Computing dependability-oriented measures on Markov chains by means of matrix functions

Ora Inizio: 
Ora Fine: 
Giulio Masetti

In the context of computing and communication system, dependability is defined as the ability to deliver service that can justifiably be trusted.
Dependability is then an umbrella term that encompasses several
attributes, such as: availability, reliability, safety, integrity, maintainability, and confidentiality. Design and analysis of models capable to express functional specifications and behavioural aspects is an important part of the system dependability justification process. 

In particular, Markov chains reflect important characteristics of
computing systems such as discrete states and memoryless property,
and then are often employed to model complex systems.
Starting from the model Markovian process, other processes can be defined to express gains or losses according to which state the Markov process is in, and then all the mentioned dependability attributes
can be evaluated as measures on these processes.

More and more complex systems are designed, and then
new techniques are required to tackle the evaluation of
dependability-oriented measures on large models.
In a world where systems comprise hundreds or thousands of interconnected components, justifying dependability is an ever increasing challenge. 

The main contribution presented in this talk is, focusing on Continuous Time Markov Chains, recasting dependability-oriented measures as the evaluation of a bilinear form where the matrix is indeed a matrix function of the infinitesimal generator matrix characterizing the Markovian stochastic process. In particular, chains with absorbing states, relevant to evaluate system reliability and connected attributes, represent one of the main challenges the modeling community has to deal with, and -- we think -- one of the cases where applying
results from the matrix functions body of knowledge can have the greater impact.