IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 20, NO. 2, FEBRUARY 2010 297 Fairness Sraegies for Wireless Resource Allocaion Among Auonomous Mulimedia Users Hyunggon Park, Member, IEEE, and Mihaela van der Schaar, Fellow, IEEE Absrac Recen research in wireless mulimedia sreaming has focused on opimizing he mulimedia qualiy in isolaion, a each saion. However, he cross-layer ransmission sraegy deployed a one saion impacs and is impaced by he oher saions, as he wireless nework resource is shared among all compeing users. Hence, efficien and fair resource managemen for auonomous wireless mulimedia users becomes very imporan. We consider qualiy-based fairness schemes based on axiomaic bargaining heory, which can ensure ha he auonomous mulimedia saions incur he same drop in mulimedia qualiy as compared o a maximum achievable qualiy for each wireless saion. Implemening his qualiy-based fairness soluion in he ime-varying channel condiion requires highcompuaional complexiy and communicaion overheads. Hence, we develop soluions ha significanly reduce he compuaional complexiy and communicaion overheads. Our simulaions show ha he proposed game-heoreic resource managemen can indeed guaranee desired uiliy-fair allocaions when wireless saions deploy differen cross-layer sraegies. Index Terms Axiomaic bargaining soluion, cross-layer opimizaion, game-heoreic mulimedia resource managemen, muliuser wireless resource managemen. I. Inroducion APLETHORA of real-ime mulimedia sreaming applicaions are saring o be deployed over emerging wireless local area neworks (WLANs) infrasrucures [1], [2]. However, he ime-varying and bandwidh-consrained wireless neworks do no provide he qualiy of service (QoS) required by he delay-sensiive and bandwidh-inensive mulimedia applicaions. To ensure he necessary QoS, recen research has focused on innovaive error resilien and bandwidh-adapive video compression, and cross-layer opimized ransmission sraegies [3], [4]. However, hese adapaion echniques have been performed in isolaion, a each mulimedia ransmier, and suffer from he imporan limiaion of no considering he ineracion (in erms of resource uilizaion) among wireless saions (WSTAs) sharing a common WLAN infrasruc- Manuscrip received June 14, 2008; revised November 11, 2008. Firs version published Sepember 9, 2009; curren version published February 5, 2010. This work was suppored by he Naional Science Foundaion (NSF) Compuer and Nework Sysems Gran 0831549, and he NSF Compuing and Communicaion Foundaions Gran 0541867. This paper was recommended by Associae Edior I. Ahmad. H. Park is wih he Deparmen of Elecronics Engineering, Ewha Womans Universiy, Seoul, Korea (e-mail: hyunggon.park@gmail.com). M. van der Schaar is wih he Deparmen of Elecrical Engineering, Universiy of California, Los Angeles, CA 90024 USA (e-mail: mihaela@ee.ucla.edu). Color versions of one or more of he figures in his paper are available online a hp://ieeexplore.ieee.org. Digial Objec Idenifier 10.1109/TCSVT.2009.2031767 1051-8215/$26.00 c 2010 IEEE ure. Emerging polling-based WLAN sandards (e.g., IEEE 802.11e [5]) ry o provide QoS o mulimedia applicaions by enabling each WSTA o reserve ime slos, i.e., ransmission opporuniies (TXOPs), where conenion-free access o he medium is provided. However, his scheme manages resources in a saic, wors-case fashion, since i does no consider he ime-varying channel condiions or video conen characerisics as well as he resuling uiliy impac for he various users [6]. To overcome his limiaion, a dynamic resource managemen scheme ha explicily considers he ime-varying video characerisics and adapive cross-layer sraegies deployed by he saions as well as he resuling mulimedia uiliies is necessary. Consequenly, he resource manager (e.g., he access poin) needs o ensure an efficien and fair resource allocaion among auonomous mulimedia users rying o maximize heir own uiliies. Fair resource allocaion sraegies among muliple compeing users have been acively researched. One of he simples fairness policies is o equally allocae resources (e.g., TXOPs in our case) among WSTAs. Alernaively, in a recenly proposed air-fairness scheme [4] for IEEE 802.11e neworks, he resources can be allocaed depending on he experienced channel condiions and he required video rae requiremens. An imporan disadvanage of hese fairness is ha hey do no consider he WSTAs uiliy impac, as uiliy funcions are usually nonlinearly increasing wih he allocaed resources. To alleviae his problem, uiliy-based allocaion has been proposed o explicily consider he derived uiliy. Proporional fairness was inroduced in [7] o allocae resources while considering he resuling uiliy. Maximizing he oal sysem uiliy [8] or maximizing he sum of logarihmic uiliies [9], [10] have been proposed as he opimal allocaions for wireless ransmission. However, his resource allocaion becomes unfair in noncollaboraive applicaions, where selfineresed and auonomous WSTAs compee for resources. Hence, exising uiliy-based fairness policies can severely penalize cerain WSTAs a he expense of oher WSTAs, which is no a desirable feaure for self-ineresed WSTAs. To address he above limiaions, we propose o solve he fair resource allocaion direcly in he mulimedia uiliy domain. We model WSTAs as auonomous and raional users compeing for available resources by proacively adaping heir cross-layer ransmission sraegies in order o maximize heir uiliies. Then, he resource manager should have he abiliy o decide how he uiliies of he auonomous users are impaced relaive o each oher based on a predeermined (agreed upon)
298 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 20, NO. 2, FEBRUARY 2010 uiliy-fair crierion. For insance, each user can be penalized an equal amoun in erms of mulimedia qualiy (i.e., incur he same drop in mulimedia qualiy as compared o is maximum achievable qualiy) by paricipaing in he resource allocaion. To ensure such a relaionship among he uiliies of he mulimedia users, we rely on a well-known game-heoreic concep axiomaic bargaining soluions [11], [12]. In his paper, bargaining problems are invesigaed based on axiomaic approaches, where a soluion ha saisfies several desirable properies (axioms) is seleced from a feasible uiliy se. The axiomaic bargaining soluions were previously proposed o resolve resource allocaion issues for various nework applicaions [13] [15]. Noe ha he axiomaic bargaining soluions do no require an ieraive bargaining process among users, bu raher hey selec a soluion from he Pareo opimal surface ha saisfies several predeermined (agreed upon) raionaliy and fairness crieria [12]. Hence, i is assumed ha users can negoiae before he game is played, and hese negoiaions can be seled by a binding agreemen represened by a se of fairness axioms [12], which represens a bargaining soluion. This is very imporan for mulimedia applicaions, which are delay-sensiive and hus canno afford o incur he delay associaed wih an ieraive resource negoiaing procedure. Several axiomaic bargaining soluions such as he Nash bargaining soluion (NBS) [11] and he Kalai Smorodinsky bargaining soluion (KSBS) [16] are differeniaed by heir unique fairness crieria. The resource managemen for neworked mulimedia applicaions based on hem is heoreically sudied in our prior work [17]. Moreover, hese approaches have been also deployed in conjuncion wih various mulimedia applicaions, such as video compression, resource managemen schemes for mulimedia sysems, video sreaming, and image processing [18] [20]. Our main conribuions are summarized as follows. To efficienly solve he problem of fair allocaion of resources among muliple wireless mulimedia users, we firs review several exising fairness policies and deermine heir performance in erms of mulimedia qualiy. Nex, we propose a new approach, based on axiomaic bargaining soluions, which enables us o define fair-allocaion rules in he uiliy domain. Specifically, we use he KSBS because is axioms can disribue he resources opimally (in a Pareo opimal sense) and fairly among auonomous WSTAs, by ensuring an equal qualiy penaly from each WSTA s maximum achievable qualiy given is curren channel condiions, conen characerisics, and cross-layer sraegies. Therefore, he KSBS can be successfully used