Sharp Bounds for the Reliability of a k-out-of-n Sys- tem Under Dependent Failures Using Cutoff Phenomenon Techniques
In this work we consider the reliability of a network where link failures are correlated. We define the reliability as the probability of the network to be working at a given time instant. Our main contribution is a collection of results giving a detailed analysis of a non-trivial scaling regime for the probability of the network being working at a certain time, as the time and size of network scales. Here we consider that the network fails when there are no links working, or more generally when less than k-out-of-n edges are working (with k close to n) like in [2] and [5]. Our results allow to study the common-cause failure models describe in [3] on networks in a realistic, relevant, yet practical, fashion: it allows to capture correlated components in the network; it allows to estimate and give error bounds for the failure probabilities of the system; and at same time only needs to specify a reduced family of parameters. Moreover, our results for the k-out-of-n failure model allow to give new scaling regimes for the probabilistic behavior of the last-ordinals in the theory of extreme values for dependent tuples. The techniques are similar to those used to estimate the asymptotic convergence profile for ergodic Markov chains [1] or [4].
[1] J. Barrera and B. Ycart Bounds for Left and Right Window Cutoffs. ALEA, Lat. Am. J. Probab. Math. Stat., 11 (2): 445–458, 2014.
[2] I. Bayramoglu and M. Ozkut. The Reliability of Coherent Systems Subjected to Marshall?Olkin Type Shocks. EEE Transactions on Reliability,, 64 (1): 435–443, 2015.
[3] U. Cherubini, F. Durante, and S. Mulinacci. Marshall – Olkin Distributions-Advances in Theory and Applications: Bologna, Italy, October 2013, volume 141. Springer, 2015.
[4] B. Lachaud and B. Ycart Convergence Times for Parallel Markov Chains. Positive systems, 169–176, 2006.
[5] T. Yuge, M. Maruyama, and S. Yanagi Reliability of a k-out-of-n Systemwith Common-Cause Failures Using Multivariate Exponential Distribution. Procedia Computer Sci- ence, 96: 968?976, 2016.
Noise sensitivity for Voronoi Percolation
Harnack inequality for degenerate balanced random walks.
We consider an i.i.d. balanced environment $\omega(x,e)=\omega(x,-e)$,genuinely d dimensional on the lattice and show that there exist a
positive constant $C$ and a random radius $R(\omega)$ with streched
exponential tail such that every non negative
$\omega$ harmonic function $u$ on the ball $B_{2r}$ of radius
$2r>R(\omega)$,
we have $\max_{B_r} u <= C \min_{B_r} u$. Our proof relies on a quantitative quenched invariance principle for the corresponding random walk in balanced random environment and a careful analysis of the directed percolation cluster. This result extends Martins Barlow’s Harnack’s inequality for i.i.d. bond percolation to the directed case.
This is joint work with N.Berger M. Cohen and X. Guo.
Sala 3, Facultad de Matemáticas, PUC
“Invariant measures of discrete interacting particle systems: Algebraic aspects”
We consider a continuous time particle system on a graph L being either Z, Z_n, a segment {1,…, n}, or Z^d, with state space Ek={0,…,k-1} for some k belonging to {infinity, 2, 3, …}. We also assume that the Markovian evolution is driven by some translation invariant local dynamics with bounded width dependence, encoded by a rate matrix T. These are standard settings, satisfied by many studied particle systems. We provide some sufficient and/or necessary conditions on the matrix T, so that this Markov process admits some simple invariant distribution, as a product measure, as the distribution of a Markov process indexed by Z or {1,…, n} (if L=Z or {1,…,n}), or as a Gibbs measure (if L=Z_n). These results are mainly obtained with some manipulations of finite words, with alphabet Ek, representing subconfigurations of the systems. For the case L=Z, we give a procedure to find the set of invariant i.i.d and Markov measures.
Construction of geometric rough paths
This talk is based on a joint work in progress with L. Zambotti (UPMC). First, I will give a brief introduction to the theory of rough paths focusing on the case of Hölder regularity between 1/3 and 1/2. After this, I will address the basic problem of construction of a geometric rough path over a given ɑ-Hölder path in a finite-dimensional vector space. Although this problem was already solved by Lyons and Victoir in 2007, their method relies on the axiom of choice and thus is not explicit; in exchange the proof is simpler. In an upcoming paper, we provide an explicit construction clarifying the connection between rough paths theory and free (nilpotent) Lie algebras. In particular, we use an explicit form of the Baker–Campbell–Hausdorff formula due to Loday in order to provide explicit expressions and bounds to achieve such a construction.
Sala de seminarios del 5to piso, CMM, 16:00 hrs.
Pontificia Universidad Católica de Chile (PUC-Chile)
Av. Vicuña Mackenna 4860, Macul,
Santiago – Chile
(+56 2) 2354 5779
Facultad de Ciencias Físicas y Matemáticas (FCFM)
Universidad de Chile
Beauchef 851, Edificio Norte, Piso 7,
Santiago – Chile