584 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 2, FEBRUARY 2008

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1 584 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 2, FEBRUARY 28 An Optiiztion Frework for Blncing Throughput nd Firness in Wireless Networks With QoS Support Ho Ting Cheng, Student Meber, IEEE, nd Weihu Zhung, Senior Meber, IEEE Abstrct Qulity-of-service (QoS) provisioning, high syste throughput, nd firness ssurnce re indispensble for heterogeneous trffic in future wireless brodbnd networks With liited rdio resources, incresing syste throughput nd intining firness re conflicting perfornce etrics, leding to nturl trdeoff between these two esures Blncing syste throughput nd firness is desired In this pper, we consider n interference-liited wireless network, nd derive generic optiiztion frework to obtin n optil reltionship of syste throughput nd firness with QoS support nd efficient resource utiliztion, by introducing the brgining floor Fro the reltionship curve, different degrees of perfornce trdeoff between throughput nd firness cn be obtined by choosing different brgining floors In ddition, our frework fcilittes cll dission control to effectively gurntee QoS of ultiedi trffic The solutions of resource lloction obtined fro the optiiztion frework chieve the Preto Optility, deonstrting efficient use of network resources Index Ters Firness, optiiztion, Preto optility, qulity-of-service (QoS), throughput I INTRODUCTION FUTURE wireless brodbnd networks re expected to support ubiquitous counictions nd obile coputing, the notion of which hs been ttrcting plethor of ttention fro cdei nd industry Ubiquitous wireless ccess cn be relized in vrious prcticl scenrios, nely hoe networking, office networking, nd city networking [1] Copred with the trditionl networking dependent on cbles, wireless network is fster in deployent with lower cost, fcilitted by voiding tie-consuing nd costly opertions such s lnd construction nd cble plceent The wireless networking prdig provides not only vible, but lso econoicl solution for both peer-to-peer pplictions nd Internet ccess In order to fully exploit proising wireless technologies, efficient nd effective rdio resource ngeent in wireless networks is crucil In wireless networks, syste throughput is usully coon perfornce etric [2] However, next-genertion wireless networks such s wireless esh networks re nticipted Mnuscript received July 25, 26; revised Deceber 5, 26, Februry 23, 27, nd My 1, 27; ccepted July 5, 27 The ssocite editor coordinting the review of this pper nd pproving it for publiction ws M Zorzi This work ws supported in prt by the Nturl Science nd Engineering Reserch Council (NSERC) of Cnd The uthors re with the Centre for Wireless Counictions, Deprtent of Electricl nd Coputer Engineering, University of Wterloo, 2 University Avenue West, Wterloo, Ontrio, Cnd N2L 3G1 (e-il: {htcheng,wzhung@bbcruwterlooc) Digitl Object Identifier 1119/TWC /8$25 c 28 IEEE to support ultiedi trffic (eg, voice, video, nd dt trffic) With heterogeneous trffic, qulity-of-service (QoS) provisioning nd firness support re lso iportnt With liited vilble rdio resources, incresing syste throughput nd intining firness re usully conflicting with ech other [3], leding to nturl trdeoff between these two perfornce esures In prticulr, blncing syste throughput nd firness with QoS support nd high resource efficiency is necessry, depending on different pplictionspecific scenrios [1] In literture, only liited work ddresses the optil reltionship of throughput nd firness [4] [13] To the best of our knowledge, there is no widely ccepted unified frework to effectively ttin different degrees of perfornce trdeoff between throughput nd firness with QoS support nd efficient resource utiliztion, which is the otivtion of this reserch With focus on interference-liited wireless networks, the key contribution of this pper is to derive unified optiiztion frework to obtin the optil reltionship (ie, trdeoff curve) of syste throughput nd firness with QoS support By introducing brgining floor, the reltionship curve is obtined by solving the optiiztion proble itertively Different degrees of perfornce trdeoff between throughput nd firness cn be chieved by siply choosing pproprite vlues of the brgining floor Given n ppliction of interest, the desired operting trdeoff cn be deterined Resource utiliztion is efficient, which is verified by ge theory, chieving Preto Optility [14] To effectively provision QoS of ultiedi trffic, cll dission control is vitl to gurntee the QoS requireents of ditted clls Whether new incoing cll should be ditted or rejected is contingent upon the wy how the resources re llocted to wireless links nd how uch resources re left in the network Contributing to the criterion of cll dission, our optiiztion frework plys n iportnt role in cll dission control so s to fcilitte the opertion of blncing throughput nd firness The rest of the pper is orgnized s follows Relted work is discussed in Section II The syste odel is given in Section III The optiiztion proble forultion is presented in Section IV Efficiency of the proposed resource lloction is ddressed in Section V Nuericl results re given in Section VI Prcticl ipleenttion issues re discussed in Section VII Finlly, conclusion is drwn in Section VIII

