arXiv subjects

Guy Latouche

Publications and source records attributed to Guy Latouche.

At least 19 recordsLinked to original sources

arXiv2025

Cut-edge centralities in an undirected graph

A centrality measure of the cut-edges of an undirected graph, given in [Altafini et al.~SIMAX 2023] and based on Kemeny's constant, is revisited. A numerically more stable expression is given to compute this measure, and an explicit expression is provided for some classes of graphs, including one-path graphs and trees formed by three or more branches. These results theoretically confirm the good physical behaviour of this centrality measure, experimentally observed in [Altafini et al.~SIMAX 2023]. Numerical tests are reported to check the stability and to confirm the good physical behaviour.

math.NA↗

arXiv2024

One-sided Markov additive processes with lattice and non-lattice increments

Dating from the work of Neuts in the 1980s, the field of matrix-analytic methods has been developed to analyse discrete or continuous-time Markov chains with a two-dimensional state space in which the increment of a level variable is governed by an auxiliary phase variable. More recently, matrix-analytic techniques have been applied to general Markov additive models with a finite phase space. The basic assumption underlying these developments is that the process is skip-free (in the case of QBDs or fluid queues) or that it is one-sided, that is it is jump-free in one direction. From the Markov additive perspective, traditional matrix-analytic models can be viewed as special cases: for M/G/1 and GI/M/1-type Markov chains, increments in the level are constrained to be lattice random variables and for fluid queues, they have to be piecewise linear. In this paper we discuss one-sided lattice and non-lattice Markov additive processes in parallel. Results that are standard in one tradition are interpreted in the other, and new perspectives emerge. In particular, using three fundamental matrices, we address hitting, two-sided exit, and creeping probabilities.

math.PR↗

arXiv2021

Strong convergence to two-dimensional alternating Brownian motion processes

Flip-flop processes refer to a family of stochastic fluid processes which converge to either a standard Brownian motion (SBM) or to a Markov modulated Brownian motion (MMBM). In recent years, it has been shown that complex distributional aspects of the univariate SBM and MMBM can be studied through the limiting behaviour of flip-flop processes. Here, we construct two classes of bivariate flip-flop processes whose marginals converge strongly to SBMs and are dependent on each other, which we refer to as \emph{alternating} two-dimensional Brownian motion processes}. While the limiting bivariate processes are not Gaussian, they possess desirable qualities, such as being tractable and having a time-varying correlation coefficient function.

math.PR↗

arXiv2021

Numerical solution of a matrix integral equation arising in Markov Modulated Lévy processes

Markov-modulated Lévy processes lead to matrix integral equations of the kind $ A_0 + A_1X+A_2 X^2+A_3(X)=0$ where $A_0$, $A_1$, $A_2$ are given matrix coefficients, while $A_3(X)$ is a nonlinear function, expressed in terms of integrals involving the exponential of the matrix $X$ itself. In this paper we propose some numerical methods for the solution of this class of matrix equations, perform a theoretical convergence analysis and show the effectiveness of the new methods by means of a wide numerical experimentation.

math.NA↗

arXiv2021

Traffic lights, clumping and QBDs

In discrete time, $\ell$-blocks of red lights are separated by $\ell$-blocks of green lights. Cars arrive at random. \ We seek the distribution of maximum line length of idle cars, and justify conjectured probabilistic asymptotics algebraically for $2\leq\ell\leq3$ and numerically for $\ell\geq4$.

math.PR↗

arXiv2021

A family of fast fixed point iterations for M/G/1-type Markov chains

We consider the problem of computing the minimal nonnegative solution $G$ of the nonlinear matrix equation $X=\sum_{i=-1}^\infty A_iX^{i+1}$ where $A_i$, for $i\ge -1$, are nonnegative square matrices such that $\sum_{i=-1}^\infty A_i$ is stochastic. This equation is fundamental in the analysis of M/G/1-type Markov chains, since the matrix $G$ provides probabilistic measures of interest. A new family of fixed point iterations for the numerical computation of $G$, that includes the classical iterations, is introduced. A detailed convergence analysis proves that the iterations in the new class converge faster than the classical iterations. Numerical experiments confirm the effectiveness of our extension.

math.NA↗

arXiv2018

Why is Kemeny's constant a constant?

