The Window Distribution of Idealized TCP Congestion Avoidance with Variable Packet Loss

Transcription

1 The Window Distribtion of Idealized TCP Congestion Avoidance with Variable Packet Loss Archan Misra Tenis J Ott archan@bellcore.com tjo@bellcore.com Bell Commnications Research 445 Soth Street Morristown, NJ 796 Abstract This aer analyzes the stationary behavior of the TCP congestion window erforming ideal congestion avoidance when the acket loss robability is not constant, bt varies as a fnction of the window size. By neglecting the detailed window behavior dring fast recovery, we are able to derive a Markov rocess that is then aroimated by a continos-time, continos state-sace rocess. The stationary distribtion of this rocess is analyzed and derived nmerically and then etraolated to obtain the stationary distribtion of the TCP window. This nmerical analysis enables s to redict the behavior of the TCP congestion window when interacting with a roter ort erforming Early Random Dro (or Random Early Detection) where the loss robability varies with the qee occancy. Keywords TCP, distribtion, variable, loss. I. INTRODUCTION In this aer, we resent a qantitative analysis of the stationary behavior of the evoltion of the TCP congestion window (cwnd) ([]) when the acket loss robability is variable and deends on the (instantaneos) window of the TCP connection. It can, ths, be viewed as a generalization of the analysis in ([2]) where the dro robability was assmed constant. The mathematical model abstracts TCP behavior into a continos cycle of congestion avoidance, acket loss and fast recovery. We disregard the details of fast recovery ([7]) of TCP and assme an idealized behavior, whereby a acket loss that occrs when the congestion window is n MSSs instantaneosly redces the congestion window (and the nmber of nacknowledged ackets) to d n e MSSs. The dynamics of window evoltion can then be catred by a discrete-time Markov rocess with state-deendent 2 conditional transition robabilities. Mathematically seaking, we consider the stochastic rocess (W n ) n=,wherew n stands for the congestion window jst after the n th good acknowledgement acket (one that advances the left marker of TCP s sliding window) has arrived at the sorce. By disregarding time-ots and the behavior dringfast recovery, this is a discrete-time Markov rocess with the following behavior: P fw n+ = w + w jw n = wg =, (w) (.) P fw n+ = w 2 jw n = wg = (w): (.2) where (w) is the acket loss robability when the congestion window is w.(the time inde n in the above eqations is referred to as ack time in this aer, since it increases only with the receit of acknowledgements.) Let the maimm vale of this This research was artially sorted by NSF Grant NCR , in sbcontract with the Pittsbrgh Sercomting Center. loss robability, over all vales of w, be denoted by ma (hence ma ). Or Markovian formlation holds when, given the crrent window size, acket losses are conditionallyindeendent of ast and ftre losses. As in ([2]), we will aroimate this rocess by a more amenable continos-sace continos-time rocess. In ([2]), which assmed a constant loss robability, the time ais was rescaled by a linear contraction with scale, and sace was rescaled by a linear contraction with scale, reslting in an effective rescaling given by W (t) = W b t c. (The time inde generated by the rescaling is called sbjective time in this aer.) The reslting analysis derived the well-known sqare-root behavior of TCP ([2],[9]): the average window of a ersistent TCP connection is of the order of. We shall also engage in similar rescalings in this aer. While or sace rescaling is still linear, the variable loss robability of or model reqires the time rescaling to be non-linear, as elained in Section 2. The window evoltion of this re-scaled rocess will be described by a differential eqation (between events of acket loss); the intervals between these acket loss events will be shown to be indeendent and eonentially distribted random variables. This differential eqation is then solved via nmerical analysis; the stationary distribtionof the TCP congestion window is aroimated from this continos rocess by aroriate corrections for the rescaling. We verified the accracy of or analysis by comaring or redictions with simlations which involved olar TCP versions (NewReno and Reno) and where ackets were randomly droed with a state-deendent loss robability. We also alied this model to redict the window distribtionof a ersistent TCP connection that interacts with a roter ort, which handles this single flow and erforms Early Random Dro (ERD) ([])). In an ERD ort, the dro robability is a fnction of the instantaneos bffer occancy (which we related to the instantaneos congestion window); we observed very good agreement between simlations reslts and analytical redictions. We also investigated the alicability of this model when the roter ort erformed Random Early Detection (RED) ([]), so that the loss robability is a fnction of the average qee length. Althogh or memoryless state-deendent Markovian loss model does not accrately reflect the comlicated effect of qee averaging in RED, ([]), srrisingly good redictions were obtained. The aer is organized as follows. In section 2, we model the Markov rocess characterizing the TCP behavior and aroimate its dynamics by an aroriate continos-time,

