Utility Proportional Fair Bandwidth Allocation: An Optimization Oriented Approach

Utility Proportional Fair Bandwidth Allocation: An Optimization Oriented Approach 61 Tobia Hark Konrad-Zue-Zentrum für Informationtechnik Berlin (ZIB), Department Optimization, Takutr. 7, 14195 Berlin, Germany hark@zib.de Abtract. In thi paper, we preent a novel approach to the congetion control and reource allocation problem of elatic and real-time traffic in telecommunication network. With the concept of utility function, where each ource ue a utility function to evaluate the benefit from achieving a tranmiion rate, we interpret the reource allocation problem a a global optimization problem. The olution to thi problem i characterized by a new fairne criterion, utility proportional fairne. We argue that it i an application level performance meaure, i.e. the utility that hould be hared fairly among uer. A a reult of our analyi, we obtain congetion control law at link and ource that are globally table and provide a utility proportional fair reource allocation in equilibrium. We how that a utility proportional fair reource allocation alo enure utility max-min fairne for all uer haring a ingle path in the network. A a pecial cae of our framework, we incorporate utility maxmin fairne for the entire network. To implement our approach, neither per-flow tate at the router nor explicit feedback beide ECN (Explicit Congetion Notification) from the router to the end-ytem i required. 1 Introduction In thi paper, we preent a network architecture that conider an applicationlayer performance meaure, called utility, in the context of bandwidth allocation cheme. In the lat year, there have been everal paper [1 7] that interpreted congetion control of communication network a a ditributed algorithm at ource and link in order to olve a global optimization problem. Even though coniderable progre ha been made in thi direction, the exiting work focue on elatic traffic, uch a file tranfer (FTP, HTTP) or electronic mail (SMTP). In [8], elatic application are characterized by their ability to adapt the ending rate in preence of congetion and to tolerate packet delay and loe rather gracefully. From a uer perpective, common to all elatic application i the requet to tranfer data in a hort time. To model thee characteritic, we reort Thi work ha been upported by the German reearch funding agency Deutche Forchunggemeinchaft under the graduate program Graduiertenkolleg 621 (MAGSI/Berlin). M. Ajmone Maran et al. (Ed.): QoS-IP 2005, LNCS 3375, pp. 61 74, 2005. c Springer-Verlag Berlin Heidelberg 2005

62 Tobia Hark Elatic Traffic Adaptive Real Time Traffic Utility Utility Bandwidth Bandwidth Fig. 1. Utilitie for elatic traffic and adaptive real-time traffic. to the concept of utility function. Following [8] and [2], traffic that lead to an increaing, trictly concave (decreaing marginal improvement) utility function i called elatic traffic. We call uch a utility function bandwidth utility ince the utility function evaluate the benefit from achieving a certain tranmiion rate. The propoed ource and link algorithm are deigned to maximize the aggregate bandwidth utility (um over all bandwidth utilitie) ubject to capacity contraint at the link. Kelly introduced in [2] the o called bandwidth proportional fair allocation, where bandwidth utilitie are logarithmic. The algorithm at the link are baed on Lagrange multiplier method coming from optimization theory, o the concavity aumption eem to be eential. A hown in [8], ome application, epecially real-time application have non-concave bandwidth utility function. A voice-over-ip flow, for intance, receive no bandwidth utility, if the rate i below the minimum encoding rate. It bandwidth utility i at maximum, if the rate i above it maximum encoding rate. Hence, it bandwidth utility can be approximated by a tep function. According to Shenker [8], the bandwidth utility of adaptive real-time application can be modeled a an S- haped utility function (a convex part at low rate followed by a concave part at higher rate) a hown in Figure 1. The paradigm of the work dealing with bandwidth utility function of elatic application in the context of congetion control i to maximize the bandwidth utilization of the network (bandwidth ytem optimum) under pecific bandwidth fairne apect (bandwidth max-min, bandwidth proportional fair). The central part of thi work i to turn the focu on fairne of uer-received utility of different application including non-elatic application with nonconcave bandwidth utility function. A uer running an application doe not care about any fair bandwidth hare, a long a hi application perform atifactory. Hence, we argue that it i an application performance meaure, i.e. the utility that hould be hared fairly among uer. To motivate thi new paradigm, we refer to the concept of utility max-min fairne introduced by Cao and Zegura in [9]. Let u conider a network coniting of a ingle link of capacity one hared by two uer. One uer tranfer data according to an elatic application with trictly increaing and concave bandwidth utility U 1 ( ). The other uer tran-

