On Buffer Limited Congestion Windo Dynamics and Packet Loss A. Fekete, G. Vattay Communication Netorks Laboratory, Eötvös University Pázmány P. sétány /A, Budapest, Hungary 7 Abstract The central result of this paper is an analytic formula describing the packet loss probability in a buffer as a function of the length of the buffer and the probability of external packet loss. This formula makes it possible to calculate the total loss along a multi-buffer, multi-link route. Also, ne types of congestion indo distributions are discovered hen the packet loss in the buffer is large. These are different from the usual Gaussian type single humped distributions and can help to develop a qualitative classification of indo distributions. TCP B Buffer 3 Link delay, D Packet size, P Loss rate, p Bandidth, C Sink Fig.. The finite buffer model. In numerical s packet size P = 5 byte and bandidth C = 56 kb/s has been used. I. INTRODUCTION Recently, a lot of study has been carried out to clarify the dynamics of TCP congestion control mechanism. The congestion indo distribution of idealized TCP has been investigated ith constant packet loss in [] and ith variable packet loss in []. Average indo size and throughput have been calculated in [3 5]. A stochastic Ito calculus has been developed in [6]. These studies considered persistent file transfers. In [7] short file transfers and the effect of slo start mechanism has been taken into account. The role of the exponential backoff mechanism in pseudo self-similar traffic has been analyzed in [8 ]. In [, ] the authors investigated a model here parallel TCP flos shared a common buffer. They supposed that the investigated common buffer as not a bottleneck buffer. The other extreme case, a severely loaded bottleneck buffer as investigated in []. While these models describe the dynamics of TCP in the presence of external packet loss quite accurately, little or no progress has been made to understand the detailed mechanism of packet loss in IP netorks. Packet loss in current netorks is generated by overloaded buffers predominantly. This is an inherent property of TCP congestion control mechanism since TCP increases its packet sending rate until packet loss occurs in one of the buffers along the route beteen the source and the destination. In this paper e take the first step to give a detailed mathematical description of the packet loss mechanism. We investigate the dynamics of TCP in the presence of a finite buffer and e derive analytic formulas for the packet loss and the congestion indo distribution. The ne formulas and distributions are validated by direct. II. THE FINITE BUFFER MODEL In this section e introduce a model for finite buffers. In our idealized model (Fig. ) one TCP injects packets into a buffer of size B [5, 6, ]. The remaining part of the netork is modeled by a fixed delay D, constant bandidth or link capacity C and random loss probability of p per packet. This probability represents the chance of link and hardare failures [3], incorrect handling of arriving packets by routers, losses and time variations due to ireless links in the path of the connection [3], and the likelihood of congestion in the buffers other than the modeled one. The buffer is large enough such that the TCPs can operate in congestion avoidance mode, but it is finite, so that packet loss can occur in it. According to the model to practically important cases ill be investigated in this paper: ) Local Area Netork (LAN) traffic, hen the bandidth delay product DC is small, only a fe packets can be out in the link and the buffer is never empty. ) Wide Area Netork (WAN) traffic, hen the bandidth delay product is large, packets are in the link and the buffer is mostly empty. From no on e ill refer to systems ith small and large bandidth delay products as LAN and WAN respectively. III. DYNAMICS OF A SINGLE TCP IN LOCAL AREA NETWORKS In this section e present an analysis of a typical LAN scenario ith a single TCP here the link delay is small and packet delay is caused mostly by buffering. The congestion indo counts the number of unacknoledged packets, and these packets can be found on the link and in the buffer. First e suppose that the link delay D is zero, and e discuss the effect of nonzero link delay later. At each packet-shift-time unit a packet is shifted from the buffer onto the link. The round trip time of a freshly sent packet ill be the time it should ait for the shifting of all previously sent packets in the system, hich is in turn measured by the congestion indo τ RTT =. Applying this equation, the congestion indo is governed by the differential equation d dτ =, () here [, B]. This equation can be solved and e obtain the time development of the congestion indo beteen losses: i (τ) = i () + τ, i [, B], ()
