21 research outputs found
Approximations for time-dependent distributions in Markovian fluid models
In this paper we study the distribution of the level at time of
Markovian fluid queues and Markovian continuous time random walks, the maximum
(and minimum) level over , and their joint distributions. We
approximate by a random variable with Erlang distribution and we
use an alternative way, with respect to the usual Laplace transform approach,
to compute the distributions. We present probabilistic interpretation of the
equations and provide a numerical illustration
Perturbation analysis of Markov modulated fluid models
We consider perturbations of positive recurrent Markov modulated fluid
models. In addition to the infinitesimal generator of the phases, we also
perturb the rate matrix, and analyze the effect of those perturbations on the
matrix of first return probabilities to the initial level. Our main
contribution is the construction of a substitute for the matrix of first return
probabilities, which enables us to analyze the effect of the perturbation under
consideration
Poisson's equation for discrete-time quasi-birth-and-death processes
We consider Poisson's equation for quasi-birth-and-death processes (QBDs) and
we exploit the special transition structure of QBDs to obtain its solutions in
two different forms. One is based on a decomposition through first passage
times to lower levels, the other is based on a recursive expression for the
deviation matrix.
We revisit the link between a solution of Poisson's equation and perturbation
analysis and we show that it applies to QBDs. We conclude with the PH/M/1 queue
as an illustrative example, and we measure the sensitivity of the expected
queue size to the initial value
General solution of the Poisson equation for Quasi-Birth-and-Death processes
We consider the Poisson equation , where
is the transition matrix of a Quasi-Birth-and-Death (QBD) process with
infinitely many levels, is a given infinite dimensional vector and is the unknown. Our main result is to provide the general solution of this
equation. To this purpose we use the block tridiagonal and block Toeplitz
structure of the matrix to obtain a set of matrix difference equations,
which are solved by constructing suitable resolvent triples
Skip-free markov processes: analysis of regular perturbations
A Markov process is defined by its transition matrix. A skip-free Markov process is a stochastic system defined by a level that can only change by one unit either upwards or downwards. A regular perturbation is defined as a modification of one or more parameters that is small enough not to change qualitatively the model.This thesis focuses on a category of methods, called matrix analytic methods, that has gained much interest because of good computational properties for the analysis of a large family of stochastic processes. Those methods are used in this work in order i) to analyze the effect of regular perturbations of the transition matrix on the stationary distribution of skip-free Markov processes; ii) to determine transient distributions of skip-free Markov processes by performing regular perturbations.In the class of skip-free Markov processes, we focus in particular on quasi-birth-and-death (QBD) processes and Markov modulated fluid models.We first determine the first order derivative of the stationary distribution - a key vector in Markov models - of a QBD for which we slightly perturb the transition matrix. This leads us to the study of Poisson equations that we analyze for finite and infinite QBDs. The infinite case has to be treated with more caution therefore, we first analyze it using probabilistic arguments based on a decomposition through first passage times to lower levels. Then, we use general algebraic arguments and use the repetitive block structure of the transition matrix to obtain all the solutions of the equation. The solutions of the Poisson equation need a generalized inverse called the deviation matrix. We develop a recursive formula for the computation of this matrix for the finite case and we derive an explicit expression for the elements of this matrix for the infinite case.Then, we analyze the first order derivative of the stationary distribution of a Markov modulated fluid model. This leads to the analysis of the matrix of first return times to the initial level, a charactersitic matrix of Markov modulated fluid models.Finally, we study the cumulative distribution function of the level in finite time and joint distribution functions (such as the level at a given finite time and the maximum level reached over a finite time interval). We show that our technique gives good approximations and allow to compute efficiently those distribution functions.----------Un processus markovien est défini par sa matrice de transition. Un processus markovien sans sauts est un processus stochastique de Markov défini par un niveau qui ne peut changer que d'une unité à la fois, soit vers le haut, soit vers le bas. Une perturbation régulière est une modification suffisamment petite d'un ou plusieurs paramètres qui ne modifie pas qualitativement le modèle.Dans ce travail, nous utilisons des méthodes matricielles pour i) analyser l'effet de perturbations régulières de la matrice de transition sur le processus markoviens sans sauts; ii) déterminer des lois de probabilités en temps fini de processus markoviens sans sauts en réalisant des perturbations régulières. Dans la famille des processus markoviens sans sauts, nous nous concentrons en particulier sur les processus quasi-birth-and-death (QBD) et sur les files fluides markoviennes. Nous nous intéressons d'abord à la dérivée de premier ordre de la distribution stationnaire – vecteur clé des modèles markoviens – d'un QBD dont on modifie légèrement la matrice de transition. Celle-ci nous amène à devoir résoudre les équations de Poisson, que nous étudions pour les processus QBD finis et infinis. Le cas infini étant plus délicat, nous l'analysons en premier lieu par des arguments probabilistes en nous basant sur une décomposition par des temps de premier passage. En second lieu, nous faisons appel à un théorème général d'algèbre linéaire et utilisons la structure répétitive de la matrice de transition pour obtenir toutes les solutions à l’équation. Les solutions de l'équation de Poisson font appel à un inverse généralisé, appelé la matrice de déviation. Nous développons ensuite une formule récursive pour le calcul de cette matrice dans le cas fini et nous dérivons une expression explicite des éléments de cette dernière dans le cas infini.Ensuite, nous analysons la dérivée de premier ordre de la distribution stationnaire d'une file fluide markovienne perturbée. Celle-ci nous amène à développer l'analyse de la matrice des temps de premier retour au niveau initial – matrice caractéristique des files fluides markoviennes. Enfin, dans les files fluides markoviennes, nous étudions la fonction de répartition en temps fini du niveau et des fonctions de répartitions jointes (telles que le niveau à un instant donné et le niveau maximum atteint pendant un intervalle de temps donné). Nous montrerons que cette technique permet de trouver des bonnes approximations et de calculer efficacement ces fonctions de répartitions.Doctorat en Sciencesinfo:eu-repo/semantics/nonPublishe
Approximations for time-dependent distributions in Markovian fluid models
In this paper we analyse Markov-modulated fluid processes over finite time intervals. We study the joint distribution of the level at time θ< ∞ and of the maximum level over [0, θ], as well as the joint distribution of the level at time θ and the minimum level over [0, θ]. We approximate θ by a random variable T with Erlang distribution and so use an approach different from the usual Laplace transform to compute the distributions. We present probabilistic interpretation of the equations and provide a numerical illustration.SCOPUS: ar.jinfo:eu-repo/semantics/publishe
Approximation for time-dependent distributions in Markovian fluid models
info:eu-repo/semantics/publishe