For many years, the IEEE 802.11 distributed coordination function (DCF) has been widely used as a dominant medium access control (MAC) protocol in wireless networks and a large number of works have been done for analyzing and improving its performance. In our earlier work, as a substitute of the IEEE 802.11 DCF, a simple MAC protocol, called the renewal access protocol (RAP), is proposed. The RAP adopts all of the legacy 802.11 standard but the backoff stage feature. Each terminal selects its backoff counter value from a fixed sized window according to a priori given selection distribution in the RAP, regardless of the packet transmission result. It is shown that, if a Poisson distribution is used as the selection distribution, then the resulting RAP achieves high short-term fairness as well as optimal throughput. In this work, we analyze the relation between delay performance and the selection distribution of the RAP. With the help of effective bandwidth theory, we derive the conditions for the selection distribution of the RAP that optimizes the queue overflow probability. We also construct the delay optimal selection distribution satisfying the optimal conditions for throughput and delay. However, we show that the use of the delay optimal selection distribution results in an extremely slow convergence to steady state compared with that of the Poisson selection distribution. Moreover, we show that the Poisson selection distribution provides near-optimal delay performance. Therefore, we conclude that the use of a Poisson selection distribution is still recommended even from the delay perspective.