How many compartments do we really need? [LTI systems]

This paper tackles the problem of assessing the compartmental nature of a system. As a first step, it is shown that this problem is strictly related to that of positive realizability, and that of determining the minimum number of compartments involved in a process is related to the problem of minimality for positive realizations. Then, necessary and sufficient conditions for a given transfer function to be that of a compartmental system of some finite order are provided. Finally, necessary and sufficient conditions for a given third order transfer function with distinct real poles to be that of a compartmental system with three compartments are provided.