Abstract. This paper discusses the shape of the RED drop function necessary to confirm the requirements

Transcription

1 On the Non-Linearity of the RED Drop Function Erich Plasser, Thomas Ziegler, Peter Reichl Telecommunications Research Center Vienna Donaucity Strasse 1, 122 Vienna, Austria plasser, ziegler, Abstract This paper discusses the shape of the RED drop function necessary to confirm the requirements for AQM design. A proper AQM design implies the avoidance of forced drops and link under-utilization as well as the applicability to a wide load range. With some examples from simulations we illustrate that the original linear RED drop function cannot cope with all of these requirements. For investigations in this paper, we use these examples to set conditions a proper drop function has to comply with. Bearing these conditions in mind, we propose two non-linear approaches for the drop function. At first, we establish a heuristic approach, where a class of functions originates, with one parameter free to define. A second approach is based on a mathematical model for the TCP window system as well as on one parameter yielded from measurements and estimations. With simulations we demonstrate that the eligible non-linear drop functions perform better in avoiding forced drops and link under-utilization and especially are applicable to a broader range of loads compared to the original linear RED function. eywords: Active Queue Management, RED, TCP, Congestion Avoidance 1. Introduction The Internet research community regards Active Queue Management (AQM) as an important mechanism to notify traffic sources about the early stages of congestion and thereby avoid the need for strong source reactions due to heavy overload [2]. The original Random Early Detection (RED) algorithm, defined in [5], can be considered to be the most popular AQM mechanism. With RED the drop probability increases linearly between two thresholds and in dependance on the average. This average is calculated as the result of the EWMA (exponentially weighted moving average) method applied to the instantaneous. The weighting factor of the EWMA method is denoted as. As a reminder the RED function is depicted in figure 1. Since the publishing of [5] significant research has been devoted to the analysis of the dynamics of RED in interaction with TCP congestion control. Several papers, [3], [4], [5], [1], [12], introduce models aimed at finding an appropriate set of the,,, parameters. The goal of a proper RED adjustment is to maintain the average between and at low oscillations and in turn help avoid forced drops (i.e. either the instantaneous exceeds the buffer size or the average exceeds ) and link under-utilization (i.e. the instantaneous decreases to zero).

2 !#" Figure 1. Original RED drop probability curve According to the literature, it turned out to be very difficult to find an appropriate parameter set to adjust RED for general deployment. The weakness of the known models is their orientation on assumed values related to the load, especially such as the round trip time ($ ), the distribution of $ and the number of flows (% ). Even if the assumptions on the input parameters fit the scenario, the applicability of the RED mechanism, adjusted according to the models, would be restricted to a small range of the assumed load values only. But in real-life deployments of RED varying and unknown load situations are typical, which basically rules out that the set of parameters found by these models is applicable to all possible scenarios. Consequently, for these models, it is impossible to find a unique set of parameters for the original RED function which satisfies realistic scenarios of varying and unknown load situations. This restriction is attributable to the linearity of the original RED function. In this paper we illustrate in simulations that a high load (which we denote as a high number of flows) requires a disproportionately higher drop probability than a low load to keep the in the same range, a requirement met only by a non-linear drop function. Based on insights, delivered by these simulations, we are able to establish necessary conditions for an appropriate non-linear drop function. First, we pursue a heuristic method and yield a class of functions, leaving one parameter to determine. Subsequently, the free parameter is defined by comparing simulations with different values of this parameter and by ruling out those values causing heavy oscillations. A second approach for the drop function is made based on a mathematical TCP window model as well as on an estimation of one parameter, yielded from simulation. With this approach we are closest to a system immanent drop function, compared to other drop functions. Additionally within this second approach, we show the possible effect of an unrealistic drop function which models a desired AQM behavior, i.e. to keep the constant independently of the load. In simulations the eligible approaches fulfill the requirements for avoiding link under-utilization and forced drops and for keeping the between and for a much broader load range than the original linear RED function. In all investigations of this paper we omit discussions about how to prevent heavy oscillations. Nevertheless, the impacts of measured oscillations are included in the resulting drop function. This paper is structured as follows: Section 2 describes the simulation topology. In section 3 the drawbacks of the original linear RED (lin-red) function are illustrated. Two examples reveal that a linear drop function is not able to cover the full load range; for that more than one parameter setting would be necessary. In section 4 we deal with the arrangement and verification of the heuristic approach. We arrange a set of criteria based on simulation results, which leads to a general model equation for the drop function with one degree of freedom, i.e. the exponent

