Iterative Water-filling for Load-balancing in

Transcription

1 Iteratve Water-fllng for Load-balancng n Wreless LAN or Mcrocellular Networks Jeremy K. Chen Theodore S. Rappaport Gustavo de Vecana Wreless Networkng and Communcatons Group (WNCG), The Unversty of Texas at Austn Abstract- Ths paper presents an effcent teratve loadbalancng algorthm for tme and bandwdth allocaton among 9 R11 T 20% User 1 T 40% access ponts (APs) and users subject to heterogeneous farness and applcaton requrements. The algorthm can be carred out 12 P 1, p22 T,_ AP 2 ether at a central network swtch wth ste-specfc propagaton predctons, or n a decentralzed manner. The algorthm T1,3*~ T2,3 AP ~~~. User 2-2, Ch22 Ch. 1 converges to maxmum network resource utlzaton from any 1,3 -.User 3 R2,3 startng pont, and usually converges n 3 to 9 teratons n varous network condtons ncludng users jonng, leavng, and movng wthn a network and varous network szes. Such a fast Fg. 1. A smple network wth 2 APs and 3 users. Dfferent thcknesses of convergence allows real-tme mplementatons of our algorthm. dashed lnes sgnfy dfferent avalable lnk capacty Ra,,. APs 1 and 2 use Smulaton results show that our algorthm has merts over other dsjont channels. T,,,S denotes the tme fracton allocated to user s over the schemes especally when users exhbt clustered patterns: Our RF channel of AP a. algorthm, when assumng multple rados at each user, acheves 48% gan of medan throughput as compared wth the max-mn far load-balancng scheme (also wth the mult-rado assumpton) balancng functonaltes perodcally send beacons wth current load, captured by the number of users, bt error rates, whle losng 14% of farness ndex; we also acheve 26% gan of medan throughput and 52% gan of farness ndex over the Strongest-Sgnal-Frst scheme (whch assumes each user has only and sgnal strengths. However, several measurement studes a sngle rado). When only a sngle rado s used, our algorthm have shown that the number of users s not a good metrc s smlar to the max-mn farness scheme, and s stll better than to determne the load [4], [5]. Balachandran et al proposes SSF wth 44% gan of 25-percentle throughput and 37% gan a better load-balancng scheme where each arrvng user of farness ndex. explctly asks for a mnmum and a maxmum bound on I. INTRODUCTION bandwdth/throughput, and a centralzed admsson control s People consder ncreasng the capacty of WLAN or mcrocellular networks by ncreasng AP densty and assgnng performed to assocate the arrvng user to an AP that s wthn the user's rado range and has the most avalable capacty proper non-overlappng frequency channels to APs. As the [1]. The work n [1] mproves the degree of load balancng number of APs to whch a user can connect ncreases, an by over 30% and user bandwdth allocaton up to 52% n algorthm that effcently assocates users to APs becomes crtcal for bandwdth and qualty of servce (QoS) management. comparson wth schemes wth lttle load balancng. The work n [6] presents a decentralzed load balancng algorthm that However, the default Strongest-Sgnal-Frst (SSF) approach can be appled to IEEE a/b/g wthout modfyng the standards whle beng transparent to end users. It was shown used n products, n whch each user chooses an AP wth the strongest sgnal, results n unevenly dstrbuted loads among APs and poor performance [1]. In order to better balance loads, vendors such as Csco, Trapeze, Aruba, Meru, and Symbol have ntroduced central swtches to have networklayer controls (e.g. load balancng and handoffs) over the AP's normal processng n physcal layers today. Ths paper presents a load-balancng algorthm that can be carred out ether n a decentralzed way wth some message exchange between APs and moble users, or at a central swtch wth ste-specfc predctons (such predctons can provde the central swtch wth detaled RF parameters, the receved sgnal-to-nose ratos (SNR), and estmate the achevable capacty for each wreless lnk; see [2]-[4] and references theren). Several heurstc load-balancng schemes have been presented. Balachandran et al [1] observed that