for auonomous WSTAs. In order o quanify he fairness achieved by several resource allocaion schemes, we inroduce a new meric, referred o as fairness comparison meric (FCM). Moreover, we develop algorihms for pracical KSBS implemenaion, which can significanly reduce he required compuaional complexiy and informaion exchange. We define a uiliy funcion, such ha heerogeneous mulimedia conens (e.g., audio, video, ec.) can be simulaneously considered in he proposed resource managemen framework. This paper is organized as follows. In Secion II, we define he uiliy funcions and briefly explain he cross-layer sraegies ha can be deployed by WSTAs. In Secion III, we propose a dynamic resource managemen and is deploymen in he sysem. In Secion IV, we describe and compare differen fairness policies for resource allocaion. In Secion V, we formulae he resource allocaion problem based on he KSBS and propose algorihms ha can significanly reduce he implemenaion complexiy. Simulaion resuls are provided in Secion VI. The conclusions are drawn in Secion VII. Several proofs in he paper are presened in Appendixes A C. II. Convenional Cross-Layer Sraegy Opimizaion We consider M compeing WSTAs ha are sreaming video conen in real-ime over he shared wireless nework. The role of he cenral resource moderaor referred o as resource manager (e.g., QoS-enabled access poin) in his paper is o divide and allocae he available TXOPs o each WSTA based on is declared raffic specificaion (TSPEC). Based on he negoiaed TSPEC, he medium access conrol (MAC) is polling he various WSTAs for a specific fracion of ime in every service inerval (SI). In his secion, we define he uiliy funcion and discuss he convenional cross-layer opimizaion sraegies. A. Mulimedia Uiliy Funcion Mulimedia users saisfacion can be improved as he disorion of mulimedia decreases. Hence, he uiliy funcion for he video coders given allocaed video rae R i o user i can be expressed as { 255 2 /D i (R i ), if R i R i,min U i (R i ) (1) 0, oherwise where R i,min is he minimum required video rae for user i and D i (R i ) is he incurred disorion given allocaed rae R i, measured as he mean square error (MSE). Noe ha he discussion below and he proposed soluion are unaffeced if he video coders [or disorion-rae (DR) model] are changed. The peak signal o noise raio (PSNR), which is a measure of video qualiy, can be expressed using he uiliy funcion, i.e., PSNR = 10 log 10 U(R). Noe ha for all video sequences, depending on he used video coder, a minimum PSNR needs o be achieved corresponding o he minimum accepable qualiy by he user [e.g., he base-layer qualiy in fine granular scalabiliy (FGS)]. This will play an imporan role in designing he proposed resource allocaion discussed in Secion V. B. Cross-Layer Sraegy Opimizaion a Each WSTA In his secion, we formulae he opimal cross-layer sraegy ha maximizes each WSTA s uiliy. We assume ha each WSTA is auonomous, and hus, each WSTA selecs is own cross-layer sraegy ha maximizes he uiliy given he conen characerisics, allocaed ime, and he experienced channel condiion [i.e., he signal o noise raio (SNR)]. We limi he cross-layer sraegies o only include he applicaion (APP)- layer prioriizaion and scheduling sraegies, MAC-layer reransmission, physical (PHY)-layer modulaion, and coding
PARK AND VAN DER SCHAAR: FAIRNESS STRATEGIES FOR WIRELESS RESOURCE ALLOCATION AMONG AUTONOMOUS MULTIMEDIA USERS 299 schemes. However, oher sraegies could also be incorporaed in his formulaion. Le s i = [phy n i, macm i, appl i ] S i be a cross-layer sraegy vecor in he feasible se of cross-layer sraegies for WSTA i, where S i = Si PHY Si MAC Si APP and Si PHY = {phy 1 i i,...,phynphy i }, Si MAC = {mac 1 i i,...,macnmac i }, and Si APP = {app 1 i i,...,appnapp i } denoe he sraegy space of PHY, MAC, and APP, respecively. For a saic resource allocaion [i.e., he ime resource is fixed as ( 1,..., M )], he opimal cross-layer sraegy is o maximize is uiliy given a TXOP allocaion i and he channel condiion (i.e., he experienced SNR) denoed by SNR i for WSTA i. Thus s i = arg max s i S i U i (R i ( i, s i )) = arg max s i S i R i ( i, s i ) i.e., WSTA i selecs he cross-layer sraegy ha maximizes R i ( i, s i ) since he TXOP allocaion i is given. Subsequenly, we ouline he seps involved in he passive cross-layer opimizaion proposed in [21]. The sraegy phy n i, n {1,...,NPHY i } represens he nh modulaion and channel coding mode exising for a WLAN sandard for WSTA i (e.g., PHY modes for he IEEE 802.11a sandard [22]). Given he channel condiion SNR i, he bi error rae (BER) when he PHY-layer sraegy phy n i of s i is deployed becomes p e (SNR i, phy n i ). Assuming independen bi error probabiliies, he packe loss probabiliy p l for WSTA i is given by p l (L i, phy n i )=1 (1 p e(snr i, phy n i ))L i (2) where L i denoes he average packe size of WSTA i in bis. For a given PHY-layer sraegy phy n i, he PHY goodpu is given by i (SNR i, phy n i )= N pk i L i (1 p l (L i, phy n i )) ( ) L i + T MAX (SNR i,phy n i ) ack + T OH N pk i + αt ex OH where MAX (SNR i, phy n i ) is he maximum achievable daa rae for he PHY-layer sraegy phy n i and T ack denoes he ime for acknowledgmen. T OH includes he shor inerframe space ime (T SIFS ) and he ime for he PHY-layer overheads (T PHYOH ), i.e., T OH =2(T SIFS + T PHYOH ). Addiional overhead is inroduced o consider he required overhead for he exernal informaion exchanges for he proposed resource managemen. Since he exernal informaion can be exchanged over each SI or a group SIs, TOH ex is considered only when he exernal informaion is exchanged, which is represened by he indicaor funcion α {0, 1}. Hence, he denominaor in (3) shows he oal required ime for ransmiing N pk i, which represens he number of packes ha can be ransmied in i, compued by N pk i = i. The PHY goodpu L i / MAX (SNR i,phy n i in (3) can be rewrien as ) T ex OH (3) )=Rphy i (SNR i, phy n MAX (SNR i, phy n i )(1 p l(l i, phy n i )) i β i,oh where ( ) β i,oh = 1+ Rphy MAX (SNR i, phy n i )(T ack + T OH ) L i + α Rphy MAX (SNR i, phy n i ) T OH ex L i N pk. (4) i The following seps involved in he cross-layer opimizaion are derived based on [6]. The MAC-layer sraegy is able o adap he reransmission per packe. In [6], i was shown ha given he packe disorion impac, he opimal packe scheduling sraegy for a scalable video coder is o ransmi he highes prioriy packe wih he maximum number of reransmissions given is delay deadline. Thus, given he PHY-layer sraegy, he maximum number of reransmission of packe v of WSTA i inasican be compued as N MAX RT i (L i,v)= MAX (SNR i, phy n i ) min(dt L i i (v), i ) 1 where i DT delay i (v) i rans (v) for he delay deadline delay i (v) of he packe v and he expeced ime insance i rans (v) ha WSTA i sars o ransmi he packe v for he firs ime. The sraegy app l i, l {1,...,NAPP i } may correspond o he adapaion of video compression parameers, packeizaion, raffic prioriizaion, and scheduling for WSTA i. The packe prioriizaion and ransmission iming of he packes (scheduling sraegy) are deermined a he APP-layer. The firs packe is ransmied a i rans. The subsequen packes, however, need o consider he expeced ransmission ime of he previous packes. Unil he packe v is successfully ransmied or he reransmission limi is reached, he average number of ransmissions can be compued as N x i (s i,n MAX RT MAX RT i (L i,v)+1 i (L i,v)) = 1 p l(l i, phy n i )N 1 p l (L i, phy n i ). (5) Hence, he average number of packes ha can be correcly ransmied during he ime i for WSTA i can be compued as N pk i ( i, s i )= { max q i L i q k=1 Nx i (s i,n MAX RT } i (L i,v k i )) MAX (SNR i, phy n i )/β i,oh where v k i denoes he kh packe of WSTA i. Therefore, he average bi rae a he APP-layer for WSTA i in ransmiing duraion i can be compued as R i ( i, s i )= Npk i ( i, s i ) L i. (7) Given he PHY-layer sraegy appropriaely seleced based on a WSTA s channel condiion, he BER (or packe error rae) is very small. Hence, we can approximae he (6)
