An Algorithmic Approach to Stochastic Bounds

Abstract

Abstract. We present a new methodology based on the stochastic ordering, algorithmic derivation of simpler Markov chains and numerical analysis of these chains. The performance indices defined by reward functions are stochastically bounded by reward functions computed on much simpler or smaller Markov chains. This leads to an important reduction on numerical complexity. Stochastic bounds are a promising method to analyze QoS requirements. Indeed it is sufficient to prove that a bound of the real performance satisfies the guarantee.

XBorne 2016: A Brief Introduction

“…But we also provide new algorithms for the stochastic bounds or the element-wise bound of the matrices, the stochastic bound or the entry-wise bounds of the steady-state distribution. These bounds are based on the algorithmic stochastic comparison of Discrete Time Markov Chain (see [10] for a survey) where stochastic comparison relations are mitigated with structural constraints on the bounding chains. More precisely, the following methods are available:…”

Stochastic Bounds with a Low Rank Decomposition

“…Vincent's algorithm [1,10] consists mainly in replacing inequalities with equalities and reordering them to obtain a constructive way to build the upper bounding matrix Q. …”

Parallelization of stochastic bounds for Markov chains on multicore and manycore platforms

“…There are some ways to reduce the number of investigated states (at the expense of some accuracy, of course) or to change the structure of the matrix (to make it more convenient to computations). Some of the methods [2,[9][10][11]16] could use the Abu-Amsha-Vincent's (AAV) Algorithm [1,12]. In previous work [3], we focused on finding stochastic bounds for Markov chains with the use only of the loop parallelism from OpenMP on Intel Xeon Phi coprocessor.…”