FUTURE wireless communication systems must provide

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1 IEEE RANSACIONS ON WIRELESS COMMUNICAIONS, VOL. 8, NO., JULY 9 Joit MS-GSC Combiig ad Dow-Lik Multiuser Diversity Schedulig Slim Be Halima, Mohamed-Slim Alouii, ad Khalid A. Qaraqe Abstract Exploitig multi-user diversity is oe solutio to efficietly use the wireless medium. he basic priciple is to assig the resources to the user experiecig the best chael coditios based o a feedback iformatio provided by differet users. I this paper, we propose three multiuser schedulig schemes which are combiatios betwee the switch ad examie trasmissio scheme i the etwork layer ad the joit miimum-selectio geeralized selectio combiig ad adaptive modulatio i the physical layer. As a first step, fairess of access is ot take ito accout i the desig of the proposed schedulig schemes. he, i a secod step, the proposed schemes are modified i order to eable them to achieve a certai short term fairess of access. I both cases, we study the performace of the proposed schemes i terms of spectral efficiecy, feedback load ad rate, ad average umber of combied ad estimated paths. Usig some selected umerical examples, the proposed schemes are compared amog themselves ad also to some recetly published schemes. Idex erms Multi-user diversity, adaptive modulatio, adaptive combiig. I. INRODUCION FUURE wireless commuicatio systems must provide high data rates to respod to the growig demads of multimedia services. O the other had, the wireless chael capacity is a scarce resource that must be used efficietly to respod to these desired requiremets. Assumig a large umber of active users i a cell, oe way to efficietly use this resource, thus extractig a certai diversity gai, is optimum schedulig betwee these multiple users experiecig differet chael coditios. I fact, i covetioal commuicatios systems GSM for example), time is divided ito fixed size time slots ad each of these time slots is assiged to a fixed user regardless of its chael coditio. However, it is highly probable that at ay give poit i time, at least oe of the other users has a better chael quality. Multi-user diversity [] [] gai ca be achieved by schedulig the user havig the best chael coditios. I order to maage priority betwee users, chael quality iformatio CQI) must be fed back to the base statio through a dedicated uplik chael. Such iformatio must be sigalled regularly i order to respod to chael time variatio. Mauscript received Jue, ; revised November, 8; accepted March, 9. he associate editor coordiatig the review of this paper ad approvig it for publicatio was M. Chiag. S. B. Halima is with the uisia Polytechic School, La Marsa, uisia. M.-S. Alouii ad K. A. Qaraqe are with the Departmet of Electrical Egieerig, exas A&M Uiversity at Qatar, Educatio City, Doha, Qatar {alouii, khalid.qaraqe}@qatar.tamu.edu). his work was supported i part by the Qatar Foudatio for Educatio, Sciece, ad Commuity Developmet ad i part by Qatar elecom. Digital Object Idetifier.9/WC.9.9 -/9$. c 9 IEEE he secod issue to be dealt with is how the chael coditios are exploited by the system. I fact, adaptive modulatio [] [] ad diversity combiig are two importat tools to respod to high data rates demads. More specifically adaptive modulatio ca achieve high spectral efficiecy over wireless chaels. he basic idea of adaptive modulatio is to match the modulatio parameters, such as costellatio size, to fadig chael coditios while maitaiig the istataeous error rate below a target value. Usually, the modulatio mode is chose based o the compariso results of received sigal stregth with several predetermied thresholds. O he other side, diversity combiig makes higher trasmissio rates possible. However, i the dowlik, diversity combiig is costraied by the limited lifetime of the user equipmet battery which has led to the adoptio of adaptive diversity combiig techiques [] [9]. Miimum-selectio geeralized selectio combiig MS-GSC) [8] comes as oe of these methods. It is based o the geeralized selectio combiig GSC) scheme. I particular, it selects the best L c paths amog the L available paths but stops the combiig operatio whe reachig a predetermied target sigal to oise ratio SNR) reducig as a cosequece the umber of active maximum ratio combiig MRC) braches ad as such the processig power cosumptio. It is clear that adaptive modulatio ca beefit from diversity combiig through the improved chael quality but the more iterestig poit is that both adaptive combiig ad adaptive modulatio use predetermied thresholds i their operatio. Based o this observatio, we combie these two cocepts ad develop ad aalyze i this paper some joit adaptive combiig ad high speed data access schemes. We proposed three schedulig schemes that are combiatios betwee the switch ad examie trasmissio SE) scheme [] i the etwork layer ad the adaptive modulatio ad MS-GSC combiig i the physical layer: ) the feedback efficiet scheme ) aimig at miimizig the feedback load to the detrimet of spectral efficiecy. ) the badwidth efficiet scheme ) aimig at achievig high badwidth efficiecy without major cosideratio of the processig power cosumptio issue, ad ) the badwidth efficiet power greedy scheme aimig at achievig the highest badwidth efficiecy with the miimum processig power cosumptio. For these three schemes uder cosideratio we evaluate their average spectral efficiecy quatified i terms of average umber of bits/s/hz), the average feedback load quatified i terms of the mea of the umber of probed users), the average feedback rate quatified i terms of the mea umber of feedback bits per schedulig operatio), ad the average