2 CHENG et l: AN OPTIMIZATION FRAMEWORK FOR BALANCING THROUGHPUT AND FAIRNESS IN WIRELESS NETWORKS WITH QOS SUPPORT 585 II RELATED WORK In the literture, utility optiiztion is tool to esure syste perfornce subject to certin constrints (eg, QoS requireents) [4] [7], where utility function is described s esure of user stisfction Proportionl firness cn be obtined by choosing suitble utility functions (eg, logrithic functions) [4] Other perfornce esure (such s throughput) cn lso be incorported into this optiiztion forultion [6] With different proble forultions (ie, utility functions), different optil solutions cn be obtined (eg, optil throughput) Pricing schees [4] cn be eployed to chieve trdeoff between throughput nd firness, to certin extent However, the utility functions used in these work y not hve ny physicl ening How to find eningful utility function with n pproprite pricing schee cn lso be probletic In ddition, ost of the current work ssues tht the utility functions cn be seprble in the dul proble [4] [7], which y not lwys be the cse, especilly for interference-liited systes such s code-division ultiple ccess (CDMA) nd ultr-widebnd (UWB) systes, ening tht pplying existing pproches (eg, [7]) generlly results in suboptil solutions In the interference-liited systes, optility should be obtined by considering ll (user) utility functions together Idel (weighted x-in) firness cn be obtined by generlized processor shring (GPS) [8], [9] or its vrints [1], where ll nodes in the network shre the totl resources With GPS, the resource llocted to ech node is dependent on its own weight, whereby ech node cn hve fir shre of resources However, the notion of weight is n bstrct concept nd the question of how to relte QoS requireents to the weight effectively reins unsolved In GPS, even though ll the weights or QoS requireents re lredy stisfied, only (weighted x-in) firness is considered Further, the throughput perfornce of GPS needs further investigtion Although soe reserch is t the trdeoff between throughput nd firness in telecounictions networks, only heuristic schees re proposed without ny optility considertion [11] [13] Thus fr, only liited reltionship of syste throughput nd firness is ddressed Joint considertion of both perfornce etrics nd hence unified frework ttining different degrees of perfornce trdeoff between the with QoS support re desired Further, with scrce rdio resources, efficient resource utiliztion is vitl, which cn be verified by ge theory [14] All these spects re ddressed in this pper III SYSTEM MODEL We consider generic syste odel which is n interference-liited wireless network We ssue tht the chnnel gins re known in dvnce or cn be estited ccurtely vi pilot sybols Let R d denote the effective bndwidth of cll on the th link, which is the iniu rte required to stisfy the QoS requireents nd depends on source trffic chrcteristics [16], [17] Let R () denote the ctul trnsission rte of the th link where = ( 1, 2,,,, M ) nd is the power scling fctor of the th link s trnsitter, ie, [, 1], nd M the totl nuber of ctive links in the network For siplicity, the ctul trnsission rte of the th link is given by [18] R () =B log 2 (1 + γ ) (1) where B is the chnnel bndwidth nd γ is the signl-tointerference-plus-noise rtio of the th link In (1), γ = G P σ n G np n n + η where P is xiu trnsit power level of the th link s trnsitter, G n is the chnnel gin fro the n th link s trnsitter to the th link s receiver, σ is the cross-correltion fctor between ny two signls, ie, σ (, 1], nd η is the bckground noise power Notice tht (1) cn be esily extended to incorporte bit-error-rte (BER) requireents, coding nd odultion schees to copute the ttinble chnnel trnsission rte (or cpcity) [19] This ttinble chnnel trnsission rte is equl to the chievble throughput t the physicl lyer In prctice, the throughput (or goodput) obtined t the ediu ccess control lyer (nd higher lyers) is usully lower thn the chnnel trnsission rte due to the overhed of pcketiztion, ediu ccess control, etc Given ll the detils of syste preters, the ctul throughput cn be coputed fro the chnnel trnsission rte In order to effectively blnce syste throughput nd firness, cll dission control is indispensble, which cn be contingent on our optiiztion frework (to be discussed in Section IV) The cll dission routine 1 is invoked whenever new cll rrives For ech incoing new cll, it is ditted s long s there exists fesible solution of the optiiztion proble, ening tht the QoS requireents (ie, in ters of effective bndwidth) of this new cll nd ll other clls in service cn be stisfied If there is no fesible solution of the optiiztion proble, the new cll is rejected due to insufficient resources vilble to eet its QoS requireents In other words, the criterion of cll dission is tntount to the fesibility of the solution of our optiiztion frework With the cll dission control in plce, the QoS requireents of ll ditted clls cn be gurnteed nd the opertion of blncing throughput nd firness cn be crried out effectively Given tht the QoS requireents cn be et, the network perfornce cn be further iproved for incresing syste throughput nd/or intining (weighted x-in) firness, utilizing the resources efficiently In this reserch, the notion of weighted x-in firness is tken s the firness perfornce of interest, which corresponds to the idel firness chieved by the GPS A sury of iportnt sybols used in this pper is given in Tble I IV OPTIMIZATION PROBLEM FORMULATION In this section, we first consider two optiiztion proble forultions, nely syste throughput optiiztion nd 1 Notice tht routing is involved in the cll dission control Prticulrly, the tsks of QoS routing include 1) route discovery; 2) cll dission control over ech link; nd 3) route repir Therefore, which link cll is to trverse long cn be deterined [15] (2)

