Analytic properties of multiple-access trees

A potentially infinite population of terminals is considered, of which a random number
X
seek reserved access to a slotted, broadcast, random-access channel. The reservation protocol is a tree search with collision detection. The time slots are assumed to be organized in nonover-lapping frames of
k
consecutive slots each, and the channel feedback is assumed to be available only at frame (as opposed to slot) boundaries. Control is exercised by varying the nodal degree at the root of the search tree according to the distribution
p
of
X
. Optimal control, and the duration of the reservation set-up time, are studied as functions of
k
and
p
.