2006 IEEE Information Theory Workshop, Punta del Este, Uruguay, March 13-17,2006. Equilibrium of Heterogeneous Congestion Control.

Transcription

1 2006 IEEE Information Theory Workshop, Punta del Este, Uruguay, March 13-17,2006 Equilibrium of Heterogeneous Congestion Control Protocols Ao Tang*, Jiantao Wang%, Steven Low*, and Mung Chiangt *Division of Engineering and Applied Science California Institute of Technology, Pasadena, CA 91125, USA tdepartment of Electrical Engineering Princeton University, Princeton, NJ 08544, USA Abstract-When heterogeneous congestion control protocols and the sum of source rates xj (t) that traverse link 1. The that react to different pricing signals share the same network, prices may represent loss probabilities, queueing delays, or the resulting equilibrium may no longer be interpreted as a quantities explicitly calculated by the links and fed back to solution to the standard utility maximization problem. We prove the sources. The function fj models a TCP algorithm that the existence of equilibrium in general multi-protocol networks t sth e function jmofesoa TCP on ts under mild assumptions. For almost all networks, the equilibria adjusts the transmission rate xj(t) of source j based on its are locally unique, and finite and odd in number. They cannot all current value and the sum of "effective prices" m (pl (t)) in be locally stable unless it is globally unique. Finally, we show that its path. The effective prices mti (Pi (t)) are functions of the if the price mapping functions that map link prices to effective l p P a t prices observed by the sources are similar, then global uniqueness on thes and thes. 1 is guaranteed. on the links and sources. When all algorithms use the same pricing signal, i.e., ml I. INTRODUCTION ml are the same for all sources j, the equilibrium properties under mild Cogsto coto protocol hav been. conditions on 91 and fj, the equilibrium of (1)-(2) exists and Cemodeled as dis- of (1)-(2) turn out to be very simple. Indeed, tributed algorithms for network utility maximization, e.g., [3],. [7], [10], [15], [4], [6]. With the exception of a few limited is unique [6]. This is proved by identifying the equilibrium of analysis on very simple topologies [9], [5], [6], [1], existing (I)-(2) with the unique solution of the utility maximization literature lterature generally assumes that all sourcesarehomogeneousproblem Her th defined eqiiru in [3] piep1lathroeflganemin that, even though they may control their rates using different 'X. p type congestion and its Lagrange dual problem [7]. tipliers, one at algorithms, they each link. This utility maximization all adapt to the problem sameltye of cngeson thus provides a simple and complete characterization of the signals, e.g., all react to loss probabilities, as in TCP Reno, eqiiru of a sigeprtcl newrln lolast or all to queueing delay, as in TCP Vegas or FAST [2]. When sources with heterogeneous protocols that react to different relatively simple dynamic behavior. When heterogeneous algorithms that use different congestion signals share the same network, the current duality W h a pricing.'. ~~~~signals share the same network, i.e., Im are framework is no longer applicable. With more congestion s s t s n different for different sources j the situation is much more control complicated. protocols being proposed and ideas of using congestion F i signals other than packet losses, including explicit feedbacks, hfor snstance, when TCP Reno and TCP Vegas or FAST share.,.x d, nt being developed in the networking community, we need a can serve as mathematically rigorous framework to understand the behavior the Lagrange multiplier at the link, and (1)-(2) can no longer be interpreted as solving the standard network of l arge-scale networks withpheterogeneous protocols.ath utility maximization problem. Basic questions, such as the purpse o ths paer i topropse sch fraewor an existence and uni ueness of e uilibrium its local and elobal some of the theoretical predictions in this paper have already quq, I. g been demonstrate experimentally in 14 stability, need to be re-examined. We focus in this paper on bee dongestrancontedexperimentocolally l in kes[14].orm the existence and uniqueness of equilibrium. Due to page limitation, all proofs are omitted and they can be found in Pl = 91l j j )) (1) II. MODEL j:lgl(j) A. Notation *rj =fj x?rj(t), EN ml (Pl(7))8 (2) A network consists of a set of L links, indexed by I1 _-r,e(i) -NlCL(j) m1(,,l ihfint aa)sc.w / fe auentto ~~~and use L to denote both the number of links and the set Here, L(j) denotes the set of links used by source j, and L ={1,...,L} of links. Each link has a price P1 as its 91 (*) models a queue management algorithm that updates the congestion measure. There are J different protocols indexed price Pl (t) at link 1, often implicitly, based on its current value by superscript j, and Ni sources using protocol j, indexed [13] /06/$ IEEE 57

