13. equilibrium of heterogeneous congestion control optimality and stability


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13. equilibrium of heterogeneous congestion control optimality and stability

  1. 1. Equilibrium of Heterogeneous Congestion Control: Optimality and Stability Ao Tang, member, IEEE, Xiaoliang Wei, member, IEEE, Steven H. Low, Fellow, IEEE, and Mung Chiang, member, IEEE Abstract—When heterogeneous congestion control protocols in Windows Vista and Window Server 2008 TCP stack [25].that react to different pricing signals (They could be different If explicit feedback is deployed, it will become possible totypes of signals such as packet loss, queueing delay etc. or feed back different signals to different users to implementdifferent values of the same type of signal such as differentECN marking values based on the same actual link congestion new applications and services. Note that in this case, thelevel) share the same network, the current theory based on utility heterogeneous signals can all be loss-based – different usersmaximization fails to predict the network behavior. Unlike in a receiving different explicit values based on the actual link losshomogeneous network, the bandwidth allocation now depends rate – or all delay-based, or a mix. Clearly, going forward, ouron router parameters and flow arrival patterns. It can be non- network will become more heterogeneous in which protocolsunique, suboptimal and unstable. In [36], existence and unique-ness of equilibrium of heterogeneous protocols are investigated. that react to different congestion signals interact. Yet, ourThis paper extends the study with two objectives: analyze the understanding of such a heterogeneous network is rudimentary.optimality and stability of such networks and design control For example, a heterogeneous network, as shown in an earlyschemes to improve them. First, we demonstrate the intricate companion paper [36], may have multiple equilibrium points,behavior of a heterogeneous network through simulations and and they cannot all be stable unless the equilibrium is globallypresent a framework to help understand its equilibrium prop-erties. Second, we propose a simple source-based algorithm to unique. This still leaves many other important properties open,decouple bandwidth allocation from router parameters and flow such as optimality and stability of equilibrium.arrival patterns by only updating a linear parameter in the In a homogeneous network, even though the sources maysources’ algorithms on a slow timescale. It is used to steer a control their rates using different algorithms, they all adaptnetwork to the unique optimal equilibrium. The scheme can be to the same congestion signal, e.g., all react to packet lossdeployed incrementally as the existing protocol needs no changeand only the new protocols need to adopt the slow timescale rate, as in the various variants of Reno and TFRC [8], or alladaption. to queueing delay, as in Vegas and FAST. For homogeneous networks, besides various detailed studies (see e.g., [27], [30]), there is already a well-developed theory, based on network I. I NTRODUCTION utility maximization, e.g. [17], [21], [23], [24], [26], [43], Congestion control in TCP (Transmission Control Protocol), that can help understand and engineer network behaviors. Infirst introduced in [11], has enabled the explosive growth particular, it is known that a homogeneous network of generalof the Internet. The currently predominant implementation, topology always has a unique equilibrium (operating point). Itreferred to as TCP Reno in this paper, uses packet loss as the maximizes aggregate utility, and the fairness associated withcongestion signal to dynamically adapt its transmission rate, it can be well predicted and controlled. More importantly, theor more precisely, its window size.1 It has worked remarkably allocation depends only on the congestion control algorithmswell in the past, but its limitations in wireless networks and in (equivalently, its underlying utility functions) but not on net-networks with large bandwidth-delay product have motivated work parameters (e.g., buffer sizes) or flow arrival patterns,various proposals some of which use different congestion and hence can be designed through the choice of end-to-endsignals. For example, in addition to loss based protocols such TCP algorithms.as HighSpeed TCP [9], STCP [19] and BIC TCP [42], schemes In contrast, we demonstrate in Section II of this paperthat use queueing delay include the earlier proposals CARD that the bandwidth allocation among heterogenous flows can[13], DUAL [39] and Vegas [3], and the recent proposal depend on both network parameters and flow arrival patterns.FAST [40]. Schemes that use one-bit congestion signal include It means that in general we cannot predict, nor control,ECN [28], and those that use multi-bit feedback include XCP the bandwidth allocation through the current design of end-[15], MaxNet [41], and RCP [6]. Indeed, recently the Linux to-end congestion control algorithms for heterogeneous net-operating system already allows users to choose from a variety works. This implies, for example, the standard “TCP friendly”of congestion control algorithms since the kernel version concept is not well defined anymore. To fully understand2.6.13, including TCP-Illinois [22] that uses both packet loss heterogeneous networks and develop ways to address theseand delay as congestion signals. Recently, compound TCP issues, we review our basic model in Section III. By iden-[33] which also uses multiple congestion signals is deployed tifying an optimization problem associated with any given 1 All our experiments and simulations use NewReno with SACK. These equilibrium point, we discuss efficiency in Section IV-A andare enhanced versions of the original Tahoe and Reno, but we will refer them fairness in Section IV-B. Study of stability then follows ingenerically as TCP Reno. Section V. Finally, we propose a general scheme to steer
