An Improved ARED Algorithm for Congestion Control of Network Transmission

Abstract

In order to achieve high throughput and low average delay in computer network, it is necessary to stabilize the queue length and avoid oscillation or chaos phenomenon. In this paper, based on Adaptive Random Early Detection (ARED), an improved algorithm is proposed, which dynamically changes the range of maximum drop probability 𝑝_max according to different network scenarios and adjusts 𝑝_max to limit average queue size 𝑞_ave in a steady range. Moreover, exponential averaging weight 𝑤 is adjusted based on linear stability condition to stabilize 𝑞_ave. A number of simulations show that the improved ARED algorithm can effectively stabilize the queue length and perform better than other algorithms in terms of stability and chaos control.

1. Introduction

In the current internet, congestion control has been a serious problem. Lack of effective management of congestion significantly affects the performance of internet, such as degradation of link utilization, more round-trip time, and even makes the network inaccessible. At the same time, various approaches have been proposed to solve the issue. These schemes can be divided into two categories. One category strengthens congestion management at the end while making few changes to internet [1, 2]. The other one aims to adapt Active Queue Management (AQM) at each router, and eventually control congestion level, such as Random Early Detection (RED) [3], Random Early Marking (REM) [4] and Virtual Queue (VQ) [5, 6]. Among them, RED is the most prominent and well-studied AQM scheme. Though RED can prevent global synchronization, reduce packet loss, and achieve high throughput, it induces oscillation or chaos to degrade the performance of the system in some network scenarios. The stability of RED gains more and more researchers’ attentions.

The main goal of AQM is to keep queue length stable in order to make a good tradeoff between high throughput and low delay. As a queue management policy, RED gateway manages queue by monitoring the average queue size 𝑞_ave. When 𝑞_ave exceeds the preset lower threshold 𝑞_min, arriving packets are randomly dropped or marked with a certain probability, so that some connections can sense the early congestion, and then adjust their window sizes to avoid serious congestion and packet lost. Once the average queue size is larger than the upper threshold 𝑞_max, the RED gateway drops/marks every arriving packet to keep 𝑞_ave below 𝑞_max. Although the RED algorithm is an effective mechanism to achieve high throughput and low delay, some researches note that RED is quite sensitive to traffic load and RED parameters [7–9], and that the average queue length exhibits nonlinear instability and chaos [10–13]. In particular, when the average queue becomes larger than the upper threshold 𝑞_max, 𝑞_ave fluctuates seriously, leading to low throughput and large packet loss rate. Many modified RED algorithms are proposed to eliminate oscillations or chaos to improve its robustness by revising the drop probability function or by adjusting control parameters.

Among these algorithms, one set of approaches are to modify the packet drop probability function to improve the stability of RED and its performances [14–18]. Particularly, Hollot et al. proposed a PI-controller (Proportional-Integral controller) to improve reaction of TCP-RED dynamics [15]. Besides that, the authors proposed LRED (Loss Ratio-based RED) in [18], which takes into account the dropping probability based on loss rate and queue length rather than average queue size. These modified algorithms outperform original RED in terms of better stability and higher throughput. However, these methods always induce additional parameters that are needed to be optimized to stabilize the queue length in different network scenarios. The setting of parameters is still an unsolved problem. Another set of algorithms does not change the basic idea of RED, but tunes the control parameters to improve the stability of the queue length [8, 9, 12, 13, 19–23]. Feng et al. have argued that there is no single set of RED parameters that can work well under different congestion scenarios. In [8], they proposed original ARED, which tunes the parameter 𝑝_max based on network traffic to keep the average queue size 𝑞_ave between the upper threshold 𝑞_max and lower threshold 𝑞_min. Floyd et al. present a revised version, also called Adaptive RED [9], which adopts AIMD mechanism to change 𝑝_max slowly. Although ARED can stabilize the queue length at a given target and achieve high throughput in some networks, ARED does not provide any systematic methods to set parameters and the target. Furthermore, the control parameter selection is based only on empirical observation and simulation analysis. As a result, it performs well in one simulation, but fails in another. Besides that, some researches propose other variants of RED, such as SARED (Stabilized ARED) [19], PD-RED [20], while others discuss the bound of 𝑝_max [21–23]. These algorithms keep the queue length stable in some network scenarios by dynamically tuning parameters, but they fail to control the fluctuation and chaos in others.

