Modeling RED with Idealized TCP Sources

Transcription

1 Moeling RED with Iealize TCP Sources P. Kuusela, P. Lassila, J. Virtamo an P. Key Networking Laboratory Helsinki University of Technology (HUT) P.O.Box 3000, FIN HUT, Finlan {Pirkko.Kuusela, Pasi.Lassila, Microsoft Research Cambrige, Unite Kingom Abstract We analyze the ynamic behavior of a single RED controlle queue interacting with a large population of iealize TCP sources, i.e., sources obeying the rules of linear increase an multiplicative ecrease. The aggregate traffic from this population is moele in terms of the epenent expecte value of the packet arrival rate which reacts to the packet loss taking place in the queue. The queue is escribe in terms of the epenent expecte values of the instantaneous queue length an of the exponentially average queue length, for which we also erive a pair of ifferential equations. This provies us with a complete moel for the ynamics of the system which we use to explore transient an equilibrium behavior. The accuracy of the moel is verifie by comparison with simulate results. Keywors: congestion control, Ranom Early Detection, TCP 1 Introuction Active queue management (AQM) algorithms have been recommene by IETF as effective mechanisms to control congestion in network routers. We consier one AQM metho, namely the RED algorithm [1], an moel the interaction between a RED controlle queue an an iealize TCP source population. A full escription of the problem consists of a close system with two interacting parts (see Figure 1.): a) a queue receiving an aggregate flow of packets an reacting ynamically to the changes in the total flow rate uner the control of the RED algorithm, an b) a population of TCP sources, each of which reacts to the packet losses at the queue as etermine by the TCP flow control mechanism. Part a) of the problem was consiere by Lassila an Virtamo [2]. Here we aim at closing the control loop back to the TCP sources escribing the interaction of parts a) an b). However, we are not moeling the entire TCP behavior, but rather the so calle congestion avoiance part of it, i.e., the part where a source increases its congestion winow linearly an ecreases it multiplicatively. λ(t) TCP source population RED controlle queue packet loss Figure 1.: Interaction between the TCP population an the RED controlle queue. xx/1

2 This paper escribes a system of couple ifferential equations for the population of TCP streams an the RED controlle queue. Such a moel can be use to explore, for example, parameter settings or stability surfaces (which are ifficult to stuy via simulation). Moreover, the moel can be extene to hanle multiple classes an ifferent roun trip s. The emphasis of this paper is to erive the moel an verify its accuracy. A follow-up paper [3] utilizes this moel in analyzing the stability of the TCP-RED interaction an showing how it epens on the system parameters. See also [3] for references on the stability of the TCP behavior. We use a moel similar to Kelly s [4] for TCP congestion avoiance behavior escribing the average arrival rate λ(t) of packets from a large collection of TCP sources. To rigorously erive the analytical moel we then assume that the arrival process to the RED controlle queue can be approximate by an inhomogeneous Poisson process with a stochastically varying arrival rate λ(t). Such an approximation is reasonable as we are moeling the aggregate traffic from a large collection of TCP sources. We emphasize on eriving equations carefully an inicating the stochastic approximations that have been mae. The accuracy of the moel is verifie by simulations. In the simulation, none of the stochastic approximations are use, an the very goo agreement with the moel shows that the stochastic assumptions are not important. Recent inepenent moeling work by Misra et al. [5] is similar to our approach in which ifferential equations are use to escribe the system variables. The work in [5] is complementary to ours; we evelop a more etaile moel for the queue ynamics, while [5] focuses more on the source