# A Sketch-Based Fine-Grained Proportional Integral Queue Management Method
## 2 Related Works  
Queue management methods play an important role in data transfer by effectively improving data transfer performance and ensuring fairness and reliability among streams as much as possible. These can be divided into early passive queue management and active queue management, which have evolved rapidly in recent years. Passive queue management (PQM) is similar to the traditional \"DropTail\". Generally, the router sets a maximum value for the queue in this algorithm. When the queue length reaches the threshold, the next packets are dropped indiscriminately until the queue length returns to the threshold [7]. With the increase in network traffic and the development of network architecture, passive queue management problems of full queues, slow storage deadlocks, and global synchronization are becoming more common [8]. Based on this, active queue management (AQM) algorithms have been proposed, which actively drop packets in router buffers before congestion occurs and send congestion feedback information to the source to effectively reduce or avoid congestion; they have become mainstream queue management methods in the industry and academia. [PERSON] and [PERSON] et al. proposed the RED [3], which is mainly based on the average length of the queue to determine whether packets are dropped or not, but the RED algorithm is very sensitive to parameter configuration and network state, which can lead to the synchronization of multiple TCPs under specific network load conditions, easily causing problems such as queue oscillation, reduced throughput, and increased delay jitter, as well as difficulties in parameter adjustment in the actual network deployment. In response to the lag problem caused by AQM algorithms that use the average queue length as a measure of network congestion, researchers have proposed some new AQM algorithms, such as BLUE [4], which adjusts the packet marking drop probability by link idle and buffer overflow conditions and manages congestion by using packet drop events and link idle events. Although the BLUE algorithm can accomplish congestion control using a small queue cache, it still suffers from parameter setting problems, especially when the number of TCP connections changes drastically, which makes the set parameters invalid and causes the queue to fluctuate drastically between packet drop and low-utilization states. [PERSON] and [PERSON] et al. proposed CoDel [5] for the bufferbloat problem to determine network congestion based on the packet delay in the queue, where packets are set a timestamp when they enter the queue, and then packet drop is judged based on the packet sojourn time in the queue. The FqCoDel [9] algorithm is the derivative version of CoDel and can effectively solve the bufferbloat problem by preprocessing different data streams and then entering the queues in different states controlled by CoDel. COBALT(the CoDel and BLUE Alternate algorithm) [10] is currently used in Linux4.19. When there is no response flow, COBALT significantly reduces the queue waiting time. Modeling the system is the basis for exploring the problems of automatic control theory. A well-known nonlinear TCP/AQM model designed based on fluid theory was proposed by [PERSON] et al. [11] and has been used ever since. [PERSON] et al. designed a PI controller based on the TCP/AQM network model [12] to provide feedback control on network congestion in AQM. The PIE [6] algorithm is also based on the controller and was proposed based on the bufferbloat problem, which combines the advantages of the easy implementation of RED and CoDel, using the rate of packets out of the queue to calculate the packet drop probability and then randomizing packet drop based on the moving trend of the delay. The parameters of the PIE algorithm are self-adjusting and can be based on the current average queuing delay without the need to perform additional processing for each packet, thus, incurring very little overhead and being easy to implement in the hardware and software. In the context of data center networks, ref.