Stability analysis of a max-min fair Rate Control Protocol (RCP) in a small buffer regime

Transcription

1 Stabiity anaysis of a max-min fair Rate Contro Protoco RCP) in a sma buffer regime Thomas Voice and Gaurav Raina Cambridge Consutants and IIT Madras Abstract In this note we anayse various stabiity properties of a max-min fair Rate Contro Protoco RCP) operating with sma buffers. We first tacke the issue of stabiity for networks with arbitrary topoogies. We prove that the max-min fair RCP fuid mode is gobay stabe in the absence of propagation deays, and aso derive a set of conditions for oca stabiity when arbitrary heterogeneous propagation deays are present. The network deay stabiity resut assumes that, at equiibrium, there is ony one botteneck ink aong each route. Lasty, in the simper setting of a singe ink, singe deay mode, we investigate the impact of the oss of oca stabiity via a Hopf bifurcation. keywords rate contro, max-min fairness, sma buffers, stabiity, bifurcation. 1 Introduction The Rate Contro Protoco RCP) [4, 7, 9] takes a radicay different approach for managing fow and congestion contro as compared to the existing Transmission Contro Protoco TCP) congestion avoidance framework. The TCP framework has, embedded in it, an impicit mechanism for detecting congestion within the network. Loss of a packet, which may be caused by the overfow of a buffer, is expected to provide the necessary feedback information. In sharp contrast, RCP aims to achieve fast fow competion times by communicating expicit rate feedback between routers and end-stations [8]. RCP is cosey reated to the expicit Contro Protoco XCP) [16], which aso proposes the use of expicit rate feedback. In fact, there is growing interest in designing scaabe agorithms for rate and congestion contro that use expicit feedback from routers; for exampe, see the recenty proposed JetMax [33] and references therein. RCP, XCP and the JetMax agorithms are intended to converge to a max-min fair resource aocation [4, 7, 16, 33]; see [22, 23, 24, 29] for a sampe of the iterature exporing various issues reated to different notions of fairness in a networking context. The performance of congestion contro agorithms is often couped with the choice of certain parameters in routers in the network. For exampe, such parameters may correspond to different choices of Active Queue Management AQM) 1

2 schemes [6, 14, 19, 21], or the size of buffers in routers [12, 26, 27]. Researchers have begun questioning the design rues for sizing buffers in routers in the Internet [3] and some recent iterature [10, 26, 32] suggests that buffers shoud indeed be much smaer as compared to the current design principes. In the ight of the separate deveopments made towards the understanding of RCP [4, 7, 9] and buffer sizing [10, 26, 32] it is natura to investigate the two, i.e., RCP and sma buffers, together. Such an investigation is aso motivated by the observation that previous buffer sizing studies have focused primariy on the impact of smaer buffers with TCP. In this note, the focus of our anaysis wi be on the imiting regime of a max-min fair RCP fuid mode operating over a communication network with sma buffers. Previous contro theoretic anaysis of RCP has focused on a singe botteneck, where the queue is modeed as a saturated integrator [4]. In our work, we assume that the size of the buffers is so sma that it is unnecessary to mode the queue expicity. Rather, with such sma buffers, at the time scae of operation of the congestion controers, it is the distribution of queue size that pays the important roe [26, 32]. It is worth stating that we do not in any way suggest exacty how sma the size of buffers ought to be. The mode we anayse is simper than the one anaysed by [4] in that we do not expicity mode the queue, but it is more invoved in the sense that our framework represents a heterogeneous network of arbitrary topoogy. In our fuid mode for RCP, apart from the mode for the queue, a other parameters are exacty the same as specified in the origina RCP agorithm [7]. The styes of anaysis we empoy have a been popuar in the study of congestion contro: for exampe, goba stabiity without propagation deays [28, 31], oca stabiity with deays [12, 14, 17, 19, 21, 28, 30], and aso an anaysis of the dynamic system when stabiity may not be guaranteed [13, 18, 25]. We now outine the essence of our theoretica contribution. First, we prove that the fuid mode of RCP is gobay stabe in the absence of propagation deays. Goba stabiity is a desirabe property, and it is reassuring that the fuid mode is gobay stabe, even if it is without propagation time deays. The next natura step is to investigate the stabiity properties of the system with time deays. To that end, we derive a set of conditions for oca stabiity when arbitrary heterogeneous propagation deays are present. The network deay stabiity resut depends on the assumption that, at equiibrium, there is ony one botteneck ink aong each route. Furthermore, in the simper setting of a singe ink, singe deay mode, we are abe to provide a sighty more in-depth study of the RCP agorithm. Using bifurcation anaysis, we investigate the impact of the oss of oca stabiity in a specia case where we show that the RCP agorithm woud aways give rise to a super-critica Hopf bifurcation. An overview of this note foows. In Section 2, we anayse some stabiity properties of RCP over networks of arbitrary topoogies. In Section 3, we investigate oca instabiity of RCP in a singe ink, singe deay mode, via a Hopf bifurcation anaysis. Finay, in Section 4 we summarise our contributions and discuss various avenues for further research. 2

