Master s Thesis. A Study on Active Queue Management Mechanisms for. Internet Routers: Design, Performance Analysis, and.

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1 Master s Thesis Title A Study on Active Queue Management Mechanisms for Internet Routers: Design, Performance Analysis, and Parameter Tuning Supervisor Prof. Masayuki Murata Author Tomoya Eguchi February 13th, 2004 Department of Information Networking Graduate School of Information Science and Technology Osaka University

2 Master s Thesis A Study on Active Queue Management Mechanisms for Internet Routers: Design, Performance Analysis, and Parameter Tuning Tomoya Eguchi Abstract In recent years, AQM (Active Queue Management) mechanisms which support the end-to-end congestion control mechanism of TCP (Transmission Control Protocol) have been actively studied by many researchers. AQM mechanisms control the queue length in a router (i.e., the number of packets in the buffer) by actively dropping arriving packets. Whereas AQM mechanisms can solve several problems of conventional Drop-Tail routers, it is known that AQM mechanisms have several problems. First, effectiveness of AQM mechanisms is heavily dependent on a choice of control parameters. For this reason, it has been variously investigated how a choice of control parameters affects the performance of AQM mechanisms. However, most of these simulation studies are simply based on a small number of simulation results. Another problem of AQM mechanisms is that AQM mechanisms cannot realize fairness among TCP connections in a general network topology where multiple routers exist. Although several AQM mechanisms for improving fairness among homogeneous TCP connections have been proposed so far, fairness issues in a general network with heterogeneous TCP connections and multiple routers have not been fully investigated. In this thesis, we focus on above two problems of AQM mechanisms. In the first part of this thesis, we quantitatively show how the performance of AQM mechanisms is affected by a setting of control parameters using multivariate analysis. We perform a great 1

3 number of simulation experiments by changing the combination of control parameters and network parameters. We then analyze thousands of simulation results statistically, and systematically investigate effects of control parameters and network parameters on the performance of AQM mechanisms. We focus on RED (Random Early Detection) which is a representative AQM mechanism as a target of our analysis. We also focus on GRED (Gentle RED), DRED (Dynamic-RED) and SRED which are improvements of RED. As a result, for instance, it becomes clear that the packet loss probability of the RED router is mostly determined by max th min th. In the second part of this thesis, we design an AQM mechanism, which operates in a general network with multiple routers and executes cooperative operation, for improving fairness among TCP connections. In our AQM mechanism, the ECN (Explicit Congestion Notification) mechanism is used. Our AQM mechanism dynamically changes the packet marking probability according to the CE (Congestion Experienced) bit in IP header of arriving packets. Using a simple steady state analysis, we show that fairness among heterogeneous TCP connections is improved in a tandem network. Also, using simulation experiments, we show that our AQM mechanism can improve fairness among TCP connections in various network environments. Keywords Active Queue Management Mechanism, RED (Random Early Detection), Performance Analysis, Parameter Tuning, Design of AQM mechanisms, Fairness 2

4 Contents 1 Introduction 8 2 Review on Active Queue Management Mechanisms RED (Random Early Detection) GRED (Gentle RED) DRED (Dynamic RED) SRED (Stabilized RED) Performance Evaluation of AQM Mechanisms by Multivariate Analysis Introduction Multiple Regression Analysis Simulation Model Effects of Control Parameters RED GRED DRED SRED Effects of Network Parameters RED GRED DRED SRED Conclusion Design of A New AQM Mechanisms Realizing Fairness among TCP connections 56 3

5 4.1 Introduction Related work Design Goals Algorithm Performance Evaluation Conclusion Conclusion 76 Acknowledgements 78 References 79 4

6 List of Figures 1 Simulation model Pairwise scatter plot of RED control parameters and packet loss probability without variable transformation Pairwise scatter plot of RED control parameters and packet loss probability with variable transformation Pairwise scatter plot of RED control parameters and average queue length Pairwise scatter plot of GRED control parameters and packet loss probability 31 6 Pairwise scatter plot of GRED control parameters and average queue length 33 7 Pairwise scatter plot of DRED control parameters and packet loss probability 35 8 Pairwise scatter plot of DRED control parameters and average queue length 36 9 Pairwise scatter plot of SRED control parameters and packet loss pability Pairwise scatter plot of SRED control parameters and average queue length Pairwise scatter plot of network parameters and RED packet loss probability Pairwise scatter plot of network parameters and RED average queue length Pairwise scatter plot of network parameters and GRED packet loss probability Pairwise scatter plot of network parameters and GRED average queue length Pairwise scatter plot of network parameters and DRED packet loss probability Pairwise scatter plot of network parameters and DRED average queue length Pairwise scatter plot of network parameters and SRED packet loss pability Pairwise scatter plot of network parameters and SRED average queue length Pseudo Code of Our Proposed Algorithm Analytic Model

