  We study the motion of a substance in a channel of a network that consists of
chain of nodes of a network (the nodes can be considered as boxes) and edges
that connect the nodes and form the way for motion of the substance. The nodes
of the channel can have different "leakage", i.e., some amount of the substance
can leave the channel at a node and the rate of leaving may be different for
the different nodes of the channel. In addition the nodes close to the end of
the channel for some (construction or other) reason may be more "attractive"
for the substance in comparison to the nodes around the entry node of the
channel. We discuss channels containing infinite or finite number of nodes and
obtain the distribution of the substance along the nodes. Two regimes of
functioning of the channels are studied: stationary regime and non-stationary
regime. The distribution of the substance along the nodes of the channel for
the case of stationary regime is a generalization of the Waring distribution
(for channel with infinite number of nodes) or generalization of the truncated
Waring distribution (for channel with finite number of nodes). In the
non-stationary regime of functioning of the channel one observes an exponential
increase or exponential decrease of the amount of substance in the nodes.
Despite this the asymptotic distribution of the substance among the nodes of
the channel in this regime is stationary. The developed theory is applied for a
study of the distrribution of migrants in countries that form migration
channels.