Utility Proportional Fair Bandwidth Allocation 63 Bandwidth Utility 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 u : utility max-min fair u u 1 max-min fair bandwidth hare Bandwidth min. video encoding rate U 1 (x 1 ) U 2 (x 2 ) 0.1 u 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 2. Utility max-min and bandwidth max-min fairne. u fer real-time video data with a non-concave bandwidth utility function U 2 ( ). Figure 2 how, how different bandwidth allocation affect the received utility. If the bandwidth i hared equally, what i referred to a max-min bandwidth allocation in thi example, uer 1 receive a much larger utility than uer 2. Converely, uer 2 would not be atified ince he doe not receive the minimum video encoding bandwidth. If we want to hare utility equally, intead of bandwidth, we would like to have a reource allocation, where the received utilitie are equal or utility max-min fair, i.e. U 1 (x 1 )=U 2 (x 2 )=u. In [9], Cao and Zegura preent a link algorithm that achieve a utility maxmin fair bandwidth allocation, where for each link the utility function of all flow haring that link i maintained. In [10], Cho and Song preent a utility max-min architecture, where each link communicate a upported utility value to ource uing that link. Then ource adapt their ending rate according to the minimum of thee utility value. In thi paper, we extend the utility max-min architecture and propoe a new fairne criterion, utility proportional fairne, which include the utility max-min fair reource allocation a a pecial cae. A utility proportional fair bandwidth allocation i characterized by the olution of an aociated optimization problem. The benefit a uer gain when ending at rate x i evaluated by anewecond order utility F (x ) and the objective i to maximize aggregate econd order utility ubject to capacity contraint. The econd order utilitie are aumed to be trictly concave, wherea the bandwidth utilitie can be choen arbitrarily. We only aume that the bandwidth utilitie are monotonic increaing in a given interval. Thi i a natural aumption ince any application will profit from receiving more bandwidth in a certain bandwidth interval. We emphaize, that our ditributed algorithm doe not need any per-flow information at the link. The feedback from link to ource doe not include overhead, uch a explicit utility value a done in [10]. It merely relie on the communication

64 Tobia Hark of Lagrange multiplier, called hadow price, from the link to the ource. Thi can be achieved by an Active Queue Management (AQM) cheme, uch a Random Early Marking (REM) [6] uing Explicit Congetion Notification (ECN) [11]. The ret of the paper i organized a follow. In the next ection, we decribe our model, the econd order utility optimization problem and it dual baed on idea of [1, 2, 5]. Given a pecific bandwidth utility, we decribe a contructive method to find the econd order utility function F ( ) that lead to a utility proportional fair reource allocation. In Section 3, we preent a tatic primal algorithm at the ource and a dynamic dual algorithm at the link olving the global optimization problem and it dual. We further preent a global tability reult for the dual algorithm baed on Lyapunov function along the line of [12]. In Section 4, we define a new fairne criterion, utility proportional fairne, and how that our algorithm achieve utility max-min fairne in equilibrium for uer haring a ingle path in the network. We further incorporate utility maxmin fairne for the entire network a a pecial cae of our framework. Finally, we conclude in Section 5 with remark on open iue. 2 Analytical Model Coniderable progre ha recently been made in bringing analytical model into congetion control and reource allocation problem [1 5]. Key to thee work ha been to explicitly model the congetion meaure that i communicated implicitly or explicitly back to the ource by the router. It i aumed that each link maintain a variable, called price, and the ource have information about the aggregate price of link in their path. In thi ection, we decribe a fluid-flow model, imilar to that in [1, 2, 5]. We interpret an equilibrium point a the unique olution of an aociated optimization problem. The reulting reource allocation i aimed to provide a fair hare of an application layer performance meaure, i.e. the utility to uer. In contrat to [1 7, 12] we do not poe any retriction on the bandwidth utility function, except for monotonicity. 2.1 Model We model a packet witched network by a et of node (router) connected by a et L of unidirectional link (output port) with finite capacitie c =(c l,l L). The et of link are hared by a et S of ource indexed by. Aource repreent an end-to-end connection and it route involve a ubet L() L of link. Equivalently, each link i ued by a ubet S(l) S of ource. The et L() ors(l) define a routing matrix { 1 if l L(), R l = 0 ele.