2 BEN HALIMA et al.: JOIN MS-GSC COMBINING AND DOWN-LINK MULIUSER DIVERSIY SCHEDULING processig power cosumptio quatified i terms of the average umber of estimated ad combied paths per scheduled user). From aother perspective, assigig the chael to the best user ca lead to the moopolizatio of the resources by a limited umber of users. herefore, i the secod part of the paper we study a algorithm that attempts to improve the fairess of access amog the users. I other words, we try to esure short-term fairess ad guaratee more frequet access to all users this has some implicatios o delay miimizatio). I our study, we evaluate the ew system performace i terms of spectral efficiecy ad feedback load. he remaider of this paper is orgaized as follows. he ext sectio gives a geeral backgroud o the chael ad system models. Sectio III is devoted to the operatio of the proposed schemes without fairess cosideratio. Sectio IV deals with the system that tries to achieve a better fairess of access betwee the scheduled users. Fially, the paper cocludes with a summary of the mai results. A. System Model II. GENERAL BACKGROUND We cosider a base statio BS) servig K users each havig L diversity braches. We cosider a discrete time implemetatio for the proposed schemes. I particular, short guard periods are periodically iserted ito the the trasmitted sigal. Durig these guard periods the BS probes each of the users i order to select the most appropriate oe accordig to the adopted schedulig scheme. Durig the probig operatio users perform a set of operatios icludig path estimatio, combied SNRs compariso, ad modulatio idex determiatio followed by CQI feedback which is the basis for the schedulig scheme. After decidig the modulatio mode, the trasmitter ad the selected receiver are cofigured accordigly throughout the subsequet data burst trasmissio. Cocerig the receiver ad due to complexity ad processig power costraits we assume that it ca ot combie more tha L c diversity paths L c <L). As such the receiver chooses the most appropriate paths to reach the desired modulatio idex uder a certai bit error rate BER) costrait. B. Chael Model o achieve multi-path diversity, we cosider a frequecy selective chael. he diversity paths represet the differet resolvable multi-paths i widebad code divisio multiple access W-CDMA or ultra wide bad UWB systems. We assume a block fadig chael, i which, the sum of the guard period ad the data burst time slot is roughly equal to the chael coherece time. hus, durig each time slot,the faded sigal amplitude remais roughly costat durig the data burst ad the previous guard period. Fially, sigals amog users are also assumed to be idepedet ad idetically distributed i.i.d.) ad the sigals of the L multi paths at each user ed are assumed to be i.i.d. ad to experiece Rayleigh type of fadig. C. Adaptive Combiig Diversity combiig requires differet operatios icludig paths estimatio ad sigal processig of the differet diversity paths. hese operatios will lead to extra power cosumptio ad therefore reduce the valuable battery lifetime of mobile termials. o remedy to such problem, differet output-snr orieted schemes [] [9] were developed. MS- GSC is oe of these schemes [8] ad is based o GSC combiig [, chapter 9]. I fact, istead of usig all the L c braches of GSC, the MS-GSC scheme uses the best paths that eable it to reach the target SNR. As a cosequece, MS- GSC ca save a cosiderable amout of processig power by keepig fewer braches active while still reachig the desired diversity gai. D. Adaptive Modulatio he process of adaptive modulatio starts by examiig the chael SNR at the receiver side. his SNR rage is divided ito N + fadig regios, each of them is associated with a particular quadrature amplitude modulatio QAM) sigal costellatio. More specifically, the divided SNR regios are γ<γ+ defied by the followig SNR thresholds: γ where γ is the SNR threshold for -QAM. Give these SNR thresholds ad give the SNR estimated at the receiver, this receiver determies the modulatio idex, such that γ ad iforms via the feedback path the trasmitter about the modulatio idex to be used. Assumig the M-QAM sigallig, the SNR thresholds for a target BER ca be determied usig the followig equatio give i [, Eq 8)] γ<γ + γ = lber ) ); =,,...,N; ) where BER is the specified BER. E. Notatios I the followig sectios, we will adopt the followig otatios ad relatios. We deote by Γ i = i γ l:l the sum of the first i ordered path SNRs γ :L γ :L... γ i:l ad its statistics are give i [, chapter 9]. We also deote by Γ k i the sum of the first i ordered path SNRs for the user k. he joit probability desity fuctio PDF) of Γ i, Γ i ad Γ Lc ca be show to be give by p Γi,Γ i,γ Lc γ i,γ i,γ Lc )=p Γi,γ i:l,γ + γ i,γ i γ i,γ Lc γ i) i Uγ Lc γ i)uγ i γ i ) U ) γ i +L c i)γ i γ i ) γ Lc U i i γi ) γi, ) where U.) is the uit step fuctio ad Γ + i = Lc l=i+ γ l:l. Fially, the joit PDF of Γ i,γ i:l ad Γ + i, p Γ i,γ i:l,γ +,is i give i [, Eq )] for L c = L adi[,eq9)]for L c <L.

