  In this paper, we analyze status update systems modeled through the
Stochastic Hybrid Systems (SHSs) tool. Contrary to previous works, we allow the
system's transition dynamics to be polynomial functions of the Age of
Information (AoI). This dependence allows us to encapsulate many applications
and opens the door for more sophisticated systems to be studied. However, this
same dependence on the AoI engenders technical and analytical difficulties that
we address in this paper. Specifically, we first showcase several
characteristics of the age processes modeled through the SHSs tool. Then, we
provide a framework to establish the Lagrange stability and positive recurrence
of these processes. Building on this, we provide an approach to compute the
m-th moment of the age processes. Interestingly, this technique allows us to
approximate the average age by solving a simple set of linear equations.
Equipped with this approach, we also provide a sequential convex approximation
method to optimize the average age by calibrating the parameters of the system.
Finally, we consider an age-dependent CSMA environment where the backoff
duration depends on the instantaneous age. By leveraging our analysis, we
contrast its performance to the age-blind CSMA and showcase the age performance
gain provided by the former.