ynamics. In aition, our term escribing the increase in the TCP winow size is more accurate. In the RED buffer moel, [5] assumes an infinite an non-emptying buffer an the moel erivation results in an aitional moeling parameter to be tune. On the contrary, our queue moel takes overflows an empty queues into account, an oes not have aitional parameters. Overall, we fin that the most accurate system moel is obtaine by using our RED moel an iviing the TCP population into homogeneous subpopulations with ientical link elays, each escribe by the Kelly s aggregate TCP moel, an then making the straightforwar moel extensions to hanle queue epenent roun trip s, outs an the network case, as is one in [5]. In a recent paper Laalaoua et al. [6] moel RED with aaptive traffic sources. The approach is to iterate the packet loss probability an the traffic intensity in tanem with solving a iffusion equation for the instantaneous queue length, thus giving the transient behavior of the queue between the iteration steps. Altman et al. [7] consier a TCP behavior together with losses that are presente by an exogenous stationary point process (an hence inepenent of the TCP winow sizes) an erive the first two moments of the TCP winow size. Also, Brown [8] provies a moel for the winow evolution of the TCP connections with ifferent roun trip s in a tail-rop queue. This paper is organize as follows: We begin by escribing the RED algorithm in Section 2. Moels for the TCP population an the RED controlle queue are erive separately in Sections 3 an 4, an then combine in Section 5. In that section we also inclue the link elays resulting in a retare functional ifferential equation (RFDE) moel, an iscuss various extensions of the moel, e.g., the possibility to hanle ifferentiate service classes with simple moifications an ealing with inhomogeneous roun trip s. The accuracy of stochastic assumptions an approximations are verifie in Section 6 where we compare the moel to a simulate system. Conclusions are given in Section 7. 2 The RED algorithm For the purpose of this paper we will use a simplifie version of the original RED algorithm, which is typical in RED analyzes an also one, e.g., in [5]. To be specific, the following simplifications are mae. We o not cover the case where the exponentially weighte average (EWA) queue xx/2

3 length is compute ifferently when the arrival occurs into an empty queue. However, as we are primarily intereste in heavily loae queues, the effect of this special hanling is of less importance. Aitionally, we o not moel the counter which measures the (in terms of packet arrivals) since the previous packet rop. (Note that the system with the counter can be approximate by a counterless system simply by using a counterless system with twice the value for the RED parameter p max, see, e.g., [1], or [2].) Our simplifie RED algorithm works as follows: Consier a queue with a finite buffer which can hol up to K packets at a. Let q n be the queue length an s n be the EWA queue length (to be efine below) at the n th arrival. For each arriving packet we compute the EWA queue length s n with s n =(1 β)s n 1 + βq n, where 0 <β<1 is an appropriate constant, typically of the orer 3. If the buffer is full, the packet is lost. If s n <T min, the arriving packet is accepte; if s n >T max the packet is iscare. In the intermeiate range T min s n T max, the packet is iscare with probability p(s n )= p max(s n T min ) T max T min, an accepte otherwise. Typical recommenations for setting the parameters are: T min =5, T max = K/2 anp max =0.1, see [9] 3 The TCP moel We aim at moeling the behavior of a TCP connection in its congestion avoiance phase. Our moel is essentially the same as in [4], however, we have altere its erivation to get a clearer iea of the approximations involve. Consier m ientical TCP sources with constant roun trip R (ranom queuing elays are not inclue). To simplify the notation, we temporarily neglect the elays in the arguments an erive the equations as if the changes to the winow sizes occurre instantaneously at