In the paper, we analyze the steady range of 𝑞_ave and its linear stability condition, and propose a novel improved ARED algorithm. The improved ARED constrains 𝑝_max in a more reasonable range according to both the steady range and the target range of ARED. What is more, the improved ARED also decreases 𝑤 to stabilize 𝑞_ave based on linear stability condition. By stabilizing 𝑞_ave within target range, the improved ARED algorithm could get high link utilization, more stable delay with less delay jitter. The rest of the paper is organized as follows. In Section 2, RED algorithm and a discrete-time TCP-RED model are introduced. Then we analyze linear stability condition of the average queue size. In Section 3, an improved ARED algorithm is proposed based on ARED algorithm, and the settings of the control parameters 𝑝_max and 𝑤 are discussed in detailed. Section 4 presents simulations results and discusses the performance of various AQM schemes. Finally, we present the conclusions in Section 5.

2. Discrete-Time TCP-RED Model

In this paper, the network topology used is a dumbbell with a single bottleneck as shown in Figure 1. There are 𝑁 connections sharing a single link with the capacity 𝐶. It is assumed that these connections are identical, long-lived, and have the same round-trip propagation delay 𝑑.

Figure 1 

Structure of the network.

2.1. Random Early Detection (RED)

RED is a recommended scheme of AQM by the Internet Engineering Task Force (IETF) [3]. The main goal of RED is to achieve low average delay and high throughput. In order to achieve the goal, the RED gateway always measures 𝑞_ave, and drops/marks the arriving packets with the probability p to notify the TCP end of the incipient congestion when 𝑞_ave>𝑞_min. Once 𝑞_ave>𝑞_max, RED gateway drops every arriving packet so that it can control the average queue size even in the absence of cooperating sources. The drop probability 𝑝 with RED can be described by the following piecewise function [3]: 𝑝=0𝑞_ave<𝑞_min,1𝑞_ave>𝑞_max,𝑞_ave−𝑞_min𝑞_max−𝑞_min𝑝_max𝑞_min<𝑞_ave<𝑞_max. (2.1) Here 𝑝_max is the drop probability when 𝑞_ave is equal to 𝑞_max, and also called the maximum drop probability. 𝑞_max and 𝑞_min are the expected upper and lower thresholds, respectively. 𝑞_ave can be got with the following equation [3]: 𝑞_ave=(1−𝑤)𝑞^′_ave+𝑤𝑞. (2.2) Here, 𝑞^′_ave is average queue size at the previous time. 𝑞 is instantaneous queue length. 𝑤 is exponential averaging weight which is limited at [0,1]. 𝑞_ave is updated at the time of packet arrival [3].

2.2. Discrete-Time Feedback Model for TCP-RED

To better understand the nonlinear instability in TCP-RED, Ranjan et al. have developed a discrete-time dynamical feedback TCP-RED model [11]. It is assumed that RED queue management mechanism with Explicit Congestion Notification (ECN) capability is implemented at each node to provide congestion signal. According to the model, the instantaneous queue size 𝑞 can be determined by some network parameters, such as the buffer size B, mean packet size M, the number of connections N, and a constant K in [1, (8/3)^0.5]. 𝑞_ave can be expressed mathematically as [11] 𝑞_ave=𝑓(𝑞^′_ave)=(1−𝑤)𝑞^′_ave𝑞_ave≥𝑞1,(1−𝑤)𝑞^′_ave+𝑤𝐵𝑞_ave≤𝑞2,(1−𝑤)𝑞^′_ave+𝑤(𝑁𝐾√𝑝−𝐶𝑑𝑀)otherwise. (2.3) Here 𝑞1=𝑝1(𝑞_max−𝑞_min)𝑝_max+𝑞_min, where 𝑝1=(𝑁𝑀𝐾/(𝑑𝐶))² is the minimum drop rate that leads to an empty queue at next time. 𝑞2=𝑝2(𝑞_max−𝑞_min)/𝑝_max+𝑞_min, where 𝑝2=(𝑁𝑀𝐾/(𝑑𝐶+𝐵𝑀))² is the maximum drop probability with full queue at next time. C is the capacity of the shared link, and d is round-trip propagation delay. The parameter p can be calculated by (2.1):

In the discrete-time model, it is noted that the average queue size is updated over time scale of approximately one RTT (round-trip time) [10, 11]. It is much longer than typical interarrival times of packets in practice. Therefore, w in the model is much larger.

