Clique-Based Utility Maximization in Wireless Mesh Networks

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1 Clque-Based Utlty Maxmzaton n Wreless Mesh Networks Algorthm, Smulaton, and Mathematcal Analyss Erwu Lu Dept. o Electrcal and Electronc Engneerng Imperal College London, Unted Kngdom Emal: erwu.lu@mperal.ac.uk Qnqng Zhang Appled Physcs Laboratory Johns Hopkns Unversty Laurel, Maryland, USA Emal: qnqng.zhang@huapl.edu Kn K. Leung Dept. o Electrcal and Electronc Engneerng Imperal College London, Unted Kngdom Emal: kn.leung@mperal.ac.uk Abstract Compared wth cellular networks, wreless mesh networks (WMNs) need more careul desgn or resource allocaton. To ths end, we develop a clque-based proportonal ar schedulng (CBPFS) algorthm or WMNs. Furthermore, we obtan a closed-orm model to quanty the throughput o connecton lnks and trac lows n mult-hop transmssons. We use the derved analytcal ramework to estmate the throughput o CBPFS and compared t wth smulatons. It s the rst tme a mathematcal model s developed to quanty the throughput o lnks and end-to-end lows n a mult-hop network where lnks are proportonally ar scheduled. Keywords- wreless mesh network; mult-hop transmsson; clque-based proportonal ar schedulng I. INTRODUCTION We consder network resource optmzaton n a wreless mesh network (WMN). A WMN conssts o rado nodes organzed n a mesh topology. Derent rom the cellular network, nodes n a WMN can communcate wth each other drectly or through ntermedate nodes. A WMN s a specal type o wreless ad-hoc network where the topology o the network s relatvely statc and nodes are wth lmted moblty [1]. We address the throughput and arness ssues n such wreless networks []. Resource allocaton mechansm reles on a sutable perormance metrc. There are two types o perormance metrcs used n resource allocaton: throughput-based perormance metrc and utlty-based perormance metrc. The ormer s oten seen n cellular networks whose obectve s to maxmze the sum (or weghted sum) o the throughput o all lnks, whle the latter s typcally used when the obectve s to exhbt some sense o arness crtera. Throughput-based perormance metrc s known to cause severe unarness among the nodes n wreless networks, and most recent work on resource allocaton has shted to a utlty-based ramework. In our context, utlty-based approach s to maxmze the aggregate utlty n the network, and we propose to use utltybased perormance metrc or resource allocaton n ths paper. In an utlty-based ramework, each lnk l s assocated a utlty uncton U l (R l ), over the lnk throughput R l. It s known Ths work was supported, n part, by Johns Hopkns Unversty, Appled Physcs Laboratory's nternal research and development unds. that by denng U(.) approprately, derent arness crtera o nterest, such as proportonal or max-mn arness, can be acheved [3][4]. Radunovc and Boudec [5] prove that the maxmn allocaton has undamental ecency problem n the lmt o long battery letme, regardless o the MAC layer, network topology, choce o routes and power constrants. Ths results n all lnks recevng the rate o the worst lnk, and leads to severe necency. For ecency, we consder proportonal arness n ths paper. We then extend the proportonal ar schedulng (PFS) algorthm so that the proposed ramework supports hgher throughput. To be specc, non-contendng lnks are grouped nto lnk clques whch are to be scheduled n a proportonally ar manner (reer to Secton II or detals). Even n cellular networks, there are ew analytcal results related to the PFS algorthm. To smply the problem, most analyses assume some knd o..d relatonshp among the lnks [6-9]. Moreover, lnear rate model and logarthm rate model are the two rate models commonly used or analyzng the perormance o PFS [6-9]. The assumpton o lnear or logarthm rate model s a reasonable modelng conventon. However, when examnng throughput perormance, t does not seem entrely satsactory to assume such smpled models [10]. Work by Telatar [11] and Smth et al. [1][13] suggest that the lnk capacty n Raylegh adng networks be better modeled by a Gaussan dstrbuton. Wth ths Gaussan rate model, Lu [10][14] conducts mathematcal analyss and provdes closed-orm expressons or the PFS throughput wthout the lmtatons mentoned above. The theoretcal results n [10][14] t wth the smulaton ones wth surprsngly hgh accuracy. Our ultmate obectve s to develop a theoretcal ramework to acltate the research on ar resource allocaton or WMNs wth multple contendng lnks and mult-hop transmssons. Towards ths end, we propose a systematc method and then derve a mathematcal model to estmate the throughput acheved or a mult-hop path n WMNs where lnks are scheduled under the proportonal ar crtera. In partcular, our contrbutons are summarzed as ollows. 