Over the years several queuing policies have been proposed to ensure fairness between competing requests at a service point. The fair queuing (FQ) algorithm due to Demers, Keshav and Shenkar (1990) is a queuing technique that attains near perfect fairness, where perfect fairness is considered to be the one attained by a fluid flow model. In a data network, the head of the line processor sharing (PS) is considered to be the most fair algorithm. It has been shown that the difference in throughput at any time, in any queue, for any arrival pattern between the FQ and the PS discipline will never exceed MAX, where MAX is the maximum packet size. This difference in throughput is taken as a metric for fairness measure of a queuing algorithm. The drawback of the FQ algorithm is its high packet processing overhead (O (log N)), where N is the number of active flows. To alleviate this problem of high computational complexity, Shreedhar and Varghese (1996) proposed a fair queuing algorithm based on the idea of deficit round robin (DRR). Although DRR reduces the packet processing overhead to O(1), its fairness measure is considerably worse (3MAX) than that of FQ (MAX). In this paper, we present a new round robin based fair queuing algorithm (FQRR) whose packet processing overhead is O(1) and fairness measure is 2MAX.