A Resource Allocation Algorithm for Users with Multiple Applications in 4G-LTE

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1 A esouce Allocato Algothm fo Uses wth Multple Applcatos 4G-LTE Ahmed Abdel-Had Electcal ad Compute Egeeg Vga Tech Algto, Vga 2223 Chales Clacy Electcal ad Compute Egeeg Vga Tech Algto, Vga 2223 Joseph Mtola III Alled Commucatos Ic. Bosto, Massachusetts 21 ABSTACT I ths pape, we cosde esouce allocato optmzato poblem fouth geeato log tem evoluto (4G-LTE wth uses ug multple applcatos. Each moble use ca u both delay-toleat ad eal-tme applcatos. I evey use equpmet(ue, each applcato has a applcatostatus dffeetato fom othe applcatos depedg o ts stataeous usage pecetage. I addto, the etwok opeatos povde subscbe dffeetato by assgg each UE a subscpto weght elatve to ts subscpto. The objectve s to optmally allocate the esouces wth a utlty popotoal faess polcy. We popose a algothm to allocate the esouces two-stages. I the fst-stage, the UEs collaboate wth the evolved ode B (eodeb that allocates the optmal ates to uses accodg to that polcy. I the secod-stage, each use allocates ts assged ate teally to ts applcatos accodg to the usage pecetage. We pove that the two-stage esouce allocato algothm allocates the optmal ates wthout eodeb kowledge of the UEs utltes. Fally, umecal esults o the pefomace of the poposed algothm ae peseted. Categoes ad Subject Descptos K.6.2 [Maagemet of Computg ad Ifomato Systems]: Istallato Maagemet pcg ad esouce allocato; C.2.1[Compute-Commucato etwoks]: etwok Achtectue ad Desg weless commucato Keywods Two-Stage esouce Allocato; Multple Utltes; Subscbe Dffeetato 1. ITODUCTIO I ecet yeas, thee has bee a apd gowth moble boadbad sevce. Ths apd gowth s both the umbe of subscbes ad the taffc of each subscbe. The weless Pemsso to make dgtal o had copes of all o pat of ths wok fo pesoal o classoom use s gated wthout fee povded that copes ae ot made o dstbuted fo poft o commecal advatage ad that copes bea ths otce ad the full ctato o the fst page. Copyghts fo compoets of ths wok owed by othes tha ACM must be hooed. Abstactg wth cedt s pemtted. To copy othewse, o epublsh, to post o seves o to edstbute to lsts, eques po specfc pemsso ad/o a fee. equest pemssos fom pemssos@acm.og. CAB 13, Octobe 4, 213, Mam, Floda, USA Copyght 213 ACM /13/$.. etwok povdes ae movg fom sgle sevce (e.g. Iteet access to multple sevce offeg (e.g. multmeda telephoy ad moble-tv [1]. Moble subscbes ae ug multple applcatos o the smat phoes, smultaeously. These dffeet applcatos ad sevces have dffeet pefomace equemets, fo example, some ae delay-toleat ad some ae eal-tme applcatos. Theefoe, they eque dffeet bt-ates ad packet delays. Due to the dffeet atue of dffeet applcatos, sevce-offeg dffeetato eeds to be take to cosdeato whe allocatg the esouces fo dffeet uses. The usage pecetage of each applcato o the UE eques a addtoal dffeetato that we call applcato-status dffeetato. I addto, etwok povdes ae ecetly povdg subscbe dffeetato [1],.e. dffeet uses equestg the same sevce eceve dffeet teatmet. Defed by the etwok povdes, subscbe dffeetato could be betwee copoate ad pvate subscbes, post- ad pe-pad subscbes, ad/o pvleged ad oamg subscbes. I [2, 3], the authos peset a optmal ate allocato algothm fo uses wth delay-toleat o eal-tme applcatos. The optmal ates ae acheved by fomulatg the ate allocato optmzato poblem a covex optmzato famewok. The authos use logathmc ad sgmodallke utlty fuctos to epeset delay-toleat ad ealtme applcatos, espectvely. I [2, 3], the ate allocato algothm gves poty to eal-tme applcatos ove delaytoleat applcatos whe allocatg esouces as the utlty popotoal faess ate allocato polcy s used. Ths esouce allocato guaatees sevce-offeg dffeetato whe allocatg esouces. I ths pape, we fomulate the esouce allocato optmzato poblem wth sevce-offeg dffeetato, applcato status dffeetato ad subscbe dffeetato that s casted a covex optmzato famewok. I ou system model, each subscbe has a subscpto weght set by the etwok. I addto, each subscbe ca u multple applcatos, each wth ts ow utlty fucto, o hs smat phoe. The applcatos ug o the phoe have dffeet applcato-status depedg o the stataeous usage pecetage ad mpotace to the subscbe. Fo example, the applcato ug o the foegoud, such a voce call, has hghe applcato-status tha the applcato ug o the backgoud, such as a automatc applcato update. Fally, the sevce-offeg dffeetato whch gves poty to eal-tme applcatos ove delay-