packet transmission epochs. The elays are introuce later in Section 5. Denote by w i (t) the winow of TCP source i at t. Corresponingly, the aggregate has a winow W (t) = i w i(t). The throughput (arrival rate), λ i (t), of source i is λ i (t) =w i (t)/r. Consier a small interval (t, t + t), or t for brevity. Let A i ( t) enote the event that there is exactly 1 arrival from source i uring t. Aitionally, given that there is a packet arrival in t from source i, letl i (t) enotetheeventthatthispacketislostanl c i (t) its complement. Our basic assumption on the traffic generate by a TCP source is that it is perioic with epenent perios, but for each TCP source the packet transmission phase is ranom. This implies that given the current winow size w i (t) ofsourcei, the probability of an arrival in t is P{A i ( t) w i (t)} = λ i (t) t = w i (t) t/r an the probability of having more than 1 arrival in t equals zero. In TCP congestion avoiance, for each successfully transmitte packet source i increases its transmission winow by 1/w i (t) (corresponing to an increase by 1 for w i (t) packets), an for each lost packet the winow is halve. We can compute the expectation of the change in the aggregate arrival rate λ(t) = i λ i(t), E[ λ(t)], by conitioning on the set of xx/3

4 winow sizes {w i (t)} an A i ( t) as follows [ ]] E[ λ(t)] = 1 R E[E[ W (t) {w i(t)}]] = [E 1 R E w i (t) {w i (t)} = 1 R E [ i [ = 1 R E i [ = t R 2 E i E[ w i (t) {w i (t)},a i ( t)] P{A i ( t) {w i (t)}} ( P{L c i (t) {w i(t)}} 1 i w i (t) P{L i(t) {w i (t)}} w ) ] i(t) wi (t) 2 R t ) ]. ( P{L c i(t) {w i (t)}} P{L i (t) {w i (t)}} w2 i (t) 2 If we further assume that the w i (t) processes are inepenent of each other an that the number of TCP sources is large, each iniviual source experiences a packet loss as etermine by the queue receiving the aggregate traffic. The conitional probability P{L i (t) {w i (t)}} correspons then to the loss probability in the queue at t, which is enote here by P L (t). Proceeing with the erivation, we obtain E[ λ(t)] = m t R 2 ( (1 P L (t)) P L (t) E[w2 i (t)] 2 We can next make a comment on the effect of the istribution of w i (t) on the coefficients above. Here we temporarily suppress the epenence from the notation. We assume that the istribution of w i is such that it scales with E[w i ], i.e. we assume that E[wi 2]=k E[w i] 2, where k is some constant. If w i is a fixe constant then we have trivially that E[wi 2]=E[w i] 2 = E[W ] 2 /m 2, i.e., k = 1. On the other han, if each TCP source in the population is assume to vary its winow between 2 3 W/m an 4 3W/m an the phases in the population are ranom, then each w i is uniformly istribute in [ 2 3 W/m, 4 3W/m]. Now large winows contribute more to the winow ecrease ue to the packet loss. Then we obtain E[wi 2 3m ]= 2W 4W 3m 2W 3m w 2 i 3m 2W w i = 28 ( ) W 2, 27 m i.e. we have k =28/27 1. Thus, this extension practically results in the same moel as above. In aition, numerical experiments with more realistic winow size istributions yiel only minor changes in the coefficient k. Hence, in this paper we use the simple assumption that k =1anE[wi 2]=E[w i] 2. Then, using the approximation E[wi 2(t)] = E[w i(t)] 2 =E[W (t)] 2 /m 2,weget E[ λ(t)] = ((1 P L (t)) mr ) 2 P L(t) E[λ(t)]2 t 2m By the linearity of the operator, E[ λ(t)] = E[λ(t)], enoting with E[λ(t)] = λ(t), an letting t 0, we arrive at t λ(t) =(1 P L (t)) m R 2 P L(t) λ 2 (t) 2m. (1) ). 4 Moel for the RED Controlle Queue The aim is to evelop a continuous moel for the transient behavior of the expectations of the two queue length processes, the exponentially average queue length s(t) an the instantaneous queue length q(t). These expectations are enote here by s(t) an q(t), respectively. ] xx/4