3 of the polynomial description of the drop function. Because of this free parameter this equation describes a class of possible drop functions. As an additive novelty, we introduce in this section the calculation of the steady state in dependance on the load, which enables us to compare the expected outcome of a drop function with realistic scenarios like ns-simulation. The equation for the steady state is derived from the analytical model for the TCP system in [11]. Simulations of ftp-traffic with different issues of the function-class enables us to determine the limit of the exponent. These simulations are used as well for the determination of the / load relation &')(*'+&%-,, to complete the second approach. Furthermore the comparison of the lin-red function is illustrated with one issue of the function-class with the exponent of two. In section 5 we introduce a second approach finding a drop function, derived from the model in [11] and the queue-size / load relation from section 4. Subsequently we establish two versions for the drop function differing in the assumption of the queue-size / load relation. The first version uses the measured queue-size / load relation in the section 4. The second version is more hypothetical, it assumes a theoretically queue-size / load relation, keeping the as constant as possible independently of the load. The purpose of the second version is to show the behavior of a theoretically best AQM design. In section 6 the original linear drop function, an appropriate chosen drop function from the first approach (with degree 2), and the two versions from the second approach are compared in simulations with ftp and realistic web traffic. 2 Simulation Environment For packet simulations we use the ns-2 simulator [14] configured with the topology depicted in figure 2. The set of parameters for all simulations is kept identical (as long as not indicated otherwise). With this topology ftp and web traffic are simulated: For ftp traffic the access hosts (h1-h4) on the left side act as sources, the access hosts (h5-h8) on the right side act as sinks. For web traffic a realistic web model is deployed [1]. The left hosts (h1-h4) represent a client requesting web pages and objects in the web page. The right hosts (h5-h8) serve as server delivering the requested data. TCP data senders are of type New-Reno. The TCP packet sizes are uniformly distributed with a mean of 5 bytes and a variance of 5 bytes. The hosts have a link capacity of 1 Mbps to the routers each. Their propagation delay to their connected router r1 or r2 is distributed between 5 ms and 1 ms. The two routers r1, r2 are connected via a bottleneck link where RED AQM is performed. The bottleneck link capacity is./(1 2 Mbps (25 packets/sec) and the propagation delay on this link is (in one direction) 354(6 272 ms. The thresholds for the linear drop function are 8(9 2 packets, :(6;72<2 packets. The buffer size is =*(6;7><2 packets. The impact of the parameter is out of scope in this paper. Nevertheless, as recommended in [13] we set the weighting parameter for the averaging procedure?(/2@, quite high to affect the drop curve not too much. The RED algorithm operates in packet mode, thus the y-axis of the figures showing the queue size indicates packets. The unit of the x-axis in the figures is seconds. The plotting of the simulations starts at 2 seconds as we are not interested in the start-up behavior of the system.

4 h1 h5 h2 h3 1Mbps, 5ms 1Mbps, 8.3ms 1Mbps, 11.6ms 1Mbps, 15ms r1 1Mbps 1ms r2 1Mbps, 5ms 1Mbps, 8.3ms 1Mbps, 11.6ms 1Mbps, 15ms h6 h7 h4 h8 Figure 2. Simulation topology 3 Illustration of Problems with the Original RED Function To illustrate that the original linear RED drop function cannot be applied to a wide load range, we refer to two simulation scenarios using ftp traffic in figure 3 and figure 4, for low load and high load, respectively. Here we denote load in terms of the number of TCP flows (% ). 1 The applied parameter setting for the simulations is default as described in section 2. The number of flows % and the maximum loss probability vary for each specific simulation as depicted in the captions. The horizontal line near the bottom in the figures indicates )(A2. Figure 3 shows in both simulations the RED over time with %B(1> flows (low load scenario). For the simulation in the left graph the maximum drop probability is set to DCE(FHGI;72, a commonly adjusted value for medium-sized load ranges as described in [12]. For the low load scenario, this results in an average periodically hitting the abscissa, which stands for significant under-utilization. Recurrent under-utilization means that in most cases the variance of the queueing delay becomes large (it obviously increases with the value of ). In this case a part of the link capacity is wasted and the AQM unnecessarily drops too many packets. If the load in the left graph of figure 3 were increased, e.g. to %J(/>72, the lin-red function would work fine with the same setting (not depicted here). For simulation in the right graph, is set 4 times lower to L =HGIM72, and, as a consequence, the is kept higher and virtually no link under-utilization occurs. We conclude to set of a lin-red function at low loads very small to avoid link under-utilization. To meet generality we refer to the slope N (see figure 1) instead of, where N ( <GO&P RQS,. If we consider a general RED drop curve which does not need to be linear and N therefore does not need to be constant, the above observation can be generalized as follows: To avoid link under-utilization and under the assumption that the amplitudes of the oscillations are in a reasonable range (e.g. the allowed range being less than 25% of ), the slope N is to be adjusted such that the mean of the average over the simulation time, should be small around the of (the term denoted as 'T, stays above, i.e. N small is referred to [12]). This implies a smooth increase of the drop function. The mean of the average over the simulation time 'T approximates a quasi infinite-time average, of the, assuming a constant load and constant N. In simulations we measure 'T as the mean over the simulation time. The term 'T is also denoted as the of the steady state of the system. We learn from the left graph that in general for small the 'T has to 1 In case of ftp-like TCP traffic, usually load is defined as the number of TCP flows divided by the bandwidth-r-product.

