Implicit ODE solvers with good local error control for the transient analysis of Markov models

Obtaining the transient probability distribution vector of a continuous-time Markov chain (CTMC) using an implicit ordinary differential equation (ODE) solver tends to be advantageous in terms of run-time computational cost when the product of the maximum output rate of the CTMC and the largest time of interest is large. In this paper, we show that when applied to the transient analysis of CTMCs, many implicit ODE solvers are such that the linear systems involved in their steps can be solved by using iterative methods with strict control of the 1-norm of the error. This allows the development of implementations of those ODE solvers for the transient analysis of CTMCs that can be more efficient and more accurate than more standard implementations.Peer ReviewedPostprint (published version

A comparison of numerical splitting-based methods for Markovian dependability and performability models

Iterative numerical methods are an important ingredient for the solution of continuous time Markov dependability models of fault-tolerant systems. In this paper we make a numerical comparison of several splitting-based iterative methods. We consider the computation of steady-state reward rate on rewarded models. This measure requires the solution of a singular linear system. We consider two classes of models. The first class includes failure/repair models. The second class is more general and includes the modeling of periodic preventive test of spare components to reduce the probability of latent failures in inactive components. The periodic preventive test is approximated by an Erlang distribution with enough number of stages. We show that for each class of model there is a splitting-based method which is significantly more efficient than the other methods.Postprint (published version

On the computation of poisson probabilities

The Poisson distribution is a distribution commonly used in statistics. It also plays a central role in the analysis of the transient behaviour of continuous-time Markov chains. Several methods have been devised for evaluating using floating-point arithmetic the probability mass function (PMF) of the Poisson distribution. Restricting our attention to published methods intended for the computation of a single probability or a few of them, we show that neither of them is completely satisfactory in terms of accuracy. With that motivation, we develop a new method for the evaluation of the PDF of the Poisson distribution. The method is intended for the computation of a single probability or a few of them. Numerical experimentation illustrates that the method can be more accurate and slightly faster than the previous methods. Besides, the method comes with guaranteed approximation relative error.Postprint (author's final draft

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

A method for the computation of reliability bounds for non-repairable fault-tolerant systems

A realistic modeling of fault-tolerant systems requires to take into account phenomena such as the dependence of component failure rates and coverage parameters on the operational configuration of the system, which cannot be properly captured using combinatorial techniques. Such dependencies can be modeled with detail using continuous-time Markov chains (CTMC’s). However, the use of CTMC models is limited by the well-known state space explosion problem. In this paper we develop a method for the computation of bounds for the reliability of non-repairable fault-tolerant systems which requires the generation of only a subset of states. The tightness of the bounds increases as more detailed states are generated. The method uses the failure distance concept and is illustrated using an example of a quite complex fault-tolerant system whose failure behavior has the above mentioned types of dependencies.Postprint (published version

METFAC-2.1 User's Guide

Technical ReportPostprint (author’s final draft

Failure distance based bounds for steady-state availability without the kwnowledge of minimal cuts

We propose an algorithm to compute bounds for the steady-state unavailability using continuous-time Markov chains, which is based on the failure distance concept. The algorithm generates incrementally a subset of the state space until the tightness of the bounds is the specified one. In contrast with a previous algorithm also based on the failure distance concept, the proposed algorithm uses lower bounds for failure distances which are computed on the fault tree of the system, and does not require the knowledge of the minimal cuts. This is advantageous when the number of minimal cuts is large or their computation is time-consuming.Postprint (published version

A failure-distance based method to bound the reliability of non-repairable Fault-tolerant systems without the knowledge of minimal cuts

CTMC (continuous-time Markov chains) are a commonly used formalism for modeling fault-tolerant systems. One of the major drawbacks of CTMC is the well-known state-space explosion problem. This work develops and analyzes a method (SC-BM) to compute bounds for the reliability of non-repairable fault-tolerant systems in which only a portion of the state space of the CTMC is generated. SC-BM uses the failure distance concept as the method described in [1] but, unlike that method, which is based on the computation of exact failure distances, SC-BM uses lower bounds for failure distances, which are computed on the system fault tree, avoiding the computation and holding of all minimal cuts as required in [1]. This is important since computation of all minimal cuts is NP-hard and the number of minimal cuts can be very large. In some cases SCBM gives exactly the same bounds as the method described in [1]; in other cases it gives less tighter bounds. SC-BM computes tight bounds for the reliability of quite complex systems with an affordable number of generated states for short to quite large mission times. The analysis of several examples seems to show that the bounds obtained by SC-BM appreciably outperform those obtained by simpler methods, eg [2], and, when they are not equal, are only slightly worse than the bounds obtained by the method in [1]. In addition, the overhead in CPU time due to computing lower bounds for failure distances seems to be reasonable.Preprin

Combinatorial methods for the evaluation of yield and operational reliability of fault-tolerant systems-on-chip

In this paper we develop combinatorial methods for the evaluation of yield and operational reliability of fault-tolerant systems-on-chip. The method for yield computation assumes that defects are produced according to a model in which defects are lethal and affect given components of the system following a distribution common to all defects; the method for the computation of operational reliability also assumes that the fault-tree function of the system is increasing. The distribution of the number of defects is arbitrary. The methods are based on the formulation of, respectively, the yield and the operational reliability as the probability that a given boolean function with multiple-valued variables has value 1. That probability is computed by analyzing a ROMDD (reduced ordered multiple-value decision diagram) representation of the function. For efficiency reasons, a coded ROBDD (reduced ordered binary decision diagram) representation of the function is built first and, then, that coded ROBDD is transformed into the ROMDD required by the methods. We present numerical experiments showing that the methods are able to cope with quite large systems in moderate CPU times.Postprint (published version

La responsabilidad en la participación: un valor cooperativo en la educación primaria

El artículo pretende destacar los principios del movimiento cooperativo, sus valores formativos y su necesaria vigencia en la actualidad. Se apuesta por el fomento de la responsabilidad en la participación en los diferentes cursos de educación primaria a partir del despliegue de diversos ejes de trabajo. Se ofrece una propuesta teoricopráctica de intervención tomando la escuela como principal ámbito de actuación concretada en diversas actividades a desarrollar en cada uno de los ciclos de una escuela de Tarragona. Los resultados de la investigación nos han puesto de manifiesto el interés, la reflexión y la sensibilidad del alumnado en temas relacionados con la responsabilidad en la participación, lo que nos insta a proseguir con optimismo para afrontar propuestas futuras