  In this work, we analyze the probabilistic cooperation of a full-duplex relay
in a multiuser random-access network. The relay is equipped with on/off modes
for the receiver and the transmitter independently. These modes are modeled as
probabilities by which the receiver and the transmitter are activated. We
provide analytical expressions for the performance of the relay queue, such as
arrival and service rates, stability conditions, and the average queue size. We
optimize the relay's operation setup to maximize the network-wide throughput
while, simultaneously, we keep the relay's queue stable and minimize the
consumed energy. Furthermore, we study the effect of the SINR threshold and the
self-interference (SI) coefficient on the per-user and network-wide throughput.
For low SINR threshold, we show under which circumstances it is beneficial to
switch off the relay completely, or switch off the relay's receiver only.