5 U U U U 2 2 [mean packets] [mean packets] time [s] time [s] Figure 3. Under-utilization with the lin-red function and low loads VXWLYZ[]\_^a`-bdcfeg^ih5jlk?`-bnmpo q rxsutvwz[]\xy`zbdcfeghx{h5jlk?`-bnml q be significantly larger than, where under-utilization repeatedly occurs despite 'T-}~ ('T(B > packets). For comparison: the average of the right graph is 'T ( 72 packets, and under-utilization has practically ceased. 2 2 [mean packets] [mean packets] time [s] time [s] Figure \ƒy`-b{c q e ghƒdh5jlk?`-bnmpo q e 4. Forced drops with the lin-red rxs t7vwz[]\ y`zbdc qfegh {h5jlk?`-bnmb{qfe function and higher loads Left: Contrary to the preceding simulations, figure 4 shows high load scenarios. Here the number of flows is increased tenfold to %ˆ( ><2, compared to figure 3. For the left graph remains the same ( LŠE(BG<;<2 ) as in figure 3. The increased % results in a high ('T{ŠE(B <; packets) which is now close to the upper limit. Forced packet drops already occur. They may cause global synchronization (see original RED paper in [5]) and may hamper ECN [7] congestion signaling. To avoid such a behavior in the case of high load situations and with the 'T close to, a relatively high drop probability is required. Therefore in the simulation for the right graph a higher LŒ (1HG 2 is implemented. Now the arranged buffer can easily absorb the load ('TlŒ(67 ; packets). We learn from these examples that the value of in the left graphs of figure 3 and figure 4 fits a medium range of loads (say, % 272 ). Extremely low load and high load scenarios cannot be covered with one single value of (or N ) for the lin-red drop function.

6 Summarizing these simulation results, we may conclude that a small N works for low loads and a large N for high loads, justifying the assumption that a non-linear drop function with increasing slope N for increasing load might well serve a wide range of loads. The next goal is therefore to derive a drop function which satisfies the following summarized conclusions transformed into conditions: The slope of the drop function should be small close to the (at low loads) in order to avoid link under-utilization. (1) The slope of the drop function should be higher near the (at high loads) to avoid forced drops and keep 'T between and. (2) 4 Heuristic Approach for Non-linear Drop Functions 4.1 Establishing Drop Functions First we concentrate on the goal to find a drop function which meets the above condition (1) on avoiding under-utilization at low loads. For all derivations for the drop function we assume to describe the steady state of the TCP system, thus we denote the argument of the approach to be 'T. We define the desired drop probability function as a general polynomial function TE(~ :& 'T], depending on the ' T, where the coefficients are to be determined. :&' T, ( T C ' T ' T For generalization of and to implement the minimum threshold in this function, as is done in lin-red (figure 1), the argument in (3) is shifted with. The following marginal conditions are to be observed: * The initial condition /(/2 for 'T š is self-evident, leading to: :&'ptœq, lžpÿ7 D # R( :& 2<, (/2 (4) * As stated in (1), the slope of the drop probability curve at should be low at low loads. Therefore we define the slope starting with zero. {& 'TœQ, lžlÿ7p D # R( {& 2<, (/2@ (5) * The upper constraint for the drop function is given in a similar manner as for the original RED function, as the drop probability should reach a defined value at the upper threshold. This constraint is also important in the context of dimensioning the buffer size for a router. :&'TyQ, lžpÿ7 ( :& Q, (ª (6) The equations (4), (5), (6) all represent reasonable boundary point conditions for :& ' T Q«). A possible further condition would be to determine the derivative (slope) at 'ptz(*. From simulation experience we know that this would not give further advantages, as long as we cannot determine the exact quantity of the optimum slope at. Even the condition in (2) can only be used qualitatively. Furthermore, at points within the curve no statements can be made. By calculating the coefficients we obtain: (3)

7 C Š From (4) we get: T +C f :& 2<, ( 2 DT (/2 (7) From (5) we get: C D & 27, 7± C (/2 C (/2 (8) From (6) we get: & Q, Š :&P Q, (ª (9) It is clear, that with these three conditions and the polynomial approach in (3) only three coefficients are determinable. This works well for a quadratic polynomial. Polynomials with higher degrees require additional conditions to be completely determined. As there are no other conditions left, we opt for the determination of polynomials with degree higher than two, to define :& 27,n² ³E :& @i :& 27,L² µ±oc{³ (ˆ2, yielding a smooth transition at 'T (Ž and strict monotonous behavior for š 'T š. Therefore, with the abbreviation - =¹ th, the corresponding coefficient for all polynomial approaches with the degree of º is: Thus, the polynomial function results in: f _( ¼Ĩ& 'TœQ, (ª & ¹» O, (1) 'TyQ ¹» We suggest to use the known methods in e.g. [5], [6], [12] for the setting of the parameters,,. The degree of the polynomial, i.e. the exponent in (11) is to be selected as free parameter, e.g. heuristically. Thus (11) provides a class of possible functions, each with an individual exponent º. The task for an AQM designer is to find that function (exponent) which operates with best performance in router implementation. Figure 5 shows four polynomial functions with different exponents ( ¼ denoted as ' -RED) but the same values for,,. As we can see, all functions start at 'T = and increase to at 'T =. This property is caused by the normalization effect of (11), which makes the functions with different exponents easily comparable to each other. At the same time, all functions satisfy the conditions in (1) and (2), which implies the inherent avoidance of link under-utilization and forced drops. Note that the lin-red function (corresponding to ' -RED) is included within this function-class. Figure 5 allows to explain the impact of the different polynomial approaches on the AQM mechanism for the case of two specific probabilities XC and. Assuming one drop function with exponent º, we see that both probabilities XC and correspond to a lower 'T as for a drop function with the degree º. In other words, the higher the degree º, the higher the originating 'T to reach a fixed 7T. The second observation we make in this figure is that a function with a higher exponent, e.g. º½(/, has a lower slope at low loads and a higher slope at high loads than a function with a lower exponent, e.g. º¾(*;. Thus the range between C and is covered by a smaller range of queue sizes from functions with higher degrees. Vice versa, if we assume the range of applied in routers to be typically between e.g. C and, a smaller range of ' T will be necessary for a drop function with a higher degree. This is evident especially in comparison to lin-red. We know from experience in simulations with lin-red that the slope of the 'T is non-linear and decreases with the load N. Because the slope of a ¼ -function increases more than (11)