7 List of Tables 1 Parameter values used in simulation Multiple regression analysis result for RED packet loss probability without variable transformation Multiple regression analysis for RED packet loss probability with variable transformation Multiple regression analysis result for RED average queue length Multiple regression analysis result for GRED packet loss probability Multiple regression analysis result for GRED average queue length Multiple regression analysis result for DRED packet loss probability Multiple regression analysis result for DRED average queue length Multiple regression analysis result for SRED packet loss pability Multiple linear regression analysis result for SRED average queue length Range of network parameters used in simulations Multiple regression analysis result for RED packet loss probability Multiple regression analysis result for RED average queue length Multiple regression analysis result for GRED packet loss probability Multiple regression analysis result for GRED average queue length Multiple regression analysis result for DRED packet loss probability Multiple regression analysis result for DRED average queue length Multiple regression analysis result for SRED packet loss pability Multiple linear regression analysis result for average queue length The CE bit marking probability and the TCP throughput in the steady state (B 1 = B 2 =0.2 [packet/ms] τ 1 = τ 2 = 10 [ms])

8 21 The CE bit marking probability and the TCP throughput in the steady state (B 1 = B 2 =0.4 [packet/ms] τ 1 = τ 2 = 10 [ms]) The CE bit marking probability and the TCP throughput in the steady state (B 1 = B 2 =0.2 [packet/ms] τ 1 = τ 2 = 20 [ms])

9 1 Introduction Congestion control is required to transfer packets efficiently in IP (Internet Protocol) network. IP protocol merely send packets from the source host to the destination host, but performs no congestion control. Since no congestion control is performed in IP network, if the number of packets transfered in a network increases, the performance of a network will seriously deteriorate [1]. Meanwhile, to transmit packets efficiently in the packet-switching network, the congestion avoidance mechanism of TCP (Transmission Control Protocol) was designed [2]. In the current Internet, most of the traffic is transmitted by TCP. Due to congestion control of TCP, the Internet has avoided congestion collapse. While congestion control of TCP is necessary and powerful, is not sufficient because conventional Drop-Tail routers in a network do not cooperate with congestion control of TCP. The Drop-Tail router has the following problems: First problem is the Lock Out phenomenon. Drop-Tail routers may prevent some TCP connections from transiting Drop- Tail routers. This Lock Out phenomenon is often the result of synchronization of the TCP packet sending rate. Another problem is Full Queues. Because Drop-Tail routers drop arriving packets when the buffer of a router is filled, the buffer of routers maintains full for the duration. This cause increases of the end-to-end packet delay. To solve problems of Drop-tail routers, the AQM mechanism was proposed [3]. The fundamental idea of AQM mechanisms is to drop packets before the buffer overflows. When TCP detects packet losses in the network, TCP decreases a packet sending rate. Since the congestion at a router is relieved due to this decrease of the packet sending rate of TCP, AQM mechanisms prevent the buffer overflow. In particular, AQM mechanisms control the queue length (the number of packets in a buffer) of a router by dropping arriving packets according to the queue length. AQM mechanisms have the following characteristics: 1. The total number of packet drops is decreased, 2. The end-to-end packet transfer delay is 8

10 decreased, 3. A Lock Out phenomenon is avoided. Since AQM mechanisms solve problems which conventional Drop-Tail routers have and improve the performance of a network, AQM mechanisms have been actively researched [3]. Especially, RED (Random Early Detection) [4], which is a representative AQM mechanism, have been warmly studied by many researchers. When a packet arrives at a router, the RED router calculates the average queue length. The RED router drops the arriving packet with the probability which is calculated from the average queue length and a configuration of control parameters. However, problems which RED has also became clear as researches were advanced. Various researches have been performed to solve problems on RED or other AQM mechanisms. The research trend of AQM mechanisms is simply summarized in what follows. The average queue length (the average number of packets in a buffer) of a RED router fluctuates according to the load of a network. Keeping the queue length small and suppressing the queueing delay small are objectives which AQM mechanisms should attain. However, it is known that if the number of TCP connections which transits a RED router increases, the average queue length of a RED router increases [5]. If the average queue length becomes large, the change of the packet drop probability of RED will become unstable. To solve this problem, GRED (Gentle RED) [6] is proposed. GRED gently changes the packet drop probability when the average queue length is large. DRED (Dynamic-RED) [7] is also an improvement of RED. DRED controls the packet drop probability according to the average queue length, the average queue length of DRED is not dependent on the number of TCP connections. Similar to DRED, SRED (Stabilized RED) [8] solves the RED s problem that the average queue length is dependent on the number of active TCP connections. Flow information is saved into the cache in a router, the number of active flows is estimated from saved information. SRED calculates the packet drop probability according to the number 9