3 8 IEEE RANSACIONS ON WIRELESS COMMUNICAIONS, VOL. 8, NO., JULY 9 III. MULIUSER SCHEDULING WIHOU FAIRNESS OF ACCESS CONSIDERAION A. Feedback Efficiet Scheme ) Mode of Operatio: Durig the guard period, the BS starts by radomly probig oe user amog the K users. his user rus the MS-GSC algorithm i order to determie the maximum costellatio size it ca reach with respect to the target BER. More specifically, at the user level, we operate as follows. First, we set γ N as the threshold SNR. If this threshold ca be reached, the receiver sets to N the modulatio idex. Otherwise, it reduces the modulatio idex ad repeats the previous operatio. his process cotiues util the user fids a acceptable modulatio idex or it fails to reach the SNR threshold correspodig to the biary modulatio. I the first case, the user feeds back b = log N) + bits idicatig the modulatio idex, where. is the floor operator. While i the latter case, it feeds back oly oe egative ackowledge NACK) bit. At the BS level, if a positive respose is received from a particular user, this user with the correspodig modulatio idex is serviced. Otherwise, it probes aother user with the same procedure described before. Fially, if the BS does ot fid a acceptable user amog the K active users, it buffers the data. ) Performace Aalysis: Average spectral efficiecy: he startig poit for the computatio of the average spectral efficiecy, η, is the followig relatios Pr[η = ] =Pr[γ Γ <γ + or...or Γ <γ ;...;Γ K <γ &γ Γ K <γ + ], =,...,N ; Pr[η = N]=Pr[Γ γ N or Γ <γ ;Γ γ N ) or... or Γ <γ ;...;Γ K <γ &Γ K γ N ]. Cosiderig the fact that the evets are mutually exclusive ad that the users are experiecig idepedet chael coditios, we ca rewrite ) as follows Pr[η = ] = K PΓLc γ )) k P ΓLc γ + ) P ΓLc γ )), =,...,N ; ) Pr[η = N]= P ΓLc γ N )) K PΓLc γ )) k. Fially, the expressio of the average spectral efficiecy, η, is give by N η= K = P ΓLc γ )) k PΓLc γ + ) P ΓLc γ )) +N P ΓLc γ N ) ) K PΓLc γ ) ) k, ) where P ΓLc.) is the cumulative distributio fuctio CDF) of Γ Lc ad which is give by [, chapter 9]. Average feedback load: he feedback load deotes the umber of users givig feedback i a schedulig operatio. he computatio of the probabilities for the feedback load, F, is give by these relatios Pr[F =] =Pr[Γ γ ], Pr[F = k] =Pr[Γ <γ ;...;Γ k <γ &Γ k γ ], k =,...,K ; Pr[F = K]=Pr[Γ <γ ;...;Γ K <γ ]. akig ito accout the idepedece betwee the radom variables, we ca rewrite these expressios as Pr[F = k] = P ΓLc γ ) ) P ΓLc γ ) ) k, k =,...,K ); Pr[F = K]= P ΓLc γ ) ) P ΓLc γ ) ) K + ) P ΓLc γ ) )K. he fial expressio of the average feedback load, F, is therefore give by F = K k P ΓLc γ ) ) P ΓLc γ ) ) k + K P ΓLc γ ) )K. 8) k= Average feedback rate per schedulig operatio: he feedback rate per schedulig operatio deotes the umber of fed back bits. I this sectio, we cosider that the modulatio idexes are coded i b bits. hus, the radom variable relative to the average feedback rate per schedulig operatio takes the values K or {b + k for k =,...,K }. O the other had, we have Pr [R = K] =Pr[Γ <γ ;...;Γ K <γ ], Pr [R = b + k] =Pr[F = k +], k =,...,K ; Pr [R = b + K ]=Pr[Γ N <γ ;...;Γ K <γ 9) &Γ K γ ]. Cosequetly, the average feedback rate per schedulig operatio, R, is R = K b + k) P ΓLc γ ) ) P ΓLc γ ) ) k ) +K P ΓLc γ )) K. ) Average umber of combied paths per scheduled user: he average umber of combied paths per scheduled user, N c,isgiveby L c N C = l Pr [N c = l], ) where K Pr [N c = l]= Pr[user k selected ad N c = l], ) ad k= N Pr[user k selected ad N c = l]= Pr[user k selected & η = = hese probabilities are calculated i what follows & N c = l]. )

4 BEN HALIMA et al.: JOIN MS-GSC COMBINING AND DOWN-LINK MULIUSER DIVERSIY SCHEDULING 9 Pr[user k selected; η = ; N c =] =Pr[Γ Lc <γ ;...;Γk <γ Lc &Γ k γ &Γ k Lc <γ+ ], =,...,N ; Pr[user k selected; η = N; N c =] =Pr[Γ Lc <γ ;...;Γk <γ Lc &Γ k γn ], Pr[user k selected; η = ; N c = l] =Pr[Γ Lc <γ ;...;Γk <γ Lc &γ Γk l ;Γk l <γ &Γ k Lc <γ+ ], =,.., N ; l =,.., L c ; ) Pr[user k selected; η = N; N c = l] =Pr[Γ Lc <γ ;...;Γk <γ Lc &Γ k l γn &Γ k l <γn ], l =,...,Lc ; Pr[user k selected; η = ; N c = L c]=pr[γ Lc <γ ;...;Γk <γ Lc &γ Γk Lc <γ+ &Γ k Lc <γ ], =,...,N ; Pr[user k selected; η = N; N c = L c]=pr[γ Lc <γ ;...;Γk <γ Lc &Γ k Lc γn ]. Based o the fact that the users are experiecig i.i.d. chael coditios, we have Pr [N c =] = K PΓLc γ ) ) k Pr[N c = l] [ N = γ + + P Γ γ N ) )], = K [ PΓLc γ ) ) k N γ + γ = p Γ,Γ,Γ Lc x, y, z)dxdydz p Γi,Γ i,γ Lc x, y, z)dxdydz) + P Γl γ N ) P Γl γ N ) )], l =,...,L c ); Pr[N c = L K c]= PΓLc γ ) ) k [ N = + γ p ΓLc,Γ Lc x, y)dxdy + P ΓLc γ N ) P ΓLc γ N ) )]. Average umber of estimated paths per user: From the mode of operatio of the MS-GSC ad the proposed scheme, it is clear that the average umber of estimated paths per user, N E, ca be writte as N E = L F. ) K B. Badwidth Efficiet Scheme ) Mode of Operatio: Durig the guard period, the BS starts by radomly probig oe user amog the K users. his user rus the MS-GSC