Cutoff points for roll call protocols in multiple access systems

A communication network with multiple users sharing a common broadcast channel is considered. A
k
-channel is defined to be a channel that can carry up to
k-1
successful transmissions simultaneously, but simultaneous transmissions by
k
or more users cause a collision and no successful transmissions take place. The question arises as to how to organize transmissions to avoid collisions or to resolve them efficiently when they occur. The most conservative approach is to use a roll call scheme, which guarantees that collisions never occur. However, other approaches may sometimes be more efficient. It is supposed that each user has the same probability
p
of wanting to transmit, and for several different models the conditions on
p
under which roll call is the most efficient scheme for a
k
-channel are determined.