1) In ths paper, we ntroduce clque-based proportonal ar schedulng (CBPFS), whch s an extenson o PFS. Snce

2 ts rst presence [3], the PFS algorthm has aroused consderable nterest (see [14] and the reerences theren). Untl now, vrtually all analytcal results on PFS are or cellular networks where lnks are non-contendng by Tme-Dvson- Multple-Access (TDMA) technque. Ths paper provdes a theoretcal ramework or the research o PFS n mult-hop or mesh networks where there are multple contendng lnks. ) Our analyss s general and covers a broad set o scenaros. CBPFS becomes PFS when used n cellular networks. The most valuable contrbuton o ths paper s a mathematcal ramework that provded quck estmates o the throughput or both lnks and mult-hop paths under the proportonal arness crtera. The rest o the paper s structured as ollows. In Secton II, we rst ntroduce concepts o clque-based schedulng n WMNs; we then present CBPFS algorthm and derve the mathematcal ramework or evaluatng the throughput o a gven lnk or low n a WMN. Closed-orm models o the throughput or both lnks and lows are also developed to acltate the mathematcal analyss. Due to space lmtaton, we do not provde detaled proos n ths conerence verson. In Secton III, we compare analytcal results wth smulaton experments to valdate the theoretcal ndngs n Raylegh adng scenaros. We conclude the paper n Secton IV. II. NETWORK MODEL A. System Model and Notatons Consder a WMN that s represented by a connected graph, G=(N,L), whch has a node set N (wth cardnalty N ), a lnk set L (wth cardnalty L ), and F source-destnaton node pars {s 1, d 1 },, {s F, d F }. We reer to a source-destnaton par {s, d } as a low, denoted by, and denote the set o lows by F. Let ζ be the end-to-end path whch s the set o wreless lnks crossed by low F and {x l [t]} the arrval process or lnk l L,.e., x l [t] s the number o packets that transmtted over lnk l n slot t. Let λ denote the average throughput o low. For any lnk l L, let C l and E[C l ] denote the capacty and average capacty, R l and E[R l ] denote the throughput and average throughput o lnk l, respectvely. At each node, a packet queue s mantaned or each destnaton. We call a queue stable ts length does not ncrease nntely. Obvously, the network s stable all queues are stable; and unstable otherwse. For the queues to be stable, t requres E[R l ] λ or any lnk l ζ. Let Λ denote the stablty regon o the network. To nd Λ, subect to resource constrants such as capacty constrants or power constrants, one needs to choose a sutable perormance metrc or resource allocaton. We assume that each lnk experences ndependent (but not necessarly dentcal) Raylegh adng n WMNs. Smlar to our prevous research on PFS, or the analyss on CBPFS, we use Gaussan rate model or lnk capacty. B. Lnk Contenton In WMNs, due to undesred ntererence and resource contenton, not all lnks n a wreless network can transmt smultaneously. For our dscusson, we assume that each node n WMNs s equpped wth one sngle antenna and operates n hal-duplex transmt/receve mode. We consder a general lnk contenton model ormulaton speced by a set o pars o lnks that contend wth each other: we say that two lnks contend ther concurrent transmssons need to access the same rado resources. There are ve knds o lnk contenton: multple lnks transmttng to the same node, multple lnks recevng rom the same node, the transmttng lnk and recevng lnk o the same node, ntra-low lnk contenton where derent lnks o the same low heavly ntererence wth each other, nterlow lnk contenton where derent lnks o derent lows heavly ntererence wth each other. The