An Analytical RED Function Design Guaranteeing Stable System Behavior

An Analytical RED Function Design Guaranteeing Stable System Behavior Erich Plasser, Thomas Ziegler Telecommunications Research Center Vienna {plasser, ziegler}@ftw.at Abstract This paper introduces the procedure for an analytical derivation of the drop probability function for the Random Early Detection queue management mechanism. The procedure is based on a model of the TCP window system and the establishment for a criterion for system stability with methods of control theory. The term stability refers to the oscillation amplitude of the steady state queue size. The outcome is a ial function of the queue size, which can be approximated with a power function of the queue size. Besides the goal of achieving low oscillations for the whole load range, this derived function shows yet other advantages. There are no parameters to adjust, which extends the applicability of RED to a wide range of load situations applying the derived function. Additionally, the under-utilization at low loads, as often experienced with the original RED function, is avoided automatically. Simulations show clear evidence that the derived function outperforms the original linear RED function.. 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 thus avoid the need for strong source reactions due to heavy overload [?]. The original Random Early Detection (RED) algorithm, defined in [?], can be considered to be the most popular AQM mechanism. With RED, the drop probability p increases linearly between two thresholds th min and th max in dependence on the average queue size q. The drop probability is constrained with a maximum value p max. The average queue size is calculated from the instantaneous queue size by using an EWMA (exponentially weighted moving This work was supported by the Austrian Kplus program and the IST project Moby Dick average) filter with the weighting factor w q. Since the publishing of [?], significant research has been devoted to the analysis of the dynamics of RED in interaction with TCP congestion control. Besides [?], several papers like [?], [?], [?], [?], [?] introduce models aimed at finding an appropriate set of the th min, th max, p max, w q parameters. Modified versions of RED are GRED [?] and ARED [?]. GRED is a fragmented extension for the RED function. Two concatenated linear RED curves with different slopes are established to meet the requirements for low and heavy loads. The problem of parameter setting remains the same as in the original RED. ARED and related forms, such as discussed in [?], adapt the maximum drop probability p max according to the load. Both RED versions do not reduce the total number of parameters. The goal of a proper RED adjustment, also aimed by the former mentioned models, is to maintain the average queue size between th min and th max at low oscillations and in turn help to avoid forced drops (i.e. either the instantaneous queue size exceeds the buffer size or the average queue size exceeds th max ) and link under-utilization (i.e. the instantaneous queue size decreases to zero). 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 (R) and the number of flows (N). Even if the assumptions on the input parameters fit the scenario, the applicability of the original RED mechanism, adjusted accordingly 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 one specific set of parameters found by these models is applicable to all possible scenarios. This restriction is caused by the linearity of the original RED function. In [?] it is illustrated that a high load requires a disproportionately higher drop probability than a low load to keep the queue size in the same range, a require- 53-346/3 $7. 23 IEEE

ment met only by a non-linear drop function. Therefore (in [?]), several approaches to obtain a non-linear drop function for RED are proposed, where, similar to the original RED function, the majority of these approaches were established by heuristics. Such non-linear functions fulfill the requirement of the queue size remaining between th min and th max for a much broader load range than the original linear RED function. Note, that the shape of such a non-linear function has to be designed carefully, otherwise heavy oscillations may occur. Building on these earlier findings, we set out in this paper to find a procedure for deriving a drop function which meets first of all the requirements for the avoidance of heavy oscillations. At the same time, this function is expected to depend solely on system immanent parameters, superseding the impractical adjustment for the th min, th max, p max parameters for different loads. As a consequence this drop function should be applicable independent on the load range. Given our emphasis placed on oscillations, we consider heavy oscillations as a state of instability which is best treated by control theory. Thereby, based on a mathematical description of the TCP window system, we determine that maximum value of the drop probability which keeps the system stable at each considered load. The result is a non-linear continuous ial function of the queue size as interpolation from the calculated drop probability values in discrete form. If we compare the performance of the original RED algorithm