2 toleat applcatos s heted the utlty popotoal faess ate allocato polcy. Ou esouce allocato algothm s pefomed twostages. The fst-stage s fo allocatg the uses ates ad ths s pefomed collaboatvely betwee the eodeb ad the UEs. The secod-stage s fo allocatg the applcatos ates ad ths s pefomed teally each UE. 1.1 elated Wok A o-covex optmzato fomulato fo mzato of utlty fuctos weless etwoks s peseted [4, ]. Both elastc ad sgmodal-lke utlty fuctos ae used. The authos peset the algothm to solve t optmally whe the dualty gap s zeo ad clude a fa allocato heustc to esue etwok stablty. A utlty -m faess esouce allocato fo uses, wth elastc ad eal-tme taffc, shag a sgle path the etwok s poposed [6]. I [7], the authos popose a utlty popotoal fa optmzato fomulato fo hgh- SI weless etwoks usg utlty -m achtectue. A compaso betwee the algothm ad the tadtoal badwdth popotoal fa algothms [8] s peseted ad a closed fom soluto that pevets oscllatos the etwok s poposed. A dstbuted powe allocato algothm fo a moble cellula system s poposed [9]. The authos used ococave sgmodal-lke utlty fuctos. The algothm povdes a appoxmato to the global optmal soluto ad theefoe could dop some uses to mze the oveall system utlzato. Theefoe, t does ot guaatee a mmum QoS fo all uses. A weghted aggegato of elastc ad elastc utlty fuctos fo each UE s poposed []. Ths aggegated utlty fuctos ae the appoxmated to the eaest stctly cocave utlty fucto fom a set of fuctos usg mmum mea-squae eo. These appoxmated utlty fuctos ae solved usg a modfed veso of the dstbuted ate allocato algothm by Fak Kelly [11]. Theefoe, the allocated ates ae appoxmatos of the optmal ates. 1.2 Ou Cotbutos Ou cotbutos ths pape ae summazed as: We peset a ovel two-stage method fo allocatg the optmal ates fo uses ug multple applcatos. I the fst-stage, the eodeb ad the UEs collaboate to allocate the optmal ate to each UE. I the secodstage, each UE teally dstbutes ts ate optmally to the dffeet applcatos ug o t. We pove that the ew two-stage esouce allocato optmzato poblem s equvalet to the oe-stage esouce allocato covex optmzato poblem that allocates ates dectly to applcatos. We peset the algothm fo solvg the two-stage optmzato poblem ad ts smulato esults. The emade of ths pape s ogazed as follows. Secto 2 pesets the poblem fomulato. Secto 3 poves that ou ovel two-stage allocated optmal ates ae equvalet to oe-stage allocated optmal ates. I Secto 4, we peset ou two-stage ate allocato algothm fo the utlty popotoal faess polcy. Secto dscusses smulato setup ad povdes quattatve esults alog wth dscusso. Secto 6 cocludes the pape. 2. POBLEM FOMULATIO We cosde sgle cell 4G-LTE moble system cosstg of a sgle eodeb ad M UEs. The ate allocated by the eodeb to th UE s gve by. Each UE has ts ow utlty fucto V ( that coespods to the applcatos ug of the UE. Ou objectve s to deteme the optmal ates the eodeb shall allocate to the UEs. We assume the use utlty fucto V ( of th UE s gve by: V ( = U α j j ( j (1 whee U j( j s the j th applcato utlty fucto, j s the ate allocated to the j th applcato, ad α j s the j th applcato usage pecetage o the th UE (.e. αj = 1. WeassumethatU j( jsastctlycocaveoasgmodallke fucto. The utlty fuctos U( have the followg popetes: U( = ad U( s a ceasg fucto of. U( s twce cotuously dffeetable ad bouded above. I ou model, we use the omalzed sgmodal-lke utlty fucto, as [9], that ca be expessed as ( 1 U( = c 1+e d (2 a( b