2 2 continos-valed rocess. In section 3, we derive the Kolmogorov differential eqation governing the stationary behavior of the generalized rocess and resent a nmerical soltion to this differential eqation. In section 4, we rovide nmerical eamles analyzing the window behavior of TCP with Early Random Dro and Random Early Detection qees and evalate the effectiveness of or nmerical techniqes in redicting TCP behavior. A. Related Work and Model Alicability There has been a fairly large body of literatre analyzing TCP window dynamics with constant dro robabilities. The sqareroot formla, which ignores the effects of TCP timeots and fast recovery, has been rigorosly derived in [2] and has also been searately reorted throgh simler analyses in [6] and [5] (the last blication also considers modifications to the formla reslting from losses of acknowledgement ackets). Ott [8] stated that the throght of ersistent TCP (bt not necessarily the congestion window) starts behaving like, when timeots become significant. The recent work by Padhye et al [3] rovidesa better estimate of throght (esecially at larger loss robabilities) by considering the effects of fast recovery and timeots in greater detail; they also asserted the deendence when timeots become significant. Kmar [4]) has resented an elaborate analysis of the erformance of different TCP versions by considering the fast recovery and timeot dynamics of each version in great detail. All these aers, however, assme a constant dro robability ; or aer differs from these aroaches in that it considers the case where the loss robability varies as a fnction of the window size. We wold also like to add a few words abot the range of loss robabilities over which or analysis alies. Both [3] and [4] have noted the imortance of timeots in analyzing the erformance of TCP versions like Tahoe, Reno and New Reno; or model, on the other hand, does not consider timeots at all. Or analysis is accrate for sch TCP versions only when loss robabilities are small enogh and the delay-bandwidth rodct (inclding the bffering delay) high enogh (aro. and above) to ensre that timeots are relatively rare events. The disroortionate imact of timeots on TCP erformance is de to the combined effects of coarse-grained timers and the integration of loss recovery mechanisms with congestion control in crrent TCP versions. When loss recovery is searated from adatation to congestion (as in SACK TCP), timeots begin to lay a relatively less imortant role and the range of loss robabilities over which or analysis holds increases. As newer TCP imlementations like SACK TCP and random dro mechanisms like RED become widesread in the Internet, or assmtions will become less restrictive. II. MODEL DESCRIPTION The TCP connection is assmed to send a large data file in the forward direction with the congestion window acting as the only constraint on the transmission of ackets. It is assmed that the connection never goes into timeot, that the receive or advertized window never limits the nmber of nacknowledged ackets, that data is always sent in eqal-sized segments (one MSS) and that acknowledgements are never lost. The receiver generates an acknowledgement for every received acket (we shall also etend the analysis to model the henomenon of delayed acknowledgements). Packet losses are assmed to be conditionally indeendent. The stochastic rocess described by the conditional robabilities in eqations (.) & (.2) alies to TCP only if the time inde corresonds to the arrival of good acks. This time, which we shall call ack time in the rest of the aer, is a ositive-integer valed variable that increments by whenever a good acknowledgement acket arrives at the sorce; it increases linearly with clock time only when the window size and rond tri times are both constant. Let the cmlative robability stationary distribtion for this rocess nder this ack time be F ack. A. Time and State-sace Rescaling To derive a more amenable continos-time continosvaled random rocess from the rocess described by eqations (.) & (.2), we rescale both the time and state-sace aes. This leads s to introdce the concet of sbjective time, whichis, roghly seaking, related to ack time throgh an invertible maing. For the case considered in [2], where the loss robability was a constant, the sbjective time was derived from ack time by linearly comressing the time scale by a factor, bysing the relation dt sbjective = :dt ack. When the loss robability is not constant bt state-deendent, a state-deendent (non-linear) scaling mst be sed. We now resent a generalized notion of sbjective time by considering a continos time stochastic rocess, X(t) with a state-deendent failre rate (). We can now derive another rocess Y ( ) from X(t) sch that an increase of dt in the time inde t of X(t) corresonds to an increment of (X(t))dt in the time inde of Y ( ). A realization of the rocess Y will ths assme the same state-sace vales as the corresonding realization of X bt at different instants of time. Sbjective time can also be thogh of as a history-and-state deendent rescaling of the base (ack) time inde. The imortance of the rocess Y ( ) lies in the fact that Y ( ) will now have a constant failre rate in its own notion of time (roved in Aendi I). The time inde,, of the rocess Y ( ) is then known as sbjective time in reference to the time inde t of the rocess X(t) and the two are related by the differential relation d = (X(t))dt (2.) Sbjective time can also be considered to be a variable stretching (or contraction) of the time inde. For the secific TCP rocess nder consideration, or qantized increment in sbjective time t is rovided by the maing t = (W n ) n (2.2) where t is the (real-valed) increment in sbjective time, n is the (integer-valed) increment in ack time and (W n ) is the loss robability associated with the vale of the window W n at ack time n. In other words, for a rocess defined nder this Readers familiar with Weighted Fair Qeing (WFQ) ([2]) may benefit from realizing that or sbjective time formlation is analogos to that the definition of virtal time in WFQ; both attemt a state-deendent rescaling of time so that the rocess of interest has an invariant behavior in the new time scale.