2 by (j, i) where j 1,... J and i =1,... Ni. The total max{z, 0}. This implies that the source rates x3 uniquely solve number of sources is N :. Ni. The L x Ni routing matrix Ri for type j sources is defined max (z) -zq UZ by Ri = 1 if source (j, i) uses link 1, and 0 otherwise. The overall routing matrix is denoted by As we will see, under the assumptions in this paper, R= [ R RJ 1 (us)11 (qj) > 0 for all the prices p that we consider, and hence we can ignore the projection [.]± and assume without Even though different classes of sources react to different loss of generality that prices, e.g. Reno to packet loss probability and Vegas/FAST to queueing delay, the prices are related. We model this qj (u) (q;') (4) relationship through a price mapping function that maps a common price (e.g. queue length) at a link to different prices A I = (e.g. loss probability and queueing delay) observed by different ' (qj) xi qi 7 Ni = sources. Formally, every link I has a price Pl. A type j source x(q) (1i (qi) j 1,... J) to denote the vector-valued reacts to the "effective price" mtn(pi) in its path, where mtn is functions composed of x2. Since q RTm(p), we often abuse a price mapping function, which can depend on both the link notation and write xj (p), xi (p), x(p). Define the aggregate and the protocol type. The exact form of m-i depends on the source rates y(p) = (Yi (p), I 1,... L) at links I as: AQM algorithm used at the link; see [14] for links with RED.' Let mnj(p) = (mni (pl), 1 1,...*L) and m(p) = (mni(pi), y(p) Rx(p) (5) 1 J). The aggregate prices for source (j, i) is defined as In equilibrium, the aggregate rate at each link is no more 'i = 7 RAml (P1) (3) than the link capacity, and they are equal if the link price is : ) strictly positive. Formally, we call p an equilibrium price, a network equilibrium, or just an equilibrium if it satisfies (from Let q-7 = (q,i = l,ni) and q =(qj,j = I1,.~J) ()() be vectors of aggregate prices. Then qi (Ri) mt (p) and q = RTm(p). P(y(p)-c) = 0, y(p) < c, p > (6) Let xi be a vector with the rate xi of source (j, i) as its *th entry, and x be the vector of t where P := diag(pi) is a t'th diagonal matrix. The goal of this entry, and x be the vector of xj W T~.. (XJ T ]Tpaper is to study the existence and uniqueness properties = [ (11)T, (z2)t (J)T ] of network equilibrium specified by (3)-(6). Let E be the equilibrium set: Source (j, i) has a utility function UW (xi) that is strictly concave increasing in its rate z. Let U MI(U,i 1,...,Ni,j F {p C ER+l P(y(p) -c) 0, y(p). c} (7) 1, J). In general, if Zk are defined, then z denotes the (column) For future use, we now define an active constraint set and vector z = (zk, Vk). Other notations will be introduced later the Jacobian for links that are actively constrained. Fix an when they are encountered. We call (c, m, R, U) a network. equilibrium price p* C E. Let the active constraint set L L(p*) C L (with respect to p*) be the set of links I at which B. Network equilibrium P* > 0. Consider the reduced system that consists only of links A network is in equilibrium, or the link prices p and in L, and denote all variables in the reduced system by c, p, y, source rates x are in equilibrium, when each source (j, i) etc. Then, since Yi (p) = cl for every I C L, we have y(p) =c maximizes its net benefit (utility minus bandwidth cost), and Let the Jacobian for the reduced system be J() (P)/p. the demand for and supply of bandwidth at each bottleneck Then link are balanced. Formally, a network equilibrium is defined &=it&j Tt( as follows. J(3RSRi.)(Ri) (p) (8) Given any prices p, we assume in this paper that the source j P rates xi are uniquely determined by where a xi (qi) ufi) qi) Oqj diag 06j (0(2)) O~~~(Xj)2,/ j where (Us) is the derivative of UW, and (U') is its. / inverse which exists since U5i1S strictly concave. Here [z] ± Ar = diag ( (10) 1One can also take the price pl used by one of the protocols, e.g. queueing dea,as the common price Pt. Inti aetecrepnigpiempig and all the partial derivatives are evaluated at the generic point function is the identity function, ml (Pl) =Pl p9 58

