IEEE rd Internationa Symposium on Persona, Indoor and Mobie Radio Communications - (PIMRC) Utiity-Proportiona Fairness in Wireess Networks G. Tychogiorgos, A. Gkeias and K. K. Leung Eectrica and Eectronic Engineering Department Imperia Coege, London SW7 AZ, UK {g.tychogiorgos, a.gkeias, kin.eung}@imperia.ac.uk Abstract Current communication networks support a variety of appications with different quaity of service (QoS) requirements which compete for its resources. This continuousy increasing competition highights the necessity for more efficient and fair resource aocation. Current Network Utiity Maximization (NUM) framework fais to achieve this target and aternative approaches cannot operate in networks that consist of wireess inks. This paper presents a NUM framework for wireess networks that shares resources according to the utiity proportiona fairness poicy. This poicy is shown to prevent rate osciations in the resource aocation process, aocate resources in a more fair manner among different types of appications and ead to the cacuation of cosed form soutions for the optima rate aocation function. Based on this poicy, a distributed rate and power aocation agorithm is proposed that gives priority to appications with greater need of resources. Finay, numerica resuts on the performance of the proposed agorithm are presented and compared against other approaches in the iterature. I. INTRODUCTION Since the semina papers of Key et a. [] and Low et a. [], the proposed Network Utiity Maximization (NUM) framework has found numerous appications in communication networks since it made cear that expressing the network resource aocation as an optimization probem can be soved by ow-compexity distributed agorithms. More specificay, they proposed an optimization probem of the form: Probem Π NUM : max r R U r (x r ) s. t. Ax C, x, where r denotes the index of the source node, x r represents the transmission rate of node r, C is a vector containing the capacities of a inks and U r (x r ) is the utiity of node r when transmitting at rate x r. In essence, the utiity represents the degree of satisfaction of a user as a function of the transmission rate. Moreover, A jr, the eement of matrix A, is when ink j ies on route r, and otherwise. The authors propose an iterative distributed agorithm where each ink in the network charges for its resources and the users determine their transmission rates according to the maximum amount they are wiing to pay. This agorithm optimizes the resource Research was partiay sponsored by the U.S. Army Research Laboratory and the U.K. Ministry of Defence and was accompished under Agreement Number W9NF---. The views and concusions contained in this document are those of the author(s) and shoud not be interpreted as representing the officia poicies of the U.S. Army Research Laboratory, the U.S. Government, the U.K. Ministry of Defence or the U.K. Government. The U.S. and U.K. Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon. aocation under two major assumptions; the utiities are a concave functions of rate and a inks have fixed capacity, e.g. are wired. These two assumptions are responsibe for a number of shortcomings of current NUM approaches, which wi be discussed in detai in the remainder of this section. Concave utiities are idea to mode appications that generate eastic traffic []. Easticity describes an appication s abiity to adapt easiy to changes in the network conditions, such as deay, throughput etc, whie sti meeting some QoS requirements. Exampes of such appications, incude FTP and HTTP [][5][], which used to generate the majority of the traffic in the internet unti recenty. However, the majority of the traffic in current networks is generated by reatime appications that are considered ineastic. Existing work modes such appications using non-concave sigmoida utiity functions [8][9][] that turn the resuting formuation into a non-convex probem. An exampe of such utiity function is shown at the top subpot in Figure. Despite the existence of anaytic methodoogy to sove or approximate the optima soution for such probems in a distributed way, this approach has significant disadvantages: The optima rate aocation function of a source node, x r (λ r ), is hard to be cacuated in a cosed form and therefore numerica gradient-based approaches must be used, which however increase convergence time and degrade accuracy. Function x r (λ r ) is discontinuous for some vaues of ink price. This causes osciations in the network that can prevent the agorithm from converging. An exampe of x r (λ r ) for a sigmoida utiity function is shown in bue at the bottom