level and sze o the contenton n a WMN s determned by the node poston, and each node s communcaton, ntererence and sensng range. In our analyss, TDMA-based schedulng s used: Tme s dvded nto small schedulng ntervals called slots. In each tme slot, the system schedules a number o non-contendng lnks to transmt smultaneously. The scheduled lnks could be derent rom slot to slot. The lnks that are to be scheduled are chosen n the current slot and the chosen lnks transmt ther packets n the next slot. C. Clque-Based Proportonal Far Schedulng (CBPFS) Reer to Fgure 1, rom the network topology represented by a graph G=(N,L) together wth the set o lows F, we generate the 1 st lnk contenton graph G LC-1 that captures the contenton among lnks n such a way that each lnk s a vertex n ths graph and two lnks that contend are adacent. Reer to the let-sde plot n Fgure. 1, there are 11 nodes, 6 actve lows 1-6, and 13 lnks -3 n the network. The rght-sde plot n Fgure 1 s the correspondng lnk contenton graph G LC-1. Accordng to the denton o lnk contenton, lnk contends wth lnks, L 6 ; lnk contends wth lnks, 1,, 3. The edge between and L 6 ndcates an ntra-low lnk contenton over low ; The edge between and ndcates an nter-low lnk contenton. G 3 8 A 1 B s 1 1 L 6 5 d 3 C 0 s 6 d d 4 3 D d 1 Fgure 1. Generaton o low contenton graph G LC From the 1 st lnk contenton graph G LC-1, we generate the 1 st clque allocaton graph G CA-1 whch s a maxmal complete sub-graph o the complement or nverse graph o the 1 st lnk contenton graph G LC-1. A clque V represents a maxmum number o concurrent lnks n the lnk contenton graph, and s smply the set o vertces n the clque allocaton graph. Reer to Fgure, the 1 st clque V 1 ={,,, 3 }. Here, the complement or nverse graph G I o a graph G s constructed n d 5 G LC-1 L

3 such a way that two vertces n G I are adacent and only they are not adacent n G. G LC-1 Fgure. Generaton o clque allocaton graph G CA-1 Reer to Fgure 3, we then remove rom G LC-1 the vertces n G CA-1 to produce the nd lnk contenton graph G LC-. Smlarly, we generate the nd clque allocaton graph G CA- and have the nd clque V ={,, }. L 6 L 6 G LC Fgure 3. Generaton o clque allocaton graph G CA-1 We repeat the above procedure to produce the K th lnk contenton graph G LC-K, the K th clque allocaton graph G CA-K and the K th clque V K, untl all lnks are n clques. Once we have allocated all lnks n clques {V, =1,,, K}, we then generate a clque schedulng graph G CS that represents each lnk clque as an entty to be scheduled. In each slot, one and only one clque V (=1,,, K) s scheduled. Once a lnk clque V s chosen to transmt accordng to the schedulng crteron, all lnks n lnk clque V wll be scheduled to transmt smultaneously. Reer to Fgure 4, there are 5 clques. When V 1 s chosen to transmt, lnks,, and 3 can transmt at the same tme. Lnk Clques: V 1 : {,,, 3 } V : {,, } V 3 : {L 6,, } V 4 : {0, 1 } V 5 : { } 3 G CS V 3 V 1 V V 4 V 5 0 L Clque-Based Schedulng Fgure 4. Generaton o clque schedulng graph G CS In lght o all the superor eatures o proportonal arness over other arness crtera, we use the concept o proportonal arness together wth the above clque-based schedulng method as our clque-based proportonal ar schedulng (CBPFS) polcy. Just as PFS n cellular networks schedules a G CA-1 G CA- 3 lnk/node at each slot, CBPFS n WMNs schedules a lnk clque at each slot. It s known that the PFS algorthm maxmzes the aggregate utlty o all lnks n a TDMA cellular network [3]. Smlarly, CBPFS maxmzes the aggregate utlty o all lnk clques n a TDMA wreless mesh network. In other words, the CBPFS algorthm s the soluton to the ollowng optmzaton problem: K = 1 ( [ ]) max ln γ. (1) t s.t. γ t] = R [ t]. () [, l, V χ [ t + 1]. (3) [ t + 1] = C, l, V 1, clque V s scheduled at next slot I[ t + 1] =. (4) 0, else R, [ t 1 1] = 1 R k C [ t] + I [ t + 1] [ t + 1] +,,. (5) where K s the total number o clques, t s the current slot, I [t+1] s the ndcator uncton o the event that lnk clque V s scheduled to transmt n tme slot t+1, R, and C, are the throughput and capacty o the th lnk l, n the th clque. Smulatons [6][8] ndcates that each lnk l, n a PFSenabled cellular