with a single parameter set in simulations, we see that this function shows significant performance improvements in scenarios with constant load as well as with varying/unknown load and round trip time distributions. The rest of the paper is organized as follows: In section 2 the mathematical model used to describe the TCP window system is introduced. The model is linearized and transformed to a transfer function in the Laplace-domain. Thereafter, the criterion for the stability range of the whole system is explained in general. Section 3 outlines how to obtain the drop function based on the TCP window model in six steps. The drop function is then compared to a simpler function (a power function) of the queue size for the purpose of easier implementation. Section 4 is divided into a subsection describing the simulation environment and a subsection presenting the simulation results. The simulations show the response of the derived function and other RED functions to FTP traffic. 2. The Marginal Stability of the TCP System As the application for control theory requires a mathematical description of the system, we use the model in [?] and [?], formulated as stochastic differential equations. This model allows to analytically investigate the behavior of TCP flows, which operate with active queue management. Note that these equations describe only a rough model, as they neglect the Slow-start and timeout effect. where dw (t) dt = R(t) W (t) 2 dq(t) dt W (t τ) p(t τ) () R(t τ) = N W (t) C (2) R(t) R(t) = q(t) C + T p (3) W... mean TCP window size of all flows [packets], q... queue length [packets], R... round trip time [sec], C...link capacity [packets per sec], T p... propagation delay [sec], N... number of TCP sessions, p... probability of packet loss, τ... time delay at which packet loss occurs Equation () describes the increase of the window size by one packet per round trip time. Each time a packet loss is detected, the window size is halved. A packet loss happens at time (t τ) as it is indicated in the right part of (). The time shift can be seen roughly as one round trip time. Equation (2) expresses the accumulation of the queue size. Equation (3) describes the round trip time as the sum of the queuing delay and the propagation delay. For stability investigations we use the definition of BIBO stability. The system is called asymptotically stable if the magnitude of the transfer function is less than one for a phase of -π in the Bode-diagram. To perform the investigation about stability, the non-linear model in (), (2), (3) has to be linearized at the value of the equilibrium state. Under the assumption of T p,c,n = const, the equilibrium state is obtained by: p = 2 and W W 2 = CR N. The overall transfer function of the linearized TCP system is denoted by: G(s) = q p = 2 C 2N e sr ( s + R )( s + 2N R 2 C ) (4) Before applying the Bode-diagram, we have to include the role of the AQM mechanism as indicated in (). To provide better insight into the interworking of the transfer function with the AQM mechanism, figure depicts the block diagram of the modeled TCP window system. The inherent negative feedback loop can be easily recognized, which makes the system generally applicable to control theory. The queue size influences the window size and vice versa. The round block represents a P-controller, which can be denoted as the AQM drop function in this case. The AQM consist of a variable factor K in analogy to the definition of the drop probability of the original RED function, p where K is constant (p = max (th max th min) q = K q). To the left of the block diagram the optional reference input parameter q ref (for a typical P-controller) is displayed. In our work we regard the pure RED function, thus q ref =. 53-346/3 $7. 23 IEEE

qr ef p = K q p G W W G 2 q is obtained by expanding (4) with s=jω and by calculating the amount of the resulting complex term. Figure : Block diagram of the TCP window system At the critical frequency ω g in the Bode diagram the phase reaches the value of -π. The condition of the system to be marginally stable is that the magnitude of the transfer function G(s) combined with the factor K must be one at the critical frequency. As the magnitude of G(s) is determinable for a specific load, it is easy to calculate the factor K g with K g G(s) =, which we denote as the highest factor K to keep the system stable. 3. Formation of a Drop Function To begin with, we have to find the critical frequency ω g and calculate the factor K g at different loads. As we see in (4), the transfer function is dependent on the load N, thus we expect a different factor K g at each different load. By interpolating the different K g s, we are thus able to present a new drop function. For the derivation the parameters of the equilibrium state are used, denoted as q, R, W, p. In packet simulations the averaged value of the measured queue size (denoted as average queue size in RED notations) over the simulation time can be used as an approximation of the equilibrium state of the queue size q. The drop function is derived in six steps: Step : The critical frequency ω g can be achieved from the initial condition of the phase of the system without load (N=). In this situation the queue size q =, and because of (3), R =T p. The phase of