3 2 RCP over arbitrary topoogy networks In this section, our objective is to anayse some stabiity properties of RCP in a sma buffer regime, over networks with arbitrary topoogies. We first show goba stabiity for the RCP fuid mode in the absence of propagation deays, and then derive a set of conditions for oca stabiity when arbitrary heterogeneous propagation deays are present. We begin with the mode description. 2.1 Mode description At the eve of theoretica abstraction, our set-up for a communication network foows the commony adopted framework [28]. We suppose that the communication network consists of an interconnection of a set of routes, S, with a set of inks, J. Each route r S represents a user of the network. Associated with each route is a set of inks which represents the path aong which that user transmits information through the network. Further, a route r has associated with it a fow rate x r t) 0, which represents a dynamic fuid approximation to the rate at which the user is sending packets aong route r, at time t. The fow rate for each r S is determined by the inks r, via the use of expicit rate feedback. Each ink J has associated with it a fow rate R t), which represents the maximum fow rate aowed for routes which pass through. Each packet which is sent through the network carries, in its header, an expicit rate feedback variabe. This variabe is initiay set to the maximum desired fow rate for r. As the packet passes through each r, if the feedback variabe is greater than R t), then ink sets it equa to R t). When the packet reaches its destination, an acknowedgement packet ack), containing the fina vaue of the expicit rate feedback variabe is returned to the origin of r, and the fow rate x r t) is updated accordingy. For each route r and ink r, we et τ r denote the propagation deay from the origin of r to, i.e. the amount of time it takes for a packet to trave from the origin to ink aong route r. Let τ r denote the propagation deay from to the origin of r, i.e. the amount of time it takes for the expicit rate feedback information from ink to reach the user who is transmitting aong route r. In RCP, a packet must reach its destination before an acknowedgement packet ack) is returned to its source. Furthermore, as we are interested in a network with sma buffers, we may safey assume that queueing deays are a negigibe component of the tota end-to-end deay. Thus for a r, τ r + τ r = τ r, the round-trip time for route r. We have the foowing mode for the end-system behaviour of RCP: for each route r S x r t) = min r R t τ r ). 1) Now, for each ink J, R t) is updated depending upon the tota aggregate fow through ink at time t. Athough the update rue is discrete, we can mode it via a fuid approximation with the foowing differentia equation [4]: for each 3

4 ink J α Ṙ t) = R t) C y t)) β ) + q t) d C d 2 C, 2) R t) where α, β are positive constants, C is the capacity at ink, d is the average round-trip time of the fows passing through ink, y t) = r: r x r t τ r ), 3) and q t) is the queue size at time t. Here we use the notation a = b) + c to mean that a = 0 if b < 0 and c 0, otherwise a = b. In the origina RCP mode [7], α = α and β = β for each J for some α and β. We have aowed these constants to vary between inks in order to study their effect on stabiity. Mode for the queue. The basis of our investigation is to study a regime where the buffers are so sma that it is inappropriate to mode the queue as a saturated integrator. This assumption simpy expresses the idea that with sma enough buffers, we may use the approximation q t) = p y t)), 4) where p ) is a continuousy differentiabe function representing the mean queue ength of ink. This is consistent with the observation in [26, 32] that in a sma buffer regime, it is the distribution of the queue size that pays the prominent roe in the dynamics of the congestion contro framework. At the eve of interest in this note, we do not propose any expicit functiona form for the mean queue ength. Different functiona forms may be suitabe candidates, but our primary focus is to investigate some stabiity properties of the dynamica system defined by 1-4). We observe that in some recent anaysis of sma buffer networks it has been shown that, under TCP congestion contro, uness fow access speeds are imited, the arriva and departure of fows to a sma buffer ink can degrade performance [20]. This study has dwet on the dynamics of TCP users arriving and departing. However, under RCP the access speed of each new fow that passes through a ink is bounded by R t). Reation to some previous work in ATM networks. We note that the RCP agorithm, basicay that given by 2), shares simiarities with some agorithms proposed in the past in the context of Asynchronous Transfer Mode ATM) networks: see [1, 2, 11, 15] for some agorithms and anaysis. The most noticeabe difference is that in prior work β = 0, and hence they do not consider the impact of feedback based on queue size. The current state of the art in the anaysis of max-min fair expicit congestion contro agorithms, with feedback deays, is [4], which proposes two forms of feedback: rate mismatch and queue size. Thus far there appears to be no definitive concusion about the excusion of feedback based on queue size and so the anaysis of system 1-4) where β > 0 4

