Proportional Fair Scheduling Algorithm in OFDMA-Based Wireless Systems with QoS Constraints

Transcription

1 30 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 1, FEBRUARY 2010 Proportonal Far Schedulng Algorthm n OFDMA-Based Wreless Systems wth QoS Constrants Tolga Grc, Chenx Zhu, Jonathan R. Agre, and Anthony Ephremdes Abstract: In ths work we consder the problem of downlnk resource allocaton for proportonal farness of long term receved rates of data users and qualty of servce for real tme sessons n an OFDMA-based wreless system. The base staton allocates avalable power and subchannels to ndvdual users based on long term average receved rates, qualty of servce QoS) based rate constrants and channel condtons. We formulate and solve a jont bandwdth and power optmzaton problem, solvng whch provdes a performance mprovement wth respect to exstng resource allocaton algorthms. We propose schemes for flat as well as frequency selectve fadng cases. Numercal evaluaton results show that the proposed method provdes better QoS to voce and vdeo sessons whle provdng more and far rates to data users n comparson wth exstng schemes. Index Terms: Heterogeneous traffc, IEEE , orthogonal frequency dvson multple access OFDMA), qualty of servce QoS), proportonal farness, resource allocaton, WMax. I. INTRODUCTION Increasng number of users demandng wreless Internet access and a growng number of wreless applcatons requre hgh speed transmsson and effcent utlzaton of system resources such as power and bandwdth. Recently, technologes lke WMax based on IEEE standard) [1] and Long Term Evoluton of 3GPP LTE) [2] are developed to address these challenges. Orthogonal frequency dvson multplexng OFDM), a multcarrer transmsson technque, s the preferred transmsson technology n next generaton broadband wreless access networks. It s based on a large number of orthogonal subcarrers, each workng at a dfferent frequency. OFDM s orgnally proposed to combat ntersymbol nterference and frequency selectve fadng. However, t also has a potental for a multple access scheme, where the subcarrers are shared among the competng users. Wthn orthogonal frequency dvson multplexng access OFDMA) framework, the resource allocated to the users comes n three dmensons: Tme slots, frequency, and power. Ths requres the scheduler to operate wth hgher degree of freedom and more flexblty, and potentally hgher multplexng capacty. Ths also ncreases the dmensonalty of the resource allocaton problem and makes the problem more Manuscrpt receved Aprl 04, 2008; approved for publcaton by Hlang Mnn, Dvson II Edtor, September 21, T. Grc s wth the Dept. of Electrcal and Electroncs Eng. of TOBB Unversty of Economcs and Technology, Ankara, Turkey, emal: tgrc@etu.edu.tr. C. Zhu and J. R. Agre are wth the Fujtsu Labs of Amerca at College Park, 8400 Baltmore Ave., Sute 302, College Park, MD, 20740, USA, emal: {chenx.zhu, jonathan.agre}@us.fujtsu.com. A. Ephremdes s wth the Department of Electrcal Engneerng and Insttute of Systems Research at Unversty of Maryland, College Park, MD, 20740, USA, emal: etony@umd.edu /10/$10.00 c 2010 KICS nvolved. We plan to develop schedulng algorthms fully takng advantage of the degree of freedom nherent to OFDMA system. Our goal s to fnd multcarrer proportonal far schemes that also satsfy heterogeneous stablty and delay requrements. There are three man ssues that need to be consdered n multple access resource allocaton. The frst one s spectral effcency, whch means achevng maxmum total throughput wth avalable bandwdth and power. In tme dvson multple access TDMA) transmsson, t s acheved by always allowng the user wth the best channel to transmt [3]. In OFDM multple subcarrers possbly experence dfferent fadng levels, whch makes spectral effcency a more complex problem. The second ssue s farness. In [3], t s studed that f the channel condtons are ndependent and dentcally dstrbuted..d.), all users eventually wll get the same servce, hence farness s mantaned [4]. On the other hand f the dstance attenuatons of users are dfferent then some users wll defntely get more servce, and some others won t get any servce. Therefore schedulng algorthms have to be proposed to provde farness among nodes. The thrd mportant ssue s satsfyng qualty of servce QoS) requrements. An example for QoS requrements can be bounds on delay or short term receved rate for real tme applcatons. In broadband wreless access systems proportonal farness mantans a good tradeoff between spectral effcency and farness. The proportonal far scheduler s proposed by Jalal et al. n [1] n the context of hgh data rate HDR) system. Ths system s orgnally desgned for data applcatonse.g., FTP and Internet). Bascally, the system s a TDMA system where a user s scheduled to transmt at each tme slot. A proportonal far schedule s such that any postve change of a user n rate allocaton must result n a negatve overall change n the system. Proportonal far resource allocaton also corresponds to an allocaton that maxmzes the sum of logarthms of receved rates [5]. Proportonal far schedulng was recently studed for multcarrer systems n [5], [6], [7], and [8]. In [5], proportonal farness formulaton s extended from sngle to multple channel systems. However t dd not nclude power control and no algorthm was proposed to fnd the optmum bandwdth allocaton. In [6], [7], and [9], proportonal far schedulng s addressed for a sngle tme nstant, that s, ther objectve was to maxmze the sum of logarthms of nstantaneous rates, rather than the long term receved rates. The recent paper [8] proposes an algorthm for proportonal farness of long term rates n OFDM, however t doesn t consder power control. Besdes n all of these works supportng real tme traffc was not addressed. The schedulng rules do not apply suffcently to dfferent QoS requrements and heterogeneous traffc. Most of the prevous work on resource allocaton for heterogeneous traffc requrements followed and extended the ap-

2 GIRICI et al.: PROPORTIONAL FAIR SCHEDULING ALGORITHM IN OFDMA proach n [10]. In sngle channel systems, largest weghted delay frst LWDF) proposed n [10] s shown to be throughput optmal. Accordng to ths scheme at each tme nstant, the user that maxmzes a combnaton of head-of-lne packet delay, averaged receved servce and current achevable rate s served. Ths scheme s extended for OFDMA-based multchannel systems n [11], [12], and [13]. In [11] average delay that can be estmated by usng average queue sze and arrval rate) s used nstead of head-of-lne delay. Usng head-of-lne delay s shown n [12] to perform better than usng average delay. All these proposed schemes assume unform power. In fact, power control can be useful n mprovng the performance of these schemes. Fnally, n [13] power control s consdered, however, power allocated to a user s proportonal to ts number of subchannels, therefore power control s stll not fully exploted. These schemes are explaned n more detal n Secton III. Our man contrbuton n ths work s formulatng a jont bandwdth/power optmzaton framework that takes more advantage of power control. OFDMA