RESOURCE allocation is an important research topic in

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1 36 IEEE TRANSACTIONS ON WIRELESS COMMNICATIONS, VOL. 6, NO., OCTOBER 27 tlty-based Resoce Allocaton n Weless Netwoks Wen-Hsng Ko and Wanjn Lao Abstact In ths pape, we stdy tlty-based maxmzaton fo esoce allocaton n the downlnk decton of centalzed weless netwoks. We consde two types of taffc,.e., best effot and had QoS, and develop some essental theoems fo optmal weless esoce allocaton. We then popose thee allocaton schemes. The pefomance of the poposed schemes s evalated va smlatons. The eslts show that optmal weless esoce allocaton s dependent on taffc types, total avalable esoce, and channel alty, athe than solely dependent on the channel alty o taffc types as assmed n most exstng wok. Index Tems tlty-based maxmzaton, esoce allocaton, weless netwoks. I. INTRODCTION RESORCE allocaton s an mpotant eseach topc n weless netwoks [-, 2-6]. In sch netwoks, ado esoce s lmted, and the channel alty of each se may vay wth tme. Gven channel condtons and total amont of avalable esoce, the system may allocate esoce to ses accodng to some pefomance metcs sch as thoghpt and faness [], [2] o accodng to the types of taffc [3]. Thoghpt and faness, howeve, ae conflctng pefomance metcs. To maxmze system thoghpt, the system wll allocate moe esoce to the ses n bette channel condtons. Ths may case ado esoce monopolzed by a small nmbe of ses, leadng to nfaness. On the othe hand, to povde faness to all ses, the system tends to allocate moe esoce to the ses n wose channel condtons so as to compensate fo the shaes. As a eslt, the system thoghpt may be degaded damatcally. The wok n [4-5] show that the system can behave ethe thoghpt-oented o fanessoented by adjstng cetan paametes. Howeve, they do not descbe how to detemne and jstfy the vale of these paametes, leavng ths tade-off nsolved. In ths pape, we focs on se satsfacton fo esoce allocaton to avod sch a thoghpt-faness dlemma. Snce t s nlkely to flly satsfy the dffeent demands of all ses, we tn to maxmze the total degee of se satsfacton. The degee of se satsfacton wth a gven amont of esoce can be descbed by the tlty fncton (), a non-deceasng fncton wth espect to the gven amont of esoce. The moe the esoce s allocated, the Manscpt eceved Decembe, 25; evsed Septembe 5, 26; accepted Decembe 22, 26. The assocate edto coodnatng the evew of ths pape and appovng t fo pblcaton was J. Ho. Ths wok was sppoted by Natonal Scence Concl (NSC), Tawan, nde a Cente Excellence Gant NSC E-2-6-PAE, and nde Gant Nmbe NSC E W.-H. Ko and W. Lao (coespondng atho) ae wth the Depatment of Electcal Engneeng and the Gadate Insttte of Commncaton Engneeng, Natonal Tawan nvesty, Tape, Tawan (emal: wjlao@nt.ed.tw). Dgtal Object Identfe.9/TWC /7$25. c 27 IEEE moe the se s satsfed. The magnal tlty fncton defned by () = d() d s the devatve of the tlty fncton () wth espect to the gven amont of esoce. The exact expesson of a tlty fncton may depend on taffc types, and can be obtaned by stdyng the behavo and feelng of ses. We leave the wok of fndng tlty fnctons to psychologsts and economsts, and focs on maxmzng the total tlty fo a gven set of tlty fnctons. We ae not the ponee to stdy tlty-based esoce management n weless netwoks. Poposals [6-] ae the examples whch ae based on the tlty fnctons of dffeent objectves fo weless netwoks. In [6], a tlty-based powe contol scheme wth espect to channel alty s poposed. In that scheme, ses wth hghe SIR vales have hghe tltes, and ths ae moe lkely to tansmt packets. Theefoe, the weless medm can be bette tlzed and the tansmsson powe can be conseved. The wok n [7] popose a tltybased bandwdth allocaton scheme, whch can adapt to channel condtons and gaantee the mnmm tlty eested by each se. In [8-9], the athos desgn a tlty-based fa