3 586 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 2, FEBRUARY 28 Sybol Definition TABLE I SUMMARY OF IMPORTANT SYMBOLS R d effective bndwidth of cll on the th link R () ctul trnsission rte of the th link Q () utility function (ie, extr throughput obtined) of the th link, ie, Q () =R () R d w weighting fctor of the th link, ie, w > P xiu trnsit power level of the th link s trnsitter G n chnnel gin fro the n th link s trnsitter to the th link s receiver power scling fctor of the th link s trnsitter, ie, [, 1] γ signl-to-interference-plus-noise rtio of the th link σ cross-correltion fctor between ny two signls, ie, σ (, 1] η bckground noise power M totl nuber of ctive links in the network B chnnel bndwidth J brgining floor, ie, J [,J ],wherej is the xiu vlue of J D devition of in {w Q () fro J, ie, D = J in {w Q () U esure of syste throughput, ie, U [, 1] V esure of firness, ie, V [, 1] W shped Jin s firness index, ie, W [, 1] weighted x-in firness optiiztion Next, the generic optiiztion proble forultion with syste throughput nd firness considertion is presented For the syste throughput optiiztion, the optiiztion proble is given by { M x R () (3) =1 subject to R () R, d 1, (4) where =( 1, 2,,,, M ) is the optiiztion vrible In fct, (3) cn be rewritten s { M ( x R () R d ) (5) =1 s the optility still intins for liner-shifted objective function [2] The syste throughput optiiztion proble (STOP) cn be rewritten s { M x Q () (6) =1 subject to Q (), 1, (7) where Q () = R () R d The physicl ening of Q () is the ount of extr resources llocted to (ie, excess throughput obtined by) the th link As coprison, Q () cn be viewed s the utility function of user in the conventionl utility xiiztion [4] [6], s Q () of user is: 1) incresing; 2) strictly concve; nd 3) twice differentible (ie, continuous) on Then, the interprettion of (6) is to chieve socil optility (ie, xiu of totl utility functions) However, in our cse, the utility functions re not seprble in the dul proble s Q () only increses with but not over Consider the following (prtil) dul proble [5]: in D(λ) (8) λ where λ = (λ 1,λ 2,, λ,, λ M ) is set of Lgrnge ultipliers nd { D(λ) = x Q ()+ λ Q () (9) Suppose tht the optility still intins when ech (user) utility function, Q (), is optiized seprtely Intuitively, ech user will siply choose the xiu power scling fctor, ie, = 1, In ost cses, due to the crossinterference (ie, σ ), the overll result will not contribute to the xiu vlue of (6), though Nsh Equilibriu is chieved [14] Optiizing individul user utility function seprtely y not lwys result in the xil solution of the STOP, s in generl fro (9), D(λ) = x = x { Q ()+ λ Q () { (1 + λ ) Q () (1) (11) x {(1 + λ ) Q () (12) Thus, the solution spce (ie, resource lloction) is not necessrily the se s those proposed in the literture (eg, [5]) Obtining the optil solution requires joint considertion of ll user utility functions, ening tht existing solutions cnnot be directly pplied For the weighted x-in firness optiiztion proble

4 CHENG et l: AN OPTIMIZATION FRAMEWORK FOR BALANCING THROUGHPUT AND FAIRNESS IN WIRELESS NETWORKS WITH QOS SUPPORT 587 (WMMFOP), the corresponding forultion is given by [21] { x in {w Q () (13) subject to Q (), 1, (14) where w is weighting fctor of the th link, ie, w >, which indictes the unwillingness of obtining extr llocted resources, ie, the sller the vlue of w, the ore eger is the th link to obtin ore extr resources The use of w is necessry for effective nd efficient resource lloction in wireless networks with heterogeneous trffic (eg, voice, video, nd dt trffic) For exple, for voice trffic, fter its effective bndwidth requireent is stisfied, llocting extr resources to it y be wsteful s the qulity of signl reception is lredy good, nd hence this trffic should be ssigned lrger weighting fctor On the other hnd, sller weighting fctor should be ssigned to dt trffic, s it dends ore throughput even though its effective bndwidth requireent is lredy et With different weighting fctors, different trffic clsses cn be differentited Notice tht the ening of the weighting fctor w of our interest is different fro tht in GPS Proposition 1: The set of fesible weighted utilities (ie, w Q (), ) in the WMMFOP hs the solidrity property [21] Proof: Suppose tht the fesible resource lloction solution is Without loss of generlity, we ssue tht there exists n n vlue such tht Q n ( ) > (ie, n > ), where n [21] For prticulr tieslot t, over which the th nd n th links re to be ctive, given the resource lloction solution, the vlues of their weighted utilities re to be w Q ( ) nd w n Q n ( ), respectively Let l t denote the length of the tieslot t, ie, l t > Siilrto[21],itis possible to prtition the slot into three inislots, nely t 1, t 2, nd t 3 with positive durtions l t1, l t2, nd l t3, respectively, such tht l t1 + l t2 + l t3 = l t Note tht the choice of how to deterine these vlues is rbitrry During t 1, the resource lloction is chosen to be the se s tht in tieslot t During t 2, the se lloction solution is kept s in t 1, except n =Duringt 3,weset n =, nd is djusted until the interference experienced by other ctive links is lrger thn tht in the originl tieslot t, if possible, otherwise, we set =1 In this new resource lloction, copred with the weighted utilities obtined in the originl resource lloction in tieslot t, ll