This letter deals with the initialization of the loss

history structure in the TFRC (TCP-Friendly Rate Control)

mechanism. This initialization occurs after the detection of the first loss event after every slowstart phase. The loss history is crucial for the algorithm since it returns the packet loss rate estimation. This estimation is used in the TFRC equation to compute the sending rate. In this letter, we propose a new method to compute the packet loss rate which is more computationally efficient and remains as accurate as the classical commonly used method. The motivation of this work is to reduce the computation

time and formulate a unified computation scheme. This method is based on the Newton’s algorithm issued from numerical analysis of the TCP throughput equation. This proposal is evaluated analytically and the results show a significant improvement in terms of the computation time

Dairaine, Laurent

Jourjon, Guillaume

Lochin, Emmanuel

English

Open Archive Toulouse Archive Ouverte

276 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 3, MARCH 2007Optimization of TFRC Loss History InitializationGuillaume Jourjon, Emmanuel Lochin, and Laurent DairaineAbstract— This letter deals with the initialization of the losshistory structure in the TFRC (TCP-Friendly Rate Control)mechanism. This initialization occurs after the detection of thefirst loss event after every slowstart phase. The loss history iscrucial for the algorithm since it returns the packet loss rateestimation. This estimation is used in the TFRC equation tocompute the sending rate. In this letter, we propose a new methodto compute the packet loss rate which is more computationallyefficient and remains as accurate as the classical commonly usedmethod. The motivation of this work is to reduce the computationtime and formulate a unified computation scheme. This method isbased on the Newton’s algorithm issued from numerical analysisof the TCP throughput equation. This proposal is evaluatedanalytically and the results show a significant improvement interms of the computation time.Index Terms— Optimization, TFRC, transport.I. INTRODUCTIONTFRC is a congestion control mechanism for unicastflows operating in a best-effort Internet environment andcompeting with TCP traffic [1]. This mechanism has beenintegrated into DCCP [2] as one of the possible congestioncontrol method. The main characteristic of TFRC is the useof a TCP equation model which provides a much lowerthroughput variation over time than TCP. As a result, it ismore suitable for multimedia applications such as audio/videostreaming or voice over IP.In addition to round trip time and received estimatedthroughput,TFRC algorithm needs an estimation of the packetloss rate. This loss estimation is computed at the receiver sideand sent periodically to the sender where the rate control isperformed. The initial packet loss estimation is crucial as itallows the sender to compute the sending rate as describedin [1] and the receiver to correctly initialize its loss historyevents. Indeed, the subsequent estimations of the packet lossrate are based on a weighted moving average using thishistory. As a consequence, the initialization of this structurehas an impact on the sending throughput and on TFRC overallperformances (for further details see section 6 in [1]).Since TFRC equation can not be analytically solved dueto the higher exponent, the method proposed in [3] is basedon a binary search process. In every iteration, this processcomputes the solution of the initialization by halving therange of study; by the end, the result will be chosen as themiddle of the last range. This method is used by all the earlyavailable TFRC implementations such as in ns-2 or DCCPManuscript received October 19, 2006. The associate editor coordinatingthe review of this letter and approving it for publication was Prof. SamuelPierre.G. Jourjon and E. Lochin are with National ICT Australia Ltd., Alexandria,NSW 1435, Australia (email: guillaume.jourjon@nicta.com.au).L. Dairaine is with the University of Toulouse.Digital Object Identifier 10.1109/LCOMM.2007.061707.implementation. To the best of our knowledge, no efficientmethod has been formulated. In this letter, we show that thebinary search is not efficient and needs a large number ofiterations to achieve the 5% accuracy required by the TFRCRFC [1]. Then, we