Abstract

A new method to compute bounds on stationary results of finite Markov processes in discrete or continuous time is introduced. The method extends previously published approaches using polyhedra of eigenvectors for stochastic matrices with a known lower and upper bound of their elements. Known techniques compute bounds for the elements of the stationary vector with respect to the lower bounds of the matrix elements and another set of bounds with respect to the upper bounds of matrix elements. The resulting bounds are usually not sharp, if lower and upper bounds for the elements are known. The new approach combines lower and upper bounds resulting in sharp bounds which are often much tighter than bounds computed using only one bounding value for the matrix elements.

Keywords: Performance analysis; Dependability evaluation; Markov processes; Stationary analysis; Numerical analysis; Bounding techniques

Article Outline

1. Introduction

2. Basic definitions and known results for stationary analysis

3. Improved stationary bounds

4. A new bounding algorithm

5. Some examples

5.1. A simple M/M/1/k model

5.2. A performability model

5.3. A nearly lumpable central server

6. Conclusions and future research

Appendix A. Outline of the proof of Theorem 3.6

References

Vitae