terative Water-fiing for Load-baancing in Wireess LAN or Microceuar Networks Jeremy K. Chen Theodore S. Rappaport Gustavo de Veciana Wireess Networking and Communications Group (WNCG), be University of Texas at Austin Abstract-This paper presents an efficient iterative oadbaancing agorithm for time and bandwidth aocation among access point4i (APs) and users subject to heterogeneous fairne4is and appication requirements. The agorithm can be car'ied out either at a centra network switch with site-specific propagation predictions, or in a decentraized manner. The agorithm converges to maximum network resource utiization from any sta'ting point, and usuay converges in 3 to 9 iterations in various network conditions incuding users joining, eaving, and moving within a network and various network sizes. Such a fast convergence aows rea-time impementations of our agorithm. Simuation resuts show that our agorithm bas merits over other schemes especiay when users exhibit custered patterns: Our agorithm, when assuming mutipe radios at each use1, achieves 48% gain of median throughput as compared with the max-min fair oad-baancing scheme (aso with the muti-radio assumption) whie osing 14% of fairness index; we aso achieve 26% gain of median throughput and 52% gain of fairness index ove.r the Strongest-Signa-First scheme (which assumes each user has ony a singe radio). When ony a singe radio is used, our agorithm is simiar to the max-min fairness scheme, and is sti better than SSF with 44% gain of 25-percentie throughput and 37% gain of fairness index.. NTRODUCTON Peope consider increasing the capacity of WLAN or microceuar networks by increasing AP density and assigning proper non-overapping frequency channes to APs. As the number of APs to which a user can connect increases, an agorithm that efficienty associates users to APs becomes critica for bandwidth and quaity of service (QoS) management. However, the defaut Strongest-Signa-First (SSP) approach used in 802.11 products, in which each user chooses an AP with the strongest signa, resuts in uneveny distributed oads among APs and poor performance [1]. n order to better baance oads, vendors such as Cisco, Trapeze, Aruba, Meru, and Symbo have introduced centra switches to have networkayer contros (e.g. oad baancing and handoffs) over the AP's norma processing in physica ayers today. This paper presents a oad-baancing agorithm that can be carried out 'either in a decentraized way with some message exchange between APs and mobie users, or at a centra switch with site-specific predictions (such predictions can provide the centra switch with deta~ed RF parameters, the received signa-to-noise ratios (SNR), and estimate the achievabe capacity for each wireess ink; see [2]-[4] and references therein). Severa heuristic oad-baancing schemes have been presented. Baachandran et a [1] observed that APs with oad- 1 This work is supported by NSF Grant AC-0305644. Fig. 1. A simpe network with 2 APs and 3 users. Different thicknesses of dashed ines signify different avaiabe ink capacity Ra 8, APs 1 and 2 use disjoint channes. T a,s denotes the time fraction aocat~d to user s over the RF channe of AP a. baancing functiomiities periodicay send beacons with current oad, captured by the number of users, bit error rates, and signa strengths. However, severa measurement studies have shown that the number of users is not a good metric to determine the oad [4], [5]. Baachandran et a proposes a better oad-baancing scheme where each arriving user expicity asks for a minimum and a maximum bound on bandwidth/throughput, and a centraized admission contro is performed to associate the arriving user to an AP that is within the user's radio range and has the most avaiabe capacity [1]. The work in [1] improves the degree of oad baancing by over 30% and user bandwidth aocation up to 52% in comparison with schemes with itte oad baancing. The work in [ 6] presents a decentraized oad baancing agorithm that can be appied to EEE 802.11 a/b/g without modifying the standards whie being transparent to end users. t was shown by exampe that the throughput of a station increases from 1.5 to 2 Mbps, and packet deays can be reduced from 450 to 8 ms. Whie the work in [1], [6] outperforms schemes with itte or no oad baancing, they are not shown to be optima. To the best of our knowedge, the ony work that achieves some form of optimaity in oad baancing is [7], which achieves max-min fair bandwidth aocation. This paper considers a network with mutipe APs and users, as depicted in Fig. 1 and tries to answer a fundamenta question: which AP(s) shoud be connected with a particuar user, and how much time shoud the specific AP(s) aocate to this user in order to achieve optima network utiization subject to heterogeneous fairness and appication requirements. Section describes the system mode and notation in detai. Section presents the formuation and an iterative agorithm for the optima aocation of channe usage time. Simuation resuts are presented in Section V. 0-7803-9392-9/06/$20.00 (c) 2006 EEE 117 Authorized icensed use imited to: University of Texas at Austin. Downoaded on December 4, 2009 at 13:11 from EEE Xpore. Restrictions appy.
.J. SYSTEM MODEL AND NOTATON We assume a muti-radio capabiity that aows mutipe channes to be received and decoded in - parae by each user (this mode has been proposed in [7]). Jt is suggested in this paper that the mutipe-radio assumption simpifies the computation to be efficient (the probem formuation is convex). Our approach can aso be used for muti-radio APs. Our agorithm aows up to an unimited number of radios on a user; however, 2 to 4 radios suffice in practice, since a user in an actua WLAN or microceuar network is usuay surrounded by at most 4 APs. We assume that users exhibit a quasi-static mobiity pattern (a mode that has been adopted in [7]) where users can move from pace to pace, but they tend to stay in the same physica paces for ong periods of time [5]. This mode aows us to consider ong-term averaged ink capacities over a time scae of about 2 seconds (denoted as TAvo), which is adequate for resource re-aocation and may- not be a noticeabe time interva for new users who are waiting to be associated with APs. The proposed oad-baancing agorithm is executed based on the predicted average capacities during every TAvo interva. Link capacities may change in successive TAvo intervas due to interference or changes in user appications or trans~ission states. The capacity Ra,s (e.g. throughput) between an AP a, and a user, s, is determined by the peak throughput for a singe ( unshared) user, and aso determined from predicted, measured, or optimized throughput estimates based on site specific information. For the case where mutipe users share a singe AP over an RF channe, the throughput between the AP, a, and a user, s, is a fraction (the time fraction of channe usage) of the ink capacity, that is, Throughputa,s = Ta,sRa,s, where Ta,s is the channe usage time between AP a and user s. During a TAvo interva, even though users may join/eave the network, or RF noise sources may emit interfering signas, the effects of these transient events on ink throughputs are quantized and samped every TAvo (e.g. bock processing is used). n the beginning of every TAvo interva, our iterative oad baancing agorithm re-adjusts the time/bandwidth resource aocation over a users and APs. The agorithm converges to optimum in merey 3 to 9 iterations regardess of network sizes, athough the computation time of each iteration grows ineary with the number of users mutipied by the number of APs controed by the switch. On a 20Hz nte Pentium computer with Windows XP, each iteration in Matab takes 30 miiseconds for a network with 36 APs and 300 users. Code impemented in assemby or C anguage woud be much faster and is very suitabe for reatime impementations of our agorithms on hardware/firmware, as contempated in [2], [3]. With the above mentioned assumptions, the rea throughput that a user experiences mainy depends on the channe usage time aocated from the APs to this user. For instance, in Fig. 1, suppose AP 1 and AP 2 aocate T1,1 = 20% and T2,1 = 40% of their time (over disjoint channes 1 and 2, respectivey) to user 1, respectivey. The tota bandwidth that user 1 obtains is b 1 = ( 0 0 R 1,1 + 1 ~ 0 R2,1; the bandwidths of users 2 and 3 can be computed in a simiar way. we consider an infinite backog of packets (fu and ready queues on every channe) for every user. Hence a user's throughput is the same as the bandwidth aocated to her. We maximize the sum utiity of throughput, which means maximizing Ei= Ui(bi) over the channe usage time in this exampe. f utiity functions are propery chosen, users wi be aocated different notions of fair aocation when the network reaches maximum sum utiity [8]. We made the