network has the same average probablty o beng scheduled (In act, ths property s also proven n our research on PFS and the detal analyses wll be n the uture papers). From the generaton o lnk clques, we know that ths nterestng property also exsts or any lnk n a CBPFSenabled WMN. Snce the same average schedulng probablty means the same number o slots allocated to any lnk, ths property (.e., equal-tme-share) makes CBPFS an ecent yet ar schedulng algorthm or WMNs. Reer to Fgure 1~3, we would lke to pont out that the network throughput could ncrease : we do not remove the vertces n V 1 to produce G LC-. (and G LC-3,, etc.,) when we have the 1 st clque V 1 ={,,, 3 } rom G LC-1. By ths, the resultng clques may overlap,.e., a lnk may belong to multple clques. Though n ths way, the network throughput could be hgher than that n CBPFS due to hgher space re-use, ths scheme rases arness ssue and s thus not desrable as a lnk belongng to multple clques wll be allocated more tme slots than other non-overlappng lnks. D. Mathematcal Analyss o CBPFS For a network topology and the set o lows, the clque schedulng graph G CS s nally generated wth the scheme descrbed n sub-secton C. We assume that there are K vertces n such graph denoted as lnk clques V 1, V, V K. Lnk clque V = {l,1, l,, l, V } (l, V L, =1,, K; =1,, V ) contans V smultaneous lnks. From sub-secton C, a lnk l k

4 belongs to one and only one lnk clques,.e., V V =,. We also have V 1 V V K = L. Let s consder the problem where these L lnks wshng to transmt data, and the rates o transmsson are randomly varyng due to channel luctuatons. We assume that channel adng keeps constant over each slot, and vares ndependently rom slot to slot. The selecton o the lnks to schedule s based on a balance between the current possble rates and arness. As descrbed n sub-secton C, the CBPFS algorthm perorms ths by comparng the rato o the capacty or each lnk clque to ts throughput tracked by an exponental movng average, whch s dened as the preerence metrc. The lnk clque wth the maxmum preerence metrc wll be selected or transmsson n the next schedulng slot. Ths s descrbed mathematcally as ollows. The end o slot t s called tme t. In next tme slot t+1, the capacty o lnk l, V s C, [t+1]. C, s a random varable whose dstrbuton depends on the characterstcs o the underlyng Raylegh adng process. For a lnk l,, ts k-pont movng average rate (.e., throughput) up to tme t s denoted by R,;k [t]. The preerence metrc o lnk clque V s denoted by Q ;k [t+1] = Σ l, V (C, [t+1])/σ l, V (R,;k [t]). For ease o exposton, n what ollows, we wll drop the subscrpt k rom the notatons R,;k [.] and Q ;k [.] where there s no conuson. Lnk clque V = arg V max Q [t+1] = arg V max Σ l, V (C, [t+1])/σ l, V (R, [t]) s scheduled to transmt, whch means all lnks l, V are scheduled to smultaneously transmt n next tme slot t+1. The movng average rate o lnk l, up to tme t+1 s updated by (5). Smlar to [3], one can prove that the CBPFS algorthm n sub-secton C s the soluton to the optmzaton problem ormulated by (1). For the analyss, our obectve s to derve a closed-orm expresson or the average throughput o lnks (and also end-to-end lows) that are scheduled under the CBPFS polcy. As we are dealng wth per-lnk schedulng and the schedulng metrc s drectly related to lnk capacty C, t s desrable to ncorporate n the analyss a stochastc model o lnk capacty n WMNs. In the ollowng, we provde accurate estmate o throughput or WMNs. As suggested n Secton I, Gaussan rate model s used or lnk capacty. We have the ollowng theorems: Theorem.1: Under CBPFS, when each lnk clque n G CS contans one and only one lnk l L (=1,, L ), assumng the capacty C o lnk l s statstcally ndependent Gaussan, the standard devaton σ o C s a monotoncally ncreasng, concave uncton o ts mean value E[C ], the average throughput E[R ] o lnk l can be approxmated by E [ R ] [ ] E C L + M yσ ρ L ( 1 [ φ( M )] ) L 1 ( y) [ φ( y) ] dy. (6) where ρ(.), φ(.) are zero mean, unt varance standard normal probablty densty uncton and dstrbuton uncton (.e., pd and cd), respectvely. Proo: Due to space lmtatons, we omt here the detals o ths proo. In act, CBPFS s PFS n ths