algorithm settig γ N as the threshold SNR which is the SNR eablig to reach the maximum costellatio size uder a certai BER costrait. If this user succeeds i providig the maximum costellatio size, it feeds back oe bit correspodig to a positive ackowledge ACK) ad thus it will be chose as the served user durig the ext data burst. Otherwise, it feeds back b = log N ) + bits iformig the BS of the maximum costellatio size it ca reach after combiig all the GSC L c braches. I the secod case, the BS probes aother user which performs the same operatio as the previous user. his procedure cotiues util the BS fids a user havig a combied SNR greater tha the γ N or all the users fail to achieve that coditio. I the latter case, the BS chooses the user with the maximum provided costellatio size, ad if there is a outage, it buffers the data. ) Performace Aalysis: Average spectral efficiecy: he differet probabilities correspodig to the differet modulatios idexes are give by Pr [η = ] =Pr[Γ Lc <γ+ ;...;Γ K Lc <γ+ &γ Γ L <γ + C or Γ Lc <γ+ ;...;Γ K Lc <γ+ &Γ Lc <γ ;Γ L γ C or...orγ Lc <γ+ ;...;Γ K Lc <γ+ &Γ Lc <γ ;...;ΓK <γ Lc ;ΓK L γ ) C ], =,...,N ; Pr [η = N]=Pr[Γ L γ N C or Γ L <γ N C ;Γ L γ N C or...or Γ L <γ N C ;...;ΓK L <γ N C ;ΓK L γ N C ]. hese relatios, whe cosiderig the idepedece betwee the users ad the fact that the evets are mutually exclusive, ca be writte as Pr [η = ] = K ) k ) K k P ΓLc γ ) P ΓLc γ + ) P ΓLc γ + ) P ΓLc γ ), ) =,...,N ; 8) ) K ) k. Pr [η = N]= P ΓLc γ N ) P ΓLc γ N ) Fially the average spectral efficiecy, η, for the scheme ca be writte as N η= K = P ΓLc γ )) k PΓLc γ + ) ) K k P ΓLc γ + ) P ΓLc γ )) +N P ΓLc γ N PΓLc γn )) k. 9) )) K Average feedback load: he startig poit to obtai the average feedback load, F, expressio is the followig relatios Pr [F =] =Pr[Γ γ N ], Pr [F = k] =Pr[Γ <γ N ;...;Γ k <γ N &Γ k γ N ], k =,...,K ; Pr [F = K]=Pr[Γ <γ N ;...;Γ K <γ N ]. hese relatios ca be rewritte as ) Pr [F =] = P ΓLC γ N ), Pr [F = k] = P ΓLc γ N ) ) P ΓLc γ N ) ) k ), k =,...,K ; Pr [F = K]= P ΓLc γ N ) ) P ΓLc γ N ) ) K + P ΓLc γ N ) )K. As a cosequece, the fial average feedback load, F, ca be writte as F = K k P ΓLc γ N ) ) P ΓLc γ N ) ) k k= +K P ΓLc γ N ) )K. ) Average feedback rate per schedulig operatio: As i the scheme, we defie b as the umber of bits ecessary to code the modulatio idices. I this case, the average feedback rate per probig operatio takes the values Kb or {kb + for k =,...,K }. he probabilities associated with the feedback rate ca be writte as Pr [R = Kb ] =Pr[Γ <γ N ;...;Γ K <γ N ], Pr[R = kb +] =Pr[F = k +], k =,...,K ; Pr[R =K )b +]=Pr[Γ N <γ N ;...;Γ K <γ N ) &Γ K γ N ].

5 IEEE RANSACIONS ON WIRELESS COMMUNICAIONS, VOL. 8, NO., JULY 9 Fially, the average feedback rate per schedulig operatio ca be writte as R = K kb +) P ΓLc γ N )) P ΓLc γ N )) k +Kb PΓLc γ N )) K. ) Average umber of combied paths per scheduled user: he average umber of combied paths per scheduled user, N c,is give by where L c N C = l Pr[N c = l], ) K Pr [N c = l]= Pr[user k is selected& N c = l], l =,.., L c ; ) k= which is give by Pr[user k is selected ad N c =]=Pr[Γ L c <γ N ;...;Γk L c <γ N &Γ k γn ], Pr[user k is selected ad N c = l] =Pr[Γ L c <γ N ;...;Γk L c <γ N ) &γ N Γk l &Γ k l <γn ], l =,...,Lc ; ad Pr [N c = L c]=pr[γ L c γ N ;Γ L c <γn &Γ L c γ N ;Γ L c <γn or Γ L c <γ N or...or Γ L c <γ N ;...;ΓK L c <γ N 8) &Γ K L c γ N ;ΓK L c <γn or {Γ L c <γ N ;...;ΓK L c <γ N &γ Γ L c <γ N or Γ L c <γ ; γ Γ L c <γ N or...or Γ L c <γ ;...;ΓK L c <γ ; γ ΓK L c <γ N )}]. As a cosequece, we have Pr[N c =] = K PΓLc γn ) ) k PΓ γ N ) ), Pr[N c = l] = K PΓLc γn ) ) k P Γl γ N ) P Γl γ N ) ), l =,...,L c ; Pr[N c = L K c]= PΓLc γn ) ) 9) k PΓLc γn ) P ΓLc γn ) ) + K PΓLc γ ) ) k PΓLc γn ) ) K k P ΓLc γ N ) P ΓLc γ ) ). Average umber of estimated paths per user: Similar to the previous sectio, we ca write the average umber of estimated paths per user, N E, as follows N E = L F. ) K C. Badwidth Efficiet Power Greedy Scheme ) Mode of Operatio: I a first step, the BS probes the users for the highest modulatio idex. I the begiig, the BS starts by radomly probig oe user amog the K users. his user rus the MS-GSC algorithm settig γ N as the threshold SNR. If this user succeeds i providig the maximum costellatio size, it feeds back oe bit correspodig to a ACK ad it is served by the BS i the ext data burst. Otherwise, it feeds back oe bit correspodig to a NACK. I the latter case, the BS probes aother user which rus also the MS-GSC algorithm with the threshold correspodig to the maximum costellatio size ad the feeds back the appropriate ackowledgemet. his operatio cotiues util the BS fids a user havig a combied SNR greater tha γ N or all the users fail to achieve that coditio. I the latter case the BS reduces the costellatio size i.e. it sets γ N as the ew threshold SNR i our case) ad repeats the probig operatio as described before for this ewly set threshold. he BS cotiues probig users util a acceptable oe with