5 With the earlier assumptions on a large TCP population an ranom phases, the aggregate stream can be approximate by a varying inhomogeneous Poisson process with rate λ(t). Strictly speaking, λ(t) is itself a stochastic process but we may assume that the fluctuations are small an approximate λ(t) E[λ(t)]. With this, the queue part of our moel consists of a RED controlle queue receiving an inhomogeneous Poisson arrival stream with a eterministic varying rate E[λ(t)] = λ(t) governe by Eq. (1). This system has been previously stuie in [2], where by using a similar technique as in the previous section, the expecte change in q(t) an s(t) uring t has been obtaine by conitioning on certain ranom variables. As a result the following approximation has been obtaine for escribing the epenent behavior of s(t) an q(t): t s(t) = λ(t)β( q(t) s(t)), ( )( ) q(t) = λ(t) 1 π K (t) 1 p( s(t)) t ( ) µ 1 π 0 (t), where π 0 an π K enote the steay state probabilities for the queue to be empty an full, respectively. Note that, in the equation for q(t) above, the first term is the expecte rate at which packets are amitte to the queue an the secon term is the expecte rate at which packets leave the queue. To evaluate the terms π 0 (t) anπ K (t) we use a quasi-stationarity approximation. The iea is to replace π 0 (t) anπ K (t) with ones that relate π 0 an π K irectly to the system variable q(t) by using known steay state relations for π 0, π K an q, i.e., in our case those of the M/D/1/K system (in [2] an M/M/1/K queue was consiere). To this en let πk (ρ) enote the steay state probability of state k in the infinite capacity M/D/1 system with loa ρ. These state probabilities can be obtaine by using, e.g., the algorithm by Virtamo []. Then the steay state probability of state k, π k (ρ), in the finite system can be obtaine from the following relations (see [11, p. 230]) πj π j (ρ) = (ρ) π0 j =0,...,K 1, (ρ)+ρg(k), G(K) π K (ρ) = 1 π0 (ρ)+ρg(k), G(K) = K 1 j=0 π j (ρ). Aitionally, we have trivially that q(ρ) = K j=0 jπ j(ρ). The steay state functions π 0 ( q) anπ K ( q) are compute by eliminating ρ from the above relations. This approximation technique is also similar to the one use in [12]. In practice, e.g., the function π 0 ( q) is best erive by computing pairs of values ( q(ρ),π 0 (ρ)) for a suitably ense iscretization of ρ proviing a iscretize version of π 0 ( q). Then one can use interpolation between the iscretization points to compute π 0 ( q) for any value of q. In this paper we use the M/D/1/K system as an example, but the above proceure applies more generally to any packet size istribution. The queue length istribution of the M/G/1 system is given by the well known Pollaczek-Khintchine formula [11, p. 230] an the istribution can be compute algorithmically as presente in, e.g., [13, p. 266]. Finally, the istribution of the M/G/1/K system is obtaine from the infinite buffer system in the same way as was one above. The applicability of the M/G/1/K approximation epens on how fast λ(t) can change. If λ(t) is slowly varying, the queue can be consiere to be in a quasi-stationary state. In our case the changes in λ(t) are riven by packet losses, which are primarily ue to the RED packet iscaring (buffer overflows shoul be rare in a RED controlle queue). The RED packet xx/5 (2)

6 iscaring probability epens on the average queue length s(t), whose scale of changes is etermine by β. In practice β is small (of the orer 3 ), an thus the process s(t) moves slowly an, also, the changes in λ(t) occur relatively slowly. Moreover, our simulations in Section 6 verify the applicability of this approximation. 5 The Moel for the Interacting System In combining the moels for the TCP population an the RED controlle queue we inclue the link elays in the system (but queuing elays are not moele an hence the elays are not epenent). In the following moel λ(t) enotes the expecte TCP throughput at the source. The acknowlegments arrive at the source with the elay R causing the upate action to the current sening rate, an the amount of upate epens on the current rate. Also, the TCP population will ajust its current sening rate base on packet losses having taken place in R/2 prior to the current. Similarly, the packets sent by the TCP population arrive at the queue after the link elay of R/2 an hence the queue will observe the R/2 elaye throughput. This gives rise to a elay