allocaton scheme to ense the same tlty vale fo each se. Howeve, lettng ses wth dffeent taffc demands to acheve an dentcal level of satsfacton may not be an effcent way of sng weless esoce. Wose, taffc whch s dffclt to be satsfed tends to consme most of the system esoce, leadng to anothe knd of nfaness. In [], a tlty-based schedle togethe wth a Fowad Eo Coecton (FEC) and an ARQ scheme s poposed. That wok gves laggng ses moe esoce and ths eslts n a smla pefomance level (.e., fxed tlty vale) fo each se. The wok n [9-2] tagets at mlt-hop weless netwoks. tlty fnctons have also been wdely sed n Intenet pcng and congeston contol []. Many bandwdth pcng schemes have been poposed fo weless netwoks [2-6]. The typcal appoach s to set a pce to ado esoce and to allocate tokens to ses. The objectve s then to maxmze the socal welfae thogh a bddng pocess. These knds of bddng schemes, whle sefl fo Intenet pcng and congeston contol, may not be pactcal fo weless netwoks. In weless envonments, the types of taffc, the nmbe of ses, and channel condtons ae all tme-vayng. It wold be vey expensve to mplement a weless bddng pocess becase the ses wold have to keep exchangng contol messages fo eal-tme bddng, and the contol potocols of the weless system wold also have to be modfed to accommodate ths pocess. Fnally, the complexty and effcency of weless bddng have not been analyzed. It s had to estmate the tme elapsed to acheve the Nash elbm. In ths pape, we stdy the weless esoce allocaton Athozed lcensed se lmted to: Natonal Tawan nvesty. Downloaded on Janay 2, 29 at 23:5 fom IEEE Xploe. Restctons apply.

2 KO and LIAO: TILITY-BASED RESORCE ALLOCATION IN WIRELESS NETWORKS 36 poblem n the downlnk of a weless netwok wth a cental contol system, sch as a cellla base staton, and attempt to maxmze the total tlty of all ses wthot a bddng pocess. We consde two common types of taffc: had QoS and best effot taffc, and popose thee allocaton algothms fo these two types of taffc, namely, ) the HQ allocaton fo had-qos taffc, 2) the elastc allocaton fo best effot taffc, and 3) the mxed allocaton fo the co-exstence of both types of taffc. These thee allocaton schemes ae all polynomal tme soltons and poved to be optmal nde cetan condtons, and n any case, the dffeence between the total tltes obtaned by o soltons and the optmal tlty ae bonded. We also develop some theoems as the geneal desgn gdelnes fo tlty-based esoce allocaton n weless netwoks. The pefomance of the poposed schemes s valdated va smlatons. The eslts show that optmal weless esoce allocaton depends on the taffc demand, total avalable esoce, and weless channel alty, athe than solely dependent on channel alty o taffc type as assmed n most exstng wok. The est of the pape s oganzed as follows. In Sec. II, thee weless allocaton schemes ae poposed and poved to be optmal nde cetan condtons. Some theoems ae developed as desgn gdelnes fo tlty-based esoce allocaton schemes. In Sec. III, the pefomance of the poposed schemes s valdated va smlatons. Fnally, the pape s conclded n Sec. IV. II. TILITY-BASED ALLOCATION SCHEMES FOR TWO TYPES OF TRAFFIC A. Poblem Statement and Defntons Sppose that thee ae n ses seved by a base staton. Let total denote the total amont of ado esoce avalable at the base staton, and, the amont of esoce to be allocated to se. ses wth the same knd of taffc may not feel the same way by gven the same amont of esoce becase the weless channel alty fo each se may not be dentcal. Let denote the channel alty 2 of se,, and =, 2, n. The smalle the vale of, the wose the channel alty. Gven an amont of esoce and channel alty, the amont of esoce actally benefcal to se s gven by θ =.LetT() denote the type of taffc of se. The tlty fncton of se s expessed by ( )= T () ( ),whee T () (.) s the tlty fncton of taffc T () and (.) s the tlty fncton fo the