  2. 2. an arbitrary heterogeneous network to the unique equilibrium congestion window W according tothat maximizes the standard weighted aggregate utility by baseRT Tupdating a linear scaler in the sources’ algorithms on a slow W ← W +α (1)timescale (Section VI). The scheme requires only local end- RT Tto-end information but does assume all flows have access to In equilibrium, each FAST flow i achieves a throughput x∗ = ia common price, which is generally true in practice since the ∗ ∗ α/qi , where qi is the equilibrium queueing delay observedcommon price can be what the incumbent dominate protocol by flow i. Hence, α is the number of packets that each FASTuses. It can be deployed incrementally as the existing protocol flow maintains in the bottleneck links along its path.needs no change and only the new protocols need to adopt In this example, one FAST flow and one Reno flow sharethe slow timescale adaption. Packet-level (ns-2) simulation a single bottleneck link with capacity of 8.3 pkts per msresults using TCP Reno and FAST are presented in Section VII (equivalent to 100Mbps with maximum packet size) and roundand Linux experiments on a realistic testbed are reported in trip propagation delay 50ms. The topology is shown in FigureAppendix IX-C to further discuss some issues that are ignored 1. The FAST flow fixes its α parameter at 50 packets.in the mathematical model. We conclude in Section VIII. We summarize here the main results that we have derived 3 5about heterogeneous congestion control in [36] and this paper: 83 de la ms 1 FAST flow 0 pk la de t/ y t/ y pk • Existence of equilibrium: Theorem 2 in [36]; ms 83 c=8.3pkt/ms 0 • Uniqueness of equilibrium. 1 25 ms one way 2 83 d la ms – Local uniqueness: Theorem 3 in [36]; pk el de t/ 0 y t/ ay 0 3pk 1 Reno flow – Global uniqueness: Theorems 7 and 12 in [36]. ms 8 4 6 • Optimality of equilibrium – Efficiency: Theorems 1 and Corollary 3 in this paper; Fig. 1. Single link example. – Fairness: Theorems 4 and 5 in this paper. • Stability of equilibrium: In all of the ns-2 simulations in this paper, heavy-tail noise – Local stability: Theorem 6 in this paper; traffic is introduced at each link at an average rate of 10% of – Special results: Theorems 12 and 13 in this paper. the link capacity.2 Figure 2 shows the result with a bottleneck • Control of heterogeneous networks: Theorem 11, Algo- buffer size B = 400 packets. In this case, FAST gets an rithms 1 and 2 in this paper. average of 2.1 pkts per ms while Reno gets 5.4 pkts per ms. Figure 3 shows the result with B = 80 packets. Since the bottleneck buffer size is smaller, the average queue is also II. T WO M OTIVATING E XAMPLES smaller. Therefore FAST gets a higher throughput of 3.4 pkts In this section, we describe two simulations to illustrate per ms and Reno gets a much lower throughput of 0.6 pkt persome peculiar throughput behavior in heterogenous networks. ms. In this case, the loss rate is fairly high and the aggregateAll simulations use TCP Reno, which uses packet loss as throughput is much lower (53.6 percent utilization) than thecongestion signal, and FAST TCP, which uses queueing delay bottleneck capacity due to many timeout events.as congestion signal. In summary, contrary to the case of homogeneous network, The first experiment (Example 1a) shows that when a bandwidth sharing between Reno and FAST depends on net-Reno flow shares a single bottleneck link with a FAST flow, work parameters in a heterogeneous network.the relative bandwidth allocation depends critically on thelink parameter (buffer size): the Reno flow achieves higher 7 FAST Reno 7 FAST Renobandwidth than FAST when the buffer size is large and smaller 6 6bandwidth when it is small. This implies that one cannot 5 5 throughput (pkt / ms) throughput (pkt / ms)control the fairness between Reno and FAST through just 4 4the design of end-to-end congestion control algorithms, since 3 3fairness is now linked to network parameters, unlike in the 2 2case of homogeneous networks. 1 1 The second experiment (Example 2a) shows that even on a 0 0 200 400 600 800 1000 1200 1400 1600 1800 0 0 200 400 600 800 1000 1200 1400 1600 1800 time(sec) time(sec)(multi-link) network with fixed parameters, one cannot control (a) A sample trajectory (b) Average behaviorthe fairness between Reno and FAST because the relativeallocation can change dramatically depending on which flow Fig. 2. FAST vs. Reno with a buffer size of 400 pkts.starts first! 2 We usually present one sample figure on the left and the summary figureA. Example 1a: dependence of bandwidth allocation on net- on the right. The sample figure shows the rate trajectory in one simulation run.work buffer size The rate value is measured every 2 seconds. The summary figure presents the rate trajectory averaged over 20 simulation runs with different random seeds. FAST [40] is a high speed TCP variant that uses delay as Each point in the summary figure represents the average throughput over aits main control signal. Periodically, a FAST flow adjusts its period of one minute. The error bars are also shown in the summary figure.