5 is sti important. We aso highight the anaysis of [5], which deveops robust controers to dea with a regime where, due to queue dynamics, the end systems have to cope with varying deays. In contrast, the focus of our anaysis is on controing the distribution of the queue size; a regime aso anaysed in [26]. A key focus of this note is a contro theoretic anaysis of the system described by 1-4), with heterogeneous feedback deays. This aspect has not previousy been anayzed, and so our anaysis presents a step forward. 2.2 Goba stabiity without propagation deays In this subsection, we show that the fuid mode of RCP is gobay stabe when the effects of propagation deays are not taken into account. This provides a check on the goba behaviour of the agorithm and compements our ater oca stabiity anaysis where we incorporate deays into the mode. We note that goba anaysis of fuid modes for congestion contro without deays foowed by a oca anaysis of the same mode with deays is a commony adopted practice in fow contro theory [28]. With RCP, one way of thinking about this is to note that the deay-free mode becomes more accurate when, for a J, the α and β terms in 2) are sma, and so the agorithm evoves on a sower timescae than propagation deays. We first deduce that with sufficienty sma α RCP is gobay stabe, and then use oca anaysis to deveop insight into when, for arger vaues of α, oca instabiities may occur. To mode RCP without the effects of deays we use 1-4) setting τ r = 0, for a r S, but eave a other parameters fixed. Ideay we woud ike to show that our deay-free mode of RCP, at equiibrium, is gobay stabe. Unfortunatey, as the foowing exampe demonstrates, we cannot expect the equiibrium points of 1-4) to be unique, or even isoated. Consider a network consisting of two identica inks,, j and one route r = {, j}. Then, since p ) is stricty increasing, there exists a unique y such that α C y) β p y) d C d 2 C = 0. By inspection, there are equiibrium points at R = y, y ) and R = y, y) for a y y. Aternativey consider the situation where and j are not competey identica, and C j > C ; then, there exists a unique y such that α j C j y ) β jp j y ) d j C j d 2 j C = 0. j Now y > y, so if x r y then Ṙj wi be stricty positive; however, if x r > y then Ṙ wi be stricty negative. Thus, for this exampe, no equiibrium point exists. Hence there may be an entire manifod of equiibrium vectors for R, or there may be no equiibrium point at a; but, the same is not true for the equiibrium vector of fow rates x. 5

6 Theorem 2.1. Suppose that xt) evoves according to 1-4) with τ r = 0 for a r S. Then there exists a unique vector x such that xt) x, as t. Proof. We prove this resut recursivey for the more genera system where, for a J, we repace 2) with Ṙ t) = R t) u t) + α C y t)) β ) + p y t)) d C d 2 C, 5) R t) where u t) 0 as t. Let N be the number of r S such that r. Set y equa to the unique vaue such that Ṙt) = 0 when y t) = y, and set R = min J y /N. We sha show that, for a r S with R = y /N for some r, x r R as t. This aows us to remove such an x r t) from the system, by repacing it with R pus a vanishing term which we incorporate into the u t), for each r. Since this wi aways remove at east one r from S, this is sufficient to prove our resut. It remains to show that x r R for r S with R = y /N, for some r. Now, for any ɛ > 0 there exists a T such that u t) < ɛ α d C, for a J, t > T. From 5) we see that, if, for t > T, for any J, R t) < y /N ) 2ɛ, then Ṙt) > δ for some δ > 0. Thus, for some T, for a t > T, J, R t) y /N ) 2ɛ. Let us assume that ɛ is sma enough so that, for any j J, with y j /N j ) > R, ) y j R > max N N ɛ. j J Let r S be such that R = y /N for some r. Let r be the set of r with R = y /N. Suppose that for some t > T, min R t) > R + 2ɛ max N j. r j J If x r t) = R j t) and j r then x r t) > R + 2ɛ max j J N j. Otherwise x r t) = R j t) with j r, but since t > T, x r > R + 2ɛ max j J N j sti hods. For any r, x r t) is sufficienty high so that y t) must be greater than N R + 2ɛ. Thus, Ṙ t) < δ for some δ > 0. Therefore, for some T, for a t > T, Hence for a t > T, min R t) R + 2ɛ max N j. r j J xr t) R 2ɛ max j J N j. Since ɛ was arbitrary, the resut foows. Note that the vector x wi be cose to that of a max-min fair aocation of fow rates, because of its construction. 6