based resource allocaton has been studed also wthout the proportonal farness objectve n [14] [19]. The works [14] and [20] propose subcarrer and bt allocaton algorthms that satsfy rate requrements of users wth mnmum total power. The authors n [15], [17], [18], and [19] address maxmzng total and weghted throughput subject to power and subcarrer constrants and do not address real tme traffc. The work n [16] ntroduces a proportonal rate constrant, where the rates of ndvdual users have to be n certan constant proportons n order to mantan farness. However, ths also doesn t guarantee any short or long term transmsson rates. The system consdered n ths work s motvated by the recent IEEE standard that defnes the ar nterface and medum access control MAC) specfcatons for wreless metropoltan area networks. Such networks ntend to provde hgh speed voce, data and on demand vdeo streamng servces for end users. IEEE standard s often referred to as WMax and t provdes substantally hgher rates than cellular networks. Besdes t elmnates the costly nfrastructure to deploy cables, therefore t s becomng an alternatve to cabled networks, such as fber optc and DSL systems [1]. In IEEE e, n order to ease the resource allocaton process the subcarrers are grouped nto subchannels. There are two man classes of subchannelzaton methods. The frst class s adaptve modulaton and codng AMC). In ths method a number of carrers adjacent on the frequency spectrum are grouped nto a band AMC subchannel. In a multpath fadng channel dfferent subchannels experence dfferent levels of fadng. Achevable rates can be maxmzed by adjustng the modulaton and codng rate accordng to the fadng level for each subchannel. The second class ncludes partal use of subchannels PUSC) and full use of subchannels FUSC). They are dversty permutaton schemes that dstrbute the sub-carrers of a subchannel pseudo-randomly n a wde frequency band. They provde frequency dversty and nter-cell nterference averagng. Ths releves the performance degradaton due to fast fadng moble envronments. PUSC s the default mode of subchannelzaton and s more sutable for moble users than AMC. Here, because of the averagng effect, we can also assume that each subchannel experences the same fadng wth respect to a user. Ths decreases the complexty of resource allocaton algorthms because the problem becomes how many subchannels nstead of whch ones. The amount of feedback s also decreased sgnfcantly because the BS doesn t need to track each subchannel separately. Lterature lacks QoS-based resource allocaton algorthms for ths scenaro, therefore we especally focus our current work on the second class of subchannels,.e., PUSC/FUSC. Our proposed scheme frst uses a varaton of LWDF schedulngpolcy tofnd the rate requrements of real tme users, whch s explaned n Secton IV. Then, solvng a constraned optmzaton problem formulated n Secton V, power and bandwdth are allocated to users n a way to maxmze proportonal farness for data users, whle satsfyng rate requrements for real tme users. Frequency selectve fadng s also consdered n Secton VI and jont subchannel and power allocaton methods are consdered n order to mprove the performance of exstng schemes. II. SYSTEM MODEL We consder a multcarrer scheme where multple access s provded by assgnng a subset of subchannels to each recever at each tme frame. Let W and P denote the total bandwdth and power, respectvely. Total bandwdth W s dvded nto K subchannels of bandwdth W sub Hz, each consstng of a group of carrers. The nose and nterference power densty s N 0, and the channel gan averaged over the entre band from the BS to user at tme t s h t), where h t) ncludes path loss, shadowng lognormal fadng) and fast fadng. As explaned n the Introducton secton we assume dstrbuted subcarrer groupng and therefore all subchannels are of equal qualty wth respect to a user. There are three classes of users. Users n the classes U D, U S,andU V demand data, vdeo streamng, and voce traffc, respectvely. Let D, S, andv be denote ther quanttes. Let U R = U S U V be the set of users demandng real-tme traffc. The system that we consder s tme slotted wth frame length T s seconds. The scheduler makes a resource allocaton decson at each tme frame. We assume that the numbers of all types of sessons are fxed throughout the system smulaton. IEEE a/e standards allow several combnatons of modulaton and codng rates that can be used dependng on the sgnal to nose rato. Here assumng constant fadng durng a frame, we model the channel as an addtve whte gaussan nose AWGN) channel. Base staton allocates the avalable power and rate among users, where p t) and w t) are the power and bandwdth allocated to user n frame t. For an SINR pt)ht) N,the 0w t) hghest order modulaton and codng scheme that guarantees a BER constrant s used [21]. In ths work, for smplcty we assume rate as a contnuous functon of SNR. The resultng SNR values can be further quantzed to the avalable values, whch s not consdered n ths work. Based on the modulaton/codng pars and correspondng SNR thresholds n the standard, t s reasonable to approxmate the optmal transmsson rate as an ncreasng and concave functon of the sgnal to nose rato. We wll adopt the Shannon channel capacty for AWGN channel as a functon for bandwdth and

3 32 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 1, FEBRUARY 2010 transmsson power assgned to user : r w t),p t)) = w t)log 2 1+β p ) t)h t). 1) N 0 w t) The reason for usng 1) s ts smplcty, and t also approxmates rate-sinr relaton n the standard wth β =0.25. The parameter 0 < β < 1 SNR gap) compensates the rate gap between Shannon capacty and rate acheved by practcal modulaton and codng technques [22]. III. BENCHMARK SCHEMES As mentoned n the Introducton, prevous works n [11], [12], [13], and [23] proposed resource allocaton schemes for OFDMA-based systems supportng heterogeneous traffc. Below, we explan these algorthms. Let h,k t) be the channel gan of user at subchannel k. A. Largest Weghted Delay Frst wth Proportonal Farness LWDF-PF) Ths s a resource allocaton scheme that combnes weghted delay based schedulng wth proportonal farness [23]. Subchannel allocaton s performed based on a utlty value, U t) = κ D HOL t) log 1+ βph ),kt). 