propose a method which computes fasterthan the binary search method.This letter is structured as follows. Section II states the prob-lem and explains our proposed method. Section III providesnumerical results and analytical analysis of them. Finally,section IV provides conclusions and perspectives.II. OPTIMIZATION OF THE LOSS RATE COMPUTATIONIn this section, we present the initialization problem andprovide the algorithm of our solution.A. Problem statementTFRC uses a TCP throughput model given by the followingequation (1).X =s(RTT ·√p·23 +RTO ·√p·278 · p · (1 + 32 · p2))(1)The sending rate (X) depends on the packet loss rate (p),the mean packet size (s) and the Round Trip Time. RTOrefers to the TCP retransmission timeout value.During the initialization phase, TFRC acts like the TCPslow start algorithm. This slow start phase can also occurduring the transfer if the RTO timeout expires [1]. This phaseis followed by a congestion avoidance phase as soon as thereceiver detects a loss. In order to compute the sending rateX , TFRC needs an estimation of the current loss event rate.This estimation is achieved using a loss history structure whichrecords the number of packets between successive loss events.From this structure, the loss rate can be computed, as describedin [1].When the slow start phase is over, the loss history hasto be initialized. The number of packets transmitted duringthe slow start phase cannot be used to estimate the lossevent rate since it does not reflect the underlying packet droprate of the connection [3]. For this reason, existing TFRCimplementations use a simple binary search process in whichthe receiver measures the receiving rate (Xmeasured) and thenstarts the estimation of the corresponding loss event rate.This estimation is performed by computing the packet lossrate that should have allowed the sender to transmit at therate Xmeasured using (1). We claim that the loss history’sinitialization can be improved with the use of a numericalanalysis based on Newton’s algorithm of the TFRC equation.1089-7798/07$25.00 © 2007 IEEEJOURJON et al.: OPTIMIZATION OF TFRC LOSS HISTORY INITIALIZATION 277B. Newton’s algorithm rate estimation methodThe binary search algorithm is well known for its easilyprogrammable properties but also for its slow convergence.Usually, numerical problems are solved using more complexalgorithms, such as the gradient method or Newton’s algorithmor primal-dual method. Because of the relatively simple equa-tion used by TFRC, we propose to set a Newton’s algorithmin order to estimate the packet loss rate. To compute thisalgorithm, during the loss history initialization phase, (1) hasto be used as a simple function of p as follows:F (p) = RTT ·√p · 23+RTO ·√p · 278· p · (1 + 32 · p2)− sXmeasured(2)Then, the problem becomes to solve F (p) = 0. (2) is apolynomial function which can be derived as follows:F ′(p) =RTT ·√232 · √p +RTO ·√278·(1.5 ·√p+112 ·p 52 ) (3)Newton’s algorithm is known to converge to the solutionin a O(n2) compared to a O(log(n)) for the binary searchalgorithm [4]. Thanks to (3), we propose using Newton’salgorithm starting with an under solution (value of p for thefirst iteration). Newton’s iterative process is performed whilethe convergence criterion is not reached and is computed asfollows:pi+1 = pi − F (pi)F ′(pi)(4)Newton’s algorithm is constrained by the existence of F (pi)and F ′(pi) for all pi ∈ [a, b], where [a, b] is the studyinterval with a = 0 and b = 1. This constraint leads us toexclude values pi ≤ 0. In our method, we claim that wehave to start the process with the value a = 10−7. Thisvalue comes from the analysis of the function F (p). Indeed,F (p) has a double concavity. This double concavity can resultin a negative value for the next iteration of the algorithm.Nevertheless, this double concavity has a stationary point forp0 = 2.84 ∗ 10−2, for all RTT , s, and Xmeasured underthe hypothesis RTO = 4RTT [1]. In an obvious way, thestationary point is no longer identical for all RTT , s, andXmeasured if RTO = 4RTT . According to this stationarypoint and the restriction p > 0 due to the square root, wehave to take a starting point inferior to p0. Thanks to theconvergence property of Newton’s algorithm [4], we knowthat it will find the result from any starting point.In order to compare the two methods, we introduce thefollowing convergence