  3. 3. 7 7 6 FAST Reno 6 FAST Reno 1-2 and link 3-4. FAST flows experience large queueing delays and are never able to fully utilize link 2-3. 5 5 throughput (pkt / ms) throughput (pkt / ms) 4 4 4 4 FAST FAST Reno Reno 3 3 3.5 3.5 3 3 2 2 throughput (pkt / ms) throughput (pkt / ms) 2.5 2.5 1 1 2 2 0 0 0 200 400 600 800 1000 1200 1400 1600 1800 0 200 400 600 800 1000 1200 1400 1600 1800 time(sec) time(sec) 1.5 1.5 (a) A sample trajectory (b) Average behavior 1 1 0.5 0.5Fig. 3. FAST vs. Reno with a buffer size of 80 pkts. 0 0 1000 2000 3000 4000 5000 6000 7000 8000 0 0 1000 2000 3000 4000 5000 6000 7000 8000 time(sec) time(sec) (a) A sample trajectory (b) Average behaviorB. Example 2a: dependence of bandwidth allocation on flow Fig. 5. Bandwidth shares of Reno and FAST when FAST starts first.arrival pattern The topology of this example is shown in Figure 4. We use 4 4RED algorithm [7] and packet marking instead of dropping. FAST FAST Reno Reno 3.5 3.5The marking probability p(b) of RED is a function of queue 3 3length b: throughput (pkt / ms) throughput (pkt / ms) 2.5 2.5  2 2  0  b≤b 1.5 1.5 1 b−b p(b) = b≤b≤b (2) 1 1  K b−b  1 b≥b 0.5 0.5 K 0 0 0 1000 2000 3000 4000 5000 6000 7000 8000 0 1000 2000 3000 4000 5000 6000 7000 8000 time(sec) time(sec)where b, b and K are RED parameters. Links 1-2 and 3-4 are (a) A sample trajectory (b) Average behaviorboth configured with 9.1pkts per ms capacity (equivalent to111 Mbps), 30 ms one-way propagation delay, and a buffer Fig. 6. Bandwidth shares of Reno and FAST when Reno starts first.of 1500 packets. Their RED parameters are (b, b, K) = (300,1500, 10000). Link 2-3 has a capacity of 13.8 pkts per ms (166 In short, bandwidth sharing in heterogeneous networksMbps) with 30 ms one-way propagation delay and a buffer size may depend on which type of TCP starts first and becomesof 1500 packets. Its RED parameters are set to (0, 1500, 10). unpredictable. There are eight Reno flows on path 1-2-3-4, utilizing allthree links, with one-way propagation delay of 90 ms. There III. M ODELare two FAST flows on each of paths 1-2-3 and 2-3-4. Both A. Notations and Assumptionsof them have one-way propagation delay of 60 ms. All FASTflows use a common α = 50 packets. Consider a network consisting of a set of L links, indexed by l = 1, . . . , L, with fixed finite capacities cl . We sometimes Path 1 Path2 abuse notation and use L to denote both the number of links (2 FAST flows) (2 FAST flows) and the set L = {1, . . . , L} of links. Each link has a price pl as its congestion measure. There are J different congestion c=9.1pkt/ms c=13.8pkt/ms c=9.1pkt/ms 1 30ms one way 2 30ms one way 3 30ms one way 4 control protocols indexed by superscript j, and N j sources (300,1500,10000) (0,1500,10) (300,1500,10000) using protocol j, indexed by (j, i) where j = 1, . . . , J and Path 3 i = 1, . . . , N j . The set of links used by source (j, i) is denoted (8 Reno flows) by L(j, i), and the total number of sources by N := j N j . The L × N j routing matrix Rj for type j sources is defined jFig. 4. Multiple equilibria scenario. by Rli = 1 if source (j, i) uses link l, and 0 otherwise. The overall routing matrix is denoted by In our simulations, one set of flows (Reno or FAST) starts at R= R1 R2 ··· RJtime zero, and the other set of flows starts at the 100th second.We presents the throughput achieved by one of the FAST flows Even though different classes of sources react to differentand one of the Reno flows. Each point in the summary figures prices, e.g. Reno to packet loss probability and Vegas/FASTrepresents the average rate over 5 minutes. Figure 5 shows to queueing delay, the prices are related. We model thisthe scenario in which FAST flows start first. Initially, FAST relationship through a price mapping function that maps aflows occupy most of the buffers in link 2-3. With the steep common “intrinsic” price (e.g. queue length) at a link toRED dropping slope in link 2-3, the Reno flows experience different prices (e.g. loss probability and queueing delay)heavy loss and have very small throughput when they join the observed by different sources. Formally, every link l has anetwork. Figure 6 shows the scenario in which Reno flows price pl . A type j source reacts to the “effective price” mj (pl ) lstart first. Initially, Reno flows maintain large queues in link in its path, where mj is a price mapping function that can l
