  We consider a class of piecewise-deterministic Markov processes where the
state evolves according to a linear dynamical system. This continuous time
evolution is interspersed by discrete events that occur at random times and
change (reset) the state based on a linear affine map. In particular, we
consider two families of discrete events, with the first family of resets
occurring at exponentially-distributed times. The second family of resets is
generally-distributed, in the sense that, the time intervals between events are
independent and identically distributed random variables that follow an
arbitrary continuous positively-valued probability density function. For this
class of stochastic systems, we provide explicit conditions that lead to finite
stationary moments, and the corresponding exact closed-form moment formulas.
These results are illustrated on an example drawn from systems biology, where a
protein is expressed in bursts at exponentially-distributed time intervals,
decays within the cell-cycle, and is randomly divided among daughter cells when
generally-distributed cell-division events occur. Our analysis leads to novel
results for the mean and noise levels in protein copy numbers, and we decompose
the noise levels into components arising from stochastic expression, random
cell-cycle times, and partitioning. Interestingly, these individual noise
contributions behave differently as cell division times become more random. In
summary, the paper expands the class of stochastic hybrid systems for which
statistical moments can be derived exactly without any approximations, and
these results have applications for studying random phenomena in diverse areas.
