On the statistical distributions of substance moving through the nodes of a channel of network

Abstract: We discuss a model of motion of substance through the nodes of a channel of a network. The channel can be modeled by a chain of urns where each urn can exchange substance with the neighboring urns. In addition the urns can exchange substance with the network nodes and the new point is that we include in the model the possibility for exchange of substance among the urns (nodes) and the environment of the network. We consider stationary regime of motion of substance through a finite channel (stationary regime of exchange of substance along the chain of urns) and obtain a class of statistical distributions of substance in the nodes of the channel. Our attention is focused on this class of distributions and we show that for the case of finite channel the obtained class of distributions contains as particular cases truncated versions of the families of distributions of Katz, Ord, Kemp, etc. The theory for the case of infinite chain of urns is presented in the Appendix.