Utility Proportional Fair Bandwidth Allocation 65 A tranmiion rate x in packet per econd i aociated with each ource. We aume, that the rate x, S lie in the interval X =[0,x max ], where x max i the maximum ending rate of ource. Thi upper bound may differ ubtantially for different application. A ubet of ource S r S tranferring real-time data, for intance, may have a maximum encoding rate x max, S r,whichcanbe much lower than the upper bound x max, S \ S r of elatic application, which are greedy for any available bandwidth in the network. Thu, ending rate of elatic application are contrained by bottleneck link in the network. Definition 1. Aratevectorx =(x, S) i aid to be feaible if it atifie the condition: x X S and Rx c. With each link l, a calar poitive congetion-meaure p l, called price, i aociated. Let y l = R l x S be the aggregate tranmiion rate of link l, i.e. the um over all rate uing that link, and let q = l L R l p l be the end-to-end congetion meaure of ource. Note that taking the um of congetion meaure of a ued path i eential to maintain the interpretation of p l a dual variable [1]. Source can oberve it own rate x and the end-to-end congetion meaure q of it path. Link l can oberve it local congetion meaure p l and the aggregate tranmiion rate y l. When the tranmiion rate of uer i x,uer receive a benefit meaured by the bandwidth utility U (x ), which i a calar function and ha the following form: where Y =[U (0),U (x max U : X Y x U (x ), )] = [u min,u max ], U (0) = u min,u (x max )=u max. Aumption 1. The bandwidth utility function U ( ) are continuou, differentiable, and trictly increaing, i.e. U (x ) > 0 for all x X, S. Thi aumption enure the exitence of the invere function U 1 ( ) overthe range [u min,u max ]. Before we preent a contructive method to generate econd order utility function, we briefly retate the overall paradigm. An optimal operation point or equilibrium hould reult in almot equal utility value for different application. The exact definition of the propoed reource allocation, i.e. utility proportional fair reource allocation, will be given below. If we want to follow thi paradigm, we mut tranlate a given congetion level of a path, repreented by q, into an appropriate utility value the network can offer to ource. We model thi utility value, the available utility, a the tranformation of the congetion meaure q by a tranformation function f (q ). Thi function i aumed to be trictly decreaing.

