Queue-Aware Optimal Resource Allocation for the LTE Downlink

Transcription

1 Queue-Aware Optmal Resource Allocaton for the LTE Downlnk Hussam Ahmed, Krshna Jagannathan, Srkrshna Bhashyam Department of Electrcal Engneerng IIT Madras, Chenna, Inda Abstract We address the problem of optmal downlnk resource allocaton n an OFDMA system, n a scenaro where very lmted channel qualty nformaton (CQI) s avalable at the basestaton. Our work s partcularly applcable n the context of the LTE downlnk, snce the feedback mechansm we consder closely resembles one of the CQI reportng modes n LTE. Specfcally, the users only report the ndces of ther best M sub-bands and an effectve CQI correspondng to these best M bands. Our polcy smultaneously performs optmal sub-band assgnment and rate allocaton, by takng nto account channel qualty as well as the queue backlogs of each user. The techncal novelty of our work les n explotng a lmt theorem on the best s reported by the users, and combnng t wthn a Lyapunov stablty framework. We show that our polcy s throughput maxmzng among all polces whch are constraned to the CQI mechansm consdered. Numercal results ndcate that n terms of throughput and average delay, our polcy compares favorably to exstng resource allocaton polces such as proportonal far. I. INTRODUCTION Orthogonal Frequency Dvson Multplexng (OFDM) s employed n most of the emergng hgh data-rate wreless cellular standards such as Long Term Evoluton (LTE) [] and IEEE 82.6 (WMAX). In ths paper, we tackle the problem of optmal resource allocaton n downlnk of an OFDMA system, n a scenaro where very lmted channel qualty nformaton (CQI) s avalable at the base-staton (BS). Our work s partcularly applcable n the context of LTE downlnk, snce the feedback mechansm we consder closely resembles one of the CQI reportng modes n LTE. In an OFDM system such as the LTE downlnk, the avalable bandwdth (say 2 MHz) s dvded nto several hundred subcarrers (e.g., 52, 24, or 248). These sub-carrers need to be allocated to multple user equpments (UEs). In practce, a resource block (RB) par consstng of 2 contguous subcarrers and 4 OFDM symbols n tme s the smallest resource allocaton unt [2]. After accountng for unusable tones, ths leaves us wth about 5 to RBs to allocate to the UEs. In order to schedule the UEs opportunstcally, the basestaton, n prncple, needs to obtan channel qualty nformaton from each UE, on each of the resource blocks. Ths s hghly mpractcal, snce t leads to an enormous amount of control overheads on the uplnk. To overcome ths, UEs n an LTE system report CQI to the base-staton n a very sparse manner. A. Related Work Varous reduced feedback mechansms have been studed n the lterature n the context of resource allocaton n OFDM downlnk. In [3], the CQI of each UE s fed back only for those sub-bands whose qualty s better than a certan threshold. The feedback overhead s even more reduced n [4], where the UEs report one-bt per sub-band whenever the channel qualty exceeds the threshold. In [5], an opportunstc feedback strategy s consdered wheren only the channel gans of prespecfed number of best sub-bands are reported. A varaton of ths polcy has been consdered n [6], [7]. In [6], the UEs feedback the average gan of the best M sub-bands and the correspondng ndces whle n [7], each UE reports an Effectve Exponental Sgnal-to-nose rato Mappng (EESM) of the best M sub-bands and ther respectve ndces. In effect, EESM translates the dfferent s on parallel channels nto a sngle effectve flat-fadng [8]. In ths paper, we assume a CQI feedback mechansm smlar to [6], [7], snce t closely resembles one of the CQI reportng modes namely, the UE-selected sub-band feedback mode defned n the LTE standards [9]. Specfcally, the UEs only report the ndces of ther best M sub-bands, where M s a small number (say 2 to 5), and the EESM correspondng to these best M bands. Downlnk resource allocaton for OFDM systems has been studed from varous perspectves n recent years. In [], resource allocaton n downlnk OFDM s posed as a utlty maxmzaton problem, whch ncludes proportonally far resource allocaton [], [2] as a specal case. The optmal power and sub-carrer allocaton are then determned usng convex dualty technques. Whle [] assumed full CQI avalablty except for an estmaton error term, [3] takes mperfect CQI nto account by factorng for outages due to erroneous CQI at the base-staton. In [4], the authors consder opportunstc resource allocaton n OFDM under varous farness constrants, and propose a Hungaran algorthm based soluton. It s worth notng that [], [3], [4] assume fully backlogged buffers (.e., that the base-staton always has data to send to the UEs), and do not consder any queung dynamcs. There s a vast lterature on optmal server allocaton to constraned queueng systems wth tme-varyng connectvtes. Most of the lterature n ths area based on the landmark papers [5], [6] whch ntroduced Lyapunov technques for A sub-band typcally conssts of one to three resource blocks.