3 3 sbjective time, time advances at a variable rate, as an increase in the ack time inde of corresonds to a state-deendent increase of (W n ) in the sbjective time inde. Ths, t(n ), the sbjective time immediately P after sending acket nmber N, is N eressed as t(n )= i= (W i). As (W n ), t is a real-valed seqence obtained by a contraction of the ack time inde. As ma #, the limiting sbjective time inde becomes a continosvariable. We shall see that, for this secific case, the rocess defined in sbjective time has a failre rate that becomes Poisson and constant only asymtotically, as ma #. If W (t) reresents the rocess W n in sbjective time t via the transformation in eqation (2.2), its samle ath between the events of acket failre can be modeled by the difference eqation W = t (W )W (2.3) As ma #, the difference eqation can be modeled by a corresonding differential eqation with increasing accracy. The differential eqation wold however, in the limit, be ill-behaved as the derivative goes to as ma #. To obtain a well behaved rocess, we also need to rescale the state sace of W (t). To rescale roerly, we assme that (W) > 8W 2,(i.e., ma the ratio between the minimm and maimm loss robabilities is niformly bonded away from ). If we then rescale the state-sace of the rocess W (t) by the mltilicative constant ma, the reslting rocess, which we call W (t), obeys the fnctional relationshi W (t) = ma W n (2.4) where n = n(t) = arg ma j : jx i= (W i ) t This continos-time and continos valed rocess W (t) will be the sbject of or stdy and analysis for the rest of the aer. Eqation (2.2) imlies that aloss robability of zero ((W n )= ) reslts in an increase in ack time bt no increase in sbjective time. Sbjective time ths loses information abot the rocess behavior dring those ack times when the system evolves deterministically withot loss; the maing in eqation (2.2) is non-invertible if (W n ) is. Later in the aer, we shall see how to correct the stationary distribtion of the rocess for ortions of the state-sace where the loss robability is ; for the time being, assme that (W n ) 6= in the region of interest. Proosition We see that as ma #, the rocess defined by eqations (.),(.2) & (2.4), converges (ath-wise) to a rocess whose window, W (t), behaves as follows: There is a Poisson rocess with intensity,the oints of which are denoted by ( n ) n=. In between the oints of this Poisson 2 The above reqirement may, in several cases, be more stringent than ractically necessary. For eamle, when the variation of loss robability with window size is very gradal, the blk of the distribtion mass will often lie arond some small vale of, say. We can then se as or sace-rescaling factor; for the rescaled rocess to be well-behaved, we then merely need (W ) to be bonded away from. rocess, the window, W, evolves according to the eqation dw dt ma = W ( ma )W (2.5) At the oints of the realization of the Poisson rocess, we have W ( + )= 2 W (, ). Proof: The roof of the differential eqation describing the window evoltion between failre events is trivial and obtained by taking aroriate limits in eqations (.), (2.2) & (2.3). The relationshi at an instant of failre also follows easily from eqations (.2) and (2.3). Note that the derivative in eqation (2.5) is always well-defined by virte of or assmtion that (W N ) > for the interval nder consideration. The roof that the instants of failre become a realization of a Poisson rocess of intensity is rovided in Aendi I. It consists of showing that as ma #, the nmber of acket transmission events in any finite interval T becomes infinitely large and the robability of loss of each transmitted acket is sch that Prob( no loss in an interval T )=e,t. The rocess defined in Proosition is an aroimation of the re-scaled TCP rocess; the aroimation becomes asymtotically accrate as the loss robabilities become smaller. For a given loss robability fnction (W ), we analyze the rescaled TCP rocess by assming that it ehibits the behavior of the limiting rocess otlined by Proosition. In other words, even for a finite loss robability, we assme that W (t) is described by the differential eqation (2.5), with an i.i.d and eonential distribtion of times between acket dros. We can ths eect the nmerical analysis otlined later to redict TCP window behavior more accrately as ma becomes smaller. B. Distribtion in (Continos) Ack Time We shall see how to comte F sbj (w), the stationary cmlative distribtion of W (t) in sbjective time, later in section III. We now consider how to correct this distribtion for the state-sace and time rescalings, introdced in eqation (2.4), assming F sbj (w) is already known. The state-sace scaling reslts in a simle linear transformation of the robability distribtion. F sbj (w) is corrected first to obtain F s (w), the cmlative stationary distribtion in sbjective time bt withot sace-rescaling by the relationshi F s (w) =F sbj ( ma w). Or desired distribtionf ack (w) can then be obtained by noting that the state-deendent rescaling of sbjective time (in eqation (2.2)) introdces a samling non-niformity in the rocess W (t). To see this non-niformity, note that an acknowledgement arrivingwhen the windowis w occies an interval of in ack time bt corresonds to an interval (w) in sbjective time: a niformly distribted samling on the sbjective time ais corresonds to a non-niform samling (with non-niformity roortionalto(w)) in the ack time frame. The samling non-niformityde to time-scaling is corrected, to obtain F ack (w), by dividing the robability density in sbjective time, df s (w), by the aroriate qantity (w). Thisis