case and one can reer to [14] or the detaled proo. Theorem.: Under CBPFS, assumng the capacty C, o each lnk l, s statstcally ndependent Gaussan, let χ =Σ l, V (C, ) denote the capacty o lnk clque V, the standard devaton σ C, o C, s a monotoncally ncreasng, concave uncton o ts mean value E[C, ], the average throughput E[R, ] o lnk l, can be approxmated by E [ R ], E K [ ] E C, E [ χ ] [ χ ] K ( 1 [ φ( M χ )] ) + yσ ρ( y) φ( y) χ where E[ ] = E[ C, ] l, V M χ K 1 [ ] dy. (7) χ s the average throughput o lnk clque V, σ χ = l σ V,, s the standard devaton o the = E χ / σ. throughput o lnk clque V, χ [ ] χ M Proo: The capacty χ =Σ l, V (C, ) o each lnk clque s statstcally ndependent Gaussan. By consderng all lnks n lnk clque V as a vrtual lnk, wth Theorem.1, we obtan the average throughput o the th vrtual lnk. Insde a vrtual lnk, all lnks transmt at the same tme slot, the average throughput o lnk l, s then proportonal to ts average capacty E[C, ]. We then have (7). One can very that Theorem.1 s a specal case o Theorem.. E. Flow Throughput n CBPFS-enabled WMNs Flows between sources and destnatons are mapped to paths, accordng to the routng algorthm (or example, shortest path routng) usng some routng metrc. In WMNs, multple lows may share a sngle lnk l. Theorem. provdes a closedorm expresson to quck evaluate the average throughput E[R l ] o any lnk l n CBPFS-enabled WMNs. Once E[R l ] s determned, one can use some algorthm to allocate E[R l ] among all lows sharng lnk l. Let τ l, denote the average data rate allocated by lnk l to low. Let N l denote the total number o lows sharng lnk l. Gven an end-to-end path ζ o low n a mult-hop network, t s known that the average end-to-end throughput λ s [15], l ζ [ τ ] λ = Mn l,. (8) Here we use a smple algorthm to share throughput among all lows on lnk l: τ l, = E[R l ]/N l,.e., all lows on lnk l have the same share o throughput. In practce, ths can be mplemented by usng lnk l to transmt data or one o the N l lows n a round-robn manner, when lnk l s scheduled.

5 Wth ths average allocaton method, we have the average end-to-end throughput o low, l ζ [ E[ R ]/ N ] λ = Mn. (9) where E[R l ] s determned by (7). l l III. SIMULATION STUDIES In ths secton, we present smulaton results that valdate our analyses. Due to space lmtatons, we would not go nto detals about the smulaton parameters and setup. For all smulatons, the CBPFS schedulng algorthm together wth the throughput sharng algorthm shown by (9) are used. To evaluate the theoretcal results presented n Secton II, E[C, ] and σ, o lnk capacty should be avalable beorehand. For example, we can estmate E[C, ] and σ, by measurng lnk l, over a perod o tme. Speccally, accordng to [1], or a sngle-nput-sngle-output (SISO), Raylegh adng lnk, we have to the ollowng orm, ( + SINR λ) e λ dλ E C W [ ] = log 1 (10) σ = W C W ( log( 1+ SINR λ) ) ( + SINR λ) log 1 e e λ λ dλ dλ (11) where W s the bandwdth, SINR s the sgnal to nerence-plusnose rato, C s the lnk capacty, E[C] and σ C are the mean value and the standard devaton o C. s 3 4 A 1 s 1 1 B L L C 0 d 3 d D Fgure 5. Network topology d In the smulaton, we use standard method to smulate Raylegh adng,.e., lnk capacty C l =W Log[1+SINR l h l ], where the channel gan h l or lnk l s a normalzed complex Gaussan random varable, W s the bandwdth. All lnks experence ndependent adng. System parameters are: d 1 d 5 bandwdth W=10MHz, path loss exponent α=.5, reerence dstance d 0 =50m, reerence SINR at d 0 s SINR d0 =5dB. For smplcty, the receved average SINR at lnk l s gven by SINR l =SINR d0 10α Log 10 [d l /r 0 ], where d l s lnk dstance. More accurate SINR models could also be used but that wll unnecessarly complcate the smulaton. All smulatons run or T=4000 schedulng slots, and we use k=500 or k-pont movng average calculaton. Reer to Fgure 5, 11 nodes are placed n an area o 300m 400m. There are 13 lnks ~3, 6 lows 1 ~ 6. We obtan 5 clques V 1 ~V 5 as shown n Fgure 4. For comparson wth the smulaton results, we use (7) and (9) to determne the theoretcal average throughput o lnks and lows, whch are presented n Table I. TABLE I. THEORETICAL AVERAGE THROUGHPUT (Mbps) V 1 V V 3 V 