Section: New Results

Analytic models

Participants : Bruno Sericola, Gerardo Rubino, Raymond Marie.

New book about Dependability Theory. Dependability metrics are omnipresent in every engineering field, from simple ones through to more complex measures combining performance and dependability aspects of systems. The new book [69] written in the team, entitled “Markov Chains and Dependability Theory” and published in 2014 by Cambridge University Press (see also http://www.amazon.fr/Markov-Chains-Dependability-Theory-Gerardo/dp/1107007577/ ), presents the mathematical basis of the analysis of these metrics. The modelling context corresponds to the most used framework, Markov models. The book describes both basic results and specialised techniques. The authors first present discrete and continuous time Markov chains before focusing on dependability measures, which necessitate the study of Markov chains on a subset of states representing different user satisfaction levels for the modelled system. Topics covered include Markovian state lumping, analysis of sojourns on subset of states of Markov chains, analysis of most dependability metrics, fundamentals of performability analysis, and bounding and simulation techniques designed to evaluate dependability measures. As stated in its abstract, the book is of interest to graduate students and researchers in all areas of engineering where the concepts of lifetime, repair duration, availability, reliability and risk are important.

Fluid models. In [77] we study congestion periods in a finite fluid buffer when the input rate depends upon a recurrent Markov process; congestion occurs when the buffer content is equal to the buffer capacity. We consider the duration of congestion periods as well as the associated volume of lost information. We derive their distributions in a typical stationary busy period of the buffer. Our goal is to compute the exact expression of the loss probability in the system, which is usually approximated by the probability that the occupancy of the infinite buffer is greater than the buffer capacity under consideration. Moreover, by using general results of the theory of Markovian arrival processes, we show that the duration of congestion and the volume of lost information have phase-type distributions.

Industrial Logistic Aspects. Motivated by the consideration of clauses of penalty, we worked again on the determination of the probability distributions of the delays of unavailability of systems on the operational sites. By considering in particular a given type of spare, we show the important role played by the possible waiting time of the change during the occurrence of a breakdown. In particular we verify that the cumulative probability distribution of the delay of unavailability possesses a relatively low tail diminution as well as a high square of cœfficient of variation. Upper and lower bounds are highlighted in the simplest case. These results allow to calculate the risk inferred by the use of clauses of penalty; for example, by proposing an expression of the expectation of the cost of penalty imposed by unit of time if any unavailability exceeding a certain threshold is penalized [62] . If the possible waiting time of the change is the obsession of the specialists of the maintenance, the consideration of stock shortages in supply chains is often underestimated when these events are rare events. A related work consisted in showing that a low probability of break can be associated with a high coefficient of variation can have a very significant consequence [54] .

We also studied the extension of our analytical method of calculation of the operational availability of a fleet of consequent systems deployed on a site and maintained by exchanges on the site of subsets (the lru for line repaired unit) in the specific case where a policy of cannibalization is implemented. We propose an approximated method which is particularly adapted to the case of systems with strong operational availability because in this case the error inferred by the approximation remains low. The developed method consists in determining the expectation of the number of blocked systems due to the lack of change, in the presence of a policy of cannibalization. This expectation is directly associated with a loss of operational availability. At present, in the presence of a policy of cannibalization, the proposed solution concerns only the systems constituted by a series of lru but the policy of cannibalization can be applied to all or part of the types of lru  [63] .