  We model the transmission of a message on the complete graph with n vertices
and limited resources. The vertices of the graph represent servers that may
broadcast the message at random. Each server has a random emission capital that
decreases at each emission. Quantities of interest are the number of servers
that receive the information before the capital of all the informed servers is
exhausted and the exhaustion time. We establish limit theorems (law of large
numbers, central limit theorem and large deviation principle), as n tends to
infinity, for the proportion of visited vertices before exhaustion and for the
total duration. The analysis relies on a construction of the transmission
procedure as a dynamical selection of successful nodes in a Galton-Watson tree
with respect to the success epochs of the coupon collector problem.