whee c = 1+eab e ab add = 1 1+e ab. So, t satsfes U( = ad U( = 1. The flecto pot of omalzed sgmodal-lke fucto s at f = b. I addto, we use the omalzed logathmc utlty fucto, as [7], that ca be expessed as U( = log(1+k log(1+k whee s the mum equed ate fo the use to acheve % utlzato ad k s the ate of cease of utlzato wth the allocated ate. So, t satsfes U( = ad U( = 1. The flecto pot of omalzed logathmc fucto s at f =. The basc fomulato of the esouce allocato poblem s gve by the followg optmzato poblem: M =1 ( U α β j j ( j j, =1 j, = 1,2,...,M, j = 1,2,...,. whee s the mum achevable ate of the eodeb, M s the umbe of UEs the coveage aa of the eodeb, ad s the umbe of applcatos ug the th UE. Coollay 2.1. The optmzato poblem (4 s a covex optmzato poblem ad thee exsts a uque tactable global optmal soluto. Poof. The objectve fucto optmzato poblem (4 gve by ( M β =1 Uα j j ( j s equvalet to the (3 (4

3 objectve fucto M =1 β αj loguj(j, so the optmzato poblem (4 ca be wtte as follows: =1 α j logu j( j β j, =1 j, = 1,2,...,M, j = 1,2,...,. Gve the poblem fomulato Secto 2, we kow that theutltyfuctosu j( jaestctlycocaveosgmodallke fuctos. Fom Lemma (III.1 [2], logu j( j s a stctly cocave fucto fo a stctly cocave o sgmodallke utlty fucto U j( j. It follows that optmzato poblem ( s covex ad as a esult optmzato poblem (4 s also covex. Theefoe, thee exsts a tactable global optmal soluto fo optmzato poblem (4. 3. TWO-STAGE OPTIMIZATIO POBLEM We dvde optmzato poblem (4 two optmzato poblems to be solved to two stages ad allocate the same optmal ates as optmzato poblem (4. I the fst-stage, the ates ae allocated to the uses by the eodeb ad the soluto s acheved collaboatvely betwee the eodeb ad the UEs. We call ths stage the eodeb-ue ate allocato (EUA stage. I the secod-stage, the ates j ae allocated to applcatos ad the allocato s doe teally the UE. We call ths stage the teal UE ate allocato (IUA stage. 3.1 EUA Optmzato Poblem EUA optmzato poblem that s solved collaboatvely betwee the eodeb ad the UEs ad ca be wtte as: M =1 V β (, =1, = 1,2,...,M. whee V = Uα j j ( j ad = { 1, 2,..., M} ad M s the umbe of UEs the coveage aea of the eodeb. I the optmzato poblem (6, sce the objectve fucto ag M =1 V β ( M sequvalettoag =1 βlog(v(, the optmzato poblem (6 ca be wtte as: β log(v ( =1, =1, = 1,2,...,M. Coollay 3.1. The optmzato poblem (6 s a covex optmzato poblem ad thee exsts a uque tactable global optmal soluto. ( (6 (7 Poof. Fom equato (1, we have that log(v ( = αj log(uj(j, ad gve the poblem fomulato Secto 2, the optmzato poblem (6 s covex (steps smla to Coollay IUA Optmzato Poblem IUA optmzato poblem s solved teally evey UE ad ca be wtte fo the th UE as follows: U α j j ( j =1 j opt, j, j = 1,2,...,. whee = { 1, 2,..., }ad opt s the ate allocated by eodebtothe th UE.Itheoptmzato poblem(8, sce the objectve fucto ag Uα j j ( j s equvalet to ag αj log(uj(j, the optmzato poblem (8 ca be wtte as: =1 α j logu j( j j opt, j, j = 1,2,...,. Coollay 3.2. The optmzato poblem (8 s a covex optmzato poblem ad thee exsts a uque tactable global optmal soluto. Poof. Gve the poblem fomulato Secto 2, the optmzato poblem (8 s covex (steps smla to Coollay Equvalece I ths secto, we show the equvalece of EUA optmzato poblem (6 ad IUA optmzato poblem (8 to optmzato poblem (4. Lemma 3.3. Fo stctly cocave o sgmodal-lke utlty fuctos U j( j, the slope of atual logathm of utlty fuctos p = S j( j = logu j( j j (p ae stctly deceasg fuc- vese fuctos j = S 1 j tos. (8 (9 ae vetble ad the Poof. Fo the stctly cocave utlty fucto U j( j case ad fom the utlty fucto popetes Secto 2, the utlty fucto s postve U j( j >, ceasg ad twce dffeetable wth espect to j. The, t follows that U j( j = U j( j j > ad U j( j = 2 U j ( j <. It j 2 follows that, we have S j( j = log(u j( j j ad S j( j j = U j ( ju j ( j U 2 j ( j U 2 j ( j = U j ( j U j ( j > <. Theefoe, S j( j of ay stctly cocave utlty fucto s stctly deceasg fucto.