2.3. Instability in TCP-RED

The interaction between TCP and RED is proved to be rather complex and nonlinear [11, 24]. In recent years, some novel approaches are proposed to better understand the complex and nonlinear phenomena in practice [25–27], which may be used to explain the nonlinear phenomena in TCP-RED. In this section, the complex behavior with RED parameters variation is analyzed based on the associated eigenvalue that is presented as follows.

If the average queue size stabilizes at a value 𝑞_ave, that is, 𝑞_ave=𝑓(𝑞_ave), the fixed point 𝑞_ave must lie in the range [q2, q1]. That can be obtained from (2.3). Considering the associated eigenvalue 𝜆 corresponding to 𝑞_ave, the linear stability condition is |𝜆|<1 [11]. |𝜆|=|𝜕𝑓(𝑞_ave)𝜕𝑞_ave|_𝑞ave|=|1−𝑤−𝑤𝑁𝐾2√𝑣(𝑞_ave−𝑞_min)^3/2|<1. (2.4) Here, 𝑣=𝑝_max/(𝑞_max−𝑞_min) means the increase factor of 𝑝. Figures 2 and 3 plot the average queue size and the associated eigenvalue for a fixed point 𝑞^∗_ave=345.1, as w varies from 0.12 to 0.2. In simulation, the parameters are the same as in [11]: 𝐶=75 Mb/s, 𝐾=(3/2)^0.5,𝑀=4, 000b,𝑑=0.1 s, 𝑁=250,𝑞_max=750 packets,𝑞_min=250 packets, 𝐵=3750 packets, 𝑝_max=0.1.

Figure 2 

Bifurcation diagram of average queue size.

Figure 3 

Absolute value of the associated eigenvalue.

From Figures 2 and 3, it is evident that for 𝑤<0.1578, 𝑞_ave converges to the fixed point 𝑞^∗_ave=345.1. At the same time, |𝜆| also stays below 1. As w increases, PDB (period doubling bifurcation) emerges from the fixed point, and then chaos appears with |𝜆|>1. It implies that if |𝜆|>1 for 𝑞_ave, the average queue size is instable at the point. In other words, the point is not a fixed one. The system oscillates and even exhibits chaos phenomenon.

Because all the parameters in (2.4) are positive, 𝜆 must be smaller than 1. The linear stability can be rewritten as 𝜆>−1, that is, 𝜆=1−𝑤−𝑤𝑁𝐾2√𝑣(𝑞_ave−𝑞_min)^3/2>−1. (2.5)

3. The Improved ARED Algorithm

In this section, we first briefly introduce ARED algorithm, and then propose a novel improved ARED algorithm. At last, we give a theoretic analysis for the proposed algorithm.

3.1. Adaptive RED

In 2001, Floyd et al. proposed a modified version of the original ARED (Adaptive RED), also called ARED [9]. To ensure high throughput and low delay, they set a target range with [𝑞_min+0.4(𝑞_max−𝑞_min), 𝑞_min+0.6(𝑞_max−𝑞_min)]. When the current average queue size is not in the range, the control parameter 𝑝_max is revised by AIMD (additive-increase multiplicative-decrease) policy, which can be expressed mathematically as: 𝑝_max=𝑝_max+𝛼,𝑞_ave>𝑞_target,𝑝_max𝑝<0.5,_max𝛽,𝑞_ave<𝑞_target,𝑝_max≥0.01. (3.1) Here, 𝑞_target is the target range. ARED changes 𝑝_max slowly, and the time scale is larger than interarrival time in original ARED. In general, the parameter interval = 0.5 seconds. Based on the principle that the change of 𝑞_ave should not be larger than the range of target in single interval [9], Floyd et al. advised to set 𝛼=min(0.01, 𝑝_max/4) and 𝛽=0.9. Besides that, 𝑝_max is constrained within the range [0.01, 0.5], so that the overall performance of RED could be still acceptable in the transition period, even though the average queue size is out of the target range. A lot of experiments demonstrate that ARED has a good performance in terms of stability and robustness [9]. However, the parameter 𝑤 and the bound of 𝑝_max are configured according to some experiment results. Therefore, ARED works well in some network scenarios but fails in others.