appropriate chael coditios is selected. I the case of a outage, the BS buffers the data. ) Performace Aalysis: Average spectral efficiecy: Based o the mode of operatio of this scheme, its average spectral efficiecy is exactly the same as the average spectral efficiecy of the scheme. Average feedback load: As for the average spectral efficiecy, the average feedback load for this scheme is give by the same relatio as the scheme. Average feedback rate per schedulig operatio: Based o the mode of operatio of this scheme, the radom variable correspodig to the umber of bits per schedulig operatio takes the values {K + k, =,...,N ; k =,...,K}. he differet probabilities associated with the feedback rate are give by Pr[R = k] Pr[R = K + k]=pr[γ L c <γ N + Pr[R = NK] =Pr[Γ L c <γ N ;...;Γk L c <γ N ;Γk L c γ N ], ;...;Γ K L c <γ N + &Γ L c <γ N ;...;Γ k L c <γ N &Γ k L c γ N ) ], =,.., N ; k =,.., K ; =Pr[Γ L c <γ ;...;ΓK L c <γ ; γ ΓK L c <γ or Γ L c <γ ;...;ΓK L c <γ ]. Cosiderig the fact that the differet radom variables are i.i.d., we have Pr[R = k] =P ΓLc γ N ) k ) P ΓLc γ N ) ), Pr[R = K + k]=p ΓLc γ N + ) K k) P ΓLc γ N ) k ) P ΓLc γ N + ) P ΓLc γ N ) ), ) =,...,N ; k =,...,K ; Pr[R = NK] =P ΓLc γ ) K ) P ΓLc γ ) P ΓLc γ ) ) +P ΓLc γ ) K. ad the resultig average feedback rate is thus R = N = k= K K + k)pr[r = K + k]. ) Average umber of combied paths per scheduled user: he average umber of combied paths per scheduled user, N c,isgiveby where Pr[N c = l] = with L c N C = l Pr [N c = l], ) N Pr[η = ; N c = l], l =,...,L c ; ) = Pr[η = ; N c = l] = where K Pr[user k selected & η = & N c = l], ) k=

6 BEN HALIMA et al.: JOIN MS-GSC COMBINING AND DOWN-LINK MULIUSER DIVERSIY SCHEDULING 9 Pr[user k selected & η = ; N c =] Pr[user k selected & η = N; N c =] Pr[user k selected & η = ; N c = l] Pr[user k selected & η = N; N c = l] Pr[user k selected & η = ; N c = L c]=pr[γ Lc <γ+ =Pr[Γ Lc <γ+ ;...;Γ K Lc <γ+ &Γ Lc <γ ;...;Γk <γ Lc &Γ k γ &Γ k Lc <γ+ ], =,.., N ; =Pr[Γ Lc <γn ;...;Γk <γ N Lc &Γ γ N ], =Pr[Γ Lc <γ+ ;...;Γ K Lc <γ+ &Γ Lc <γ ;...;Γk <γ Lc &Γ k l γ ;Γk l <γ &Γ k ) Lc <γ+ ], =,.., N ; =Pr[Γ Lc <γn ;...;Γk <γ N Lc &Γ l γ N ;Γ l <γ N ], &Γ Lc <γ ;...;Γk Lc ;...;Γ K Lc <γ+ <γ &γ Γk Lc <γ+ &Γ k Lc <γ ], =,.., N ; Pr[user k selected & η = N; N c = L c]=pr[γ Lc <γn ;...;Γk <γ N Lc &Γ Lc γ N ;ΓLc <γn ]. Fially, whe we cosider the fact that the users experiece i.i.d. chael coditios, it ca be show that Pr [N c =] = N Pr [N c = l] = N K = + γ PΓLc γ ) ) k PΓLc γ+ ) ) K k + P Γ γ N ) ) K γ K = + K p Γ,Γ,Γ Lc x, y, z)dxdydz PΓLc γn ) )k, PΓLc γ ) ) k PΓLc γ+ ) ) K k + p Γi,Γ i,γ Lc x, y, z)dxdydz PΓLc γn ) ) k P Γl γ N ) P Γl γ N ) ), l =,..., ; Pr[N c = L N K c]= PΓLc γ ) )k P ΓLc γ+ ) ) K k = + γ p ΓLc,Γ Lc x, y)dxdy + P ΓLc γ N ) P ΓLc γ N ) ) K 8) PΓLc γn ) )k. Average spectral efficiecy bit/s/hz) 8 MRC HSE VSE Average SNR per Path db) Fig.. Average spectral efficiecy versus the average SNR for the MRC,,, HSE,, ad VSE schemes whe L=, L c =, K=, N=8, ad BER =.. Average umber of estimated paths per user HSE VSE Average SNR per Path db) Fig.. Average umber of estimated paths per user versus the average SNR for the,, HSE,, ad VSE schemes whe L=, L c =, K=, N=8, ad BER =.. D. Numerical Results I this sectio, we make a compariso betwee the three proposed schemes as well as a compariso betwee these ew schemes ad some recetly published schemes. I Fig., equatios ), 9), [, Eq )], ad [, Eq )] are plotted, the MRC curve is obtaied usig simulatio. Cocerig Fig., equatios ), ), [, Eq 8)], ad [, Eq )] are plotted. Equatios 8), ad ) are plotted i Fig.. I Fig., equatios ), ), ad ) are represeted, the MRC curve is represeted usig simulatio. Fially, we plot i Fig. equatios ), ), ad ). Fig. shows that the spectral efficiecy of the scheme is lower tha the spectral efficiecy of the horizotal switch ad examie trasmissio HSE) []. Fig. shows that the average umber of estimated paths per user for the scheme is greater tha that of the vertical switch ad examie trasmissio VSE) []. I fact, accordig to the mode of operatio of MS-GSC, it is ecessary to estimate ad rak all the L available paths wheever we probe a ew user, cotrary to what is performed i the VSE which will estimate just as may paths as eeded but ot ecessarily combie the best oes). Fig. shows also that the ad schemes achieve the same spectral efficiecy ad the best possible i compariso with HSE ad. However, this comes at the expese of a high feedback load ad umber of estimated paths per user as a illustratio i Fig. ad Fig.. Seeig that the ad schemes provide the maximum spectral efficiecy, we also compare them with a scheme havig the same mode of operatio but which adopts MRC istead of MS-GSC. I other words, whe a user is probed, it performs full L-paths MRC combiig operatio i order to determie the maximum spectral efficiecy that ca be reached. As it is show i Fig., the performace of the three schemes are ot very differet i terms of spectral efficiecy. However, as illustrated i Fig., the ad schemes offer cosiderable savigs i terms of umber of combied paths whe compared to MRC. I additio it is clear from Fig. that the scheme reduces cosiderably the umber of combied paths compared to the scheme ad is as such iterestig from a battery power efficiecy perspective. Fially, we ca see from Fig. that the scheme achieves best performace from a feedback perspective but this comes at the expese of a cosiderable loss i terms of spectral efficiecy, as illustrated i Fig.. able I summarizes the key results preseted i this sectio.