criterion:∣∣∣∣Xmeasured −XcomputedXmeasured∣∣∣∣  α (5)where Xmeasured is the rate measured by the receiver,Xcomputed is the rate computed with (1) and α is the com-putation accuracy criterion (also called stopping criterion).According to [1], this accuracy is recommended to be at leastTABLE ISUMMARY OF THE NUMBER OF ITERATIONS FOR BOTH ALGORITHMSaverage min max standard deviationBinary search 21.7379 1 29 4.1656998Newton 5.00034 3 13 0.0688248TABLE IIEXAMPLE OF α′ VALUERTT Xmeasured s α′100ms 1Mbit/s 8Kbits 0.0006 (n = 11)400ms 2Mbit/s 8Kbits 0.000015 (n = 17)5%, meaning that when Xcomputed equals to Xmeasured moreor less 5% the process stops. The 5% tolerance is introducedfor two main reasons: firstly (1) is difficult to invert, andsecondly the numerical computation cost of p [1]. The nextsection will evaluate the convergence pace of both algorithms.III. NUMERICAL RESULTS AND INTERPRETATIONA. Numerical resultsThis section presents the numerical results of our proposedloss rate computation method compared to the classical binarysearch method. We have implemented these two algorithmsin C++ and evaluated analytically the number of iterations tocompute p for given X and RTT values. All the computationshave been performed on a Pentium IV processor. The study isperformed over a large scale of bandwidth and delay values,with an RTT ranging from 1ms to 1000ms and a bandwidthranging from 1KB/s to 100MB/s.Table I shows the summary of the number of iterationsrequired to reach the 5% accuracy in this context.The results concerning the binary search are expected asthis method tends to converge with a number of iterations n[4] with:n =⌈log2(|b− a|α)⌉(6)where [a, b] is the study interval with a = 0 and b = 1, α is theconvergence criterion and x denotes the ceiling of x (i.e.,the smaller integer ≥ x). Nevertheless in (6), the convergencecriterion α is supposed to be function of the iterative parameter(in our case p). In this study the convergence criterion isfunction of Xcomputed. In order to explain the number ofiterations needed by the binary search, we have to use theequation (7).n =⌈log2(|b− a|α′)⌉(7)where α′ is the equivalent convergence criterion on p for α =0.05. For the same reason, as it is impossible to solve directlythe equation, the translation from α to α′ cannot be done witha simple function. In order to illustrate this translation, we givetwo examples as shown in Table II.Our large range of numerical experiments tends to showthat there is a correlation between RTT , Xmeasured and thenew accuracy on p. Indeed, in our experiments, the number ofiterations needed to compute the binary search increase withthe RTT and Xmeasured.278 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 3, MARCH 20070 0.5 1 1.5 2 2.5 3 3.50 1 2 3 4 5 6 7 8 9  10EfficiencyMbytes/s RTT=400msRTT=30msRTT=1msFig. 1. Comparison of binary search and Newton’s algorithm efficiency.The results concerning Newton’s algorithm are also linkedto the properties of this algorithm. Indeed, Newton’s algorithmconverges monotonically from any starting point [4]. Never-theless, there is no general inferior or superior boundary forNewton’s algorithm.Obviously, the number of iterations is linked to the com-putation time needed by both methods. However, even ifNewton’s algorithm requires a stable number of iterations,we need to verify if it is more computationally efficient. Inthe next section, we therefore compare the computation timeof both algorithms. We focus on the range where the binarysearch algorithm is efficient.B. Interpretation and discussionIn this section, we study the efficiency of Newton’s algo-rithm over the binary search algorithms in terms of compu-tation time. We define the efficiency criterion  as the ratioof the binary search algorithm computation time (tdicho) toNewton’s algorithm computation time (tNewton): =tdichotNewton(8)The results are shown in Fig. 1. When the efficiency criterionis lower than one it corresponds to a better efficiency of thebinary search algorithm.In Fig. 1, we represent the same study case as in Table I.We focus on the smaller range of bandwidth than in Table Ias this range is less favorable to Newton’s algorithm. In thisstudy range, our study shows that Newton’s algorithm is moreefficient for more than 95% of the cases, as efficient for lessthan 4% and less efficient for less than 1% of the cases.In