here i () is the congestion indo value after the i th packet loss event. Since one packet is shifted out from the buffer onto the link in each time unit, and each packet might get lost ith the same probability p, in our continuous fluid model the inter-loss time intervals (τ i ) are independent, exponentially distributed random variables ith mean /p and probability distribution p(τ i ) = p exp( pτ i ), i Æ. (3) Note the folloing important property of this distribution: Suppose that a certain length of time has been elapsed since a packet loss. Then the probability distribution of the time interval remaining until the next packet loss is still given by (3) regardless of the elapsed time. We no derive the formula hich connects consecutive indo values before losses. Let s denote by bli the indo value immediately before the i th loss event and by i( bli ) the distribution of bli. If the value of the random variable τ i is small enough, the next bli+ can be obtained from (). Hoever, if τ i is so large that the indo ould gro above the buffer size then packets ill be dropped at the buffer, and the indo ill be set to B. Accordingly, a mapping can be given hich connects the consecutive bli values: f τi ( bli ) = bli /4 + τ i if τ i < B bli /4, B if τ i B bli /4. Such a ay, the time elapsed until the next packet loss might be smaller than τ i if τ i is too large. Due to the property of the distribution (3) noted above, at the succeeding application of (4) the next τ i+ time interval can be dran from distribution (3) again. The next indo distribution i+ ( bli+ ) can no be calculated by the Perron Frobenius operator, L, of the mapping: i+ ( bli+ ) = L i( bli ) = (5) = B δ ( bli+ f τi ( bli ) ) i( bli )p(τ i )dτ i d bli here δ() is the Dirac-delta distribution and e averaged over the distribution (3). The integration in τ i can be carried out. The stationary distribution ( bl ) is the fix point of (5). The stationary distribution ( bl ) can be ritten in the form (4) ( bl ) = A(p, B)δ( bl B) + f( bl ), (6) here f : Ê Ê continuous regular function and A(p, B) is a constant. For all n Æ e define the folloing functions: f n : ] B/ n+, B/ n] Ê; f(), (7) The pieceise solution of (5) on the disjoint intervals can be ritten in the form n f n ( bl ) = Ck n (p, B)p ble p bl 4k /. (8) k= Substituting (6), (7) and (8) into (5), recursive formulas for the constants Ck n(p, B) can be derived. The initial term C (p, B) of the iteration can be expressed by A(p, B) hich can be calculated [4] self-consistently from (5). With the folloing abbreviations: c L = k=, F(x) = e x/8 e x/ and 4 k G(x) = k= F ( x/4 k) k i= 4 e obtain i A ( pb ) = c L c L F ( pb ) G( pb ). (9) The numerical value of c L is approximately.454. Note that in this equation the parameters appear only ( in the pb combination. Consequently the constant C ) k n pb also depends on the pb product only, not separately on p or B. The pb expression is the control parameter in our model. Systems in hich external packet losses and buffer sizes differ, but the pb product is the same, are similar in the sense that they can be described ith the same constants Ck n. A. The Interpretation of A ( pb ) and the Effective Loss No e present a brief explanation of the meaning of A ( pb ). If e calculate the average time elapsed beteen to packet-loss events, e obtain τ = A( pb ) p () The meaning of this simple expression becomes clearer if e recognize that / τ is the probability of packet loss per unit time p : p = τ = p A(pB ). () This equation can also be ritten as p = ( A ( pb )) p and A ( pb ) can be interpreted as the ratio of the number of packets hich lost on the link and the total amount of lost packets. Similarly, A ( pb ) is the ratio of the number of packets hich are lost in the buffer to the total number of packet losses. The possibility that this ratio can be estimated from our model is the main result of this section. This interpretation and the exact knoledge of the form of A ( pb ) allos us to treat buffer-losses as if they ere link-losses. It also makes possible to calculate the total loss along a multi-buffer, multi-link route. According to () the measured p/p expression should be equal to A ( pb ) and it should not depend on p and B separately, only on the pb product. In order to verify () e implemented our model in Netork Simulator v..b7a (ns) [5] and carried out a number of s ith different B and p parameter values in the pb parameter range. The link capacity, link delay and packet size ere fixed in the study of LAN and only buffer size B and loss probability p ere varied. During the comparison of the s and the formulas e have to take into account that in reality the system can store more packets than the actual buffer size. For example the receiver is processing one packet, and one acknoledgment packet is traversing back to the sender during the file transfer increasing the maximum number of unacknoledged packets in the system by to. In other ords, the TCP behaves as if the buffer ould be bigger than it really is.
