Bounding steady-state availability models with group repair and phase type repair distributions

We propose an algorithm to obtain bounds for the steady-state availability using Markov models in which only a small portion of the state space is generated. The algorithm is applicable to models with group repair and phase type repair distributions and involves the solution of only four linear systems of the size of the generated state space, independently on the number of “return” states. Numerical examples are presented to illustrate the algorithm and compare it with a previous bounding algorithm.Postprint (published version

Tight steady-state availability bounds using the failure distance concept

Continuous-time Markov chains are commonly used for dependability modeling of repairable fault-tolerant computer systems. Realistic models of non-trivial fault-tolerant systems often have very large state spaces. An attractive approach for dealing with the largeness problem is the use of pruningmethods with error bounds. Several such methods for computing steady-state availability bounds have been proposed recently. This paper presents a new method which exploits the failure distance concept to bound more efficiently the behavior in the non-generated state space. It is proved that the bounding method gives tighter bounds than previous methods. Numerical analysis shows that the new bounds can be significantly tighter.Postprint (published version

Comparing Markov Chains: Aggregation and Precedence Relations Applied to Sets of States, with Applications to Assemble-to-Order Systems

International audienceSolving Markov chains is, in general, difficult if the state space of the chain is very large (or infinite) and lacking a simple repeating structure. One alternative to solving such chains is to construct models that are simple to analyze and provide bounds for a reward function of interest. We present a new bounding method for Markov chains inspired by Markov reward theory: Our method constructs bounds by redirecting selected sets of transitions, facilitating an intuitive interpretation of the modifications of the original system. We show that our method is compatible with strong aggregation of Markov chains; thus we can obtain bounds for an initial chain by analyzing a much smaller chain. We illustrate our method by using it to prove monotonicity results and bounds for assemble-to-order systems

Computing bounds on steady state availability of repairable computer systems

Abstract. One of the most Important performance measures for computer system designers is system avadabdlty. Most often, Markov models are used m representing systems for dependability/availability analysls. Due to complex interactions between components and complex repair policles, the Markov model often has an Irregular structure, and closed-form solutions are extremely difficulty to obtain. Also, a realistlc system model often has an unmanageably large state space and lt qulcfdy becomes impractical to even generate the entire transition rate matrix. In this paper, we present a methodology that can (i) bound the system steady state awdilabiIity and at the same time, (u) drastically reduce the state space of the model that must be solved. The bounding algorlthm is lteratwe and generates a part of the transition matrix at each step. At each step, tighter bounds on system availability are obtained. The algorithm also allows the size of the submodel, to be solved at each step, to be chosen so as to accommodate memory limitations. This general bounding methodology prowdes an efficient way to evaluate dependabdlty models with very large state spaces without ever generating the entu-e transition rate matrw