11 of flows. These AQM mechanisms solve the problem which the average queue length of a RED router fluctuates according to the load of a network. The performance evaluation of AQM mechanisms. Various performance evaluations of AQM mechanisms using analytical techniques have been researched. In [9], the authors have analyzed how the number of TCP connections and the round-trip time affect the stability of an AQM mechanism using the control theory. In [10], the authors have modeled TCP and RED, and have shown that if the round-trip time and the bandwidth of a bottleneck link become large, the network which consists of TCP and RED becomes unstable. In these researches, only a very simple network is modeled and analytic techniques cannot be applied to a large scale and complicated network. Various performance evaluation by simulation experiments have also been researched. In [11], simulation experiments have been performed changing four control parameters of RED, and it has been clearly shown how control parameters affect the response time. In [12], simulations have been performed for the performance evaluation of RED in heterogeneous environments. In these simulation researches, effects of control parameters on the performance of AQM mechanisms have been analyzed from a small number of simulation result. Control parameter tuning. The designer of RED showed a guideline for a configuration of control parameters of RED [13]. However, in [5], the authors have shown that the optimal parameter configuration of RED depends on the number of TCP connections which transits a router. In [11], the authors have indicated the method of RED control parameters tuning from simulation results when the response time is regarded as the performance metric. In [14], the authors have shown effects of control parameters of RED on a throughput from some simulation results. However, in these researches, the method 10

12 of the control parameter tuning is discussed from a small number of simulation result. The method of control parameters tuning is not fully shown from results which are analyzed quantitatively or mathematically how each control parameter affects on the performance of AQM mechanisms. Unfairness among TCP connections. In a general network where multiple routers exist, the bandwidth allocation for TCP connections satisfy F h A fairness [15]. Generally, it is thought that the bandwidth allocation for TCP connections should satisfy Max-Min fairness. SRED and FRED (Fair RED) [16] which realize fairness among TCP connections by the support of an AQM mechanism were proposed. SRED stores information for every flow into cache, and changes a packet drop probability for every flow based on information. FRED prepares multiple queues for every flow and improves fairness by applying a different packet drop probability for every flow. However, AQM mechanisms for a general network where multiple routers exist is not fully examined. Although various researches on AQM mechanisms have been performed, there are some problems which have not been solved today. If the method of a control parameter tuning of AQM mechanisms is discussed from a small number of simulation result, an efficient parameter tuning cannot be performed. In the network where multiple bottleneck routers exist, unfairness among TCP connections exists. Based on problems on AQM mechanisms described above, we deal with the following problems in this thesis: We quantitatively show that effects of control parameters of AQM mechanisms on the performance. If the impact of control parameters on the performance of AQM mechanisms can be quantitatively shown, we can indicate a suitable parameter tuning. We also want to improve fairness among TCP connections in a general network where multiple routers exist. In the first part of this thesis, we quantitatively analyze effects of control parameters 11

13 of AQM mechanisms on the performance using a multivariate analysis. We perform thousands of simulations changing control parameters and network parameters. We statistically analyze simulation results using a multiple regression analysis, which is one of several multivariate analysis methods. We quantitatively show the impact of control parameters and network parameters on performance metrics (the packet loss probability and the average queue length) of AQM mechanisms. The simple guideline of control parameter tuning is shown from obtained analytic results. In the second part of this thesis, we design a new AQM mechanism which cooperates multiple routers. Fairness among TCP connections is improved by our proposed AQM mechanism. We use the ECN (Explicit Congestion Notification) mechanism [17, 18]. The packet marked probability is estimated for every flow at a router, and the packet marking probability is changed for every flow from acquired information. Using a simple steady state analysis, we show that unfairness among TCP connections resulting from the difference of the number of hops is improved by our proposed AQM mechanism. The rest of this thesis is as follows. In Section 2, we shows the outline of algorithms of AQM mechanisms which are analysis objects in this thesis. In Section 3, the performance evaluation of AQM mechanisms are performed using a multivariate analysis. In Section 4, we design a new AQM mechanism which improves unfairness among TCP connections resulting from the difference of the number of hops. We evaluate the performance of our proposed AQM mechanism by a simple steady state analysis. In Section 5, we describes the conclusion of this thesis and discuss future works. 12

14 2 Review on Active Queue Management Mechanisms 2.1 RED (Random Early Detection) RED randomly drops arriving packets with a probability, which is determined by its average queue length (i.e., the average number of packets in the router s buffer) [4]. Let q be the current queue length. When a packet arrives at the RED router, the estimated value of the average queue length, q, is updated as q (1 w q )q w q q (1) where w q is a control parameter, which is the weight of a low pass filter. RED determines the packet loss probability p b based on the average queue length q as p b = 0 if q<min th 1 if q max th (2) max p ( q min th max th min th ) if min th q<max th where min th is the minimum threshold, max th is the maximum threshold, and max p is the maximum packet loss probability. These are control parameters of RED. Finally, RED randomly drops arriving packets with the probability p a defined by p a = p b 1 count p b (3) where count is the number of packets that have arrived at the RED router since the last packet dropping. The RED router does not distinguish TCP connections, and drops all packets identically with the same packet loss probability p a. Since the packet loss probability p a is determined from RED control parameters (i.e., w q, min th, max th,and max p ), changing RED control parameters directly affects its performance metrics such as the average queue length. 13