3.2. The Improved ARED Algorithm

Since ARED adjusts 𝑝_max by monitoring whether 𝑞_ave is within the target range, it is sensitive to the change of 𝑞_ave. Besides that, it causes oscillation when w is improper. In the section, we propose a novel ARED, named improved ARED, to reduce the parametric sensitivity and traffic sensitivity. The algorithm is described in Algorithm 1.

Every interval seconds:  if $(q_{\text{ave}}>q_{\text{target}}\hspace{0.17em}\&\&\hspace{0.17em}{p}_{\max \hspace{0.17em}}<p_{\max \hspace{0.17em}\_}u)$    $p_{\max \hspace{0.17em}}=p_{\max \hspace{0.17em}}+\alpha$;  else    if $(q_{\text{ave}}<q_{\text{target}}\hspace{0.17em}\hspace{0.17em}\&\&\hspace{0.17em}{p}_{\max \hspace{0.17em}>}{p}_{\max \hspace{0.17em}\hspace{0.17em}\_}l)$      $p_{\max \hspace{0.17em}}=p_{\max \hspace{0.17em}}\times \beta$;    if $(q_{\text{ave}}\in q_{\text{target}})$     Calculate $\lambda$ by(2.4)      if ($\lambda <-1\mathrm{\hspace{0.17em}\hspace{0.17em}}\&\&\mathrm{\hspace{0.17em}\hspace{0.17em}}w>w_{\min \hspace{0.17em}\hspace{0.17em}}$)        $w=w\times \gamma$; Parameters: $p_{\max \hspace{0.17em}\_}u$: the upper threshold of $p_{\max \hspace{0.17em}}\text{;}$ p1/0.6 $p_{\max \hspace{0.17em}\_}l$: the lower threshold of $p_{\max \hspace{0.17em}}\text{;}$ p2/0.4 $\gamma$: decrease factor of $w$ $w_{\min \hspace{0.17em}\hspace{0.17em}}$: the lower threshold of $w\text{;}$$1-\exp \hspace{0.17em}\hspace{0.17em}(-1/(10Cd/M))$ Other parameters are the same as those with Adaptive RED

Algorithm 1 

Improved ARED algorithm

There are two main differences between ARED and the improved ARED: (1) 𝑝_max is constrained in the range [p2/0.4, p1/0.6] in the improved ARED, instead of the range [0.01, 0.5] in ARED, (2) w is also adjusted in the improved ARED, when the linear stability condition is not satisfied and 𝑞_ave stays in the steady range.

3.3. Maximum Drop Probability 𝑝_max

Base on the discussion in Section 2.3, one can see that 𝑞_ave is stable only when it is in the range [q2, q1] and 𝜆 at the point is below −1. In order to stabilize 𝑞_ave, it is necessary to ensure 𝑞_ave∈[𝑞2,𝑞1]. If the target range in ARED is constrained in the steady range [q2, q1], 𝑝_max could be adjusted effectively to keep 𝑞_ave in the range. In the case, the two following inequalities are obtained: 𝑞2<𝑞_min+0.4(𝑞_max−𝑞_min),𝑞1>𝑞_min+0.6(𝑞_max−𝑞_min). (3.2)

Substituting 𝑞1=𝑝1(𝑞_max−𝑞_min)/𝑝_max+𝑞_min and 𝑞2=𝑝2(𝑞_max−𝑞_min)/𝑝_max+𝑞_min into (3.2), we obtain the range of 𝑝_max: 𝑝20.4<𝑝_max<𝑝10.6. (3.3)

By the way, the drop mechanism with ARED can keep 𝑞_ave within the steady range [q2, q1], so the stability of 𝑞_ave depends on the value of 𝜆 from (2.5). Figure 4 shows the drop probability with the improved ARED algorithm.

Figure 4 

Drop probability of the improved ARED algorithm.

