# A Sketch-Based Fine-Grained Proportional Integral Queue Management Method
## 3 System Model
### Queue Management
As the experiment proceeds, the S-PIE algorithm starts to gradually decrease and is lower than the average captain of the RED algorithm, indicating that the differentiated packet drop strategy used by the S-PIE algorithm can quickly relieve network congestion, maintain a lower average queue length, and effectively reduce delay. The average queue length represents the backlog level of the queue, which directly affects the packet queuing waiting time in the buffer, that is, it directly affects queue delay. Figure 5a,b show the RTT of the monitored packets in the two scenarios, respectively. As can be seen from the figure, although S-PIE adds the Sketch-based traffic statistics module and the specific operation adds the insertion operation of Sketch after the packet is queued and the query operation of Sketch before queuing, S-PIE still shows a better RTT performance. In Scenario 1, during the simulation of RED, CoDel, PIE, and S-PIE, the RTT values achieved were 81.34, 85.97, 85.51, and 80.68 ms, respectively. It is noteworthy that S-PIE demonstrated a reduction of 0.08%, 6.15%, and 5.65% in average RTT compared to RED, CoDel, and PIE, respectively. Moving on to Scenario 2, the average RTT values for these four algorithms were measured at 81.84, 85.53, 87.05, and 80.95 ms. Impressively, S-PIE exhibited a decrease of 1.09%, 5.36%, and 7.01% in average RTT compared to RED, CoDel, and PIE, respectively. Moreover, in networks 0-2 s, when packets arrive in large numbers, the packet RTTs of RED, CoDel, and PIE have large peaks and fluctuations, whereas S-PIE is more stable. 2. Throughput Throughput refers to the number of packets received at the receiving end per unit time, and a higher throughput represents the actual transmission capability of the link. In the throughput comparison experiment, we first considered Scenario 1, where there is a significant difference in traffic distribution, and whether the link can allocate reasonable resources for small traffic. Then, by comparing the performance of the average queue length, we found that S-PIE can maintain a shorter average queue length, so we want to test whether S-PIE will cause excessive throughput performance drop in this case. Figure 4: Comparison of average queue length in different scenarios. Figure 5: Comparison of RTT in different scenarios. Figure 6 shows a comparison of the four algorithms in Scenario 1 in terms of throughput at different time periods. Scenario 1 makes a clear distinction between traffic, with 50 nodes simulating sending large flows and 50 nodes simulating sending small flows. As can be seen from the figure, at the 5-s mark, the cumulative bandwidth allocation for large and small flows stands approximately at 40.35, 41.78, 41.33, and 41.18 Mbps for the RED, CoDel, PIE, and S-PIE algorithms, respectively. After the simulation time reaches 5 s, the link utilization reached approximately 80%. Combined with the average queue-length situation in Figure 4a, the network reaches a congested state, and the RED, CoDel, and PIE algorithms have a clear difference in throughput between large and small traffic in this scenario; that is, large flows take up more bandwidth. S-PIE gradually approached the throughput of small flows to the throughput of large flows after 5 s, i.e., a more equitable bandwidth utilization was achieved between large and small flows. Figure 7 shows a comparison of the throughput of the four algorithms under Scenario 2 for different time periods. In the beginning 0-5 s, the link utilization was still low, and the throughput grew and stabilized as the simulation time was normal, and the sender continued to send packets. Figure 8 shows the average throughput comparison of the four algorithms for multiple experiments. As can be seen from the figure, the S-PIE algorithm maintains a throughput performance similar to the other three AQM algorithms under the same network environment.