We consider a distributed random channel access policy in a cognitive radio network where multiple secondary users (SUs) contend for spectrum usage over available multiple primary user (PU) channels. In the random channel access policy each SU stochastically determines whether to access a channel based on the access probability (AP) which is adapted to the sensing results. It is then important to obtain the optimal AP values from the SUs' throughput performance perspective.
In our analysis, we assume that all SUs acquire a common channel state information that is potentially erroneous. We analyze the throughput of an arbitrary SU and the packet collision probability of a PU due to the interference by SUs. By considering a requirement on the packet collision probability of a PU, we rigorously derive an explicit expression on the optimal AP values that maximize the throughput of an SU under the requirement for PUs. In the derivation we show that the multi-dimensional optimization problem with AP values can be converted into a one-dimensional optimization problem, so that the optimal AP values can be easily found. Numerical examples are provided to validate our analysis and to investigate the performance behavior of the optimal channel access policy.