  Although the concepts of age, survival and transit time have been widely used
in many fields, including population dynamics, chemical engineering, and
hydrology, a comprehensive mathematical framework is still missing. Here we
discuss several relationships among these quantities by starting from the
evolution equation for the joint distribution of age and survival, from which
the equations for age and survival time readily follow. It also becomes
apparent how the statistical dependence between age and survival is directly
related to either the age-dependence of the loss function or the survival-time
dependence of the input function. The solution of the joint distribution
equation also allows us to obtain the relationships between the age at exit (or
death) and the survival time at input (or birth), as well as to stress the
symmetries of the various distributions under time reversal. The transit time
is then obtained as a sum of the age and survival time, and its properties are
discussed along with the general relationships between their mean values. The
special case of steady state case is analyzed in detail. Some examples,
inspired by hydrologic applications, are presented to illustrate the theory
with the specific results.