  4. 4. depend on both the link and the protocol type. The exact form bandwidth cost, and network links adjust bandwidth pricesof mj depends on the AQM algorithm used at the link; see l according to the utilization of the links:(2) for links with RED. Let mj (p) = (mj (pl ), l = 1, . . . L) l −1 +and m(p) = (mj (pl ), j = 1, . . . J). The aggregate prices for xj qi i j = Uij j qisource (j, i) is defined as j pl (t) = ˙ yl (p(t)) − cl =: fl (p(t)) (5) qi = Rli mj (pl ) j l (3) l Remark: There are different fluid models in the literature. For j example, the “primal algorithm” has dynamics at sources whileLet q j = = 1, . . . , N j ) and q = (q j , j = 1 . . . , J) (qi , i T the congestion signal at links depends on the instantaneousbe vectors of aggregate prices. Then q j = Rj mj (p) and arrival rate or even both arrival rate and queue state. One Tq = R m(p). is refered to e.g., [5], [18], [21], [31] for related discussion Let xj be a vector with the rate xj of source (j, i) as its i and justification. The main issues of heterogeneous congestionith entry, and x be the vector of xj : control (multiple equilibria, optimality loss and asymmetric Ja- T cobian which may lead to instability) remain the same for both x = (x1 )T , (x2 )T , . . . , (xJ )T the primal and dual models. In other words, the difficulty dueSource (j, i) has a utility function3 Uij (xj ) that is strictly con- to heterogeneity is the same for various dynamical models. For icave increasing in its rate xj . Let U = (Uij , i = 1, . . . , N j , j = example, if there are two marking functions p1 (t) = gl (yl (t)) l 1 i 2 21, . . . , J). and pl (t) = gl (yl (t)) at the same link l as in the primal model, With the above notation, we refer to (c, m, R, U ) as a then these functions g serve the role of m functions definednetwork, where (in general) z denotes the (column) vector above and aggregate rate yl becomes the intrisic measure ofz = (zk , ∀k). The following basic assumptions are adopted, as congestion as pl defined before. The results and techniquesin [36] that studies the existence and uniqueness of equilibrium developed in this paper should be useful for analyzing otherfor heterogeneous protocols. models. −1A1: Utility functions Uij are strictly concave increasing, and Under the assumptions in this paper, Uij j qi > 0 for twice continuously differentiable in their domains. Price all the prices p that we consider, and hence we can ignore the mapping functions mj are continuously differentiable projection [·]+ and assume, without loss of generality, that l and strictly increasing with mj (0) = 0. −1 l xj qi i j = Uij j qi (6)A2: For any > 0, there exists a number pmax such that if pl > pmax for link l, then (6) is nothing but the “response function” of TCP which deter- mines source rate based on its observed end-to-end congestion xj (p) < i j for all (j, i) with Rli = 1 signal. These are mild assumptions. Concavity and monotonicity In equilibrium, the aggregate rate at each link is no moreof utility functions are often assumed in network pricing for than the link capacity, and they are equal if the link price iselastic traffic. The assumption on mj means that sources to strictly positive. Formally, we call p an equilibrium price (or lobserve the fluctuation as link congestion (pl ) rises and falls, as a network equilibrium or an equilibrium) if it satisfies (fromthey must in order to control congestion. Assumption A2 says (3), (6), (4))that when pl is high enough, then every source going through P (y(p) − c) = 0, y(p) ≤ c, p ≥ 0 (7)link l has a rate less than , modeling the basic intuition incongestion control. where P := diag(pl ) is a diagonal matrix. When all sources react to the same price, then the equilib- rium described by (3), (4), (6) and (7) is the unique solutionB. Network Model of the following utility maximization problem defined in [17] As usual, we use xj q j = xj qi , i = 1, . . . , N j and j and its Lagrange dual [23]: ix(q) = xj q j , j = 1, . . . , J to denote the vector-valued max Ui (xi ) (8) x≥0functions composed of xj . Since q = RT m(p), we often abuse i inotation and write xj (p), xj (p), x(p). Define the aggregate subject to Rx ≤ c (9) isource rates y(p) = (yl (p), l = 1, . . . , L) at links l as: where we have omitted the superscript j = 1. The strict y j (p) = Rj xj (p), y(p) = Rx(p) (4) concavity of Ui guarantees the existence and uniqueness of the optimal solution of (8)–(9) as well as the global convergence We consider the “dual algorithm” [17], [23] 4 where sources of the dual algorithm.select transmission rates that maximize their utility minus For heterogeneous case, the utility maximization problem no longer underlies the equilibrium described by (3), (4), (6) 3 Most TCP variants proposed or deployed can be shown to implicitly and (7). The current theory cannot be directly applied andmaximize some strictly concave increasing utility functions [24]. Here we substantial difficulties had to be overcome when exploringtake this reverse-engineering view and use utility function to represent theexact form of congestion protocol. even some basic questions such as existence and uniqueness 4 Delay is omitted for simplicity. of equilibrium [36].