300 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 20, NO. 2, FEBRUARY 2010 average number of packe ransmissions (including reransmissions) given in (5) as follows: N x 1 i (s i,nmax RT i (L i,v)) 1 p l (L i, phy i ) (8) where s i is he opimal cross-layer sraegy, phy i and N MAX RT i denoes he corresponding PHY-layer and MAC-layer sraegies for WSTA i. Thus, he average number of packes ha can be correcly ransmied during he TXOP allocaion i can be approximaed as N pk i ( i, si ) Rphy MAX (SNR i, phy i ) L i βi,oh (1 p l (L i, phy i ) i where βi,oh represens β i,oh in (4) wih he seleced PHYlayer sraegy phy i. Therefore, he approximae average video bi rae (i.e., bi rae a he APP-layer) can be expressed as R i ( i, si )=Npk i ( i, si )L i = i (SNR i, phy i ) i. (9) III. Dynamic Resource Managemen and Cross-Layer Opimizaion The convenional resource managemen sraegy discussed in he previous secion is inefficien because i does no consider he ime-varying source and channel characerisics, and he cross-layer sraegies adaped by WSTAs are passively opimized. To address his limiaion, we propose a dynamic resource managemen framework which explicily considers ha he WSTAs adap heir cross-layer ransmission sraegies in real-ime. The proposed sysem is able o deermine fair wireless resource allocaion sraegies across auonomous WSTAs, despie he informaionally disribued naure of he problem. This is achieved by allowing WSTAs o dynamically exchange informaion abou heir uiliies and resource requiremens depending on heir insananeous channel and source characerisics. The dynamic resource allocaion means ha he TXOP allocaion is repeaedly divided every SI or group of SIs depending on he channel condiion, cross-layer sraegy, and used fairness policy. The resul of he resource allocaion is represened by he TXOP allocaion =( 1,..., M ), where i (0 i ) denoes he allocaed TXOP o WSTA i and M i. We model his TXOP allocaion problem as a game played by WSTAs hrough adaping heir crosslayer sraegies and hus, operaing a differen qualiy levels. To enable he proposed dynamic resource managemen, each WSTA will need o deermine is exernal informaion and ransmi i o he resource manager. Noe ha his exernal informaion is he sraegy wih which a WSTA plays he resource managemen game, and i will be discussed in Secion IV. Self-ineresed and auonomous WSTA i ries o obain as much TXOP allocaion i as possible, while simulaneously selecing he opimal cross-layer sraegy o maximize is uiliy. In he proposed dynamic resource managemen approach, he allocaed TXOP i is a funcion of he cross-layer sraegies of Fig. 1. Achievable feasible qualiy ses for wo WSTAs. WSTAs 1 and 2 ransmi Foreman and Coasguard sequences [common inermediae forma (CIF)] under he channel SNRs of 18 db and 23 db, respecively. PHY i denoes he PHY mode choice of WSTA i. The duraion of SI is 100 ms (i.e., = 100 ms) in his example. oher WSTAs since all WSTAs are sharing he limied resource (i.e., ). The corresponding join TXOP allocaion and crosslayer sraegy opimizaion problem is expressed as [si, i ] = arg max U i (R i ( i (S i, s i ), s i )) s i S i, 0 i (S i,s i ) = arg max R i ( i (S i, s i ), s i ) s i S i, 0 i (S i,s i ) where S i = M k=1,k i S k. We illusrae how he uiliy se is affeced by his resource allocaion and cross-layer sraegy. Fig. 1 shows he achieved qualiy ses for he simple case of wo WSTAs. In Fig. 1, we observe ha he qualiy derived by one WSTA impacs he qualiy ha can be derived by he oher WSTA due o he ime resource sharing. Moreover, we observe ha differen cross-layer sraegies induces differen feasible uiliy (or qualiy) ses, where WSTA 1 has a fixed PHY mode, bu WSTA 2 is able o deploy wo differen PHY modes. As a resul, differen feasible qualiy ses are formed depending on he cross-layer sraegies deployed by WSTA 2, hereby showing ha if one WSTA adops a beer cross-layer sraegy, an improved uiliy se can be formed (i.e., a superse of he original se). Hence, for example, if a WSTA having a limied compuaional power canno opimize is cross-layer sraegy, he WSTA can be penalized based on he proposed resource managemen sraegy. This will be analyically invesigaed in Secion V. Based on hese examples, we can conclude ha an efficien algorihm is required o allocae he ime resources fairly and opimally given he compeiive muliuser nework, and a he WSTA side, he cross-layer sraegies need o be opimized as his significanly impacs he video performance. To address his requiremen, we propose o implemen he following dynamic resource allocaion framework a he resource manager side. 1) Session Iniializaion: Prior o he acual video ransmission, he resource manager announces he deployed fairness policy F and collecs basic informaion abou every WSTA,
PARK AND VAN DER SCHAAR: FAIRNESS STRATEGIES FOR WIRELESS RESOURCE ALLOCATION AMONG AUTONOMOUS MULTIMEDIA USERS 301 Fig. 2. Overall sysem for he proposed dynamic resource managemen framework. e.g., ypes of mulimedia applicaions, encoder ypes, ypes of ransmied mulimedia sreams, minimum qualiy and olerable delays, used packe lenghs, ec. This informaion can be used o idenify he corresponding uiliy funcions. Subsequenly, for each SI or group of SIs, he resource manager performs he following seps. 2) Polling and Collecing Informaion: The resource manager polls WSTAs and collecs from hem he exernal informaion (ψ 1,..., ψ M ), which depends on he deployed resource allocaion scheme and is exemplified in deail in Secion IV for he various fairness policies. We denoe as he se of possible exernal informaion. Noe ha various algorihms lead o differen and hus o various ransmission overheads. In his paper, we assume ha he overhead is negligible. 3) Allocaing Time Resources: The resource manager decides he nonnegaive ime resource allocaion ( 1 (S 1, s 1 ),..., M (S M, s M )) based on he colleced exernal informaion and he deployed fairness policy F : R M + defined as F(ψ 1,...,ψ M )=( 1 (S 1, s 1 ),..., M (S M, s M )). 4) Polling WSTAs: Based on he deermined ime resource allocaion ( 1 (S 1, s 1 ),..., M (S M, s M )), he WSTAs are polled. Imporanly, in he above resource allocaion, we assume ha he WSTAs ruhfully declare heir exernal informaion. 1 The wireless sysem framework is shown in Fig. 2. Noe ha uilizaion of some blocks and parameers depends on deployed fairness policy, which will be discussed in Secions IV and V. The following seps summarize how he WSTAs inerac wih he resource manager. 1) Session Iniializaion: Every WSTA sends he basic informaion o he resource manager and lisens o he announced fairness policy. Subsequenly, for each SI or group of SIs, WSTAs perform he following seps. 2) Deploying Cross-Layer Sraegies: Based on he channel condiion, every WSTA deploys he opimal cross-layer sraegy ha maximizes is own uiliy. 3) Deermining and Announcing Exernal Informaion: Every WSTA decides which exernal informaion (ψ 1,...,ψ M ) should be ransmied based on he fairness policy of he resource manager. More deails on he exernal 1 This is an implici assumpion used in all MAC wireless resource managemen implemened oday. This assumpion migh no always be rue, and incenives or penalies migh be hen needed for he WSTA o declare heir exernal informaion correcly. In his case, mechanism design echniques could be used [23] o provide incenives o WSTAs. informaion are discussed in Secion IV. This informaion is announced o he resource manager when i is polled. 4) Transmiing Daa: Every WSTA sars o ransmi when i is polled by he resource manager. Various algorihms can be adoped for video sreaming [3]. In summary, every WSTA decides and declares he exernal informaion based on he fairness policy deployed in he resource manager. Based on he declared exernal informaion, he resource manager deermine a resource allocaion. IV. Exising Fairness Policies and Limiaions In his secion, we review exising fairness policies for resource managemen and highligh how hese policies can be deployed in he discussed dynamic resource managemen and wha are heir limiaions for mulimedia ransmission. A. Maximum Toal Sysem Qualiy (MTSQ) If here are no fairness consrains, maximizing he oal sysem qualiy represens a suiable resource allocaion soluion [23]. The TXOP allocaion =(1,..., M ) in MTSQ sysems is expressed as = arg max 10 log 10 U i (R i ( i, si )) M i where 10 log 10 U i (R i ( i, si )) is PSNR of WSTA i. This opimizaion problem can be solved by sandard convex opimizaion mehods since each PSNR is eiher a linear (e.g., he FGS video coder) or concave (e.g., H. 264 video coder) funcion wih respec o he video rae. The TXOP allocaion = (1,..., M ) in one SI by his sraegy is denoed by F MTSQ (ψ 1,...,ψ M ), where he exernal informaion is ψ i =(SNR i, si ) for all i. The limiaion of his sraegy for compeiive neworks is ha he individual WSTAs qualiies are no explicily considered. B. Equal Time Allocaion (ETA) The ETA sraegy is he simples resource allocaion scheme. The available ime on a channel is equally divided and allocaed o WSTAs. Hence, he corresponding TXOP allocaion in one SI is expressed as = F ETA (ψ 1,...,ψ M )=( /M,..., /M) where ψ i = 0 (i.e., no exernal informaion is required) for all i. While his allocaion seems o be fair, i can be very inefficien in erms of he achieved qualiy, since i allocaes he resources wihou considering he video qualiy, which depends on video characerisics, channel condiions, and deployed cross-layer sraegies. C. Air Time Allocaion (ATA) For he ATA sraegy [4], he available ime on a channel is proporionally divided o he required ime for achieving each WSTA s insananeous rae requiremen. This resource allocaion can be expressed as
