In this paper, we formulate the optimal channel access of secondary user system in cognitive radio networks as the total reward maximization by Partially Observable Markov Decision Processes (POMDPs). Match Ability is proposed that takes allowable channel transmission rate and bit-error-rate (BER) required by receiving end into account. A heuristic greedy algorithm for channel access is proposed that is based on the local maximum match ability. By combining the utility function of queuing delay with local match ability, we improve the global average delay fairness. It is demonstrated by simulation results that as the average arrival rate increases, the large stable region of low delay can be maintained by adjusting utility function of the average waiting time under the condition that all the control information time overhead is not considered in the centralized global fairness control.