4 4 achieved by the transformation df ack (w) = dfs(w) (w) R dfs() () (2.6) C. A Generalized Process The analysis sed to derive the stationary distribtionof W (t) is alicable to a more general class of rocesses. For eamle, any arbitrary rocess with a state-deendent failre rate can be redced to a rocess with a constant Poisson failre rate by moving to an aroriate sbjective time. Ths, we do not lose generality by considering only rocesses with constant failre rates. Consider a general rocess W (t), described by the differential eqation dw = (2.7) dt q(w ) in between the instants of failreof a Poisson rocess with rate ; let q be a well-behaved fnction (finitely many discontinities) sch that q(w) > 8 w. At the instants of failre of the Poisson rocess, the rocess evoltion is given by W (t + )=A(W (t, )), where A(w) : [; )! [; ) is a strictly increasing fnction of w sch that A(w) <w;8w >;A() =. Since A is strictly increasing, it has an inverse fnction a(w), sch that a(a(w)) = w and a(w) >w;8w >. The analysis resented later can be easily etended to consider this whole class of rocesses. For the TCP-secific case at hand, we have A(w) = 2 w (so that a() =2), the intensity of the Poisson rocess is and the rate fnction q(w ( W ma )W )=. ma In the net section, we shall formlate and solve the Kolmogorov eqation for this generalized rocess. Or nmerical eamles will, however, solely deal with the TCP-secific rocess for simlicity. III. THE STATIONARY KOLMOGOROV EQUATION AND ITS SOLUTION In this section we obtain the stationary distribtion of the rocess, defined in section II, whose behavior is described by the eqation dw(t) dt = q(w(t)) in between the oints of a Poisson rocess of rate. At the oints of the Poisson rocess, W (t) is obtained by W (t + ) = A(W (t, )); leta() be the inverse fnction of A(). Proosition 2 The stationary cmlative distribtion F sbj () of the rocess in section II.C satisfies the differential eqation df sbj () d = q()(f sbj (a()), F sbj ()) (3.) Proof: If F sbj (; t) is the cmlative distribtion fnction at (sbjective) time t, then the distribtions at times t and t + t can be related as F sbj ( t + ;t+ t) = q() F sbj (; t) + t(f sbj (a()), F sbj ()) The first term in the RHS of the above eqation asserts that the rocess cannot increase by more than q() t in an interval of time t while the second term considers the robability of loss events that wold case the rocess vale to redce below at time t + t. Since the stationary distribtion F sbj () is invariant in t, we get the reslting differential eqation df sbj () = q()(f sbj (a()), F sbj ()) (3.2) d We were nable to obtain a closed form analytical soltion for this differential eqation. We however rovide an oen-form analytical eression for F sbj () that translates into a raidly converging nmerical techniqe for evalating the cmlative distribtion. In assing, we note that the aroimation of the TCP rocess reslts in the differential eqation df sbj () = q()(f sbj (2), F sbj ()) (3.3) d which will be sed in the nmerical tests to be resented later. A. Soltion of the Eqation Let G be the comlementary distribtion fnction defined by the relation G() =, F sbj (). Eqation (3.) is eqivalent to the eqation dg() + q()g() =q()g(a()) (3.4) d with the Rbondary conditions G() =, G() =. Let Q() = q()d and define G() =H()e,Q() where H() is an arbitrary fnction (to be evalated). H() is then seen to obey the differential eqation H() =H(z), z q()e Q() G(a())d (3.5) Now, sose that lim " H() eists and is eqal to H. H will eist only if the tail of the comlementary distribtion decays as e,q(). By evalating the behavior of eqation (3.4) for very large (where G(a()) can be considered to be with negligible error), we can easily see that this henomenon of eonential decay is indeed tre. Now, by letting z "in eqation (3.5) and noting that G(a()) = e,q(a()) H(a()),wehave H() = H, q()e (Q(),Q(a())) H(a())d (3.6) with the bondary conditions H() =andh() = H. By defining J() as J() = q()e Q(),Q(a()) = q()e,r a() q()d, eqation (3.6) redces to H() = H, J ()H(a())d (3.7) By iterated eansion, H() can be shown to obey the relation H() = H X k= (,) k k,fold z } { > ::: k>k,