ifferential equation (or RFDE): t s(t) = λ(t R/2)β( q(t) s(t)), t q(t) = λ(t R/2)(1 π K ( q(t)))(1 p( s(t))) µ(1 π 0 ( q(t))), (3) t λ(t) = P L(t R/2) λ(t) λ(t R)+(1 P L (t R/2)) m λ(t R) 2m R 2, λ(t) P L (t) =p( s(t)) + π K ( q(t)) p( s(t))π K ( q(t)). Here we have taken a symmetric elay structure, but any other elay structure can be hanle with obvious moifications. Also, note that in the TCP moel in Section 3, P L enote the probability of the packet loss observe by the TCP population. In the above system, packets may be roppe ue to the RED amission control or the buffer overflow. Hence we approximate the packet loss by P L = p( s) +π K ( q) p( s)π K ( q) (see [2] for iscussion on inepenence assumptions). Aitionally, observe that in the last ifferential equation in (3), λ(t) appears with two arguments: λ(t R) represents the upate rate at t, whereas λ(t) arises from the amount of change to the current rate. After the TCP population activates at t 0, the queue ynamics activate at t 0 +R/2 whereas the TCP congestion ynamics activate at t 0 + R. Functions q, s :[t 0,R/2] R an λ :[t 0,R] R are given as initial ata. Also, note that when there are no packet losses, the slope of the linear increase in the throughput is slightly less than m/r 2 ue to the term λ(t R)/ λ(t), a fact also seen in the simulations. 5.1 Two traffic classes We illustrate how the above moel can be extene to ifferent traffic classes following the treatment in [14]. Assume that we have a population of m TCP sources, each with a constant R, an the throughput of the aggregate in enote by λ 1. In aition there is some UDP traffic with intensity λ 2. For simplicity both traffic sources are assume to sen packets of equal size an as a local approximation the packet arrival intensity from each source is assume to be Poissonian. Also the weight parameter β in the RED algorithm is assume to be the same in each class. We refer to the TCP traffic as class 1 traffic an the UDP traffic is enote by class 2. As in Section 4 we can write ifferential equations escribing the evolution of EWA queue lengths s i in each class: s i /t = λ i β ( q i s i ). As the packet size is the same in each class, we may consier the queue with the aggregate arrivals from both classes an enote the expecte queue length xx/6

7 by q. In the queue we may label inepenently each packet in class i, i =1, 2, with probability λ i (1 p i ( s i ))/( λ 1 (1 p 1 ( s 1 )) + λ 2 (1 p 2 ( s 2 ))), where we have taken into account that the RED rops p 1 an p 2 will thin the arrival processes. Hence the expecte queue length in class i satisfies q i = λ i (1 p i ( s i )) λ 1 (1 p 1 ( s 1 )) + λ 2 (1 p 2 ( s 2 )) q. The evolution of the total queue is given by t q =(1 π K( q)) ( λ1 (1 p 1 ( s 1 )) + λ 2 (1 p 2 ( s 2 )) ) µ (1 π 0 ( q)). Finally, the TCP population is escribe by t λ 1 = P L,1 2m λ 2 +(1 P L,1 ) m R 2, where P L,1 = p 1 ( s 1 )+π K ( q) p 1 ( s 1 )π K ( q). (Note that there is no ODE for UDP traffic.) This moel views classes 1 an 2 uncouple in the sense that the ranom packet rop in a class is base on the EWA queue length of that class only. The RIO algorithm [15], in which class 1 has priority an class 2 is amitte into the queue base on the exponentially average total queue length, is obtaine easily by replacing p 2 ( s 2 )withp 2 ( s 1 + s 2 )inabove. 5.2 Approximating variable roun trip s In our moel the simplest way to approximate a TCP population with variable roun trip s is to partition the total TCP population into n sets so that in each set i, m i TCPs have the same roun trip R i. Each subpopulation i can be moele an treate as TCP class i in the previous generalization. No priority structure between the classes is introuce if the classes are uncouple an the RED parameters are the same in each class. This moeling paraigm is scalable; even a large number of ifferential equations can be solve in a reasonable. 