# A Sketch-Based Fine-Grained Proportional Integral Queue Management Method
## 3 System Model
### Queue Management
Combined with the average queue-length implementation in Figure 4b, it is shown that S-PIE can maintain a shorter average buffer queue length with essentially no reduction in algorithm throughput; that is, it can achieve a lower queue delay. Figure 6: Comparison of throughput by time period (Scenario 1). Figure 7: Comparison of throughput by time period (Scenario 2). 3. Fairness refers to the fairness of the different flows in a network. The AQM algorithm should guarantee fairness between different flows to avoid some flows taking up too many network resources and affecting other flows. Figure 9a shows a comparison of the fairness of the four algorithms under Scenario 1. As can be seen from the figure, the fairness of each algorithm gradually improves as the network stabilizes. When there is a significant difference in the size of the data streams, the original classical algorithms RED, CoDel, and PIE all show poor fairness performance. Among these, PIE has the worst fairness in this scenario, and RED can achieve relatively better fairness. The S-PIE algorithm can achieve a higher fairness. In Scenario 1, at the 5-s mark, which represents the period of maximal network instability and diminished fairness across all algorithms, the fairness indices for RED, CoDel, PIE, and S-PIE are recorded approximately as 0.67, 0.64, 0.52, and 0.86, respectively. Amidst this context of comparatively reduced fairness indices, the S-PIE algorithm achieves enhancements of approximately 28.90%, 33.74%, and 63.68% relative to the RED, CoDel, and PIE algorithms, respectively. As illustrated in Figure 9a, within Scenario 1, the RED, CoDel, PIE, and S-PIE algorithms demonstrate network stability by the 20-s mark, concurrently showcasing optimized fairness indices of approximately 0.97, 0.94, 0.92, and 0.99, respectively. In comparison to the RED, CoDel, and PIE algorithms, the fairness index of the S-PIE algorithm was elevated by approximately 2.07%, 4.78%, and 7.17%, respectively. Combined with the comparison of the throughput performance, it can be seen that under Scenario 1, large flows occupy most of the bandwidth, and small flows are bandwidth-constrained. In the case of congestion, the original active queue management algorithms RED, CoDel, and PIE choose to adopt the same packet-drop policy for small and large flows, that is, small flows are dropped with the same drop probability as large flows, so small flows are more affected, making the bandwidth occupation less bandwidth consumption and lower throughput. On the other hand, the S-PIE algorithm drops large flows punitively when the network becomes congested, thus, effectively alleviating the bandwidth resource-sharing problem on bottleneck links and ensuring fairness among data flows. Figure 9b shows a comparison of the fairness of the algorithms under Scenario 2. As can be seen from the figure, similar to Scenario 1, the fairness indices of all four algorithms exhibit an increasing trend as the simulation time increases. However, compared to Scenario 1, which clearly distinguishes between large and small flows, each algorithm in Scenario 2 exhibits a higher fairness performance than in Scenario 1. Similarly, as depicted in Figure 9b, under the conditions of Scenario 2, the average fairness indices for the entire simulation period are measured approximately at 0.91, 0.89, 0.87, and 0.94 for the RED, CoDel, PIE, and S-PIE algorithms, respectively. Relative to the RED, CoDel, and PIE algorithms, the fairness of the S-PIE algorithm was elevated by approximately 2.51%, 5.16%,

Figure 8: Comparison of throughput of multiple experiments (Scenario 2). and 8.20%. By comparing the fairness indices of active queue-management algorithms in different network congestion environments, it was demonstrated that the packet drop strategy of S-PIE can effectively alleviate network congestion and ensure higher fairness among data flows. (4) Packet drop

Figure 10a,b show the passive and active packet drops for each of the four active queuing algorithms, respectively.