5 5 J ( ) :::J( k )d k :::d (3.8) Aendi III rovides a roof that the above infinite sm indeed converges to a limit when the fnction q() is non-decreasing in ; this condition holds for the TCP rocess whenever the dro robability is a non-decreasing fnction of the window size. B. Nmerical Comtation Reeated sbstittion in eqation (3.7) offers a nmerical techniqe for evalating H(). As H() tends to a limit as ", it can be treated as a constant beyond a certain vale er (chosen sch that the reslting error in comting H() is at most a small vale ). We can then obtain an aroimation for H() by setting the vale of H() beyond er to be a constant and comting H() between (; er ). After the algorithm converges, we can divide by H() to satisfy the bondary conditions H() =;H() = H. The comlete nmerical rocedre for comting F sbj () is as follows:. Choose a small ositive constant ( >), which indicates the accracy of the comtation. R 2. Find er sch that J() d. er 3. Let B () = forall and let B i () = ; 8 > er ; 8 i: R 4. Also comte K() = J () d for A( er) er. Denote K(A( er )) by. 5. For all vales of i,letb i () =, K(); for A( er ) er. 6. Reeat the following iteration in the range (;A( er )) ntil the fnction converges below a secified bond: B i () =, A(er) J() B i, () d, : 7. Let the final soltion be denoted by B(). 8. Renormalize B() = B() B() to satisfy the necessary bondary conditions. B() is then the nmerical estimate for H(). 9. The comlementary robability distribtion is then obtained as G() = B()e,Q() (3.9). Comte F sbj () from F sbj () =, G(). C. Correcting for Lossless Evoltion As noted in section II.A, the rescaled TCP rocess in sbjective time cannot catre the dynamics of the window evoltion when the loss robability is (as sbjective time freezes dring these eochs). From a samle ath oint of view, the infinite derivative in eqation (2.5) (Proosition ) and the zero time increment in eqation (2.2) imly that whenever the TCP rocess (in sbjective time) enters an interval in the state-sace corresonding to loss, it instantaneosly jms from the lower to the er end of the interval. In this sbsection, we show how F ack () for the TCP rocess, obtained from the maing in eqation (2.6), can be corrected to incororate the dynamics of the lossless evoltion; the corresonding correction for the generalized rocess is then straightforward. The correction for the density f ack () in ack time (after the correction for state-sace rescaling has been comleted) is comted by the level crossing rincile which eqates the rate at which the rocess evolves to the right of a vale to the rate at which the rocess transitions to the left. For the TCP rocess, this reslts in the eqality f 2 ack () = ()df ack (): (3.) This follows by noting that at a oint, TCP evolves to the right at a rate while it moves to the left at the rate governed by the loss rate in the interval (; 2). By first obtaining the vales of F ack () (to a scaling constant) in the regions with non-zero loss robability,we can correct the soltion for regions with zero loss robability sing the eqation (3.) 3. (If F ack () in the RHS of eqation (3.) is nknown for any, it follows that () = also; the nknown region may ths be left ot of the comtation.) The nmerical recie for correcting the distribtion for the lossless region is.. For the region(s) where () =, comte the density f ack () sing the level crossing relation f ack () = 2 ()f ack ()d (3.) 2. Renormalize f ack () by R f ack ()d over the entire statesace to ensre a well formed robability distribtion fnction f ack (). IV. SIMULATIONS AND RESULTS We now resent nmerical eamles to comare or analytical reslts with those obtained via simlations. The simlations were carried ot with the TCP Reno and NewReno versions in the ns simlator ackage. Althogh these versions differ in their fast recovery mechanisms and in the freqency of timeots, the erformance of the two versions was fond to be almost identical for the relatively low loss environments stdied in or simlations. To obtain adeqate statistical confidence, simlationreslts were obtained by averaging over rns with mltile seeds; each rn comrised at least 6 acket transmissions. While the entire simlation rocess wold take,5 mintes, the nmerical comtation over a fairly fine grid ( oints) took only abot 3 secs (on a tyical workstation). A. TCP with Simle State-Deendent Loss The reslts in Fig. corresond to the case when the acket dro robability deends directly on the window size. We 3 The level-crossing eqation (3.) is actally valid for the entire range of the TCP rocess state-sace (and not jst where () is ). It can easily be seen that the eqation (3.3) (in sbjective time) is eqivalent to eqation (3.) (in ack time) by noting that the following set of relations: ()df ack () = K:dF s(); f ack () = K:fs() ; F () s() =F sbj ( ma); andf s() = maf sbj ( ma), wherek is a normalizing constant. The elaborate rescalings and comtationsin sbjective time in this aer are necessarysimly becase there does not aear to be a simle way of solving eqation (3.) directly over the entire state sace! Another way of looking at the sbjective time formlation is therefore to think of it merely as a change of measre that relaces eqation (3.) with the more tractable version in eqation (3.3).