3 C. Current theory: J = 1 A. Multiple equilibria: examples In this subsection, we briefly review the current theory In a single-protocol network, if the routing matrix R has for the case where there is only one protocol, i.e., J = 1, full row rank, then there is a unique active constraint set L and explain why it cannot be directly applied to the case of and a unique equilibrium price p associated with it. If R does heterogeneous protocols. not have full row rank, then equilibrium prices p may be non- When all sources react to the same price, then the equi- unique but the equilibrium rates x(p) are still unique since the librium described by (3)-(6) is the unique solution of the utility functions are strictly concave. following utility maximization problem defined in [3] and its In contrast, the active constraint set in a multi-protocol Lagrange dual [7]: network can be non-unique even if R has full row rank (Example 2). Clearly, the equilibrium prices associated with max ui (xi) (11) different active constraint sets are different. Moreover, there can be multiple equilibrium prices associated with the same subject to Rx < c (12) active constraint set (Example 1). where we have omitted the superscript j 1. The strict Example 1: unique active constraint set but uncountably concavity of Ui guarantees the existence and uniqueness of many equilibria the optimal solution of (11)-(12). The basic idea to relate In this example, we assume all the sources use the same the utility maximization problem (11)-(12) to the equilibrium utility function equations (3)-(6) is to examine the dual of the utility max- 1.2 imization problem, and interpret the effective price m1l(pl) UI'(i) (13) as a Lagrange multiplier associated with each link capacity 2 constraint (see, e.g., [7], [10], [6]). As long as ml(pl) > 0 Then the equilibrium rates xi of type j sources are determined and ml (0) = 0, one can replace Pl in (6) by ml (pl). The by the equilibrium prices p as resulting equation together with (3)-(5) provides the necessary and sufficient condition for xi(p) and ml (pi) to be primal and X 1 - (RI Tmi (p) dual optimal respectively. where 1 is a vector of appropriate dimension whose entries This approach breaks down when there are J > 1 types are all Is. We use linear price mapping functions: of prices because there cannot be more than one Lagrange multiplier at each link. In general, an equilibrium no longer m, (p) KIP maximizes aggregate utility, nor is it unique. However, as where Ki are L x L diagonal matrices. Then the equilibrium shown in Theorem 1, existence of equilibrium is still guar- rate vector of type j sources can be expressed as anteed under the following assumptions: )T X A1: Utility functions Ui' are strictly concave increasing, and 1 - (Ri Kip twice continuously differentiable in their domains. Price When only links with strictly positive equilibrium prices are mapping functions ml are continuously differentiable in included in the model, we have their domains and strictly increasing with ml (0) = 0. A2: For any c > 0, there exists a number Pmax such that if y(p) = 3 Rixi(p) c Pi > Pmax for link 1, then j=1 z(p) <ce for all (j, i) with Rl = 1 Substituting in xi(p) yields These are mild assumptions. Concavity and monotonicity of z Ri (Ri )TKip ZE Rjl-c utility functions are often assumed in network pricing for j=1 j=1 elastic traffic. Moreover, most TCP algorithms proposed or deployed turn out to have strictly concave increasing utility which is a linear equation in p for given Ri, Ki, and c. It has functions; see e.g. [6]. The assumption on ml preserves the a unique solution if the determinant is nonzero, but has no or relative order of prices and maps zero price to zero effective multiple solutions if price. Assumption A2 says that when Pl is high enough, then every source going through link I has a rate less than c. minutes det ( Ri (Ri )TKi 0 for presentation. j=1 Theorem 1. Suppose Al and A2 hold. There exists an equi- When J = 1, i.e., there is only one protocol, and R1 has librium price p* for any network (c, m, R, U). full row rank, det(r1(r1)tk1) > 0 since both R1 (R1 )T and K1 are positive definite. In this case, there is a unique III. REGULAR NETWORKS equilibrium price vector. When J =2, there are networks Theorem 1 guarantees the existence of network equilibrium, whose determinants are zero that have uncountably many We now study its uniqueness properties. equilibria. See [12] for an example where R does not have J T J 59