the system is equated to the phase margin set to -π ( 8 o ). Equation (6) is the initial version of (5) to calculate ω g. ϕ = π = arctan( ω gr 2 C 2N ) arctan(ω gr ) (ω g R ) (5) π 2 =arctan(ω g R )+(ω g R ) (6) By transforming (6) and substituting for R =T p, we obtain ω g =.863 T p and with T p =.2 sec, ω g = 4.3 rad. Step 2: In this step the round trip time R = R (N,C,ω g ) is determined by varying the load N =-N max in discrete steps. We increase the load until p= is reached. A closed form for R could not be found because of the high order of R, not even by substitution with a rough approximation of tan(ωr ). Thus we have to determine the round trip time R numerically. Step 3: Now, all necessary values are determined to calculate the maximum value of K g = G(j ω g). Equation (7) K g = ( ω2 g R3 C 2N ) 2 ( + ω 2 g R 2 RC 2N +) 2 C 3 R 3 4N 2 (7) Step 4: The steady state queue size corresponding to the load is easily obtained by a slight transcription of (3) into q =(R T p ) C and using R from step 2. Step 5: So far we have obtained two functions in dependency of the load (q (N), K g (N)). For deployment of RED the AQM mechanism calculates the drop probability as a function of the average queue size. Instead the dependency of q and K g on the queue size is required, which is the goal of this step 5. This can be easily reached by assigning the discrete values of q (N) and K g (N), which may also be regarded as vectors, to a discrete vector K g (q ). In contrast to the original RED function, where K g is constant for all load situations, this derivation denotes K g as a non-linear function of q. Step 6: Finally, we attain the desired function p (q )by integrating K g over q. The motivation behind is that q is modeled as a harmonic signal with an amplitude K and a frequency ω. As a small variation of the queue size provokes a small variation on the drop probability, we conclude K g is the slope of the drop probability function. This is consistent with simulation experience stating that the slope of the loss probability curve is responsible for oscillations. To achieve a continuous function, we now approximate the resulting discrete function p(q ) with a ial. The approximations where the continuous function best fits the discrete function is of order three, where we do not emphasize to find the optimum approximation. The value of the drop function reaches the maximum p= at q =36. Because p= is never reached in operation the buffer size need not to be that large as 36 packets. The resulting drop function with C=25 and T p =.2 is: p = 3.96e 6 q 6 2.9e 3 q 5 +.24e q 4 (8).49e 8 q 3 +4.26e 6 q 2 8.27e 5 q +.9e 3 For an easier implementation of (8) in routers we compare the derived function with several simple non-linear approaches in [?]. These approaches have the form of q p = p max ( th max th min ) i, where the exponent i and p max are to be defined. For normalization reasons th max is fixed with the largest queue size in the derived drop function, here q =36 for p max =, th min =. Through this comparison, we see that the derived drop function with the parameters C=25, Tp=.2 is best approximated by an exponent of the queue size of i=3. This approximation simplifies implementation in routers avoiding the calculation of the ial of high order. 53-346/3 $7. 23 IEEE

In this paper we address the effects of variations of C and T p only in a few words, because they have no effect on the procedure of this derivation and because of space limits. Detailed results will be presented in subsequent research outcomes. It is a fact, that large delay times can steer a system in an instable state easier than small delay times. Therefore we chose T p as relatively high to meet the requirement of the system remaining in the stable state for a large range of occurring T p s. It is assumed that an ISP operator is able to estimate an average T p for incoming flows which can be adjusted once. For the variation of the bottleneck capacity C it is an important fact that if C is increased/decreased in the same relation as the considered load N max, the minimum window size remains the same as in this example. Accordingly, the derived shape of the drop function will be the same. Note, that also the scale of the queue size will change with the same ratio as C and N max. The approximation curve remains proportional to the cubic of the queue size. In general, the link capacity does not change in router deployment, thus the drop curve has to be calculated only once. loss probability.9.8.7.6.5.4.3.2. derived function q 3 5 5 2 25 3 35 4 queue size Figure 2: Derived drop function in comparison with the cubic function In figure 2 we see the resulting derived drop function compared with the normalized cubic RED function. The derived drop function guarantees the prevention of heavy oscillations for all considered loads, and additionally, as the factor K g is determined to be maximal (and so is p), the function pursuits keeping the queue size, and hence the delay low. Arguments that for a low queue size there is a high possibility of under-utilization can be responded by considering the shape of the curve at this queue size in figure 2. On one hand, at low load the function features a flat slope (small drop probability), meeting exactly a requirement for avoiding under-utilization as proposed in [?]