APs wth load- 'Ths work s supported by NSF Grant ACI by example that the throughput of a staton ncreases from 1.5 to 2 Mbps, and packet delays can be reduced from 450 to 8 ms. Whle the work n [1], [6] outperforms schemes wth lttle or no load balancng, they are not shown to be optmal. To the best of our knowledge, the only work that acheves some form of optmalty n load balancng s [7], whch acheves max-mn far bandwdth allocaton. Ths paper consders a network wth multple APs and users, as depcted n Fg. 1 and tres to answer a fundamental queston: whch AP(s) should be connected wth a partcular user, and how much tme should the specfc AP(s) allocate to ths user n order to acheve optmal network utlzaton subject to heterogeneous farness and applcaton requrements. Secton II descrbes the system model and notaton n detal. Secton III presents the formulaton and an teratve algorthm for the optmal allocaton of channel usage tme. Smulaton results are presented n Secton IV /06/$20.00 (c) 2006 IEEE 117

2 II. SYSTEM MODEL AND NOTATION We assume a mult-rado capablty that allows multple channels to be receved and decoded n parallel by each user (ths model has been proposed n [7]). It s suggested n ths paper that the multple-rado assumpton smplfes the computaton to be effcent (the problem formulaton s convex). Our approach can also be used for mult-rado APs. Our algorthm allows up to an unlmted number of rados on a user; however, 2 to 4 rados suffce n practce, snce a user n an actual WLAN or mcrocellular network s usually surrounded by at most 4 APs. We assume that users exhbt a quas-statc moblty pattern (a model that has been adopted n [7]) where users can move from place to place, but they tend to stay n the same physcal places for long perods of tme [5]. Ths model allows us to consder long-term averaged lnk capactes over a tme scale of about 2 seconds (denoted as TAVG), whch s adequate for resource re-allocaton and may not be a notceable tme nterval for new users who are watng to be assocated wth APs. The proposed load-balancng algorthm s executed based on the predcted average capactes durng every TAVG nterval. Lnk capactes may change n successve TAVG ntervals due to nterference or changes n user applcatons or transmsson states. The capacty Ra,s (e.g. throughput) between an AP a, and a user, s, s determned by the peak throughput for a sngle (unshared) user, and also determned from predcted, measured, or optmzed throughput estmates based on ste specfc nformaton. For the case where multple users share a sngle AP over an RF channel, the throughput between the AP, a, and a user, s, s a fracton (the tme fracton of channel usage) of the lnk capacty, that s, Throughputa,s = Ta,sRa,s, where Ta,s s the channel usage tme between AP a and user s. Durng a TAVG nterval, even though users may jon/leave the network, or RF nose sources may emt nterferng sgnals, the effects of these transent events on lnk throughputs are quantzed and sampled every TAVG (e.g. block processng s used). In the begnnng of every TAVG nterval, our teratve load balancng algorthm re-adjusts the tme/bandwdth resource allocaton over all users and APs. The algorthm converges to optmum n merely 3 to 9 teratons regardless of network szes, although the computaton tme of each teraton grows lnearly wth the number of users multpled by the number of APs controlled by the swtch. On a 2GHz Intel Pentum computer wth Wndows XP, each teraton n Matlab takes 30 mllseconds for a network wth 36 APs and 300 users. Code mplemented n assembly or C language would be much faster and s very sutable for realtme mplementatons of our algorthms on hardware/frmware, as contemplated n [2], [3]. Wth the above mentoned assumptons, the real throughput that a user experences manly depends on the channel usage tme allocated from the APs to ths user. For nstance, n Fg. 1, suppose AP 1 and AP 2 allocate T1,1 = 20% and T2,1 = 40% of ther tme (over dsjont channels 1 and 2, respectvely) to user 1, respectvely. The total bandwdth that user 1 obtans s b1 = 20 RI, + l40r2,1; the bandwdths