p/p.8.6.4 A ( p(b + b) ) B = 3 B = 4 B = 5 B = 6 B = 7.5.45.4.35.3.5...5. 4 6 8 p(b + b) Fig.. Comparison of the theoretical function A p(b + b) and the p/p expression obtained from numerical s..5 3 4 5 6 7 8 (a) pb =. In order to treat this problem e assumed that e have to set the congestion indo limit to B + b and e must replace B ith B + b on our expressions: p/p = A ( p(b + b) ). () Number of s ere carried out for different buffer sizes B, and the unknon parameter b as fitted ith the help of (). It as found that b is constant and practically independent of B. The parameter b as set to the average of the estimated values obtained from s, b =.5354. Simulation results are shon in Fig., here e compared the theoretical A ( p(b + b) ) and the measured p/p. We got an almost perfect match. At the end of this subsection e sho the estimated effective loss p in the pb limit. This is the extreme bottleneck buffer case, hen packets get lost only in the buffer. It is easy to verify that p = 8 3B holds in the pb limit. This formula has been calculated in [6] for the same bottleneck scenario. This is a further indication that our calculation is correct. Since e obtained the above expression as a limit of our model, our ork can be vieed as a generalization of previous studies. So far e presented the analytic form of the distribution of before-loss values of the congestion indo. In practice the distribution of the congestion indo at an arbitrary moment is relevant. We calculated this distribution, and e obtained = f() A(pB ), (3) It can be seen that the final distribution is proportional to the regular part of the before-loss distribution, and there is no Dirac-delta distribution in (3). To verify our results e carried out s by ns. During the s at least, loss events ere recorded to obtain statistically reliable results. We varied the pb control parameter through the pb range and the buffer size B beteen 3 and 7. The actual buffer size has been shifted to B + b in accordance ith Subsection III-A. For demonstration e compare the results and our model in Fig. 3 at pb =. and 4. parameter values and at B = 7 buffer size..5.45.4.35.3.5..5..5 3 4 5 6 7 8 (b) pb = 4. Fig. 3. Comparison of results and theoretical model at buffer size B = 7. i(), i N denote pieces of the congestion indo distribution i() = f i ()/ A pb. For the interpretation of the results e have consider to minor effects: ) after a packet loss TCP does not increase its cnd instantly, but keeps it at a constant value for a small amount of time generating small plateaus in the time evolution of cnd. We supposed in the calculations that TCP increases its indo immediately. So, these plateaus ere filtered out during the preparation of the histograms in Fig. 3. ) TCP does not simply take the half of the congestion indo, but it also takes its integer part. This and the discrepancy beteen the fluid and the discrete model makes it necessary to shift the histogram to the right along the -axis by units. Keeping in mind the above empirical corrections, e obtain very impressive agreement beteen the s and the analytical calculations. B. Discussion of the Model Traditionally it is assumed that the congestion indo distributions in systems ith random packet loss are single humped Gaussian like functions []. Hoever this is not the case in general in systems here finite buffers dominate the in-