the links excluding the th nd n th links hve the se or higher weighted utilities in t 1 nd t 2, respectively In t 3, their weighted utilities cn be higher, the se, or lower, depending on the vlue of Since the prtitioning into t 1, t 2, nd t 3 is entirely rbitrry, it is possible to choose their lengths l t1, l t2, nd l t3 so tht there exist sll ɛ (> ) nd ɛ n (> ) such tht the weighted utility of the th link increses by t ost ɛ nd the weighted utility of the n th link decreses by t ost ɛ n, while the rest of the ctive links hve the se or higher weighted utilities Let w Q (ã) nd w n Q n (ã) denote the newly obtined weighted utilities of the th link nd the n th link, respectively We cn now cquire the following inequlities: w Q ( ) < w Q (ã) < w Q ( )+ɛ nd w n Q n ( ) ɛ n < w n Q n (ã) <w n Q n ( ) Therefore, the vlue of w Q () cn be incresed by t ost ɛ by decresing the vlue of w n Q n () by t ost ɛ n And, w Q (ã) w n Q n (ã) w k Q k (ã) = + w Q ( ) w n Q n ( ) w k Q k ( ) ɛ + k,n ɛ ṇ ɛ ḳ (15) where ɛ k >, k, n Since ll w Q (ã),, belong to the fesible set (ie, Q (ã), ), by the definition of solidrity [21], the set of fesible weighted utilities in the WMMFOP hs the solidrity property In fct, the WMMFOP cn be trnsfored into [2] x (16) subject to w Q () L, (17) Therefore, the WMMFOP cn be rewritten s follows: x (18) subject to w Q () L, 1, (19) With the solidrity property, t the optil point, ll w Q (),, re to be equl, nd the xiu vlue of L is unique in WMMFOP [21] Let J nd â denote the optil solutions (ie, xil L nd optil ) ofthewmmfop If the constrints w Q () J,, re dded to the STOP, the odified STOP becoes { M x Q () (2) =1 subject to w Q () J, 1, (21) Proposition 2: The optil solution â obtined fro the WMMFOP is lso the optil solution for the odified STOP Proof: Suppose tht there exists nother solution ã such M =1 Q (ã) > M =1 Q (â) It ens tht there tht exists soe such tht w Q (ã) >J within the fesible region In the WMMFOP, if for soe, w Q (ã) >J, J cn be incresed by decresing the vlue of ã or incresing the vlue of ã n or both, for n, within the fesible region until it reches the xil vlue, sy J However, it contrdicts to the stteent tht J is the xil vlue obtined fro the WMMFOP Therefore, no such solution ã exists The optil solution â obtined fro the WMMFOP is lso the optil solution for the odified STOP With the new constrint set, the solution obtined fro the odified STOP not only chieves weighted x-in firness,

5 588 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 2, FEBRUARY 28 but lso optiizes the syste throughput Therefore, to bridge the syste throughput nd firness perfornce esures together, we introduce preter clled brgining floor, denoted by J, where J [,J ] nd J is solution (ie, the xil vlue) of the WMMFOP Motivted by Proposition 2, we propose the generic optiiztion proble (GOP), which is given by { M x Q () (22) =1 subject to w Q () J, 1, (23) Clerly, the solutions of the GOP for xil syste throughput re obtined when J =while tht for xil (weighted x-in) firness when J = J, where J is obtined fro the WMMFOP In this reserch, our focus is not to solve the GOP Insted, we eploy it s unified frework for deducing the optil reltionship between syste throughput nd firness with QoS support Let be the optil solution obtined fro the GOP Proposition 3: The syste throughput (ie, M =1 Q ( )) is non-incresing function of brgining floor J Proof: When J increses (decreses), the fesible region of in the GOP shrinks (expnds) For J 1 J 2 J,let 1 nd 2 denote the optil solutions of the GOP with J 1 nd J 2, respectively The fesible region of of the GOP with J 2 is only subset of tht with J 1 Thus, M =1 Q ( 1) M =1 Q ( 2) nd hence the syste throughput does not increse with the vlue of J Corollry 1: The iniu vlue of w Q ( ) (ie, in {w Q ( )) is non-decresing function of J Proof: Let J be the solution of WMMFOP nd 1 be the optil solution of the GOP with J 1, where J 1 J For J 1 <J 2 J, consider the following two cses: Cse 1:If the solution 1 is fesible for the GOP with J 2, fro Proposition 3, 1 is lso the optil solution for the GOP with J 2 Hence, the iniu vlue of w Q ( ) is the se for the GOP with J 1 nd J 2 Cse 2:If the solution 1 is not fesible for the GOP with J 2, it ens tht there exists soe such tht w Q ( 1) <J 2 nd hence in {w Q ( 1) < J 2 Thus, 1 is n infesible solution for the GOP with J 2 In ddition, suppose tht 2 is the optil solution for the GOP with J 2, ie, w Q ( 2) J 2, Hence, in {w Q ( 2) J 2 > in {w Q ( 1) By cobining the bove two cses, in {w Q ( 2) in {w Q ( 1) nd hence the iniu vlue of w Q ( ) (ie, in {w Q ( )) does not decrese with the vlue of J Theore 1: The syste throughput (ie, M =1 Q ( )) does not increse with J, but the iniu vlue of w Q ( ) (ie, in {w Q ( )) does not decrese with J Proof: By Proposition 3 nd Corollry 1, it is proved Corollry 2: A reltionship between the syste throughput nd weighted x-in firness perfornce cn be chieved by solving the GOP with different vlues of J Proof: Fro Theore 1, for J =the solution obtined fro the GOP corresponds to the xil syste throughput, while for J = J the solution obtined fro the GOP corresponds to the xil weighted x-in firness perfornce When J increses fro zero, the solution obtined fro the GOP cn refer to decresed syste throughput nd incresed firness perfornce Therefore, the perfornce trdeoff nd hence desired reltionship between the syste throughput nd weighted x-in firness perfornce cn be chieved by solving the GOP with different vlues of J, ie, J [,J ] Fro the perspective of network design, with liited ount of resources, iproving firness perfornce will reduce the syste throughput, which tches with the perspective of Corollry 2 With different vlues of J, the trdeoff curve of syste throughput nd firness cn be obtined Thus, the GOP should be solved with different vlues of J itertively The itertive procedure to obtin the trdeoff curve is described below: Step 1: Find J by solving the WMMFOP; Step 2: Set J =nd solve the GOP, whereby the obtined solution corresponds to the xil throughput perfornce; Step 3: Increse J by δj nd solve the GOP gin; Step 4: Repet Step 3 until J = J, which corresponds to the xil firness Through the bove procedure, different sets of the optil solution, the corresponding reltionships of syste throughput nd firness, nd hence the desired trdeoff cn be obtined by suitbly selecting the vlue of J Notice tht fter the trdeoff curve is procured, the choice of the J vlue usully depends on the purpose of the ppliction of interest nd/or the prerogtive of the syste designer With the fixed vlue of J, the existence of the optil solution of the GOP (ie, the desired trdeoff) is ssured due to the cll dission control in plce, discussed in Section III In cse tht the vlue of J is llowed to chnge rndoly when the syste is in use, conservtive pproch cn be eployed: whenever new cll rrives, we only check the fesibility condition on the solution of the WMMFOP (ie, the GOP with J = J ) The new cll is ditted if there exists solution, or rejected otherwise Since the fesible region does not shrink nd hence the solution obtined initilly will not becoe infesible when the vlue of J decreses, this prevents the ongoing trnsissions fro being dropped In this cse, the se cll dission control is eployed, gurnteeing the QoS requireents of ll clls in service with ny vlue of J V EFFICIENCY EVALUATION BY GAME THEORY In this section, we show tht our solutions obtined fro the GOP given by (22) (23) chieve efficient use of resources In ge theory, efficient resource utiliztion is deterined by the concept of Preto Optility [14] Definition 1: An ction profile b =(b 1,b 2,, b M ) is sid to be Preto optil if nd only if there exists no other

6 CHENG et l: AN OPTIMIZATION FRAMEWORK FOR BALANCING THROUGHPUT AND FAIRNESS IN WIRELESS NETWORKS WITH QOS SUPPORT 589 ction profile b such tht for soe, Y ( b) >Y (b ) nd Y n ( b) Y n (b ),forn, where Y ( ) is pyoff of user in the context of ge theory In words, n ction profile (or resource lloction) is Preto optil if there exists no other ction profile tht kes soe user(s) better off without king the other user(s) worse off Proposition 4: The optil solution of the GOP is Preto optil Proof: Given the ction profile or optil solution (ie, resource lloction) obtined fro the GOP, denote Q ( ) or w Q ( ) s the pyoff of the th link Consider nother ction profile ã Forsoe, ifq (ã) >Q ( ), then Q n (ã) <Q n ( ) for soe n, s either ã > or ã n < n or both According to Definition 1, the optil solution obtined fro the GOP chieves the Preto Optility Fro the perspective of ge theory, the resources re efficiently utilized for incresing syste throughput nd/or intining firness In other words, for the reltionship of syste throughput nd firness, every point (ie, resource lloction) on the trdeoff curve (discussed in Section VI) is Preto optil, utilizing the resources efficiently VI NUMERICAL RESULTS This section presents nuericl results on: 1) syste throughput nd firness perfornce versus the vlue of J in the GOP; nd 2) the desired reltionship (ie, trdeoff curve) of syste throughput nd firness In the nuericl nlysis, we siply solve the GOP by n exhustive serch with n increent size of δ =1 Suppose tht there re I itertions in the itertive procedure for the throughput nd firness trdeoff curve Let i be the optil solution obtined fro the GOP in the i th itertion The esure of syste throughput in the i th itertion (ie, i I) is given by ( M ) { =1 Q M ( i ) in i I =1 Q ( i ) U = { M { x i I =1 Q M ( i ) in i I =1 Q ( i ) (24) where U [, 1], ie, the lrger the vlue of the syste throughput, the lrger the vlue of U Let D i = J in {w Q ( i ), where J is the solution of the WMMFOP nd D i represents the devition of the iniu vlue of w Q ( i ) ong ll M links fro J, ie, the lrger the vlue of D i, the poorer the weighted x-in firness perfornce The esure of weighted x-in firness in the i th itertion is given by x i I {D i D i V = (25) x i I {D i in i I {D i where V [, 1], ie, the lrger the vlue of D i, the sller the vlue of V V indictes the firness perfornce of the worst link In the literture, Jin s firness index [22] is widely eployed s esure of network-wise firness perfornce Let JFI i be the Jin s firness index in the i th itertion, where JFI i = ( w Q ( i )) 2 M (w Q ( i )) 2 (26) TABLE II SYSTEM PARAMETERS FOR THE NUMERICAL ANALYSIS FOR THE CASE OF EQUAL WEIGHTING FACTORS Preter Vlue M 4 B 1 η 1 σ 1 P 1, w 1, (2,1,5,1) J 329 R d 1,, R d 4 The shped Jin s firness index in the i th itertion is given by JFI i in i I {JFI i W = (27) x i I {JFI i in i I {JFI i where W [, 1], ie, the lrger the vlue of Jin s firness index, the lrger the vlue of W In this nuericl nlysis, we consider four ctive links in the network, ie, M =4 The fding coefficient of link is odeled s coplex Gussin rndo vrible with zero en nd unit vrince The chnnel gin trix G, ie, G =[G n ] M M, used for the nuericl nlysis is rndoly generted nd norlized, which is given below: G = (28) Under the condition tht the