66 Tobia Hark Aumption 2. The tranformation function f ( ) decribing the available utility of a path ued by ender i aumed to be a continuou, differentiable, and trictly decreaing function of the aggregate congetion meaure q, i.e. f (q ) < 0 for all q 0 and S. Thi aumption i reaonable, ince the more congeted a path i, the maller will be the available utility of an application. The main idea i, that each uer hould end at data rate x in order to match it own bandwidth utility with the available utility of it path. Thi lead to the following equation: U (x )=[f (q )] umax u, S, (1) min w, if a w b where [w] b a := min{max{w, a},b} = a, if w<a b, if w>b. Note that the utility a ource can receive i bounded by the minimum and maximum utility value u min and u max. Hence, the ource rate x are adjuted according to the available utility f (q ) of their ued path a follow: x = U 1 ([f (q )] umax ), S. (2) u min Aource S react to the congetion meaure q in the following manner: if the congetion meaure q i below a threhold q <q min := f 1 (u max ), then the ource tranmit data at maximum rate x max := U 1 (u max ); if q i above a threhold q >q max := f 1 (u min ), the ource end at minimum rate x min := U 1 (u min ); if q i in between thee two threhold q Q := [q min,q max ], the ending rate i adapted according to x = U 1 (f (q )). Lemma 1. The function G (q )=U 1 ([f (q )] umax u ) i poitive, differentiable, min and trictly monotone decreaing, i.e. G (q ) < 0 on the range q Q,andit invere G 1 ( ) i well defined on X. Proof. Since U ( ) i defined on X, U 1 ( ) i alway nonnegative. Since f ( ) i differentiable over Q,andU 1 ( ) i differentiable over Y, the compoition G (q )=U 1 (f (q )) i differentiable over Q. We compute the derivative uing the chain rule: G (q )=U 1 (f (q ))f (q ). The derivative of the invere (f (q )) can be computed a U 1 U 1 (f (q )) = 1 U 1 (U (f (q ))) > 0. With the inequality f ( ) < 0, we get G (q ) < 0, q Q. Hence, G (q )i trictly monotone decreaing in Q,oitinvereG 1 (x )exitonx. 2.2 Equilibrium Structure and Second Order Utility Optimization In thi ection we tudy the above model at equilibrium, i.e. we aume, that rate and price are at fixed equilibrium value x,y,p,q.fromtheabove model, we immediately have the relationhip: y = Rx, q = Rp.

Utility Proportional Fair Bandwidth Allocation 67 In equilibrium, the ending rate x X, S atify: x = U 1 ([f (q)] umax )=G (q ). (3) u min Since q repreent the congetion in the path L(), the ending rate will be decreaing at higher q, and increaing at lower q. Now we conider the invere G 1 (x ) of the above function on the interval X, and contruct the econd order utility F (x ) a the integral of G 1 (x ). Hence, F ( ) ha the following form and property: F (x )= G 1 (x )dx with F (x )=G 1 (x ). (4) Lemma 2. The econd order utility F ( ) i a poitive, continuou, trictly increaing, and trictly concave function of x X. Proof. Thi follow directly from Lemma 1 and the relation F (x )=G 1 (x )= 1 G (q ) < 0. The contruction of F ( ) lead to the following property: Lemma 3. The equilibrium rate (3) i the unique olution of the optimization problem: max F (x ) q x. (5) x X Proof. The firt order neceary optimality condition to problem (5) i: F (x )=q G 1 (x )=q x = U 1 ([f (q )] umax ) u min Due to the trict concavity of F ( ) onx, the econd order ufficient condition i alo atified completing the proof. The above optimization problem can be interpreted a follow. F (x )ithe econd order utility a ource receive, when ending at rate x,andq x i the price per unit flow the network would charge. The olution to (5) i the maximization of individual utility profit at fixed cot q and exactly correpond to the propoed ource law (2). Now we turn to the overall ytem utility optimization problem. The aggregate price q enure that individual optimality doe not collide with ocial optimality. An appropriate choice of price p l,l L mut guarantee that the olution of (5) alo olve the ytem utility optimization problem: max F (x ) (6) x 0 S.t. Rx c. (7)

