Franck Quessette

Graphs and Stochastic Automata Networks

We show how some graph theoretical arguments may be used to reduce the complexity of the computation of the steady-state distribution of Markov chain. We consider the directed graph associated to a Markov chain derived from a Stochastic Automata Network (SAN). The structural properties of the automata are used to establish new various results. First, we establish the complexity of the resolution for Stochastic Automata Networks with a sparse matrix representation of the automata. This results are used to compare simple SAN (i.e. without functions) with methods which generates a sparse representation of Markov chains (i.e. Markovian Petri Nets for instance) on some examples.

An open tool to compute stochastic bounds on steady-state distributions and rewards

We present X-Bounds, a new tool to implement a methodology based on stochastic ordering, algorithmic derivation of simpler Markov chains and numerical analysis of these chains. The performance indices defined by reward functions are stochastically bounded by reward functions computed on much simpler or smaller Markov chains obtained after aggregation or simplification. This leads to an important reduction on numerical complexity. Typically, chains are ten times smaller and the accuracy may be good enough.

Computing the steady-state distribution of g-networks with synchronized partial flushing

We have shown in [5,4,3] that G-networks with synchronized partial flushing still have a product form steady-state distribution. These networks may have very complex dynamics where an arbitrary number of customers leave an arbitrary number of queues at the same time. The network flow equation are non linear and the usual approaches to solve them fail. We present here a new numerical algorithm which is based on a transform of the G-network to a classical G-network with triggers. We show that the flow equation are transformed by a classical elimination procedure. This new result puts more emphasis on the importance of flow equations following the approach recently proposed by Gelenbe in [2].

Multiple class G-networks with jumps back to zero

We consider multiple class G-networks of processor sharing queues with negative customers which destroy all the customers in a queue. We prove that these networks have a product form solution for steady-state distribution. These networks may have some applications in reliability or performability as the negative customers may clearly model breakdown of computer or communication systems. We also show that under simple assumptions these networks always have a stationary distribution.<<ETX>>

Research article: Automated prediction of three-way junction topological families in RNA secondary structures

We present an algorithm for automatically predicting the topological family of any RNA three-way junction, given only the information from the secondary structure: the sequence and the Watson-Crick pairings. The parameters of the algorithm have been determined on a data set of 33 three-way junctions whose 3D conformation is known. We applied the algorithm on 53 other junctions and compared the predictions to the real shape of those junctions. We show that the correct answer is selected out of nine possible configurations 64% of the time. Additionally, these results are noticeably improved if homology information is used. The resulting software, Cartaj, is available online and downloadable (with source) at: http://cartaj.lri.fr.

An Algorithmic Game-Theory Approach for Coarse-Grain Prediction of RNA 3D Structure

We present a new approach for the prediction of the coarse-grain 3D structure of RNA molecules. We model a molecule as being made of helices and junctions. Those junctions are classified into topological families that determine their preferred 3D shapes. All the parts of the molecule are then allowed to establish long-distance contacts that induce a 3D folding of the molecule. An algorithm relying on game theory is proposed to discover such long-distance contacts that allow the molecule to reach a Nash equilibrium. As reported by our experiments, this approach allows one to predict the global shape of large molecules of several hundreds of nucleotides that are out of reach of the state-of-the-art methods.

Multiple class G-networks with iterated deletions

Abstract We present a new type of multiclass generalized networks of queues with a steady-state product form solution. At its arrival into a queue, a negative customer (or a signal) starts an iteration. At each step of the iteration, a customer is deleted according to a probability which may depend on the type of customer. The iteration stops when the deletion fails. We study networks with multiple classes of positive customers, one class of signals and three service disciplines: FIFO, LIFO/PR and PS.

Bin packing with fragmentable items

We consider a variant of the Bin Packing Problem dealing with fragmentable items. Given a fixed number of bins, the objective is to put all the items into the bins by splitting them in a minimum number of fragments. This problem is useful for modeling splittable resource allocation. In this paper we introduce the problem and its complexity. We give and prove several properties then we present various approximation algorithms and specially a 6 5 -approximation algorithm.

Quasi-Birth-and-Death Processes with Level-Geometric Distribution

A special class of homogeneous continuous-time quasi-birth-and-death (QBD) Markov chains (MCs) which possess level-geometric (LG) stationary distribution is considered. Assuming that the stationary vector is partitioned by levels into subvectors, in an LG distribution all stationary subvectors beyond a finite level number are multiples of each other. Specifically, each pair of stationary subvectors that belong to consecutive levels is related by the same scalar, hence the term level-geometric. Necessary and sufficient conditions are specified for the existence of such a distribution, and the results are elaborated in three examples.

Modelling buffer admission mechanisms using stochastic automata networks

The stochastic automata networks formalism is an attractive technique to model complex systems with interacting components. Each component of the system is modelled by a single automaton; interactions between components are modelled by labels on the arcs which may represent synchronization and state-dependent transitions. Every automaton is associated with some matrices which allow to build the transition matrix of the underlying Markov chain, using tensor algebra. To illustrate this methodology, we introduce two buffer policies which could be used inAtm switching node. Every policy manages two priority levels which have distinct cell loss requirements. The first buffer policy is based on the push-out mechanism : a high priority cell replaces a low priority cell when the buffer is full. The second policy causes the discarding of all the low priority cells when the user transmits a request to send a burst of cells. In both studies, we compute the loss probabilities of each type of cells under various assumptions.RésuméLe formalisme des réseaux d’automates stochastiques est un modèle attractif pour représenter des systèmes complexes comportant plusieurs composants ayant des interactions entre eux. Chaque composant est modélisé par un automate dont les arcs sont étiquetés par des probabilités et des synchronisations portant sur plusieurs automates. A chaque automate et à chaque transition sont associées des matrices qui permettent grâce à l’algèbre tensorielle de dériver la matrice de transition de la chaîne de Markov sous-jacente. En guise d’illustration, les auteurs présentent deux stratégies de gestion de tampons qui pourraient être utilisées dans un nœudAtm. Chaque stratégie gère deux niveaux de priorité différents qui exigent des taux de perte différents. La première stratégie est basée sur un mécanisme d’écrasement où une cellule de haute priorité remplace une cellule de basse priorité quand le tampon est plein. La seconde stratégie entraîne la destruction de toutes les cellules de basse priorité lorsque l’utilisateur envoie à chaque nœud une requête de transmission d’une rafale importante de cellules. Dans les deux cas, ils calculent les probabilités de perte des cellules de chaque type sous différentes hypothèses.