In recent years, the reduced cost and increased capacity of memory have resulted in a growing number of buffers in switches and routers. Consequently, today's networks suffer from bufferbloat, a term that refers to excess frame buffering resulting in high latency, high jitter, and low throughput. Although ring aggregation is an efficient topology for forwarding traffic from multiple, widely deployed user nodes to a core network, a fairness scheme is needed to achieve throughput fairness and avoid bufferbloat, because frames are forwarded along ring nodes. N Rate N+1 Color Marking (NRN+1CM) was proposed to achieve per-flow fairness in ring aggregation networks. The key idea of NRN+1CM is to assign a color that indicates the dropping priority of a frame according to the flow-input rate. When congestion occurs, frames are selectively discarded based on their color and the frame-dropping threshold. Through the notification process for the frame-dropping threshold, frames are discarded at upstream nodes in advance, avoiding the accumulation of a queuing delay. The performance of NRN+1CM was analyzed theoretically and evaluated with computer simulations. However, its ability to avoid bufferbloat has not yet been proven mathematically. This paper uses an M(n)/M/1/K queue model to demonstrate how bufferbloat is avoided with NRN+1CM's frame-dropping threshold-notification process. The M(n)/M/1/K queue is an M/M/1/K queuing system with balking. The state probabilities and average queue size of each ring node were calculated with the model, proving that the average queue size is suppressed in several frames, but not in the most congested queue. Computer simulation results confirm the validity of the queue model. Consequently, it was logically deducted from the proposed M(n)/M/1/K model that bufferbloat is successfully avoided with NRN+1CM independent of the network conditions including the number of nodes, buffer sizes, and the number and types of flows.