type of taffc descbed by T () (.) bt takng nto accont the channel alty of se. The magnal tlty These algothms can help detemne and valdate the paamete settngs of exstng schedles. Most of the exstng weless schedles focs on the desgn of schedlng systems whch povde sch pefomance gaantees as delay bonds of packets o faness among ses, bt leave the weghts of the ees (o ses) ndecded. Ths mssng component can be povded by optmal esoce allocaton schemes as poposed n ths pape. 2 The channel alty paamete s povded by the lowe layes to ndcate the popoton of effectve tansmsson wth a gven amont of ado esoce. Snce may be tme-vaant, the poposed algothm shold be opeated peodcally to adapt to the cent channel condton. Fo smplcty, bt wthot loss of genealty, we wll teat as a constant fo each se n the est of the pape. M Fg.. Fg. 2. M (a) Had QoS Bandwdth The tlty fnctons of two types of taffc. M Bandwdth (a) Had QoS (b) Best effot Bandwdth Bandwdth (b) Best effot The magnal tlty fnctons of two types of taffc. fncton of (.) s d T ()( ) d = T () ( ), and that of T () (.) s T () (.). O objectve s to maxmze n ( ), sbject to n total and. An optmal allocaton fo n ses wth total avalable esoce total s defned as follows. Note that the optmal allocaton may not be ne n the system. Defnton 2.: A esoce allocaton R = {, 2,, n } fo n ses s an optmal allocaton f fo all feasble allocatons R a = {, 2,, n }, n (R ) (R a ), whee (R ) = ( ) and (R a )= n j ( j ). j= Defnton 2.2: A nt-step tlty fncton step () efes to a tlty fncton whose step ( M )= f = M,and step () =,othewse,whee step () = dstep() d. Defnton 2.3: A concave tlty fncton concave () efes to a tlty fncton whose concave () > and concave () < fo all, whee concave() = dconcave() d and concave() = dconcave() d. By defnton, a nt-step fncton s a dscete fncton, and a concave tlty fncton s a non-deceasng and contnos fncton wth espect to esoce. Fg. 2 plots the magnal tlty fnctons fo the two types of taffc shown n Fg. Moe temnology sed n ths pape s defned as follows. Defnton 2.4: R = {, 2, n } s a fll allocaton f R = total. Defnton 2.5: Fo R = {, 2, n },allses wth > (.e., those who ae allocated esoces) ae efeed to as allocated ses and all ses j wth j =(.e., those who ae not allocated esoces) ae nallocated ses. Defnton 2.6: R = {, 2, n } s magnally fa f t satsfes the followng two condtons: ) Each allocated se (.e., wth > ) nr mst have the same magnal tlty vale. 2) Fo each nallocated se j (.e., wth j =)nr, ts magnal tlty vale j () cannot exceed ( ),whch s the magnal tlty of each allocated se. Athozed lcensed se lmted to: Natonal Tawan nvesty. Downloaded on Janay 2, 29 at 23:5 fom IEEE Xploe. Restctons apply.

3 362 IEEE TRANSACTIONS ON WIRELESS COMMNICATIONS, VOL. 6, NO., OCTOBER 27 Defnton 2.7: Fo a magnally fa allocaton R = {, 2, n }, the magnal tlty vale of each allocated se n R s eal, and s efeed to as the allocated magnal tlty m (R). Ths, fo each allocated se n R, ( )= m (R), and fo each nallocated se j, j () < m (R). B. HQ Allocaton fo Had QoS Taffc Sppose that thee ae n ses n the ee, all wth had QoS taffc. Let esde denote the esdal esoce n the system. The esoce allocaton algothm desgned fo ses whose tlty fnctons ae all nt-step fnctons s efeed to as the HQ allocaton and the otpt s denoted by R HQ = {, 2, n }. Gven the total avalable esoce n the system total, the channel alty and tlty fncton T () (.) fo all se, R HQ can be obtaned as follows. ) Intalze, =, 2,,n; esde total. 2) Sot all ses n the ee n descendng ode of M M. 3) Repeat Steps (4) and (5) ntl the ee becomes empty. 