V. Ramaswami

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.

Fluid Flow Models and Queues—A Connection by Stochastic Coupling

We establish in a direct manner that the steady state distribution of Markovian fluid flow models can be obtained from a quasi birth and death queue. This is accomplished through the construction of the processes on a common probability space and the demonstration of a distributional coupling relation between them. The results here provide an interpretation for the quasi-birth-and-death processes in the matrix-geometric approach of Ramaswami and subsequent results based on them obtained by Soares and Latouche.

Transient Analysis of Fluid Flow Models via Stochastic Coupling to a Queue

Abstract Markovian fluid flow models are used extensively in performance analysis of communication networks. They are also instances of Markov reward models that find applications in several areas like storage theory, insurance risk and financial models, and inventory control. This paper deals with the transient (time dependent) analysis of such models. Given a Markovian fluid flow, we construct on the same probability space a sequence of queues that are stochastically coupled to the fluid flow in the sense that at certain selected random epochs, the distribution of the fluid level and the phase (the state of the modulating Markov chain) is identical to that of the work in the queue and the phase. The fluid flow is realized as a stochastic process limit of the processes of work in the system for the queues, and the latter are analyzed using the matrix-geometric method. These in turn provide the needed characterization of transient results for the fluid model.

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>>

Modeling and characterization of large-scale Wi-Fi traffic in public hot-spots

Server side measurements from several Wi-Fi hot-spots deployed in a nationwide network over different types of venues from small coffee shops to large enterprises are used to highlight differences in traffic volumes and patterns. We develop a common modeling framework for the number of simultaneously present customers. Our approach has many novel elements: (a) We combine statistical clustering with Poisson regression from Generalized Linear Models to fit a non-stationary Poisson process to the arrival counts and demonstrate its remarkable accuracy; (b) We model the heavy tailed distribution of connection durations through fitting a Phase Type distribution to its logarithm so that not only the tail but also the overall distribution is well matched; (c) We obtain the distribution of the number of simultaneously present customers from an Mt/G/∞ queuing model using a novel regenerative argument that is transparent and avoids the customarily made assumption of the queue starting empty at an infinite past; (d) Most importantly, we validate our models by comparison of their predictions and confidence intervals against test data that is not used in fitting the models.

An experimental evaluation of the matrix-geometric method for the GI/PH/1 queue

A GI/PH/1 queue is a single server queueing model with general interarrival time distribution and phase type service time distribution. It provides a unified framework to include a large number of special cases of single server models that are commonly used. A class of algorithms for computing the performance measures for such queues goes under the name “ matrix-geometric solution ” and has been implemented as a FORTRAN package by V. Ramaswami. Reported here are the results of a numerical experiment with that package. Using a carefully selected set of examples, we address many issues related to the computational complexity and numerical accuracy of the matrix-geometric method. Among other things, some insights into modeling service times using two moment approximations are also obtained. The examples reflect a wide variety of characteristics in the interarrival and service time distributions and may therefore be of independent interest as possible test problems to evaluate the performance of algorithms fo...

Steady State Analysis of Finite Fluid Flow Models Using Finite QBDs

The Markov modulated fluid model with finite buffer of size β is analyzed using a stochastic discretization yielding a sequence of finite waiting room queueing models with iid amounts of work distributed as exp (nλ). The n-th approximating queue’s system size is bounded at a value qn such that the corresponding expected amount of work qn/(nλ) → β as n → ∞. We demonstrate that as n → ∞, we obtain the exact performance results for the finite buffer fluid model from the processes of work in the system for these queues. The necessary (strong) limit theorems are proven for both transient and steady state results. Algorithms for steady state results are developed fully and illustrated with numerical examples.

Transient Analysis of Fluid Models via Elementary Level-Crossing Arguments

An analysis of the time-dependent evolution of the canonical Markov modulated fluid flow model is presented using elementary level-crossing arguments.

The PH/PH/1 queue at epochs of queue size change

The PH/PH/1 queue is considered at embedded epochs which form the union of arrival and departure instants. This provides us with a new, compact representation as a quasi-birth-and-death process, where the order of the blocks is the sum of the number of phases in the arrival and service time distributions. It is quite easy to recover, from this new embedded process, the usual distributions at epochs of arrival, or epochs of departure, or at arbitrary instants. The quasi-birth-and-death structure allows for efficient algorithmic procedures.