Abstract — We develop a continuous-time Markov chain model of a dependability system operating in a randomly changing environment and subject to probabilistic cascading failures. A cascading failure can be thought of as a rooted tree. The root is the component whose failure triggers the cascade, its children are those components that the root’s failure immediately caused, the next generation are those components whose failures were immediately caused by the failures of the root’s children, and so on. The amount of cascading is unlimited. We consider probabilistic cascading, in the sense that the failure of a component of type i causes a component of type j to fail simultaneously with a given probability, with all failures in a cascade being mutually independent. Computing the infinitesimal generator matrix of the Markov chain poses significant challenges because of the exponential growth in the number of trees one needs to consider as the number of components failing in the cascade increases. We provide a recursive algorithm generating all possible trees corresponding to a given transition, along with an experimental study of an implementation of the algorithm on two examples. The numerical results highlight the effects of cascading on the dependability of the models. Index Terms — availability, reliability modeling, Markov processes, trees, cascading failures.

Abstract — We develop a continuous-time Markov chain model of a dependability system operating in a randomly changing environment and subject to probabilistic cascading failures. A cascading failure can be thought of as a rooted tree. The root is the component whose failure triggers the cascade, its children are those components that the root’s failure immediately caused, the next generation are those components whose failures were immediately caused by the failures of the root’s children, and so on. The amount of cascading is unlimited. We consider probabilistic cascading, in the sense that the failure of a component of type i causes a component of type j to fail simultaneously with a given probability, with all failures in a cascade being mutually independent. Computing the infinitesimal generator matrix of the Markov chain poses significant challenges because of the exponential growth in the number of trees one needs to consider as the number of components failing in the cascade increases. We provide a recursive algorithm generating all possible trees corresponding to a given transition, along with an experimental study of an implementation of the algorithm on two examples. The numerical results highlight the effects of cascading on the dependability of the models. Index Terms — availability, reliability modeling, Markov processes, trees, cascading failures.

Abstract—The increasing reliance on computer technology nowadays has resulted in a rapidly growing need to build reliable and fault resistant computer-based systems. Computer system reliabilities are conventionally modeled and analyzed using techniques such as fault tree analysis (FTA) and reliability block diagrams (RBD), which provide static representations of system reliabilities. A recent extension to RBD, called dynamic reliability block diagrams (DRBD), provides a framework for modeling dynamic reliability behaviors of computer-based systems. However, analyzing a DRBD model in order to locate and identify design errors, such as a deadlock error or a faulty state, is not trivial when done manually. A feasible approach to verifying a DRBD model is to develop a formal model of the DRBD, and analyze it using programmatic methods. In this paper, we first define a reliability markup language (RML) that can be used to formally describe DRBD models. Then we present an algorithm that automatically converts a DRBD model into a colored Petri net (CPN). We use a case study to illustrate the effectiveness of our approach and demonstrate how system properties of a DRBD model can be verified using an existing Petri net tool. Our approach is compositional and provides a potential solution to automated verification of DRBD models. Index Terms—System reliability, reliability block diagram (RBD), dynamic RBD (DRBD), extensible markup language (XML), colored Petri nets (CPN), formal modeling and analysis, automated verification, deadlock detection. 1

Abstract—Computer system reliability is conventionally modeled and analyzed using techniques such as fault tree analysis (FTA) and reliability block diagrams (RBD), which provide static representations of system reliability properties. A recent extension to RBD, called dynamic reliability block diagrams (DRBD), defines a framework for modeling dynamic reliability behavior of computer-based systems. However, analyzing a DRBD model in order to locate and identify design errors, such as a deadlock error or faulty state, is not trivial when done manually. A feasible approach to verifying it is to develop its formal model, and then analyze it using programmatic methods. In this paper, we first define a reliability markup language (RML) that can be used to formally describe DRBD models. Then we present an algorithm that automatically converts a DRBD model into a colored Petri net (CPN). We use a case study to illustrate the effectiveness of our approach and demonstrate how system properties of a DRBD model can be verified using an existing Petri net tool. Our formal modeling approach is compositional, thus it provides a potential solution to automated verification of DRBD models. Index Terms—System reliability, reliability block diagram (RBD), extensible markup language (XML), colored Petri net (CPN), time Petri net, formal modeling and analysis, automated verification, deadlock detection. API BNF