  5. 5. IV. O PTIMALITY Corollary 2. All equilibrium points are Pareto efficient. As we have shown in [36], for heterogeneous congestion Pareto efficiency can be viewed as a necessary requirementcontrol networks, equilibrium cannot be characterized by (8)– for an efficient allocation. An equilibrium is optimal if it is(9) anymore. In this section, we further investigate the devia- Pareto efficient and maximizes (possibly weighted) aggregatetion of optimality in terms of both efficiency and fairness. This utility. As shown in (8)-(9), for the homogeneous case, theanalysis provides insights on networks with heterogeneous equilibrium is indeed optimal. For the heterogeneous case,congestion signals, for example, how to define inter-protocol Theorem 1 implies a bound on the loss in optimality, as thefairness. It also motivates the algorithm design in section VI. following corollary states. Corollary 3. Assume all utility functions are nonnegative, i.e.,A. Efficiency U (x) ≥ 0. Suppose the optimal aggregate utility is U ∗ and We first make the following key observation, which moti- ˆ U is the achieved aggregate utility at an equilibrium (ˆ) of a xvates other results on optimality and algorithm development. network with heterogeneous protocols. ThenTheorem 1. Given an equilibrium p∗ , there exists a positive Uˆ γminvector γ(p), such that the equilibrium rate vector x∗ (p) is the ∗ ≥ (17) U γmaxunique solution of following problem: where γmin and γmax are any lower and upper bounds of max γi Uij (xj ) j i (10) j j γi 5 , i.e., γmin ≤ γi ≤ γmax . x≥0 i,j subject to Rx ≤ c (11) Proof. Assume x is one of the solutions of (10)-(11), then ˆ max γi Uij (xj ) = j j xi ˆ γi Uij (ˆj ) ≤ γmax U (18) x i i,j i,jProof. The KKT (Karush-Kuhn-Tucker) optimality conditions On the other hand,for (10)-(11) are: max j γi Uij (xj ) ≥ γmin max Uij (xj ) = γmin U ∗ (19) j γi Uij (xj ) = i j Ril pl for all (i, j) (12) x i,j i x i,j i l Uˆ γmin pT (Rx − c) = 0 (13) Combining the two equalities above, we get U∗ ≥ γmax Rx − c ≤ 0 (14) It has been well known that price can serve as the “in-where the (x, p) are the primal-dual variables. We now claim visible hand” to coordinate competing users and realize op-these conditions are satisfied with equilibrium rates and prices timal resource allocation. That however requires two basic(x∗ , p∗ ) by choosing assumptions. The first assumption is that users are all price j ∗ takers. If instead they are noncooperative game players, there j l Ril pl will be efficiency loss. Such “price of anarchy” was recently γi = j j ∗ (15) l Ril ml (pl ) bounded from above for both routing [29] and congestionTo see this, note (13) and (14) are conditions for equilibrium. control [14]. The second assumption is the homogeneity ofAfter substituting (15) into (12), we have price, which does not hold in networks with more than one type of congestion control signals. Our result above attempts Uij (xj∗ ) = j Ril mj (p∗ ) l (16) to quantify the “price of heterogeneity” in congestion control. i l lThat is consistent with equations (3) and (6) that are used to B. Fairnessdefine equilibrium. In this subsection, we study fairness in networks shared by Unlike the homogenous case where the equilibrium maxi- heterogeneous congestion control protocols. Two questions wemizes aggregate utility i Ui (xi ), in the heterogeneous case, address are: how the flows within each protocol share amongan equilibrium x(p∗ ) maximizes a weighted aggregate utility themselves (intra-protocol fairness) and how these protocols j j i,j γi Ui (xi ), where the weight depends on the equilib- share bandwidth in equilibrium (inter-protocol fairness). Therium itself. Theorem 1 characterizes this underlying convex results here generalize the corresponding theorems in [35].optimization problem that an equilibrium solves. It further 1) Intra-protocol fairness: As indicated by (8)–(9), when amotivates the algorithm in section VI. Since this optimization network is shared only by flows using the same congestionproblem itself depends on the equilibrium, it cannot be used to signal, the utility functions describe how the flows sharefind equilibrium directly, nor does it guarantee existence and bandwidth among themselves. When flows using differentuniqueness properties as in the single-protocol case [36]. congestion signals share the same network, this feature is still As stated by the celebrated first fundamental theorem of preserved “locally” within each protocol.welfare economics, assuming a homogeneous price signal,any competitive equilibrium is Pareto efficient. As a direct 5 Both γmin and γmax can be bounded using mj . For example, for a ˙l network with both loss based and delay based protocols and assuming REDcorollary of Theorem 1, the same holds for networks with is used, the slopes of RED at different links can be used to compute γminheterogeneous price signals. and γmax .