4) Pop ot se who s now at the head of the ee. 5) If esde > M,then = M ; esde = esde. The tlty fncton fo se wth had QoS taffc s descbed by T () () = M f ( M ),wheef (.) s a nt-step fncton, s the channel alty of ths se, M s the knd of QoS taffc, M s the pefeed amont of esoce to be allocated. The allocaton le of the HQ allocaton s to assgn esoces to ses n descendng ode of M M, sbject to k total, whee k s the lagest vale satsfyng ths constant, as llstated n Fg. 3. The allocaton poblem fo ses wth abtay nt-step tlty fnctons can be poved to be NP-complete. The pefomance of the HQ allocaton can be poved to be close to the optmm. When the tlty fnctons of all ses ae dentcal, the HQ allocaton can be poved to be optmal. Lemma 2.: The allocaton poblem s NP-complete f the tlty fnctons fo all ses ae abtay nt-setp fnctons. Poof: Ths poblem can be edced fom the / knapsack poblem, an NP-complete poblem. Consde a knapsack wth capacty c > and n tems. Each tem has a vale of v > and a weght of w >. The poblem s then to fnd a selecton of tems that maxmze n δ v sbject n to δ w c, wheeδ =f the tem s selected, and, othewse. Theefoe, any nstance of the knapsack can be edced to an nstance of o poblem by sbstttng total = c, = v f step ( w ) and =, fo =, 2,,n. Snce an optmal solton to o poblem s also a solton to the gven knapsack poblem, the knapsack poblem s a specal case of o poblem, t follows that o poblem s NP-had. Next, t can be obseved that o poblem s an NP poblem becase any gven solton can be vefed as a feasble solton and bonded at a gven tlty vale n polynomal Fg. 3. Assgnment 2 k- k M M M 2 2 M 2 M k k M k M k k M k Allocaton odeng of k ses n the HQ allocaton tme. Snce o poblem s both n NP and NP-had, t s an NP-complete poblem. Theoem 2.2: (R HQ ) (R op ) max,wheer HQ and R op ae the poposed solton and the optmal solton, espectvely, of ths HQ allocaton poblem, (x) s the total tlty of all ses fo solton x, and max s the maxmm tlty vale taken ove all ses,.e., max = max { M k }. k n Poof: Let p denote the ognal HQ allocaton poblem, whch s NP-complete as poved n Lemma 2., and let p denote the poblem by elaxng the ntegty constant of poblem p (.e., each se n p can be seved factonally). To bette ndcate the optmal solton to each poblem, we let R p op and R p op epesent the optmal soltons to poblems p and p, espectvely. Snce p s a maxmzaton poblem, we have (R p op ) (Rp op ) (Rp HQ ),wheerp HQ means R HQ. The optmal solton to p, R p op, s obtaned by sotng QoS ses n the ee n deceasng ode of M / M as n R p HQ. The dffeence between Rp op and R p HQ s only n those ses factonally seved n R p op. It follows that (R p HQ )+ x >(R p op), whee x s the tlty vale of the nallocated se whose M M n R p HQ s the lagest. Ths, x max.snce(r HQ )+ max (R p HQ )+ x > (R p op) (R p op), we obtan (R p HQ ) >(Rp op) max. Theoem 2.3: The HQ allocaton solton R HQ s an optmal allocaton f the nt-step tlty fnctons step () of all n ses n the ee ae dentcal. Poof: Let M denote the nt-step tlty fncton fo all ses. The nealty n Theoem 2.2 can be ewtten as n M + M >(R p op) (R p op) n M,.e.,n + > (R p op ) M (Rp op ) M n. Snce both (Rp op ) M and n ae ntegal, (R p op )=n M. Theefoe, R HQ mst be optmal. Theoem 2.4: The tme complexty of R HQ s O(n log n). Athozed lcensed se lmted to: Natonal Tawan nvesty. Downloaded on Janay 2, 29 at 23:5 fom IEEE Xploe. Restctons apply.

4 KO and LIAO: TILITY-BASED RESORCE ALLOCATION IN WIRELESS NETWORKS 363 ( ) ( ) ( ) ( ) ( ) ( ) R R2 bp Poof: In the elastc allocaton, ) BE s the magnal tlty vale at whch pont the total esoce has been flly allocated, and 2) fo all allocated ses n R elastc, BE < () and ( )= BE ; fo all nallocated ses j, BE j (). Wth concave tlty fnctons,.e., fncton wth () > fo all, t can be easly poved by contadcton that an optmal allocaton fo elastc ses,.e., R elastc,mst be fll. Smlaly, by sng contadcton agan, we can pove that R elastc mst also be magnally fa. Snce all ses tlty fnctons ae nceasng, only one allocaton can be both fll and magnally fa. Theefoe, f an allocaton s fll and magnally fa, tmstber elastc. Theoem 2.6: The tme complexty of