of users 2 and 3 can be computed n a smlar way. we consder an nfnte backlog of packets (full and ready queues on every channel) for every user. Hence a user's throughput s the same as the bandwdth allocated to her. We maxmze the sum utlty of throughput, whch means maxmzng E3 I U (b) over the channel usage tme n ths example. If utlty functons are properly chosen, users wll be allocated dfferent notons of far allocaton when the network reaches maxmum sum utlty [8]. We made the assumpton that all APs are under the control of a network swtch. However, some rogue APs or RF nose sources may emt nterferng sgnals n the coverage area of the controlled APs. In ths case, some controlled APs or overlay sensors can detect sgnals from rogue APs. Wth detected sgnal parameters and ste specfc knowledge, poston locaton technques can locate the rogue APs [2], [3]. Then, AP channel assgnments are changed so that the APs near the rogue APs operate at orthogonal RF channels n order to elmnate most nterference from rogue APs. Then, the swtch wll predct SNR and lnk capactes between users and controlled APs usng ste specfc models for the rogue locatons and transmt propertes, and apply our algorthm to fnd the optmal resource allocaton accordngly. Ths paper assumes the frequency band of each AP has been properly assgned [2], [3], and focuses on fndng the optmal bandwdth/tme allocaton n a fully-controlled network. Wth an assgned allocated frequency channel, each AP serves ts user by tme sharng. The fracton of tme resource dedcated for payload transmssons between users and an AP, a, over an RF channel s denoted as Tfrac (0 < T frac < 1) (e.g., t ranges from 59% to 88% n a). The subscrpt a n Tfrac s used, snce the payload tme fractons may dffer from AP to AP. We suppose that each user shares her utlty functon to all the APs that transmt sgnals strong enough to reach her. Then, each AP allocates ts tme resource (over ts assgned RF channel) to users based on the nformaton of the utlty functons of all the users wthn ts coverage area, based on ste specfc knowledge [2]-[4]. In ths paper, utlty functons are assumed to be concave, contnuously dfferentable, and strctly ncreasng [9] for smplcty of analyss. Let n and m denote the numbers of APs and of users, respectvely. We use a or s as ndex when referrng to a specfc AP or user, and use j or as dummy ndces of APs or users when performng a summaton. User s s sad to be wthn the coverage of AP a f Ra,s > 0; otherwse, Ra,s = 0. Each entry n the rate matrx can be predcted from a ste-specfc predcton engne [2]-[4]. Wthn a unt tme perod, suppose AP a allocates a tme fracton Ta,s (over the assgned RF channel of AP a) to user s (0 < Ta,s < 1). The actual bandwdth that user s gets from AP a s Ta,sRa,s. III. MAXIMUM SUM UTILITY WITH TIME ALLOCATION The optmal AP-user assocaton can be formulated as the sum utlty maxmzaton problem n (1) over tme resources 118

3 from APs on dfferent RF channels to users. max E U( Tj,Rj,) j subject to Ta, < Tarac,Va, over Ta,s > 0,Va, s. (1) It s hard to fnd a closed-form expresson of the optmal channel usage tme allocaton for (1). Nevertheless, f the optmzaton s over the tme resources of only a sngle AP (over one channel), assumng the other APs' tme allocatons are fxed, closed-form expressons for each AP's optmal tme allocaton have closed-form expressons, shown n (11) whch are solutons to formulaton (3). Theorem 3.1 dscussed below shows that the orgnal multple AP problem n (1) reaches the optmum f and only f the tme allocaton from every AP smultaneously has the closed-form expressons as n (11). Hence, the optmzaton of the multple-ap problem can be done by successvely optmzng each AP's tme resources, as presented n Algorthm 1 as an effcent teratve algorthm. Our dervaton and proofs extend [10] to a wde class of utlty functons (beyond logarthmc) for dfferent degrees of farness and applcaton needs. The sole constrant n (1) means that the total channel usage tme used at each AP s upper bounded. The objectve s to maxmze the network utlty E U(j Tj,Rj,). Mo and