2 resource allocaton. Subsequently, these Lyapunov methods, whch explctly take queue lengths nto account for makng resource allocaton decsons, have been appled n varous contexts ncludng hgh-speed swtches [7], satelltes [8], wreless [9], and optcal networks [2]. In addton to beng nherently throughput maxmzng, Lyapunov based resource allocaton polces can also been used to ensure Qualty of Servce (QoS) metrcs such as delay guarantees [2], [22] and farness [23]. In the above Lyapunov based resource allocaton polces, the resource allocaton decson s based on the UE s channel qualty as well as queue backlogs, and these are typcally assumed to be avalable perfectly and nstantaneously at the base staton. In contrast, [24] proposes a throughput optmal resource allocaton algorthm under delayed channel nformaton; ther polcy utlzes the condtonal expectaton of the channel qualty, gven the delayed measurements. In [25], a cross layer resource allocaton polcy whch maxmzes the throughput under delayed CQI and takes nto account the channel outage event s proposed. There has also been recent work on low-complexty dynamc resource allocaton for OFDM [26], [27] to ensure low delay, but these papers do not consder sparse CQI feedback. B. Our Contrbutons In the present paper, we propose a queue-aware resource allocaton polcy for the OFDM downlnk that s optmzed for the specfc form of the CQI avalable at the base-staton. As descrbed earler, we assume that the UEs only report the ndces of ther best M sub-bands, where M s a small number, and the EESM correspondng to these best M subbands. We develop a sub-band assgnment and rate allocaton algorthm whch s throughput maxmzng under ths CQI scenaro, when the total number of sub-bands s large. In other words, our algorthm s guaranteed to keep the queueng system stable for all traffc rates that can be stablzed by any resource allocaton polcy whch s constraned to ths CQI scenaro. One of the techncal contrbutons of the paper les n obtanng an explct characterzaton of the outage probablty on each of the M reported sub-bands. In order to obtan the outage probablty expresson, we explot a Gumbel lmt theorem on the jont dstrbuton of the best M sub-bands, whch subsequently leads to an explct expresson for the condtonal densty, gven the EESM. It s worth commentng that the Gumbel weak lmt s an attractor for the extremal values of a farly large famly of dstrbutons [28], so that our work does not crucally depend on the assumpton that the sub-band gans are..d. Raylegh dstrbuted. Another dstngushng feature of our resource allocaton polcy s that t naturally decouples for each sub-band, and does not ental solvng any computatonally ntensve matchng problems [4], [26]. II. SYSTEM MODEL Consder a downlnk system wth one BS and K UEs. The BS mantans a separate queue correspondng to each UE. Tme s slotted, and the queue correspondng to th UE receves exogenous arrvals accordng to a random process. We denote the amount of data that enters queue durng tme slot t by A (t), and the queue length correspondng to the th UE durng slot t by Q (t). We assume that the arrval process A (t) s..d. from slot to slot, wth mean λ and a fnte second moment. We assume that the channel between the BS and th UE s a frequency selectve Raylegh fadng channel. We remark that ths Raylegh fadng assumpton s not crucal to our work, but t makes exposton easer. OFDM transmsson wth N c sub-carrers s used. The for the th UE on the j th subcarrer follows an exponental dstrbuton. The average for the th UE s denoted as γ ave,. We assume that the downlnk channel gans of the UEs are not known to the BS unless the UEs feedback ther CQI to the BS. Ths corresponds to a scenaro where the uplnk and the downlnk channels are not recprocal, or a scenaro where the UEs are not transmttng any data on the uplnk, so that recprocty (even f present) cannot be exploted. In order to reduce feedback overhead, we assume that the sub-carrers are grouped nto N sub-bands n such a way that the channel can be approxmated as flat-fadng n each