4 Fo the sgmodal-lke utlty fucto U j( j case, the utlty fucto of the omalzed sgmodal-lke fucto s gve by equato (2. Fo < j <, we have the fst ad secod devatve as S j( j = j 1 d j(1+e a j( j b j 2 j 2 a jd je a j( j b j + aje a j( j b j (1+e a j( j b j > a 2 jd je a j( j b j S j( j = c j (1 d j(1+e a( j b j + a2 je a j( j b j (1+e a j( j b j 2 <. It follows that S j( j of ay sgmodal-lke utlty fucto s stctly deceasg fucto. As a esult, S j( j of all the utlty fuctos Secto 2 ae stctly deceasg fuctos. Theefoe, S j( j fuctos ae vetble ad j = S 1 j ae stctly deceasg fuctos. Coollay 3.4. The th use optmal ate allocated by optmzato poblem (6 s equal to the th use aggegated applcatos ates j allocated by optmzato poblem (4. Poof. Fom optmzato poblem (, we have the Lagaga: L T( j = ( =1 β 2 α j logu j( j p T( j +z =1 ( whee z s theslack vaable adp T s thelagage multple whch coespods to the total pce pe badwdth fo the M chaels (.e. shadow pce [2]. So, we have L T( j j U j( j = β α j pt = (11 U j( j U j( j p T = β α j = fj(j (12 U j( j j = f 1 j (p T (13 = j = f 1 j (p T. (14 Fom optmzato poblem (7, we have the Lagaga: L S( = ( β logv ( p S( +z ( =1 =1 whee z s the slack vaable. So, we have L S( V = β ( ps = (16 V ( p S = β V ( V ( (17 usg logv ( = αj loguj(j fom equato (1 ad = j we have logv ( j logv ( = ( j logu j( j =α j j j V ( U j( j V ( =αj U j( j substtutg (17 we have α j logu j( j (18 U j( j p S = β α j = pt = fj(j (19 U j( j = j = f 1 j (p T (2 so the shadow pces p T ad p S of optmzato poblem (4 ad (6 ae equal ad so ae the ates of equato (14 ad (2. Coollay 3.. The j th applcato th use optmal ate j allocated by optmzato poblem (8 s equal to the j th applcato th use optmal ate j allocated by optmzato poblem (4. Poof. Fom optmzato poblem (9, we have the Lagaga: L I( j = ( α j logu j( j p I( j opt +z (21 whee z s the slack vaable ad p I s the Lagage multple whch coespods to the teal pce pe badwdth fo the total th use applcatos (.e. teal shadow pce. L I( j j U j( j = α j pi = (22 U j( j U j( j β p I = β α j = fj(j (23 U j( j usg costats of equato (9 the we have opt = j = f 1 j (β p I (24 fom equato (2 the p T = β p I. Fom equato (12 ad (23, the optmal ates j allocated by optmzato poblem (8 ae equal to optmal ates j allocated by optmzato poblem (4. Theoem 3.6. Optmzato poblems (6 ad (8 ae equvalet to optmzato poblem (4. Poof. It follows fom Coollay 3.4 ad 3. that optmzato poblems (6 ad(8 ae equvalet to optmzato poblem (4. 4. ALGOITHMS The optmal ates ae allocated two-stages. I the fst-stage, EUA algothm allocates the uses ates. I the secod-stage, IUA algothm allocates the applcatos ates j.

