Relay is useful in wireless networks to assist the sources to deliver packets to the destinations. In this paper, the effect of full-duplex relay with random access is evaluated. Based on the model of two sources, two destinations, and one full-duplex relay with two queues, we obtain the analytical expressions for significant parameters, such as the arrival rate and the service rate for queues at the relay. Then, we evaluate the per-user throughput and the per-user average delay as functions of the signal-to-interference-plus-noise ratio (SINR) threshold and the self-interference cancelation coefficient through both analysis and simulation. We also consider the case where the two queues at the relay have different priorities and evaluate the effect of different transmit probabilities on the system performance. Our results reveal that the transmit probabilities and the transmit powers have an enormous impact on the self-interference, the average size and the empty probability of the queues. Specially, the queues tend to be stable when increasing the transmit probabilities and the transmit powers.