New results on a generalized coupon collector problem using Markov chains

We study in this paper a generalized coupon collector problem, which consists in determining the distribution and the moments of the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we obtain expressions of the distribution and the moments of this time. We also prove that the almost-uniform distribution, for which all the non-null coupons have the same drawing probability, is the distribution which minimizes the expected time to get a fixed subset of distinct coupons. This optimization result is extended to the complementary distribution of that time when the full collection is considered, proving by the way this well-known conjecture. Finally, we propose a new conjecture which expresses the fact that the almost-uniform distribution should minimize the complementary distribution of the time needed to get any fixed number of distinct coupons.Comment: 14 page

Occupation Times in Markov Processes

We consider, in a homogeneous Markov process with finite state space, the occupation times that is, the times spent by the process in given subsets of the state space during a finite interval of time. We first derive the distribution of the occupation time of one subset and then we generalize this result to the joint distribution of occupation times of different subsets of the state space by the use of order statistics from the uniform distribution. Next, we consider the distribution of weighted sums of occupation times. We obtain the forward and backward equations describing the behavior of these weighted sums and we show how these equations lead to simple expressions of this distribution

Optimization results for a generalized coupon collector problem

We study in this paper a generalized coupon collector problem, which consists in analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the non-null coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one, which stochastically maximizes the time needed to collect a fixed number of distinct coupons. An computer science application shows the utility of these results.Comment: arXiv admin note: text overlap with arXiv:1402.524

Uniform Node Sampling Service Robust against Collusions of Malicious Nodes

International audienceWe consider the problem of achieving uniform node sampling in large scale systems in presence of a strong adversary. We first propose an omniscient strategy that processes on the fly an unbounded and arbitrarily biased input stream made of node identifiers exchanged within the system, and outputs a stream that preserves Uniformity and Freshness properties. We show through Markov chains analysis that both properties hold despite any arbitrary bias introduced by the adversary. We then propose a knowledge-free strategy and show through extensive simulations that this strategy accurately approximates the omniscient one. We also evaluate its resilience against a strong adversary by studying two representative attacks (flooding and targeted attacks). We quantify the minimum number of identifiers that the adversary must insert in the input stream to prevent uniformity. To our knowledge, such an analysis has never been proposed before

A Finite Buffer Fluid Queue Driven by a Markovian Queue

We consider a finite buffer fluid queue receiving its input from the output of a Markovian queue with finite or infinite waiting room. The input flow into the fluid queue is thus characterized by a Markov modulated input rate process and we derive, for a wide class of such input processes, an approach for the computation of the stationary buffer content of the fluid queue and so for the computation of the stationary overflow probability. This approach leads to a numerically stable algorithm for which the precision of the result can be specified in advance

Analysis of a large number of Markov chains competing for transitions

International audienceWe consider the behavior of a stochastic system composed of several identically distributed, but non independent, discrete-time absorbing Markov chains competing at each instant for a transition. The competition consists in determining at each instant, using a given probability distribution, the only Markov chain allowed to make a transition. We analyze the first time at which one of the Markov chains reaches its absorbing state. When the number of Markov chains goes to infinity, we analyze the asymptotic behavior of the system for an arbitrary probability mass function governing the competition. We give conditions for the existence of the asymptotic distribution and we show how these results apply to cluster-based distributed systems when the competition between the Markov chains is handled by using a geometric distribution

Convergence speed of a link-state protocol for IPv6 router autoconfiguration

This report presents a model for the NAP protocol, dedicated to the auto-configuration of IPv6 routers. If the auto-configuration of hosts is defined by IPv6 and mandatory, IPv6 routers still have to be manually configured. In order to succeed in new networking domains, a full auto-configuration feature must be offered. NAP offers a fully distributed solution that uses a link state OSPFv3-like approach to perform prefix collision detection and avoidance. In this report, we present a model for NAP and analyze the average and maximum autoconfiguration delay as a function of the network size and the prefix space size

Modélisation et Évaluation des Attaques Ciblées dans un Overlay Structuré

Session Sécurité RéseauInternational audienceDans cet article, nous nous intéressons aux attaques ciblées dans le cadre des systèmes pair-à-pair large échelle. Ces attaques ont pour but d'affaiblir les nœuds ciblés de manière à diminuer leur capacité à fournir ou à utiliser des services de l'overlay. Pour se prémunir de telles attaques, nous tirons parti du clustering de l'overlay sous-jacent. Cela permet de mettre en place un système de churn induit préservant la répartition aléatoire des identifiants des nœuds dans l'overlay et ainsi rendre impossible toute prédiction de l'adversaire quant à celle-ci. Nous montrons qu'en randomisant légèrement les opérations élémentaires de l'overlay, ainsi qu'en introduisant des temps de séjour adaptés, l'effet de ces attaques ciblées est sensiblement amoindri, et la propagation des effets de l'attaque à l'ensemble du système est évitée

On cover times of Markov chains

We consider the cover time of a discrete-time homogenous Markov chain, that is the time needed by the Markov chain to visit all its states. We analyze both the distribution and the moments of the cover time and we are interested in exact results instead of asymptotic values of the mean cover time which are generally considered in the literature. We first obtain several general results on the hitting time and the cover time of a subset of the state space both in terms of distribution and moments. These results are then applied to particular graphs namely the generalized cycle graph, the complete graph and the generalized path graph. They lead to recurrence or analytic relations for the distribution and the mean value of their cover times