8 .6 loss probability p General curve parameters: pmax =.5 thmax = 2 packets thmin = 2 packets.2.1 q 3 RED q 5 RED [packets] Figure 5. Polynomial approaches À of degree 1, 2, 3 and 5. at the lin-red function for higher load, we conclude that the higher the degree º of the non-linear function, the higher 'T and the more constant 'T in dependence of the load. As a consequence, curves with a higher exponent tend to stabilize the resulting ' T (smaller variation and therefore less variation in queuing delay). Of course, there must be a limit for improving the stabilizing behavior by increasing the exponent, otherwise we would end up with the Drop Tail behavior, to which ¼ in (11) tends for º Á. The limit is given with the increase of the amplitudes of the oscillations. Remarkably, the difference between the ¼ s is smaller at the higher drop value than at XC. This reveals that in application the difference between two function will be more visible at lower loads. Due to the normalization effect the characteristics can only differ strongly at one end of the curve and this is at low loads. We summarize the observations of the properties of the function class for ¼ in the following with three conclusions: * All functions ¼ are a rough and simple approach for generating a drop function. All of these curves support RED AQM to avoid link under-utilization. * For application of these functions in RED AQM, the parameters 7Âp 7Âp can be adjusted with the same well-known methods (see section 1) as for lin-red. * For all load situations the resulting 'T is expected to be larger at higher degree approaches, causing higher delay. But this is negligible, as long as the is of a reasonable order, and yields a major advantage, i.e. one setting for,, fitting a broader range of flows than is the case for the lin-red function. 4.2 The Steady State Queue Size of the Heuristic Functions After established possible drop functions in a closed form, we investigate the properties of these functions if integrated in a mathematical model of the TCP window system. The model we use is introduced in [11] and coarsely reprinted in the appendix. It describes the window size and the

9 T Ö T Q % as a function of time. For the dependant variables we can find an equilibrium point (steady state) which we denote with $_T (round trip time), 7T (drop probability), ÃÄT (window size) and ' T (, which corresponds to the approximated ' T from measurements in the sections above). From the angle of steady state in the model we use the steady state loss probability 7T (see appendix) as the initial expression for the theoretical investigations: 7T ( ;R%. $ The goal is to find an expression for calculating the 'pt in dependence of the load N applied to the TCP system. Because the drop function ¼ (11) are derived for the steady state as is (12) as well, they can be equated, which gives: Inserting &$_T( 3¼4 'T( ;%. $ ; (ª 'TœQ ¹ 'fg7., from (28) and solving for 'T yields: Æ Å Æ Ç ¹ %.3¼4 'T (12) (13) (14) Note that the limiting characteristic of the function ¼ for º) Á is the well-known Drop Tail, in which case 'T (ª. This can be verified in (14) as well as anticipated in figure 5. Equation (14) has in general no analytical solution, but closed forms exist for ºS(, ºÈ(É; and º-(~Ê, whereof we show here the closed forms for º-(Ž and º-(~;. To solve equations for other exponents º in (14), numerical methods must be used. Sometimes numerical methods have disadvantages. This may be the case if e.g. in (14) a series of values of 'T in dependence of % have to be calculated. Therefore the initial values for the solution are to be set differently for each %. For this example, increasing % requires increasing initial values. If the increase of the initial value does not fit quite well the problem, the series of solution may show discontinuities. With (14) calculations of the can be done analytically, instead of simulating it, which is an important novelty of this paper. The characteristic equation for 'T from where the numerical and the analytical solution has to start is: ;R¹» &'TœQ, &.Ë3¼4 'TD, (/2 (15) To present a closed form for 'T for the original lin-red function we apply (15) with º =1 and solve for 'T, where the coefficients Ö and Ø ' TÍÌ ÎÏ <±7ÐÒÑ Ó5Ô ( Õ5Ö Å Q are given as ÕØ Å Q ; 354. (16)