15 2.2 GRED (Gentle RED) GRED is an improvement of RED (Random Early Detection) [6]. RED drastically changes the packet drop probability to 1.0 when the average queue length is large. Hence, when the average queue length is large, the queue length become unstable. GRED prevents the queue length from becoming unstable by gently changing the packet drop probability. In what follows, we briefly explain how GRED determines its packet drop probability p b. GRED determines the packet drop probability p b based on the average queue length q as p b = 0 if q<min th max p ( q min th max th min th ) if min th q<max th (1 max p )( q max th max th )max p if max th q<2 max th 1 if q>2 max th where min th is the minimum threshold, max th is the maximum threshold, max p is the maximum packet drop probability, and all are control parameters of GRED. 2.3 DRED (Dynamic RED) RED has a problem that the average queue length is dependent on the number of active TCP connections. DRED solves this problem by using a feedback control, which adjusts the packet drop probability in proportion to its average queue length [7]. DRED is therefore able to stabilize the queue length at the target value without being dependent on the number of TCP connections. DRED uses a fixed sampling interval, and the packet drop probability is updated every sampling interval. In what follows, we focus on a packet that arrives at the router in the n-th sampling interval. First, DRED obtains the error signal e(n) as e(n) = q(n) T 14

16 where q(n) is the current queue length and T is the target queue length. Next, the filtered error signal ê(n) is updated as ê(n) =(1 β)ê(n 1) βe(n) (4) where β is the DRED s control parameter, which specifies the weight of an exponential averaging. Finally, using ê(n), DRED determines the packet drop probability p d (n) as [ { } ] p d (n) = min max p d (n 1) αê(n) B, 0,θ (5) where B is the buffer size of the router, α is the DRED s control parameter specifying the feedback gain for the packet drop probability, and θ is the upper-bound of the packet drop probability. The packet drop probability p d (n) is updated every sampling interval, and DRED does not drop a packet if the current queue length is less than a predefined threshold L for maintaining high resource utilization. 2.4 SRED (Stabilized RED) In RED, the average queue length depends on the number of TCP connections. Moreover, RED does not distinguish misbehaving TCP flows, which will not reduce their transmission rates after packet losses. For solving these problems, SRED estimates the number of active TCP connections in a statistical manner, and determines the packet drop probability according to the estimated number of TCP connections [8]. For preventing unfairness caused by misbehaving TCP flows, SRED uses a different (i.e., large) packet drop probability for misbehaving TCP flows. For estimating the number of active TCP connections, SRED uses zombie list. The zombie list maintains information on each TCP connection. Each entry of the zombie list consists of a flow identifier, a counter, and a time stamp. When a packet arrives at the router, SRED compares a randomly chosen entry from the zombie list with the entry 15

17 corresponding to the arriving packet. If these entries coincide, the counter in the entry is incremented by one. Otherwise, the entry is probabilistically replaced by the information on the arriving packet with probability p. With the zombie list, SRED estimates the number of active TCP connections. For distinguishing misbehaving TCP flows, the zombie list is also used. See [8] for the details of SRED. We briefly explain the packet dropping algorithm of SRED. First, SRED compares a randomly chosen entry from the zombie list with the entry corresponding to the arriving packet. We focus on the n-th arriving packet. If these entries coincide, H(n) is set to one. Otherwise, H(n) is set to zero. The probability P (n) that the zombie list contains the entry for the arriving packet is estimated by P (n) = (1 α) P (n 1) αh(n) (6) where α is the SRED s control parameter, which specifies the weight of an exponential averaging. Next, in proportion to the current queue length q, the packet drop probability p sred (q) is updated for every packet arrival as p sred (q) = p max if 1 3 B q<b 1 4 p max if 1 6 B q<1 3 B 0 if 0 q< 1 6 B (7) where B is the buffer size of a router, and p max is the SRED s control parameter, which limits the upper-bound of the packet drop probability. Finally, SRED randomly drops an arriving packet with the probability p zap defined by ( p zap = p sred (q) min 1, ( 1 H(n) ) P (n) ) 1 (256 P (n)) 2 (8) 16