  6. 6. Theorem 4. Given an equilibrium (ˆ, p), let cj := Rj xj be x ˆ ˆ ˆ Let X := { x| xj (µ) ≤ xj ≤ xj (µ), µ ≥ 0, Rx ≤ c}. Xthe total bandwidth consumed by flows using protocol j at includes all possible rates of flows using protocol j if theyeach link. The corresponding flow rates xj are the unique ˆ were given strict priority over other flows or if others weresolution of: given strict priority over them, and all rates in between. In Nj this sense X contains the entire spectrum of inter-protocol max Uij (xj ) subject to Rj xj ≤ cj ˆ (20) fairness among different protocols. The next result says that i j x ≥0 i=1 every point in this spectrum is achievable by an appropriate choice of parameter µ.Proof: Since (ˆj , pj ) ≥ 0 is an equilibrium, from (3) to (7), x ˆ Let S(µ) denote the set of equilibrium rates of flowswe have when the protocol parameter is µ. Clearly, equilibrium is Uij xj ˆi = Rli pj j ˆl for i = 1, ..., N f characterized by (3), (4), (7) and (21). l Theorem 5. For every link l, assume there is at least one typeThis, together with (from the definition of cj ) ˆ J flow that only uses that link. Given any x ∈ X, there exists an µ ≥ 0 such that x ∈ S(µ). Rli xj ≤ cj , j ˆi ˆl pj ˆl Rli xj − cj j ˆi ˆl = 0, ∀l Proof: Given any x ∈ X, the capacity for all type J flows is i i c − k=J Rk xk . Since Rx ≤ c (for all coordinates), we haveforms the necessary and sufficient condition for xj and pj to ˆ ˆ c − k=J Rk xk ≥ RJ xJ , which is greater than or equal to 0.be optimal for (20) and its dual respectively. Hence the following utility maximization problem solved by flows of type J is feasible: Note that in Theorem 4, the “effective capacities” cj are ˆnot preassigned. They are the outcome of competition among max UiJ (xJ ) i xJ ≥0flows using different congestion prices and are related to inter- iprotocol fairness, which we now discuss. subject to RJ xJ ≤ c − Rk xk 2) Inter-protocol fairness: Even though flows using differ- k=Jent congestion signals individually solve a utility maximization Let pJ be the associated Lagrange multiplier vector. Byproblem to determine their intra-protocol fairness, they in the assumption that every link has at least one single-linkgeneral do not jointly solve any predefined convex utility type J flow, we know pJ > 0 for all l. Choose µj = l imaximization problem. Here we provide a feasibility result, Rli mj ((mJ )−1 (pJ )) jwhich says any reasonable inter-protocol fairness is achievable l l l (Ui ) (xj ) j l . It can be checked that all equations iby linearly scaling congestion control algorithms. that characterize an equilibrium (3), (4), (7) and (21) are Assume flow (j,i) has a parameter µj with which it chooses i satisfied.its rate in the following way: In general, one can view Theorem 1 as defining fairness of −1 1 flows using heterogeneous protocols and can conclude that xj i j qi = Uij qj j i (21) µi price mapping functions (router parameters) affect fairness (supported by Example 1a). Clearly, if one can choose priceOur main result here says that for a network with J protocols, mapping functions, one can achieve any predefined fairness.given any desirable bandwidth allocation across protocols, More interestingly, Theorem 5 implies that given any reason-there exists a µ vector such that one of the resulting equilibria able fairness among flows using different congestion signals,achieves the given bandwidth partition. Before stating the in terms of a desirable rate allocation x, there exists a protocoltheorem, we first characterize the feasible set of predefined parameter vector µ that can achieve it without changingbandwidth allocation. parameters inside the network. In section VI, we will discuss Assume that except for j = J, flow (j, i) has parameter µj . i distributed algorithms to compute a particular µ, which willOr equivalently, we can define µJ = 1. The equilibrium rates i result in the optimal bandwidth allocation.xj clearly depend on parameter µ. For j = 1, 2, ...J − 1, letxj (µ) be the unique rate vector of flows using protocol j if V. S TABILITYthere were no other protocols in the network, i.e., xj (µ) solvesthe following problem. For general dynamical systems, a globally unique equilib- rium point may not even be locally stable [16], [32]. In this Nj section, we focus on the stability of heterogeneous congestion max µj Uij (xj ) i i subject to Rj xj ≤ c j x ≥0 control protocols, which dictates whether an equilibrium can i=1 manifest itself experimentally or not [35]. For general net-Let xj (µ) be the unique rates of type j flows if network works, it is shown that once the degree of heterogeneity iscapacity were (c − k=j Rk xk )+ and no other protocols are properly bounded, the equilibrium is not only unique as shownin the network, i.e., xj (µ) solves the following problem. in [36] but also locally stable. Stronger results for some special Nj cases can be found in the Appendix IX-A.max j µj Uij (xj ) i i subject to Rj xj ≤ (c − Rk xk )+ We now state the general result on local stability. It essen-x ≥0 tially says that if the similarity condition on price mapping i=1 k=j