302 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 20, NO. 2, FEBRUARY 2010 1 M = = 1,MAX M,MAX where i,max denoes he required ime o achieve he insananeous rae requiremen of WSTA i. Noe ha boh i,max and i are a funcion of he insananeous rae requiremen and he deployed cross-layer sraegy. This policy is equivalen o allocaing rae o WSTAs proporionally, according o each WSTA s insananeous rae requiremens denoed by R i,max for WSTA i, i.e., R 1 ( 1, s1 ) = = R M( M, sm ) (10) R 1,MAX R M,MAX where R i denoes he achievable rae given he TXOP allocaion of he resource manager. Using (9), his can be equivalenly expressed as a funcion of he TXOP allocaion 1 (SNR 1, phy 1 ) 1 = = Rphy M (SNR M, phy M ) M (11) R 1,MAX R M,MAX Noe ha R i,max is only a funcion of he desired qualiy level and video characerisics and no of he deployed cross-layer sraegy. Since he resource manager has already received he basic informaion during he iniializing session, each WSTA needs o only send he channel condiion and he opimal cross-layer sraegy as is exernal informaion. The TXOP allocaion by he ATA policy is denoed by F ATA (ψ 1,...,ψ M ), where ψ i = (SNR i, si ) for all i. The TXOP allocaion mus saisfy (11) and M i =. D. Generalized Processor Sharing (GPS) The GPS sraegy is inroduced in [24] and used as a fair scheduling soluion in several applicaions [4], [25]. For he GPS sraegy, he ime resources can be allocaed proporionally o each WSTA s insananeous rae requiremen. This resource allocaion can be expressed as 1 M = =. (12) R 1,MAX R M,MAX Since he ime resources are allocaed only proporionally o he insananeous rae requiremens, he resource allocaion is independen of he exernal informaion. The TXOP allocaion by he GPS policy is denoed by F GPS (ψ 1,...,ψ M ), where ψ i = 0 for all i. The TXOP allocaion mus saisfy (12) and M i =. Noe ha he ATA and GPS can have limiaions on resource allocaions in uiliy domain for mulimedia applicaions due o he nonlineariy of uiliy funcions even hough hey can be fair (proporional) soluions in resource domain. E. Nash Bargaining Soluion The NBS, which was originally inroduced by Nash [11], can be deployed o divide resources opimally (in he Pareo opimal sense) o WSTAs based on is fairness axioms. For he NBS, he ime allocaion vecor can be deermined such ha he resuling uiliies are maximizing he Nash produc, which is he produc of uiliies. Hence, he TXOP allocaion by he NBS policy can be expressed as = F NBS (ψ 1,...,ψ M ) where he TXOP allocaion maximizes he Nash produc defined as n (U i(r i (i, s i )) d i), or equivalenly n log(u i(r i (i, s i )) d i), while saisfying he resource consrain M i =. The disagreemen poin d =(d 1,...,d M ) will be discussed in he nex secion. Hence, he NBS can be inerpreed as he maximizer of he sum of he logarihmic uiliy funcions, and hus, i can be used for collaboraive WSTAs o achieve he maximum sysem performance (e.g., for collecion of collaboraive users such as cameras in a surveillance applicaions). Therefore, he NBS does no provide a fair resource allocaion for self-ineresed and auonomous WSTAs. F. Proporional Fairness (PF) The noion of PF was firs inroduced in [7], and i proposes a fair soluion in uiliy domain. This fairness noion is used o allocae resources in several applicaions (e.g., [9] and [10]). In [7], i has been shown ha if each user s uiliy funcion is logarihmic, hen he soluion for maximizing he sum of uiliy funcions leads o a proporional fair allocaion. By considering he uiliy funcions o be a logarihm of he video raes, he soluion ha maximizes he sum of he uiliy funcions is he proporional fair allocaion of video raes. Thus, he TXOP allocaion is a proporional fair soluion if = arg max log R i ( i, s i ). (13) M i An ineresing propery of he PF can be obained when he uiliy is se o be he video rae requiremen. In his specific case, he soluion of (13) is exacly he same as he ETA, i.e., =( /M,..., /M) (see Appendix A), and hus F PF = F ETA wih ψ i = 0 for all i. Alernaively, he uiliy funcions in he PF can be considered as a logarihm of he video uiliy funcions [e.g., he uiliy funcions defined in (1)]. In his case, he soluion can be expressed as = arg max M i log U i (R i ( i, si )) = arg max PSNR i. (14) M i Thus, in his case, he PF soluion becomes exacly he same as he MTSQ, i.e., F PF = F MTSQ wih ψ i =(SNR i, si ) for all i. As we discussed in ETA and MTSQ, hese fairness policies can resul in inefficien resource allocaions for auonomous mulimedia users. Moreover, i should be noed ha he PF is a special case of he NBS when he disagreemen poin is he origin as we discussed in Secion IV-E, i.e., F PF = F NBS if d = 0 for he NBS. Hence, he PF also does no provide a fair resource allocaion for auonomous WSTAs as he NBS can only be used for maximizing he sysem performance for collaboraive WSTAs. V. Proposed Uiliy-Fairness based on KSBS As menioned in he inroducion, for a fair allocaion of resources among auonomous WSTAs, i is essenial ha we
PARK AND VAN DER SCHAAR: FAIRNESS STRATEGIES FOR WIRELESS RESOURCE ALLOCATION AMONG AUTONOMOUS MULTIMEDIA USERS 303 consider he relaive impac beween he resuling qualiies of mulimedia users. We argue ha he resource managemen has he following properies. 1) I should be Pareo opimal. 2) I should reward he users effor o increase heir uiliies given a cerain resource allocaion by efficienly adaping heir cross-layer sraegies. 3) I should no be biased oward any paricular user. 4) I should lead o he same resource allocaion independenly of he uiliy calibraion. Specifically, he resource managemen should se he qualiy drop o be he same among users. Alernaively, bias can be induced by weighing he qualiy drop according o he imporance of users [17]. These uiliy-based fairness properies can be achieved by he fairness axioms of he well-known KSBS [16]. A. Required Componens of Mulimedia WSTAs for he KSBS In his secion, we idenify he required elemens for he KSBS. The noaion X i represens he achievable uiliy for WSTA i (i.e., X i = U i ( )) and he vecor inequaliy x y represens componen-wise inequaliy (i.e., x i y i for all i) hroughou his paper. An axiomaic bargaining soluion for a bargaining problem (S, d) where a feasible uiliy se S and he disagreemen poin d, is a funcion F : (S, d) R M such ha F(S, d) S. Hence, i is necessary o idenify he feasible uiliy se S and he disagreemen poin d for mulimedia WSTAs in order o deploy he KSBS. 1) Feasible Uiliy Se: A feasible uiliy se S is he se of all uiliy pairs ha every WSTA can joinly form given all possible TXOP allocaions. Thus, he feasible uiliy se can be expressed as { S = (U 1 (R 1 ( 1, s 1 )),...,U M (R M ( M, s M ))) } i SI, s i S i for all i. (15) The feasible uiliy se needs o be a leas comprehensive for he KSBS [26]. Definiion 1 (d-comprehensive Se): Given a poin d R M and a se S R M, he se S is d-comprehensive if d x y and y S implies x S. Proposiion 1: The feasible uiliy se S is d-comprehensive. Proof: See Appendix B. 2) Disagreemen Poin: A se of minimum achievable uiliies for all WSTAs is referred o as he disagreemen poin (d). The disagreemen poin can be achieved when WSTAs do no reach an agreemen in a negoiaion