10 ' Š Š 3 Ø 3 Õ Š Õ Ö ( M. ;Ú Š 4 ;<ÊH D3 with the denotation Ù ( 4. ;<ÊH ;. Š 3 4 Š Ù % ;Ù % ( Q ÝÜÝP 4l ( Û. Þ. 354D. 2<MÙ % M Š 3 4.Ù % 4 Q For presenting the closed form for 'T for the ' ; Û D354]. Q Û 3 4. Ù % MOHÙ % Œ (17) (18) -RED function we solve (14) for º =2. ' TÍÌ Ç ±7ÐÒÑ Ó¼Ô ( Q ; &.Ë354yQ, &.3¼4, Ê ¹» Eß7% ; (19) In Figure 6 we show the results from the calculations with the measurements of the s in simulations. We selected the function lin-red, ' -RED (both calculated with (16) and (19)) and -RED (which was calculated numerically) for comparison. The parameters used are described in section 2 with the exception of, which is set à(á27@ to be comparable to the example in figure 4. The load range is % The calculated ends at a of 'T)(â;<272 packets, because of 8(â;<272 packets. Theoretically calculated data values for the above ( ;72<2 packets are irrelevant and are truncated therefore. The curves obtained with simulation in the graph are denoted as ns- function (e.g. ns-lin-red). The curves for the measured data in simulations represent the steady state 'T averaged over the simulation time ns q 3 RED.27 [packets] 15 1 q 3 RED ns Rtt [sec] q 3 RED.23 5 ns N Figure 6. Left: Calculated and measured ã ä N Right: Calculated round trip time r ä The comparison of measured data and calculated data in the left graph allows to evaluate the accuracy of the TCP model. We see that the divergence between the model and the measurements increases with the number of flows. Up to about %J( ><2 flows, the deviation is linear; it increases afterwards. This load (%B(/><2 ) corresponds to a window size per flow of.$_t GH%J(67@ packets.

11 Š x ƒ Below this limit of the window size (above the corresponding load) this model lacks in accuracy. If we want to keep the model in the accuracy range for higher loads, we have to increase the bottleneck capacity to reach or surpass this window size. The reason for the inaccuracy is that TCP Slow-start, which is neglected in the model, occurs at higher loads more often than at lower loads. Thus, the real average window size and the queue size are lower than in their modeled counterparts. Consequently all of the following model-based results should be interpreted accordingly. The right graph of figure 6 shows the calculated round trip time $_T, which obviously has the same shape as the. This presentation was chosen to depict the order of magnitude of $_T at each load level to help the reader to reproduce the results here easily occurrence 1 q 3 RED [packets] Figure 7. Distribution density of lin-red, ã -RED, ã nowing the steady state 'T as a function of the load % -RED (where all other parameters are assumed to be constant) we are able to present the distribution of s for these loads as a supplementation to figure 6. Figure 7 shows the density function of the s for the original lin-red function, ' -RED and ' -RED function. We see the decrease of the standard deviation at higher exponent of the function. It is an indication for stabilizing the over all loads. At the same time, the 'T is shifting towards higher values. Now we investigate the behavior of the at different values of the exponent and in more detail. The theory implies that a higher exponent causes a higher 'pt, shown in figure 5. Subsequently we compare this theory with measurements in simulations. The graphs in figure 8 show the dependency of the on the exponent of the function ¼ and. The graphs shown here are those with an extreme setting of (lowest: =.1, highest: =1.). Simulations within this range of are not shown here, but bear resemblance to the depicted graphs. Each of the settings for is performed with the following series of the exponents: j=1., 1.5, 2., 2.5, 3., 3.5, 4., 5., 7.. The load is increased in steps of ¹:%J( > up to a high value of % (/Ê72<2. These simulations reveal distortions of a supposed shape beginning with an exponent of º8( 3. The distortion emerge due to the calculation process of to obtain 'T, caused by high oscillations. The distortions, i.e. the high oscillation amplitudes become certainly larger for curves with higher exponents. To address a practical point of view, AQM mechanisms should not be rated on the

12 @ 2 pmax=1. 2 pmax= j=7. j= j=4. j=3.5 j=3. j= j= j= j= N N Figure 8. steady state ã ä of different heuristic approaches amount of oscillations only, but if the amplitudes of oscillations exceed a critical value, jitter becomes improperly large. Because of the boundaries of the lower and upper thresholds the determination of an averaged 'T becomes senseless. Thus we can conclude to denote that curve in figure 8 as the curve with the highest exponent, which shows a yet quite smooth shape of 'T. We denote the upper limit of applicable exponents in figure 8 and equation (11) with º =3. Curves with higher exponents of this approach imply improperly high oscillations. Note that the slope at the beginning of the graphs in figure 8 increases with the exponent, because the drop probability decreases with the exponent for the same low load. The second observation we make from the figure 8 and for the function with º =3 is that can roughly approximate the / load relation 'T:(B'pT & %z, easily. In all of the variations of (including all others not shown here) we experienced that the shape for the lin-red function is approximately linear, applicable up to %É(å;<272 in figure 8 right graph. We denote the relation 'S( '+& %z, for the lin-red function with 'S( æi%, where the slope varies from æª( 27@ M (for =.1) to ær(/2@ (for =1.). The characteristic of the functions with º½}ç are to suppose only, because of the distortions and are not considered here. All other functions (with º where º need not to be integer) may be approximated with ' (éè¼ê %. The factor è will adapt the approximation to a different exponent as well to a different. The range for the exponent ºà(/ lays between è«(6> (for =.1) and è8(/> (for =1.). 4.3 Comparison of lin-red and ' -RED with ftp Flows As a final step for this heuristic approach we compare the average (from the EWMA method) of the lin-red function and the ' -RED function (being a member of the functions-class) -RED function shows the trend for all other reliable (º-š1 ) non-linear functions with. The ' higher exponent. The simulations are made with low, medium and high loads. The parameters used here equal those in the right graph of figure 3 ( (âgo 2 ). The graphs in figure 9 demonstrate the behavior of the original RED function. The graphs in figure 1 correspond to figure 9 and show the behavior of the ' -RED function with the same load. The particular advantage of non-linear functions is seen at low loads where the ' -RED function