18 3 Performance Evaluation of AQM Mechanisms by Multivariate Analysis 3.1 Introduction For solving problems of conventional Drop Tail routers, researches on AQM (Active Queue Management) mechanisms have been actively performed in the last several years [3]. AQM mechanisms control the queue length (i.e., the number of packets in the router s buffer) by actively discarding arriving packets before the router s buffer becomes full. For instance, one of typical AQM mechanisms called RED (Random Early Detection) [4] randomly drops an arriving packet with a probability, which is calculated from four RED s control parameters. The RED router does not distinguish TCP connections; i.e., it uses the same packet dropping discipline for all TCP connections. Hence, the implementation of the RED router is simple. However, RED has several drawbacks. In particular, it is known that the effectiveness of RED is heavily dependent on a choice of control parameters [4, 16]. Since it is complicated to analyze both TCP and RED simultaneously, most researches on RED have been performed based on simulation experiments. For instance, simulation experiments are repeated by changing RED control parameters, and effects of RED control parameters on its response time are investigated in [11]. In [14], the effect of each RED control parameter on its throughput is investigated from several simulation results. However, these simulation studies are simply based on a small number of simulation results, and effects of RED control parameters on its performance have not been investigated in a systematic way. Moreover, another problem of RED that its average queue length in steady state depends on the number of active TCP connections has been reported in [4, 5]. Hence, in the literature, several variants of RED GRED (Gentle RED) [6], DRED (Dynamic RED) [7], and SRED (Stabilized RED) [8] have been proposed for solving problems of 17

19 RED. GRED is an improvement of RED by using an ad hoc approach [6]. In RED, when the average queue length becomes large, the packet drop probability is changed drastically. Hence, RED has a problem that the queue length becomes unstable when the average queue length is large. GRED solves this problem by gently changing the packet drop probability when the average queue length is large. Although extensive studies on RED have been performed by many researchers, the performance of GRED has not been fully investigated. DRED is also an improvement of RED [7]. DRED solves the RED s known problem that the average queue length in steady state is dependent on the number of active TCP connections. DRED dynamically adjusts its packet drop probability in proportion to its average queue length. So, in DRED, the average queue length is not dependent on the number of TCP connections. However, similar to RED, the performance of DRED is significantly affected by a setting of its control parameters such as α, β, T,andL [7]. For example, simulation experiments show that α and L are directly related to its queue length and packet loss probability. However, effects of those control parameters on DRED s performance (e.g., the average queue length and the packet loss probability) have not been quantitatively investigated. Similar to DRED, SRED solves the RED s problem that the average queue length is dependent on the number of active TCP connections [8]. The key idea of SRED is to estimate the number of active TCP connections using a small cache called zombie list. SRED determines its packet drop probability in proportion to the estimated number of active TCP connections. Hence, in SRED, the average queue length is almost independent of a setting of control parameters [8]. However, the performance of SRED has not been fully evaluated, and effects of SRED s control parameters on its performance have not been clarified. In this thesis, we therefore systematically analyze the performance of four AQM mech- 18

20 anisms RED, GRED, DRED, and SRED by applying the multivariate analysis for a large number of simulation results. Namely, we first run a great number of simulation experiments by changing the combination of control parameters of AQM mechanisms. We then analyze thousands of simulation results statistically. In this thesis, we investigate effects of control parameters of four AQM mechanisms on their performance by applying the multivariate analysis; i.e., one of several performance metrics (i.e., the packet loss probability and the average queue length) is chosen as a response variable, and control parameters are chosen as predictor variables. Of several multivariate analysis methods, we utilize the multiple regression analysis. We also investigate the effect of network parameters (i.e., the number of TCP connections, the bottleneck link bandwidth, and the propagation delay) on the performance of RED, GRED, DRED, and SRED using multivariate analysis. We present a few guidelines for configuring control parameters of those AQM mechanisms. The organization of this section is as follows. In Section 3.2, we show the outline of the multiple regression analysis, which is one of representative multivariate analysis methods. We then briefly explain how the multiple regression analysis is applied for evaluating performance of AQM mechanisms. In Section 3.3, we explain our simulation model and parameter setting used in our simulation experiments. In Section 3.4, we present multivariate analysis results and discuss, in particular, how control parameters of AQM mechanisms are related to their performance metrics. In Section 3.5, we disucss how network parameters are related to the performance of AQM mechanisms. Finally, in Section 3.6, we summarize this paper and discuss future works. 3.2 Multiple Regression Analysis Multivariate analysis is a technique for statistically analyzing observed data for investigating correlation among multiple factors. Multivariate analysis is capable of systematically handling a huge amount of data. In this thesis, we use a multiple regression analysis, which 19