5 4.1 EUA Algothm I ths secto, we peset the fst-stage of esouce allocato whee the ates ae allocated to the UEs. The algothm s dvded to a UE algothm show Algothm (1 ad a eodeb algothm show Algothm (2. The algothm s a modfcato of the dstbuted algothms [2]. Algothm 1 UE Algothm Sed tal bd w (1 to eodeb loop eceve shadow pce p S( fom eodeb f STOP fom eodeb the Calculate allocated ate opt else Solve ( = ag ( = w ( p S ( β logv ( p S( Sed ew bd w ( = p S( ( to eodeb ed f ed loop I Algothm (1 ad (2, each UE stats wth a tal bd w (1 whch s tasmtted to the eodeb. The eodeb calculates the dffeece betwee the eceved bd w ( ad the pevously eceved bd w ( 1 ad exts f t s less tha a pe-specfed theshold δ. We set w ( =. If the value s geate tha the theshold, eodeb calculates the M=1 w ( shadow pce p S( = ad seds that value to all the UEs. Each UE eceves the shadow pce to solve the ate that mzes logβ V ( p S(. That ate s used to calculate the ew bd w ( = p S( (. Each UE seds the value of ts ew bd w ( to eodeb. Ths pocess s epeated utl w ( w ( 1 s less tha the theshold δ. Algothm 2 eodeb Algothm loop eceve bds w ( fom UEs {Let w ( = } f w ( w ( 1 < δ the STOP ad allocate ates (.e opt to use else M=1 w Calculate p S( = ( Sed ew shadow pce p S( to all UEs ed f ed loop 4.2 IUA Algothm I ths secto, we peset the secod-stage of esouce allocato whee the ates j ae allocated teally the UE to ts applcatos. The algothm s show Algothm (3. The UE uses the allocated ate the fst-stage opt ad solves the mzato poblem that s gve by =. Fally, the ag (αj loguj(j pij+piopt UE allocates the ates j to the coespodg applcatos.. SIMULATIO ESULTS Algothm (1, (2 ad (3 wee appled to vaous logathmc ad sgmodal-lke utlty fuctos wth dffeet paametes MATLAB. The smulato esults showed covegece to the optmal global pot. I ths secto, we Algothm 3 Iteal UE Algothm loop eceve opt fom eodeb {by Algothm (1 ad (2} Solve = ag (αj loguj(j pij + piopt { = { 1, 2,..., }} Allocate j to the j th applcato ed loop Uj(j U 11 (Sg a =,b = U 21 (Sg a = 3,b = 2 U 31 (Sg a = 1,b = 3 U 12 (Log k = U 22 (Log k = 3 U 32 (Log k = Fgue 1: The applcatos utlty fuctos U j( j. peset the smulato esults of sx utlty fuctos, smla to [2, 3], show Fgue 1, coespodg to thee UEs wth aggegated utltes show Fgue 2. We use thee omalzed sgmodal-lke fuctos that ae expessed by equato (2 wth dffeet paametes, a =, b = whch s a appoxmato to a step fucto at ate = (e.g. VoIP, a = 3, b = 2 whch s a appoxmato of a adaptve eal-tme applcato wth flecto pot at ate = 2 (e.g. stadaddeftovdeosteamg, ada = 1, b = 3 also s a appoxmato of a adaptve eal-tme applcato wth flecto pot at ate = 3 (e.g. hgh defto vdeo steamg. These sgmodal-lke utlty fuctos ae ug Use (.e UE 1, 2, ad 3, espectvely. We use thee logathmc fuctos that ae expessed by equato V( j V 1 = U α Uα V 2 = U α Uα V 3 = U α Uα Fgue 2: The aggegated utlty fuctos V ( coespodg to the th use.

