  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.