4 full row rank. We provide here an example with J = 3 where Flows (1,1) and (1, 2) have identical utility function U1 and R still has full row rank. source rate x1, and flow (2, 1) has a utility function U2 and The network is shown in Figure 1 with three unit-capacity source rate x links, Cl 1. There are three different protocols with the Links 1 and 3 both have capacity cl and price mapping functions ml (p) = p and m2(p) for protocols 1 and 2 respec- \ X1 ) 71 \1X2 ) X3 Jtively. Link 2 has capacity c2 and price mapping functions T / m (p) = p and m2(p). In [14], we prove that when assumption Al holds, the Kx2 4 > X2 > network shown in Figure 2 has at least two equilibria provided: 1) C1< C2 < 2c1; 2) for j 1, 2, (UJ)'(x) (x possibly oc, if and only Fig. 1. Example 1: uncountably many equilibria. if xi 0. 3) for I 1, 2, ml2(pl) > p2 as Pi > -1, and satisfy corresponding routing matrices 2m2((Ul)'(c2 - cl)) < (U2)'(2c, - c2) T~~~~~~~~~~~~~~~~~~~ = R3T(1,1,1)T < mt2((ul)'(c2 -Cl)) B. Regular networks The linear mapping functions are given by Examples 1 and 2 show that global uniqueness is generally K1 = I, K2 =diag(5, 1, 5), K3 = diag(1, 3, 1) not guaranteed in a multi-protocol network. We now show, however, that local uniqueness is basically a generic property It is easy to calculate that of the equilibrium set. We present our main results on the 3 [7 4 1 structure of the equilibrium set here, providing conditions for ZRi(Ri)TKi the equilibrium points to be locally unique, finite and odd in i=1 L J number, and globally unique. which has determinant 0. Using the utility function defined in Consider an equilibrium price p* C E. Recall the active wh1ch~ ~~ ~ ~ ~ ~ ~ ~ ~~~~ontan set Leemnndefne by1n phe Thet equnchonu prine pnfo (13), we can check that the following are equilibrium prices constraint set L defined by pi. The equilibrium price p* for for all ec [0,1/24]: the links in L is a solution of I P2 /4-2,e YP (14) Pi = p3 I1/8+,e = The corresponding rates are By the inverse function theorem, the solution of (14), and 1 = = hence the equilibrium price p*, is locally unique if the Jacobian 7/83 - c =2 3/4 + 2c matrix J(p*) = &/&p is nonsingular at p*. We call a network i = x2 = 1/8-3e xi 4e (c, m, R, U) regular if all its equilibrium prices are locally All capacity constraints are tight with these rates. Since there unique. next result shows that almost all networks are regular, is ao-nf aee lkt ate is an one-link flow at every link, therac nthe e con ntaset and that regular networks have finitely many equilibrium isny uqueli. a. prcces. This justifies restricting our attention to regular netmany equilibria. porics. works. Example 2: multiple active constraint sets each with a Theorem 2. Suppose assumptions Al anda2 hold Given any unique equilibrium price mapping functions m, any routing matrix R and utility Consider the symmetric network in Figure 2 with 3 flows. functions U, There are two protocols in the network with the following 1) the set of link capacities cfor which not all equilibrium xi xi prices are locally unique has Lebesgue measure zero in 2) the number of equilibria for a regular network > (c, m, R, U) is finite. xi For the rest of this subsection, we narrow our attention to Fig. 2. Example 2: two active constraint sets. networks that satisfy an additional assumption: A3: Every link 1 has a single-link flow (j, i) with routing matrices ()(cl) >0O. 10 Assumption A3 says that when the price of link 1 is small 11 R1=, R2=(,,1Tenough, the aggregate rate through it will exceed its capacity. 1 This ensures that the active constraint set contains all links [ 60

5 and facilitates the application of Poincare-Hopf theorem by denote the identity permutation. Then a-va. See [8] for avoiding equilibrium on the boundary (some Pi = 0). 2 more details. Finally, denote dmi /dpi by Th. Since all the equilibria of a regular network have nonsingu- Theorem 6. Suppose assumptions AJ-A3 hold If for any lar Jacobian matrices, we can define the index I(p) of p C E ' ~~~~~~~~~vector j C {1,.,J}' L and a as I any permutations a, k, r in P) if det (J(p)) >0 {1,.,L}L, PI if det (J(p)) < 0 Tih[k(j)], + j7m[n(j)], > J7[J (j)], (15) Then, we have Theorem 3. Suppose assumptions AJ-A3 hold Given any then the equilibrium of a regular network is globally unique. regular network, we have IV. CONCLUSION J(p) =(1) LWhen sources sharing the same network react to different S pee pricing signals, the current duality model no longer explains where L is the number of links. the equilibrium of bandwidth allocation. We have introduced a mathematical formulation of network equilibrium for multi- We give two important consequences of this theorem. protocol networks and studied several fundamental properties, Corollary 4. Suppose assumptions AJ-A3 hold. A regular such as existence, local uniqueness, number of equilibria, and network has an odd number of equilibria. global uniqueness. We prove that equilibria exist, and are almost always locally unique. The number of