The IEEE 802.16 standard adopts a centralized resource-allocation mechanism to poll nodes with three polling modes, i.e., the unicast, multicast, and broadcast polling modes, depending on the residual bandwidth of the base station (BS). In the multicast and broadcast polling modes, the BS uses the truncated binary exponential backoff (TBEB) algorithm as the contention-resolution process (CRP) for mobile nodes to access the wireless network while the traffic load increases up to saturation. The random contention-based TBEB used in the initial ranging and bandwidth request suffers from high collision probability under a high traffic load. TBEB thus significantly degrades the grade of service (GoS), particularly while performing handoff in the mobile (IEEE 802.16e) or multihop relay WiMAX (IEEE 802.16j) networks. Additionally, WiMAX's TBEB exhibits two critical problems. First, TBEB neglects the node priority (i.e., the new or handoff node) and the service flow class (i.e., the real-time or nonreal-time service class); thus, it cannot achieve the optimal network revenue. Second, all different-priority connections immediately transit to the same minimum contention windows (CWs) after winning contentions and then easily lead to high collision probability at succeeding contentions. Therefore, this paper first proposes an efficient contention-resolution algorithm that consists of three key contributions: 1) providing the adaptive minimum-maximum backoff-value algorithm (AM^2) to partition collision domains; 2) supporting a dynamic waiting-penalty algorithm (DWP) for successful contentions; and 3) differentiating the decrements of CWs (DDW) to avoid collisions, even though the contentions randomly choose the same CW value. Second, we model the proposed approach as a discrete-time Markov chain model and then mathematically analyze several important metrics, i.e., the collision probability, access delay, GoS, and throughput. Numerical results indicate that the analytical results are very close to the simulation results, which justify the accuracy of the analytical model. Additionally, the proposed approach outperforms IEEE 802.16 and all compared approaches in collision probability, delay, GoS, and network throughput.