addition, we propose to study the correlation between thenumber of iterations and the number of CPU cycles needed toreach the 5% of accuracy. In order to explore the way thesetwo characteristics interact, we proceed to an algorithmic studyof both algorithms. We present in Table III the number ofelementary operations in both algorithms and the theoreticalnumber of CPU cycles for every operation1.Usually, the division and the square root need the highestnumber of CPU cycles. We see that Newton’s algorithm1According to Intel PIV documentation.TABLE IIILIST OF THE NUMBER OF ELEMENTARY OPERATIONS(n = number of iterations)+ ∗ / sqrtBinary search 4n+ 4 8n+ 8 2n+ 2 nNewton 5n 14n 3n nNumber of CPU cycles 0.5 15 70 70TABLE IVWORST CASE STUDYIterations CPU cyclesBinary search 30 10222Newton 13 6402.5needs more elementary operations per iteration. But as shownpreviously, it also needs less iteration for a given accuracy.Next we study these two algorithms for the worst case. Forthe binary search, the worst case is when the result of thecomputation is outside the range [0, 1]2. In this case, themaximum number of iterations should be fixed a-priori as astatic variable. According to results obtained and summarizedin Table I, this variable is equal to 30. The Newton’s worstcase has been evaluated to 13 iterations by our computationscheme. In these conditions the worst case study results arepresented in Table IV.We show that the number of cycles needed in both cases islargely in favor of Newton’s algorithm.IV. CONCLUSIONIn this letter, we present an efficient method to computethe packet loss rate of the TFRC equation. We show that thismethod outperforms the 5% accuracy in terms of number ofiterations and computation time. The initial results presentedin this letter show the efficiency of the proposed method. Dueto its low computation needs, it is particularly well-suited tomobile devices with low processing power. Furthermore, dueto the low standard deviation of the number of iterations, itis well-suited to real time processing. In future work, we willevaluate the performance implications of using this mechanismin a TFRC implementation.ACKNOWLEDGMENTWe thank Jean-Luc Lamotte for his advice concerning thealgorithmic study of both algorithms. We would like to thankanonymous reviewers and Se´bastien Ardon for their valuableremarks.REFERENCES[1] M. Handley, S. Floyd, J. Padhye, and J. Widmer, “TCP friendly ratecontrol TFRC: protocol specification,” IETF, Request for Comments.[2] E. Kohler, M. Handley, and S. Floyd, “Datagram congestion controlprotocol (DCCP),” IETF, Request for Comments 4340, Mar. 2006.[3] J. Widmer, “Equation-based congestion control,” Ph.D. dissertation, Uni-versity of Mannheim, Feb. 2000.[4] W. Gautschi, Numerical Analysis: An Introduction. Boston: Birkha¨user,1997.2These rare cases have been voluntary excluded from the Table I.

Datagram congestion control protocol (DCCP),” IETF, Request for Comments 4340,

Equation-based congestion control,”

Numerical Analysis: An Introduction. Boston: Birkh¨ auser,

TCP friendly rate control TFRC: protocol speciﬁcation,” IETF, Request for Comments.

Optimization of TFRC loss history initialization

International audienceThis letter deals with the initialization of the loss history structure in the TFRC (TCP-Friendly Rate Control) mechanism. This initialization occurs after the detection of the first loss event after every slowstart phase. The loss history is crucial for the algorithm since it returns the packet loss rate estimation. This estimation is used in the TFRC equation to compute the sending rate. In this letter, we propose a new method to compute the packet loss rate which is more computationally efficient and remains as accurate as the classical commonly used method. The motivation of this work is to reduce the computation time and formulate a unified computation scheme. This method is based on the Newton's algorithm issued from numerical analysis of the TCP throughput equation. This proposal is evaluated analytically and the results show a significant improvement in terms of the computation time

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Optimization of TFRC Loss History Initialization

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Optimization of TFRC loss history initialization

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