In this paper we propose an improved BDD approach to the network reliability analysis, that allows the user to compute an exact solution or an approximation based on reliability bounds when network complexity makes the former solution practically impossible. To this purpose, a new algorithm for encoding the connectivity graph on a Binary Decision Diagram (BDD) has been developed; it reduces the computation memory peak with respect to previous approaches based on the same type of data structure without increasing the execution time, and allows us also to derive from a subset of the minpaths/mincuts a lower/upper bound of the network reliability, so that the quality of the obtained approximation can be estimated. Finally, a fair and detailed comparison between our approach and another state of the art approach presented in the literature is documented through a set of benchmarks.

This paper focuses on a sub-class of Dynamic Fault Trees (DFTs), called Priority Dynamic Fault Trees (PDFTs), containing only static gates, and Priority Dynamic Gates (Priority-AND, and Functional Dependency) for which a priority relation among the input nodes completely determines the output behavior. We define events as temporal variables, and we show that, by adding to the usual Boolean operators new temporal operators denoted BEFORE and SIMULTANEOUS, it is possible to derive the structure function of the Top Event with any cascade of Priority Dynamic Gates, and repetition of basic events. A set of theorems are provided to express the structure function in a sum-of-product canonical form, where each product represents a set of cut sequences for the system. We finally show through some examples that the canonical form can be exploited to determine directly and algebraically the failure probability of the Top Event of the PDFT without resorting to the corresponding Markov model. The advantage of the approach is that it provides a complete qualitative description of the system, and that any failure distribution can be accommodated.

The automation of power grids by means of supervisory control and data acquisition (SCADA) systems has led to an improvement of power grid operations and functionalities but also to pervasive cyber interdependencies between power grids and telecommunication networks. Many power grid services are increasingly depending upon the adequate functionality of SCADA system which in turn strictly depends on the adequate functionality of its communication infrastructure. We propose to tackle the SCADA risk analysis by means of different and heterogeneous modeling techniques and software tools. We demonstrate the applicability of our approach through a case study on an actual SCADA system for an electrical power distribution grid. The modeling techniques we discuss aim at providing a probabilistic dependability analysis, followed by a worst case analysis in presence of malicious attacks and a real-time performance evaluation.

The paper discusses a Dynamic Markovian Agent Model obtained by adding mobility to a recently introduced new formalism suitable for the analysis of large scale systems, composed by a population of interacting entities, called Markovian Agents (MA). The differential equations describing the evolution of the MA density in time and space are derived, and their numerical solution is briefly sketched. An application to the analysis of the flow of vehicles in a road tunnel is discussed, together with the evaluation of the probability of collision against a fixed obstacle.

Power management in battery operated sensor networks, is a hot topic addressed by many ongoing researches. One of the most commonly employed technique consists in turning on and of the power of the radio unit to reduce the power consumption. In this paper we exploit the modelling power of Interacting Markovian agent to evaluate the performance of four different on-off strategies. A Markovian agent (MA) is an entity whose behaviour is described by a continuous time Markov chain (CTMC) and that is able to interact with other MA's by sending and receiving messages. A perceived message may induce a state transition in an MA according to a perception function than can depend on the geographical location of the MA's, on the message routing strategy and on the transmission property of the medium. We represent each sensor by an MA and we show how MA's can interact to produce the collective behaviour of the sensor network.

This paper presents a software tool allowing the automatic analysis of a dynamic fault tree (DFT) exploiting its conversion to a dynamic Bayesian network (DBN). First, the architecture of the tool is described, together with the rules implemented in the tool, to convert dynamic gates in DBNs. Then, the tool is tested on a case of system: its DFT model and the corresponding DBN are provided and analyzed by means of the tool. The obtained unreliability results are compared with those returned by other tools, in order to verify their correctness.

In order to cope efficiently with the dependability analysis of redundant systems with replicated units, a new, more compact fault-tree formalism, called Parametric Fault Tree (PFT), is defined. In a PFT formalism, replicated units are folded and indexed so that only one representative of the similar replicas is included in the model. From the PFT, a list of parametric cut sets can be derived, where only the relevant patterns leading to the system failure are evidenced regardless of the actual identity of the component in the cut set. The paper provides an algorithm to convert a PFT into a class of High-Level Petri Nets, called SWN. The purpose of this conversion is twofold: to exploit the modeling power and flexibility of the SWN formalism, allowing the analyst to include statistical dependencies that could not have been accommodated into the corresponding PFT and to exploit the capability of the SWN formalism to generate a lumped Markov chain, thus alleviating the state explosion problem. The search for the minimal cut sets (qualitative analysis) can be often performed by a structural T-invariant analysis on the generated SWN. The advantages that can be obtained from the translation of a PFT into a SWN are investigated considering a fault-tolerant multiprocessor system example.