68 Tobia Hark Thi problem i a convex program, imilar to the convex program in [1, 5, 7], for which a unique optimal rate vector exit. For olving thi problem directly global knowledge about action of all ource i required, ince the rate are coupled through the hared link. Thi problem can be olved by conidering it dual [7]. 3 Dual Problem and Global Stability In accordance with the approach in [1], we introduce the Lagrangian and conider price p l,l L a Lagrange multiplier for (6),(7). Let L(x, p) = F (x ) p l (y l c l )= F (x ) q x + p l c l S l L S l L be the Lagrangian of (6) and (7). The dual problem can be formulated a: min V (q )+ p l c l, (8) p l 0 S l L where V (x )=max F (x ) q x, x X. (9) x 0 Due to the trict concavity of the objective and the linear contraint, at optimal price p, the correponding optimal x olving (9) i exactly the unique olution of the primal problem (6),(7). Note that (5) ha the ame tructure a (9), o we only need to aure that the price q given in (5) correpond to Lagrange multiplier q givenin(9). A hown in [7], a traightforward method to guarantee that equilibrium price are Lagrange multiplier i the gradient projection method applied to the dual problem (8): { d dt p γ l (p l (t))(y l (t) c l ) if p l (t) > 0 l(t) = γ l (p l (t))[y l (t) c l ] + (10) if p l (t) =0, where [z] =max{0,z} and γ l (p l ) > 0 i a nondecreaing continuou function. Thi algorithm can be implemented in a ditributed environment. The information needed at the link i the link bandwidth c l and the aggregate tranmiion rate y l (t), both of which are available. In equilibrium, the price atify the complementary lackne condition, i.e. p l (t) are zero for non-aturated link and non-zero for bottleneck link. We conclude thi ection with a global convergence reult of the dual algorithm (8) combined with the tatic ource law (5) uing Lyapunov technique along the line of [12]. It i only aumed that the routing matrix R i noningular. Thi guarantee that for any given q S there exit a unique vector (p l,l L ) uch that q = l L p l. Theorem 1. Aume the routing matrix R i noningular. Then the dual algorithm (10) tarting from any initial tate converge aymptotically to the unique olution of (6) and (7). The proof of thi theorem can be found in [13]. For further analyi of the peed of convergence, we refer to [1].

4 Utility Proportional Fairne Utility Proportional Fair Bandwidth Allocation 69 Kelly et al. [2] introduced the concept of proportional fairne. They conider elatic flow with correponding trictly concave logarithmic bandwidth utility function. A proportional fair rate vector (x, S) i defined uch that for any other feaible rate vector (y, S) the aggregate of proportional change i non-poitive: y x 0. x S Thi definition i motivated by the aumption that all uer have the ame logarithmic bandwidth utility function U (x ) = log(x ). By thi aumption, a firt order neceary and ufficient optimality condition for the ytem bandwidth optimization problem max U (x ).t. Rx 0 x 0 x 0 i U (x x )(y x )= y x 0. x S S Thi condition i known a the variational inequality and it correpond to the definition of proportional fairne. Before we come to our new fairne definition, we retate the concept of utility max-min fairne. It i imply the tranlation of the well known bandwidth maxmin fairne applied to utility value. Definition 2. Aetofrate(x, S) i aid to be utility max-min fair, if it i feaible, and for any other feaible et of rate (y, S), the following condition hold: if U (y ) >U (x ) for ome S, then there exit k S uch that U k (y k ) <U k (x k ) and U k (x k ) U (x ). Suppoe we have a utility max-min fair rate allocation. Then, a uer cannot increae it utility, without decreaing the utility of another uer, which receive already a maller utility. We further apply the above definition to a utility allocation