do evolution. No e are in the position to investigate the qualitatively different types of congestion indo distributions. The distribution is non-zero only on the [, B] interval, and e can classify the histograms according to the monotonity of the last segment of the pieceise distribution, = f ()/ ( A ( pb )). It can be shon that the folloing three situations are possible: pb : the last segment is monotonously increasing in this parameter range (see Fig. 3(a)). The total packet loss is dominated by the buffer in this range. < pb < 4 : the maximum of the last segment is beteen B/ and B The buffer and external losses are comparable in this case. 4 pb < + : the last segment is monotonously decreasing in this case. In this parameter range the buffer plays a limited role. The congestion indo is mostly limited by the external loss(see Fig. 3(b)). We have assumed in the previous analysis that the link delay is zero. We can easily extend the validity of the results for situations here the link delay is non-zero. In this case in addition to the buffer, N l = CD/P number of packets and acknoledgments can be found on the link here C, D and P are the link capacity, the link delay and the packet size respectively. The congestion indo limit in this situation must be set to the total number of packets B + N l in the system and the link can be treated as a part of the buffer. IV. DYNAMICS OF A SINGLE TCP IN WIDE AREA NETWORKS In this section e present the analysis of an idealized WAN scenario here buffering delay is small compared to the link delay. In an ideal WAN scenario the buffer size ould be zero. Hoever, in ns simulator the buffer must be set to a small finite number B, otherise the packets cannot pass over the buffer in the simulator. In WAN situation the round-trip time of a freshly injected packet is approximately constant, τ RTT N l, if e neglect the buffering delay, and the development of the congestion indo can be given by bli (τ i ) = bli () + N l τ i. bli [, B + N l ], (4) here /λ is the average time beteen external losses. The Perron Frobenius equation (5) can be solved, and the stationary distribution ( bl ) can be ritten in the form (6) again. No the pieceise components of the regular part of the stationary distribution, f n : ] (B + N l )/ n+, (B + N l )/ n] Ê; bl f( bl ), can be given by f n ( bl ) = n Ck n (λn l(b + N l ))λn l e λn l bl k. (6) k= With the ne definitions of c W = k=, F(X) = k e X/ e X and G(X) = k= F(X/k ) k i= get A(X) = i e c W c W F( X) G( X), (7) here X = λn l (B + N l ), and the numerical value of c W is approximately 3.467. With the help of A(λN l (B + N l )) the coefficients C n k (λn l(b + N l )) can be calculated recursively. The average time elapsed beteen to consecutive packet losses can be calculated similarly to (). The probability of packet loss per unit time λ can be expressed by the inverse of this average: λ = τ = λ A(λN l (B + N l )). (8) According to (8) the measured λ/λ should be equal to A(λN l (B + N l )). While both λ and λ are loss probabilities per unit time, their ratio is simply the ratio of the external and the total packet loss: p/p = λ/λ. The value of λ appearing in A(λN l (B + N l )) is the number of the packets lost on the link divided by the time TCP actively transferred packets. This time is the total time minus the inactive periods (plateaus in cnd) after packet losses discussed at the end of Section III- A. Simulation results are shon in Fig 4, here e compared the theoretical A(λN l (B + N l )) and the measured p/p. The measured data point fit A(λN l (B + N l )) almost perfectly. In s the link capacity, packet size and buffer size ere fixed, and only the link delay D and the external packet loss probability p ere altered. The corresponding mapping, hich connects the consecutive bli values and analogous to (4), can be given by.8 A(λN l(b + N l)) B + N l = 63 f τi ( bli ) = { bli + τi τ RTT B + N l if τ i < τ RTT (B + N l bli ), if τ i τ RTT (B + N l bli ). The difficulty of the WAN scenario is that the distribution of times beteen packet losses cannot be given by a simple formula like (3) exactly. There are periods, hen there is no queue in the buffer, and packets cannot get lost. In order to handle the model analytically, e assumed that the distribution of times beteen consecutive losses can be approximated ith an effective exponential distribution p(τ i ) = λe λτi, (5) p/p.6.4. 3 4 5 6 7 8 λn l (B + N l ) Fig. 4. Comparison of the theoretical function A(λN l (B + N l )) and the p/p expression obtained from numerical s.