7 2.3 Loca stabiity with propagation deays In this subsection, we derive conditions for the oca stabiity of 1-4) when propagation deays are present. Our resut depends on the assumption that there is ony one botteneck ink aong each route; that is, for each r S there is ony one r such that x r t) = R t) at equiibrium. For each r S, J, we et x r be the equiibrium vaue of x r t) and R be the maximum of x s for a s S such that s. For each J, we et y be the unique vaue such that Ṙt) = 0 whenever y t) = y. We can assume, without oss of generaity, that for a J y = r: r x r. If this does not hod, then when the system is cose to equiibrium, R t) wi increase for a t. If R t) is initiay much bigger than x r for a r S, then it wi have no effect on a ocay stabe system and can be ignored. The proof of oca stabiity proceeds as a recursive argument. From our assumptions, for each r S there is a unique j r such that x r = R j. Thus for a r S, r, whenever the system is cose to equiibrium, either x r t) = R t τ r ), or ese x r < R and R t τ r ) has no effect on x r t). Starting with the J such that R is minima, we recursivey isoate each J and mode the effect of the rest of the system on R t) as a perturbation. For each J, the stabiity of the isoated systems for j J with R j < R impies that this perturbation is vanishing. We initiay restrict our attention to the singe ink case, J = {}, and we consider the foowing generaisation of 2) ) ) + Ṙ t) = R t) f y t) N R + u t), 6) R t) where y t) is defined as in 3), f ) is an increasing differentiabe function with f 0) = 0 and u t) 0 as t. When the recursive argument is compete, the vector ut) represents the behaviour of R j t) for j J such that R j < R. We examine oca stabiity for the singe ink case 6) by considering what happens if R t) moves away from R. We show that under the conditions provided, as R t) gets cose to moving the furthest away from R, R t) must turn back towards R. As R t) can ony reach smaer and smaer distances away from R, convergence is ensured. Theorem 2.2. Consider the case where the network consists of ony a singe ink, J =, with fow rates foowing 6). If f 0)R τ r < 1, r S then for a ɛ, there exists a and u such that, if R t) R a for a t 0 and u t) < u for a t, then R t) R ɛ for a t and R t) R as t. 7

8 Proof. Let τ be max r S τ r, the maximum round-trip time. Suppose for some t, γ 1, a and u, R t) R = γa, R t ) R a and u t ) < u a for a t [t 2τ, t]. Then for a t [t τ, t], Ṙt ) R + a ) ) u a + f 0) S a + oa ). Thus for a r S So R t τ r ) R t) τ r R f 0) S a + τ r R + a ) a u + oa ). 7) Ṙ t) R t) = f 0) S R y t) ) u t) + oa ) = f 0) S R R t) ) u t) + f 0) r S = f 0) S γa + δ, ) R t) R t τ r ) + oa ) where δ f 0)R τ r f 0) S a + S τ R + a ) a u + oa ) r S = f 0) 2 S 2 R d a + S τ R + a ) a u + oa ). Since f 0) S R d < 1, there exists a vaue of a < ɛ, u and γ < 1 such that, if a a then δ is guaranteed to be ess than γf 0) S a. In which case Ṙt) must have the same sign as R R t). We can take u = u a, with a as given above, and if R t) R a for a t 0 and u t) < u for a t, then R t) R ɛ for a t. Furthermore, for a a < a, there exists a time T such that, for a t > T, u t) < u a. From the above anaysis, we know that after T, if R t) is beyond a of R then it wi converge to R at a rate of γ. Thus there is some T > T such that R t) R a for a t > T. Since a was arbitrary, R t) R as required. We now proceed with our argument by recursivey appying the above Theorem 2.2 to each ink in the genera network case. Theorem 2.3. For system 1-4) the equiibrium point R is ocay asymptoticay stabe provided that, for each J, α + β p y ) ) d C d 2 C R τ r < 1. 8) r: r,r =x r Proof. Let δ be the minimum of R R j /2 for j J. If for a t and a J, R t) R < δ then each J wi evove according to ) ) Ṙ t) = R t) f t) y eq + u t), 9) y eq 8