21 is one of several multivariate analysis techniques. Using the multiple regression analysis, we can analyze effects of multiple predictor variables (i.e., affecting factors) on a response variable (i.e., an influenced factor). In what follows, we briefly explain how the multiple regression analysis is applied to performance evaluation of AQM mechanisms. The multiple regression analysis is a method of statistically analyzing effects of predictor variables on a response variable [19]. Predictor variables are parameters that should affect the response variable, and the response variable is a parameter that should be changed when predictor variables are changed. When applying the multiple regression analysis, both predictor variables and the response variable must be chosen appropriately. If these variables are chosen inappropriately, the result of the multiple regression analysis would be meaningless although it might seem to be plausible. For instance, the multiple regression analysis might indicate the correlation between predictor variables and the response variable, even though the response variable is independent of predictor variables. Thus, before performing the multiple regression analysis, it is necessary to check the correlation between these variables for choosing predictor variables and the response variable appropriately. In this thesis, we therefore use a pairwise scatter plot [20], which is a set of scatter plots for each variable pair, for checking the correlation between variables. In the multiple regression analysis, a regression equation representing the average relationship between the response variable and predictor variables is obtained from a set of measured predictor variables and the response variable. Provided that the response variable y has linearity regarding predictor variables x i, a regression equation for the response variable is given by linear combination of n predictor variables x i. y = β 0 β 1 x 1... β n x n ɛ (9) where the coefficient of a predictor variable, β i, is called a regression coefficient, which indicates how much the predictor variable x i affects the response variable y. Intheabove 20

22 equation, ɛ is called a residual. We introduce ŷ as the estimated value of the response variable from measured predictor variables and the regression equation, i.e., ŷ y ɛ. In the multiple regression analysis, by assuming that the mean of residuals is zero and that the distribution of residuals follows the normal distribution, β i is determined from m sets of measured response variable and predictor variables. Once the regression equation is obtained appropriately, effects of predictor variables on the response variable can be investigated from regression coefficients β i. In the multiple regression analysis, the smaller the residual is, the better the accuracy of the regression equation is. As an index for measuring the accuracy of the regression equation, R 2 (multiple R-squared) is widely used in the multiple regression analysis. R 2 is defined from the ratio of residual variance to response variable s variance. More specifically, R 2 is defined as R 2 = 1 mi=1 ɛ 2 i mi=1 (ŷ i ȳ) 2 = mi=1 (y i ȳ) 2 mi=1 (ŷ i ȳ) 2 (10) where ȳ is the mean of m response variables. In the above equation, y i,ŷ i,andɛ i represent y, ŷ, and ɛ calculated from ith set of measured response variables and predictor variables. R 2 takes a value between 0 and 1, and it indicates how well the response variable is explained by predictor variables. If R 2 is large, it suggests that the response variable is well explained by predictor variables. Conversely, if R 2 is small, it suggests that the response variable cannot be explained by predictor variables. In Section 3.4 and Section 3.5, we analyze effects of control parameters of AQM mechanisms and network parameters on their performance metrics using the multiple regression analysis. We choose one of performance metrics of AQM mechanisms (i.e., the packet loss probability and the average queue length) as a response variable, and either control parameters of AQM mechanism or network parameters as predictor variables. We first obtain a great number of simulation results by diversely changing either control parame- 21

23 ters of AQM mechanisms or network parameters. From simulation results, we then have a pairwise scatter plot for different response variables. The pairwise scatter plot illustrates relations between each variable pairs as a scatter plot. We next apply the multiple regression analysis to simulation results. For measuring the accuracy of the multiple regression analysis, R 2 will be used. When R 2 is close to zero, it implies that the multiple regression analysis is not successful, and that some factors other than predictor variables chosen affect the response variable. On the other hand, when R 2 is close to one, it implies that the multiple regression analysis is successful so that effects of either control parameters of AQM mechanisms or network parameters can be estimated from the regression coefficients. 3.3 Simulation Model Figure 1 shows the simulation model used in this paper, which consists of five TCP connections and two AQM routers. Both AQM routers are either RED, GRED, DRED, or SRED. In this configuration, the link between two AQM routers is the bottleneck. Unless explicitly stated, we use parameter values shown in Tab ms 2 ms 50 ms RED Router 1.5 Mbps RED Router 10 Mbps 10 Mbps Source Host Destination Host Figure 1: Simulation model. Every simulation runs for 30 seconds. We use each simulation result of the last 5 22

24 Table 1: Parameter values used in simulation. Bottleneck link bandwidth 1. 5 [Mbit/s] Propagation delay of the bottleneck link 50 [ms] Packet Size 1,000 [byte] Buffer Size 100 [packet] seconds for calculating performance metrics of the AQM mechanisms such as the average queue length, the throughput and the packet loss probability. Under these conditions, in Section 3.4, we have obtained thousands of simulation results for different RED, GRED, DRED and SRED control parameters. Moreover, in Section 3.5, we have obtained for different combinations of network parameters. Each simulation is run for 30 seconds. We use the last 5 seconds of simulation results for obtaining performance metrics of the AQM mechanisms such as the average queue length, the throughput, and the packet loss probability. 3.4 Effects of Control Parameters In this section, we present analytic results of the multiple regression analysis for RED, GRED, DRED, and SRED when their control parameters are chosen as the predictor variables RED Packet Loss Probability The pairwise scatter plot when the packet loss probability of RED is chosen as the response variable is shown in Fig. 2. This figure shows, for example, that there is almost linear relation between w q and the packet loss probability, and that the relation between max th and the packet loss probability is nonlinear. In the multiple regression analysis, it is assumed that the relation between predictor variables and the 23