subpot of Figure. The heuristics proposed in iterature to resove these osciations offer approximations that in some cases can be far from the optima soution. Despite the fact that the utiity function U r (x r ) is defined for rates within the range [ ] x min r,x max r, ony a sma part of this range can be achieved. This restricts the appicabiity of such approaches in practica probems. For exampe, the rate for the utiity of Figure takes vaues within the range [, ] but the feasibe range region (shown in back) is restricted ony to either zero or vaues within [.5, ]. The traditiona NUM formuation maximizes the aggregate utiity in the network. Moreover, it has been shown [] and their percentage is expected to increase as forecasted in [7] where λ r is the aggregate price in the network 978--7-59-//$. IEEE 89
U r (x r ) x r * (λ r ).8... User Utiity function 5 7 8 9 Data 8 Fig.. Optima Rate Aocation Utiity Feasibe Region Optima Rate Feasibe Region.5..5..5 Aggregate Link Price λ r The feasibe rate region of a sigmoida utiity function that the resuting bandwidth aocations foows the so-caed (bandwidth) proportiona fairness. Whie this type of fairness seems to perform we when a users foow the same utiity, this approach is responsibe for some contradictory behaviors in cases that users foow different utiities, i.e. when users have different QoS needs. In such cases, proportiona fairness favors users which require ow rate to achieve high utiity. As pointed out first in [], a bandwidth proportiona fair optimization agorithm favors users with ow demand, i.e. those with rapidy increasing utiity function. This happens because aocating a unit of rate to a utiity with arge derivative eads to arger increase in the aggregate utiity than when aocating to users with high demand, i.e. with sma vaue of utiity derivative. To resove this contradictory behavior, authors in [] define a new type of fairness, caed utiity proportiona fairness. According to that, a bandwidth aocation x =[x,x,...,x R ] T is utiity proportiona fair, if it is feasibe and for any other feasibe aocation x, x r x r U r (x. () r R r) The utiity proportiona fairness can be achieved if the utiity function of each user is transformed according to: xr U r (x r )= m r U r (y) dy, m r x r M r, () where m r and M r are the minimum and maximum transmission rates for user r respectivey, and the objective function of Probem Π NUM is changed to r R U r (x r ). Authors in [] propose a distributed agorithm to sove Probem Π NUM in order to achieve utiity proportiona fairness in wired networks shared by various types of appications. However, current communication networks are often consisted of wireess networks, whose capacity is not constant but depends on the interference. This need highights the necessity of extending the current utiity proportiona fairness framework to be abe to adjust ink powers according to the channe conditions in the network. Motivated by the aforementioned shortcomings of current bandwidth and utiity proportiona fairness mechanisms in wireess networks, this paper makes the foowing contributions: Proposes a utiity proportiona fair optimization formuation for high-sinr wireess networks. Utiity proportiona fairness can prevent the osciations, caused when a utiity function is non-concave, aow the use of the fu range of possibe rate vaues and cacuate the optima rate. Derives anaytica soutions for the optima rate aocation function for a number of widey used appication types. Proposes a distributed utiity proportiona fair agorithm to jointy optimize transmission powers and data rates in high-sinr wireess networks. The rest of the paper is organized as foows. First, Section II presents a utiity proportiona fair formuation for high- SINR wireess networks and proposes a distributed gradient agorithm to cacuate the optima resource aocation. Consequenty, Section III provides cosed form soutions of the optima rate for a number of appication types and discusses how these formuas can be used to prevent osciations. Section IV presents numerica resuts iustrating the convergence and performance of the proposed approach and Section V concudes our current work and outines our future research pan. II. PROBLEM FORMULATION This chapter focuses on the deveopment of an optimization formuation for wireess networks that achieves utiity proportiona fairness whie taking into account the interference among wireess inks and the different QoS requirements of various appications. A. Network Mode Consider a muti-hop wireess network where each node can operate either as traffic source, destination or reay that just forwards traffic