process, obaining heir minimum uiliies from a game. Hence, if raional WSTAs join he resource managemen game, hey expec o achieve higher uiliies han he disagreemen poin. Thus, he disagreemen poin can be expressed as d = ( ( ) Xmin 1,... ),XM min = min X 1,...,min X M S. X S X S This disagreemen poin plays a very imporan role in rae allocaion for video WSTAs. As we discussed in Secion II-A, based on differen video characerisics and/or semanic imporance, ha may be varying over ime, he minimum uiliy (qualiy) requiremens of various WSTAs can be differen (i.e., he minimum accepable PSNR for various video sequences is differen). Hence, he resource manager guaranees he minimum uiliy requiremen for each WSTA, by correspondingly adjusing he disagreemen poin. For insance, he disagreemen poin can be deermined based on he base-layer qualiy (i.e., minimum accepable qualiy) for he MPEG-4 FGS video coder or H. 264, which needs o be saisfied when ransmiing video sequences. This feaure can be suppored by he proposed KSBS, which is essenial for mulimedia applicaions sreamed over ime-varying channels. The decision of he disagreemen poin can be deermined and communicaed during he session iniializaion. For he simpliciy of he noaion, we assume in he remainder of he paper ha he disagreemen poin coincides wih he origin (i.e., d = 0) of he uiliy domain, as he feasible uiliy se can be correspondingly ranslaed based on he minimum uiliy requiremen as proven in [26]. 3) Fairness Properies of KSBS: The KSBS gives a unique Pareo opimal soluion ha fulfills he fairness axioms proposed in [16]. A general inerpreaion of hese axioms for mulimedia applicaion is also provided in [17]. For mulimedia WSTAs, he fairness axioms of individual monooniciy saes ha increasing he maximum achievable uiliy in a direcion favorable o WSTA i always benefis WSTA i. Formally, given anoher feasible uiliy se S, if S S, d = d, and max X S,X d X k =max X S,X d X k for all k {1,...,M}\{i}, hen [F(S, d )] i [F(S, d)] i.for example, le (S, d) and (S, d) be wo bargaining problems, where S S and he maximum achievable uiliies of all WSTAs are he same excep WSTA i. Individual monooniciy saes ha he WSTA i gains more uiliy in (S, d) han in (S, d) by he KSBS. Based on he effec of differen cross-layer sraegies on each WSTA, his propery provides a srong moivaion o deploy he opimal cross-layer sraegy for auonomous mulimedia WSTAs since he individual monooniciy guaranees o improve one WSTA s uiliy if i adops a beer cross-layer sraegy. This propery is shown in Proposiion 2. Proposiion 2: When resources are allocaed based on he KSBS, if one WSTA deploys a beer cross-layer sraegy given a channel condiion, his always benefis his WSTA. Proof: See Appendix C. B. The KSBS for Mulimedia WSTAs For he bargaining problem (S, d) idenified in Secion V-A, he KSBS X = F(S, d) = (X1,...,X M ) for M WSTAs saisfies [16] X = F(S, d) =d + λ MAX (X MAX d) (16) where X MAX =(X 1 MAX,...,XM MAX ) for Xl MAX = max X S, X d X l, l =1,...,M,isheideal poin and λ MAX = max λ {λ d + λ(x MAX d) S}. The KSBS for mulimedia WSTAs can be inerpreed as [17]
304 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 20, NO. 2, FEBRUARY 2010 Algorihm 1 KSBS Implemenaion Require: WSTA characerisics informaion, exernal informaion ψ i =(si, SNR i) for all i. 1: Idenify he feasible uiliy se S and he disagreemen poin d given he exernal informaion 2: Compue he KSBS for (S, d); (X1,...,X M )=F(S, d) 3: Compue he TXOPs; i = R 1 i (Ui 1 (Xi )) for all i 4: Poll WSTAs based on he allocaed TXOPs PSNR 1 +10 log 10 α 1 = = PSNR M +10 log 10 α M (17) where PSNR i (PSNR i MAX PSNR i ) denoes he qualiy decrease (or drop) from WSTA i s maximum achievable qualiy and α i is he bargaining power assigned o WSTA i. Hence, he resource allocaion based on he fairness provided by he KSBS is suiable for auonomous mulimedia WSTAs since he qualiy drops from heir maximum achievable qualiies for all WSTAs are he same (if all bargaining powers are he same). Noe ha differen bargaining powers can be assigned o he differen users based on heir mulimedia characerisics (e.g., higher moion, ec.), and i leads o he adjused qualiy drop. The bargaining powers can be deermined based on several rules for mulimedia [17]. The TXOP allocaion process in one SI based on he KSBS can be expressed as = F KSBS (ψ 1,...,ψ M ) =(R 1 U 1 (18) F U R)(ψ 1,...,ψ M ) where a composie funcion of f and g is denoed by f g(x) = f (g(x)) and F is he KSBS. U and R denoe a se of uiliy funcions and a se of rae funcion, i.e., U = (U 1 ( ),...,U M ( )) and R = (R 1 ( ),...,R M ( )). The exernal informaion is ψ i =(SNR i, si ) for all i. The involved seps are shown in Algorihm 1. C. Fairness Comparison in Terms of Mulimedia Qualiy In his secion, we compare he fairness achieved by differen resource managemen sraegies discussed in his paper. To quanify how fairly he resources are allocaed by he resource manager, we inroduce a new FCM. We define he FCM such ha i can compare he performances achieved by differen resource managemen sraegies. By defining he social uiliy funcion for a resource managemen sraegy F and he corresponding resource allocaion F as { U sys ( F ) = max PSNR i MAX PSNR i ( i) } 1 i M = max { PSNR i( i )} (19) 1 i M he fairness of differen resource managemen sraegies can be compared in erms of maximum qualiy drop. U sys ( F ) defined in (19) represens he larges qualiy drop among WSTAs in he nework. Thus, he social opimal sraegy F can be deermined such ha i minimizes he maximum qualiy drop, i.e., F = arg min U sys ( F ) (20) F F where F denoes a se of available resource managemen sraegies. In he considered resource managemen sraegies, F = F KSBS because he KSBS resuls in he same qualiy drop among WSTAs. Noe ha he definiion of funcion U sys ( F ) in (19) is moivaed by an egaliarian social welfare funcion discussed in [27]. For he social uiliy funcion in (19), he raio beween U sys ( F ) for F F and U sys ( FKSBS ) can be used as a similar meric o he price of anarchy [28], which measures he fairness achieved by differen resource managemen sraegies, because F KSBS is he socially opimal sraegy. Specifically, he raio, denoed by FCM F, is defined as FCM F = U sys( F ) U sys ( FKSBS ). (21) Noe ha FCM F becomes larger if sraegy F resuls in a resource allocaion ha leads o a larger qualiy drop among WSTAs. This meric can be furher exended by considering oher social uiliy funcions (e.g., oal aggregaed uiliy, ec.), which can emphasize oher aspecs of opimaliy as well as fairness. Alernaively, he reference social uiliy funcion U sys ( FKSBS ) deermined based on he KSBS can also be generalized by inroducing bargaining powers. The FCM for various resource managemen sraegies are quanified in Secion VI-A. D. Low-Complexiy Implemenaions for he KSBS In he preceding secions, we formulae he bargaining problem and provide he inerpreaion of he KSBS given a fixed channel condiion. However, he channel condiion is ime-varying even in he case when he WSTAs are no mobile. To successfully consider he ime-varying channel condiion, he complexiy required for deploying he KSBS needs o be considered. 2 In his secion, we design algorihms for he KSBS in order o reduce he required compuaional complexiy. 1) Exernal Informaion Exchanges in Every SI: If channel condiion is ime-varying, he opimal sraegy is o deploy he KSBS o every SI, i.e., repeaedly apply Algorihm 1 in every SI. However, as discussed, i requires high-compuaional complexiy o obain he KSBS (including he formaion of he feasible uiliy se) in every SI. More specifically, if he resource manager considers quanized service inervals wih sep size ( ), hen ( / ) M uiliy poins in he feasible uiliy se need o be idenified. Thus, he compuaional complexiy C(M) required for he KSBS can be expressed as C(M) = P ( / ) M, where P is a posiive consan. This implies ha he ime required for resource allocaion based on he KSBS increases exponenially wih respec o he number of users in he nework. Hence, he required complexiy in oal during he ransmission ime T can be expressed as ( ) T T M SI C 1 (M) = C(M) =P. 2 We assume ha he required overheads for exchanging he exernal informaion are negligible, as hey can be expressed wih a few byes and can be augmened in TSPEC.
