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Resilient packet ring (RPR), which is standardized as IEEE 802.17, is a new medium access control (MAC) protocol for high-speed metro-area ring networks. RPR supports spatial reuse and, therefore, maintaining fairness among different nodes in accessing the ring bandwidth is a challenging task in RPR. In order to achieve fairness among nodes, a fairness algorithm is employed at each RPR node. When a node is not given access to the ring it becomes congested. In this case, the fairness algorithm calculates and advertises a fair rate to all upstream nodes contributing to the congestion point. Consequently, the congested node will be able to add its local traffic to the ring. In this paper, we develop an analytical model for fair rate calculation in the standard RPR fairness algorithm in the parking lot scenario. We first ignore the link propagation delay and model the system using a non-linear discrete-time low-pass filter. We, then, consider the link propagation delay and develop a more realistic model. We verify our model by simulation results and analyze the convergence of the fairness algorithm. Further, the effect of various parameters on the convergence time is investigated. Finally, we determine the low-pass filter coefficient in order to ensure that the convergence time of the algorithm in the parking lot scenario is within its minimum range.