equilibria is Notice that Corollary 4 implies the existence of equilibrium, almost always finite and must be odd. Finally the equilibrium Although we have this via theorem 1 in a more general setting, is globally unique if the price mapping functions are similar. this simple corollary shows the power of Theorem 3. The next result provides a condition for global uniqueness. REFERENCES We say that an equilibrium p* C E is locally stable if the [1] L. Grieco and S. Mascolo. Performance Evaluation and Comparison corresponding Jacobian matrix J(p*) defined in (8) is stable, of Westwood+, New Reno, and Vegas TCP Congestion Control ACM thats,eeryigenvluefj *= * y p SIGCOMM Computer Communications Review,34(2):25-38, April 2004 that is, every eigenvalue of J(p*) =Ay(p* )/0p has negative [2] D. Wei, C. Jin, S. H. Low and S. Hegde FAST TCP: motivation, archireal part. tecture, algorithms, performance. to appear in IEEEIACM Transactions on Networking, Corollary 5. Suppose assumptions AJ-A3 hold. The equi- [3] F. Kelly, A. Maoulloo, and D. Tan. Rate control for communication librium of a regular network is globally unique if and only networks: shadow prices, proportional fairness and stability. Journal of if every equilibrium in E index (-I)'. the Operational Research Society, 49: ,1998. If every equlibrium point In F has an Index (-1i)L In [4] S. Kunniyur and R. Srikant. End-to-end congestion control: utility particular, if all equilibria are locally stable, then E contains functions, random losses and ECN marks. IEEEIACM Transactions on exactly one point. Networking, 11(5): , Oct [5] K. Kurata, G. Hasegawa, and M. Murata. Fairness comparisons between Finally we reveal that, under assumptions A1-A3, if the TCP Reno and TCP Vegas for future deployment of TCP Vegas. Proj... ~~~~~~~~ceedings of INET, price mapping functions prc M3are mp1n unhnsm similar, then the equilibrium [6cednsoINT20. of a regular network is globally unique. IEEEIACM Trans. on Networking, 11(4): , August aesmla, hn h eullru [6] S. H. Low. A duality model of TCP and queue management algorithms. To state the result concisely, we need the notion of permuta- [7] S. H. Low and D. Lapsley. Optimization flow control, I: basic algorithm tion. We call a vctor Laperutationifeach and convergence. IEEEIACM Transactions on tion.we cllavectr a- (~i~.. o~) apermtatin ifeach Networking, 7(6): , December a 01 (7i is distinct and takes value in {1,... L}. Treating a- as a [8] C. Meyer. Matrix Analysis and Applied Linear Algebra. SIAM, mapping a: {1,..., L}I {1,..., L}, we let a-' denote [9] J. Mo, R. La, V Anantharam, and J. Walrand. Analysis and comparison its*uique nvereperutaton.franvectra aa) of TCP Reno and Vegas. Proceedings of IEEE Infocom, March ItS unique Inverse permutation. For any vector a C XRL, ta-(a) [10] J. Mo and J. Walrand. Fair end-to-end window-based congestion control. denotes the permutation of a under a, i.e., [a-(a)]i = a,,,. IEEEIACM Transactions on Networking, 8(5): , October If a C {1,... L}'L is a permutation, then a-(a) is also a [11] A. Simsek, A. Ozdaglar and D. Acemoglu. Generalized Poincare-Hopf permutation write*aainstead. 1 = (I Theorem for compact nonsmooth regions. submitted for permnutaton and we often a-a instead. Let l (1,..., L) publication, [12] A. Tang, J. Wang, S. H. Low and M. Chiang. Network Equilibrium 21t is recently shown in [11] that A3 is not necessary and one can of Heterogeneous Congestion Control Protocols. Proceedings of IEEE generalize Theorem 3 to Infocom, [13] A. Tang, J. Wang, S. H. Low and M. Chiang. Equilibrium of Het- ( l))l(p) I(p) = 1 erogeneous Congestion Control Protocols. Caltech CS Technical Report pee CaltechCSTR: , April edu/caltechcstr: where where L(p) _p)is the number oflinsoftheactveonstainseassciaedith of links of the active constraint set associated with [14] A. Tang, J. Wang, S. Hedge and S. H. Low. Equilibrium and fairness equilibrium p. Clearly, if L(p) -L, it reduces to Theorem 3. This generalized of networks shared by TCP Reno and Vegas/FAST. Telecommunication theorem also allows [11] to conclude the number of equilibria is odd (and Systems, 30(4): , December therefore existence) without A3. In this paper, although A3 is imposed, all [15] H. Yaiche, R. Mazumdar, and C. Rosenberg. A game theoretic results can be viewed as with respect to a fixed active constraint set with framework for bandwidth allocation and pricing in broadband networks. appropriate modifications. In particular, the global uniqueness result theorem IEEE/ACM Transactions on Networking, 8(5), October directly apply without A3 since J has a similar structure as J except with a smaller dimension. 61