6 6 achieve this effect by assing a TCP connection throgh a single qee with negligible link roagation and transmission delay (all otstanding ackets are ths effectively resident in the qee), and indeendently droing each arriving acket with a robability that varies with the qee occancy. The dro robability in this eamle increases linearly with qee occancy. It can be seen that the simlated behavior offers ecellent agreement with the nmerical rediction in this eamle. For comarison roses, we inclde the distribtion redicted by the sqare-root formla in ([2]) assming a constant dro robability; the constant vale of the dro robability was taken to be the dro robability corresonding to the mean TCP window size obtained via simlation. As eected, the sqare-root aroimation redicts a mch larger variation in the window size than the tre distribtion Stationary Distribtion of TCP Window in Ack Time _ma=.2, B*RTT~=, min_th=2, ma_th=275 State-deendent Loss (Simlation) State Deendent Loss (Theory) SqRoot Formla Figre : TCP Window Evoltion and State-Deendent Loss B. Predicting TCP behavior with Qee Management Techniqes One of the goals of or analysis is to redict the window distribtion of a ersistent TCP flow when it interacts with roter qee management mechanisms like Early Random Dro (ERD) and Random Early Detection (RED), where the acket dro robability is not constant bt varies with the qee occancy. In the resent aer, we only model the simle case where the ersistent TCP connection is the only flow throgh the roter ort; we are crrently investigating etensions to mltile TCP flows. While both ERD and RED involve variable dro robabilities that deend on the qee occancy, they have significant differences (discssed in Aendi II), of which the two most imortant are: The dro robability in RED is a fnction of the movingaveraged qee occancy while the dro robability in ERD is a fnction of the instantaneos qee length. Even if he qee occancy remains constant, RED increases the dro robability for every acceted acket (which we call dro biasing) reslting in inter-dro gas that are niformly generated; ERD dros each acket with the same dro robability, reslting in inter-dro gas that are geometrically distribted. These differences make RED mch harder to model than ERD: the se of averaged qee occancies to determine dro robabilities destroys the state-deendent loss model (the dro robability is then a fnction of the ast state behavior) while the dro biasing fnctionality negates the indeendent acket dro assmtion. We circmvent these roblem by (simlistically) assming that the dro robability deends only on the instantaneos qee length and that each acket is droed indeendently. We ths ignore the effect of qee averaging in RED; we shall however resent a simle correction to accont for the effect of dro biasing. B. Relating the Loss Probability to Qee Occancy As already stated, we assme that the loss robabilityis determined by the instantaneos qee occancy (for both RED and ERD); the loss robability for a given TCP window is derived by relating the qee occancy to the TCP window. Neglecting the eriods of fast recovery, the nmber of nacknowledged ackets in flight, when the window is W n, eqals bw n c,orinan aroimate sense, W n.ifb(kts/sec) is the service rate of the (bottleneck) qee and the rond-tri delay (ignoring the qeing delay) is RT T (sec), then B:RT T ackets are necessary to fill the transmission ie. Assming that this ie is always fll 4, the occancy of the qee is given by the residal nmber of nacknowledged ackets, so that we have Q n = W n, B:RT T (4.) For or eeriments, the loss fnction is given by the traditional model of RED behavior, i.e., (Q) =forq min th, Q,minth math,minth ma (Q) = ma for Q ma th and (Q) = for min th <Q<ma th. The loss robability as a fnction of the window size is then given by (W, B:RT T ) 5. While the above model cannot catre the qee averaging fnction of RED, we can make a simle correction to aroimate the effect of dro biasing in or model. We note that for a given vale of dro robability, the niform distribtion of inter-dro gas in RED imlies that the average ga is ;the 2 geometric distribtion of gas (reslting from an indeendent loss model) imlies an average ga of. For the RED simlations, we accordingly modify or analytical dro fnction sch that or average agrees with that of RED, i.e., for a given qee occancy q, wemake model (q) =2 red (q). B.2 Eerimental Reslts Illstrative reslts of or validation eeriments are rovided in figres 2 and 3, which lot the nmerically redicted cmlative distribtion of the TCP window against that obtained from simlations. Figre 2 shows that or nmerical analysis rovides an ecellent match with simlation when the qee imlements the ERD algorithm. The distribtion redicted by the sqareroot formla is also rovided for comarison. Figre 3 consists of two grahs, the to one for a RED qee with B:RT T 4 This assmtion holds only if the bffer never nderflows (which, in trn, can hold only if the time taken by the bffer to drain min th ackets is longer than RT T. 5 The reader will note that the ack arriving at the sorce at time n (when the window is W n) corresondsto a acket generateda rond-tri time earlier when the window was W n ; the loss robability of the acket acked at n shold ths be (W n ). However, as cwnd increases by a maimm of segment in a rond-tri time W n W n, so that the loss robability of the acket acked at n can be assmed to be (W n) with negligible error.