  7. 7. functions that guarantees uniqueness [36] is satisfied, the P -matrix by Lemma 7. By similar arguments in [36], it isunique equilibrium is also locally stable. In particular, if for enough to show det(M (−J )) > 0, which will be done in theany l all mj are the same, then (22) is satisfied and the l remainder of the proof.equilibrium is locally stable. This certainly agrees with our Following [36], let π denote an L-bit binary sequenceknowledge on the homogeneous case. that represents the path consisting of exactly those links k We call a vector σ = (σ1 , . . . , σL ) a permutation if each for which the kth entries of π are 1, i.e., πk = 1. Letσl is distinct and takes value in {1, . . . , L}. Treating σ as a Π(k, l) := {π|πk = πl = 1} be the set of paths that contain j jmapping σ : {1, . . . , L} → {1, . . . , L}, we let σ −1 denote both links k and l. Let Iπ = {i|Rli = 1 if and only if πl = 1}its unique inverse permutation. For any vector a ∈ L , σ(a) be the set of type j sources on path π, possibly empty. Letdenotes the permutation of a under σ, i.e., [σ(a)]l = aσl . −1If a ∈ {1, . . . , L}L is a permutation, then σ(a) is also a j j ∂ 2 Uij rπ = rπ (p) = − (23)permutation and we often write σa instead. Let l = (1, . . . , L) j ∂(xj )2 i i∈Iπdenote the identity permutation. Then σl = σ. Finally, denote j jdmj /dpl by mj . l ˙l where rπ is zero if Iπ is empty. Denote by 1(a) the indicator function that is 1 if the assertion a is true and 0 otherwise.Theorem 6. If for any vector j ∈ {1, . . . , J}L and any Definepermutations σ, k, n in {1, . . . , L}L , L L L L µ(j) := mjl ˙l (24) [k(j)]l [n(j)]l [σ(j)]l ml ˙ + ml ˙ ≥ ml ˙ (22) l=1 l=1 l=1 l=1 L j ρ(j, π) := rπll (25)then the equilibrium of a regular network is locally stable. l=1Proof. For a real matrix A, if all its principle minors are For any permutation k, Define L+ = {l|kl = l} and L− = k kpositive, A is called a P -matrix [34]. If aii ≥ 0, aij ≤ 0, then {l|kl = l}. We then haveA is called an M -matrix. Clearly if a P -matrix is symmetric,then it is positive definite and hence stable. However, the Ja- det(M (−J )) = G(j, π) ρ(j, π) (26)cobian matrix in our problem is not symmetric when multiple j πprotocols exist, which is the main difficulty in proving stability. where the last summation in (26) is over the vectorBefore getting into the main proof, we state three lemmas. One index π = (π 1 , . . . , π L ) that takes value in the setis referred to [1] for other related results. { all L-bit binary sequences }L . l = (1, . . . , L) denotes theLemma 7. If A is a P -matrix and also an M -matrix, then identity permutation, and “π ∈ Π(k, l)” is a shorthand forall its eigenvalues have positive real parts. “π l ∈ Π(kl , l), l = 1, . . . , L”. and − Let e be the column vector e = [1, 1, · · · , 1]T . G(j, π) := 1(π ∈ Π(k, l)) sgnk(−1)|Lk | µ(j)(27) kLemma 8. If A is an M-matrix and all its eigenvalues havepositive real parts, then there is an D = diag[d1 , · · · , dn ], Then let Θ0 be the largest subset of the set of all possibledi > 0 for all i, such that D−1 ADe > 0. In other words, A (j, π)’s that is permutationally distinct, i.e., no vector in Θ0is strictly diagonally dominant. is a permutation of another vector in Θ0 . We then have For a matrix A, we define its comparison matrix M (A) = det(M (−J (p))) = H(j, π) ρ(j, π) (28) (j,π)∈Θ0(mij ) by setting mii = |aii |, and mij = −|aij | if i = j.Clearly M (A) is an M -matrix. The following lemma pointsout a simple yet important fact that relates diagonal dominance H(j, π) = 1(σ(π) ∈ Π(k, l))T (29)property of A with positive diagonal entries and that of M (A). σ∈Σ(j,π) kLemma 9. Suppose all diagonal entries of A are positive. If wherethere is a an D = diag[d1 , · · · , dn ], di > 0 for all i, such that − T = sgnk(−1)|Lk | µ(σ(j))D−1 M (A)De > 0, then D−1 ADe > 0, i.e., A is also strictlydiagonally dominant. and Σ(j, π) is the largest subset of the set of all permutations σ that generates distinct σ(j, π). We now state the proof of Theorem 6. We need to show We now use (29) to derive a sufficient condition underall eigenvalues of −J have positive real parts, where J is the which H(j, π) are nonnegative for all permutationally distinctJacobian of equilibrium equations (J = ∂y/∂p) evaluated at (j, π). The main idea is to show that for every negative termequilibrium. It is enough to show −J is strictly diagonally in the summation in (29), either it can be exactly cancelled bydominant and by Lemma 9 we only need to show M (−J ) is a positive term, or we can find two positive terms whose sumstrictly diagonally dominant since all diagonal entries of −J has a larger or equal magnitude under the given condition.are positive (each link has at least one flow using it). Using Theorem 6 is then directly implied by the following Lemma,Lemma 8, it suffices to show that M (−J ) is positive stable, whose proof is provided in the Appendix IX-B.which then can be reduced to check whether M (−J ) is a