2) R t) D HOL t) s the head of lne packet delay of user. The parameter κ s a postve constant that reflects the QoS prorty of the user/sesson. If QoS requrement s defned as delay volaton probablty, P D >D max ) <δ, where D max s the delay constrant and δ s the probablty of exceedng ths constrant typcally 0.05), then the constant κ can be defned as κ = logδ) D [11], [23], and ths weghted delay metrc s used max n channel allocaton n sngle-channel tme-sharng systems, and referred to as largest weghted delay-frst LWDF) scheme [10], [11], [23]. Here, R t) s the average receved rate whch s updated as, R t +1)=α R t)+1 α )r t) 3) where 0 <α < 1 s typcally close to one. In the proportonal far scheme T =1/1 α ) s the length of the sldng tme wndow and average rate s computed over ths tme wndow at each frame. Ths way we consder both current rate as well as rates gven to the user n the past. Observed at tme t, the hghest consderaton s gven to the current rate rt), and the rates receved at the past t 1, t 2, carry dmnshng mportance. Includng R t) provdes proportonal farness. The LWDF-PF method can be summarzed as follows: 1. Intalze w =0,r =0, Ω =,, where w, r,and Ω are the number subchannels, rate and set of subchannels for user. 2. for k=1:k 2-1. Select user =arg max U t) :q t)>r T s where q t) s the number of bts watng to be transmttedtouser at tme t Allocate subchannel k to user, Ω =Ω {k} Update receved rate as r W sub log 2 1+ βph ),kt) k Ω 2-4. Update queue sze as q t) q t) W sub log 2 1+ βph ),kt) end Step 2 avods excessve allocaton of resources to users and prevents underutlzaton. For flat fadng channels, snce h,k = h, k the receved rate s updated as r = Ω W sub log 2 1+ βph t) ) where Ω s the cardnalty of the set Ω. B. Channel-Aware Queue-Aware Schedulng wth Jont Subchannel and Power Allocaton CAQA-JSPA) Ths algorthm proposed n [13] extends LWDF n two ways: 1) The power allocated to each user s stll proportonal to the number subchannels, however a power gven to a user can be optmally dstrbuted to ts subcarrers to maxmze receved rate wth the gven resources. Subchannels are stll allocated one by one. Let w be the aggregate amount of bandwdth allocated to user. The power allocated to ths user s P w /W. Ths tme receved rate r s calculated by allocatng ths amount of power optmally to the w /W sub subcarrers. Ths power optmzaton s repeated as each new subchannel s allocated to a user. The receved rate for user at each step s therefore found by solvng r =max p,k k Ω W sub log 2 s.t. k Ω p,k P w /W. 1+ βp ),kh,k t) Ths s solved by waterfllng. For the case of flat fadng or dstrbuted subcarrer groupng the optmal allocaton s the unform allocaton, therefore ths extenson does not provde any mprovement for ths case. 2) As each subchannel s allocated and receved rates are updated, head of lne packet delay D HOL t) s estmated based on the current receved rate and arrval tmes of the unserved packets [13]. The followng metrc s used at step k, HOL D t) U,k t) = κ R,k t) log 1+ βph ),kt) where R,k t) = α R,k t 1) + 1 α )W sub log βp,k t)h,kt) N 0W ) s the average servce that user receved from subchannel k. Here, p,k t) s the power allocated to user and subchannel k at tme t. Forflat fadng channels usng separate

4 GIRICI et al.: PROPORTIONAL FAIR SCHEDULING ALGORITHM IN OFDMA average rates for subchannels also don t make any dfference because each subchannel s equal and subchannels are allocated randomly once the number of subchannels to be allocated to each user s determned. The max delay utlty MDU) algorthm proposed n [11] uses mean queue sze nstead of head of lne delay. Smulaton results show that CAQA algorthm [12] acheves better performance than MDU, therefore t s not taken as a benchmark n ths work. The algorthms explaned above are proposed for systems wth random packet arrvals and delay constrants. Best effort traffc such as Web browsng or FTP, on the other hand have very loose or no delay constrants. Besdes, we assume that data traffc source adjusts ts transmsson rate to sute the servce rate and t can always use any bandwdth assgned to t. We can assume that transmsson queue of data traffc s never empty. Under such assumptons there s no delay for data traffc. In order to be able to use metrc 2) for data traffc we wll assume a fxed e.g., one second) delay for data sessons. IV. PROPOSED SCHEME Our prmary am s to fnd a schedulng scheme that supports data traffc as well as delay senstve real-tme traffc. Our approach s based on the assumpton of frequency-flat fadng. Ths mples that each subchannel s equal wth respect to a user. Based on ths assumpton we consder total bandwdth as a contnuously dvsble quantty. Our soluton for resource allocaton conssts of, Determnng the rate constrants for users demandng real tme traffc. Formulatng and solvng an optmzaton problem that ams proportonal farness for data users subject to rate constrant for real tme users and total power/bandwdth constrants. Quantzng the resultant bandwdth to nteger multples of subchannel bandwdth. We wll frst formulate the proportonal farness objectve for data users. A. Proportonal Farness for Data Traffc It s proven n [4] by Tse that a proportonal far allocaton for a sngle carrer system also maxmzes the sum of the logarthms of average user rates N =1 log R where N s the number of users and R s the average receved rate of user. In a sngle carrer system proportonal farness s acheved by schedulng at each frame t, auser accordng to: =argmax r t) R t). 4) Here r t) s the nstantaneous transmttable rate to user at the current frame. R t) s the average data rate that user receves over tme. At each frame the average rate s updated accordng to 3). So ths method mantans farness n the long run, whle tryng to schedule the user wth the best channel at each frame. Proportonal far resource allocaton problem n OFDMA systems was modeled prevously n [6] and [7] as maxmzng the sum of logarthms of nstantaneous rates. In [7], t was studed for flat fadng multchannel systems and formulated as a jont power and bandwdth optmzaton problem as N =1 log r w,p )) subject to power constrant N =1 p P, bandwdth constrant N =1 w W and p,w 0,, where r w,p ) s the nstantaneous rate functon n 1). In [7], effcent and low complexty algorthms are proposed to solve the above optmzaton problem. Some algorthms were also proposed for the same objectve n [6] and [9]. However ths objectve functon ams proportonal farness only n a sngle tme slot as opposed to long term. In fact, data users do not need farness n a very short term. Far allocaton n a few seconds e.g., thousand frames) of tme wndow s enough, snce t takes that much tme to download a webpage. Ths gves a degree of freedom and facltates the use of tme dversty n order to maxmze the data rate and proportonal farness and t should be exploted. Long term proportonal farness n multchannel systems was formulated n [5] and a soluton was proposed n the recent work [8]. Here the authors propose a subchannel and tme slot allocaton scheme n order to maxmze the sum of logarthms of long term receved rates. They use the followng greedy approach. At each tme frame the long term average receved rates up to the begnnng of the current slot are gven. The average rates are computed usng a movng average formula smlar to 3). The problem at each frame s to determne the current rates that maxmze the log-sum of movng averages. In ths work we follow a smlar approach and consder the maxmzaton of the followng objectve functon. Crt)) = N log α R t 1) + 1 α )r w t),p t))). =1 After some rearrangements the objectve for data users becomes: max N pt),wt) =1 = max pt),wt) N log α + 1 α ) )r w t),p t)) R t 1) α + 1 α ) )r w t),p t)). 5) R t 1) The novelty of our proposed scheme s that we also consder power optmzaton n addton to subchannel. Besdes we nclude rate constrants for the voce and vdeo streamng users as a constrant, whch wll be explaned below. B. Real Tme Traffc Proportonal farness objectve n 5) ams at provdng farness to data users. On the other hand, real tme traffc hasmore strct delay and packet loss requrements, whch can be translated nto strct nstantaneous rate requrements. In ths work we propose to use a varaton of the LWDF-PF algorthm n computng the rate requrements for real tme users. The only dfference s that as new subchannels are allocated to user after Step 2-3 average receved rate R t) s updated as R t+1) = α R t)+1 α )r t).thsavodsexcessve allocaton to a user. LWDF-type of algorthms have good performance, however they can be mproved by usng power control. Let w, U R be the amount of bandwdth gven to each real