the algothm elastc allocaton s O(n). Poof: Snce the opeaton at each step at most takes tme O(n), the tme complexty of the elastc allocaton algothm s O(n). Rtotal Fg. 4. An example of the elastc allocaton. Fst, the magnal tlty fnctons of ses and j ae nveted (.e., () and j ()). Then, the two nveted magnal tlty fnctons ae smmed,.e., Σ () = (). Fnally, the elatonshp among BE, total,, and 2 s decomposed. Poof: The tme complexty of R HQ can be expessed by a fncton of the nmbe of ses n the netwok. Snce the complexty of Step (2) (sotng) s O(n log n) and ths teaton domnates the opeaton, the oveall complexty of R HQ s O(n log n). C. Elastc Allocaton fo Best Effot Taffc We next consde the best effot taffc. The esoce allocaton algothm fo ses wth concave tlty fnctons () s efeed to as the elastc allocaton and the otpt s denoted by R elastc = {, 2, n }. Gven the total avalable esoce total, the channel alty and magnal tlty fncton () fo each se, R elastc can be obtaned as follows. ) Fo each se, deve (), the nveted fncton of (). 2) Deve Σ () by smmng p () ove all ses,.e., Σ () = (). 3) Fnd Σ (), the nveted fncton of Σ (). 4) Fnd BE, whch s eal to Σ ( total ). 5) Fo all, =, 2,,n, f BE < (), then = ( BE ) ; else =. The allocaton le of ths scheme s to ) deve the aggegated tlty fncton fom the nvese fnctons of all ses, 2) calclate the allocated magnal tlty fom the aggegated tlty fncton, and 3) detemne fo each se. As an example, Fg. 4 llstates the elastc allocaton algothm wth two best effot taffc. Theoem 2.5: The elastc allocaton R elastc s optmal f the tlty fnctons fo all ses ae concave tlty fnctons. D. A Mxte of Had QoS and Best Effot Taffc Fnally, we consde the co-exstence of QoS and best effot taffc n the system, whch s efeed to as mxed allocaton and the otpt of whch s denoted by R mx = {, 2, n }. Let BE denote the amont of esdal esoce to be gven to best effot taffc, and Δ, the tlty gan by allocatng esoce to QoS se. Othe notatons eman the same as n the HQ and the elastc allocatons. Gven the total avalable esoce total, the channel alty and magnal tlty fncton () fo each se, R mx can be obtaned as follows. ) Intalze, =, 2,,n ;and BE total. 2) Sot all QoS ses n descendng ode of M M,and stoe them n the ee. 3) Fo each best effot se j, deve j () fom j (); Fnd Σ () by smmng p j () ove all ses j; Fnd Σ (), the nveted fncton of Σ (). 4) If the ee s not empty, then pop ot the QoS se at the head of the ee; else go to Step (8). 5) Fo the popped se : f BE M,then = M ; else =;gotostep(4). BE 6) Δ = M BE Σ ()d. 7) If (Δ > ), then BE = BE ;gotostep(4); else =;gotostep(8); 8) If Σ ( BE ) < j (), then j = j ( Σ ( BE )); else j =. The allocaton le of ths mxed allocaton s to: ) allocate esoce to the fst k QoS ses at the soted ee, and 2) then allocate the esdal bandwdth (.e., total QoS )toall best effot ses based on the elastc allocaton.thevaleofk s detemned based on the eement that thee s sffcent esoce fo ths QoS se and the tlty gan Δ k s postve (.e., BE k > ). We have poved n Lemma 2. that the allocaton poblem fo had QoS taffc s NP-complete. It follows that the Athozed lcensed se lmted to: Natonal Tawan nvesty. Downloaded on Janay 2, 29 at 23:5 fom IEEE Xploe. Restctons apply.

5 364 IEEE TRANSACTIONS ON WIRELESS COMMNICATIONS, VOL. 6, NO., OCTOBER 27 k 2 QoS ses 2 8 Fg. 5. k 2 BE, k BE,k BE,2 BE, An example of mxed allocaton. BE, total Bandwdth Best Effot ses allocaton poblem fo the co-exstence of best effot and had QoS taffc s also NP-complete. Agan, the poposed mxed allocaton algothm can acheve a pefomance lowe bonded by (R op ) max. When all QoS ses have an dentcal tlty fncton, the mxed allocaton can be poved to be optmal. Theoem 2.7: Fo the mxed allocaton poblem, (R mx ) (R op ) max,wheer mx s the poposed solton, R op s the optmal solton, (x) s the total tlty of all ses fo solton x, and max