We prove that the large deviation principle holds for a class of
processes inspired by semi-Markov additive processes. For the
processes we consider, the sojourn times in the phase process need
not be independent and identically distributed. Moreover the
state selection process need not be independent of the sojourn
times. We assume that the phase process takes values in a finite set and
that the order in which elements in the set, called states, are
visited is selected stochastically. The sojourn times determine
how long the phase process spends in a state once it has been
selected. The main tool is a representation formula for the sample
paths of the empirical laws of the phase process. Then, based on
assumed joint large deviation behavior of the state selection and
sojourn processes, we prove that the empirical laws of the phase
process satisfy a sample path large deviation principle. From this
large deviation principle, the large deviations behavior of a
class of modulated additive processes is deduced. As an illustration of the utility of the general results, we provide
an alternate proof of results for modulated Lévy processes. As a
practical application of the results, we calculate the large deviation
rate function for a processes that arises as the International
Telecommunications Union's standardized stochastic model of two-way
conversational speech.