7 7 and the bottom one with B:RT T = 5. The to grah isolates the effect of aroimating the RED averaging rocess from the erformance obtained when this aroimation is combined with the assmtion of a fll ie (eqation (4.)). The two grahs show, somewhat srrisingly, that the nmerical redictions (with the correction for dro biasing) rovide fairly close agreement with the simlated distribtion when the qee imlements RED. The closeness of the fit is somewhat neected since the averaging effect in RED qees tyically last over 5 ackets; we eected this memory to significantly degrade the accracy of or modeling _ma=.2, min_th=2, ma_th=275 Stationary Distribtion of TCP Window (with RED) and State_Deendent Loss RED/ No Delayed ack(simlation) RED/ No Delayed acks (Theory) RED/ Delayed ack(simlation) RED/ Delayed acks (Theory) C. Incororating Delayed Acknowledgements Stationary Distribtion of TCP Window (with RED) in Ack Time Or model of TCP window evoltion has so far assmed that TCP receivers generate an acknowledgement for every arriving acket. Many imlementations, however, se delayed acknowledgements to slow the rate of window eansion or alleviate congestion on the reverse link. We can model this artifact by noting that if the receiver sends one ack for every K ackets received, then the TCP window grows from W to W + W for every K ackets transmitted. An aroimation to this behavior is achieved by sosing that the TCP window grows by only =K th of its vale for every acket transmitted i.e. by modifying the window evoltion eqation to W n+ = W n + K:Wn. Nmerical reslts verify the effectiveness of this correction in acconting for the henomenon of delayed acknowledegments. The grahs in figre 3 contain the comarisons between analysis and simlations when a TCP connection erforming delayed acknowledgements is combined with the RED qee management algorithm while figre 4 shows the comarisons when a TCP erforming delayed acknowledgements interacts with the ERD qee management algorithm. For the ERD qee, we also rovide the theoretical distribtion obtained by alying the correction for delayed acknowledgements in the sqare-root formla [2] _ma=.2, B*RTT=5, min_th=2, ma_th=275 Stationary Distribtion of TCP Window (with ERD) in Ack Time ERD(Simlation) ERD (Theory) SqRoot Formla Figre 2: Behavior of TCP Window with Early Random Dro (and Eternal Delay) _ma=.2, B*RTT=5, min_th=2, ma_th=275 RED/ No Delayed ack(simlation) RED/ No Delayed acks (Theory) RED/ Delayed ack(simlation) RED/ Delayed acks (Theory) Figre 3: Behavior of TCP Window with Random Early Detection (with & withot eternal delay and delayed acks) Stationary Distribtion of TCP Window (with Delayed Acks) (throgh an ERD ort) in Ack Time _ma=.2, B*RTT=5, min_th=2, ma_th=275 ERD(Simlation) with delayed acks ERD (Theory) with delayed acks SqRoot Formla with delayed acks Figre 4: Behavior of TCP Window with Delayed Ack and Early Random Dro V. CONCLUSION In this aer, we resent a techniqe for analyzing and redicting the window distribtion of a ersistent TCP connection over a ath where acket losses occr with variable (state-deendent) robability and where acket dros are conditionally ncorrelated events. The key to an effective nmerical rocedre for redicting the distribtionis the change in the inde to sbjective time, which is a history-deendent rescaling of the time inde.

8 8 Comarisons with simlation reslts sggest that this techniqe is fairly accrate in redicting obtaining TCP window behavior of crrent TCP versions when the loss robabilities are low and timeots are relatively rare events. In articlar, we find that this analysis can be sed to redict the window behavior of a single ersistent TCP flow interacting with qee management algorithms like Early Random Dro and Random Early Detection. While the accracy of the redictions was eected for Early Random Dro, the fit for the case of Random Early Detection was srrisingly good. We are crrently attemting to tilize this aroach to etend the analysis to the case of mltile ersistent TCP connections interacting with ERD/RED qees. By obtaining reasonably accrate distribtions of individal TCP windows, we hoe to derive an accrate characterization of the comosite qee dynamics (inclding its mean and variance). APPENDICES I. POISSON NATURE OF PACKET DROP EVENTS We rove here that the sbjective time formlation reslts in an inter-loss interval that is eonentially distribted with mean and is indeendent of ast and ftre intervals. For the TCP rocess nder consideration, this roerty is asymtotically tre as the loss robabilities tend towards. Let s find the robability P fx i >Tg, i.e., the robability at least sbjective time T elases between the (i, ) th acket loss and the i th acket loss. We renmber the ackets: acket j (temorarily) denotes the j th acket after the one that was the i th loss. Since the congestion window is increasing after the i th loss, there eists with robability a (random!) N sch that N <T N+: The robabilities j are also random. The robability of interest is that none of the