  Stochastic dynamics of several systems can be modeled via piecewise
deterministic time evolution of the state, interspersed by random discrete
events. Within this general class of systems, we consider time-triggered
stochastic hybrid systems (TTSHS), where the state evolves continuously
according to a linear time-varying dynamical system. Discrete events occur
based on an underlying renewal process (timer), and the intervals between
successive events follow an arbitrary continuous probability density function.
Moreover, whenever the event occurs, the state is reset based on a linear
affine transformation that allows for the inclusion of state-dependent and
independent noise terms. Our key contribution is derivation of necessary and
sufficient conditions for the stability of statistical moments, along with
exact analytical expressions for the steady-state moments. These results are
illustrated on an example from cell biology, where deterministic synthesis and
decay of a gene product (RNA or protein) is coupled to random timing of
cell-division events. As experimentally observed, cell-division events occur
based on an internal timer that measures the time elapsed since the start of
cell cycle (i.e., last event). Upon division, the gene product level is halved,
together with a state-dependent noise term that arises due to randomness in the
partitioning of molecules between two daughter cells. We show that the TTSHS
framework is conveniently suited to capture the time evolution of gene product
levels, and derive unique formulas connecting its mean and variance to
underlying model parameters and noise mechanisms. Systematic analysis of the
formulas reveal counterintuitive insights, such as, if the partitioning noise
is large then making the timing of cell division more random reduces noise in
gene product levels.