to its neighbors. We define the transmission rate vector r =[r,r,...,r M ] T which incudes the transmission rates of a M source nodes in the wireess network. Moreover, we define the ink as the tupe (T,R ), where T is the transmitting and R the receiving node, respectivey. We aso define p =[p,p,...,p L ] T as the vector which incudes the transmission powers of the L inks. The wireess channe is modeed as foows. Let G be a matrix of size L L, where G km, with k, m,,...,l, represents the path oss coefficient for the path between the transmitter of ink k and the receiver of ink m. The eements of the path oss matrix G depend on the physica characteristics of the wireess inks. As expained earier, each source node i is associated with a utiity function U i (r i ). The utiity function of a user represents the degree of satisfaction that a user enjoys when sending at a specific rate. In other words, the user utiity function refects the Quaity of Experience (QoE) of a user when data content is deivered at a specific rate. This QoE cannot be determined precisey for each user but prior work in the iterature [][8] has identified approximate forms/shapes for 8
various appications, such as HTTP, FTP and video streaming appications. Finay, we aso associate each wireess ink with a convex cost function V (p ). This function represents the cost of using the imited power resources of the wireess channe. The incorporation of this cost function eads towards more energy efficient resource aocations. B. Optimization Probem The network performance optimization is formuated as the foowing maximization probem: M L Probem Π MWN :max U i (r i ) γ V (p ) r,p i= = M s. t. α i r i C (p), inks i= where U i (r i ) is the transformed utiity function given by () for rate r i, parameter α i is one if the traffic of user i is passing through ink, and zero otherwise. The rates r i, with i,,...,m, and powers p, with,,...,l, are positive quantities and γ is a positive weighting parameter. The capacity of a ink foows Shannon s capacity formua, C (p) =B og ( + SINR ) and is a function of the Signa to Noise pus Interference Ratio (SINR) at the receiver of the ink. This formua is a non-concave function of powers and this might prevent any gradient based agorithm from converging to the optima power vector. However, under the assumption that SINR, the formua C (p) =B og (SINR ) can provide a sufficienty accurate approximation of ink capacity. Moreover, this function is a concave function of powers []. For the remainder of this paper, we assume high SINR environments and therefore the ink capacity C (p) wi be cacuated using this approximation. Duaity Theory [] provides an efficient methodoogy to sove optimization probems distributedy. For this reason, one shoud initiay form the Lagrangian function as M { L(r, p, λ) = Ui (r i ) r i λ i} () + i= L λ B og p G p k G k + n γ L V (p ), = k where λ i = L = α iλ is the aggregate price of user i to send a unit of rate through the network. It is evident from () that Probem Π MWN consists of two subprobems couped by the dua variabe vector λ. The first one determines the optima rate to maximize the net revenue of the source node, whie the second determines the transmission power of the inks. Consequenty, according to duaity theory every source i can cacuate its optima rate ri (λ) using ri (λ) = arg max [ U i (r i ) r i λ i]. () The power and dua variabes can be cacuated iterativey using: = λ (t) =λ (t ) δ λ (t) (5) λ p (t) =p (t ) + δ p (t), () p where δ λ (t) and δ p (t) are sma positive step sizes and = B og λ p G p k G k + n M α i r i (7) k = γv (p )+ p λ m m [ λ i= p n() G m P k m G kmp k + n m ]. (8) Equations ()-() constitute a joint prima-dua distributed agorithm, which wi be described in detai in the next section aong with how utiity proportiona fairness can ead to the cacuation of cosed form soutions for (). III. THE PRICE-BASED RATE ALLOCATION FUNCTION The existence of various types of user appications compicates the process of cacuating the optima rate aocation of a user for a specific aggregate price. According to optimization theory [], the optima rate wi be at the point where the first derivative of the objective function diminishes and therefore ri (λ) =U ( i λ i ). (9) In the traditiona NUM framework U i ( ) =U i ( ), where U i ( ) is the utiity function of user i as defined earier. The optima rate can be cacuated using (9) ony if the utiity function is concave function of rates. If U i ( ) is partiay convex and partiay concave, as with sigmoida utiities, the first derivative cannot be inverted since it is not a one-to-one function. For sigmoida