solution exists in the WMMFOP, we consider two cses: 1) ll the weighting fctors re equl; nd 2) the weighting fctors re different A Equl Weighting Fctors In this cse, the syste preters re given in Tble II, where J is coputed by solving the WMMFOP We follow the itertive procedure described in Section IV nd obtin the nuericl results given in Tble III First, we study the behviors of the syste throughput esure U nd the firness esure V with different vlues of J, which is given in Fig 1 As entioned in Section IV, the trdeoff curve is obtined by solving the GOP itertively, strting t the xil syste throughput nd ending t the xil firness As expected, in Fig 1, U decreses fro the xiu vlue (ie, U = 1) to the iniu vlue (ie, U =) with J, while V increses fro the iniu vlue (ie, V = ) to the xiu vlue (ie, V = 1) with J, which shows tht incresing syste throughput nd intining firness re conflicting with ech other The shped Jin s firness index is lso plotted for coprison Fro the results shown in Tble III, the iniu weighted utility vlue, in {w Q (), increses with J, s expected Note tht the firness perfornce esure of V nd tht of W re different (ie, V for the worst-link firness perfornce while W for the network-wise firness perfornce); however, in the cse of equl weighting fctors, the

7 59 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 2, FEBRUARY 28 TABLE III NUMERICAL RESULTS FOR THE CASE OF EQUAL WEIGHTING FACTORS J Q 1 () Q 2 () Q 3 () Q 4 P () Q() in {w Q () U V W =[ 1, 2, 3, 4 ] [1, 46, 64, 14] [1, 5, 66, 18] [1, 52, 61, 21] [1, 54, 57, 24] [1, 6, 61, 29] [1, 62, 56, 32] [1, 61, 45, 33] [1, 65, 44, 37] [1, 74, 5, 44] [1, 73, 4, 45] [1, 82, 44, 52] [1, 87, 42, 57] [1, 91, 38, 61] [1, 95, 34, 65] [94, 1, 33, 7] [78, 1, 33, 71] U: shped throughput esure V: shped firness esure W: shped Jin s firness index V W U 5 U, V, W J V, W Fig 1 The syste throughput esure nd the firness esures ginst the vlue of J with equl weighting fctors Fig 2 The reltionship of syste throughput nd firness with equl weighting fctors generl trend of both curves grees with ech other Thus, both firness esures tch with the x-in firness perfornce, s both V nd W increse with in {w Q (), in generl Consider the trend of ech utility function For every link, its utility vlue (ie, Q () of the th link) ginst the vlue of J is given in Tble III For sll J, the links with sller effective bndwidths usully obtin lrger utility vlues (eg, Q 3 ()) It is intuitive tht those links with sller effective bndwidths hve ore freedo to increse their throughputs thn other links with lrger effective bndwidths As J increses, the utility vlues of those links with sller effective bndwidths decrese for the ske of chieving certin level of firness However, with different chnnel gins, soe link, sy the th link, with sll effective bndwidth y be forced to use sll vlue of so tht only sll vlue of Q () is chieved, for exple, Q 4 () in our exple Fro (28), G 24 =1, ening tht the interference ipct fro the 4 th link to the 2 nd link is significnt In order to eet ll the effective bndwidth requireents, the 4 th link cn only use sll vlue of 4, which results in sll vlue of Q 4 () Nonetheless, the utility vlues of ll links converge to the se vlue when the condition of xil firness is et (ie, weighted x-in firness) s ll the weighting fctors re equl Notice tht the discrepncies in Tble III re erely due to the discrete exhustive serch used in the nuericl nlysis The desired trdeoff curve of syste throughput nd firness perfornces is shown in Fig 2 The curve is bit concve in shpe, ening tht in nerly unfir sitution (ie, V ), unit decrese in syste throughput gives lrger rginl iproveent in weighted x-in firness perfornce At ner-xil firness point (ie, V 1), lrger decrese of syste throughput is required to further increse the firness esure Fro this curve, different de-

8 CHENG et l: AN OPTIMIZATION FRAMEWORK FOR BALANCING THROUGHPUT AND FAIRNESS IN WIRELESS NETWORKS WITH QOS SUPPORT 591 TABLE IV SYSTEM PARAMETERS FOR THE NUMERICAL ANALYSIS FOR THE CASE OF DIFFERENT WEIGHTING FACTORS Preter Vlue M 4 B 1 η 1 σ 1 P 1, (w 1,, w 4 ) (1,2,4,8) R d 1,, R4 d (2,1,5,1) J 9249 U, V, W U: syste throughput esure V: firness esure W: shped Jin s firness index grees of perfornce trdeoff between syste throughput nd firness cn be found by suitbly choosing the vlue of J The shped Jin s firness index is lso plotted for reference This trdeoff curve is undoubtedly useful for effective nd efficient resource lloction With ppliction-specific constrints (such s firness or throughput requireents), desired trdeoff point cn be obtined fro this reltionship curve nd hence the corresponding resource lloction cn be deduced J Fig 3 The syste throughput esure nd the firness esures ginst the vlue of J with different weighting fctors 1 9 V W 8 B Different Weighting Fctors For the cse of different weighting fctors, the detils of the syste preters for the nuericl nlysis re given in Tble IV nd the nuericl results re given in Tble V In Fig 3, the curve of the syste throughput esure U nd tht of the firness esure V with different vlues of J re plotted, nd the trdeoff curve of syste throughput nd firness perfornces is shown in Fig 4 For U nd V, the trends of these two curves in Fig 3 nd Fig 4 re ore or less the se s those shown in Fig 1 nd Fig 2, respectively For W, however, the curve of W devites ore wy fro tht of V in the cse of different weighting fctors In fct, using W in this cse is less effective to ccurtely indicte the iproveent of