13 ë 1 5 mean [packets] ë mean [packets] mean [packets] 15 ë time [s] time [s] time [s] Figure 9. Simulations with lin-red and loads: N=2, N=5, N= ë 1 5 mean [packets] ë mean [packets] mean [packets] 15 ë time [s] time [s] time [s] x Figure 1. Simulations with ã -RED and loads: N=2, N=5, N=15 yields significantly higher s than lin-red and stays therefore above the state of underutilization. We can also note that the difference of the ' T from one to another load is smaller for ' -RED than for lin-red. To give an example: The result of ( ' T at % =15) minus ( ' T at % =5) is smaller using the ' -RED function than at lin-red. The difference is especially smaller at low loads, because the value of the derivative of the ' -RED function is much smaller than for lin-red at these loads. For the ' -RED function we can observe that ' T varies less than for lin-red concerning the loads in this example. At higher loads, we see similarities to the behavior in ' T of lin-red. Because of the normalization for high loads there must be similar s. Concerning oscillation amplitudes the ' -RED function behaves roughly equal to but not worse than lin-red over the whole load range. 5 Alternative Approach for a Drop Function For the second approach we use the mathematical description introduced in [11], which also is used in section 4.2. From (29) and (3) we derive (2) as a system inherent drop function. T ( ;7% &.?354 ' T], (2) Note, that in (2) the drop probability depends on the load % and the ' T. To establish only, we need to find the relation of ' T ( 'pt & %z,. In section 4.2 we solved this by equating (2) with our non-linear approach. Here we take two different paths to establish the relation ' T ( ' T & %z,. Records from measurements Theoretical implication for an optimum AQM T as a function of %

14 Û Õ. 3 Õ. Š 3 Š At the end of this section we establish a virtual drop function to show how a 'T( 'T & %-, relation must look like to obtain a linear RED function. 5.1 Records from measurements If we assume the 'pt (J'T&%-, relation being linear as 'T:(Jæìß %, the drop function would be a function of the square of the 'T, (with the inverse function % ( 'TDG<æ ), using (2). But as we see in figure 8, if the drop probability would be a square of the, the relation %J( %í& 'T], would not be linear any more. Instead, if we approximate the 'T( 'T & %z, relation with 'T ( ê èîß%, this model is valid for approximations of all eligible non-linear functions (' -RED) in figure 8. A higher factor è would assume a higher drop tendency and is to compare with higher non-linearity. We denote the resulting drop function in equation (22) as the Tl -function (based on measurements). Figure 11 shows that the function in (22) rises slower, remains quite flat for low queue sizes and causes lower oscillations than the discussed ' -RED function in section 4.1. For comparison we normalized the curves ' -RED and Tl -function to Tì(~ at 'Tì( ;72<2 packets, which gives è8( M. 7Tp ( ;7' T Œ è Œ &.?354 'ptd, ( è Œ ;7' ' T (ªèïß ê % (21) T Q Ê.?354]'T 4 Q 4 &.?354 Œ Š 'T], &.3¼4 'T], (22) The function Tp is the most likely system inherent drop function for the TCP system we can find in our approach. The shape meets the conditions (1) (2) based on our experiments and stays under the limit for the exponent ºïš. 5.2 Theoretical implication for an optimum AQM In this approach for the 'Ţ ( 'T&%-, relation we aim to a constant independent from the load, imitating an optimum AQM. We model this 'Tï(*'T&%-, relation by a logarithmic function, whose slow increase emphasizes a desired small variability of the. Besides, a logarithmic function is that mathematical model nearest to the unwanted Drop Tail mechanism, but just allowing burst to be absorbed. The drop function is denoted as TlP (based on theoretical assumption): 'T( ð)ßñ ònó C T&%-, 7Tp ( Çõô ž ;ß 2 ö &.?354 'T], (23) The normalization of this curve in figure 11 (T =1 at 'T =2 packets) results in ð8(/ 7Ê7@ ;<; Conditions to result in a lin-red function Because we rely on the mathematical model of the TCP system in this section, the different approach of the 'T ( 'T&%-, relation led to two non-linear drop functions. In the same way we can ask how the 'Tà(1'T&%-, relation must look like if the result of the drop function would be linear