25 response variable is linear. If it is nonlinear, the multiple regression analysis cannot be applied. In such a case, it is necessary to perform some variable transformation. In what follows, for demonstrating effectiveness of variable transformation, we first show the result without variable transformation. Table 2 shows the result of the multiple regression analysis when RED control parameters are chosen as predictor variables and the packet loss probability is chosen as the response variable. In this table, a column named regression coefficient shows coefficients of predictor variables of the regression equation, and a column named standardized regression coefficient shows regression coefficients normalized by standard deviations of predictor variables and the response variable. Using these standardized regression coefficients, it becomes possible to compare effects of multiple predictor variables, which originally have different distributions. A column named t-value is the result of t-test, which investigates whether the distributions of residuals is changed by removing one of predictor variables from the regression equation. A column named P-value shows the probability that the distribution of residuals is the same when one of predictor variables is removed from the regression equation. A column named R 2 shows how much influence a predictor variable has on the response variable. From Tab. 2, one can find that all P-values of predictor variables are zero. This suggests that from the multiple regression analysis, all predictor variables affect the response variable. However, R 2 is small (i.e., 0.50), which implies that the regression equation cannot predict the response variable so accurately. Hence, in what follows, we improves the accuracy of the regression equation by performing variable transformation. In general, an appropriate variable transformation method should be chosen from many transformation methods by observing how the pairwise scatter plot is affected by the chosen variable transformation method. From Fig. 2, it seems that a logarithmic transformation is appropriate. On the contrary, there is a paper [21] where the average queue length of the RED router 24

26 is analytically derived, so that we can choose a variable transformation method based on this analytic result. Namely, in the paper [21], equilibrium values of the TCP window size w and the average queue length of the RED router q in steady state are derived as w = ( ) maxth min th 3N max p (q min th ) 1 1 (11) 2 q = Nw Bτ (12) where N is the number of TCP connections, B is the bandwidth of the RED router, and τ is the two-way propagation delay. Note that the analytic model used in the paper [21] is the same with our simulation model. As can be seen from Eq. (2), the packet loss probability p b is determined by the average queue length. Moreover, by focusing on the first term of the right side of Eq. (11), it can be found that the average queue length is determined by the product of min th, max th min th,andmax p. In other words, the average queue length of the RED router is approximately expressed by the summation of logarithmically transformed variables: log min th,log(max th min th ), and log max p. In the followings, we therefore use log w q,logmin th,log(max th min th ), and log max p as predictor variables. The pairwise scatter plot in this case is shown in Fig. 3. By performing variable transformation, linear relation can be observed between, for example, the predictor variable log(max th min th ) and the packet loss probability. The result of the multiple regression analysis is shown in Tab. 3. It can be found that the value of R 2 is 0.66, which is larger than 0.50 of the previous case (see Tab. 2). From absolute values of standardized regression coefficients, one can find that the standardized regression coefficient of log(max th min th ) is the largest (i.e., 0.35). The second largest is the standardized regression coefficient of log min th (i.e., 0.09), which is much smaller than This means that the packet loss probability of the RED router is mostly determined by max th min th. On the other hand, it can be found that neither the maximum packet loss probability max p nor the weight of the low pass filter w q largely affect the packet loss probability. This is because operation of 25

27 Figure 2: Pairwise scatter plot of RED control parameters and packet loss probability without variable transformation the RED router becomes unstable when max th min th is too small [21], and many packet losses occur at the RED router regardless of the maximum packet loss probability max p. Average Queue Length We next show the analytic result when the average queue length of RED is chosen as the response variable. In what follows, because of space limitation, if variables transformed is necessary, we only show the result with logarithmically transformed predictor variables. The pairwise scatter plot when the average queue length is chosen as the response variable is shown in Fig. 4. As can be seen from this figure, there exists almost linear relation between RED control parameters and the average queue length. Table 4 shows the result of the multiple regression analysis. One can find that the value of R 2 is large (i.e., 0.88), which indicates that the accuracy of the regression equation is good. By focusing on absolute values of standardized regression coefficients, it can be found that max th is the largest (i.e., 0.84). Absolute values of max p (-0.33), 26