. On the other hand, a high load forces a disproportional high drop probability to keep the queue size in a range compared to low load. We already mentioned this requirement in section. Another advantage of the derived drop function is its applicability for the whole load range considered in the derivation, which even does not exclude the theoretical limit of infinite load. All higher loads requiring higher loss probability as one are treated with p= anyway. 4. Verification with FTP and Web Traffic Simulations 4.. Environment For packet simulations we use the ns-2 simulator [?] configured with the topology depicted in figure 3. In our investigation we simulate FTP and Web traffic but due to constraints in space in this paper we show the results with FTP traffic only. For FTP traffic the access hosts (h -h 4 ) on the left side act as sources, the access hosts (h 5 -h 8 ) on the right side act as sinks. The access links to the routers each have a capacity of Mbps. The propagation delay to their connected router r or r 2 is distributed with a mean of ms and a variation of 5 ms at maximum. The two routers r, r 2 are connected to the bottleneck link where RED AQM is performed. The bottleneck link capacity is C= Mbps (25 packets/sec) and the propagation delay on this link is T p = ms in one direction. The thresholds at which the drop function starts and ends to work are: th min = packets, th max =36 packets. The buffer size is B=4 packets. The weighting parameter is set to w q =., as recommended in [?]. h h2 h3 h4 Mbps, 5ms Mbps, 8.3ms Mbps,.6ms Mbps, 5ms r Mbps ms Mbps, 5ms Mbps, 8.3ms Mbps,.6ms Mbps, 5ms Figure 3: Simulation topology The RED algorithm operates in packet mode. TCP data senders are of type New-Reno. The TCP packet sizes are uniformly distributed with a mean of 5 bytes between an interval of [45,55] bytes, The simulations with FTP traffic are run for 2 seconds simulation time. We have concisely verified that confidence intervals are small for all scenarios. For these simulations the load is varied from N min =2, 5,, 2, 3 to N max =4 FTP flows. r2 h5 h6 h7 h8 53-346/3 $7. 23 IEEE

4.2. Simulation Results To compare the performance of different drop functions we chose the original linear RED function to show the advantage of a non-linear function against the linear function. We took two versions of the linear function with extremal values for p max : p max =. (indicated as lin ) for the lower case, p max =. (indicated as lin ) for the higher case. Additionally we simulate the approximated cubic curve (indicated as ) to show the slight differences to the derived function in (8). Both linear RED functions and the cubic function are adjusted with th min = and th max =36. For the derived ial function (indicated as ) no parameters are necessary. 3 25 lin lin simulation period, providing a metric for link-underutilization. Underflows occur if a packet arrives and the instantaneous queue size is zero. 4. the coefficient of variation of the queue size, averaged over the whole simulation period (coeff. of var. = standard deviation / mean) ; this is a normalized measure of the amplitudes of the oscillations, we want to keep in control. underflows [packets].8.6.4.2.8.6.4 2 x 6 lin lin average queue size [packets] 2 5 5 5 5 2 25 3 35 4 Figure 4: Average queue size - FTP traffic forced drops [packets] 2.5 x 5 2.5.5 lin lin 5 5 2 25 3 35 4 Figure 5: Number of forced drops - FTP traffic We use the simulations to extract four measures:. the mean of the average queue size calculated over the whole simulation period and corresponds to q in section 2; this measure shows the characteristics of the queue size q at different loads; 2. the forced drops accumulated over the whole simulation period; forced drops are undesirable as they disable ECN [?] and may cause global synchronization, 3. the occurred underflows accumulated over the whole.2 5 5 2 25 3 35 4 Figure 6: Number of underflows - FTP traffic The figure 4 shows the mean value of the average queue size. If we consider the queue size only, an ideal AQM performs with a mean queue size to be as constant as possible over a broad load range. It can be observed that the performance of the cubic and the derived function is very similar and that both functions manage to keep the average queue size reasonably high in low load scenarios and significantly smaller than the maximum threshold for the linear functions (th max =36 packets) in the high load scenarios. In case of low load, the derived and the cubic functions have a higher q than the linear functions for the simulations. The linear function with p max =. exhibits the highest dependency on the load and converges to th max in case of high load. The figure 5 illustrates that the linear function with p max =. is the only function with forced drops. This is in line with our expectations because with p max =. the slope is low, allowing the queue size to be very high. The figure 6 shows the number of underflows. Because the linear function with p max =. forces a low queue size in general, this function show the highest number of underflows. In comparison the cubic and the derived function best in this metric. Finally, the figure 7 illustrates the coefficient of variation of the instantaneous RED queue size as a metric for the normalized amplitude of queue size oscillations. Also in this measure the cubic and the derived function show a good performance. No linear function is able to stabilize the queue for all load situations. Looking at the figure 4 we observe that the lin - function shows the highest value for q at high loads. Thus the slightly better performance of the lin -function for 53-346/3 $7. 23 IEEE