3.4. Exponential Averaging Weight 𝑤

The inequality (2.5) shows that 𝜆 depends on control parameters (𝑞_max, 𝑞_min, w, 𝑝_max), system parameter N, the constant K and 𝑞_ave. Since the number of connections N is out of control, a network manager usually changes control parameters to stabilize 𝑞_ave. The thresholds 𝑞_min and 𝑞_max refer to the tradeoffs between throughput and delay, and they are constants in general. Thus, they are set in advance. The control parameters 𝑝_max is adjusted by ARED scheme. Therefore, w becomes the only parameter to be chosen to achieve the stability of 𝑞_ave. Since 𝜆 is a monotony decrease function of w from (2.5), 𝜆>−1 can be ensured by decreasing w when 𝜆 is below −1. In other words, the linear stability of 𝑞_ave can be obtained by decreasing w.

On the other hand, if w is too small, the average queue size cannot reflect the change of instantaneous queue size, so that RED gateway cannot detect the early congestion. In order to avoid too mall of w, a lower threshold 𝑤_min=1−exp(−1/(10𝐶𝑑/𝑀)) is used [9]. Besides that, multiplicative-decrease policy is used to reduce 𝑤 quickly.

4. Simulations

The instability of RED schemes comes from its parametric sensitivity and traffic sensitivity. In other words, RED scheme is very sensitive to control parameters (w, 𝑞_min) and system parameters (N, d). In this section, we test the performance of the improved ARED, the gentle RED, and ARED when these parameters vary in a large range. The three algorithms use the discrete-time TCP-RED model for simulation, and are evaluated by the stability of 𝑞_ave. The parameters in simulation are the same as in [11]: 𝐶=75 Mb/s, 𝐾=(3/2)^0.5,𝑀=4,000b,𝑑=0.1s,𝑁=250,𝑞_max=750 packets,𝑞_min=250 packets,𝐵=3750 packets, 𝑤=0.15, 𝑝_max=0.1, 𝛾=08 except special descriptions. Besides that, because the instant of observation is about one RRT [10, 11], 𝑝_max is updated after every 5 iterations. It is roughly 0.5 seconds for both the improved ARED and ARED.

4.1. Control Parameters (𝑤,𝑞_min)

In the subsection, we test the parametric sensitivity of various algorithms. The plots in Figure 5 present the average queue size when w varies from 0.12 to 0.34 for the gentle RED, ARED and the improved ARED, respectively. Figure 5(a) shows that for gentle RED, 𝑞_ave can be stabilized at 345.1 with a small w. As w increases, 𝑞_ave becomes instable, and then PDB appears. When w is larger than 0.164, 𝑞_ave exhibits chaos phenomenon. ARED cannot stabilize 𝑞_ave within the target range until 𝑤>0.334 as shown in Figure 5(b). It implies that ARED is more stable than gentle RED when w varies. However, it also leads to PDB and chaos. From Figure 5(c), it is found that the improved ARED keeps 𝑞_ave stable at a point within target range when w is set at different values. The experiments show that the robustness of the improved ARED is better than Adaptive RED and gentle RED in terms of w.

(a) Gentle RED algorithm

(b) ARED algorithm

(c) Improved ARED algorithm

Figure 5 

Average queue size $q_{\text{ave}}$ versus w.

In simulations, ARED can stabilize 𝑞_ave with a larger w compared with gentle RED because 𝑝_max with ARED is adjusted dynamically. However, when w is larger than some critical values, 𝑞_ave is more sensitive to the change of queue length, and thus changes fast. As a result, ARED gateway frequently adjusts 𝑝_max that leads to oscillation. In the same case, the improved ARED decreases w according to the linear stability condition besides adjusting 𝑝_max. Therefore, the performance of the improved ARED outperforms ARED.

Figure 6 demonstrates the change of the average queue size for various algorithms with the variation of 𝑞_min. From these figures, it is evident that gentle RED does a poor job in terms of stability due to the constant 𝑝_max. However, both ARED and the improved ARED perform better in eliminating oscillations and chaos when the lower threshold changes from 250 to 550. When 𝑞_min is near 500 packets, ARED exhibits oscillations as shown in Figure 6(b). That is because the target range [𝑞_min+0.4(𝑞_max−𝑞_min), 𝑞_min+0.6(𝑞_max−𝑞_min)] shrinks as 𝑞_min increases, which means that 𝑞_ave stays within the range with a smaller probability. Under this condition, 𝑝_max fluctuates frequently, so 𝑞_ave fluctuates too. However, the improved ARED changes the value of w based on linear stability condition to reduce the effect of the queue length on 𝑞_ave, and then reduce the fluctuation of 𝑝_max. By doing so, the improved RED degrades the sensitivity of 𝑞_min to some extent, and improves the stability of 𝑞_ave compared with ARED and gentle RED, as shown in Figure 6(c).