s the maxmm tlty vale taken ove all ses,.e., max = max { M k }. k n Poof: Ths theoem can be poved by elaxng the constant n the ognal poblem as n Theoem 2.2. Theefoe, the dffeent n the total tlty between R mx and the elaxed poblem s bonded by max,.e., (R mx ) (R op ) max. Theoem 2.8: The mxed allocaton R mx s an optmal allocaton fo taffc mxed wth dentcal nt-step fnctons and abtay concave fnctons. Poof: Consde the fst k QoS ses n the soted ee, whee k s the maxmm possble vale satsfyng k total. The set of all possble optmal allocatons s then: R, R,, R,, R k,wheer s the allocaton n whch the fst QoS ses ae allocated esoce n an amont of j = M / j, j =, 2,,, and the esdal bandwdth s all allocated to best effot ses. The tlty gan Δ, =, 2,,k, s expessed by Δ = (R ) (R )= BE M Σ ()d. BE Snce QoS ses ae soted n deceasng ode of the, ths leads to that s allocated n nceasng ode of, esltng n Δ Δ + (as shown n Fg. 5). Ths, the allocaton R n [R, R, R 2,, R,, R k ] s the optmal allocaton, whee s the lagest vale satsfyng Δ >. E. Implementaton Isse In pactce, we can avod sch opeatons as nvetng nonlnea fnctons, nvetng the smmatons of nonlnea fnctons, and nvetng non-lnea fnctons by a sees of smmatons and smple ntepolatons. Sppose max s the maxmm magnal tlty vale allowed by the system. We can patton the ange of the Fg (a) (b) θ Resoce allocaton n R HQ = {, 2, n}. vales n the y-axs fo any (contnos) magnal tlty fncton, say (.), nto vales spaced eally apat,.e., k = max k max M,wheeM s a tnable npt paamete whch detemnes the nmbe of steps n total, and k =,,,M.Wethenlet k = ( k ), k =,,,M,n a one dmensonal fxed length aay Γ=[,, 2,... M ]. Snce (.) s a deceasng fncton, thee s a one-to-one coespondence between k and k. Ths method elmnates the nveson pocess and can obtan any eed vales (.e., () and ()) by ntepolaton. Fo a gven vale, the system can fnd the vale k satsfyng k < k+.by ntepolaton, () = k max M k k+ k. Smlaly, fo any gven vale, () = k +( k+ k ) k k+ k, whee k s the vale satsfyng k+ < k. Ths appoach can also edce the comptatonal ovehead of the smmaton fncton Σ (), whch can be calclated by R Σ = [ j j j ( ), ( ),, ( M )] gven that thee ae j tlty fnctons (, 2,, j ) to be aggegated. Lkewse, the ovehead fo the ntegal of Σ () can also be edced. In ths way, the comptatonal complexty of fndng () and () s only O(M) becase t takes at most M steps to fnd the vale k fo k k< k+. The complexty of the smmaton s O(jM) becase fo k =,,,M, t takes j steps to calclate j ( k ). Athozed lcensed se lmted to: Natonal Tawan nvesty. Downloaded on Janay 2, 29 at 23:5 fom IEEE Xploe. Restctons apply.

6 KO and LIAO: TILITY-BASED RESORCE ALLOCATION IN WIRELESS NETWORKS Allocated Bandwdth 5 _QoS 25 _BE Fg. 8. An example of mxed allocaton. Fg (a) (b) θ Resoce allocaton n R elastc = {, 2, n}. III. PERFORMANCE EVALATION In ths secton, we condct smlatons to evalate the pefomance of o allocaton algothms. We consde QoS taffc, best effot taffc, and the con-exstence of both. The smlaton paametes ae descbed as follows. Fo QoS taffc, the tlty fncton s a nt-step fncton wth a = and M =,.e., QoS () =f ( ); fo best effot taffc, BE () = e /.Thevaleof s andomly geneated by a nfom dstbton ove [, ]. We then mease the dstbtons of and θ nde dffeent vales of total. Fg. 6 shows the vales of and θ fo QoS taffc n the HQ allocaton. Fg. 6(a) depcts that when total s small, the system tends to allocate moe esoce to the ses n bette channel condtons; bt as total nceases, the amont of esoce allocated to ses s stll fxed ethe at a o, becase the tlty fncton s a nt-step fncton. Fg. 6(b) shows that the actal amont of esoce obtaned by each se, f allocated, s dentcal,.e., at the θ. Fg. 7 shows the vales of and θ fo best effot taffc n the elastc allocaton. Fg. 7(a) depcts that when total s small, the system tends to allocate moe esoce to the ses