4 V 5 3 L From Table I, we note that lows 1, 3 and 4 have the same average end-to-end throughput. Ths ndcates that lnk s the bottleneck o these three lows. Network provder can allocate more resource to lnk.to resolve such ssue. Preerably, one can use more ntellgent throughput sharng algorthm to mprove end-to-end throughput wthout the need or precous rado resource. As a drecton or uture work, ntellgent throughput sharng methods other than the one suggested by (9) wll ntegrate wth the CBFPS ramework. The smulated throughput and the theoretcal average throughput o lnks are plotted n Fgure 6. For ease o presentaton, only,, and L 6 are shown. The analytcal results provde very good estmates o the smulaton ones. Throughput (Mbps) L1:Smulaton L3:Smulaton L4:Smulaton L6:Smulaton L1:Analyss L3:Analyss L4:Analyss L6:Analyss Schedulng slot Fgure 6. CBPFS Perromance: Smulaton and Analyss

6 Fgure 7 depcts the number o slots allocated to each lnk n our 4000-slot smulaton. Obvously, each lnk has the same share o tme slots n CBPFS. Ths property makes CBPFS a promsng schedulng method or WMNs. Number o Schedulng Slots Lnk No. Fgure 7. The number o slots allocated to each lnk IV. CONCLUSIONS The PFS algorthm s proposed or cellular networks; n ths paper, we proposed a Clque-Based PFS (CBPFS) scheme, where lnks are grouped nto lnk clques whch are proportonally ar scheduled to acheve maxmzed utlty. CBPFS s a PFS extenson and can be used n both cellular networks and WMNs. For the proposed CBPFS algorthm, ths paper also analyzed ts perormance and provded closed-orm expressons to evaluate the throughput o both lnks and lows n Raylegh adng envronments. The theoretcal analyses match qute well wth smulatons. For uture research, we wll analyze CBPFS n varous adng envronments and consder ntellgent throughput sharng algorthms wth CBPFS. REFERENCES [1] I. F. Akyldz, X. Wang, and W. Wang, Wreless mesh networks: a survey, Computer Networks, vol. 47, No. 4, March 005, pp [] J. Tang, G. Xue, and W. Zhang, Cross-layer desgn or end-to-end throughput and arness enhancement n mult-channel wreless mesh networks, IEEE Trans. Wreless Commu., vol.6, no.10, October. 007, pp [3] F. Kelly, "Chargng and Rate Control or Elastc Trac", Eur. Trans. on Telecommun., February 1997, pp [4] X. Huang and B. Bensaou, On max-mn arness and schedulng n wreless ad-hoc networks: Analytcal ramework and mplementaton, n Proc. ACM MobHoc, Long Beach, Calorna, October 001. [5] B. Radunovc and J-Y L. Boudec, Rate perormance obectves o mult-hop wreless networks, n Proc. IEEE INFOCOM, Hong Kong, March 004. [6] J. M. Holtzman, "Asymptotc analyss o proportonal ar algorthm," n Proc. IEEE PIMRC, San Dego, CA, 001, pp [7] H. J. Kushner and P. A. Whtng, "Asymptotc Propertes o Proportonal-Far Sharng Algorthms: Extensons o the Algorthm," n Proc. o the Annual Allerton Conerence on Communcaton, Control and Computng, vol. 41, 003, pp [8] S. Borst, "User-level perormance o channel-aware schedulng algorthms n wreless data networks," n Proc. INFOCOM, San Francsco, 003, pp [9] J-G. Cho and S. Bahk, "Cell-Throughput Analyss o the Proportonal Far Scheduler n the Sngle-Cell Envronment," IEEE Trans. Veh. Technol., vol. 56, no., pp , March 007. [10] E. Lu and K. K. Leung, Far resource allocaton under raylegh and/or rcan adng envronments, n Proc. IEEE PIMRC, Cannes, France, September 008. [11] I. E. Telatar, "Capacty o mult-antenna Gaussan channels," European Transactons on Telecommuncatons, Vol.10, No.6, pp , November/December [1] P. J. Smth and M. Sha, "On a gaussan approxmaton to the capacty o wreless MIMO systems," n Proc. IEEE ICC, New York, Aprl 00, pp [13] P. J. Smth, S. Roy, and M. Sha, Capacty o MIMO systems wth sem-correlated lat adng, IEEE Trans. Inorm. Theory, vol. 49, no. 10, October 003. [14] E. Lu, Q. Zhang, and K. K. Leung, Opportunstc resource allocaton or requency-selectve adng, mult-carrer systems wth arness constrants, n Proc. IEEE ICC, Dresden, Germany, June 009. to appear. [15] Y. Gao, D-M. Chu, and J. C. S. Lu, Determnng the end-to-end throughput capacty n mult-hop networks: methodology and applcatons, SIGMetrcs/Perormance, Sant Malo, France, June 006.