PARK AND VAN DER SCHAAR: FAIRNESS STRATEGIES FOR WIRELESS RESOURCE ALLOCATION AMONG AUTONOMOUS MULTIMEDIA USERS 305 Algorihm 2 Channel condiion or video characerisics driven exernal informaion exchanges for WSTA Require: Channel condiion variaion hreshold δ c, video characerisics variaion hreshold δ q, previous and curren channel condiion: SNR i and SNR i, previous and curren video characerisics: Vi and V i, previous TXOP allocaion i. 1: loop 2: if SNR i SNR i δ c hen 3: Find he bes cross-layer sraegy given SNR i ; si 4: Reques new TXOP allocaion: send ψ i = (SNR i, si ) o resource manager 5: New TXOP based on he KSBS (Algorihm 1) by he resource manager; i 6: else if V i Vi δ q hen 7: Reques new TXOP allocaion: send mulimedia characerisics informaion o resource manager 8: New TXOP allocaion based on he KSBS (Algorihm 1) by he resource manager; i 9: else 10: Updae TXOP allocaion; i := i 11: end if 12: end loop Algorihm 3 Ieraive mehod for he KSBS wih no exernal informaion exchange Require: TXOP adjusmen sep. 1: loop 2: Received qualiy drop for previous allocaed TXOP allocaion ( 1,..., M); PSNR =( PSNR 1,...,PSNR M). 3: Compue qualiy drop change; P =( PSNR PSNR ) 4: Adjus TXOP allocaion based on P; i = i +[ P] i for all i. 5: end loop 2) Channel Condiion or Video Characerisics Driven Exernal Informaion Exchanges: Small variaion of he channel condiion or he video characerisics for WSTAs does no induce significan changes in heir seleced cross-layer sraegies or achievable qualiies. Hence, here will be small changes in he feasible uiliy se as well as he resuling KSBS. Hence, he compuaional complexiy for he TXOP allocaion can be significanly reduced only by exchanging he exernal informaion and compuing he KSBS when channel condiion or video characerisics changes significanly. WSTAs keep esimaing he channel condiion and he video characerisics. When he channel condiion variaion is larger han he predeermined hreshold δ c, or he video characerisics variaion is larger han he hreshold δ q, hey are allowed o reques a new TXOP allocaion. Noe ha hreshold δ c and δ q can be adapively adjused by he resource manager by considering he visual impac of WSTAs on he achieved qualiy. The required seps for WSTAs are presened in Algorihm 2. Noe ha his algorihm needs o be implemened by he WSTAs. As shown in Algorihm 2, he required compuaional complexiy in he resource manager can be esimaed as ( ) M SI C 2 (M) =m C(M) =P m where m ( T/ ) represens he number of SIs where WSTAs reques new TXOP allocaions during ransmission. Therefore, he complexiy reducion by deploying Algorihm 2 as compared o he approach in Secion V-D1 is given by C 2 (M) C 1 (M) = P m ( / ) M P T/ ( / ) M = m T/ 1. Thus, he complexiy reducion achieved based on Algorihm 2 becomes significan if he channel condiions or he video characerisics of he WSTAs do no considerably change, i.e., smaller value of m. 3) Qualiy Drop as Exernal Informaion: The required compuaional complexiy can be furher reduced by exchanging he informaion abou he qualiy drop, insead of exchanging he exernal informaion or he video characerisics informaion for he KSBS. Since he exernal informaion or he video characerisics informaion is no exchanged, he resource manager canno compue he KSBS. However, he resource manager can use he qualiy drop informaion for each WSTA o obain a soluion o he KSBS. This algorihm is developed based on he inerpreaion of he KSBS for he mulimedia shown in (17). In every SI, each WSTA compues is own qualiy drop for he given TXOP allocaion and send his informaion o he resource manager. Then, he resource manager can adjus is TXOP allocaion, such ha WSTAs can achieve he same qualiy drop (or adjused qualiy drop based on he bargaining powers). This soluion significanly reduces he compuaional complexiy, as he resource manager does no need o compue he KSBS direcly. This TXOP allocaion algorihm is presened in Algorihm 3. Noe ha in Algorihm 3 can be adapively predeermined based on applicaions. This algorihm requires only a consan compuaional complexiy. Hence C 3 (M) =P M where P is a posiive consan. Thus, he complexiy reducion based on Algorihm 3 as compared o he approach in Secion V-D1 is given by C 3 (M) C 1 (M) = P M P T/ ( / ) M 1 (22) for M 2. Therefore, he complexiy reducion can be significanly improved as he number of WSTAs in a nework increases. VI. Simulaion Resuls In his secion, we firs show simulaion resuls comparing he KSBS wih he oher soluions described in Secion IV. Then, we show he effec of bargaining powers in he KSBS. For a simulaion seup, each WSTA is assumed o have he abiliy o choose he opimal cross-layer sraegy given he channel SNR. Based on his informaion, he resource manager allocaes he available ime resources o he WSTAs during each SI. The parameer values for he DR models of he differen coders are deermined based on he H.264 video coder and he FGS video coder.