PhFit, a new phase-type fitting tool is presented in this paper. PhFit allows for approximating distributions or set of samples not only by continuous but by discrete phase-type distributions as well. The implemented algorithms separate the fitting of the body and the tail part of the distribution which results in satisfactory fitting also for heavy-tail distributions. Moreover, PhFit allows the user to choose the distance measure according to which the fitting is performed. The tool is equipped with a graphical interface that visualizes the goodness of the provided approximation from several aspects.

This paper introduces a unified approach to phase-type approximation in which the discrete and continuous phase-type models form a common model set. The models of this common set are assigned with a non-negative real parameter, the scale factor. The case when the scale factor is strictly positive results in discrete phase-type distributions and the scale factor represents the time elapsed in one step. If the scale factor is 0, the resulting class is the class of continuous phase-type distributions. Applying the above view, it is shown that there is no qualitative difference between the discrete and the continuous phase-type models. Based on this unified view of phase-type models one can choose the best phase-type approximation of a stochastic model by optimizing the scale factor.

Petri nets represent a useful tool for performance, dependability, and performability analysis of complex systems. Their modeling power can be increased even more if nonexponentially distributed events are considered. However, the inclusion of nonexponential distributions destroys the memoryless property and requires to specify how the marking process is conditioned upon its past history. We consider, in particular, the class of stochastic Petri nets whose marking process can be mapped into a Markov regenerative process. An adequate mathematical framework is developed to deal with the considered class of Markov Regenerative Stochastic Petri Nets (MRSPN). A unified approach for the solution of MRSPNs where different preemption policies can be defined in the same model is presented. The solution is provided both in steady-state and in transient condition. An example concludes the paper.

Non-Markovian models allow us to capture a very wide range of circumstances in which it is necessary to model phenomena whose times to occurrence is not exponentially distributed. Events such as timeouts in a protocol, service times at a machine performing the same task on each part, and memory access or instruction execution in a low-level h/w or s/w model, have durations which are constant or with a very low variance. Phase-type distributions can be used to approximate a non-exponential, but they increase the size of the state space.The analysis of stochastic systems with non-exponential timing is of increasing interest in the literature and requires the development of suitable modeling tools. Recently, some effort has been devoted to generalize the concept of Stochastic Petri Nets (SPN), by allowing the firing times to be generally distributed.A particular case of non-Markovian SPN, is the class of Deterministic and SPN (DSPN) [1]. A DSPN is a non-Markovian SPN where, in each marking, at most one transition is allowed to have a deterministic firing time with enabling memory policy.A new class of stochastic Petri nets has recently been defined [2, 3] by generalizing the deterministic firing times of the DSPN to generally distributed firing times. The underlying stochastic process for these classes of Petri nets is a Markov Regenerative Process (MRGP). This observation has opened a very fertile line of research aimed at the definition of solvable classes of models whose underlying marking process is an MRGP, and therefore referred to as Markov Regenerative Stochastic Petri Nets (MRSPN).Some of the results in this filed will be described in the session. In particular, Ciardo investigates stochastic confusion by defining the selection probability for transitions attempting to fire at the same time. German introduces the "method of supplementary variables" for the derivation of state equations describing the transient behavior of the marking process. Puliafito describes how, under some constraints, concurrent enabling of several generally distributed timed transitions is allowed. Bobbio and Telek discuss how age memory policy can be included to capture preemptive mechanisms of the resume (prs) type.

An aggregation method for computing transient cumulative measures of large, stiff Markov models is presented. The method is based on classifying the states of the original problem into slow, fast-transient, and fast-current states. The authors aggregate fast-transient states and fast-recurrent states so that an approximate value to the desired cumulative measure can be obtained by solving a nonstiff set of linear differential equations defined over a reduced subset of slow states only. Several examples are included to illustrate how stiffness arises naturally in actual queuing and reliability models, and to show that cumulative measures provide a better characterization of the time-dependent system behavior.>

Petri nets in which random delays are associated with atomic transitions are defined in a comprehensive framework that contains most of the models already proposed in the literature. To include generally distributed firing times into the model one must specify the way in which the next transition to fire is chosen, and how the model keeps track of its past history; this set of specifications is called an execution policy. A discussion is presented of the impact that different execution policies have on semantics of the mode, as well as the characteristics of the stochastic process associated with each of these policies. When the execution policy is completely specified by the transition with the minimum delay (race policy) and the firing distributions are of the phase type, an algorithm is provided that automatically converts the stochastic process into a continuous time homogeneous Markov chain. An execution policy based on the choice of the next transition to fire independently of the associated delay (preselection policy) is introduced, and its semantics is discussed together with possible implementation strategies.>

An approximation algorithm for systematically converting a stiff Markov chain into a nonstiff chain with a smaller state space is discussed in this paper. After classifying the set of all states into fast and slow states, the algorithm proceeds by further classifying fast states into fast recurrent subsets and a fast transient subset. A separate analysis of each of these fast subsets is done and each fast recurrent subset is replaced by a single slow state while the fast transient subset is replaced by a probabilistic switch. After this reduction, the remaining small and nonstiff Markov chain is analyzed by a conventional technique.