6 Moel valiation Here we illustrate the accuracy of our ifferential equations by comparing against simulate results an we use the moel to explore equilibrium surfaces as functions of the system parameters. Simulation is use to valiate the accuracy of the moel, i.e., that the stochastic assumptions are not critical. In the simulation none of the stochastic approximations have been use. Also, each TCP source is a perioic source with a varying sening rate reacting inepenently to successful packet transmissions or ranom packet rops accoring to the congestion avoiance rate control rules. The only assumptions that have been retaine in our simulations are the omission of the counter, that the RTT is a fixe constant, an that TCP congestion control works in the congestion avoiance phase. 6.1 Simulation implementation To be specific, our simulation has been implemente in the following manner. The sources moel the behavior of a greey FTP source population transmitting constant length packets. Hence, each TCP source i has its own congestion winow w i (as oppose to our moel), an R is a common constant for all sources. The iea is to escribe the source as a flui process with instantaneous rate w i /R, an sen a packet at those instants of where the cumulative flui attains an integer value. Thus each source has the following behavior i) it tries to sen w i packets uring R an ii) it tries to space them in a smooth manner, i.e., the packets are xx/7

8 E@qD E@sD Figure 2.: Evolution of λ(t), q(t) an s(t) (unit of is 00 on the x-axis). not sent in one burst, as may happen with the real TCP. Each sent packet arrives into the queue after a elay of R/2. Upon arrival at the queue, an acknowlegement is sent back to the source whether the packet was accepte or iscare (either by RED control or buffer overflow). The acknowlegement reaches the source after a elay of R/2. The source either increases its congestion winow by 1/w i, i.e., sets w i w i +1/w i, if the packet was accepte or halves the congestion winow, i.e., sets w i w i /2. Aitionally, to avoi the sources from being synchronize right from the start of the simulation, each source is initialize with a winow size of 1 but the sources are turne on, i.e., transmit their first packets, at a which is uniformly istribute in the range [0,R]. Thus, the aggregate traffic into the queue consists of a superposition of perioic sources with a varying sening rate. In the numerical examples the following system parameters will remain constant. The buffer size K = 75, the service 1/µ = 1, R=00 an β =0.002 (as is suggeste in the literature). Also, the unit of is taken to be 1/µ. ChoosingR to be 00 s the average packet transmission correspons to a real life system where, assuming the packet size to be 500 bytes an the link spee to be 155 Mbit/s, R is ( bit)/155 Mbit/s 26 ms. Aitionally, in the analytical moel we start the system with initial ata q(t) = s(t) =0for0 t R/2 an λ(t) =m/r for 0 t R which correspons to setting q 0 = s 0 =0anw i = 1 for each source in the simulation. The simulate results have been obtaine by averaging over 0 inepenent simulation runs. 6.2 TCP examples Here we illustrate the accuracy of our moel with two examples. The examples have been chosen to emonstrate a case where the queue reaches equilibrium in a nice manner an a case where severe oscillations are encountere. In both examples we fix the number of TCP sources m = 0 an T max = K/2 =37.5. (Note that a larger TCP population is expecte to improve the accuracy of the analytical moel.) First we look at an example where we set the other RED parameters to T min =0anp max =. The results are shown in Figure 2. for the evolution of λ(t) (left figure), for q(t) (milefigure)anfor s(t) (right figure). In all the figures in this section, the soli lines show the results obtaine from the RFDE system (3) an the ashe lines correspon to simulate results. From the figures we can see that the results from the RFDE system match well with the simulate results an the equilibrium values are quite close. This also verifies, among other stochastic approximations, that our assumption in the moel on the locally Poisson nature of the aggregate traffic is applicable an gives accurate results, even when the actual traffic consists of a superposition of non-poisson sources. Changing T min = 5 an increasing p max =0.5 results in strong oscillations (see Figure 3.). Even in this case, we can observe rather goo agreement between the analytical an the simulate results. The only iscrepancy is that apparently the simulate system contains elements (phase mixing) that ampen the oscillations an prolong the oscillation perios, which oes not take place in the solutions of the RFDE system. As a continuation of the work reporte here, we have erive an submitte for publication xx/8