of a ingle path. Definition 3. Conider a ingle path in the network denoted by a et of adjacent link (l L p ).AumeaetofuerS Lp S hare thi path, i.e. L() =L p for S Lp. Then, the et of rate x, S i aid to be path utility max-min fair if the rate allocation on uch a path i utility max-min fair. Now we come to our propoed new fairne criterion, baed on the econd order utility optimization framework. Definition 4. Aume, all econd order utilitie F ( ) are of the form (4). A rate vector (x, S) i called utility proportional fair if for any other feaible rate vector (y, S) the following optimality condition i atified:

70 Tobia Hark S F (x x )(y x )= G 1 (x )(y x ) S = S f 1 (U (x ))(y x ) 0 (11) The above definition enure, that any proportional utility fair rate vector will olve the utility optimization problem (6), (7). If we further aume, all uer have the ame tranformation function f( ) =f ( ), S, then we have the following propertie of a utility proportional fair rate allocation, which are proven in Appendix A. Theorem 2. Suppoe all uer have a common tranformation function f( ) and all econd order utility function are defined by (4). Let the rate vector (x X, S) be proportional utility fair, i.e. the unique olution of (6). Then the following propertie hold: (i) The rate vector (x X, S) i path utility max-min fair. (ii) If q 1 Q 1, q 2 Q 2 and q 1 q 2 for ource 1, 2,then U 1 (x 1 ) U 2 (x 2 ). (iii) If ource 1 ue a ubet of link that 2 ue, i.e. L( 1 ) L( 2 ),and U 1 (x 1 ) <u max 1,thenU 1 (x 1 ) U 2 (x 2 ). It i a well-known property of the concept of proportional fairne that flow travering everal link on a route receive a lower hare of available reource than flow travering a part of thi route provided all utilitie are equal. The rationale behind thi i that thee flow ue more reource, hence hort connection hould be favored to increae ytem utility. Tranferring thi idea to utility proportional fairne, we get a imilar reult. Flow travering everal link receive le utility compared to horter flow, provided a common tranformation function i ued. If thi feature i undeirable, ince the path a flow take i choen by the routing protocol and beyond the reach of the ingle uer, the econd order utilitie can be modified to compenate thi effect. We how that an appropriate choice of the tranformation function f ( ) will aure a utility max-min bandwidth allocation in equilibrium. Theorem 3. Suppoe all uer have the ame parameter dependent tranformation function f (q,κ)=q 1 κ, S, κ>0. The econd order utilitie F (x,κ), S are defined by (4). Let the equence of rate vector x(κ) =(x (κ) X, S) be utility proportional fair. Then x(κ) approache the utility max-min fair rate allocation a κ. The proof of thi theorem can be found in Appendix B. 5 Concluion We have obtained decentralized congetion control law at link and ource, which are globally table and provide a utility proportional fair reource allocation in equilibrium. Thi new fairne criterion enure that bandwidth utility

Utility Proportional Fair Bandwidth Allocation 71 value of uer (application), rather than bit rate, are proportional fair in equilibrium. We further howed that a utility proportional fair reource allocation alo enure utility max-min fairne for all uer haring a ingle path in the network. A a pecial cae of our model, we incorporate utility max-min fairne for all uer haring the network. To the bet of our knowledge, thi i the firt paper dealing with reource allocation problem in the context of global optimization, that include non-concave bandwidth utility function. We are currently working on n-2 (Network Simulator) implementation of the decribed algorithm. Firt imulation reult are promiing. An open iue and challenge i to deign the feedback control interval for real-time application. There i clearly a tradeoff between the two conflicting goal: tability (delay) and minimal packet overhead (i.e. multicat) in the network. Neverthele, we