In their 1960 book on finite Markov chains, Kemeny and Snell established that a certain sum is invariant. The value of this sum has become known as {\it Kemeny's constant}. Various proofs have been given over time, some more technical than others. We give here a very simple physical justification, which extends without a hitch to continuous-time Markov chains on a finite state space. For Markov chains with denumerably infinite state space, the constant may be infinite and even if it is finite, there is no guarantee that the physical argument will hold. We show that the physical interpretation does go through for the special case of a birth-and-death process with a finite value of Kemeny's constant. Keywords: Kemeny's constant; discrete-time Markov chains; continuous-time Markov chains; passage times; deviation matrix.

math.PR↗

arXiv2018

Analysis of fluid flow models

Markov-modulated fluids have a long history. They form a simple class of Markov additive processes, and were initially developed in the 1950s as models for dams and reservoirs, before gaining much popularity in the 1980s as models for buffers in telecommunication systems, when they became known as fluid queues. More recent applications are in risk theory and in environmental studies. In telecommunication systems modelling, the attention focuses on determining the stationary distribution of the buffer content. Early ODE resolution techniques have progressively given way to approaches grounded in the analysis of the physical evolution of the system, and one only needs now to solve a Riccati equation in order to obtain several quantities of interest. To the early algorithms proposed in the Applied Probability literature, numerical analysts have added new algorithms, improved in terms of convergence speed, numerical accuracy, and domain of applicability. We give here a high-level presentation of the matrix-analytic approach to the analysis of fluid queues, briefly address computational issues, and conclude by indicating how this has been extended to more general processes.

math.PR↗

arXiv2017

The time-dependent expected reward and deviation matrix of a finite QBD process

Deriving the time-dependent expected reward function associated with a continuous-time Markov chain involves the computation of its transient deviation matrix. In this paper we focus on the special case of a finite quasi-birth-and-death (QBD) process, motivated by the desire to compute the expected revenue lost in a MAP/PH/1/C queue. We use two different approaches in this context. The first is based on the solution of a finite system of matrix difference equations; it provides an expression for the blocks of the expected reward vector, the deviation matrix, and the mean first passage time matrix. The second approach, based on some results in the perturbation theory of Markov chains, leads to a recursive method to compute the full deviation matrix of a finite QBD process. We compare the two approaches using some numerical examples.

math.PR↗

arXiv2017

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.

math.PR↗

arXiv2016

General solution of the Poisson equation for Quasi-Birth-and-Death processes

We consider the Poisson equation $(I-P)\boldsymbol{u}=\boldsymbol{g}$, where $P$ is the transition matrix of a Quasi-Birth-and-Death (QBD) process with infinitely many levels, $\bm g$ is a given infinite dimensional vector and $\bm u$ 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 $P$ to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples.

math.NA↗

arXiv2016

Feedback control: two-sided Markov-modulated Brownian motion with instantaneous change of phase at boundaries

We consider a Markov-modulated Brownian motion $\{Y(t), ρ(t)\}$ with two boundaries at $0$ and $b > 0$, and allow for the controlling Markov chain $\{ρ(t)\}$ to instantaneously undergo a change of phase upon hitting either of the two boundaries at semi-regenerative epochs defined to be the first time the process reaches a boundary since it last hits the other boundary. We call this process a flexible Markov-modulated Brownian motion. Using the recently-established links between stochastic fluid models and Markov-modulated Brownian motions, we determine important characteristics of first exit times of a Markov-modulated Brownian motion from an interval with a regulated boundary. These results allow us to follow a Markov-regenerative approach and obtain the stationary distribution of the flexible process. This highlights the effectiveness of the regenerative approach in analyzing Markov-modulated Brownian motions subject to more general boundary behaviors than the classic regulated boundaries.

math.PR↗

arXiv2016

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.

math.NA↗

arXiv2015

Slowing time: Markov-modulated Brownian motion with a sticky boundary

We analyze the stationary distribution of regulated Markov modulated Brownian motions (MMBM) modified so that their evolution is slowed down when the process reaches level zero --- level zero is said to be {\em sticky}. To determine the stationary distribution, we extend to MMBMs a construction of Brownian motion with sticky boundary, and we follow a Markov-regenerative approach similar to the one developed in past years in the context of quasi-birth-and-death processes and fluid queues. We also rely on recent work showing that Markov-modulated Brownian motions may be analyzed as limits of a parametrized family of fluid queues. We use our results to revisit the stationary distribution of the well-known regulated MMBM.

math.PR↗

arXiv2014

Lyapunov exponents for branching processes in a random environment: The effect of information

We consider multitype Markovian branching processes evolving in a Markovian random environment. To determine whether or not the branching process becomes extinct almost surely is akin to computing the maximal Lyapunov exponent of a sequence of random matrices, which is a notoriously difficult problem. We define dual processes and we construct bounds for the Lyapunov exponent. The bounds are obtained by adding or by removing information: to add information results in a lower bound, to remove information results in an upper bound and we show that to add more information gives smaller lower bounds. We give a few illustrative examples and we observe that the upper bound is generally more accurate than the lower bound.

math.PR↗

arXiv2014

Approximations for time-dependent distributions in Markovian fluid models

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

math.PR↗

arXiv2014

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.

math.PR↗

arXiv2014

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.

math.PR↗