9 E@qD E@sD Figure 3.: Evolution of λ(t), q(t) an s(t) (unit of is 00 on the x-axis). E@λD E@qD E@sD Figure 4.: Evolution of λ(t), q(t) an s(t) (unit of is 00 on the x-axis). (see Kuusela et al. [3]) the stability analysis of the system giving, not only sufficient, but also necessary conitions for the asymptotic stability. Note that the stability analysis one in Hollot et al. [16] as a continuation of [5], uses a ifferent technique, an provies only sufficient conitions for their system. For the examples shown here our stability analysis verifies the visual observation of the first example being stable an the secon one unstable. 6.3 Uncontrolle UDP traffic an the TCP steay state Next we explore the effect of uncontrolle UDP traffic on the steay state of the system. In this case, the UDP traffic is here moele as a pure Poisson source with arrival rate λ =. The rest of the system parameters are: m = 0 (same as earlier), p max =, T min =0an T max = 40. The system was starte from an empty state, an at the UDP source was turne on an shut off at The results are shown in Figure 4. for λ(t) (left figure), for q(t) (mile figure), an for s(t) (right figure). Here λ(t) is the aggregate arrival rate from the TCP population an the UDP traffic. Again we can observe a goo corresponence an the results show how the system first reaches an equilibrium, how it is affecte by the UDP pulse an how the equilibrium is reache after the pulse isappears. 6.4 Equilibrium surfaces Here we plot some equilibrium surfaces of the system (3) as functions of the system parameters. We consier a buffer of size 40 with RED parameters T min =ant max = 30. The RED rop parameter p max is taken as a free parameter that varies in [0.01, 0.35]. The equilibrium epens on the TCP population parameters only through TCPpar = m/r, which is allowe to vary in [0.05, 0.7]. Figure 5. (left figure) illustrates the queue size at the equilibrium as a function of p max an TCPpar. We point out that if the RED rop probability is small but the TCP population is aggressive (either m is large or R is small) the system (3) oes not have an equilibrium as the queue tens to fill up to T max an then oscillates at values slightly below T max. In aition, even if an equilibrium exists it is not necessarily a stable one (see [3]). The surface of the expecte gooput of the TCP population is illustrate in Figure 5. (right figure). Note that the surfaces clearly illustrate the problematic TCP feature (from the system esigners point of view) that the system performance epens on the number of TCP connections, which xx/9

10 p max TCPpar p max TCPpar Figure 5.: The queue size at equilibrium (left figure) an the gooput of the TCP population at equilibrium (right figure). is also iscusse in [17]. If one wishes to set a fixe target queue length q t [T min,t max ] (which also coincies with the exponentially average queue length) to be achieve at the equilibrium, the relationship ( m R = PL 1 π0 ( q t ) ) 2(1 PL )(1 π K ( q t )) (1 p( q t )), where P L = p( q t )+π K ( q t ) p( q t )π K ( q t ), can be use to explore, e.g., how p max has to be ajuste to compensate the changes in m/r. The above relationship has been erive from (3) at the equilibrium state. 