7 IEEE RANSACIONS ON WIRELESS COMMUNICAIONS, VOL. 8, NO., JULY 9 Average feedback load 9 8 Average SNR per Path db) Fig.. Average feedback load versus the average SNR for the,, ad schemes whe L=, L c =, K=, N=8, ad BER =.. Average feedback rate per schedulig operatio 8 Average SNR per Path db) Fig.. Average feedback rate per schedulig operatio versus the average SNR for the,, ad schemes whe L=, L c =, K=, N=8, ad BER =.. Average umber of Combied paths MRC Average SNR per Path db) Fig.. Average umber of combied paths per scheduled user versus the average SNR for the,,, ad MRC schemes whe L=, L c =, K=, N=8, ad BER =.. IV. ACHIEVING FAIRNESS BEWEEN HE SCHEDULED USERS A. Motivatio I the three multiuser dow lik schedulig schemes proposed i the previous sectio, the operatio mode is based o probig the users i a sequetial maer. More specifically, ABLE I SUMMARY OF HE KEY PERFORMANCE FEAURES OF HE PROPOSED SCHEMES Schedulig scheme Key performace features Low spectral efficiecy Low feedback load Low umber of estimated paths High power savigs Low feedback rate High spectral efficiecy High feedback load High umber of estimated paths Worst performace i terms of power savigs High feedback rate High spectral efficiecy High feedback load High umber of estimated paths Good performace i terms of power savigs Highest feedback rate accordig to the feedback iformatio, the BS selects the appropriate user havig acceptable chael coditios eablig at least BPSK trasmissio) which is the case for the scheme or havig the best chael coditios which is the case for the ad schemes. Examiig the mode of operatio of the schemes discussed before, there is o guaratee i terms of fairess of access. I fact, with such schemes, some users may ed up beig deprived from service after several schedulig rouds ad subjected as such to the well kow starvatio problem. Based o this observatio, we desig a scheduler that is able to esure a certai short term fairess betwee the i.i.d. users, while simultaeously employig opportuistic schedulig strategies to icrease the total system throughput by selectig users with high-quality chaels whe possible. Differet fair schedulig schemes such as the proportioal fair algorithm [] were desiged to achieve fairess betwee o i.i.d. users. I this paper, the proposed system is based o the three schedulig schemes preseted i the previous sectio ad adopts the idea of ehaced equal access EEA) schedulig policy suggested i []. More specifically, i the three schemes suggested i sectio whe a user is scheduled i a time slot it is removed from the schedulig operatio i the ext roud util all the remaiig users have bee scheduled. herefore, with EEA all the users are served i the miimum possible time. However, this improvemet of fairess comes at the expese of a certai degradatio i the performaces of the system maily i terms of spectral efficiecy. hus, we study i the followig how the performace of our proposed schemes are modified whe EEA is adopted. I particular, we derive ew expressios for the average spectral efficiecy per time slot ad the average feedback load per time slot. hese results are validated usig Mote-Carlo simulatios. B. Mode of Operatio of he Proposed Scheme he mode of operatio of this scheme is based o the idea of EEA schedulig policy suggested i [].o achieve the desired fairess of access betwee the differet users the proposed scheme operates as follows:. I the first time slot, the BS performs the probig operatio accordig to oe of the schemes preseted i sectio

8 BEN HALIMA et al.: JOIN MS-GSC COMBINING AND DOWN-LINK MULIUSER DIVERSIY SCHEDULING III o the K activeavailableusers.. Oce oe user user ) is selected, i the ext time slot, this user is removed from the pool of the users to be scheduled ad the BS performs the same previous operatio o the K remaiig users util selectig aother user user ) amog them.. he BS repeats the previous operatios for the ext time slots util the K users are scheduled. Whe adoptig the proposed scheme we are sure to achieve a eve repartitio of resources betwee the differet users. Moreover, such a scheme, ad uder acceptable chael coditios, guaratees a maximum iter-access time of K time slots. Note that we refer by acceptable chael coditios to the fact that i each time slot of K cosecutive time slots we have at least oe of the probed users able to reach the BPSK threshold SNR. C. Performace Aalysis he schedulig process of the system is modeled as a discrete-state