weighted x-in firness perfornce though the iniu weighted utility vlue, in {w Q (), increses with J, shown in Tble V Nonetheless, the Jin s firness index is network-wise firness esure, insted of esuring the worst-link firness perfornce, though the generl trend of the curve grees with the weighted x-in firness perfornce Notice tht the fluctutions in the grph re prtly becuse of the discrete exhustive serch used in the nuericl nlysis With different weighting fctors, the lloction solution in the GOP is not the se s tht with the equl weighting fctor As entioned, the sller the vlue of w,theore eger is the th link to obtin ore extr resources At the lst row of Tble V, the vlue of Q 1 () is the lrgest while tht of Q 4 () is the sllest, s w 1 <w 2 <w 3 <w 4 In fct, with different weighting fctors, different trffic clsses cn be differentited Therefore, the use of w is crucil for further iproving the effectiveness nd efficiency of resource lloction in wireless networks with heterogeneous trffic (eg, voice, video, nd dt trffic) U V, W Fig 4 The reltionship of syste throughput nd firness with different weighting fctors VII DISCUSSIONS ON PRACTICAL IMPLEMENTATION In this section, soe issues on prcticl ipleenttion re discussed For wireless networks with centrlized control, centrl controller (such s bse sttion) essentilly collects requests fro wireless nodes, kes decisions on cll dission, nd lloctes resources to wireless links The globl network infortion (ie, network resources, QoS requireents of clls, nd chnnel conditions) re vilble t the centrl controller, thereby leding to n esier prcticl ipleenttion to obtin (ner-)optil resource lloction by solving the GOP In contrst, for wireless networks with distributed control (without centrl coordintion), cquiring globl network infortion by essge exchnges is not desired, cusing considerble ount of overhed Thus, ech node usully hs its locl network infortion only Node clustering is vible pproch, where the whole network is divided into clusters Within ech cluster, clusterhed erely

9 592 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL 7, NO 2, FEBRUARY 28 TABLE V NUMERICAL RESULTS FOR THE CASE OF DIFFERENT WEIGHTING FACTORS J Q 1 () Q 2 () Q 3 () Q 4 P () Q() in {w Q () U V W =[ 1, 2, 3, 4 ] [1, 46, 64, 14] [1, 47, 6, 15] [1, 49, 58, 16] [1, 54, 63, 18] [1, 56, 59, 19] [1, 57, 55, 2] [1, 59, 52, 21] [1, 64, 55, 23] [1, 63, 47, 23] [1, 65, 44, 24] [1, 72, 49, 27] [1, 77, 5, 29] [1, 76, 42, 29] [1, 76, 36, 29] [1, 75, 29, 29] [1, 76, 24, 3] dels with the network ctivity of its own neighborhood [23] Ech clusterhed cn run the GOP loclly by treting the inter-cluster interference s portion of the intr-cluster interference, yet leding to suboptil solution Whether resource lloction solution obtined is optil or suboptil depends on the lgorith design Although our frework is universl in the sense tht it cn be pplied to ny syste odel which is interference-liited (eg, CDMA systes), one drwbck is tht solving this optiiztion proble generlly involves high coputtionl coplexity Soe (suboptil) lgoriths with low coplexity re preferred for the ske of prcticl ipleenttion To tckle non-trivil optiiztion proble (ie, non-convex in nture) such s the GOP, two pproches re coonly used: 1) Convex pproxition The originl optiiztion proble is relxed to convex optiiztion proble The optil solution of the relxed proble cn be chieved by clssicl ethods such s grdient ethods nd interior-point ethods [24] In [7], it is proved tht under high signl-to-interference-plusnoise rtio pproxition, the objective function of the GOP cn be forulted to be concve function (by verifying tht the Hessin trix is negtive definite) nd hence the GOP becoes convex optiiztion proble A rich body of literture exists on the theory of convex optiiztion whereby fst, siple, nd robust prcticl lgoriths cn be devised (eg, grdient-bsed itertions) [7], [24]; nd 2) Interpreting Krush-Kuhn-Tucker (KKT) conditions [24] The necessry conditions for the optil solution cn be verified the by KKT conditions The KKT conditions for the GOP re s follows: M Q () Q () + α w = λ μ, =1 (29) w Q () J, (3) α (J w Q ()) =, (31) λ ( 1) =, (32) μ =, (33) α,λ,μ, (34) 1, (35) where α is the Lgrnge ultiplier for the constrint (3), nd λ nd μ re the Lgrnge ultipliers for the constrint (35) Optil nd/or heuristic lgoriths cn be deduced bsed on the structure of the KKT conditions or conceived s ethods for solving the KKT conditions [24], [25] For exple, cobining (29) nd (33), we hve λ (1 + α w ) Q () nd hence λ (1 + α w ) Q () M n M Q n () =, (36) n Q n (), (37) Therefore, if is positive, equlity holds for (37), which is the necessry condition of being positive, providing soe guidelines for the design of resource lloction lgorith To hndle lrge ount of decision vribles, dt structure plys n iportnt role in deterining the coplexity For instnce, tree ipleenttion together with sorting nd/or serching cn fcilitte to bring lower coputtionl coplexity to the lgorith [26] In this pper, the gol is to develop unified frework to blnce throughput nd firness Nonetheless, the ctul syste perfornce with prcticl ipleenttion needs further investigtion Devising prcticl lgoriths nd ctul syste

