The Resilient Packet Ring (RPR), defined under IEEE 802.17, has been proposed as a high-speed backbone technology for metropolitan area networks. RPR is introduced to mitigate the underutilization and unfairness problems associated with the current technologies SONET and Ethernet, respectively. The key performance objectives of RPR are to achieve high bandwidth utilization, optimum spatial reuse on the dual rings, and fairness. The challenge is to design an algorithm that can react dynamically to the traffics in achieving these objectives. The RPR fairness algorithm is comparatively simple, but it poses some critical limitations that require further investigation and remedy. One of the major problems is that the amount of bandwidth allocated by the algorithm oscillates severely under unbalanced traffic scenarios. These oscillations are barrier to achieving spatial reuse and high bandwidth utilization. DVSR was another algorithm proposed to solve the fairness issue with no oscillation at the steady state, but at the expense of a high computational complexity O(NlogN), where N is the number of nodes in the ring. In this paper, we propose the Low Complexity Distributed Bandwidth Allocation (LCDBA) algorithm to allocate bandwidth fairly to RPR nodes with a very low computational complexity O(1) that will converge to the exact max-min fairness in a few round trip times with no oscillation at the steady state.