# Computing performability measures in Markov chains by means of matrix functions

## TL;DR

This paper presents a novel matrix function approach for efficiently computing performability measures in Markov chains, enabling faster calculations and sensitivity analysis for large-scale models.

## Contribution

It introduces a new formulation translating performability measure computation into matrix function evaluations, improving efficiency and scalability over existing methods.

## Key findings

- Outperforms state-of-the-art commercial solvers on large examples
- Enables sensitivity analysis of measures with respect to model parameters
- Provides a comprehensive theoretical framework for the new approach

## Abstract

We discuss the efficient computation of performance, reliability, and availability measures for Markov chains; these metrics, and the ones obtained by combining them, are often called performability measures. We show that this computational problem can be recasted as the evaluation of a bilinear forms induced by appropriate matrix functions, and thus solved by leveraging the fast methods available for this task. We provide a comprehensive analysis of the theory required to translate the problem from the language of Markov chains to the one of matrix functions. The advantages of this new formulation are discussed, and it is shown that this setting allows to easily study the sensitivities of the measures with respect to the model parameters. Numerical experiments confirm the effectiveness of our approach; the tests we have run show that we can outperform the solvers available in state of the art commercial packages on a representative set of large scale examples.