Pierre Semal

Bounds for the Positive Eigenvectors of Nonnegative Matrices and for their Approximations by Decomposition

Cet article traite des vecteurs propres positifs des matrices irreductibles non reparties qui sont simplement caracterisees par une limite superieure donnee λ sur leur rayon spectral et par une matrice L donnee de limites inferieures pour leurs elements. Analyse du cas particulier des matrices stochastiques et on donne un exemple numerique

Computable Bounds for Conditional Steady-State Probabilities in Large Markov Chains and Queueing Models

A method is presented to compute, for the steady-state conditional probabilities of a given Markov subchain, the best lower and upper bounds derivable from the submatrix of transition probabilities between the states of that subchain only. The bounds can be improved when additional information, even fragmentary, on the entire chain is available. When the submatrix has a special structure, analytical expressions of the bounds can be obtained. The method is shown to be useful and economical to bound performance measures in large nonproduct-form queueing network models of computer communication systems.

Refinable bounds for large Markov chains

A method to bound the steady-state solution of large Markov chains is presented. It integrates the concepts of eigen-vector polyhedron and of aggregation and is iterative in nature. The bounds are obtained by considering a subset only of the system state space. This makes the method specially attractive for problems which are too large to be dealt with by traditional methods. The quality of the bounds depends on the locality of the system which is studied: when the system spends most of its time in a small subset of states, tight bounds can be obtained by considering this subset only. Finally, the bounds are refinable in the sense that the tightness of the bounds can be improved by enlarging the subset of states which is considered. The method Is illustrated on a model of a repairable fault-tolerant system with 16 million states. Tight bounds on its availability are obtained by considering less than 0.1 percent of its state space. >

Probability masses fitting in the analysis of manufacturing flow lines

The analysis of manufacturing systems with finite capacity and with general service time distributions is made of two steps: the distributions have first to be transformed into tractable phase-type distributions, and then the modified system can be analytically modelled. In this paper, we propose a new alternative in order to build tractable phase-type distributions, and study its effects on the global modelling process. Called “probability masses fitting” (PMF), the approach is quite simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval, leading to a discrete distribution. PMF shows some interesting properties: it is bounding, monotonic, refinable, it approximates distributions with finite support and it conserves the shape of the distribution. With the resulting discrete distributions, the evolution of the system is then exactly modelled by a Markov chain. Here, we focus on flow lines and show that the method allows us to compute upper and lower bounds on the throughput as well as good approximations of the cycle time distributions. Finally, the global modelling method is shown, by numerical experiments, to compute accurate estimations of the throughput and of various performance measures, reaching accuracy levels of a few tenths of a percent.

Two Bounding Schemes for the Steady-State Solution of Markov Chains

Two different techniques for bounding the steady-state solution of large Markov chains are presented. The first one is a verification technique based on monotone iterative methods. The second one is a decomposition technique based on the concepts of eigen-vector polyhedron. The computation of the availability of repairable fault-tolerant systems is used to illustrate the performances and the numerical aspects of both methods.

Monotone iterative methods for Markov chains

This paper deals with monotone iterative methods for the computation of the steady-state probability vector of irreducible Markov chains. The emphasis is laid on verification techniques aiming at deriving, from an approximation, true bounds on the solution.

A Tight Bound on the Throughput of Queueing Networks with Blocking

In this paper, we present a bounding methodology that allows to compute a tight bound on the throughput of fork-join queueing networks with blocking and with general service time distributions. No exact models exist for queueing networks with general service time distributions and, consequently, bounds are the only certain information available. The methodology relies on two ideas. First, probability mass fitting (PMF) discretizes the service time distributions so that the evolution of the modified system can be modelled by a discrete Markov chain. Second, we show that the critical path can be computed with the discretized distributions and that the same sequence of jobs offers a bound on the original throughput. The tightness of the bound is shown on computational experiments (error on the order of one percent). Finally, we discuss the extension to split-and-merge networks and the approximate estimations of the throughput.

Iterative Algorithms for Large Stochastic Matrices

Markov chains have always constituted an efficient tool to model discrete systems. Many performance criteria for discrete systems can be derived from the steady-state probability vector of the associated Markov chain. However, the large size of the state space of the Markov chain often allows this vector to be determined by iterative methods only. Various iterative methods exist, but none can be proved a priori to be the best. In this paper, we propose a practical measure which allows the convergence rate of the various existing methods to be compared. This measure is an approximation of the modulus of the second largest eigenvalue of the iteration matrix and can be determined a priori. The model of a queueing network is used as an example to compare the convergence of several iterative methods and to show the accuracy of the measure.

Computable Dependability Bounds for Large Markov Chains

A new method to bound the steady-state solution of large Markov chains is presented. The method integrates the concepts of eigenvector polyhedron and of aggregation. It is specially suited for Markov chains with high locality and very large state spaces.