Finally, the distribution of the congestion indo at an arbitrary moment as calculated, and as the result of the calculations e get = f() A(λN l (B + N l )) (9) similarly to the LAN case. As a demonstration of formula (9) e carried out WAN s here maximum N l = 6 TCP and acknoledgment packets could be on the link simultaneously and the buffer size as set to B = 3. The parameter λ has been determined as the ratio of the external losses and the active TCP time. The external loss rate as varied in the pn l (B + N l ) range, all other parameters ere fixed. The theoretical distributions and histograms obtained from ns s can be seen in Fig. 5. Our model gives a qualitatively correct description of the WAN situation..5.45.4.35.3.5..5..5.5.45.4.35.3.5..5..5 3 4 5 6 7 (a) λn l (B + N l ) =.7398 3 4 5 6 7 (b) λn l (B + N l ) = 3.4 Fig. 5. Comparison of results and theoretical model. The link could carry maximal N l = 6 number of TCP and ACK packets. i(), i N denote pieces of the congestion indo distribution i() = f i ()/ ( A(λN l (B + N l ))). V. CONCLUSIONS In this paper e investigated the TCP congestion avoidance algorithm in netorks here the finite buffer size limits the maximal achievable congestion indo size. This ork is the first attempt to treat mathematically the buffer-limited behavior of TCP. The stationary probability distribution of the congestion indo process as presented in LAN and WAN situations. The calculations ere validated by computer s and it as shoed that our analysis agrees ith the s properly. The most important development accomplished by this study is that the total loss felt by the TCP, including the buffer and the external packet loss, can be predicted from the netork parameters. The presented analytical expression, A(x), can be computed numerically ithout difficulty and the total loss can be calculated by a simple formula. It as also shon that A(x) and the coefficients that appear in the probability distributions depend only on a certain combination of the parameters. This combination is the control parameter in our model. Netorks ith the same control parameter are equivalent in the sense that the same portion of the total packet loss occur at the buffer, and the coefficients are the same in the distribution function. We hope that this model can be used as the base element of a more complex netork and it can be extended to more complicated systems. REFERENCES [] T. J. Ott, J. H. B. Kemperman, and M. Mathis, The stationary behavior of ideal TCP congestion avoidance, in Proceedings of IEEE INFOCOM 99, Ne York, 999. [] Archan Misra and Teunis Ott, The indo distribution of idealized TCP congestion avoidance ith variable packet loss, in INFOCOM 99, March 999. [3] T. V. Lakshman and Upamanyu Madho, The performance of TCP/IP for netorks ith high bandidth-delay products and random loss, IEEE/ACM Transactions on Netorking, June 997. [4] Eitan Altman, Konstantin Avrachenkov, and Chadi Barakat, Stochastic model of TCP/IP ith stationary random losses, in ACM SIGCOMM,. [5] Patric Bron, Resource sharing of TCP connections ith different round trip times, in INFOCOM,. [6] V. Misra, W. Gong, and D. Tosley, Stochastic differential equation modeling and analysis of TCP indosize behavior, in Proc. Performance 99, Istambul, Turkey, october 999. [7] N. Cardell, S. Savage, and T. Anderson, Modeling TCP latency, in IEEE Infocom, Tel-Aviv, Israel, March. [8] L. Guo, M. Crovella, and I. Matta, TCP congestion control and heavy tails, Tech. Rep. BUCS-TR--7, Computer Science Dep., Boston University,. [9] D. R. Figueiredo, B. Liu, V. Mishra, and D. Tosley, On the autocorrelation structure of TCP traffic, Tech. Rep. -55, Dep. of Computer Science, University of Massachusetts, Amherst, November. [] A. Fekete and G. Vattay, Self-similarity in bottleneck buffers, in Globecom, November. [] A. Misra, T. Ott, and J. Baras, The indo distribution of multiple TCPs ith random queues, in IEEE GLOBECOM, December 999. [] A. Fekete, G. Vattay, and A. Veres, Improving the / p la for single and parallel TCP flos, in 7th International Teletraffic Congress, December. [3] J. Bolot, End-to-end packet delay and loss behavior in the Internet, in ACM SIGCOMM, 993. [4] A. Fekete and G. Vattay, On buffer limited congestion indo dynamics and packet loss, Tech. Rep., Communication Netorks Laboratory,, http://cnl.elte.hu/staff/fekete/. [5] Ucb/lbnl/vint netork simulator ns (version ), http://mash.cs.berkeley.edu/ns/. [6] M. Mathis, J. Semke, J. Mahdavi, and T. Ott, The macroscopic behavior of the TCP congestion aviodance algorithm, Computer Communications Revie, vol. 7, pp. 3, July 997.