5 34 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 1, FEBRUARY 2010 Lw, p,,λ w,λ r )= U D + U R 1 α )w log 2 1+ p α + λ r R n w ) + P p + λ w W w log 2 1+ p ) ) r c. 12) n w w tme sesson as a result of applyng the scheme descrbed above. Then the rate constrants for each real tme sesson s calculated as r c = w log 2 1+ βph ) t), U R. 6) We wll use r c as the rate constrant n the constraned optmzaton problem to be defned n Secton V. Solvng that problem wll help us acheve the same rate wth less resources, whch wll leave more resources for the data users. Ths way we propose a jont power/bandwdth optmzaton scheme that can be used on top of LWDF-type of unform power schemes n order to mprove the performance 1. Delay constraned voce and vdeo sessons have lower rate and they have to be served despte possbly bad channel condtons, therefore they are lmted by power. On the other hand, delay tolerant data sessons are only scheduled when ther SINR s good, therefore they are bandwdth lmted. Our jont power and bandwdth allocaton framework tends to gve more power to the former and more bandwdth to the latter types of sessons. By comparng t drectly to LWDF and CAQA, t wll be easy to observe the effects of jont power/bandwdth control on the performance. V. JOINT DATA AND REAL TIME RESOURCE ALLOCATION In ths secton we combne the proportonal far schedulng objectve n 5) and real tme user rate requrements defned n 6) and propose an adaptve power and bandwdth allocaton APBA) scheme. We formulate a constraned optmzaton problem, where the objectve functon s 5) and the constrants are the total power/bandwdth constrants and the rate requrements for real tme sessons. Let n = N0 βh. The resultng optmzaton problem s as follows: 2 Fnd p, w ) = arg max p,w U D 1 α )w log 2 1+ p α + R n w ) 1 Please note that the proposed soluton s not exactly optmal. Frst of all we make a contnuous bandwdth assumpton as explaned n Secton IV. Secondly we determne rate constrants based on results of a LWDF-type unform power subchannel allocaton scheme, whch may not be optmal. Besdes the noton of optmalty s unclear when there are contradctory objectves of maxmzng proportonal farness for data users and provdng QoS for real tme sessons. However, our approach s based on solvng a convex optmzaton problem, whch s whyweusethewordsoptmzaton, oroptmal throughout the paper. 2 Here, p, w,andn are the values at tme t. The tme ndex s not shown for convenence. 7) subject to w log 2 1+ p n w p P 8) w W 9) ) r c, U R 10) p,w 0, U D U R. 11) Here U R s the set of real tme sessons wth rate greater than zero. A. Soluton to the Constraned Optmzaton Problem The objectve functon 7) s an ncreasng functon of w, p), therefore the maxmum s acheved only when the constrants 8), 9), and 10) are all met wth equalty. For ths reason, we can replace these nequaltes wth equaltes n the dscusson below. Lemma 1: The problem n 7), 8), 9), 10), and 11) s a convex optmzaton problem. Proof: In the Appendx. Actually there s no guarantee that a soluton can be found that satsfes 10) for all users. The rate requrements for real tme users can be too hgh that t may be mpossble to satsfy wth the gven channel condtons. To start wth, we assume that the problem s feasble. We wll dscuss about how to detect nfeasblty of the problem and what to do n that case n the next secton. We can wrte the Lagrangan of the problem as [24] n 12) Takng the dervatves of Lp, w,,λ w,λ r ) wth respect to p, w for all users, we get the followng: For users U D : Lp, w,,λ w,λ r ) p =0 p,w ) = 1/n ln 2) )) ) R α + w log 2 1+ p n w 1+ p n w Lp, w,,λ w,λ r ) w =0 p,w ) ) 1+ p n w log λ w = ln 2 1+ p n w 1+ p n w ) R α + w log 2 ) p n w 1+ p n w 13) )) 14)

6 GIRICI et al.: PROPORTIONAL FAIR SCHEDULING ALGORITHM IN OFDMA where α = α 1 α. By dvdng 14) wth 13) we can wrte for all U D : λ w =Λ x = n 1 + x )log1+x ) x ) 15) where x = p n w denotes the optmal effectve SINR, whch s the SINR multpled by the SINR gap parameter β. For users U R : Lp, w,,λ w,λ r ) p =0 p,w ) λ r = 1 1 n ln 2 1+ p n w Lp, w,,λ w,λ r ) w =0 p,w ) λ w λ r = 1 ) log 1+ p ln 2 n w p n w 1+ p n w. 16) 17) Combnng 16) and 17) dvdng λw )forall U R agan gves: λ w =Λ x = n 1 + x )log1+x ) x ). 18) By wrtng 18) we can elmnate λ r s from the problem. It s worth notng that we get the same relaton between Λ x /n and x for all users 15) and 18)). At ths pont, t s convenent to defne the functon f x x) as: Then, we have f x x) =1+x)log1+x) x. 19) x Λ x )=fx 1 Λ x /n ), U D U R. 20) Lemma 2: Effectve SINR x Λ x )) s a monotonc ncreasng functon of Λ x for users U D U R. Proof: The proof s n Appendx. From 13) and 15) for data users we can wrte, for U D : w,λ w )= p,λ w )= [ 1 ] + ln 2n 1 + fx 1 λw ))R α log1 + fx 1 [ 1 λw ))1 + fx 1 ln 2n 1 + fx 1 λw log1 + fx 1 λw λw ))n 21) ] +f 1 ))R α x ))1 + fx 1 λw )) λw ). 22) The [ ] + =max0,.) operator n 22) and 22) guarantees that w,p 0 for all users. For users U R we have: Lp, w,,λ w,λ r ) λ r =0 p,w ) ) r c = w log 1+ p n w, U R. 23) Usng 18) and 23), we fnd, w,λ w )= log 1 + fx 1 24) λw )) p,λ w )= rc f x 1 λw )n log 1 + fx 1 λw )). 25) Total power and bandwdth constrant equatons are Lp, w,,λ w,λ r ) =0 p,w ) P = p λ p,λ w) 26) U D U R Lp, w,,λ w,λ r ) λ w =0 p,w ) W = w λ p,λ w). 27) Gven and λ w we can compute the power and bandwdth for all users usng 22) 25). We need to fnd the rght λ p and λ w so that the power and bandwdth constrants are satsfed. Let S p,λ w ) and S w,λ w ) be the total bandwdth and total power functons. The problem s fndng λ w and λ p such that S w λ p,λ w)= S p λ p,λ w)= r c w λ p,λ w)=w 28) p λ p,λ w)=p. 29) In fact because of the convexty of the problem ts optmal soluton s unque and t s the λ w and λ p values that satsfy the total power and bandwdth constrants. Before fndng those values we have to frst fnd out f the problem s feasble. The procedure for fndng t s explaned n the Appendx. If the problem s not feasble then the BS chooses the user that consumes the most power probably the worst channel condton) and decreases ts rate to half r c rc /2). If rc <r0 r0 s the bt arrval rate for the sesson) then r c s set to zero. The BS doesn t drectly decrease ts rate to zero, because the user wth bad channel condton probably needs urgent servce otherwse t wouldn t have been gven resource n the ntal subchannel allocaton phase), therefore the BS stll tres to gve a chance to that users. Ths rate decreasng procedure s repeated untl the problem s feasble. Fndng the two optmal Lagrange multplers requre a two dmensonal search. Ellpsod method [25] s a subgradentbased optmzaton method that s a generalzaton of bnary search for multple dmensons. It s especally useful n optmzng non-dfferentable functons. Our problem ncludes such non-dfferentable functons because of the exstence of the [ ] + operator n power/bandwdth evaluatons. Brefly, the ellpsod method nvolves fndng an ntal ellpsod that s guaranteed to nclude the optmal values. Then, at each step of the algorthm a subgradent s found and the volume of the ellpsod s decreased untl t s small enough. The subgradents that are used and the ellpsod updates are explaned n the Appendx. Below, we prove some propertes of the optmal Lagrange multplers