Failure distance based bounds of dependability measures

El tema d'aquesta tesi és el desenvolupament de mètodes de fitació per a una classe de models de confiabilitat basats en cadenes de Markov de temps continu (CMTC) de sistemes tolerants a fallades.Els sistemes considerats a la tesi es conceptualitzen com formats per components (hardware o software) que fallen i, en el cas de sistemes reparables, són reparats. Els components s'agrupen en classes de forma que els components d'una mateixa classe són indistingibles. Per tant, un component és considerat com a una instància d'una classe de components i el sistema inclou un bag de classes de components definit sobre un cert domini. L'estat no fallada/fallada del sistema es determina a partir de l'estat no fallada/fallada dels components mitjançant una funció d'estructura coherent que s'especifica amb un arbre de fallades amb classes d'esdeveniments bàsics. (Una classe d'esdeveniment bàsic és la fallada d'un component d'una classe de components.)La classe de models basats en CMTC considerada a la tesi és força àmplia i permet, per exemple, de modelar el fet que un component pot tenir diversos modes de fallada. També permet de modelar fallades de cobertura mitjançant la introducció de components ficticis que no fallen per ells mateixos i als quals es propaguen les fallades d'altres components. En el cas de sistemes reparables, la classe de models considerada admet polítiques de reparació complexes (per exemple, nombre limitat de reparadors, prioritats, inhibició de reparació) així com reparació en grup (reparació simultània de diversos components). Tanmateix, no és possible de modelar la reparació diferida (és a dir, el fet de diferir la reparació d'un component fins que una certa condició es compleixi).A la tesi es consideren dues mesures de confiabilitat: la no fiabilitat en un instant de temps donat en el cas de sistemes no reparables i la no disponibilitat en règim estacionari en el cas sistemes reparables.Els mètodes de fitació desenvolupats a la tesi es basen en el concepte de "distància a la fallada", que es defineix com el nombre mínim de components que han de fallar a més dels que ja han fallat per fer que el sistema falli.A la tesi es desenvolupen quatre mètodes de fitació. El primer mètode dóna fites per a la no fiabilitat de sistemes no reparables emprant distàncies a la fallada exactes. Aquestes distàncies es calculen usant el conjunt de talls mínims de la funció d'estructura del sistema. El conjunt de talls mínims s'obté amb un algorisme desenvolupat a la tesi que obté els talls mínims per a arbres de fallades amb classes d'esdeveniments bàsics. El segon mètode dóna fites per a la no fiabilitat usant fites inferiors per a les distàncies a la fallada. Aquestes fites inferiors s'obtenen analitzant l'arbre de fallades del sistema, no requereixen de conèixer el conjunt de talls mínims i el seu càlcul és poc costós. El tercer mètode dóna fites per a la no disponibilitat en règim estacionari de sistemes reparables emprant distàncies a la fallada exactes. El quart mètode dóna fites per a la no disponibilitat en règim estacionari emprant les fites inferiors per a les distàncies a la fallada.Finalment, s'il·lustren les prestacions de cada mètode usant diversos exemples. La conclusió és que cada un dels mètodes pot funcionar molt millor que altres mètodes prèviament existents i estendre de forma significativa la complexitat de sistemes tolerants a fallades per als quals és possible de calcular fites ajustades per a la no fiabilitat o la no disponibilitat en règim estacionari.The subject of this dissertation is the development of bounding methods for a class of continuous-time Markov chain (CTMC) dependability models of fault-tolerant systems.The systems considered in the dissertation are conceptualized as made up of components (hardware or software) that fail and, for repairable systems, are repaired. Components are grouped into classes, the components of the same class being indistinguishable. Thus, a component is regarded as an instance of some component class and the system includes a bag of component classes defined over a certain domain. The up/down state of the system is determined from the unfailed/failed state of the components through a coherent structure function specified by a fault tree with basic event classes. (A basic event class is the failure of a component of a component class.)The class of CTMC models considered in the dissertation is quite wide and allows, for instance, to model the fact that a component may have different failure modes. It also allows to model coverage failures by means of introducing fictitious components that do not fail by themselves and to which uncovered failures of other components are propagated. In the case of repairable systems, the considered class of models supports very complex repair policies (e.g., limited repairpersons, priorities, repair preemption) as well as group repair (i.e., simultaneous repair of several components). However, deferred repair (i.e., the deferring of repair until some condition is met) is not allowed.Two dependability measures are considered in the dissertation: the unreliability at a given time epoch for non-repairable systems and the steady-state unavailability for repairable systems.The bounding methods developed in the dissertation are based on the concept of "failure distance from a state," which is defined as the minimum number of components that have to fail in addition to those already failed to take the system down.We develop four bounding methods. The first method gives bounds for the unreliability of non-repairable fault-tolerant systems using (exact) failure distances. Those distances are computed using the set of minimal cuts of the structure function of the system. The set of minimal cuts is obtained using an algorithm developed in the dissertation that obtains the minimal cuts for fault trees with basic event classes. The second method gives bounds for the unreliability using easily computable lower bounds for failure distances. Those lower bounds are obtained analyzing the fault tree of the system and do not require the knowledge of the set of minimal cuts. The third method gives bounds for the steady-state unavailability using (exact) failure distances. The fourth method gives bounds for the steady-state unavailability using the lower bounds for failure distances.Finally, the performance of each method is illustrated by means of several large examples. We conclude that the methods can outperform significantly previously existing methods and extend significantly the complexity of the fault-tolerant systems for which tight bounds for the unreliability or steady-state unavailability can be computed