28 Table 2: Multiple regression analysis result for RED packet loss probability without variable transformation predictor variable regression coefficient standardized regression coefficient t-value P-value intercept w q min th max th max p R min th (0.14), and w q (-0.02) become small in this order. This suggests that, at least in our simulation configuration, the maximum threshold max th is the dominant factor. This phenomenon can be explained as follows. As can be seen from Eq. (2), RED discards all arriving packets when the average queue length is larger than max th. Thus, the average queue length of RED is mostly upper-bounded by the maximum threshold max th,which explains why max th is the dominant factor of the average queue length GRED Packet Loss Probability Figure 5 shows the pairwise scatter plot displaying the relation among GRED control parameters and the packet loss probability. Table 5 shows the result of the multiple regression analysis for the packet loss probability of GRED. One can find from absolute values of standardized regression coefficients that values of min th, max p, max th and w q become small in this order. This shows that the magnitude of effects of max p and max th on the packet loss probability is about 2/3 and 1/2 of that of min th.by 27

29 Figure 3: Pairwise scatter plot of RED control parameters and packet loss probability with variable transformation Figure 4: Pairwise scatter plot of RED control parameters and average queue length 28

30 Table 3: Multiple regression analysis for RED packet loss probability with variable transformation predictor variable regression coefficient standardized regression coefficient t-value P-value intercept log w q log min th log(max th min th ) log max p R comparing Tab. 6 and Tab. 5, one can find that absolute values of standardized regression coefficients of min th, max p, max th,andw q are almost the same. Average Queue Length Figure 6 shows the pairwise scatter plot displaying the relation among GRED control parameters and its average queue length. First, we focus on absolute values of the standardized regression coefficients. One can find that the standardized regression coefficient of min th is the largest. The values of max p, max th,andw q become small in this order. This means that effects of min th, max p,andmax th on the average queue length become small in this order. These results can be explained as follows. Since GRED does not drop a packet when the average queue length is less than min th,the minimum value of the average queue length is determined by min th. On the other hand, absolute values of standardized regression coefficients show that magnitude of effects of max p and max th on the average queue length is the half of that of min th. In addition, the value of the standardized regression coefficient of w q is very small (i.e., -0.02), and this 29

31 Table 4: Multiple regression analysis result for RED average queue length predictor variable regression coefficient standardized regression coefficient t-value P-value intercept w q min th max th max p R shows that w q hardly affects the average queue length. In general, for avoiding buffer overflow and underflow, it is desirable that the average queue length is stabilized at an appropriate value. To realize this, min th should be configured so that the buffer underflow can be prevented. Then, max p and max th should be configured so that buffer overflow can be prevented. By comparing with analytic results for RED, it can be found that in RED, max th has the largest impact on the average queue length, whereas in GRED, min th has. This is because GRED improves RED s problem that the packet drop probability becomes one when the average queue length is larger than max th. Namely, this implies that RED s simulation results or analytic results cannot be used to configure control parameters of GRED DRED Packet Loss Probability Figure 7 shows the pairwise scatter plot displaying the relation among DRED control parameters and the packet loss probability. Table 7 shows the result of the multiple regression analysis for the packet loss probability. By focusing on 30

32 wq minth maxth maxp lossrate Figure 5: Pairwise scatter plot of GRED control parameters and packet loss probability absolute values of standardized regression coefficients, one can find that the absolute value of T is the largest, and then values of α, L, andβ become small in this order. Standardized regression coefficients of α, β, and L are very small (i.e., less than 1/9 of the standardized regression coefficient of T ). Average Queue Length Figure 8 shows the pairwise scatter plot displaying the relation among DRED control parameters and the average queue length of DRED. Table 8 shows the result of the multiple regression analysis. By focusing on absolute values of standardized regression coefficients in Tab. 8, one can find that the absolute value of T is the largest, then values of α, β, andl becomes small in this order. Note that the absolute value of the standardized regression coefficient of α, β, andl is less than 1/9 of that of T. As we have explained in Section 2.3, T is the target queue length of DRED so that it should have direct impact on the average queue length. However, the correlation between T and the average queue length in Fig. 8 shows that the average queue length of DRED is not 31

33 Table 5: Multiple regression analysis result for GRED packet loss probability predictor variable regression coefficient standardized regression coefficient t-value P -value intercept w q min th max th max p R always equal to T ; i.e., the average queue length is scattered around T. On the other hand, standardized regression coefficients of α and β are small. This is because, as can be seen from Eqs. (4) and (5), α and β determine the DRED s transient characteristics, but do not affect steady state characteristics such as the average queue length. From these observations, we conclude that the control parameter T has the largest impact on DRED s performance metrics (i.e., the packet loss probability and the average queue length) whereas other control parameters have little impact. Hence, when configuring DRED s control parameters, only the control parameter T should be chosen carefully so that neither buffer overflow nor buffer underflow occurs SRED Packet Loss Probability Figure 9 shows the pairwise scatter plot displaying the relation among SRED control parameters and the packet loss probability. Table 9 shows the result of the multiple regression analysis for the packet loss probability. By focusing on absolute values of standardized regression coefficients, the absolute value of the standardized 32