(a) Gentle RED algorithm

(b) ARED algorithm

(c) Improved ARED algorithm

Figure 6 

Average queue size $q_{\text{ave}}$ versus $q_{\min \hspace{0.17em}}$.

4.2. System Parameters (𝑁,𝑑)

In this subsection, we test the stability and robustness of different algorithms under different network load N and different round-trip propagation delay d.

Figure 7 shows the average queue size when N increases from 100 to 1500. From Figure 7(a), it is evident that the system is difficult to stabilize 𝑞_ave with small N. As N increases, the system is more stable and 𝑞_ave stabilizes at a larger value. These results are consistent with that in [28]. However, if N continues to increase without constrain, there exists a value of N with 𝑞_ave near 2𝑞_max, which corresponds to the drop probability of 1 for gentle RED. Once N exceeds the value, 𝑞_ave oscillates [8]. Therefore, the system can stabilize 𝑞_ave only if N varies in some range when other parameters are determined. Figure 7(b) shows that 𝑞_ave of ARED is stable when 𝑁∈[520,1080]. However it increases when 𝑁>1080 and oscillates when 𝑁<520. In the simulations, the improved ARED can keep 𝑞_ave stable in the target range with 𝑁∈[100,1500], as shown in Figure 7(c).

(a) Gentle RED algorithm

(b) ARED algorithm

(c) Improved ARED algorithm

Figure 7 

Average queue size $q_{\text{ave}}$ versus N.

For small N, ARED and the improved ARED decrease 𝑝_max to reduce the effect of single source on traffic so they can achieve good stability. For large N, a lot of packets need to be dropped to trigger enough congestion signals with a large drop probability. For the improved ARED, both p1 and p2 will increase accordingly, and the bounds of 𝑝_max will be enlarged as shown in (3.3). Therefore, the improved ARED can provide enough congestion signal by adapting a larger 𝑝_max. However, the upper threshold of 𝑝_max is set to 0.5 for ARED, so it is ineffective to provide a large enough drop probability with 𝑁>1080. Therefore, the improved algorithm can get better performance in the simulations.

The plots in Figure 8 show the average queue size with respect to the round-trip propagation delay. For gentle RED, larger d results in instability as shown in Figure 8(a), because it takes more time to transmit an early congestion signal. After the ends receive the congestion signals, a lot of packets have been sent out, caused more serious congestion with a long queue length. On the other hand, when the system recovers from congestion, the ends may still decrease sending rate, due to the delay arrival of congestion signals sent at the previous time. In this case, the average queue size will be very small and the network resource cannot be fully utilized. In the experiments, both ARED and the improved ARED can reduce aggressiveness of drop mechanism through decreasing 𝑝_max that makes 𝑞_ave enter into stability slowly. Figures 8(b) and 8(c) also verify that both algorithms are effective in control of oscillation and chaos.

(a) Gentle RED algorithm

(b) ARED algorithm

(c) Improved ARED algorithm

Figure 8 

Average queue size $q_{\text{ave}}$ versus d.

5. Conclusions

In this paper, we study the instability of TCP-RED based on a discrete-time feedback system model and propose an improved ARED algorithm in which bounds of the maximum drop probability 𝑝_max are optimized and the lower threshold of the exponential averaging weight w is adjusted based on linear stability condition. Simulations based on the discrete-time model show that the improved ARED algorithm can stabilize 𝑞_ave effectively in the target range with various values of control parameters (w, 𝑞_min) and system parameters (N, d). Compared with the gentle RED and ARED, the improved ARED algorithm is less sensitive to parameters, and performs better than the two schemes in terms of stability. The simulation results imply that the improved ARED can achieve high throughput and less delay.

Acknowledgments

The work was supported by National Natural Science Foundation of China (Grant no. 60573125, 60703112, and 60873264).