306 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 20, NO. 2, FEBRUARY 2010 TABLE I Resource Allocaion Based on Fairness Policies Scenario SNR [db] Sraegy Time Allocaion [ms] PSNR [db] PSNR i [db] FCM F MTSQ [38.0, 62.0] [38.1039, 30.3372] [4.1927, 3.8262] 1.0045 ETA [50.0, 50.0] [39.2957, 28.9654] [3.0008, 5.1979] 1.2453 ATA [15.3, 84.7] [34.1583, 32.9298] [8.1383, 1.2335] 1.9498 1 [28, 23] GPS [15.9, 84.1] [34.3283, 32.8599] [7.9682, 1.3034] 1.9090 KSBS (0.5, 0.5) [38.2, 61.8] [38.1225, 29.9893] [4.1740, 4.1740] 1.00 KSBS (0.2, 0.8) [16.5, 83.5] [34.4741, 32.3615] [7.8224, 1.8018] KSBS (0.6, 0.4) [46.6, 53.4] [38.9765, 29.0823] [3.3201, 5.0810] MTSQ [98.2, 1.8] [42.2272, 23.3025] [0.0694, 2.7845] 1.5035 ETA [50.0, 50.0] [39.2957, 24.7081] [3.0008, 1.3789] 1.6203 ATA [4.4, 95.6] [28.7519, 26.0376] [13.5447, 0.0494] 7.3136 2 [28, 13] GPS [15.9, 84.1] [34.3283, 25.7016] [7.9682, 0.3853] 4.3025 KSBS (0.5, 0.5) [65.3, 34.7] [40.4445, 24.2349] [1.8520, 1.8520] 1.00 KSBS (0.2, 0.8) [21.7, 78.3] [35.6604, 25.4714] [6.6361, 0.6155] KSBS (0.6, 0.4) [85.7, 14.3] [41.6263, 23.6558] [0.6703, 2.4312] TABLE II Consecuive Bargaining Over Time-Varying Channel Scenario Sraegy SNR [db] PSNR i [db] PSNR Improvemen [%] PSNR i 1 [s 1 s 2 s 3 s 4 s 5 ] [10 18 21 25 28] [28.14 29.97 30.70 18.85 18.85] 7.51 db 2 [s 1 s 2 s 3 s4 s 5 ] [10 18 21 25 28] [29.27 31.10 31.83 29.27 29.37] [4.0 3.8 3.7 55.3 55.8] 6.38 db 3 [s 1 s 2 s 3 s 4 s 5 ] [23 20 21 25 28] [29.33 31.16 31.83 29.27 29.37] [0.2 0.2 0.0 0.0 0.0] 6.38 db 4 [s1 s 2 s 3 s 4 s 5 ] [23 20 21 25 28] [38.01 31.84 31.89 29.33 29.43] [29.6 2.2 0.2 0.2 0.2] 6.32 db A. Comparison wih Exising Fairness Policies In his secion, we compare he resource allocaions discussed in Secion IV. We assume ha here are wo WSTAs in he sysem. For he ATA policy, we used he 35 db qualiy level, which is considered as a desired video qualiy level for mos videos, and hus, he R 1,MAX and R 2,MAX in (10) are deermined o saisfy his qualiy level. For he KSBS, we use several bargaining powers. We assume ha is 100 ms and he channel SNR is fixed in his duraion. The packe lengh has a maximum lengh of 500 B. The simulaion resuls are shown in Table I for wo channel condiion scenarios. The channel SNRs for WSTAs 1 and 2 are 28 db and 23 db in scenario 1, and 28 db and 13 db in scenario 2. From he simulaion resuls for he wo scenarios, we found ha he various fairness crieria of he resource manager derive disinc resource allocaions, resuling in differen achieved video qualiies. In boh scenarios, hough he MTSQ policy leads o he maximum sum of he PSNRs, he qualiy difference beween WSTAs is significanly large. This is unfair for noncollaboraive muliuser ransmission. This unfairness is increased when he channel condiion is furher degraded (see scenario 2 in Table I). The ETA policy achieves a fair allocaion in erms of he ime resources, bu i is inefficien in he uiliy domain. This policy achieves neiher he highes sum of PSNR nor a similar video qualiy level for WSTAs. The GPS policy leads o proporional allocaion in he ime resources. Though his policy adaps based on he video sequences characerisics o some exen, i is independen of he channel condiions and i does no consider he uiliy explicily. The ATA policy enables he WSTAs o achieve a similar qualiy level. However, i should be noed ha his policy is unfair as i severely penalizes he WSTA experiencing a beer channel condiion. This unfairness becomes worse when he WSTA channel condiion furher degrades (see he qualiy drop). Hence, his policy is unfair and undesirable for mulimedia applicaions. However, he KSBS allocaes he resources such ha WSTAs achieve he same qualiy penaly (see he qualiy drop). The fairness achieved by differen resource allocaion schemes are quanified based on he FCM defined in (21). As discussed in Secion V-C, he value of FCM increases for a resource managemen sraegy F as i resuls in larger qualiy drops among WSTAs. This can be verified from he resuls in Table I. Noe ha even if in some cases he fairness policies provide a similar resource allocaion, only he KSBS can enable he implemenaion of differen fairness crieria based on he video qualiy experienced by he WSTAs by inroducing bargaining powers. For example, if he goal of he resource managemen is o achieve a similar qualiy level for he WSTAs, bargaining powers around (0.2, 0.8) can be used. Alernaively, if he goal of he resource managemen is o maximize he oal sum of PSNR values, bargaining powers around (0.6, 0.4) can be used. B. Ineracion Among WSTAs In his secion, we show simulaion resuls o quanify he impac of one WSTA s cross-layer sraegy on he oher WSTAs uiliies. For he simulaion, we assume ha here are five WSTAs ransmiing differen CIF video sequences a 30 Hz, i.e., Foreman (WSTA 1), Coasguard (WSTAs 2 and 3), and Mobile (WSTAs 4 and 5) encoded by he wavele video coder. The resource manager deploys he KSBS for he resource allocaion. The simulaion resuls for four scenarios are presened in Table II. In scenarios 1 3, we can observe he impac of he deployed cross-layer sraegies given a channel condiions. In scenario 2, WSTAs 4 and 5 deploy beer cross-layer sraegies
PARK AND VAN DER SCHAAR: FAIRNESS STRATEGIES FOR WIRELESS RESOURCE ALLOCATION AMONG AUTONOMOUS MULTIMEDIA USERS 307 Fig. 3. Achieved qualiy based on Algorihm 2. (a) Large variaion of channel condiion. (b) Small variaion of channel condiion. Fig. 4. Achieved qualiy based on Algorihm 3. (a) Large variaion of channel condiion. (b) Small variaion of channel condiion. han hose in scenario 1 given he same channel condiion, leading o beer qualiy improvemen for hem due o he individual monooniciy of he KSBS. In scenarios 2 and 3, all WSTAs mainain heir cross-layer sraegies (i.e., hey do no opimize heir cross-layer sraegies for he change of channel condiions) even hough he channel condiions are improved. In his case, we observe ha here is almos no uiliy improvemen for all of hem. In scenario 4, WSTAs 1 and 2 deploy opimized cross-layer sraegies han hose in he scenario 3 given he channel condiion, which resuls in an improved qualiy for hem as well. From hese simulaion resuls, we can conclude ha he cross-layer ransmission sraegies wih which WSTAs play he resource managemen game are very imporan and have an essenial impac on boh he individual qualiy of he WSTAs as well as heir impac on he uiliy of he compeing WSTAs. C. Comparison of Proposed Algorihms In Secion V-D, several algorihms for he efficien KSBS implemenaions are developed. We quanify he performance of he algorihms focusing on he achieved qualiy and he required complexiy. In he simulaions, we assume ha he = 100 ms and channel condiion varies over ime for WSTAs [6]. Fig. 3 highlighs he achieved qualiy and he channel adapaion of a WSTA based on he proposed Algorihm 2. When he channel condiion changes significanly [Fig. 3(a)], i.e., more han he hreshold δ c, he WSTAs reques new TXOP allocaions o he resource manager more ofen, hence he qualiy achieved by Algorihm 2 coincides a imes wih hose obained by he KSBS. Thus, Algorihm 2 does no provide considerable gain in erms of he compuaional complexiy when channel condiion changes significanly. However, if he channel condiion varies slowly [Fig. 3(b)], he WSTAs based on Algorihm 2 do no frequenly reques new TXOP allocaions, which can reduce he compuaional complexiy associaed wih he resource allocaion implemenaion. Hence, we can observe ha here is a small qualiy gap (a mos 0.5 db PSNR) beween he qualiies achieved by he KSBS a every SI and Algorihm 2. We assume ha if no exernal informaion is exchanged, he TXOP allocaion is fixed from he beginning of he ransmission. Similarly, Fig. 4 highlighs he achieved qualiy and he channel adapaion of a WSTA based on he proposed Algorihm 3. Noe ha his algorihm does no require he direc compuaion of he KSBS, which can significanly reduce he compuaional complexiy. We can observe ha his algorihm can provide a quie similar performance in erms of he achieved qualiy o he KSBS a every SI. Therefore, we conclude ha he proposed algorihms can provide pracical soluions for implemening he KSBS wih significanly less compuaional complexiy and informaion exchanges.