7 36 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 1, FEBRUARY 2010 λ p, λ w and Λ x = λ w λ that wll be useful n fndng an ntal p ellpsod. Lemma 3: The followng nequaltes hold: Λ x Λ mn x = mn Λ x Λ max x = max mn UD U R Λ max { P n f x n W { P n f x n W )} )} 30) 1 { n 1 + fx 1 } 31) Λ mn x /n ))R α λ w λ max p x. 32) Proof: Proof s n the Appendx. The Lemma above gves us upper bounds for λ p and λ w and they are useful n defnng the ntal ellpsod. After we fnd λ w and Λ p, we compute the optmal bandwdth and power values for all nodes. We wll refer ths scheme as APBA. Ths scheme has complexty of ON), because at each step of the teraton SNR values has to be computed for gven Lagrange multplers. In fact, t s less complex than that because only few of real tme users have nonzero rate requrements at frame. Note that the complexty s ndependent of the number of subchannels. B. Bandwdth Allocaton for Unform Power Ths s a smpler alternatve to jont power-bandwdth allocaton. The rate functon for user becomes w log 2 ) 1+ P n W. We can modfy the problem 7) 11) by usng ths rate functon and excludng the power constrant n 8). By usng standard technques the optmal bandwdth values can be found by solvng these set of equatons, + w λ w )= 1 R α ), U D 33) λ w log 2 1+ P n W r c w λ w )= ), U log 2 1+ P R 34) n W W = w λ w ). 35) Snce bandwdth s a monotonc functon of λ w, optmal bandwdths can be found by bnary search. Powers can be calculated as p = P w W, U D U R. We wll refer to ths scheme as adaptve bandwdth allocaton ABA). C. SINR/Bandwdth Quantzaton and Reshufflng In practce, bandwdth allocaton s n terms of nteger number of subchannels. Hence, we have to apply the followng resource shufflng procedure Quantze the bandwdth values to the nearest number of subchannel. Quantze to one subchannel f t s less than that. If the total bandwdth s greater than W,thenfnd the node that has the largest ncrease n bandwdth due to quantzaton and decrease ts one subchannel. If there s no such node left, then fnd user wth the hghest bandwdth and decrease by one subchannel. If total bandwdth s smaller than W,thenfnd the node wth the hghest decrease n bandwdth due to quantzaton and ncrease ts subchannels by one. If there s no such node left, then fnd user wth best channel condton and gve t one more bandwdth. VI. FREQUENCY SELECTIVE CHANNELS Most of the proposed OFDMA resource allocaton schemes n the lterature consder frequency selectve fadng and propose subchannel-by-subchannel allocaton schemes n order to explot frequency and multuser dversty. Optmal solutons n such a scenaro wth our objectves of farness and QoS s prohbtvely complex. However, power control can stll be used to mprove the performance of of LWDF-type of schemes. In ths case we dvde the problem nto two stages as the allocaton for real-tme and data users. The scheme that we use s summarzed as follows Determne user rates and number of subchannels allocated to each user by applyng an LWDF-type scheme. Based on the above nformaton allocate subchannels n order to satsfy above rates wth mnmum power. Calculate the powers for the real tme users. Keep ther allocatons fxed and reallocate the resdual subchannels and power to the data users. For the frst step we apply the classcal LWDF-PF algorthm explaned n [23], except that we update average receved rates as α R +1 α )r after allocatng each subchannel. Ths avods excessve allocaton to users. After the frststepweob- tan set of subchannels Ω allocated to user and total rate r c = k Ω W sub log βph,k N 0W sub ) for all users, where h,k s the channel gan for user at subchannel k. Gven the number of subchannels w = Ω ) and rate constrants r c found n the frst step, we wll reallocate the subchannels to satsfy those rate constrants wth mnmum power. The authors n [20] propose a lnear programmng-based soluton to determne the subchannels allocated to each user n order to mnmze the total power gven the rate constrants and number of subchannels for each user. Even ths soluton s of hgh complexty, therefore a greedy scheme called Vogel s method s used and almost the same performance s acheved wth much lower complexty. Once the subchannels are determned, waterfllng can be used n order to satsfy rate constrants wth mnmum power. In the second step we apply the Vogel s method explaned as follows [20]: 1. Set number of subchannels w = Ω, bts per symbol r S = c W sub w, = 1,,N. Then, ntalze Ω = {1, 2,,K} and Ω =,. 2. Set subchannel costs as C,k = N0W sub βh,k 2 S 1),, k. Sort subchannel costs for each user n ascendng order, whch s denoted by C Now C,1 s the cost of the lowest cost subchannel for user ). 3. Calculate user penaltes as P = C,w +1 C,1,. Ths reflects the opportunty cost of not makng an mmedate allocaton to a user. If the current best avalable channels) are very good, then the penalty of not allocatng them to that user s hgh.

8 GIRICI et al.: PROPORTIONAL FAIR SCHEDULING ALGORITHM IN OFDMA Table 1. Smulaton parameters. Parameter Value Cell radus 1.5km User dstances 0.3,0.6,0.9,1.2,1.5 km Total power P) 20 W Total bandwdth W) 10 MHz Frame length 1msec Voce traffc CBR 32 kbps Vdeo traffc kbps FTP fle 5MB AWGN p.s.d.n 0 ) -174 dbm/hz Pathloss exponent γ) 3.5 ψ DB Nμ ψdb,σ ψdb ) N0dB,8dB) Coherent tme fast/slow) 5 msec/400 msec.) PathlossdB, d n meters) log 10 d + ψ db 4. Repeat untl w =0, 5. Fnd user =argmax :w>0 P 6. Fnd subchannel k =argmn k Ω C,k. Update Ω Ω {k }, Ω Ω/{k }, w = w 1 and C,k =,. SortC agan and update penaltes as n Steps 2 and end Repeat At each repetton of lnes 4 to 7, ths algorthm allocates a subchannel to a user, so t ends n K steps. The subchannels are allocated to users n a way that requres close-to-mnmum power to satsfy the rates obtaned n the frst step. In the thrd step, gven the subchannel allocatons we further optmze powers. Frst, optmze power for real tme users by usng waterfllng. Then, we reallocate resdual subchannels Ω ) and power P ) among the data users. Usng the proportonal farness objectve n 7) s stll not practcal for the frequency selectve case. Consderng that 1 α s small, the approxmaton ln1 + x) x for small x can be used to convert t to the followng max Ω,p,k 1 R U D U D k Ω W sub log 2 1+ βh ) ),kp,k 36) sub k Ω p,k P 37) Ω Ω j =,, j U D 38) U D Ω Ω. 39) Ths s a weghted sum rate maxmzaton