Statistical distributions connected to motion of substance in a channel of a network

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 extension of the model with respect to the previous research is that we consider also the possibility for exchange of substance among the urns 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 for the substance in the channel. Our attention is concentrated on this class of distributions and we show that for the case of finite channel the class of distributions contains as particular cases truncated versions of the families of distributions of Katz, Ord, Kemp, etc. Several characteristics of the obtained class of distributions are calculated. The theory for the case of chain containing infinite number of urns is presented in the Appendix. (C) 2019 Elsevier B.V. All rights reserved.