308 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, VOL. 20, NO. 2, FEBRUARY 2010 VII. Conclusion In his paper, we have aimed a addressing he problem of fair allocaion of resources among muliple wireless mulimedia users. We review several exising fairness policies, analyze heir performance in erms of he resuling mulimedia qualiy, and discuss heir limiaions. We also propose an uiliy-based fairness soluion (KSBS) ha enables every WSTA o experience he same qualiy drop from is maximum achievable qualiy. In he simulaion resuls, we show ha he KSBS provides a fair resource allocaion for mulimedia applicaions. Moreover, we quanify he impac of one WSTA s cross-layer sraegies on oher WSTAs achievable qualiy. Finally, we show ha he pracical soluions, which significanly reduce he compuaional complexiy, can provide a similar performance o he KSBS when channel condiion or video characerisics are changing. Appendix A We show ha he PF is he ETA if he uiliy funcions are se o be a logarihm of he video raes. The opimizaion problem in (13) is = arg max log R i ( i, s i ) M i = arg max M i [ log i (SNR i, phy i ) i To simplify he noaion, we subsiue a i for he erm i (SNR i, phy i ). Then, he objecive funcion is expressed as M log a i i = log M a i i. Since he logarihmic is a nondecreasing funcion, his opimizaion problem is equivalen o M M (1,..., M )=argmax i a i (23) ( 1,..., M ) SI where M i and i 0. Using he well-known relaionship beween arihmeic and geomeric mean and he fac ha M i =,wehave ( i M M i SI ]. ) 1 M. (24) The equaliy holds when 1 / = = M /. Since a i is consan for all i if he cross-layer sraegy and he channel condiion are given, he soluion of he opimizaion problem (i.e., he TXOP allocaion by he proporional fairness) is (1,..., M )=(/M,..., /M), which is he ETA. Thus, he achieved rae of WSTA i by he proporional fairness is a i i = a i /M = i (SNR i, phy i )/M. Noe ha similar proof was derived from [29]. Inversely, we can show ha he ETA saisfies he proporional fairness crieria [7]. A vecor of rae (a 1 /M,...,a M /M) is proporionally fair if i is feasible and if for any oher feasible vecor of rae x =(a 1 1,...,a M M SI ) saisfies x i a i /M a i /M 0. (25) For any oher feasible vecor of rae, his is rue because x i a i /M M a i /M = a i i a i /M a i /M = M i M M M =0. (26) Appendix B Proof of Proposiion 1 Le S be he feasible uiliy se for a cerain cross-layer sraegy and here is a given poin d R M. Suppose ha d x y and y S. Using he definiion of uiliy funcions, we can express x and y wih respec o feasible TXOP allocaions ( 1,..., M ) and ( 1,..., M ), where M i and M i, as follows: x =[U 1 (R 1 ( 1 )),,U M (R M ( M ))] T, y =[U 1 (R 1 ( 1 )),,U M(R M ( M (27) ))]T. Noe ha he rae in (27) is only a funcion of ime no a cross-layer sraegy because he feasible uiliy se S is already formed by a se of cross-layer sraegies. Since he uiliy funcion U i (R i ( i )) is a nondecreasing for a rae R i ( i ), he following inequaliies are equivalen: x T y T [U 1 (R 1 ( 1 )),...,U M (R M ( M ))] [U 1 (R 1 ( 1 )),...,U M(R M ( M ))] [R 1 ( 1 ),...,R M ( M )] T [R 1 ( 1 ),...,R M( M )]. From he rae R i ( i ) in (7), we have he following equivalen inequaliies: R i ( i ) R i ( i )= Npk i ( i ) N pk i ( i )= i i. Hence, x y [ 1,..., M ] T [ 1,..., M ]T, and herefore, x S, since y S. Appendix C Proof of Proposiion 2 Le S be he feasible uiliy se formed by he WSTAs crosslayer sraegies {s 1,...,s M }, where s i =[phy n i i, mac m i i, app l i i ]. Then, he se S is expressed as { } S = (U 1 (R 1 ( 1, s 1 ))),...,(U M (R M ( M, s M ))) i where U i ( ) and R i ( ) are he uiliy funcion and he average video bi rae funcion defined in (7), respecively. Le s j = [phy n j j, mac m j j, app l j j ] be anoher cross-layer sraegy, which is available o WSTA j and enables i o achieve a higher uiliy. Hence, he resuling feasible uiliy se S is expressed as S = {(U 1 (R 1 ( 1, s 1 ))),...,(U j (R j ( j, s j ))),..., (U M (R M ( M, s M ))) i }. (28) Based on he cross-layer sraegies {s 1,...,s j,...,s M}, a larger feasible uiliy se can be formed, S S. By he axiom of individual monooniciy of he KSBS, he achieved uiliy for WSTA j is always improved.
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Hyunggon Park (S 08 M 09) received he B.S. degree (magna cum laude) in elecronics and elecrical engineering from he Pohang Universiy of Science and Technology, Pohang, Gyungbuk, Souh Korea, in 2004, and he M.S. and Ph.D. degrees in elecrical engineering from he Universiy of California, Los Angeles (UCLA), in 2006 and 2008, respecively. Currenly, he is an Assisan Professor wih he Deparmen of Elecronics Engineering, Ewha Womans Universiy, Seoul, Korea. In 2008, he was an Inern a he IBM T. J. Wason Research Cener, Hawhorne, NY, and he was a Senior Researcher wih he Signal Processing Laboraory (LTS4), Swiss Federal Insiue of Technology, Lausanne, Swizerland from 2009 o 2010. His research ineress include game-heoreic approaches for disribued resource managemen (resource reciprocaion and resource allocaion) sraegies for muliuser sysems, and muliuser ransmission over wireless/wired/peer-opeer neworks. Dr. Park received he Graduae Sudy Abroad Scholarship from he Korea Science and Engineering Foundaion during 2004 2006, and received he Elecrical Engineering Deparmen Fellowship a UCLA in 2008. Mihaela van der Schaar (M 98 SM 04 F 10) received he M.S. and Ph.D. degrees in elecrical engineering from he Eindhoven Universiy of Technology, Eindhoven, he Neherlands, in 1996 and 2001, respecively. She is currenly an Associae Professor wih he Deparmen of Elecrical Engineering, Universiy of California, Los Angeles. Her research ineress include mulimedia communicaions, neworking, processing, and sysems. Prof. van der Schaar received he Naional Science Foundaion Career Award in 2004, he Bes Paper Award from he IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY in 2005, he Okawa Foundaion Award in 2006, he Inernaional Business Machines Faculy Award in 2005, 2007, and 2008, and he Mos Cied Paper Award from he European Associaion for Signal Processing: Image Communicaions Journal in 2006. She was an Associae Edior for IEEE TRANSACTIONS ON MULTIMEDIA, IEEE SIGNAL PROCESSING LETTERS, IEEE TRANS- ACTIONS ON CIRCUITS AND SYSTEMS FOR VIDEO TECHNOLOGY, and SIGNAL PROCESSING MAGAZINE among ohers. She holds 30 graned U.S. paens and hree ISO awards for her conribuions o MPEG video compression and sreaming inernaional sandardizaion aciviies.