9 where u t) represents the effect of R j t) for a R j < R, and y eq t) = x r t τ r ), r: r,x r=r with y eq equa to the equiibrium vaue of y eq t). For each J, we have f w) = α w + y d C C ) + β p y + w) d 2 C. By definition, 9) is an exampe of 6) for the reduced network formed by and a routes r S with r and x r = R. Furthermore, 8) is precisey the condition of Theorem 2.2 for this reduced system. Suppose R t) R < δ for a J, for a t. By differentiabiity of p ), for any u we can find an ɛ such that if Rj t) R j < ɛ for a t and for a j such that R j < R, then u t) < u for a t. This aows us to prove oca stabiity recursivey. We begin with the J such that R is maxima. We appy Theorem 2.2 to find conditions under which R t) converges to R and R t) R < δ for a t. This gives us a and u, where the initia conditions of R t) shoud be within a of R and u t) shoud tend to zero and aways be bounded by u. We can find an ɛ < δ so that if Rj t) R j < ɛ for a t and Rj t) converges to R j for a j, then u t) < u for a t and u t) converges to 0. Hence we set R t) [ R a, R + a ] for t 0 as our initia condition for, remove from the network and repeat this process for δ = ɛ. Since we remove one ink each time, eventuay we wi find suitabe initia conditions for the entire network. Note that each ink does not necessariy need to keep track of which fows are under its contro in order to meet 8). The condition hods if, for a J, α + β p y ) ) y d d 2 d p C < 1, where d p is the average per packet round-trip time. Here we say that the roundtrip time of a packet is that of the fow which has transmitted it. Thus d p = 1 x r τ r. y r: r This suggests that perhaps it shoud be d p, and not d, in the RCP controer 2). If for each J we et γ be such that p y ) = γ p y )/y, then p y ) = γ α d β C y y < γ α d β, thus 8) is ensured if, for a J, α = d d p 1 + γ ). 10) 9

10 REMARK 1. This parameter choice scheme is attractive, because it is fairy decentraized and ony requires oca information. Each ink ony needs to determine or estimate the average round-trip times for packets passing through that ink. In current specifications of RCP, sources actuay measure their own round-trip times and advertise them to inks in packet headers. REMARK 2. The network deay stabiity resut depends on the weak assumption that, at equiibrium, each route has ony one botteneck ink. That is, at equiibrium, for each route there is a unique ink with minima per-fow bandwidth avaiabe. Note that a network with mutipe botteneck inks may be transformed into one with ony a singe botteneck ink, for exampe, by making a sma change in the parameters β, J. Preiminary numerica resuts suggest that Theorem 2.3 may hod in genera; but finding an anaytic resut to confirm this, or a counter-exampe to disprove it, remains an open probem. From a practica networking context, there is growing interest in deveoping agorithms that may be used in conjunction with an expicit congestion contro architecture [9, 16, 33]. However, it is imperative to have theoretica guidance to steer a proposed set of agorithms towards a successfu practica impementation. This note represents an advance towards that goa. In the next section we investigate the impact of the oss of oca stabiity in RCP. 3 Loca bifurcation anaysis In the choice of parameters for any congestion contro proposa, a key aim is to ensure that they ead to a stabe equiibrium. Most proposas for congestion contro, for exampe see [28], ead to the anaysis of noninear time deayed dynamica systems. For such noninear systems, typicay sufficient conditions for oca stabiity guide parameter choices. Foowing oca stabiity, a oca bifurcation theoretic anaysis can make us comfortabe in running the system cose to the edge of stabiity. We first recapituate a resut about the oss of oca stabiity in a noninear retarded functiona differentia equation, and then use it to anayse the RCP fuid mode. Foowing the anaysis in [25] we outine a oca Hopf bifurcation resut for the foowing noninear deay equation ut) = η ξ x ut τ) ± ξ xy ut)ut τ) ), 11) where η, τ, ξ x, ξ xy > 0. The parameter η has been intentionay introduced to just tip the above equation over the edge of oca) stabiity. This exogenous, non-dimensiona, parameter wi act as our bifurcation parameter. We now state the foowing Theorem about equation 11): 10