utiities, one shoud use aternative methods with a negative impact on the agorithm convergence speed. Such an aternative coud be a gradient based iterative equation of the form: r i (t +)=r i (t)+δ r (t) () r i where δ r (t) is a positive step size and L(r,p,λ) r i is the gradient of the agrangian function with respect to r i. However, such an approach wi not aways converge to the goba optimum. In fact, according to the necessary and sufficient condition in [5], the agorithm wi converge ony if () is continuous around the optima price vector λ. If this condition does not hod, there can be osciations in the network that wi prevent the agorithm from converging. In such case, an osciation resoving heuristic, such as the ones presented in [8] and [], is necessary to ensure stabiity but at the cost of oosing optimaity. Considering utiity proportiona fairness, however, by using the transformation of (), the probem becomes convex even for sigmoida utiities and () aways satisfies the condition in []. This means that the iterative equation () wi be abe to cacuate the optima soution but more importanty this aso aows to cacuate a cosed form soution for (9) directy. 8
TABLE I THE OPTIMAL RESOURCE ALLOCATION FUNCTION FOR WIDELY USED TYPES OF APPLICATIONS Appication Type User Utiity Function Optima Rate Aocation Function HTTP U i (r i )= og( r i ) r min ri (λ) =r min ( ) r max λ i r min og( rmax ) r min FTP U i (r i )= og(r i+) og(r max +) r i (λ) =(r max +) λ i Video Streaming U i (r i )= +exp( α(r i r β)) i (λ) = α β og(λi ) α When the user utiity function is transformed according to (), the first derivative can be easiy cacuated as: U i (r i )= U i (r i ). () Eq. () is invertibe as ong as it is continuous and monotonic, which are both true for any concave utiity and any sigmoida utiity that foows the shape shown in Figure. Hence, combining (9) and (), we find that the optima rate is given by: ( ) ri (λ) =U i λ i. () Therefore, it is possibe to cacuate a cosed form soution for () for any utiity function that satisfies these two properties. This is a significant advantage of the utiity proportiona fairness approach which eads to the deveopment of agorithms that cacuate the optima soution even for non-concave utiities and converge significanty faster than the traditiona approaches. Based on the anaysis above, we derive the optima rate aocation for browsing, fie transfer and video streaming appications using the suggested utiity functions in [] and [8], when utiity proportiona fairness is appied. These optima rate aocation functions are demonstrated in Tabe I. r min and r max represent the minimum and maximum transmission rate of a user, and parameters α and β are caibration parameters of the sigmoida utiity. An important observation here is that the continuity of () for a aggregate prices aso impies that when using utiity proportiona fairness a rates within the range [ r min,r max] are feasibe contrary to what happens with bandwidth proportiona fairness where ony a sma part of the tota rate range is feasibe, as iustrated in Figure. This shows that the optimization agorithm has the robustness to adjust to any changes in the ink prices and take advantage of the fu range of the avaiabe rate region in order to maximize user satisfaction in the network. Having formuated the proportiona fair optimization probem for wireess networks and derived cosed forms of the optima rate aocation functions for some of the most common appications, the next step is to deveop a distributed agorithm to jointy optimize transmission powers and data rates in the aforementioned wireess network. The iterative equations ()-() can be used to create a distributed agorithm to jointy optimize rates, powers and prices with minimum information exchange between users. This agorithm consists of two parts; Agorithm is carried Agorithm Optima Rate Cacuation At time t, source i =,...,M: : receives the aggregate price λ i (t); : cacuates the optima rate, ri (λ), using () or the formuas in Tabe I; : starts transmitting at time t + at rate ri (λ); out by each source node and Agorithm in each ink. This joint agorithm is an extension of the standard gradientbased agorithm and wi converge to the optima soution for sufficienty sma vaues of the step sizes δ λ (t) and δ p (t) [], since Probem Π MWN has been convexified using the utiity transformation of () and the High-SINR Shannon capacity approximation formua. Regarding the information exchange of the agorithm, users need to know the aggregate ink price λ i. This can be either stored in the ACK packets sent by the destination to the source node, or, if the ink price is viewed