Approximations for time-dependent distributions in Markovian fluid models

In this paper we study the distribution of the level at time $\theta$ of Markovian fluid queues and Markovian continuous time random walks, the maximum (and minimum) level over $[0,\theta]$, and their joint distributions. We approximate $\theta$ by a random variable $T$ 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

The morphing of fluid queues into Markov-modulated Brownian motion

Ramaswami showed recently that standard Brownian motion arises as the limit of a family of Markov-modulated linear fluid processes. We pursue this analysis with a fluid approximation for Markov-modulated Brownian motion. Furthermore, we prove that the stationary distribution of a Markov-modulated Brownian motion reflected at zero is the limit from the well-analyzed stationary distribution of approximating linear fluid processes. Key matrices in the limiting stationary distribution are shown to be solutions of a new quadratic equation, and we describe how this equation can be efficiently solved. Our results open the way to the analysis of more complex Markov-modulated processes.Comment: 20 page; the material on p7 (version 1) has been removed, and pp.8-9 replaced by Theorem 2.7 and its short proo

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

Extinction probabilities of branching processes with countably infinitely many types

We present two iterative methods for computing the global and partial extinction probability vectors for Galton-Watson processes with countably infinitely many types. The probabilistic interpretation of these methods involves truncated Galton-Watson processes with finite sets of types and modified progeny generating functions. In addition, we discuss the connection of the convergence norm of the mean progeny matrix with extinction criteria. Finally, we give a sufficient condition for a population to become extinct almost surely even though its population size explodes on the average, which is impossible in a branching process with finitely many types. We conclude with some numerical illustrations for our algorithmic methods

Shift techniques for Quasi-Birth and Death processes: canonical factorizations and matrix equations

We revisit the shift technique applied to Quasi-Birth and Death (QBD) processes (He, Meini, Rhee, SIAM J. Matrix Anal. Appl., 2001) by bringing the attention to the existence and properties of canonical factorizations. To this regard, we prove new results concerning the solutions of the quadratic matrix equations associated with the QBD. These results find applications to the solution of the Poisson equation for QBDs

Two-dimensional fluid queues with temporary assistance

We consider a two-dimensional stochastic fluid model with $N$ ON-OFF inputs and temporary assistance, which is an extension of the same model with $N = 1$ in Mahabhashyam et al. (2008). The rates of change of both buffers are piecewise constant and dependent on the underlying Markovian phase of the model, and the rates of change for Buffer 2 are also dependent on the specific level of Buffer 1. This is because both buffers share a fixed output capacity, the precise proportion of which depends on Buffer 1. The generalization of the number of ON-OFF inputs necessitates modifications in the original rules of output-capacity sharing from Mahabhashyam et al. (2008) and considerably complicates both the theoretical analysis and the numerical computation of various performance measures

Les ateliers d’artistes dans l’écosystème montréalais : Une étude de localisation

Cet article évalue l’importance des ateliers d’artistes à Montréal tout en interrogeant leurs logiques de localisation. L’analyse des ateliers inscrits aux registres de la Ville de Montréal entre 1996 et 2005 montre une forte concentration dans les quartiers centraux, une tendance au regroupement dans quelques édifices-phares et un fléchissement important de l’offre depuis l’an 2000. Elle révèle également une logique de localisation qui répond moins à celle des nouveaux arrondissements qu’à la structure linéaire de l’ancienne base industrielle de Montréal. La conclusion discute les politiques publiques pouvant favoriser une meilleure insertion des artistes dans la ville.This article looks at artist studios as they recently developed in Montréal following a new policy by the City of Montréal to encourage the transformation of “abandoned” industrial buildings into new spaces for the arts. The study documents an important concentration of these work-spaces in the central districts of Montréal and a trend towards increasing amalgamation in selected buildings along the old rail network. Stable from 1996 until 2000, the number of studios has drop significantly in 2005 following the recovery of the real estate market. The article discusses the consequences of this linear rather than concentric distribution on the possibility of establishing an official “cultural district” in the center of Montréal