believe that thi framework ha a great potential in providing real-time ervice for a growing number of multimedia application in future network. Reference 1. S. H. Low and D. E. Lapley: Optimization Flow Control I. IEEE/ACM Tran. on Networking 7 (1999) 861 874 2. F. P. Kelly, A. K. Maulloo and D. K. H. Tan: Rate Control in Communication Network: Shadow Price, Proportional Fairne, and Stability. Journal of the Operational Reearch Society 49 (1998) 237 52 3. R. J. Gibben and F. P. Kelly: Reource pricing and the evolution of congetion control. Automatica (1999) 1969 1985 4. S. H. Low: A duality model of TCP flow control. In: Proceeding of ITC Specialit Seminar on IP Traffic Meaurement, Modeling and Management. (2000) 5. S. H. Low, F. Paganini, J. Doyle: Internet Congetion Control. IEEE Control Sytem Magazine 22 (2002) 6. S. Athuraliya, V. H. Li, S. H. Low and Q. Yin: REM: Active Queue Management. IEEE Network 15 (2001) 48 53 7. S. H. Low, F. Paganini, J. C. Doyle: Scalable law for table network congetion control. In: Proceeding of Conference of Deciion and Control. (2001) 8. S. Shenker: Fundamental Deign Iue for the Future Internet. IEEE JSAC 13 (1995) 1176 88 9. Z. Cao, E. W. Zegura: Utility max-min: An application-oriented bandwidth allocation cheme. In: Proceeding of IEEE INFOCOM 99. (1999) 793 801 10. J. Cho, S. Chong: Utility Max-Min Flow Control Uing Slope-Retricted Utility Function. Available at http://nety.kait.ac.kr/publication (2004) 11. S. Floyd: TCP and Explicit Congetion Notification. ACM Comp. Commun. Review 24 (1994) 10 23 12. F. Paganini: A global tability reult in network flow control. Sytem and Control Letter 46 (2002) 165 172 13. T. Hark: Utility Proportional Fair Reource Allocation - An Optimization Oriented Approach. Technical Report ZR-04-32, Konrad-Zue-Zentrum für Informationtechnik Berlin (ZIB) (2004)

72 Tobia Hark Appendix A Proof of Theorem 2: To (i): if ource S Lp hare the ame path, they receive the ame aggregate congetion feedback in equilibrium q p = q, S Lp. Two cae are of interet. (a) Suppoe for all ource the following inequality hold: f(q p ) <u max, S Lp. Hence, all ource adapt their ending rate according to the available utility f(q p )=U (x ). Thi correpond to the trivial cae of path utility max-min fairne, ince all ource receive equal utility. (b) Suppoe a et Q S Lp receive utility U (x )=u max <f(q p ). We prove the theorem by contradiction. Aume the utility proportional fair rate vector (x, S) i not path utility max-min fair with repect to the path L p.by definition, there exit a feaible rate vector y, S with U j (y j ) >U j (x j ) for j S Lp \ Q (12) uch that for all k S Lp \ (Q {j}) withu k (x k ) U j (x j ) the inequality U k (y k ) U k (x k ) (13) hold. In other word, we can increae the utility of a ingle ource rate U j (x j ) to U j (y j ) by increaing the rate x j to y j without decreaing utilitie U k (y k ),k S Lp \ (Q {j}) which are already maller. We repreent the rate increae of ource j by y j = x j + ξ j,whereξ j > 0 will be choen later on. Here again, we have to conider two cae: (b1) Suppoe, there exit a ufficiently mall ξ j > 0thatwedonothaveto decreae any ource rate of the et {y k,k S Lp \(Q {j})} to maintain feaibility. Hence, we can increae the ytem utility while maintaining feaibility. Thi clearly contradict the proportional fairne property of x. (b2) Suppoe, we have to decreae a et of utilitie (U k (y k ),k K), which are higher then U j (x j ), i.e. U k (y k ) <U k (x k )withu k (y k ) >U j (x j ),k K S Lp \ (Q {j}). Thi correpond to decreaing the et of rate y k = x k ξ k, k K with ξ k ξ j. Due to the trict concavity of the objective function of (6), we get the following inequalitie: F j(x j )=f 1 (U j (x j )) >f 1 (U k (y k )) = F k(y k ), k K S Lp \ (Q {j}). Due to the continuity of F ( ), S, wecanchooeξ j with y j = x j + ξ j uch that F j(x j + υ j ξ j ) >F k(y k ) for all k K S Lp \ (Q {j}) and υ j (0, 1).

