attached device and to the available charge, respectively. We develop a tailored numerical algorithm for the computation of the distribution of the consumed energy and show how different workload patterns influence the overall lifetime of a battery.

Abstract-Gossip protocols are designed to operate in very large, decentralised networks. A node in such a network bases its decision to interact (gossip) with another node on its partial view of the global system. Because of the size of these networks, analysis of gossip protocols is mostly done using simulations, that tend to be expensive in computation time and memory consumption.We employ mean-field approximation for an analytical evaluation of gossip protocols. Nodes in the network are represented by small identical stochastic models. Joining all nodes would result in an enormous stochastic process. If the number of nodes goes to infinity, however, mean-field analysis allows us to replace this intractably large stochastic process by a small deterministic process. This process approximates the behaviour of very large gossip networks, and can be evaluated using simple matrix-vector multiplications.

Abstract-Business and social life have become increasingly dependent on large-scale communication and information systems. A partial or complete breakdown as a consequence of natural disasters or purposeful attacks might have severe impacts. Survivability refers to the ability of a system to recover from such disaster circumstances. Evaluating survivability should therefore be an important part of communication system design. In this paper we take a model checking approach toward assessing survivability. We use the logic CSL to phrase survivability in a precise manner. The system operation is modelled through a labelled CTMC. Model checking algorithms can then decide automatically whether the system is survivable. We illustrate our method by evaluating the survivability of the Google file system using stochastic Petri nets.

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