Linking age, survival and transit time distributions

TL;DR

This paper develops a comprehensive mathematical framework linking age, survival, and transit time distributions, highlighting their relationships, dependencies, and symmetries, with applications in hydrology and other fields.

Contribution

It introduces a unified approach to derive relationships among age, survival, and transit time distributions, including their joint distributions and symmetries, in a general and steady state context.

Findings

Derived equations for joint age-survival distribution.

Established relationships between age, survival, and transit time.

Demonstrated applications with hydrologic examples.

Abstract

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.