problem subject to resdual total power constrant and subject to the subchannel allocaton constrants No two user can share the same subchannel and subchannels should be allocated only from Ω ). Ths s a nonconvex problem, however usng Lagrange dual relaxaton technques near-optmal soluton s found n [18]. The soluton s especally optmal as number of subchannels ncreases. VII. PERFORMANCE EVALUATION For the numercal evaluatons we dvde the users to 5 classes accordng to the dstances, 0.3, 0.6, 0.9, 1.2, 1.5 km. At each dstance level one ffth of the total number of users exst. We use the parameters n Table 1. Table 2. OFDMA-related parameters. Parameter Value Nomnal channel bandwdth W =10MHz FFT sze N FFT = Number of used subcarrers N used = 840. Samplng factor n s = F s/w =8/7 Samplng frequency F s = n W/ = MHz Subcarrer spacng Δf = F s/n FFT = Hz Used bandwdth N used Δf = MHz Useful symbol tme T b =1/Δf = μs Guards perod rato 1/8 OFDM symbol tme T s =1+1/8) T b = msec Subchannelzaton mode DL-PUSC Tones per subchannel 24 Subchannel bandwdth W sub =24 Δf = KHz Number of subchannels K =30 Table 2 summarzes the OFDMA-related parameters used n ths smulaton and ther dervatons. Here FFT sze means the number of samples n the Fast Fourer Transformaton. Number of used subcarrers N used s smaller than N FFT because the outer carrers n a subchannel do not carry modulaton data. We wll compare our algorthm wth the benchmark LWDF- PF [23] and CAQA-JSPA [13] schemes. Delay exceedng probablty s taken as δ =0.05 for all users. Delay constrant for voce and vdeo users are 0.1 and 0.4 seconds, respectvely. For LWDF-PF algorthm we assume that the delay constrant for data users s 1 second and buffer length s nfnte. We assume a constant HOL delay of 1 second for the data sessons. Flter values are α =0.999, 0.995, 0.98 for data, streamng and voce sessons. Performance crtera are as follows. We wll observe the total throughput for all data users. For data users we wll also observe total log-sum rate Ct) = U D log R. For real tme users we wll measure the outage probablty, whch means the percentage of the transmtted packets that volate ther delay constrant. A. Flat Fadng A.1 Increasng Number of Real Tme Users Fgs. 1 and 2 show the effects of ncreasng the number of real tme sessons on outage and data throughput performance. We assume that half of the real tme users are voce and vdeo streamng users. In Fg. 1, we observe that the proposed APBA scheme acheves the best throughput performance. APBA acheves 25% performance mprovement wth respect to benchmark algorthms especally when the number of real tme users s large. The reason s that the proposed algorthm provdes more power and less bandwdth to users wth worse channel condtons therefore avods the waste of bandwdth. The performance of the unform-power ABA scheme s almost dentcal to the benchmarks, whch agan proves that power control s the process that makes the dfference. We also observe that the performances of the two benchmarks are almost dentcal, where CAQA-JSPA acheves slghtly more throughput Fg. 1) but more outage Fg. 2). We also see the performance mprovement n terms of log-sum of throughput. Snce t nvolves logarthms, the performance dfferences can t be hgh n numbers. Even small dfferences are mportant n ths case. In Fg. 2, we see the outage performances. We can observe that proposed APBA algorthms acheves less outage n addton

9 38 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 1, FEBRUARY 2010 a) b) Fg. 3. Total data throughput for ncreasng number of data users S=V=40). Fg. 1. Data user performance vs. number of real tme users D=20, S=V): a) Total data throughput and b) log-sum throughput for ncreasng number of real tme users. a) a) b) b) Fg. 2. Real-tme user performance vs. number of real tme user D=20, S=V): a) Outage probablty for voce and b) outage probablty for vdeo streamng sessons for ncreasng number of real tme users. Fg. 4. Real tme user performance vs. number of data users S=V=40): a) Outage for vdeo and b) Outage for voce for ncreasng number of data users. to better throughput. Compared to the proposed scheme benchmarks have 25% hgher outage for vdeo when number of real tme users s large. The mprovement n voce sessons s even more. A.2 Increasng Number of Data Users In Fgs. 3 and 4, we see the effects of ncreasng number of data users on data throughput and delay. From Fg. 3 t can be observed that data throughput s an ncreasng concave func- ton of total number of data users. As number of users ncrease, multuser dversty s n effect, however ts effect dmnshes as number of users ncreases. We also observe that there s a constant 15% performance dfference between proposed APBA and benchmarks. We can conclude that proposed scheme provdes more mprovement when number of real tme users s hgh. Although t s not ncluded here, APBA has also better performance n terms of log-sum of throughput. Fg. 4 shows that ths ncrease n throughput s obtaned wthout sacrfcng the outage performance.

10 GIRICI et al.: PROPORTIONAL FAIR SCHEDULING ALGORITHM IN OFDMA a) b) assumptons lead to smplfed resource allocaton polces n case of heterogeneous traffc requrements. We developed an optmzaton approach n order to provde proportonal farness for data users and satsfy delay requrements of real tme users such as voce and vdeo streamng. We solved the optmzaton problem and developed an algorthm that fnds the optmal bandwdth and power gven to each user. Our problem formulaton also explots tme dversty for data users by consderng the long term averaged receved rate n optmzaton. The proposed scheme can be used on top of exstng largest weghted delay frst type of resource allocaton polces and mprove ther performance by jont power/bandwdth control. Fnally we numercally showed that when compared wth the well known LWDF- PF and channel/queue aware CAQA) schemes n the lterature, our scheme both provdes better proportonal farness for data sessons and provdes better QoS for real tme sessons. IX. APPENDIX I A. Proof of Concavty of Objectve Functon c) Fg. 5. Performance vs. number of real tme users for the case of frequency selectve fadng S=V, D=40): a) Data throughput, b) vdeo outage, and c) voce outage. B. Frequency Selectve Fadng Fg. 5 shows the performance for the case of frequency selectve fadng. Frst of all, here we observe that frequency selectvty especally mproves the real tme sesson outage performance sgnfcantly. We see that the proposed resource allocaton scheme for frequency selectve fadng systems can provde 8 15% and more than 2 Mbps) performance mprovement n terms of total data throughput. The percentage mprovement ncreases as the number of real tme users ncrease. The mprovement wth respect to LWDF-PF s less n frequency selectve case than n the flat fadng case. Ths s because explotng the frequency dversty n frequency selectve fadng provdes better SINRs for the users. The rate s a logarthmc functon of SINR therefore power control provdes dmnshng mprovement as SINR ncreases. We also observed that LWDF-PF performs better than CAQA-JSPA, although the latter