Utility Proportional Fair Bandwidth Allocation 73 Comparing the aggregate econd order utilitie of the rate vector x and y uing the mean value theorem, we get: F (x ) F (y )= (F k (x k ) F k (y k )) + F j (x j ) F j (y j ) S S = (F k (y k + ξ k ) F k (y k )) + F j (x j ) F j (x j + ξ j ) = (F k (y k )+F k(y k + υ k ξ k )ξ k F k (y k )) +F j (x j ) (F j (x j )+F j (x j + υ j ξ j )ξ j = F k (y k + υ k ξ k )ξ k F j (x j + υ j ξ j )ξ j ξ k max (F k (y k + υ k ξ k )) F j (x j + υ j ξ j )ξ j ξ j (max (F k(y k + υ k ξ k )) F j(x j + υ j ξ j )) < 0, υ j (0, 1), υ k (0, 1),k K. The lat inequality how that x i not the optimal olution to (6). Thu, x cannot be utility proportional fair. Thi contradict the aumption and prove that x i path utility max-min fair. To (ii):aume q 1 Q 1, q 2 Q 2 and q 1 q 2 for ource 1, 2. Applying (1) to given q 1,q 2,wehavef(q 1 )=U 1 (x 1 ) f(q 2 )=U 2 (x 2 ) becaue of the monotonicity of f( ). To (iii): From L( 1 ) L( 2 ) it follow, that q 1 q 2. Since the available utility f( ) i monotone decreaing in q and the bandwidth utility U 1 (x 1 ) <u max 1 of uer 1 i not bounded by it maximum value, it follow, that f(q 1 )=U 1 (x 1 ) [f(q 2 )] umax 2 u = U min 2 (x 2 ). 2 Appendix B Proof of Theorem 3: Since all element of the equence x(κ) olve (6) ubject to (7), the equence i bounded. Hence, we find a ubequence x(κ p ),p N +, uch that lim κp = x. We how, that thi limit point x i utility max-min fair. The uniquene of the utility max-min fair rate vector x will enure that every limit point of x(κ) i equal x. Thi prove the convergence of x(κ) tox. Since all uer S ue the ame tranformation function f (q )=q 1 κ, S, the econd order utility and it derivative applied to the rate vector x (κ) have the following form: F (x (κ)) = U (x (κ)) κ F dx (κ) with x (κ) = U (x (κ)) κ, S.

74 Tobia Hark We aume that the limit point x =(x X, S) i not utility max-min fair. Then we can increae the bandwidth utility of a uer j while decreaing the utilitie of other uer k K S \{j} which are larger than U j (x j ). More formal, it exit a rate vector y =(y X, S) and an index j S with U j (y j ) > U j (x j ), j S and U k (y k ) < U k (x k )withu k (y k ) > U j (x j )fora ubet k K S \{j}. Wechooeκ 0 o large that for all element of the ubequence x(κ p )withκ p >κ 0 the inequalitie U j (y j ) >U j (x j (κ p ),j S, and U k (y k ) <U k (x k (κ p )) with U k (y k ) >U j (x j (κ p ))foraubetk K S \{j} hold. With the inequality U j (x j (κ p )) < U k (x k (κ p )), k K, wecanchooe κ 1 >κ 0 large enough uch that U j (x j (κ p )) κp >C U k (x k (κ p )) κp, (14) for all k K, κ p >κ 1,andC>0 an arbitrary contant. Hence, there exit a κ 1 large enough that the following inequality hold: U j (x j (κ p )) κp > (x k (κ p ) y k ) max }{{} U k(x k (κ p )) κp,κ p >κ 1. (15) >0 We evaluate the variational inequality (11) given in the definition of utility proportion fairne for the candidate rate vector (y X, S) andκ p >κ 1. F x (κ p ) (x (κ p ))(y x (κ p )) = U (x (κ p )) κp (y x (κ p )) S S = U j (x j (κ p )) κp (y j x j (κ p )) + U k (x k (κ p )) κp (y k x k (κ p )) >U j (x j (κ p )) κp (y j x j (κ p )) max U k(x k (κ p )) κp (x k (κ p ) y k ) > 0, uing (15). Hence, the variational inequality i not valid contradicting the utility proportional fairne property of x(κ p ).