  Continuous-time Markov chains (CTMCs) are popular modeling formalism that
constitutes the underlying semantics for real-time probabilistic systems such
as queuing networks, stochastic process algebras, and calculi for systems
biology. Prism and Storm are popular model checking tools that provide a number
of powerful analysis techniques for CTMCs. These tools accept models expressed
as the parallel composition of a number of modules interacting with each other.
The outcome of the analysis is strongly dependent on the parameter values used
in the model which govern the timing and probability of events of the resulting
CTMC. However, for some applications, parameter values have to be empirically
estimated from partially-observable executions. In this work, we address the
problem of estimating parameter values of CTMCs expressed as Prism models from
a number of partially-observable executions. We introduce the class parametric
CTMCs -- CTMCs where transition rates are polynomial functions over a set of
parameters -- as an abstraction of CTMCs covering a large class of Prism
models. Then, building on a theory of algorithms known by the initials MM, for
minorization-maximization, we present iterative maximum likelihood estimation
algorithms for parametric CTMCs covering two learning scenarios: when both
state-labels and dwell times are observable, or just state-labels are. We
conclude by illustrating the use of our technique in a simple but non-trivial
case study: the analysis of the spread of COVID-19 in presence of lockdown
countermeasures.
