Conditional steady-state bounds for a subset of states in Markov chains

The problem of computing bounds on the conditional steady-state probability vector of a subset of states in finite, ergodic discrete-time Markov chains (DTMCs) is considered. An improved algorithm utilizing the strong stochastic (st-)order is given. On standard benchmarks from the literature and other examples, it is shown that the proposed algorithm performs better than the existing one in the strong stochastic sense. Furthermore, in certain cases the conditional steady-state probability vector of the subset under consideration can be obtained exactly. Copyright 2006 ACM

Computing Quantiles in Markov Reward Models

Probabilistic model checking mainly concentrates on techniques for reasoning about the probabilities of certain path properties or expected values of certain random variables. For the quantitative system analysis, however, there is also another type of interesting performance measure, namely quantiles. A typical quantile query takes as input a lower probability bound p and a reachability property. The task is then to compute the minimal reward bound r such that with probability at least p the target set will be reached before the accumulated reward exceeds r. Quantiles are well-known from mathematical statistics, but to the best of our knowledge they have not been addressed by the model checking community so far. In this paper, we study the complexity of quantile queries for until properties in discrete-time finite-state Markov decision processes with non-negative rewards on states. We show that qualitative quantile queries can be evaluated in polynomial time and present an exponential algorithm for the evaluation of quantitative quantile queries. For the special case of Markov chains, we show that quantitative quantile queries can be evaluated in time polynomial in the size of the chain and the maximum reward.Comment: 17 pages, 1 figure; typo in example correcte

Componentwise bounds for nearly completely decomposable Markov chains using stochastic comparison and reordering

Cataloged from PDF version of article.This paper presents an improved version of a componentwise bounding algorithm for the state probability vector of nearly completely decomposable Markov chains, and on an application it provides the first numerical results with the type of algorithm discussed. The given two-level algorithm uses aggregation and stochastic comparison with the strong stochastic (st) order. In order to improve accuracy, it employs reordering of states and a better componentwise probability bounding algorithm given st upper- and lower-bounding probability vectors. Results in sparse storage show that there are cases in which the given algorithm proves to be useful. © 2004 Elsevier B.V. All rights reserved

Loss rates bounds for IP switches in MPLS networks

International audienc

COMPUTING THE BOUNDS ON THE LOSS RATES J.-M. FOURNEAU

Abstract: We consider an example network where we compute the bounds on cell loss rates. The stochastic bounds for these loss rates using simple arguments lead to models easier to solve. We proved, using stochastic orders, that the loss rates of these easier models are really the bounds of our original model. For ill-balanced configurations these models give good estimates of loss rates

Bounding the loss rates in a multistage ATM switch

International audienc

Stochastic Bounds Applied to the End to End QoS in Communication Systems

End to end QoS of communication systems is essential for users but their performance evaluation is a complex issue. The abstraction of such systems are usually given by multidimensional Markov processes whose analysis is very difficult and even intractable, if there is no specific solution form. In this study, we propose an algorithm in order to automatically derive aggregated Markov processes providing upper and lower bounds on performance measures. We applied the algorithm to the analysis of an open tandem queueing network with rejection in order to derive performance measure bounds. Parametric aggregation schemes have been proposed in order to compute bounds on loss probabilities and end to end mean delays. Therefore a tradeoff between the accuracy of the bound and the size of considered Markov chains is possible.ou