Walrand have proposed a class of utlty functons that capture dfferent degrees of farness and model applcatons wth heterogeneous needs parameterzed by q [8]: U(b) { (1 gb, 1 b C (0,oo). (2) log b, f q= I The parameter q has an ndex because each user may have a dfferent applcaton/farness requrement. Ths famly of utlty functons s concave, contnuously dfferentable, and strctly ncreasng [8]. The sum of concave functons s stll a concave functon; hence, problem (1) s convex snce a concave functon s to be maxmzed over a convex constrant set [9]. The work n [8] shows that f q -) oo, the formulaton n (1) becomes a specal case that acheves max-mn farness, as studed n [7]. Wthn every TAVG, R remans constant after block processng, and the optmal sum utlty and T wll be determned accordngly. Suppose the sum utlty optmzaton s performed over the channel usage tme resources of only AP a, Ta, = [Ta,I, Ta,2,..., Ta,m], assumng that the tme allocatons from the other APs to users are fxed. Then the formulaton n (1) s reduced to max E U (Ta,Ra, + Ca,) subject to ZTa, < T frac, over Tars > 0 Vs, (3) where Ca, Tj,Rj, are fxed. j:j7a Denote by A,a the Lagrange multpler for the constrant n (3). Then, the Lagrangan [9] s gven by L(Ta,, NAa) = U(Ta,Ra,+Ca,)- a (E Ta,-Taac). (4) Snce utlty functons U5(.) are ncreasng, t s natural to exhaust the tme resource for maxmzng sum utlty [9]; therefore, at the maxmum of (3), we have Y Ta, Tarac Then, the suffcent and necessary optmalty condtons (KKT condtons) [9] for (3) can be wrtten as: Ra,sUs(Ta,sRa,s + Ca,s) = Aa f Ta,s > 0, Vs (5) < Aa f Ta,s = 0, Vs (6) E Ta, = Tfrac (7) Tars > O: Vs; Aa > (8) It s obvous that no tme s allocated to lnks wth zero capacty (.e. Ta,s = 0 f Ra,s = 0). Therefore, we focus on dervng the optmal Ta,s for Ra,s > 0. For general utlty functons, the optmal tme fracton can be derved from (5): Ta,s {R us ( Aa) Ca,s a,s Ra,s Ra,sI Whle closed-form solutons of Tars do not exst for general utlty functons, they can be obtaned for the famly of utlty functons n (2), for whch (5) becomes AL _ Ra,s Aa (10) &Ta,s (Ta,sRa,s + Cars )qs Equatng (10) wth zero gves the optmal tme allocaton: I -1) T A (- q's )R( qs a,s = a a, s Ca,s Ra,s l l (9) (11) In (9) and (11), the notaton {x}+ s needed because Ta,s s nonnegatve: {x}+ = x f x > 0 and {x}+ = 0 otherwse. By substtutng (11) or (9) nto ZTa, = Tlrac n (7), Na for each AP a can be numercally solved [9], [10]. In each teraton of our algorthm, fndng the tme resources of each AP requres solvng a sngle-varable (Aa) polynomal equaton wth m terms; hence, the tme complexty of each teraton s O(nm). If the parameter q, = 1, the expresson of Ta,s n (11) s the water-fllng expresson, where the constant a` s known as the water-fllng level [10]. Theorem 3.1: {Ta,s, a, s} s an optmal soluton to (1) f and only f {Ta,, Ta,2,..., Ta,m} s the soluton n (11) for AP a wth the tme allocaton from the other APs {Tjj, Vj 74 a, Vs} fxed, for all a = 1, 2,..., n. (The proof s omtted as t s a natural extenson of Theorem 1 n [10].) As descrbed n Theorem 3.1, the tme allocatons from each AP to users can be solved by (11), assumng tme allocatons from the other APs are fxed. Hence, the optmal tme allocaton for the multple-ap optmzaton problem (1) can be found by an teratve algorthm (see Algorthm 1). Theorem 3.2: Algorthm 1 results n an optmal sum utlty and causes {Ta,s, Va, s} to converge to an optmal tme 119

4 Algorthm 1 An teratve algorthm to solve (1) Gven a rate matrx {Ra,s,,Va, s}. Start wth a vald tme allocaton {Ta,s: Va, s}. repeat for each AP a = 1,2,...,n do Compute {Ca,s Vs} by (3). Compute {Ta,s,Vs} by (11) or (9). end for untl the sum utlty converges Output {Ta,s: Va, s}. allocaton for Formulaton (1). (The proof can be extended from the proof of Theorem 2 n [10]. Note that the optmum tme allocaton {Ta,s, Va, s} may not be unque.) Algorthm 1 can be carred out n a decentralzed manner: each AP a computes the optmal tme allocaton {Ta,s,Vs} only for those users who are n the coverage of ths AP. For the computaton of each user's Ta,s, a