n bette channel condtons. Snce the tlty fncton of best-effot taffc satsfes () > fo all, all ses wll demand esoce all the tme. Ths, the vale of fo each se nceases wth the vale of total. Fg. 7(b) shows that ses wth lage always eslt n lage θ. Fg. 8 shows the vales of QoS and BE, as a fncton of total fo mxed taffc n the mxed allocaton. Snce the QoS taffc s allocated esoces n dscete amonts, when one moe QoS se s seved, the vale of BE wll dop. Ths effect becomes moe pononced when total s lage, mplyng that the emanng nallocated QoS ses have lowe and need moe esoces to compensate fo the bad channel altes. De to the space lmtatons, we do not nclde the dstbtons of and θ fo the mxed allocaton n ths pape. The eslts can be fond n [8], the chaactestcs of whch ae smla to those n Fgs. 6 and 7. In Fg. 9, dffeent esoce allocaton schemes ae compaed wth the poposed allocaton schemes. The compason s based on the scheme popsoed n [4], whch allocates ado esoce popotonally based on facto α. Dependng on the settng of the vale α, the system can be tned to wok wth dffeent pefomance metcs. The cve denoted thoghpt s fo α =, whch gves moe esoces to the ses n bette channel condtons, theeby leadng to a lage system thoghpt. The cve denoted faness s fo α =, gvng all ses an dentcal vale of θ =. The cve denoted fxed s fo α =, whch povdes the same amont of esoce to all ses. Note that the schemes poposed n [8-9] ae the examples of the faness scheme (.e., α = ), and the GR+ scheme n [] s an example of the thoghpt scheme. Fg. 9(a) compaes the poposed HQ allocaton wth dffeent allocaton schemes, and Fg. 9(b) compaes the poposed elastc allocaton wth dffeent allocaton schemes. Note that the axs of total n Fg. 9(b) s n the logathmc scale. The eslts show that the thoghpt-fst scheme has a hghe total tlty when total s small, bt the fxed allocaton one s close to the poposed scheme as total nceases. Fnally, when total becomes vey lage, the faness-fst scheme can acheve the hghest tlty. IV. CONCLSION In ths pape, we stdy tlty-based maxmzaton fo esoce allocaton n nfastcte-based weless netwoks. We develop some essental theoems fo tlty-based esoce Athozed lcensed se lmted to: Natonal Tawan nvesty. Downloaded on Janay 2, 29 at 23:5 fom IEEE Xploe. Restctons apply.

7 366 IEEE TRANSACTIONS ON WIRELESS COMMNICATIONS, VOL. 6, NO., OCTOBER 27 tlty tlty Fg (a) QoS taffc HQ Thoghpt Fxed Faness (b) Best effot taffc Elastc Thoghpt Fxed Faness tlty compason wth dffeent esoce allocaton schemes. management. Then, thee polynomal tme esoce allocaton algothms ae poposed fo two types of tlty fnctons. We pove that, n any case, the dffeence between the total tltes obtaned by o poposed soltons and the optmal tlty s bonded, and nde cetan condtons, all these thee schemes can acheve the maxmm total tlty (.e., optmal). Fom the smlaton eslts, we fnd that dffeent types of taffc ee dffeent knds of schemes to acheve optmal allocaton. In addton, when total s small, the system tends to allocate moe esoces to the ses n bette channel condtons,.e., thoghpt-oented; howeve, when total s abndant, the system becomes faness-oented, meanng that even wth the same taffc, the pefeence tendency between thoghpt and faness can stll dffe. Ths leads s to conclde that exstng channel-dependent-only esoce schemes and schedles cannot povde optmal allocaton n weless netwoks. To the best of o knowledge, ths s the fst wok to addess all the thee sses togethe. REFERENCES [] D. Angeln and M. Zoz, On the thoghpt and faness pefomance of heteogeneos downlnk packet taffc n a locally centalzed CDMA/TDD system, n Poc. IEEE VTC-Fall 22. [2] A. Fska et al., Pefomance of WCDMA hgh speed packet data, n Poc. IEEE VTC 22-Spng. [3] Y.Cao,V.O.K.L,andZ.Cao, Schedlng delay-senstve and besteffot taffc n weless netwoks, n Poc. IEEE ICC 23. [4] Z. Jang and and N. K. Shankaanaayana, Channel alty dependent schedlng fo flexble weless esoce management, n Poc. IEEE Globecom 2. [5] W.