Markov process Markov chai) where the umber of states at each schedulig process is equal to the time slot order umber l. We defie λi) as λi) = P ΓLc γ ) ) i, 9) ad we itroduce the probability Θk, l) that i the l th time slot the umber of users is equal to k which is give i [] by l λk)), k = K; [ λk +)) Θk +,l ) Θk, l)= ] ) +λk)θk, l ), K l )) <k<k; ) l λk i), k = K l ). i= Average spectral efficiecy per time slot: Based o the probability that i the l th time slot the umber of users is equal to k, the average spectral efficiecy for the l th time slot ca be writte as η K, if l =; ηl) = l ) η K j ) ΘK j ),l), if <l K; j= where η k is give by N k P ΓLc γ )) i = i= P ΓLc γ + ) P ΓLc γ )) +N P ΓLc γ N )) k PΓLc γ )) i,, η k = i= N k P ΓLc γ )) i P ΓLc γ + ) ) ) k i = i= P ΓLc γ + ) P ΓLc γ )) +N P ΓLc γ N )) k PΓLc γ N )) i. / i= Fially, the average spectral efficiecy per time slot, η, for L time slots ad K users ca be writte as η = L L ηl). ) Average feedback load per time slot: Similarly to what was doe before ad based o the probability that i the l th time slot the umber of users is equal to k, the average feedback load per time slot, F,forK users ad L time slots is F = L F l), ) L where F K, if l =; F l)= l F K j ) ΘK j ),l), if <l K, ) j= ad k i P ΓLc γ )) P ΓLc γ )) i i= +k P ΓLc γ F k = )) k, ; k i ) P ΓLc γ N )) P ΓLc γ N )) i i= +k P ΓLc γ N)) k, /. Probability that K Users Have Access to the Chael i K ime Slots: he evaluatio of such probability gives us a idea about the rage of SNRs i which the maximum time of access betwee two servig operatio for all the users is equal to K time slots. he probability of havig all the K users served i K time slots is give by P = K PΓLc γ )k). ) k= D. Numerical Results I this sectio, we preset some umerical ad simulatio results of the ew spectral efficiecy ad feedback load of the system. We compare the ew performaces of the schedulig schemes studied i sectio III after adoptio of the EEA idea whe schedulig users. Fig. shows the average spectral efficiecy per time slot for the system. It is clear that the cost of achievig a better fairess betwee the scheduled users is some loss i spectral efficiecy. I fact, i the medium SNR regio, the spectral efficiecy per time slot is reduced cosiderably for the ad schemes but for the scheme, there is practically o loss i terms of spectral efficiecy which is explaied by the fact that such a scheme does ot take ito cosideratio spectral efficiecy as the primary goal. Fig. illustrates the average feedback load per time slot for the system. he remark to be made is that the EEA reduces the feedback load per time slot. Ideed, Fig. shows that the feedback load is reduced by almost half of its origial value i the low to medium SNR regios but remais the same i the high SNR regio this is due to the fact that i these regios this value is the optimal oe). Such a tred ca be explaied by reductio i the umber of users to be scheduled as the schedulig process progresses whe EEA is adopted.

9 IEEE RANSACIONS ON WIRELESS COMMUNICAIONS, VOL. 8, NO., JULY 9 Average spectral efficiecy per chael bit/s/hz) 9 8 / without EEA / with EEA without EEA with EEA Average SNR per Path db) Probabilty of havig satisfactio of K users i K Is... Average SNR db) Fig.. Average spectral efficiecy per time slot versus the average SNR for the three schemes with EEA ad without EEA whe L =, L =, L c =, K =, N =8,adforBER =.. Fig. 8. Probability of havig access of all the K users i K time slots versus the average SNR for the three schemes whe L =, L =, L c =, K =, N =8,adBER =.. Average Feedback load per time slot 9 8 / / with EEA with EEA Average SNR db) Fig.. Average feedback load per time slot versus the average SNR for the three schemes with EEA ad without EEA whe L =, L =, L c =, K =, N =8,adforBER =.. Fially, Fig. 8 shows that startig from db, all the K users are served i K time slots, whe L = K. Such tred ca be explaied by the fact that i this SNR regio, there is a high probability to have at least oe user with a SNR higher tha the BPSK threshold SNR. his also meas that i this case, the maximum waitig time to have a other access to the chael is K time slots. o coclude, the previous aalysis shows that the performace pealties of the EEA i terms of spectral efficiecy do ot seem sigificat compared to the gai achieved i terms of feedback load ad probability of fair access. V. CONCLUSION We preseted i this paper three multiuser schedulig schemes based o the MS-GSC combiig. We evaluated their performace ad compared this performace to that of the HSE, VSE ad MRC schemes. It was show, that the achievesa miimumfeedback load, while the ad schemes achieve the best spectral efficiecy. It was also show that there is ot a sigificat differece betwee the, ad MRC schemes i terms of spectral efficiecy. However, the two former schemes clearly outperform MRC i terms of average combied paths with a certai advatage to the scheme. he proposed schemes were used i cojuctio with EEA to achieve better short term fairess of access betwee the i.i.d. users. I this case, the etwork scheduler removes the scheduled user i the ext time slots util servig all the available users i the cell. It was show that i good chael coditios, EEA guaratees for each user to have a aother access to the chael after maximally K time slots. It was also show that this improvemet i fairess is accompaied by a feedback load gai, but at the expese of a reductio i average spectral efficiecy per time slot. REFERENCES [] D. Gesbert ad M.