Jérôme Morio

On the Sustainability of the Extreme Value Theory for WCET Estimation.

Measurement-based approaches with extreme value worst-case estimations are beginning to be proficiently considered for timing analyses. In this paper, we intend to make more formal extreme value theory applicability to safe worst-case execution time estimations. We outline complexities and challenges behind extreme value theory assumptions and parameter tuning. Including the knowledge requirements, we are able to conclude about safety of the probabilistic worst-case execution estimations from the extreme value theory, and execution time measurements.

A survey of rare event simulation methods for static input–output models

Abstract Crude Monte-Carlo or quasi Monte-Carlo methods are well suited to characterize events of which associated probabilities are not too low with respect to the simulation budget. For very seldom observed events, such as the collision probability between two aircraft in airspace, these approaches do not lead to accurate results. Indeed, the number of available samples is often insufficient to estimate such low probabilities (at least 10 6 samples are needed to estimate a probability of order 10 - 4 with 10% relative error with Monte-Carlo simulations). In this article, one reviewed different appropriate techniques to estimate rare event probabilities that require a fewer number of samples. These methods can be divided into four main categories: parameterization techniques of probability density function tails, simulation techniques such as importance sampling or importance splitting, geometric methods to approximate input failure space and finally, surrogate modeling. Each technique is detailed, its advantages and drawbacks are described and a synthesis that aims at giving some clues to the following question is given: “which technique to use for which problem?”.

Revising Measurement-Based Probabilistic Timing Analysis

The Measurement-Based Probabilistic Timing Analysis (MBPTA) computes probabilistic Worst-Case Execution Time (pWCET) estimates with the Extreme Value Theory (EVT) from measurements of tasks execution time. In this work we approach the MBPTA open problems by proposing guidelines for a formal and correct EVT application. Statistical analyses of measurement of execution times, the EVT parameters selection, the pWCET reliability and the worst-case guarantees are investigated in order to provide the best possible MBPTA. The MBPTA enhancements proposed are validated with test cases from both artificially time randomized real-time systems and non-time randomized real-time systems.

An island particle algorithm for rare event analysis

Estimating rare event probability with accuracy is of great interest for safety and reliability applications. In this paper, we focus on rare events which can be modeled by a threshold exceedance of a deterministic input–output function with random inputs. Some parameters of this function or density parameters of input random variables may be fixed by an experimenter for simplicity reasons. From a risk analysis point of view, it is not only interesting to evaluate the probability of a critical event but it is also important to determine the impact of such tuning of parameters on the realization of a critical event, because a bad estimation of these parameters can strongly modify rare event probability estimations. In the present paper, we present an example of island particle algorithm referred to as sequential Monte Carlo square (SMC2). This algorithm gives an estimate of the law of random phenomena that leads to critical events. The principles of this statistical technique are described throughout this article and its results are analysed on different realistic aerospace test cases.

Optimisation of interacting particle systems for rare event estimation

a b s t r a c t The interacting particle system (IPS) is a recent probabilistic model proposed to estimate rare event probabilities for Markov chains. The principle of IPS is to apply alternatively selection and mutation stages to a set of initial particles in order to estimate probabilities or quantiles more accurately than with usual estimation techniques. The practical issue of IPS is the tuning of a parameter in the selection stage. Kriging-based optimisation strategy with a low simulation cost is thus proposed in order to minimise the probability estimate relative error. The efficiency of the proposed strategy is demonstrated on different test cases.

Reliability-based sensitivity estimators of rare event probability in the presence of distribution parameter uncertainty

Abstract This paper aims at presenting sensitivity estimators of a rare event probability in the context of uncertain distribution parameters (which are often not known precisely or poorly estimated due to limited data). Since the distribution parameters are also affected by uncertainties, a possible solution consists in considering a second probabilistic uncertainty level. Then, by propagating this bi-level uncertainty, the failure probability becomes a random variable and one can use the mean estimator of the distribution of the failure probabilities (i.e. the “predictive failure probability”, PFP) as a new measure of safety. In this paper, the use of an augmented framework (composed of both basic variables and their probability distribution parameters) coupled with an Adaptive Importance Sampling strategy is proposed to get an efficient estimation strategy of the PFP. Consequently, double-loop procedure is avoided and the computational cost is decreased. Thus, sensitivity estimators of the PFP are derived with respect to some deterministic hyper-parameters parametrizing a priori modeling choice. Two cases are treated: either the uncertain distribution parameters follow an unbounded probability law, or a bounded one. The method efficiency is assessed on two different academic test-cases and a real space system computer code (launch vehicle stage fallback zone estimation).

Probabilistic analysis of cache memories and cache memories impacts on multi-core embedded systems

Task execution is heavily affected by the different elements composing real-time systems. Modeling and analyzing such effects would allow reducing the pessimism lying behind the worst-cases. A measurement-based probabilistic approach is developed in order to characterize cache behavior with probabilistic average and worst-case profiles. The approach applies also statistics for studying the impact that different system configurations have on the profiles as well as for evaluating the impact of caches on task execution times. The quality of the probabilistic models is verified through test cases with benchmark tasks running on non time-randomized multi-core real-time systems.

Probabilistic Safety Analysis of the Collision Between a Space Debris and a Satellite with an Island Particle Algorithm

Collision between satellites and space debris seldom happens, but the loss of a satellite by collision may have catastrophic consequences both for the satellite mission and for the space environment. To support the decision to trigger off a collision avoidance manoeuver, an adapted tool is the determination of the collision probability between debris and satellite. This probability estimation can be performed with rare event simulation techniques when Monte Carlo techniques are not enough accurate. In this chapter, we focus on analyzing the influence of different simulation parameters (such as the drag coefficient) that are set for to simplify the simulation, on the collision probability estimation. A bad estimation of these simulation parameters can strongly modify rare event probability estimations. We design here a new island particle Markov chain Monte Carlo algorithm to determine the parameters that, in case of bad estimation, tend to increase the collision probability value. This algorithm also gives an estimate of the collision probability maximum taking into account the likelihood of the parameters. The principles of this statistical technique are described throughout this chapter.

Tuning of adaptive interacting particle system for rare event probability estimation

Abstract The interacting particle system (IPS) for rare event algorithm has been well mathematically formulated, with a wide variety of results on the estimation accuracy of the probability of rare event. Despite this theoretical point of view, the practical side of this algorithm has not been handled completely. Indeed, a tuning parameter has a significant influence on the effective algorithm performance. Moreover, the choice of a good parameter value often proves to be fastidious and may decrease the usefulness of the IPS algorithm. Therefore, we propose a statistical technique in order to make the IPS algorithm fully adaptive. We derive this strategy for threshold exceedance probability estimation and for the estimation of probability density function tail. The performances of the proposed method have been studied on a toy case and on two more complex estimation problems in optical fiber and financial engineering.

Analysis of extreme aircraft wake vortex circulations

Because of its take-off, an aircraft generates a wake essentially made of a pair of counter-rotating vortices notably characterized by their total circulation. In this test case, we analyze the evolution of wake vortex total circulation with the extreme value theory. It consists in estimating the probability that a wake vortex circulation exceeds a given threshold for a fixed set of CMC samples