  Stochastic models such as Continuous-Time Markov Chains (CTMC) and Stochastic
Hybrid Automata (SHA) are powerful formalisms to model and to reason about the
dynamics of biological systems, due to their ability to capture the
stochasticity inherent in biological processes. A classical question in formal
modelling with clear relevance to biological modelling is the model checking
problem. i.e. calculate the probability that a behaviour, expressed for
instance in terms of a certain temporal logic formula, may occur in a given
stochastic process. However, one may not only be interested in the notion of
satisfiability, but also in the capacity of a system to mantain a particular
emergent behaviour unaffected by the perturbations, caused e.g. from extrinsic
noise, or by possible small changes in the model parameters. To address this
issue, researchers from the verification community have recently proposed
several notions of robustness for temporal logic providing suitable definitions
of distance between a trajectory of a (deterministic) dynamical system and the
boundaries of the set of trajectories satisfying the property of interest. The
contributions of this paper are twofold. First, we extend the notion of
robustness to stochastic systems, showing that this naturally leads to a
distribution of robustness scores. By discussing two examples, we show how to
approximate the distribution of the robustness score and its key indicators:
the average robustness and the conditional average robustness. Secondly, we
show how to combine these indicators with the satisfaction probability to
address the system design problem, where the goal is to optimize some control
parameters of a stochastic model in order to best maximize robustness of the
desired specifications.