  8. 8. Lemma 10. Suppose for any j ∈ {1, . . . , J}L and permuta- Algorithm 1 Two timescale control schemetions σ, k, n in {1, . . . , L}L , we have for a regular network 1) Every source chooses its rate by j −1 qi (t) xj (t) = (U ) i µj (t) ; µ(k(j)) + µ(n(j)) ≥ µ(σ(j)) i 2) Every source updates its µj by iThen, for all (j, π) ∈ Θ0 , H(j, π) ≥ 0. mj (pl (t+T )) µj (t + T ) = µj (t) + κj i i i l∈L(j,i) l − µj (t) i l∈L(j,i) pl (t+T ) VI. S LOW TIMESCALE UPDATE where κj is the stepsize for flow (j, i) and T is large iA. Motivation enough so that the fast timescale dynamics among x and p can reach steady state. As pointed out in Corollary 2, all equilibria are Paretoefficient. However, based on analysis in section IV, largeefficiency loss may occur and no guarantee on fairness can beprovided. This motivates us to turn from analysis to design, Parameter w enables us to control fairness and to achieveand develop a readily implementable control mechanism that any desired fair bandwidth allocation. Moreover, Theorem 11“drives” any network with heterogeneous congestion control suggests Algorithm 1 as a two-timescale scheme to controlprotocols to a target operating point with a fair and efficient the operating point of networks with heterogenous congestionbandwidth allocation. Our target equilibrium is the maximizer control protocols. The essential idea in Algorithm 1 is thatof some weighted aggregate utility. The first step is to set up by reacting to the same price (pl (t)) on slow timescale, itthe existence and uniqueness of such a solution. is guaranteed to reach the optimal equilibrium in the long run. Yet the algorithm allows sources to react to their ownTheorem 11. For any given network (c, m, R, U ), for any effective prices mj (pl (t)) on fast timescale. This flexibility on ipositive vector w, there exists a unique positive vector µ such timescales is important in practice when, for example, the linkthat, if every source scales their own prices by µj , i.e., i prices pl are loss probability that are hard to reliably estimate −1 1 on the fast timescale. The slow timescale algorithm only xj = Uij i mj (pl ) l (30) updates a linear scaler (µj ), which is readily implementable, µj i i e.g., this corresponds to updating a parameter α in FAST; seethen, at equilibrium(x, p), x solves Section VII. In general, one can always choose mj (pl ) = pl l for a particular j, say j = 1. Then µ1 = 1. This is desirable for i 1 max Uij (xj ) i (31) incremental deployment as only new protocols need to adapt x≥0 wj (i,j) i while the current Reno (j = 1) does not. subject to Rx ≤ c (32)Moreover, B. Numerical Examples mj (pl ) Throughout this subsection, we provide some numerical 1 l∈L(j,i) l µj i = j results to further validate the effectiveness of Algorithm 1. wi l∈L(j,i) pl For simplicity we choose w = 1, i.e., we attempt to maximize the aggregate utility.Proof. We claim that the optimality conditions of (31) and (32) Example 3: L=3 with multiple equilibriaare the same as equations that characterize the equilibrium of We use the following example that has multiple equilibria [36].the above system ((3), (30), (4) and (7)). Capacity constraints, The network is shown in Figure 7 with three unit-capacitynonnegativity, and complementary slackness are obviously the links, cl = 1. There are three different protocols with thesame. We only need to check the relation between rates and corresponding routing matricesprices at equilibrium. Those are T 1 1 0 µj Uij (xj ) = mj (pl ) (33) R1 = I, R2 = , R3 = (1, 1, 1)T i i l 0 1 1 l∈L(j,i)and The price mapping functions are linear: mj (pl ) = kl pl where l j j 1 l∈L(j,i) ml (pl ) µj = i j (34) K 1 = I, K 2 = diag(5, 1, 5), K 3 = diag(1, 3, 1) wi l∈L(j,i) pl Utility functions of sources (j, i) areCombining them, we get j 1 βi (xj )1−αi /(1 − αi ) j j j if αi = 1 j Uij (xj ) = i pl (35) Uij (xj , αi ) = i j j i j j wi βi log xi if αi = 1 l∈L(j,i) j jwhich is the relation between x and p specified by the with appropriately chosen positive constants αi and βi [36].optimality conditions of problem (31)-(32). On the other hand, These utility functions can be viewed as a weighted version ofgiven x and p that satisfy (35), one can always define µ by the α-fairness utility functions proposed in [26]. Parameters µj i(34), and (33) will also be satisfied. are updated every 20 time units. We show that starting from