Delay distributions of slotted ALOHA and CSMA

We derive the closed-form delay distributions of slotted ALOHA and nonpersistent carrier sense multiple access (CSMA) protocols under steady state. Three retransmission policies are analyzed. We find that under a binary exponential backoff retransmission policy, finite average delay and finite delay variance can be guaranteed for G<2S and G<4S/3, respectively, where G is the channel traffic and S is the channel throughput. As an example, in slotted ALOHA, S<(ln2)/2 and S<3(ln4-ln3)/4 are the operating ranges for finite first and second delay moments. In addition, the blocking probability and delay performance as a function of r_max (maximum number of retransmissions allowed) is also derived.