10 CHENG et l: AN OPTIMIZATION FRAMEWORK FOR BALANCING THROUGHPUT AND FAIRNESS IN WIRELESS NETWORKS WITH QOS SUPPORT 593 opertions (such s ediu ccess control) tilored for our syste with ccurte trffic odels is left for future work VIII CONCLUSION In this pper, we propose the unified optiiztion frework for interference-liited wireless networks, whereby the optil reltionship curve of syste throughput nd firness cn be obtined Different degrees of perfornce trdeoff between syste throughput nd firness cn be chieved, by suitbly djusting the vlue of the brgining floor QoS support is ssured with the help of the cll dission control bsed on the fesibility of the solution of the GOP Fro the perspective of ge theory, the resource lloction solutions chieve the Preto Optility, deonstrting efficient use of network resources ACKNOWLEDGEMENT The uthors wish to thnk the nonyous reviewers for their helpful reviews nd suggestions which iproved the qulity nd presenttion of this pper REFERENCES [1] H T Cheng, H Jing, nd W Zhung, Distributed ediu ccess control for wireless esh networks, Wireless Coun Mobile Coputing, vol 6, no 6, pp , Sep 26 [2] R Jurdk, C V Lopes, nd P Bldi, A survey, clssifiction nd coprtive nlysis of ediu ccess control protocols for d hoc networks, IEEE Coun Surveys Tutorils, vol 6, no 1, pp 2 16, 24 [3] H Luo, J Cheng, nd S Lu, Self-coordinting loclized fir queueing in wireless d hoc networks, IEEE Trns Mobile Coputing, vol 3, no 1, pp 86 98, Jn Mr 24 [4] F P Kelly, A Mulloo, nd D K H Tn, Rte control in couniction networks: Shdow prices, proportionl firness nd stbility, J Opertionl Reserch Society, vol 49, no 3, pp , 1998 [5] S H Low nd D E Lpsely, Optiiztion flow control I: Bsic lgorith nd convergence, IEEE/ACM Trns Networking, vol 7, no 6, pp , Dec 1999 [6] D Julin, M Ching, D O Neill, nd S Boyd, QoS nd firness constrined convex optiiztion of resource lloction for wireless cellulr nd d hoc networks, in Proc IEEE INFOCOM, June 22, vol 2, pp [7] M Ching, Blncing trnsport nd physicl lyers in wireless ultihop networks: Jointly optil congestion control nd power control, IEEE J Select Ares Coun, vol 23, no 1, pp , Jn 25 [8] A K Prekh nd R G Gllger, A generlized processor shring pproch to flow control in integrted services networks: The singlenode cse, IEEE/ACM Trns Networking, vol 1, no 3, pp , June 1993 [9], A generlized processor shring pproch to flow control in integrted services networks: The ultiple node cse, IEEE/ACM Trns Networking, vol 2, no 2, pp , 1994 [1] S Lu, V Bhrghvn, nd R Sriknt, Fir scheduling in wireless pcket networks, IEEE/ACM Trns Networking, vol 7, no 4, pp , Aug 1999 [11] G Miklos nd S Moln, Fir lloction of elstic trffic for wireless bse sttion, in Proc IEEE Globeco, 1999, vol 3, pp [12] J Dnjnovic, A Jin, T Chen, nd S Srkr, Scheduling the CDMA2 reverse link, in Proc IEEE VTC, 22, vol 1, pp [13] B Wng, K C Chu, nd H-P Schwefel, Resource lloction for non-reltie dt users in UMTS uplink, in Proc IEEE WCNC, Mr 23, vol 2, pp [14] G Owen, Ge Theory, 3rd ed Sn Diego: Acdeic Press, 21 [15] A Abdrbou nd W Zhung, A position-bsed QoS routing schee for UWB obile d hoc networks, IEEE J Select Ares Coun, vol 24, no 4, pp , Apr 26 [16] F P Kelly, Notes on effective bndwidths, in Stochstic Networks: Theory nd Applictions Oxford, UK: Oxford University Press, 1996 [17] D Wu nd R Negi, Effective cpcity: A wireless link odel for support of qulity of service, IEEE Trns Wireless Coun, vol 2, no 4, pp , July 23 [18] C E Shnnon, A theticl theory of counictions, Bell Lbs Technicl J, vol 27, pp , 1948 [19] A Goldsith, Wireless Counictions Cbridge, UK: Cbridge University Press, 24 [2] H M Wgner, Principles of Opertions Reserch Upper Sddle River, NJ: Prentice Hll, Inc, 1975 [21] B Rdunović nd J Y Le Boudec, Rte perfornce objectives of ultihop wireless networks, IEEE Trns Mobile Coputing, vol 3, no 4, pp , Oct Dec 24 [22] R Jin, A Durresi, nd G Bbic, Throughput firness index: An explntion, ATM Foru Docuent Nuber: ATM Foru/99-45, Feb 1999 [23] J Yu nd P Chong, A survey of clustering schees for obile d hoc networks, IEEE Coun Surveys Tutorils, vol 7, no 1, pp 32 48, First Qtr 25 [24] S Boyd nd L Vndenberghe, Convex Optiiztion Cbridge, UK: Cbridge University Press, 24 [25] J Hung, R A Berry, nd M L Honig, Distributed interference copenstion for wireless networks, IEEE J Select Ares Coun, vol 24, no 5, pp , My 26 [26] M A Weiss, Dt Structures nd Algorith Anlysis in C Menlo Prk, CA: Addison Wesley, 1997 Ho Ting Cheng (S 5) received the BEng nd MPhil degrees in Infortion Engineering fro The Chinese University of Hong Kong (CUHK) in 23 nd 25, respectively He is reserch ssistnt nd currently working towrd his PhD degree t the Deprtent of Electricl nd Coputer Engineering, University of Wterloo, Cnd His reserch interests include distributed resource lloction, wireless esh networking, nd couniction theory Weihu Zhung (M 93-SM 1) is Professor in the Deprtent of Electricl nd Coputer Engineering, University of Wterloo, Cnd She is co-uthor of the textbook Wireless Counictions nd Networking (Prentice Hll, 23) Her current reserch interests include wireless counictions nd networks, nd rdio positioning Dr Zhung is co-recipient of Best Pper Awrd fro IEEE ICC 27 nd Best Student Pper Awrd fro IEEE WCNC 27 She received the Outstnding Perfornce Awrd in 25 nd 26 fro the University of Wterloo for outstnding chieveents in teching, reserch, nd service, nd the Preier s Reserch Excellence Awrd (PREA) in 21 fro the Ontrio Governent for deonstrted excellence of scientific nd cdeic contributions She is the Editor-in-Chief of the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, nd n Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, theeurasip Journl on Wireless Counictions nd Networking, ndtheinterntionl Journl of Sensor Networks