Guy Latouche

Introduction to matrix analytic methods in stochastic modeling

Preface Part I. Quasi-Birth-and-Death Processes. 1. Examples Part II. The Method of Phases. 2. PH Distributions 3. Markovian Point Processes Part III. The Matrix-Geometric Distribution. 4. Birth-and-Death Processes 5. Processes Under a Taboo 6. Homogeneous QBDs 7. Stability Condition Part IV. Algorithms. 8. Algorithms for the Rate Matrix 9. Spectral Analysis 10. Finite QBDs 11. First Passage Times Part V. Beyond Simple QBDs. 12. Nonhomogeneous QBDs 13. Processes, Skip-Free in One Direction 14. Tree Processes 15. Product Form Networks 16. Nondenumerable States Bibliography Index.

A logarithmic reduction algorithm for quasi-birth-death processes

Quasi-birth-death processes are commonly used Markov chain models in queueing theory, computer performance, teletraffic modeling and other areas. We provide a new, simple algorithm for the matrix-geometric rate matrix. We demonstrate that it has quadratic convergence. We show theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.

Finite birth-and-death models in randomly changing environments

An efficient computational approach to the analysis of finite birth-and-death models in a Markovian environment is given. The emphasis is upon obtaining numerical methods for evaluating stationary distributions and moments of first-passage times.

Efficient Algorithmic Solutions to Exponential Tandem Queues with Blocking.

Stable queuing systems consisting of two groups of servers, having exponential service times, placed in tandem and separated by a finite buffer, are shown to have a steady-state probability vector of matrix-geometric form. The queue is stable as long as the Poisson arrival rate does not exceed a critical value, which depends in a complicated manner on the service rates, the numbers of servers in each group, the size of the intermediate buffer and the unblocking rule followed when system becomes blocked. The critical input rate is determined in a unified manner.For stable queues, it is shown how the stationary probability vector and other important features of the queue may be computed. The essential step in the algorithm is the evaluation of the unique positive solution of a quadratic matrix equation.

Matrix-analytic methods for fluid queues with finite buffers

The application of matrix-analytic methods to the resolution of fluid queues has shown a close connection to discrete-state quasi-birth-and-death (QBD) processes. We further explore this similarity and analyze a fluid queue with finite buffer. We show, using a renewal approach, that the stationary distribution is expressed as a linear combination of two matrix-exponential terms. We briefly indicate how these terms may be computed in an efficient and numerically stable manner.

Risk processes analyzed as fluid queues

This paper presents the Laplace transform of the time until ruin for a fairly general risk model. The model includes both the classical and most Sparre-Andersen risk models with phase-distributed claim amounts as special cases. It also allows for correlated arrival processes, and claim sizes that depend upon environmental factors such as periods of contagion. The paper exploits the relationship between the surplus process and fluid queues, where a number of recent developments have provided the basis for our analysis.

A general class of Markov processes with explicit matrix-geometric solutions

SummaryWe consider a class of Markov chains for which the stationary probability vector, when it exists, is of the matrix-geometric form. The essential step in the computational algorithm usually is the evaluation of a matrixR. We consider two general cases for which that matrix is explicitly determined.ZusammenfassungIn der Bedienungstheorie treten Markovketten auf, deren Übergangsmatrizen blocktridiagonal sind. Die stationären Verteilungen lassen sich unter zusätzlichen Voraussetzungen mit Hilfe einer ResolventenmatrixR ausdrücken. Sie ist im allgemeinen als Lösung einer inR quadratischen Matrixgleichung erhältlich. Wir beweisen, daß in zwei Sonderfällen die MatrixR jeweils einer linearen Gleichung genügt und leiten diese Gleichung her. Damit wird die ResolventenmatrixR leichter zugänglich. In beiden Fällen wird zugelassen, daß die MatrizenR keine endliche Reihenanzahl haben. Das Auftreten beider Sonderfälle wird durch Beispiele aus der Bedienungstheorie belegt.

QUEUES WITH PAIRED CUSTOMERS

Queueing systems with a special service mechanism are considered. Arrivals consist of two types of customers, and services are performed for pairs of one customer from each type. The state of the queue is described by the number of pairs and the difference, called the excess, between the number of customers of each class. Under different assumptions for the arrival process, it is shown that the excess, considered at suitably defined epochs, forms a Markov chain which is either transient or null recurrent. A system with Poisson arrivals and exponential services is then considered, for which the arrival rates depend on the excess, in such a way that the excess is bounded. It is shown that the queue is stable whenever the service rate exceeds a critical value, which depends in a simple manner on the arrival rates. For stable queues, the stationary probability vector is of matrix-geometric form and is easily computable. QUEUEING THEORY; SERVICES BY PAIRS; EXCESS; COMPUTATIONAL PROBABILITY

ALGORITHMS FOR INFINITE MARKOV CHAINS WITH REPEATING COLUMNS

We consider Markov chains on an infinite state space, with a lower block-Hessenberg transition matrix. If the columns repeat as in the case of the GI/PH/1 queue, then the stationary probability vector is matrix-geometric. The major difficulty in computing the stationary vector resides in the determination of the solution of a non-linear matrix equation.

Performance of an ATM switch: simulation study

A parametric model is proposed for the cell arrival traffic on the input lines of a switch. The cell stream alternates between active and silent periods, and general distributions are allowed for the lengths of these periods. The traffic is represented by a discrete Markov chain model obtained through a simple moment matching procedure. The cell arrivals at in input port of the switch are typically non-Poisson and exhibit periodicities. The simulation experiments illustrate that each parameter included in the model has a significant impact on the performance of the switch (as measured by cell-loss probabilities, average cell delay, and steady-state queue length distribution).<<ETX>>