7 Conclusions We have evelope a ynamical moel to escribe the evolution of the TCP aggregate an the RED controlle queue. The TCP sources were iealize to operate only in the congestion avoiance phase. It is worth mentioning that this approach can be easily applie to other aitive-increase-multiplicative-ecrease schemes by changing the ifferential equation for the throughput λ accoringly. In aition, the moel of source an queue interaction can be use to explore, for example, various parameter settings or stability surfaces. Moreover, this approach can be easily extene to stuy the interaction of other traffic behaviors with RED, such as TCP variants, equation-base controls, rate aaptive traffic etc. Because the mathematical moel is in terms of expecte values of the source an queue variables, an equivalent simulation stuy woul require averaging over a large number of replications, making it easily an infeasible approach. In eriving the ifferential equation moel several stochastic assumptions an approximations were necessary, in particular, the aggregate packet arrivals were approximate by an inhomogeneous Poisson process. We point out that our main interest is the congestion control aspect of the complete system, an partial justification to such approximation is the observation by Veres an Boa [18]: several statistical tests inicate that the aggregate TCP traffic entering through a bottleneck buffer is short-range epenent. Moreover, our careful simulation verification, in which neither TCP winow nor stochastic assumptions were use, showe that the approximations were well justifie. xx/

11 The aim of this paper was to carefully erive an then valiate the moel. Aitionally, in [3] we have successfully performe the stability analysis of the system presente here, eriving not only sufficient, but also necessary conitions for the system to be asymptotically stable. This follow-up paper contains several illustrative figures that show how the system parameters affect the system performance. References [1] S. Floy an V. Jacobson, Ranom Early Detection gateways in congestion avoiance, IEEE/ACM Transactions on Networking, vol. 1, no. 3, pp , [2] P. Lassila an J. Virtamo, Moeling the ynamics of the RED algorithm, in Proceeings of QofIS 00, September 2000, pp [3] P. Kuusela, P. Lassila, an J. Virtamo, Stability of TCP-RED congestion control, submitte for publication, available at [4] F. Kelly, Mathematical moelling of the Internet, available at uk./~frank/ (also an extene version in Mathematics Unlimite 2001 an Beyon, Springer Verlag, 2000), [5] V. Misra, W. Gong, an D. Towsley, A flui-base analysis of a network of AQM routers supporting TCP flows with an application to RED, in Proceeings of ACM SIGCOMM, August 2000, pp [6] R. Laalaoua, S. Jȩruś, T. Atmaca, an T. Czachórski, Diffusion moel of RED control mechanism, in in Proceeings of International Conference on Networking, July , Colmar, France. [7] E. Altman, K. Avrachenkov, an C. Barakat, A stochastic moel of TCP/IP with stationary ranom losses, in Proceeings of ACM SIGCOMM, 2000, pp [8] Brown P., Resource sharing of TCP connections with ifferent roun trip s, in Proceeings of IEEE INFOCOM, 2000, pp [9] S. Floy et al., Discussions on setting parameters, REDparameters.txt, November [] J. Virtamo, Numerical evaluation of the istribution of unfinishe work in an M/D/1 system, Electronics Letters, vol. 31, no. 7, pp , [11] D. Gross an C. Harris, Funamentals of queueing theory, Wiley, 3r eition, [12] Wang W., D. Tipper, an Banerjee S., A simple approximation for moeling nonstationary queues, in Proceeings of IEEE INFOCOM, March 1996, pp [13] H. C. Tijms, Stochastic Moels: An Algorithmic Approach, Wiley, [14] P. Kuusela an J. T. Virtamo, Moeling RED with two traffic classes, in Proceeings of NTS-15, August 2000, pp , available at com2/. [15] D. Clark an W. Fang, Explicit allocation of best-effort packet elivery service, IEEE/ACM Transactions of Networking, vol. 6, no. 4, pp , August [16] C. Hollot, V. Misra, D. Towsley, an W-B. Gong, A control theoretic analysis of RED, to appear in Proceeings of IEEE INFOCOM 2001, xx/11

12 [17] T. J. Ott, T. V. Lakshman, an L. H. Wong, SRED: Stabilize RED, in Proceeings of IEEE INFOCOM, March [18] A. Veres an M. Boa, The chaotic nature of TCP congestion control, in Proceeings of IEEE INFOCOM, March 2000, pp xx/12