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Abstract:
Linear compartmental models are commonly used in different areas of science,
particularly in modeling the cycles of carbon and other biogeochemical elements.
The representation of these models as linear autonomous compartmental systems
allows different model structures and parameterizations to be compared. In particular,
measures such as system age and transit time are useful model diagnostics. However,
compact mathematical expressions describing their probability distributions remain to
be derived. This paper transfers the theory of open linear autonomous compartmental
systems to the theory of absorbing continuous-timeMarkov chains and concludes that
the underlying structure of all open linear autonomous compartmental systems is the
phase-type distribution. This probability distribution generalizes the exponential distribution
from its application to one-compartment systems to multiple-compartment
systems. Furthermore, this paper shows that important system diagnostics have natural
probabilistic counterparts. For example, in steady state the system’s transit time
coincides with the absorption time of a related Markov chain, whereas the system age
and compartment ages correspond with backward recurrence times of an appropriate
renewal process. These relations yield simple explicit formulas for the system diagnostics
that are applied to one linear and one nonlinear carbon-cycle model in steady state.
Earlier results for transit-time and system-age densities of simple systems are found
to be special cases of probability density functions of phase-type. The new explicit
formulas make costly long-term simulations to obtain and analyze the age structure
of open linear autonomous compartmental systems in steady state unnecessary.