assumption that a APs are under the contro of a network switch. However, some rogue APs or RF noise sources may emit interfering signas in the coverage area of the controed APs. n this case, some controed APs or overay sensors can detect signas from rogue APs. With detected signa parameters and site specific knowedge, position ocation techniques can ocate the rogue APs [2], [3]. Then, AP channe assignments are changed so that the APs near the rogue APs operate at orthogona RF channes in order to eiminate most interference from rogue APs. Then, the switch wi predict SNR and ink capacities between users and controed APs using site specific modes for the rogue ocations and transmit properties, and appy our agorithm to find the optima resource aocation accordingy. This paper assumes the frequency band of each AP has been propery assigned [2], [3], and focuses on finding the optima bandwidth/time aocation in a fuy-controed network. With an assigned aocated frequency channe, each AP serves its user by time sharing. The fraction of time resource dedicated for payoad transmissions between users and an AP, a, over an RF channe is denoted as T!rac (0 ::::; T!rac ::::; 1) (e.g., it ranges from 59% to 88% in 802.a). The subscript a in TJrac is used, since the payoad time fractions may differ from AP to AP. We suppose that each user shares her utiity function to a the APs that transmit signas strong enough to reach her. Then, each AP aocates its time resource (over its assigned RF channe) to users based on the information of the utiity functions of a the users within its coverage area, based on site specific knowedge [2]-[4]. n this paper, utiity functions are assumed to be concave, continuousy differentiabe, and stricty increasing [9] for simpicity of anaysis. Let n and m denote the numbers of APs and of users, respectivey. We use a or s as index when referring to a specific AP or user, and use j or i as dummy indices of APs or users when performing a summation. User s is said to be within the coverage of AP a if Ra,s > 0; otherwise, Ra 8 = 0. Each entry in the rate matrix can be predicted from a ste-specific prediction engine [2]-[4]. Within a unit time period, suppose AP a aocates a time fraction T a,s (over the assigned RF channe of AP a) to user s (0::::; T a,s ::::; 1). The actua bandwidth that user s gets from AP a is T a,sra,s. MAXMUM SUM UTLTY WTH TME ALLOCATON The optima AP-user association can be formuated as the sum utiity maximization probem in (1) over time resources 118 Authorized icensed use imited to: University oftexas at Austin. Downoaded on December 4, 2009 at 13:11 from EEE Xpore. Restrictions appy.
from APs on different RF channes to users. max """'U (""'T R ) L..,; ~ L..,; :;,~ :;,t subject to LTa,i ~ T!rac, Va, overt a,s ;::: 0, Va, s. (1) t is hard to find a cosed-form expression of the optima channe usage time aocation for (1). Nevertheess, if the optimization is over the time resources of ony a singe AP (over one channe), assuming the other APs' time aocations are fixed, cosed-form expressions for each AP's optima time aocation have cosed-form expressions, shown in (11) which are soutions to formuation (3). Theorem 3.1 discussed beow shows that the origina mutipe AP probem in (1) reaches the optimum if and ony if the time aocation from every AP simutaneousy has the cosed-form expressions as in (11). Hence, the optimization of the mutipe-ap probem can be done by successivey optimi~ing each AP's time resources, as presented in Agorithm 1 as an efficient iterative agorithm. Our derivation and proofs extend [10] to a wide cass of utiity functions (beyond ogarithmic) for different degrees of fairness and appication needs. The soe constraint in (1) means that the tota channe usage time used at each AP is upper bounded. The objective is to maximize the network utiity Li Ui(Lj Tj,iRj,i). Mo and Warand have proposed a cass of utiity functions that capture different degrees of fairness and mode appications with heterogeneous needs parameterized by qi [8]: The parameter Qi has an index i because each user i may have a different appication/fairness requirement. This famiy of utiity functions is concave, continuousy differentiabe, and stricty increasing [8]. The sum of concave functions is sti a concave function; hence, probem (1) is convex since a concave function is to be maximized over a convex constraint set [9]. The work in [8] shows that if qi ---+ oo, the formuation in (1) becomes a specia case that achieves max-min fairness, as studied in [7]. Within every TAvc, R remains constant after bock processing, and the optima sum utiity and T wi be determined accordingy. Suppose the sum utiity optimization is performed over the channe usage time resources of ony AP a, T a, = [T a,, T a,2,..., T a,m], assuming that the time aocations from the other APs to users are fixed. Then the formuation in (1) is reduced to max L Ui(Ta,iRa,i + Ca,i) i subject to LT a,i :::; T!rac' over T a,s ~ 0 v s, (3) where ca,i = L Tj,iRj,i are fixed. j:jia Denote by Aa the Lagrange mutipier for the constraint in (3). Then, the Lagrangian [9] is given by L(Ta,e, Aa) = L Ui(Ta,iRa,i+Ca,i)-Aa(LTa,i-T!rac). (4) Since utiity functions Us ( ) are increasing, it is natura to exhaust the time resource for maximizing sum utiity [9]; therefore, at the maximum of (3), we have Ei Ta,i = T~rac. Then, the sufficient and necessary optimaity conditions (KKT conditions) [9] for (3) can be written as: Ra.,sU8(Ta,sRa,.' +Ca.,.'!)= Aa if Ta, 8 > 0, Vs (5) < Aa if Ta, 8 = 0, Vs (6) ~T. = Tfrac L..J a,z a (7) Ta,8 ~ 0, Vs; Aa > 0. (8) t is obvious that no time is aocated to inks with zero capacity (i.e. T a,s = 0 if Ra,s = 0). Therefore, we focus on deriving the optima T a,s for Ra,s > 0. For genera utiity functions, the optima time fraction can be derived from (5): T = {-1-u'-(~) _ ca,s }+ a,s Ra s 8 Ra s Ra 8 ' Whie cosed~form soutions of T a,s do not exist for genera utiity functions, they can be obtained for the famiy of utiity functions in (2), for which (5) becomes ol 8Ta 1 8 Equating (10) with zero gives the optima time aocation: (-_.!._) (..L-1) Ca 8 T -,\ }+ qs R qs 1 a 1 8- { a a 1 8 - R a 1 8 (9) (10) (11) n (9) and (11), the notation {x}+ is needed because Ta 18 is nonnegative: { x} + = x if x ~ 0 and { x} + = 0 otherwise. By substituting (11) or (9) into 'Ei T a,i = T!rac in (7), Aa for each AP a can be numericay soved [9], [10]. n each iteration of our agorithm, finding the time resources of each AP requires soving a singe-variabe (.Aa) poynomia equation with m terms; hence, the time compexity of each iteration is 0( nm). f the parameter q 8 = 1, the expression of T a,s in (11) is the water-fiing expression, where the constant.a; 1 is known as the water-fiing eve [10]. Theorem 3.1: {Ta, 8, Va, s} is an optima soution to (1) if and ony if {Ta,, Ta,2,..., Ta,m} is the soution in (11) for AP a with the time aocation from the other APs {Tj, 8 : Vj =f a, V s} fixed, for a a = 1, 2,..., n. (The proof is omitted as it is a natura extension of Theorem 1 in [10].) As described in Theorem 3.1, the time aocations from each AP to users can be soved by (11), assuming time aocations from the other APs are fixed. Hence, the optima time aocation for the mutipe-ap optimization probem (1) can be found by an iterative agorithm (see Agorithm 1). Theorem 3. 2: Agorithm 1 resuts in an optima sum utiity and causes {Ta, 8, Va, s} to converge to an optima time 119 Authorized icensed use imited to: University of Texas at Austin. Downoaded on December 4, 2009 at13:11 from EEE Xpore. Restrictions appy.
Agorithm 1 An iterative agorithm to sove (1) Given a rate matrix {Ra,s, 'Va, s }. Start with a vaid time aocation {Ta, 8, 'Va, s}. repeat for each AP a = 1, 2,..., n do Compute {ca,s, \s} by (3). Compute {Ta., 8, \s} by (11) or (9). end for unti the sum utiity converges Output {Ta,s, 'Va, s}, aocation for Formuation (1). (The proof can be extended from the proof of Theorem 2 in [10]. Note that the optimum time aocation {Ta,s, 'Va, s} may not be unique.) Agorithm 1 can be carried out in a decentraized manner: each AP a computes the optima time aocation {T a,s, V s} ony for those users who are in the coverage of this AP. For the computation of each user's Ta 8, a constant ca 8 needs to be known, which in turn requir~s the knowedg~ of the bandwidth that this user s receives from APs other than AP a. n a reaistic WLAN setup, a user is under the coverage of no more than 4 APs; hence, the computation of ca,s at each user is efficient. APs sequentiay perform such decentraized computing. When the sum utiity converges, a contro message may be sent to APs to stop the decentraized computing. V. SMULATON RESULTS n this section, we compare the throughput and fairness performance of our maximum utiity (denoted as MaxUti) scheme with the max-min fairness scheme in [7], denoted as Ma.xMin, and the Strongest-Signa-First scheme in current 802.11 impementations. We consider a simpified scenario of free-space propagation mode where no obstaces exist in the vicinity of APs. t is cear that our agorithm can utiize site specific information, which wi be considered in future work. We consider different percentages (between 1% and 5%) of users joining, eaving, or moving within the network; hence, the ink capacities change over time. We sampe R for every TAva, and within this time interva, R is fixed. Two kinds of user distributions, namey unifonn and custer (or hotspot), are considered. First, users are uniformy distributed in a 600 meters by 600 meters square that encompasses the 36 APs. Second, we consider that a hotspot at the center attracts more peope: users are distributed in a circe-shaped area centered at the midde of the APs with a radius of 250 meters. Users are randomy ocated on this circe based on their uniformy generated poar coordinates (the distance from the center and the poar ange are uniformy distributed between (0, 250) and (0, 21r), respectivey). From the viewpoint of the Cartesian coordinate, the user density is higher near the center than near the circumference of the circe. Each point on the figures is an average over 100 independent runs. n the SSF case, each user (whose transceiver can hande ony a singe channe) associates with the strongest AP, and then each AP eveny distributes its time resources to the associated users. Simuations show 120 that the number of iterations (mosty between 3 and 9) does not grow with the number of users. Our agorithm converges quicky even for arge networks. Figs. 2 and 3 show the medians and the 25-percenties of user throughputs, respectivey. Tabe presents fairness indices (see [11] for this metric) for cases with 400 users; scenarios with different number of APs and users are omitted, since their fairness index vaues are simiar to those in Tabe. Both MaxUti and MaxMin assume that each user has mutipe radios. For fair comparisons with SSF, we aso compute singeradio resuts by propery rounding muti-ap time aocation; MaxMin-R denotes the resuts produced by the rounding method in [7]. The MaxUti-R resuts were obtained by a different rounding method: we first compute norma mutiperadio time aocation; then, if any user indeed uses mutipe APs, this user simpy chooses the AP that suppies her with the most bandwidth. Finay, if any AP has any time resource remained not aocated, this AP aocates the remaining time proportionay to its associated users. For exampe, if the rate matrix R = [ ~ ~ ~! ] and a users' utiity parameters, q, are 1, then the optima time fraction (aowmg. mutt. ra d' tos ) 1s. T = [ 0.417 0.417 0.166 0 0 0 0. 375 0.. 625 Each user chooses ony one singe AP; then the time rna-. b T [ 0.417 0.417 0 0 trtx ecomes = 0 0 0.375 0.625. Then, since the first AP has time fraction (16.6%) remained, the remaining time is proportionay distributed to users 1 and 2; finay the time matrix for the singe-radio case is T = 0.5 0.5 0 0 [ 0 0 0.375 0.625. A trade-off between throughput and fairness can be seen in muti-radio cases MaxUti and MaxMin. Our MaxUti has very good performance in custer case: in Fig. 2(b), MaxUti exhibits about 48% higher median throughput over MaxMin whie sacrificing ony 14% of fairness as in Tabe. t is because MaxMin tends to achieve absoute fairness (its fairness index is amost 100% as in Tabe ) by sacrificing throughput (giving more time resource to users with poor ink capacities). Our MaxUti trades throughput with fairness; even in uniform case in Fig. 2(a), MaxUti yieds 9% higher median throughput than MaxMin whie osing 2% of fairness as in Tabe. Our agorithm, with mutipe radios at each user, outperforms SSF by 26% and 52% in terms of median throughput and fairness index, respectivey, as in Fig. 2(b) and Tabe. Surprisingy, the singe-radio scheme MaxUti-R yieds worse median throughput than SSF, mainy because our rounding method (as presented in the numerica exampe above) makes users choose stronger APs, thereby causing unbaanced oads on APs. The rounding method in [7] may be modified to be imposed upon MaxUti for better rounding performance; this is a subject for future research. Nevertheess, MaxUti R yieds simiar 25-percentie user throughputs as MaxMin-R, and is 44% and 17% higher than SSF in custer and uniform cases, respectivey (as seen in Fig. 3). Moreover, Tabe indicates that SSF has poor fairness indices as compared Authorized icensed use imited to: University of Texas at Austin. Downoaded on December 4, 2009 at 13:11 from EEE Xpore. Restrictions appy.