6 ( ( 2 ( 3 ( Fgue 3: The uses ates covegece ( wth umbe of teatos fo = (EUA algothm. (3 wth = ad dffeet k paametes whch ae appoxmato fo delay toleat applcatos (e.g. FTP. We use k = {, 3,.}. These logathmc utlty fuctos ae ug Use 1, 2, ad 3, espectvely. We set β = 1. Let α = {α 11,α 12,α 21,α 22,α 31,α 32} be the set of applcato-status weghts..1 Covegece Dyamcs fo = I the followg smulatos, we set =, applcatostatus weghts α = {.1,.9,.,.,.9,.1}, ad umbe of teatos = 4. I Fgue 3, we show the allocated ates of dffeet uses wth the umbe of teatos. Ths s the soluto of optmzato poblem (6 usg EUA algothm. The use ates ae used to solve optmzato poblem (8 usg IUA algothm to acheve the optmal applcatos ates. Fgue 4, we show the allocated applcato ates j fo each use wth the umbe of teatos. Ths soluto s equvalet to solvg optmzato poblem (4..2 Fo I the followg smulatos, we set δ = 3 ad the eodeb total ate takes values betwee ad wth step of. I Fgue, we show the fal uses ates wth dffeet eodeb total ate. Ths s the soluto of optmzato poblem (6 usg EUA algothm. Fgue 6, we show the fal applcatos ates j of dffeet uses wth dffeet eodeb total ate. Ths s the soluto of optmzato poblem (8 usg IUA algothm..3 Sestvty to chage α I the followg smulatos, we set δ = 3 ad the total achevable ate of the eodeb =. We measue the sestvty of the chage the usage pecetages (coespodg to applcato-status dffeetato of the applcato ug the UEs. The uses swtch betwee 1j( 2j( 3j( ( 12 ( (a The applcato ates 1j( of the 1 st use. 21 ( 22 ( (b The applcato ates 2j( of the 2 d use (c The applcato ates 3j( of the 3 d use. 31 ( 32 ( Fgue 4: The applcatos ates covegece j( wth umbe of teatos fo = (IUA algothm. the applcatos wth the followg usage pecetages {.1,.9,.,.,.9,.1} ; t 4 {.,.,.3,.7,.2,.8} ;4 < t 8 α(t = {.1,.9,.9,.1,.9,.1} ;8 < t 12 {1.,.,.9,.1,.8,.2} ;12 < t 16 {.,.,.9,.1,.8,.2} ;16 < t 2. ( I Fgue 7, we show the uses ates covegece wth tme fo the chagg usage pecetages gve by α(t. 6. COCLUSIO I ths pape, we poposed a ovel two-stage appoach fo esouce allocato 4G-LTE. I the fst-stage, eodeb collaboates wth the UEs to allocate the ates to all the UEs ts coveage aea. I the secod-stage, each UE teally allocates ates to ts applcatos. We poved that ths esouce allocato s optmal ad that t s equvalet to the dect allocato of ates to applcatos by eodeb. Ou poposed algothm takes to cosdeato sevce-offeg dffeetato (eal-tme ad delay-toleat applcatos,

7 use1 1 use2 2 use j Fgue : The uses ates ae the soluto to optmzato poblem (6 fo dffeet values of. applcato-status dffeetato (usage pecetage of evey applcato wth a UE ad subscbe dffeetato (subscbes poty wth a etwok. We showed though smulatos that ou two-stage algothm coveges to the optmal ates tme Fgue 7: The use ates covegece wth the chage applcatos usage pecetages α(t. 7. EFEECES [1] H. Ekstom, QoS cotol the 3GPP evolved packet system, Commucatos Magaze, IEEE, vol. 47, pp , febuay 29. [2] A. Abdel-Had ad C. Clacy, A Utlty Popotoal Faess Appoach fo esouce Allocato 4G-LTE, submsso. [3] A. Abdel-Had ad C. Clacy, A obust Optmal ate Allocato Algothm ad Pcg Polcy fo Hybd Taffc 4G-LTE, PIMC, Fgue 6: The applcatos ates j ae the soluto to optmzato poblem (8 usg optmal use ates of optmzato poblem (6 o the soluto of optmzato poblem (4 fo dffeet values of. [4] G. Tychogogos, A. Gkelas, ad K. K. Leug, A ew dstbuted optmzato famewok fo hybd ad-hoc etwoks, GLOBECOM Wokshops, pp , 211. [] G. Tychogogos, A. Gkelas, ad K. K. Leug, Towads a fa o-covex esouce allocato weless etwoks, PIMC, pp. 36 4, 211. [6] T. Haks, Utlty popotoal fa badwdth allocato: A optmzato oeted appoach, QoS-IP, pp ,. [7] G. Tychogogos, A. Gkelas, ad K. K. Leug, Utlty-popotoal faess weless etwoks., PIMC, pp , IEEE, 212. [8] T. adagopal, T.-E. Km, X. Gao, ad V. Bhaghava, Achevg mac laye faess weless packet etwoks, Poceedgs of the 6th aual teatoal cofeece o Moble computg ad etwokg, MobCom, (ew Yok, Y, USA, pp , ACM, 2. [9] J.-W. Lee,.. Mazumda, ad. B. Shoff, Dowlk powe allocato fo mult-class weless systems, IEEE/ACM Tas. etw., vol. 13, pp , Aug.. []. L. Kule, esouce Allocato fo Smat Phoes 4G LTE Advaced Cae Aggegato, Maste Thess, Vga Tech, ov [11] F. Kelly, A. Maulloo, ad D. Ta, ate cotol commucato etwoks: shadow pces, popotoal faess ad stablty, Joual of the Opeatoal eseach Socety, vol. 49, 1998.