-S. Alouii, Selective multi-user diversity, i Proc. IEEE Iteratioal Symp. o Sigal Processig ad Iformatio echology ISSPI ), Dresde, Germay, Dec., pp.. [] B. Holter, M.-S. Alouii, G. E. Øie, ad H.-C. Yag, Multiuser switched diversity trasmissio, i Proc. IEEE Veh. echol. Cof. VC ), vol., Los Ageles, CA, USA, Sept., pp. 8. [] Y. A. Harthi, A. ewfik, ad M.-S. Alouii, Multiuser diversity with quatized feedback, IEEE ras. Wireless Commu., vol., pp., Ja.. [] M.-S. Alouii ad A. J. Goldsmith, Adaptive modulatio over Nakagami fadig chaels, Kluwer J. Wireless Commuicatios, vol., pp. 9, May. [] A. Goldsmith ad S.-G. Chua, Adaptive coded modulatio for fadig chaels, IEEE ras. Commu., vol., pp. 9, May 998. [] K. Hole, H. Holm, ad G. E. Øie, Adaptive multidimesioal coded modulatio over flat fadig chaels, IEEE J. Select. Areas Commu., vol. 8, pp. 8, July. [] M.-S. Alouii ad H.-C. Yag, Miimum estimatio ad combiig geeralized selectio combiig MEC-GSC), i Proc. IEEE It. Sym. Iform. heory ISI ), Adelaide, Australia, Sept., pp.. [8] H.-C. Yag, New results o ordered statistics ad aalysis of miimumselectio geeralized selectio combiig GSC), IEEE ras. Wireless Commu., vol., pp. 8 88, July. [9] H.-C. Yag ad M.-S. Alouii, MRC ad GSC diversity combiig with a output threshold, IEEE ras. Veh. echol., vol., pp. 8 9, May. [] M. K. Simo ad M.-S. Alouii, Digital Commuicatio over Geeralized Fadig Chaels: A Uified Approch to Performace Aalysis. New York, NY: Joh Wiley & Sos,. [] H.-C. Yag, N. Belhaj, ad M.-S. Alouii, Performace aalysis of joit adaptive modulatio ad diversity combiig over fadig chaels, IEEE ras. Commu, vol., pp. 8, Mar.. [] S. Choi, M.-S. Alouii, K. A. Qaraqe, ad H.-C. Yag, Soft hadover overhead reductio by RAKE receptio with figer replacemet, i Proc. IEEE Iteratioal Cof. o Commu. Systems ICCS ), Sigapore, Nov.. [] K.-H. Park, Y.-C. Ko, ad M.-S. Alouii, Joit adaptive combiig ad multiuser dow-lik schedulig, IEEE ras. Veh. echol., vol., pp , Sept. 8.

10 BEN HALIMA et al.: JOIN MS-GSC COMBINING AND DOWN-LINK MULIUSER DIVERSIY SCHEDULING [] P. Viswaath, D. N. C. se, ad R. Loria, Opportuistic beamformig usig dump atea, IEEE ras. Iform. heory, vol., pp. 9, Jue. [] M. Johasso, Diversity-ehaced equal access-cosiderable throughput gais with -bit feedback, i Proc. IEEE Workshop o Sigal Processig Advaces i Wireless Commuicatios SPAWC ), Lisbo, Portugal, July, pp.. [] Y. A. Harthi, A. ewfik, ad M.-S. Alouii, Multiuser diversityehaced equal access with quatized feedback i multicarrier OFDM systems, i Proc. IEEE Vehicular echology Coferece, VC - Fall), Dallas, X, Sept., pp. 8. Slim Be Halima was bor i Sfax, uisia. He received the egieerig Diploma from the uisia Polytechic School, La Marsa, uisia, i. He received the master degree i Digital elecommuicatio Systems from Pierre et Marie Curie Uiversity, Paris, Frace, i 8. Mohamed-Slim Alouii S 9, M 98, SM, F 9) was bor i uis, uisia. He received the Ph.D. degree i electrical egieerig from the Califoria Istitute of echology Caltech), Pasadea, CA, USA, i 998. He was a Associate Professor with the departmet of Electrical ad Computer Egieerig of the Uiversity of Miesota, Mieapolis, MN, USA. Sice July, he has bee with the Electrical ad Computer Egieerig Program of the exas A&M Uiversity at Qatar, Educatio City, Doha, Qatar, where his curret research iterests iclude the desig ad performace aalysis of wireless commuicatio systems. Khalid A. Qaraqe M 9-S ) was bor i Bethlehem. Dr Qaraqe received the B.S. degree i EE from the Uiversity of echology, Baghdad i 98, with hoors. He received the M.S. degree i EE from the Uiversity of Jorda, Jorda, i 989, ad he eared his Ph.D. degree i EE from exas A&M Uiversity, College Statio, X, i 99. From 989 to Dr Qaraqe has held a variety positios i may compaies ad he has over years of experiece i the telecommuicatio idustry. Dr Qaraqe has worked for Qualcomm, Ead Desig Systems, Cadece Desig Systems/ality Corporatio, SC, SBC ad Ericsso. He has worked o umerous GSM, CDMA, WCDMA projects ad has experiece i product developmet, desig, deploymets, testig ad itegratio. Dr Qaraqe joied the departmet of Electrical Egieerig of exas A&M Uiversity at Qatar, i July, where he is ow a seior associate professor. Dr Qaraqe research iterests iclude commuicatio theory ad its applicatio to desig ad performace, aalysis of cellular systems ad idoor commuicatio systems. Particular iterests are i the developmet of WCDMA ad broadbad wireless commuicatios ad diversity techiques.