5 5 2 25 3 35 4 coefficient of variation.4.2.8.6.4.2 lin lin Figure 7: Coefficient of variations - FTP traffic high loads in figures 7 can be explained by the normalization effect using the coefficient of variation calculation. Observing the performance of the linear functions only, we see immediately that there does not exist a single linear drop function which is able to deal with all load situations. Although the linear function with p max =. features few underflows (see figure 6) and minimal oscillations (figure 7) its performance concerning forced drops in case of high load is poor (figure 5), because q is significantly the highest (figure 4) of all compared functions. On the other hand, the linear function with p max =. performs reasonably well in case of forced drops (figure 5), but performs poorly in low load scenarios at the measure of underflows (figure 6). At high loads even the coefficient of variation is relatively high (figure 7), because of the very small queue size q (figure 4) at the whole load range. The cubic and the derived functions are very similar in their behavior combining performance in all scenarios as best. 5. CONCLUSIONS In this paper we introduced an algorithm for analytically deriving a RED drop function which guarantees low oscillations for all loads. The derivation is based on applying stability investigations using a mathematical model for the TCP window system. Heavy oscillations are compared to an instable state of the system. Consequently, by means of control theory that shape of the drop function is determined to keep the system stable. The resulting function is non-linear and can be described by a ial expression. The advantage of this function lies not only in avoiding heavy oscillations but also in avoiding link under-utilization at low loads. The applicability of the derived function is independent of the load range, no parameters are to be adjusted. Compared to the original linear drop function applicability is extended by far. For implementation the shape of the derived function can be approximated with a normalized power function of the queue size. Our example with realistic system parameters gives an approximation function of the cubic of the queue size. The effort to implement the approximated cubic function is not much higher compared to the linear function. Verifications in simulation with FTP traffic show the advantages of the derived and the approximated cubic functions against the original linear shape using the measures of the mean queue size, forced drops, link-under-utilization and coefficient of variation of the queue size. As a result the derived function causes minimum dependency of the queue size on the load, and minimum link-under-utilization and forced packet drops. Rating the functions in a combination of the above measures the derived function performs best over the entire load range. Acknowledgement: We want to thank Christoph Mecklenbraeuker (ftw.) for his useful hints to handle the Matlab tool for non-linear equation solving. References [] NS Simulator Homepage: www.isi.edu/nsnam/ns/. [2] B. Braden et al. Recommendations on queue management and congestion avoidance in the internet. RFC 239, 998. [3] W. Feng, D. Kandlur, D. Saha, and K. Shin. A selfconfiguring red gateway. Infocom, 999. [4] V. Firoiu and M. Borden. A study of active queue management for congestion control. [5] V. Firoiu and M. Borden. A study of active queue management for congestion control. Infocom, 2. [6] S. Floyd. Tcp and explicit congestion notification. ACM Computer Communication Review, 24(5), October 994. [7] S. Floyd. Recommandations on using the gentle variant of red. http://www.aciri.org/floyd/red/gentle.html, March 2. [8] S. Floyd, R. Gummadi, and S. Shenker. Adaptive red: An algorithm for increasing the robustness of red s active queue management. under Submission, http://www.aciri.org/floyd/red.html, 2. [9] S. Floyd and V. Jacobson. Random early detection gateways for congestion avoidance. IEEE/ACM Transactions on Networking, pages 397 43, August 993. [] C. Hollot, V. Misra, D. Towsley, and W. Gong. A control theoretic analysis of red. IEEE Infocom, 2. [] G. Iannaccone, C. Brandauer, T. Ziegler, C. Diot, S. Fdida, and M. May. Comparison of tail drop and active queue management performance for bulk-data and web-like internet traffic. IEEE ISCC, July 2. [2] V. Misra, W. Gong, and D. Towsley. Fluid-based analysis of a network of aqm routers supporting tcp flows with an application to red. Proceedings of ACM / Sigcomm, 2. [3] E. Plasser and T. Ziegler. On the non linearity of the red drop function. ICCC, NS Simulator Homepage: www.isi.edu/nsnam/ns/, August 22. [4] T. Ziegler. On averaging for active queue management congestion avoidance. IEEE ISCC, 22. [5] T. Ziegler, S. Fdida, and C. Brandauer. A quantitative model of red with tcp traffic. IEEE/ACM IWQoS, May 2. 53-346/3 $7. 23 IEEE