15 Q Å Õ Õ x Û drop probability p p m.1 p th q [packets] Figure 11. Model related drop functions compared with ã -RED (7T½(*N 'T ). We expect that the result shows the suitability of the linear function. From (2) we get: If we calculate 'T( 'T& %z, we obtain: ' T ø Å Õ ;7.3¼4 %9ø &.3¼4 'T], ê 'T (24) ù E(6 27Mß%1ßO;ß ;7. Š 3 Š 4 M % (25) The equation (25) shows that the 'T increases faster in dependency with the load % than a linear function. This characteristic is in conflict with measurements in figure 8 and does not support the AQM requirement to keep the constant. From this point of view, the linear approach is not desirable. 6 Verification of Selected Functions with ftp and Web Traffic Simulations We compare the performance of the original lin-red function, the ' -RED function of section 4.1 and the two functions 7Tp and TlP from section 5. These functions are normalized at the queue size 'TE(92<2 and the drop probability TE(127@ ;<>. The normalization corresponds to :(62@P> for lin-red, to E(~H@P2 for ' -RED, to > for the 7Tp -function, and to ðz( Ê7;7@ M for the TlP -function. The buffer size is 3 packets. The thresholds are ( ;72<2 and ( 2 packets. Subsequent simulations employ two types of traffic; everlasting ftp TCP flows and realistic web traffic. Web traffic is created according to SURGE, which is a state-of-the-art model for web traffic. The user think times are Pareto distributed, the inter-object times are Weibull distributed, the number of objects per web-page is Pareto distributed and file sizes consist of a mixed Log- Normal/Pareto distribution. For further details of this model we refer to [1]. Simulations are run for 2 seconds in case of ftp traffic and 1 seconds in case of web traffic.

16 All depicted simulations show the following four characteristics of the compared functions: The steady state 'T as a property of how the corresponding function differs from the desired property of a constant for all loads. The coefficient of variation of the instantaneous as a metric for the amplitudes of oscillations. The throughput as indication if there is a function which does not allow to pass the typical throughput due to its shape. The number of underflows as a measure of an overextended loss function and how often the hits the abscissa. A underflow is often denoted as zero event. Obviously, the number of zero events is an indicator for link under-utilization. 9 8 p m p th p m p th steady state q [packets] coefficient of variation number of flows N Figure 12. ftp-traffic: Left: ã ä number of flows N Right: coefficient of variation p m p th 18 x p m p th throughput [packets] number of underflows number of flows N number of flows N Figure 13. ftp-traffic: Left: throughput Right: underflows All of these characteristics are computed over the whole simulation time. Although forced drops are part of conditions in our conditions, we abandon to show the corresponding graphs because all

17 records for forced drops of our simulations are zero. This shows that the setting of the parameters for the drop functions was done accordingly in accordance to avoid forced drops. The figures 12 and 13 show simulations with ftp-traffic. The load is varied from very low to very high by increasing the number of ftp flows (N=2, 5, 1, 15, 2, 3, 4). The buckling shape of the steady state ' T of the logarithmic function ( TlP ) is caused from high oscillations in the corresponding zone. Hence, we see in the right graph of figure 12 that the coefficient of variation is highest for that logarithmic function. It is remarkable that in the graph for the coefficient of variation at low load the lin-red function has the highest values of all functions. This can be ascribed to the low of the lin-red function at low loads. 1 9 p m p th p m p th steady state q [packets] coefficient of variation number of websessions Figure 14. Web-traffic: Left: ã ä number of websessions Right: coefficient of variation 3 25 p m p th 6 x 15 5 p m p th throughput [packets] number of underflows number of websessions number of websessions Figure 15. Web-traffic: Left: throughput Right: underflows In the search for a general statement for the performance of the investigated drop functions we can see that differences especially in low load scenarios are exhibited. As shown in the left part of figure 12, 'T is significantly smaller for the linear function than for the others. This causes numerous zero events (see figure 13, right part) and thus severe throughput degradation, as shown

18 in the left part of figure 13. The logarithmic function and the Tl function are superior in keeping 'T high in case of low load resulting in high throughput and a small number of zero events. By rating the drop functions, the only function outstanding is the lin-red function. In all calculated parameters this function has a slightly negative performance at low loads. The 7Tp -function seems to be the best compromise. The simulation with web traffic are shown in the figures 14 and 15, where the number of web sessions is varied from 1, 1, 3, 5 to 1 sessions. Here we see less differences in performance between the investigated functions. In case of low load, 'T and the throughput mainly depend on the temporal characteristic of the Web traffic model. 7 CONCLUSION This paper investigates the proper shape of the RED drop function. Although RED was introduced several years ago, this is the first paper which emphasizes on approaches to find a proper drop function. The need for such an investigation was evident because the original linear drop function does not perform well within a wide range of loads. The drop functions supposed in this paper origin from two different approaches. The first approach uses conditions which are derived from requirements a linear drop function has to comply with at different loads. The result is a class of functions with one free parameter, determinable from simulations. The suitable functions are of the form of a power of the steady state, where the range of exponents from 2 to 3 seems to be appropriate, but the exponent of 3 represents the upper limit because of emerging oscillations. As an intermediate result and another novelty of this paper, the calculation of the as a function of the load by considering the drop function is shown. A closed form for the linear drop function as well for the squared drop function is given. By this means it is possible to determine the accuracy of the mathematical model compared with the simulated reality. The second approach to find a drop function is based on a mathematical model and the characteristic of the / load relation. The characteristic of the / load relation is established in two versions leading to two versions of drop functions of this approach. The first version measures the / load relation from simulations and gives a drop function with lower slope at low loads compared to the squared function. The second version uses an unrealistic / load relation assuming the being constant as possible independent of the load. The motivation for this version is to show the impact of a desired AQM behavior. We compared the derived functions and the original RED-function in simulations with ftp and web traffic. With ftp traffic the linear function shows disadvantages in variation, throughput and number of underflows. The realistic function of the second approach performs best. The graphs with web traffic simulation do not give enough information for a rating of the drop function. The investigation of drop functions applied to web traffic is certainly an open scientific field. Another way to use these derived non linear drop functions would be to parameterize the existing GRED function. Vice versa, if the derived function should emerge as to difficult to implement, this function could be approximated by the GRED function. The oscillating behavior of drop functions is not in the scope of this paper, but as the most drop functions arrive to an oscillating behavior at some conditions, the discussion of conditions for preventing heavy oscillations is topic of our ongoing investigations.