also ncludes power control. Ths may be because, n CAQA-JSPA average receved rate for each subchannel s consdered separately. Ths does not reflect the actual rate that the user receved. Besdes power control n JSPA does not make a sgnfcant dfference because a user stll receves power proportonal to number of channels t s allocated. VIII. CONCLUSIONS In ths work we consdered the problem of resource allocaton n OFDMA based downlnk wreless multple access systems wth flat fadng or dstrbuted subcarrer groupng. Such The reward functon Cw n, p n )= log α R +1 α )w log 2 1+ p )) n w U 40) s a concave functon of w and p for all U D. Proof: If we take the Hessan H r of rp, w) =w log 2 1+ p nw ) [ ] 1 w p Hr = p + nw) 2, 41) p p2 w we see that t s negatve defnte, therefore the functon r s strctly concave. Multplyng t wth a constant 1 α), addng t wth a constant αr preserves concavty. Logarthm of a concave functon s also concave. Lnear combnaton 40) of concave functons are concave too; therefore reward functon s a concave functon of w and p for all U D. B. Convexty of the Feasble Set The feasble set of power and bandwdth levels w, p) defned by 8), 9), 10), and 11) defnes a convex set. Proof: Consder two power-bandwdth vectors w 1, p 1 ) and w 2, p 2 ) that are n the feasble set. Now let us consder power-bandwdth vector λw 1 +1 λ)w 2,λp 1 +1 λ)p 2 ). It s clear that ths vector satsfes the feasblty constrants n 11). Now consder a user U R. Ths user has a rate constrant r c n 10). If w1,p1 ) and w2,p2 ) both satsfy constrant 10): p1 rw 1,p 1 )=w 1 log n w 1 )=r c 42) rw 2,p 2 )=w 2 log p2 n w 2 )=r c, U R 43)

11 40 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 1, FEBRUARY 2010 From the concavty of the Shannon capacty wth respect to w and p, we can wrte λ =1 λ): rλw 1 + λw 2,λp 1 + λp 2 ) =λw 1 + λw 2 )log 2 1+ λp1 + ) λp2 n λw 1 + λw2 ) 44) rλw 1 + λw 2,λp 1 + λp 2 ) ) ) λw 1 log 2 1+ p1 n w 1 + λw 2 log 2 1+ p2 n w 2 45) rλw 1 + λw 2,λp 1 + λp 2 ) λr c + λr c = r c. 46) a) Hence the power bandwdth values λw 1 + λw2,λp1 + λp2 ) also satsfy the rate constrants for users U R. C. Proof of Lemma 2 Proof: Effectve SINR x Λ x )) s a monotonc ncreasng functon of Λ x for users U D U R. df xx) Takng the dervatve dx = log1 + x) > 0, therefore f x x) s an ncreasng functon of x. Therefore ts nverse fx 1 Λ x /n ) s also an ncreasng functon of Λ x. For a gven values of Λ x /n, fx 1 Λ x /n ) can be found by applyng Newton s method typcally n as few as three teratons wth 0.1% accuracy.) D. Feasblty of the Soluton In the prevous secton we stated that there exsts a soluton to the problem, f the rate constrants are feasble. We now consder how to detect an nfeasble problem and what to do n that case. If the problem s feasble.e. f the avalable power and bandwdth s enough to satsfy rate requrements of real tme sessons), then there exsts a Λ x = n f x x ), U R so that the followng nequaltes hold: r 0 log 1 + f 1 W, 47) U R x Λx n )) r 0f x 1 Λx n )n log 1 + fx 1 P. 48) Λx n )) U R We know for all users U D U R that x = fa 1 Λ x /n ) s x ncreasng n Λ x. The expresson log1+x ) s strctly ncreasng n x. From these propertes we can deduce that the left hand sde LHS) of 47) s a decreasng and LHS of 48) s an ncreasng functon of Λ x. Let Λ 0 x be the smallest Λ x value that satsfes the nequalty 47. There exsts such a Λ 0 x > 0 because LHS of 47) s a strctly decreasng functon whch s nfnty for Λ x =0and zero for Λ x =. The problem s feasble f and only f RHS of 48) s smaller than P. : If the LHS of 48) s smaller than P then both feasblty condtons 47) and 48) hold, therefore the problem s feasble. : If the problem s feasble, then there exsts Λ x such that both 47) and 48) hold. Now let s assume that LHS of 48) s greater than P, then because of the monotoncty of the functons t s also greater for all Λ x Λ 0 x. Note that the b) Fg. 6. Convergence of the total power and bandwdth usng Ellpsod Method: a) Total power and b) total bandwdth. LHS of 47) s greater than W for Λ x < Λ 0 x. Ths means that there s no Λ x such that both 47) and 48) hold and the problem s nfeasble. Ths s a contradcton, therefore the LHS of 48) s smaller than P. The property holds. E. Proof of Lemma 3 We can prove the nequaltes for the optmal Λ x usng contradcton. Suppose that Λ x > {n f x P/n W )}, max U D U R, then fx 1 Λ P x/n ) > n,, W from the monotoncty property. Then, the total power P n W s greater than U D U w R = P, whch contradcts wth the power constrant, therefore the upper bound s proven. For the lower bound assume that Λ x < mn {n f x P/n W )}, U D U R, U D U R then fx 1 Λ P x/n ) < n W,, from the monotoncty property. Then, the total power s smaller than U D U w P R n W = P. Ths s not optmal because proportonal farness metrc can be ncreased by usng the resdual power, therefore the lower bound s also proven. For a feasble problem, bandwdth w of at least one data user should be greater than zero. From 22) we can wrte ths as 1 Λ mn mn UD {n 1 + f 1 x Λ x /n ))}. By wrtng x nstead of Λ x and rearrangng, we can wrte 32). From Λ x = λw we can easly wrte 32). F. Implementaton of the Ellpsod Method The optmal Λ=[Λ w ] T s guaranteed to be contaned n Λ max =[Λ max w λ max p ] T. The ntal value s chosen as Λ 0 = 0.5 Λ max. The square matrx P 0 s defned as A =0.5 dag[λ max ] 2 ). Then, λ λ 0 ) T A 0 1 λ λ 0 ) 1 forms the ntal ellpsod. At each step, subgradent d s chosen.

12 GIRICI et al.: PROPORTIONAL FAIR SCHEDULING ALGORITHM IN OFDMA If λ n w < 0 or λ n p < 0 then d = [ λ n ] +. If λ n w,λ n p > 0 then d = [W w λ n w,λ n p )P p λ n w,λ n p )] T. Updates are as follows [25]: g = d/ d T Ad λ n+1 = λ n 1 3 An g A n+1 = 4 3 A n 2 3 An g g T A n) Fg. 6 llustrates the characterstcs of the sum-powers S p λ w, ) and sum-bandwdth S w λ w, ) for 20 data users and 20 real tme users at one pont n tme. From the graph we see that ndeed sum-power and sum-bandwdth converges to the constrant values. Because of the convexty of the problem convergence always occurs and the obtaned Lagrange multplers gve the optmal power and bandwdth values. [21] IEEE Standard for Local and Metropoltan Area Networks, Part 16: Ar Interface for Fxed and Moble Broadband Wreless Access Systesm, Amendment 2: Physcal and Medum Access Control Layers for Combned Fxed and Mobel Operaton n Lcensed Bands and Corrgendum 1, IEEE, Feb [22] X. Qu and K. Chawla, On the performance of adaptve modulaton n cellular systems, IEEE Trans. Commun., pp , June [23] G. Song, Cross-layer resource allocaton and schedulng n wreless multcarrer networks, Ph.D. Dssertaton, Georga Insttute of Technology, Apr [24] L. Vanderberghe and S. Boyd, Convex Optmzaton. Mar [25] S. Boyd, Ellpsod Method. Stanford Unversty Class Notes, [Onlne]. Avalabe: REFERENCES [1] C. Eklund, R. B. Marks, K.L. Stanwood, and S. Wang, IEEE standard : A techncal overvew of the wreless MAN ar nterface for broadband wreless access, IEEE Commun. Mag., June [2] H. Ekstrom, A. Furuskar, J. Karlsson, M. Meyer, S. Parkvall, J. Torsner, and M. Wahlqvst, Techncal solutons for the 3G Long-Term Evoluton, IEEE Commun. Mag., pp , Mar [3] R. Knopp and P.A. Humblet, Informaton capacty and power control n sngle-cell multuser communcatons, n Proc. IEEE ICC, [4] D. Tse, Forward lnk multuser dversty through rate adaptaton and schedulng. submtted to IEEE J. Sel. Areas Commun., [5] H. Km and Y. Han, A proportonal far schedulng for multcarrer transmsson systems, IEEE Commun. Lett., pp , Mar [6] G. Song and G. L, Utlty-based resource allocaton and schedulng n OFDM-based wreless broadband networks, IEEE Commun. Mag., Dec [7] C. Zhu and J. Agre, Proportonal-far schedulng algorthms for OFDMAbased wreless systems, Preprnt, Fujtsu Labs, [8] M. Kaneko, P. Popovsk, and J. Dahl, Proportonal farness n multcarrer system wth multslot frames: Upper bound and user multplexng algorthms, IEEE Trans. Wreless Commun., pp , Jan [9] T. Nguyen and Y. Han, A proportonal farness algorthm wth QoS provson n downlnk OFDMA systems IEEE Commun. Lett., pp , Nov [10] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, and P. Whtng, Provdng qualty of servce over a shared wreless lnk, IEEE Commun. Mag., pp , Feb [11] G. Song, Y. Y. L, L. J. Cmn, and H. Zheng, Jont channel-aware and queue-aware data schedulng n multple shared wreless channels, n Proc. IEEE WCNC, Mar. 2004, pp [12] P. Parag, S. Bhashyam, and R. Aravnd, A subcarrer allocaton algorthm for OFDMA usng buffer and channel state nformaton, n Proc. IEEE VTC, Sept. 2005, pp [13] C. Mohanram and S. Bhashyam, Jont subcarrer and power allocaton n channel-aware queue-aware schedulng for multuser OFDM, IEEE Trans. Wreless Commun., pp , Sept [14] C. Y. Wong, R. S. Cheng, K. B. Letaef, and R. D. Murch, Multuser subcarrer allocaton for OFDM transmsson usng adaptve modulaton, n Proc. IEEE VTC, May 1999, pp [15] W.Rheeand J.M.Coff, Increase n capacty of multuser OFDM system usng dynamc subchannel allocaton, n Proc. IEEE VTC, May 2000, pp [16] Z. Shen, J. G. Andrews, and B. L. Evans, Adaptve resource allocaton n multuser OFDM systems wth proportonal rate constrants, IEEE Trans. Wreless Commun., pp , Nov [17] H. Km, Y. Han, and S. Km, Jont subcarrer and power allocaton n uplnk OFDMA systems, IEEE Commun. Lett., pp , June [18] K. Seong, M. Mohsen, and J. M. Coff, Optmal resource allocaton for OFDMA downlnk systems, n Proc. IEEE ISIT, July 2006, pp [19] J. Huang, V. Subramanan, R. Agrawal, and R. Berry, Downlnk schedulng and resource allocaton for ofdm systems, n Proc. 40th Annual Conference on Informaton Scences and Systems, Mar. 2006, pp [20] I. Km, I.-S. Park, and Y. H. Lee, Use of lnear programmng for dynamc subcarrer and bt allocaton n multuser OFDM, IEEE Trans. Veh. Technol., pp , July Tolga Grc receved hs B.S. degree from Mddle East Techncal Unversty, Ankara Turkey n 2000 and Ph.D. degree from Unversty of Maryland, College Park n 2007, both n Electrcal Engneerng. He has been a research assstant at Insttute of Systems Research from 2000 to In 2005 he has spent sx months as an ntern at the Intellgent Automaton Inc, Rockvlle, MD, USA. He was a research assstant at the Fujtsu Labs, College Park MD, USA n He s currently an assstant professor at TOBB Unversty of Economcs and Technology, Ankara, Turkey. Hs research nterests nclude broadband cellular wreless access networks, satellte systems, and wreless ad hoc networks; network functons lke multple access, routng, broadcastng, resource allocaton; optmzaton of performance objectves lke farness, energy effcency, and qualty of servce. Chenx Zhu receved hs B.S. from Tsnghua Unversty n Bejng, Chna n 1993 and Ph.D. from the Unversty of Maryland, College Park n From 2001 to 2004 he was wth Flaron Technologes Inc. n Bedmnster, New Jersey workng on flash-ofdm, an OFDMA-based moble broadband wreless access network. Snce November 2004 he has been a research scentst at Fujtsu Laboratores of Amerca, where he has worked on wreless-lan, WMAX, and LTE. Jonathan Agre s Vce Presdent and General Manager at the Fujtsu Laboratores of Amerca located n College Park, Maryland where he has helped to establsh the Trusted Cybersystems Research Center. The center s focused on research to ncrease the trustworthness of computer and communcaton systems. Through the center, he s also actve n several standardzaton actvtes such as the WMAX Forum, the Trusted Computng Group, and the INCITS M1 Bometrcs group. Currently, he s workng on securty of ad hoc networks and standardzaton of vascular bometrcs. He obtaned the Ph.D. n Computer Scence from the Unversty of Maryland, n 1981 n performance modelng of dstrbuted systems. He has over 70 techncal artcles and papers publshed n conferences, journals and books and over 35 patent applcatons. Pror to jonng Fujtsu, he was at the Jet Propulson Laboratory JPL) workng n advanced communcaton research and at the Rockwell Internatonal Scence Center specalzng n dstrbuted systems for sensor networks and real-tme factory control systems.

13 42 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 12, NO. 1, FEBRUARY 2010 Anthony Ephremdes holds the Cyntha Km Professorshp of Informaton Technology at the Electrcal and Computer Engneerng Department of the Unversty of Maryland n College Park where he holds a jont appontment at the Insttute for Systems Research, of whch he was among the foundng members n He obtaned hs Ph.D. n Electrcal Engneerng from Prnceton Unversty n 1971 and has been wth the Unversty of Maryland ever snce. He has held varous vstng postons at other Insttutons ncludng MIT, UC Berkeley, ETH urch, INRIA, etc) and co-founded and co-drected a NASA-funded Center on Satellte and Hybrd Communcaton Networks n He has been the Presdent of Pontos, Inc, snce 1980 and has served as Presdent of the IEEE Informaton Theory Socety n 1987 and as a member of the IEEE Board of Drectors n 1989 and He has been the General Char and/or the Techncal Program Char of several techncal conferences ncludng the IEEE Informaton Theory Symposum n 1991 and 2000, the IEEE Conference on Decson and Control n 1986, the ACM Mobhoc n 2003, and the IEEE Infocom n 1999). He has served on the Edtoral Board of numerous journals and was the Foundng Drector of the Farchld Scholars and Doctoral Fellows Program, a Unversty-Industry Partnershp from 1981 to He has receved the IEEE Donald E. Fnk Prze Paper Award n 1991 and the frst ACM Achevement Award for Contrbutons to Wreless Networkng n 1996, as well as the 2000 Fred W. Ellersck MILCOM Best Paper Award, the IEEE Thrd Mllennum Medal, the 2000 Outstandng Systems Engneerng Faculty Award from the Insttute for Systems Research, and the Krwan Faculty Research and Scholarshp Prze from the Unversty of Maryland n 2001, and a few other offcal recogntons of hs work. He also receved the 2006 Aaron Wyner Award for Exceptonal Servce and Leadershp to the IEEE Informaton Theory Socety. He s the author of several hundred papers, conference presentatons, and patents, and hs research nterests le n the areas of Communcaton Systems and Networks and all related dscplnes, such as Informaton Theory, Control and Optmzaton, Satellte Systems, Queueng Models, Sgnal Processng, etc. He s especally nterested n Wreless Networks and Energy Effcent Systems.