as the ink deay, it can be impicity measured by the packet queuing deay in the network. Then, the power cacuation process requires that a ink knows the channe conditions of neighboring nodes. This information can be easiy obtained from the ower ayers of the protoco stack with no additiona signaing overhead. The use of the cost function V (p ) affects convergence of the power contro probem as we. When the weighting factor γ is zero, there can be a case where equation (8) is aways positive. This woud cause the distributed agorithm to increase transmission powers indefinitey and therefore the agorithm wi not converge. This was further justified in [] as foows. Consider an arbitrary wireess network and an iteration t of the optimization process where the power vector is p = [p, {, L}] and the rate vector r = [ri t,i {, N}]. If at the next iteration the powers are increased by a sma percentage, et ɛ, the resuting power vector wi be ˆp =(+ɛ) p and( the capacity of each ink ) wi now (+ɛ) p become C (ˆp) =B og G >C (+ɛ) k p kg k +n (p). This increase of the ink capacity woud aso resut in higher network utiity and therefore the distributed agorithm wi continue increasing powers indefinitey. However, if γ>, the optimization agorithm wi reach a point where any further increase in the transmission power woud not resut in an increase at the aggregate utiity and therefore the agorithm wi converge to a specific power vector. In existing work, such cases are often prevented by assuming a maximum transmission power. Such an assumption, woud be reasonabe in practica systems, but is not appicabe 8
Agorithm Link Price Cacuation At time t, a ink =,...,L: : cacuates the incoming aggregate rate; : cacuates the new price using (5); : cacuates the new power using (); : sends the new price λ (t +)to a sources that are using ink and starts transmitting using p (t +); Fig.. 7 8 7 8 9 5 Network Topoogy Exampe 5 on the theoretica anaysis of the probem since it creates artificia convergence points []. Therefore, using the cost function V (p ) is a more natura way of assuring both energy efficiency and convergence of the distributed power contro agorithm. IV. NUMERICAL RESULTS The utiity proportiona fairness () approach was appied to various network scenarios in MATLAB, an exampe of which is the network topoogy shown in Figure for iustration purposes. The wireess network consists of source nodes, intermediate nodes and a set of destination nodes. The simuation setup consisted of a variety of types of appications, incuding FTP, HTTP and mutimedia appications. This dictated the use of different utiity functions, concave or sigmoida, according to the type of appication. A appications were modeed using the utiities of Tabe I for various vaues of parameters. More specificay for the exampe of Figure, source nodes - and 5 serve reatime appications, whereas source nodes and serve eastic appications modeed by concave utiities. The path oss coefficients G were significanty arger than these of the interfering channes, i.e. terms G k for k =,...,Land k, in order to aow the use of the high-sinr channe capacity approximation formua with ow approximation error. The performance of the approach is compared against the traditiona bandwidth proportiona fairness () [8] approach used in prior work in order to show that can successfuy avoid the occurence of rate osciations and can ead to fair aocation of resources when heterogeneous appications compete. During the BFP optimization process, the sef-reguating heuristic [8] was used in order to resove any osciations that might occur. Figure shows the convergence of both the objective function of the optimization probem and the utiity functions of sources. When is used, users and foow a sigmoida utiity and start to osciate after about 8 iterations, as the spikes indicate. The sef-reguating heuristic 9 User Utiity Objective function.5.5 Convergence of Utiities U, U, U, U in U, U in U in U in 5.5.5.5 Convergence of Objective function.5 8 Fig.. Convergence of Utiity and Objective Functions removes them from the optimization process and therefore their utiity is. The remaining users compete for a the network resources which eads to higher individua utiities for these users. On the other hand, there are no osciations when is used and the resuting rate aocation eads to the same degree of satisfaction for a sources. In genera, gives priority to users with higher rate requirements whie aocates more rate to users that are satisfied easier in an attempt to achieve higher aggregate utiity in the network. For exampe, at the fina rate aocation in, a the eastic appications are aocated some rate whie ony two out of the four mutimedia appications are aowed to transmit. The convergence of the rate aocation of the first four sources for both and approaches is iustrated in Figure. It is evident that for the osciations occuring at the rate aocation of sources and cause spikes in the aocations of the rest as we, whie in rate are converging Source 7 5 8 8 8 Fig.. Source Convergence of Rate Aocation 5.5.5.5 Source 8.5.5.5 Source 8 8