constant Ca,s needs to be known, whch n turn requres the knowledge of the bandwdth that ths user s receves from APs other than AP a. In a realstc WLAN setup, a user s under the coverage of no more than 4 APs; hence, the computaton of Ca,s at each user s effcent. APs sequentally perform such decentralzed computng. When the sum utlty converges, a control message may be sent to APs to stop the decentralzed computng. IV. SIMULATION RESULTS In ths secton, we compare the throughput and farness performance of our maxmum utlty (denoted as MaxUtl) scheme wth the max-mn farness scheme n [7], denoted as MaxMn, and the Strongest-Sgnal-Frst scheme n current mplementatons. We consder a smplfed scenaro of free-space propagaton model where no obstacles exst n the vcnty of APs. It s clear that our algorthm can utlze ste specfc nformaton, whch wll be consdered n future work. We consder dfferent percentages (between 1% and 5%) of users jonng, leavng, or movng wthn the network; hence, the lnk capactes change over tme. We sample R for every TAVG, and wthn ths tme nterval, R s fxed. Two knds of user dstrbutons, namely unform and cluster (or hotspot), are consdered. Frst, users are unformly dstrbuted n a meters by meters square that encompasses the 36 APs. Second, we consder that a hotspot at the center attracts more people: users are dstrbuted n a crcle-shaped area centered at the mddle of the APs wth a radus of 250 meters. Users are randomly located on ths crcle based on ther unformly generated polar coordnates (the dstance from the center and the polar angle are unformly dstrbuted between (0, 250) and (0, 2w), respectvely). From the vewpont of the Cartesan coordnate, the user densty s hgher near the center than near the crcumference of the crcle. Each pont on the fgures s an average over 100 ndependent runs. In the SSF case, each user (whose transcever can handle only a sngle channel) assocates wth the strongest AP, and then each AP evenly dstrbutes ts tme resources to the assocated users. Smulatons show that the number of teratons (mostly between 3 and 9) does not grow wth the number of users. Our algorthm converges quckly even for large networks. Fgs. 2 and 3 show the medans and the 25-percentles of user throughputs, respectvely. Table I presents farness ndces (see [11] for ths metrc) for cases wth 400 users; scenaros wth dfferent number of APs and users are omtted, snce ther farness ndex values are smlar to those n Table I. Both MaxUtl and MaxMn assume that each user has multple rados. For far comparsons wth SSF, we also compute snglerado results by properly roundng mult-ap tme allocaton; MaxMn-R denotes the results produced by the roundng method n [7]. The MaxUtl-R results were obtaned by a dfferent roundng method: we frst compute normal multplerado tme allocaton; then, f any user ndeed uses multple APs, ths user smply chooses the AP that supples her wth the most bandwdth. Fnally, f any AP has any tme resource remaned not allocated, ths AP allocates the remanng tme proportonally to ts assocated users. For example, f the 5 3 rate matrx R [ ] and all users' utlty parameters, q, are 1, then the optmal tme fracton (allow- ~0.17 ng mult rados) s T Each user chooses only one sngle AP; then the tme matrx becomes T [ Then, [ j snce the frst AP has tme fracton (16.6%) remaned, the remanng tme s proportonally dstrbuted to users 1 and 2; fnally the tme matrx for the sngle-rado case s T A trade-off between throughput and farness can be seen n mult-rado cases MaxUtl and MaxMn. Our MaxUtl has very good performance n cluster case: n Fg. 2(b), MaxUtl exhbts about 48% hgher medan throughput over MaxMn whle sacrfcng only 14% of farness as n Table I. It s because MaxMn tends to acheve absolute farness (ts farness ndex s almost 100% as n Table 1) by sacrfcng throughput (gvng more tme resource to users wth poor lnk capactes). Our MaxUtl trades throughput wth farness; even n unform case n Fg. 2(a), MaxUtl yelds 9% hgher medan throughput than MaxMn whle losng 2% of farness as n Table I. Our algorthm, wth multple rados at each