first N ackets are lost. Since N is random, this robability eqals X n= P fn = ng E[ ny j= (, j )jn = n]: (I.) As long as (with robability one) maf j ; j ng is almost zero, the eression (I.) is close to X n= P fn = nge[e,p n j= j jn = n]: P N Since <T, j= j N+, we see that as long as (for eamle) E[maf j ; j N + g] # ; (I.2) P fx i >Tg!e,T. Since the above roof is also indeendent of the size of the acket that cased the i-th acket loss, we see that if condition (I.2) holds 6, the inter-loss intervals (in sbjective time) are not only eonentially distribted, bt also indeendent of ast and ftre intervals. This establishes the fact that the loss events are realizations of a Poisson rocess of intensity in sbjective time. 6 Note that the condition (I.2) may hold even if ma is not small. All we II. DIFFERENCES BETWEEN ERD AND RED In this aendi, we discss the differences between the Early Random Dro (ERD) and the Random Early Detection (RED) algorithms, which are imortant in nderstanding the alicability of or loss model. The imortant differences are: RED oerates on the average (and not the instantaneos) qee length. The dro robability,, is ths a fnction of the weighted average (Q avg ) of the qee occancy i.e., is a fnction not jst of Q n bt of (Q n ;Q n, ;Q n,2 ;:::) with an eonential decay. Q avg closely mirrors the instantaneos occancy only if the qee varies slowly. To revent large inter-dro drations, RED increases the dro robability for every acceted acket. (This roerty, which we call dro biasing, is achieved by sing a variable, cnt, which increases with every sccessive acceted acket; the tre droing robability is then given (Q) by, cnt:(q). This reslts in an inter-dro eriod that is niformly distribted between (;:::;b(q) c), as oosed to the indeendent dro model in ERD which reslts in geometrically distribted inter-dro eriods. Some RED configrations have a shar discontinity in dro robability: when the average qee eceeds ma th, (Q) becomes so that all incoming ackets are deterministically droed. This contrasts with or assmtion that random dro occrs throghot the entire range of the bffer occancy. Or analysis alies to RED qees only if the TCP rocess almost never bilds qees that eceed ma th. III. PROOF OF CONVERGENCE OF H() R To see that H() in eqation (3.8) indeed converges to a limit, let s define C() by C() = J ()d. Now, assme that there eists a >, sch that A() 8 (i.e., a() ).This is a stronger reqirement than A() <; in the case of the TCP model, = 2. Now since q() is a non-decreasing fnction of, a() q()d q()d, q()d so that a() q()d, q()d (III.3) Hence, C() d where =, C() q()e,q() d (III.4) (, ) q()e,q() d (III.5) (, ) e,q() (III.6) This shows that C() is er bonded by C(). (Note that for the case of TCP, = 2and =, so that C() =; in other cases, C() is some finite vale.) Now, consider a random variable with density f() = J() C() and let X ;X k ;:::;X k be k i.i.d really need is the congestion window almost always stays small enoghfor (w) to be small.

9 9 realizations of this random variable and let X();X(2);:::;X(k) be the order statistic. Then, k,fold z } { > ::: k>a(k,) J ( ) :::J( k )d k :::d = fc()g k k! Prob(X(j) >a(x(j,)) for j(2;:::;k)jx j >; 8 j): (III.7) Hence, if we denote the sm of the first l terms in the RHS of eqation (3.8) as H l (), we see that H(), H l () H X j=l C() j which roves that H() is indeed convergent. j! ; (III.8) REFERENCES [] V Jacobson, "Congestion Avoidance and Control", SIGCOMM 988. [2] T Ott, M Matthis and J Kemerman, "The Stationary Behavior of Idealized Congestion Avoidance", ft://ft.bellcore.com/b/tjo/tcpwindow.s, Agst 996. [3] J Padhye, V Firoi, D Towsley and J Krose, "Modeling TCP Throght: a Simle Model and its Emirical Validation", Proceedings of Sigcomm 98,Setember 998. [4] A Kmar,"Comarative Performance Analysis of Versions of TCP in a Local Network with a Lossy Link", IEEE/ACM Transactions on Networking, Agst 998. [5] T V Lakshman, U Madhow and B Ster, "Window-based Error Recovery and Flow Control with a Slow Acknowledgement Channel: a Stdy of TCP/IP Performance", Proceedings of Infocom 97, Aril 997. [6] S Floyd, "Connections with Mltile Congested Gateways in Packet- Switched Networks Part : One-way Traffic", Comter Commnication Review, Vol.2, No.5, October 99. [7] V Jacobson, "Modified TCP congestion avoidance algorithm", Aril 3, 99, end2end-interest mailing list. [8] T Ott, "Is Random Early Dro fair?", ft://ft.isi.ed/end2end/end2endinterest-996.mail (Nov. 9, 996). [9] M Mathis, J Semke, J Mahdavi and T Ott, "The Macroscoic Behavior of the TCP Congestion Avoidance Algorithm", Comter Commnications Review, Jly 997. [] E Hashem, "Analysis of Random Dro for Gateway Congestion Control", MIT-LCS-TR-56. [] S Floyd and V Jacobson, "Random Early Detection Gateways for Congestion Avoidance", IEEE/ACM Transactions on Networking, Agst 993. [2] A Parekh and R Gallager, A Generalized Processor Sharing Aroach to Flow Control in Integrated Services Networks: The Single-Node Case, IEEE/ACM Transactions on Networking, Jne 993