Hédi Nabli

Performability analysis: a new algorithm

We propose, in this paper, a new algorithm to compute the performability distribution. Its computational complexity is polynomial and it deals only with nonnegative numbers bounded by one. This important property allows us to determine truncation steps and so to improve the execution time of the algorithm.

Asymptotic solution of stochastic fluid models

We study the asymptotic behavior of fluid models with input and output rates modulated by a finite state irreducible Markov chain. Several methods have been proposed for the computation of the stationary distribution of buffer content. In this paper, we propose a new method based on recurrence relations for general fluid models. The main advantage of this method is its numerical stability, since it only deals with a non-increasing sequence of positive numbers bounded by 1. We apply this method to a stochastic fluid model and we interpret the numerical results.

An overview on the simplex algorithm

In this paper, the simplex algorithm and its variants are investigated. First, we define a new concept called formal tableau, which leads to derive easily the dual solution from the latest primal table; without any distinction between the original variables and the slack ones. Second, we propose a new method for initializing the simplex algorithm. Unlike the two-phase and the big-M methods, our technique does not involve artificial variables. The computational results reveal that this new method is very favorable especially when the number of artificial variables is significant. Finally, this method will be combined with the notion of formal tableau leading naturally to a second new approach.

Transient and Asymptotic Analysis of General Markov Fluid Models

In this paper, transient and asymptotic behaviors of general Markov fluid models are studied and analyzed. The input and output rates are assumed to be modulated by a finite state irreducible Markov process, which can admit states with zero effective input rate. The main advantage of the proposed methods is their accuracy and their numerical stability. For the transient solution, properties of stationary detection lead to reduce considerably the computational complexity of the algorithm. As for the asymptotic solution, it is derived from the transient ones. We apply these methods to a general Markov fluid model and we interpret the numerical results.

Performability analysis for degradable computer systems

Abstract Degradable performance of fault-tolerant computer systems has given rise to considerable interest in mathematical models for combined evaluation of performance and reliability. Most of these models are based upon Markov processes. Several methods have been proposed for the computation of the probability distribution of performability upon an interval of time [0, t ]. In this paper, we present a new algorithm based on the uniformization technique to compute this distribution for block degradable models. The main advantage of this method is its low polynomial computational complexity and its numerical stability, since it only deals with a nonincreasing sequence of positive numbers bounded by 1. This important property allows us to determine new truncation steps which improve the execution time of the algorithm. We apply this method to a degradable computer system.

Time to stationarity for general Markov fluid models

In this paper, we analyse the buffer content behaviour for general Markov fluid models. Departing from a uniformization-based solution in transient regime, we deduce the steady-state buffer content distribution and we give a simple expression measuring a time to stationarity. Moreover, for a fixed period of time, an upper bound for the absolute difference between the transient solution and the asymptotic one is proposed. As an application example, a heterogeneous fluid-flow model is considered and analysed with our proposed method. Copyright

GoP-based fluid Markovian modelling of video traffic

The constant evolution of video standards and technologies and the important growth of the amount of video data in networks leads to a need to build new models for video sources. In this paper, we specify a Markovian fluid model based on GoPs (Group of Pictures) that creates a video source descriptor. This descriptor can be used to compute the loss rate observed when transmitting the video via the network. We show also how to build an artificial video traffic having the same statistical characteristics of the original source.

Comparison of computation methods for the steady-state markov modulated fluid queues

The Markov-modulated fluid models are mathematical tools widely used in telecommunication networks modeling. These models represent the discrete entities of the network traffic as a continuous fluid whose rates depend on a Markov process. The study of the fluid models shows that they are governed by a linear differential system. Many techniques are used to solve these equations such as spectral analysis, Laplace transforms, orthogonal polynomials and recurrence relations. The purpose of this work is to study mathematically the resolution techniques and to compare them in term of computational complexity, accuracy and stability.

Performability measure for acyclic Markovian models

Abstract Continuous-time Markov processes with a finite-state space are generally considered to model degradable fault-tolerant computer systems. The finite space is partitioned as ∪ m i =1 B i , where B i stands for the set of states which corresponds to the configuration where the system has a performance level (or reward rate) equal to τ i . The performability Y t is defined as the accumulated reward over a mission time [0, t ]. In this paper, a renewal equation is established for the performability measure and solved for both “standard” and uniform acyclic models. Two closed form expressions for the performability measure are derived for the two types of models. Furthermore, an algorithm with a low polynomial computational complexity is presented and applied to a degradable computer system.

Short communication: Uniqueness of asymptotic solution for general Markov fluid models

We investigate the asymptotic workload distribution of fluid models with input and output rates which are modulated by an irreducible Markov process. An analytical solution is proposed in [H. Nabli, Asymptotic solution of stochastic fluid models, Perform. Eval. 57 (2004) 121-140] for general Markov fluid model. This probability distribution is controlled by a linear differential system with specific boundary conditions. The solution uniqueness is proved under a conjecture in the field of linear algebra. This conjecture is valid for all particular fluid models considered in the literature. In this paper, we will prove this conjecture for general fluid models. The numerical computation of the stationary distribution will be also discussed.