smoothy to the optima soution. Finay, Figure 5 shows the convergence of the power aocation for inks to. It is evident from the peaks around iteration 9 that the existence of osciations in the approach affects the convergence of powers as we, whereas in the powers converge smoothy to their optima vaues. In addition, it is cear that the different aocation poicy between and aso eads to different vaues of transmission powers due to the difference in the traffic passing through each ink...8....8... Link 5 5 Fig. 5. Link 5 5.... Link 5 5.... Convergence of Transmission Power Aocation Link 5 5 V. CONCLUDING REMARKS This paper discussed how utiity proportiona fairness can be used to resove many of the shortcomings of traditiona NUM approaches in wireess networks. More specificay, we proposed a utiity proportiona-fair optimization formuation for high-sinr wireess networks and deveoped a joint distributed rate and power aocation agorithm to sove this probem. In addition, it was shown that the use of utiity proportiona fairness aows the cacuation of cosed form soutions for the optima rate aocation for a wide range of popuar appications, prevents osciations in the network and assures that a appications wi be treated equay in terms of the rate aocation. Our approach was aso simuated and compared against the traditiona bandwidth proportiona fair approach. The focus of our future research wi be twofod. First, we intend to examine more types of appications and cacuate cosed form soutions for the optima rate aocation and, second, we wi examine aternative approximations of the non-concave Shannon s capacity formua that wi be efficient for both high and ow SINR environments. REFERENCES [] F. P. Key, A. Mauoo, and D. Tan, Rate contro in communication networks: Shadow prices, proportiona fairness and stabiity, Journa of the Operationa Research Society, pp. 7 5, 998. [] S. Low and D. Lapsey, Optimization fow contro. i. basic agorithm and convergence, IEEE/ACM Transactions on Networking, vo. 7, no., pp. 8 87, December 999. [] W. Staings, Data and Computer Communications, 9th ed. Pearson Custom Pubishing,. [] D. Paomar and M. Chiang, A tutoria on decomposition methods for network utiity maximization, IEEE Journa on Seected Areas in Communications, vo., no. 8, pp. 9 5, August. [5] D. P. Paomar and M. Chiang, Aternative distributed agorithms for network utiity maximization: Framework and appications, EEE Transactions on Automatic Contro, vo. 5, no., pp. 5 9, December 7. [] C. Liu, L. Shi, and B. Liu, Utiity-based bandwidth aocation for tripepay services, in Universa Mutiservice Networks, 7. ECUMN 7. Fourth European Conference on, feb. 7, pp. 7. [7] Cisco, Cisco visua networking index: Forecast and methodoogy, 5, Cisco Systems Inc., Tech. Rep., June. [Onine]. Avaiabe: http://www.cisco.com/en/us/soutions/coatera/ ns/ns55/ns57/ns75/ns87/white paper c-58.pdf [8] J.-W. Lee, R. R. Mazumdar, and N. B. Shroff, Non-convex optimization and rate contro for muti-cass services in the internet, IEEE Journa on Seected Areas in Communications, vo., no., pp. 87 8, August 5. [9] P. Hande, S. Zhang, and M. Chiang, Distributed rate aocation for ineastic fows, IEEE/ACM Transactions on Networking, vo. 5, no., pp. 5, December 7. [] G. Tychogiogos, A. Gkeias, and K. K. Leung, Towards a fair nonconvex resource aocation in wireess networks, in IEEE PIMRC, Toronto, Canada, September. [] W.-H. Wang, M. Paaniswami, and S. H. Low, Appication-oriented fow contro: Fundamentas, agorithms and fairness, IEEE/ACM Transactions on Networking, vo., no., pp. 8 9, December. [] M. Chiang, To ayer or not to ayer: baancing transport and physica ayers in wireess mutihop networks, in IEEE INFOCOM, vo., march, pp. 55 5 vo.. [] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press,. [] D. P. Bertsekas, Noninear Programming. Athena Scientific, 999. [5] G. Tychogiogos, A. Gkeias, and K. K. Leung, A new distributed optimization framework for hybrid ad-hoc networks, in IEEE GobeCom Workshop on Heterogeneous, Muti-Hop, Wireess and Mobie Networks, Houston, USA, December. [] J. S. P. V. Hutson and M. J. Coud, Appications of Functiona Anaysis and Operator Theory. Academic Press, 98. 8