11 Theorem 3.1. A necessary and sufficient condition for oca stabiity is ηξ x τ < π/2, and treating η as the bifurcation parameter, the first oca Hopf bifurcation occurs with period 4τ at η = η c, where η c ξ x τ = π/2. Further, as the Hopf condition is just vioated, the equation wi aways undergo a super-critica Hopf bifurcation where the ampitude of the stabe bifurcating soutions wi be proportiona to ξ x 20πη ηc ). 3π 2 ξ xy 3.1 RCP: singe ink, singe deay mode In our bifurcation anaysis, for the sake of simpicity, we sha eave the term that represents feedback based on queue size out of the mode by taking β = 0. Consider the foowing singe ink, singe deay, RCP fuid mode α ) Ṙt) = ηrt) Cτ C yt)), 12) where yt) = s Rt τ) is the aggregate oad at the ink and η is the nondimensiona bifurcation parameter. Define ut) = Rt) R, and take a Tayor expansion of 12) to obtain ut) = η α τ α ut τ) η ut)ut τ). τr Theorem 3.1 eads us to the foowing concusions about equation 12). A necessary and sufficient condition for oca stabiity is ηα < π/2, and treating η as the bifurcation parameter, the first oca Hopf bifurcation occurs with period 4τ at η = η c, where η c α = π/2. If the Hopf condition is just vioated, the equation wi aways undergo a super-critica Hopf bifurcation where the ampitude of the stabe bifurcating soutions wi be proportiona to 20πη ηc ) R. 3π 2 We emphasise that equation 12) cannot produce a sub-critica Hopf bifurcation; but, in the bifurcation anaysis, we have omitted any additiona noninear effects that woud arise if β > 0 in the RCP mode. In the next section, we summarise our contributions in this note and outine some avenues for further research. 11

12 4 Contributions and avenues for further research As RCP aims for max-min fairness, it is appropriate to first anayze a mode that embodies the origina formuation [7], as we did in this note. We first proved that the RCP fuid mode is gobay stabe in the absence of propagation deays. Then, we derived a set of conditions for oca stabiity when arbitrary heterogeneous propagation deays are present. The network deay stabiity resut depends on the weak assumption that, at equiibrium, there is ony one botteneck ink aong each route. An interesting avenue for research woud be to show if this resut may hod in genera, or to provide a counter-argument to disprove it. The stabiity conditions have the attractive property of being fairy decentraized, and so RCP is potentiay suitabe for depoyment in very arge networks. Observe that the sma buffer regime has enabed us to tacke the question of stabiity for arge networks with arbitrary topoogies. Lasty, in a singe ink, singe deay mode, we investigate the impact of the oss of oca stabiity in a specia case β = 0) where we show that the RCP agorithm woud aways give rise to an innocuous-ooking super-critica Hopf bifurcation. We now outine some natura avenues for research. Choice of parameters. We ony considered the imiting case of very sma buffers. The choice of β impacts the rate at which the queue is drained. Sma vaues of β drain the queue sowy; so with sma β and for arge enough buffers it is appropriate to mode the queue as a saturated integrator. With arge vaues of β, the queue may drain fast enough to necessitate the deveopment of finer queueing theoretic modes. Such queuing modes woud certainy contribute to the noninearity in the RCP dynamica system, and hence affect the stabiity of the time deayed network. Further, the resuts from oca bifurcation theory coud aso be subte. In our bifurcation theoretic anaysis we set β = 0, which removes any noninearity that may arise from feedback based on queue size. In the above modeing regimes, it woud be important to understand the impact of the RCP parameters α and β) on stabiity, instabiity and rate of convergence. Fairness and stabiity. A parameterized famiy of α-fair rate aocations was introduced in [23]. The parameter α ies in the range 0, ), and the cases α 0, α = 1 and α correspond respectivey to an aocation which achieves maximum throughput, is proportionay fair or is max-min fair. In this note we ony considered a max-min fair rate aocation; as originay specified [7]. An important direction for research is to incorporate different notions of fairness into the RCP framework, and anayse some stabiity properties of such networks. Expicit congestion contro. A key question is the scae of the step-change in rate that is necessary, at a resource, to accommodate new fows. Another reevant concern is how resources may be shared between TCP users and those using a rate controed architecture. 12

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