user, outperforms SSF by 26% and 52% n terms of medan throughput and farness ndex, respectvely, as n Fg. 2(b) and Table I. Surprsngly, the sngle-rado scheme MaxUtl-R yelds worse medan throughput than SSF, manly because our roundng method (as presented n the numercal example above) makes users choose stronger APs, thereby causng unbalanced loads on APs. The roundng method n [7] may be modfed to be mposed upon MaxUtl for better roundng performance; ths s a subject for future research. Nevertheless, MaxUtl- R yelds smlar 25-percentle user throughputs as MaxMn-R, and s 44% and 17% hgher than SSF n cluster and unform cases, respectvely (as seen n Fg. 3). Moreover, Table I ndcates that SSF has poor farness ndces as compared 120

5 ,,,1.%Xt 0) tm axutl axutl-r axmn axmn-r SF 60 3MaxUtl wmaxutl-r 3MaxMn FMaxMn-R SSF (a) user dstrbuton U) 0.9; ) 0.6 X ) 0.3- r n 3MaxUtl MaxUtl-R 3MaxMn >MaxMn-R 8SSF U) Q O0.5 c 0 a) ' 0.4 (an (a) user dstrbuton emaxutl \emaxutl-r emaxmn axm n-r (b) ed user dstrbuton LC C\ 20 (b) ed user dstrbuton Fg. 2. The medan of user throughput. Fg. 3. The 25-percentle of user throughput. TABLE I FAIRNESS INDEX (CF. [11]) OF USER THROUGHPUT ALLOCATION FOR TWO KINDS OF USER DISTRIBUTIONS (CLUSTER AND UNIFORM) IN A NETWORK WITH 36 APS AND 400 USERS. (UNIT: %) MaxMn MaxMn-R MaxUtl MaxUtl-R SSF wth all other schemes (37% lower than MaxUtl-R n cluster case, for example). In summary, our method, MaxUtl-R, outperforms SSF n terms of 25-percentle throughput and farness ndex wth small sacrfce of medan throughput. V. CONCLUSIONS We fnd analytcal expressons for the optmal channel usage tme allocaton and present a fast teratve algorthm to acheve the optmum. Smulaton results show that when users are clustered, our utlty maxmzaton formulaton yelds substantal throughput gan over both the max-mn scheme n [7] and the SSF scheme, whch s currently beng used n WLAN products. When users are unformly dstrbuted n space, our max utlty scheme s smlar as the scheme n [7], and acheves better farness than SSF. Regardless of the number of APs or users n a network, the convergence of the sum utlty s fast n varous network condtons such as users jonng, leavng, or movng wthn the network. Therefore, the teratve algorthm has good scalablty and can be mplemented n real tme. REFERENCES [1] A. Balachandran, P. Bahl, and G. M. Voelker, "Hot-spot congeston relef n publc-area wreless networks," n Proc. Fourth IEEE Workshop on Moble Computng Systems and Applcatons, June 2002, pp [2] T. S. Rappaport and R. R. Skdmore, "System and method for predctng network performance and poston locaton usng multple table lookups," U.S. Patent Appl., no , Dec [3] "System and method for automated placement or confguraton of equpment for obtanng desred network performance objectves and for securty, RF tags, and bandwdth provsonng," U.S. Patent Appl., no , Nov [4] C. Na, J. K. Chen, and T. S. Rappaport, "Measured traffc statstcs and throughput of IEEE lb publc WLAN hotspots wth three dfferent applcatons," IEEE Trans. Wreless Commun., to appear. [5] D. Kotz and K. Essen, "Analyss of a campus-wde wreless network," n Proc. the Eghth Annual Int. Conf: on Moble Computng and Networkng (MobCom). ACM Press, September [6] H. Velayos, V. Aleo, and G. Karlsson, "Load balancng n overlappng wreless LAN cells," n Proc. IEEE Internatonal Conference on Communcatons, vol. 7, June 2004, pp [7] Y. Bejerano, S.-J. Han, and L. E. L, "Farness and load balancng n wreless LANs usng assocaton control," n Proc. ACM MobCom, Sept. 2004, pp [8] J. Mo and J. Walrand, "Far end-to-end wndow-based congeston control," IEEE/ACM Trans. Networkng, vol. 8, no. 5, pp , Oct [9] S. P. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge Unversty Press, [10] W. Yu, W. Rhee, S. Boyd, and J. M. Coff, "Iteratve water-fllng for Gaussan vector multple-access channels," IEEE Trans. Inform. Theory, vol. 50, no. 1, pp , Jan [11] R. Jan, D. Chu, and W. Hawe, "A quanttatve measure of farness and dscrmnaton for resource allocaton n shared computer systems," DEC, Research Report TR-301,