19 References [1] P. Barford, M.E. Crovella, Generating representative Workloads for Network and Server Performance Evaluation, Performance 98/ACM Sigmetrics 98, 1998 [2] B. Braden, D. Clark, J. Crowcroft, B. Davie, S. Deering, D. Estrin, S. Floyd, V. Jacobson, G. Minshall, C. Partridge, L. Peterson,. Ramakrishnan, S. Shenker, J. Wroclawski, L. Zhang, Recommendations on Queue Management and Congestion Avoidance in the Internet, RFC 239, [3] W. Feng, D. andlur, D. Saha, and. Shin. A Self-Configuring RED Gateway. Infocom, March [4] V. Firoiu, M. Borden, A Study of Active Queue Management for Congestion Control, Infocom 2. [5] S. Floyd, V. Jacobson, Random Early Detection Gateways for Congestion Avoidance, IEEE/ACM Transactions on Networking, August 1993, p [6] S. Floyd, Recommandations on using the gentle variant of RED, March [7] S. Floyd, TCP and Explicit Congestion Notification, ACM Computer Communication Review, V. 24 N. 5, October 1994, p [8] C. Hollot, V. Misra, D. Towsley, W. Gong, A Control Theoretic Analysis of RED, Infocom 21. [9] C. Hollot, V. Misra, D. Towsley, and W. Gong, On designing improved controllers for AQM routers supporting TCP flows, Infocom 21. [1] G. Iannaccone, C. Brandauer, T. Ziegler, C. Diot, S. Fdida, M. May, Comparison of Tail Drop and Active Queue Management Performance for bulk-data and Web-like Internet Traffic, ISCC 21, Hammameth, Tunesia, July 21. [11] V. Misra, W. Gong, D. Towsley, Fluid-based Analysis of a Network of AQM Routers Supporting TCP Flows with an Application to RED, ACM / Sigcomm, 2. [12] T. Ziegler, S. Fdida, C. Brandauer, A quantitative Model of RED with TCP Traffic, IEEE/ACM IWQoS, arlsruhe, May 21. [13] T. Ziegler, On Averaging for Active Queue Management Congestion Avoidance, ISCC 22, Italy. [14] NS Simulator Homepage,

20 æ ú æ ( Ã % ; T A Appendix: Introduction to the Analytical Model of TCP With the Stochastic Differential Equations (26), (27) in [11] describing the TCP-window system arose the ability to investigate the behavior of TCP flows operating with Active Queue Management in analytical ways. All derivations of this appendix are made in detail in [8], [9]. The TCP system is modeled as follows: where with 3¼4 Ã1&P w, ú $ï&p w, '+&P w, ú Q Ã~&P w, % $ï&, $ï&p w, ( ( TCP window size [packets] ' ( round trip time [sec]. ( propagation delay [sec] % ( probability of packetloss ü Ã~&P ûq ü5, ;7$ï&P ýq ü5, a&p ûq ü5, (26) Ã~&,aQ. (27) þ ² ³ 354 (28) ( [packets] ( link capacity [packets per sec] ( number of TCP sessions ( delay time Equation (28) denominates the round trip time RTT as the sum of the queuing delay and the propagation delay. The equilibrium state of a system is obtained by setting the derivative terms of the corresponding differential equations to zero, and by leaving the time dependency (e.g. Ã~&, Ã8T ). The remaining variables, hence describe the values for the equilibrium state. Under the assumption of 3¼4DÂH.ÂH%B( ÿò L, the equations for the equilibrium state (29) and (3) are obtained. Ã (/2 7T ( 'ì(/2 ÃÄT(.ç$ T After linearization of the system equations around the equilibrium point we obtain: Ã1&P w, ( Q ;7% $ T. '+&P w, ( æfã~&p w,aq % $_T æiã1&p w,aq (29) (3) $_T. ;% æw& ûq $ï&p w,µ, (31) $_T æi'+&, (32)