UnHorm Uniform emaxut -$-MaxUtii-R BMaxMin ~MaxMin-R -*SSF 200 300 400 (a) Uniform user distribution Custer (/) 0. ------------~--------------r- ------------- -------------..2'!! BMaxUti ::2! 0.8 ---- -------+------------+---------- -$-MaxUtii-R - "5!! BMaxMin _@-0. 7 - ------- --+------------+---------- ~MaxMin-Rg>! -*"SSF ~ 0.6 - -- ----- 1 - -----------r..------------:------------- +"" ii 0.5 -------- -- ~----------- : } :::3 : 0 04 ------------- -- ---- : ---~------------- &1.! 'g 0,3 -------------~------------- - ::2! 0.2!:- 200 300 400 (b) Custered user distribution (/) _g. 300 400 500 (a) Uniform user distribution Custer :2 s! : 8MaxUti 0 - T T -$-MaxUtii-R. ;!! HMaxMin e 0.5 ----- --+----------+--------- ~MaxMin-R i 0.4 --- ------- : --------J-------- -;:~~~------- 0 ~ 0.3 ----------- ; - -----,------------ ~! 8?. 0.2 -----------+----------+------- 1 O (\J 200 300 400 600 (b) Custered user distribution Fig. 2. The median of user throughput. Fig. 3. The 25-percentie of user throughput. TABLE FARNESS NDEX (CF. [11]) OF USBR THROUGHPUT ALLOCATON FOR TWO KNDS OF USER DSTRBUTONS (CLUSTER AND UNFORM) N A NETWORK WTH 36 APS AND 400 USERS. (UNT:%) with a other schemes (37% ower than MaxUti-R in custer case, for exampe). n summary, our method, MaxUti-R, outperforms SSF in terms of 25-percentie throughput and fairness index with sma sacr~fice of median throughput. V. CONCLUSONS We find anaytica expressions for the optima channe usage time aocation and present a fast iterative agorithm to achieve the optimum. Simuation resuts show that when users are custered, our utiity maximization formuation yieds substantia throughput gain over both the max-min scheme in [7] and the SSF scheme, which is currenty being used in WLAN products. When users are uniformy distributed in space, our max utiity scheme is simiar as the scheme in [7], and achieves better fairness than SSF. Regardess of the number of APs or users in a network, the convergence of the sum utiity is fast in various network conditions such as users joining, eaving, or moving within the network. Therefore, the iterative agorithm has good scaabiity and can be impemented in rea time. REFERENCES [1] A. Baachandran, P. Bah, and G. M. Voeker, "Hot-spot congestion reief in pubic-area wireess networks," in Proc. Fourth EEE Workshop on Mobie Computing Systems and Appications, June 2002, pp. 70-80. [2] T. S. Rappaport and R. R. Skidmore, "System and method for predicting network performance and position ocation using mutipe tabe ookups,'' U.S. Patent App., no. 20040259555, Dec. 2004. [3] --, "System and method for automated pacement or configuration of equipment for obtaining desired network performance objectives and for security, RF tags, and bandwidth provisioning," U.S. Patent App., no. 20040236547, Nov. 2004. [4] C. Na, J. K. Chen, and T. S. Rappaport, "Measured traffic statistics and throughput of EEE 802.b pubic WLAN hotspots with three different appications," EEE Trans. Wireess Commun., to appear. [5] D. Kotz and K. Essien, "Anaysis of a campus-wide wireess network," in Proc. the Eighth Annua nt. Conf. on Mobie Computing and Networking (MobiCom). ACM Press, September 2002. [6] H. Veayos, V. Aeo, and G. Karsson, ''Load baancing in overapping wireess LAN ces," in Proc. EEE nternationa Conference on Communications, vo. 7, June 2004, pp. 3833-3836. [7] Y. Bejerano, S.-J. Han, and L. E. Li, "Fairness and oad baancing in wireess LANs using association contro," in Proc. ACM MobiCom, Sept. 2004, pp. 315-329. [8] J. Mo and J. Warand, "Fair end-to-end window-based congestion contro," EEEACM Trans. Networking, vo. 8, no. 5, pp. 556-567, Oct. 2000. [9] S. P. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [10] W. Yu, W. Rhee, S. Boyd, and J. M. Cioffi, "terative water-fiing for Gaussian vector mutipe-access channes,'' EEE Trans. nfomj. Theory, vo. 50, no. 1, pp. 145-152, Jan. 2004. [11] R. Jain, D. Chiu, and W. Hawe, "A quantitative measure of fairness and discrimination for resource aocation in shared computer systems," DEC, Research Report TR-301, 1984. 121 Authorized icensed use imited to: University of Texas at Austin. Downoaded on December 4 1 2009 at 13:11 from EEE Xpore. Restrictions appy.