## Section: New Results

### Analytic models

Participants : Bruno Sericola, Gerardo Rubino, Raymond Marie, Laura Aspirot.

Fluid models are powerful tools for evaluating the performance of packet telecommunication networks. By masking the complexity of discrete packet based systems, fluid models are in general easier to analyze and yield simple dimensioning formulas. Among fluid queuing systems, those with arrival rates modulated by Markov chains are very efficient to capture the burst structure of packet arrivals, notably in the Internet because of bulk data transfers. By exploiting the Markov property, very efficient numerical algorithms can be designed to estimate performance metrics such the overflow probability, the delay of a fluid particle or the duration of a busy period. In [76] , we analyze the transient behaviour of a fluid queue driven by a general ergodic birth and death process using spectral theory in the Laplace transform domain. These results are applied to the stationary regime and to the busy period analysis of that fluid queue.

In [36] , another type of fluid model is considered. We present preliminary results on the analysis of a Machine Repairman Model when the number of machines goes to infinity. The analysis is based on identifying appropriate fluid limits of the associated stochastic processes. We are currently working on the analysis of the speed of the convergence of these stochastic processes towards their fluid limits.

In [19] , we present an approximate method for the transient analysis of stiff ctmc . The origin of the method is due to S. M. Ross who proposed to approximate the transient probability at a deterministic time $t$ by the value of the transient probability at a random time $X$ where $X$ is an Erlang random variable having expectation $t$. The major contributions of the paper are the use of new numerical techniques to solve the basic equations of the original method and the extension of the method to reward measures. We also conduct an experimental evaluation of the resulting errors using non-trivial examples.

In [86] , we presented an extension of robdd s that is able to accommodate certain dependencies among their (Boolean) variables. In particular, this extension shows evidence of being applicable to evaluating the dependability (reliability, availability) of systems whose structures are representable by a Boolean function. This extension consists of three main parts. The first part is the notion of a phratry with its associated new definitions and constraints. The second part consists of the adaptation and complementation of the original rules used in the construction of robdd s. The final part concerns additional custom-made steps needed to determine the functional valuations that are specific to solving measure in question.

The survivability of a system being its ability to function during and after a failure, we developed in [63] a model to study the power distribution in smart grids during the (transient) period that starts after a failure till the system fully recovers. The proposed model bridges power flow modeling of reactive power compensation with performability/survivability modeling of automation distribution networks. We use a Markov chain to characterize the phased recovery of the system after a failure. Then, we associate with each state of the Markov chain a set of corresponding rewards to characterize the active and reactive power supplied and demanded in that state. We connect the survivability model with an availability model, to produce a generalization of the System Average Interruption Duration Index (SAIDI) and the Customer Average Interruption Duration Index (CAIDI), which are two of the most important power grid reliability metrics. The survivability model allows us to obtain closed form expressions for the SAIDI and related metrics.

In [62] , we consider the case of important systems located on operational sites far away from logistic support forces, either because the operational site is in an inhospitality place, or because it is not profitable to maintain a dedicated team on the operational site. Due to the importance of the systems, some service level agreement has been signed, including conditional financial clauses. To take into account such a situation, a preventive maintenance is realized according to projected calendars. The paper shows that, given that the life-times of equipments are supposed to be Erlang-$k$ distributed, it is optimal to realize a preventive maintenance, as long as the ratio of the two intervention costs ${C}_{p}/{C}_{c}$ is lower than the ratio $(k-1)/k$, ${C}_{p}$ being the cost of a preventive maintenance intervention and ${C}_{c}$ being the cost of a curative maintenance intervention (because of excessive delay, there is a significant penalty associated with each curative maintenance intervention). The methodology to compute the optimal value of the period ${T}^{*}$ and the corresponding optimal cost per time unit are presented, for a given value of the ratio ${C}_{p}/{C}_{c}$. An extended version of this work has been accepted for publication in a journal ([26] ).

The study [60] focuses on the determination of the probability distributions of two random variables, the asymptotic “up-time” and “down-time” of a system for the sake of potential “Service Level Agreement”. In these new generation agreements, penalizations can be enforced for a too long “down-time” or for a too short “up-time”. First, we determine the probability distributions of the two random variables “up-time” and “down-time”, for a system with a general structural function. Second, we point out the importance of rare events such as the backorders in the contribution of a large tail distribution of the down-time. Respectively, we exhibit the importance of redundant structures and also of sub-system hyper-exponential lifetimes in the existence of short up-times, with respect to the mean up-time value of the system.

The study [61] deals with the determination of spares of systems of systems of the same type (such as fleet of aircraft, fleet of ship). For a multi-site workshop and multi-level of repair organization, we present an optimization algorithm using the criteria of expected number of backorders as local objective. With respect to a previous algorithm based only on the criteria of the global availability of the system, the new algorithm is, for large maintenance systems, very efficient, in terms of execution time and in of data manipulation.

The study [41] concerns the performance evaluation of crisis management systems with respect to the dimensioning of the system. By definition, a crisis has no steady state and the study must be done on the transient behavior. A faithful model was built (in altarica ) and solved thanks to simulation. Our own participation was mainly to determine the number of objects to create such that the simulation ends successfully with a high probability, before running out of available objects.

Last, in [54] we continue the exploration of the concept of duality proposed by Anderson, applied to the analysis of the transient behavior of queueing systems. This work analyzes the transient distribution of the number of customers in a Restart Markovian queue, where together with “typical” customers other signals arrive to the queue having as a consequence the removal of all the customers present in the system.