-T. Chen, K.-C. Shh, and J.-L. Chang, Flexble packet schedlng fo alty of sevce povsonng n weless netwoks, n Poc. ICPADS, Dec. 22. [6] M. Xao, N. B. Shoff, and E. K. P. Chong, A tlty-based powecontol scheme n weless cellla systems, IEEE/ACM Tans. Netw., vol., no. 2, 23. [7] Y. Cao and V. O. K. L, tlty-oented adaptve QoS and bandwdth allocaton n weless netwoks, n Poc. IEEE ICC 22. [8] G. Banch and A. T. Campbell, A pogammable MAC famewok fo tlty-based adaptve alty of sevce sppot, IEEE J. Sel. Aeas Commn., vol. 8, no. 2, pp , 2. [9] R.-F. Lao and A. T. Campbell, A tlty-based appoach to anttatve adaptaton n weless packet netwoks, ACM WINET, vol. 7, no. 5, pp , Sept. 2. [] X. Gao, T. Nandagopal, and V. Bhaghavan, Achevng applcaton level faness thogh tlty-based weless fa schedlng, n Poc. IEEE Globecom 2. [] F. P. Kelly, Chagng and ate contol fo elastc taffc, Eopean Tans. Telecommn., Jan [2] V. A. Ss, B. Bscoe, and D. Songhst, Economc models fo esoce contol n weless netwoks, n Poc. IEEE PIMRC 22, Lsbon, Potgal, Sept. 22. [3] V. A. Ss, Resoce contol fo elastc taffc n CDMA netwoks, n Poc. IEEE MOBICOM 22. [4] V. A. Ss and C. Cocobets, Resoce contol fo loss-senstve taffc n CDMA netwoks, n Poc. IEEE INFOCOM 24. [5] P. Mabach and R. Bey, Downlnk esoce allocaton and pcng fo weless netwoks, n Poc. IEEE INFOCOM 22. [6] P. L, R. Bey, M. L. Hong, and S. Jodan, Slow-ate tlty-based esoce allocaton n weless netwoks, n Poc. IEEE Globecom, Nov. 22. [7] S. Shenke, Fndamental desgn sses fo the fte Intenet, IEEE J. Sel. Aeas Commn., vol. 3, no. 7, pp , 995. [8] W. H. Ko and W. J. Lao, tlty-based optmal esoce allocaton n weless netwoks, n Poc. IEEE Globecom 25. [9] K.-D. W and W. Lao, Flow allocaton n mlt-hop weless netwoks: a coss-laye appoach, to appea n IEEE Tans. Weless Commn., 27. [2] L. Chen, S. H. Low, and J. C. Doyle, Jont congeston contol and meda access contol desgn fo weless ad hoc netwoks, n Poc. IEEE Infocom, Mam, FL, Mach 25. Wen-Hsng Ko was bon on Febay 6, 98, n Tachng, Tawan. He eceved the B.S. degee n electcal engneeng fom Natonal Tawan nvesty, Tape, Tawan, n 22, whee he s cently wokng towad the Ph.D. degee n the Gadate Insttte of Electoncs Engneeng. He joned Intenet Reseach Laboatoy, NT, n 22. Hs eseach nteests nclde weless esoce management, netwok economcs and 82.6 WMAX netwoks. Wanjn Lao eceved the BS and MS degees fom Natonal Chao Tng nvesty, Tawan, n 99 and 992, espectvely, and the Ph.D. degee n Electcal Engneeng fom the nvesty of Sothen Calfona, Los Angeles, Calfona, SA, n 997. She joned the Depatment of Electcal Engneeng, Natonal Tawan nvesty (NT), Tape, Tawan, as an Assstant Pofesso n 997. Snce Agst 25, she has been a fll pofesso. He eseach nteests nclde weless netwoks, mltmeda netwoks, and boadband access netwoks. D. Lao s cently an Assocate Edto of IEEE Tansactons on Weless Commncatons and IEEE Tansactons on Mltmeda. She seved as the Techncal Pogam Commttee (TPC) chas/co-chas of many ntenatonal confeences, ncldng the Ttoal Co-Cha of IEEE Infocom 24, the Techncal Pogam Vce Cha of IEEE Globecom 25 Symposm on Atonomos Netwoks, and the Techncal Pogam Co-Cha of IEEE Globecom 27 Geneal Symposm. D. Lao has eceved many eseach awads. Papes she co-athoed wth he stdents eceved the Best Stdent Pape Awad at the Fst IEEE Intenatonal Confeences on Mltmeda and Expo (ICME) n 2, and the Best Pape Awad at the Fst IEEE Intenatonal Confeences on Commncatons, Ccts and Systems (ICCCAS) n 22. D. Lao was the ecpent of K. T. L Yong Reseache Awad honoed by ACM n 23, and the ecpent of Dstngshed Reseach Awad fom Natonal Scence Concl n Tawan n 26. She s a Seno membe of IEEE. Athozed lcensed se lmted to: Natonal Tawan nvesty. Downloaded on Janay 2, 29 at 23:5 fom IEEE Xploe. Restctons apply.