sub-band. Further, we consder the best M feedback mechansm smlar to [7], where each UE reports () the EESM correspondng to ts best M( N) sub-bands, and () the ndces of those sub-bands. Let γ j (t) be the on the jth sub-band for the th UE n slot t and γ () (t), γ (2) (t)... γ (N) (t) be the ordered sub-band s for th UE n descendng order. The EESM for the best M sub-bands correspondng to the th UE n slot t, denoted γ eff, s defned by γ eff (t) = η ln M e γ (j) (t) η, () M where η s a parameter that depends on the modulaton and codng scheme (MCS). Snce we are consderng a downlnk problem, the BS s assumed to know the nstantaneous queue lengths Q (t) for all the UEs. III. PROBLEM FORMULATION In ths secton, we develop a mathematcal formulaton of the optmal resource allocaton problem. As mentoned earler, the followng nformaton s assumed to be avalable wth the BS durng tme-slot t (for smplfed notaton, we omt t): () The EESMs γ eff = [γ eff, γ eff 2,..., γ eff K ] () The ndex sets I = [I, I 2,..., I K ] () The queue length vector Q = [Q, Q 2,..., Q K ]. where I s the set of ndces reported by the th UE and s gven by I = {, 2,... M } where j s the ndex of the j th best sub-band of the th UE. Gven ths nformaton, our am s to come up wth a resource allocaton polcy whch

3 can maxmze throughput whle keepng all queues at the BS stable. In order to make ths statement precse, we develop some termnology and notaton. A resource allocaton polcy performs the followng two operatons n each slot. Sub-band assgnment: For each sub-band j that s reported by at least one UE, the polcy determnes a unque UE to assgn the sub-band. (Recall that a sub-band can be allocated to at most one UE due to nterference consderatons, whereas a UE can be allocated multple sub-bands). Rate allocaton: Gven that j th sub-band s assgned to th UE, determne the rate r,j at whch data transmsson wll take place on j th sub-band. From now on, we use the notaton [, j] for the th UE - j th sub-band par. In the nterest of smplcty, we restrct our attenton to polces whch allocate equal power to all scheduled sub-bands, although our framework can be modfed to nclude optmal power allocaton for dfferent sub-bands. To be precse, defne Î = K = I as the set of all dstnct subbands reported by at least one UE, and let M = Î denote the number of such dstnct sub-bands. Assume that the BS has a power budget of P for transmssons durng each slot. Then, the base staton allocates power P/M to each sub-band. Let C,j be the nstantaneous capacty of [, j]. Under the above assumptons, we have C,j = log 2 ( + P ) M γj. (2) For a relable communcaton over a sub-band, the rate assgned to [, j], r,j, should not exceed C,j. Gven γ eff and the ndex sets I, we say [, j] s n outage f the rate allocated to [, j] s greater than C,j. The outage probablty for [, j] when the assgned rate s r,j s defned as follows: { } γ eff P,j (r,j ) = P C,j < r,j, I. (3) We defne a natural metrc, namely goodput, as the average successfully transmtted rate over a sub-band [29]. The goodput for [, j] when the assgned rate s r,j s defned as follows: G,j (r,j ) = r,j ( P,j (r,j )). (4) Next, we brefly revew the queueng dynamcs and stablty consderatons of the queueng system at the BS. A. Stablty consderatons The queue evoluton equaton for the th UE can be wrtten as Q (t + ) = max{q (t) µ (t), } + A (t), (5) where A (t) and µ (t) are arrval and servce processes of the th UE queue. Here, µ (t) s the amount of data served from the th UE queue durng slot t, and can be wrtten as µ (t) = a,j r,j H,j (t). In the above expresson, a,j denotes the fracton of tme the j th sub-band s allocated to the th UE durng slot t. Clearly a,j. (6) = Later, we wll show that our optmal polcy allocates a subband to at most one UE durng each tme-slot. Next, H,j (t) s an ndcator random varable whch takes a value whenever the transmsson through [, j] durng slot t s successful and otherwse. Thus, P {H,j (t) = } = P N,j (r,j). In the sprt of [3], we say that the queueng system at the BS s strongly stable f for each UE, lm sup T T T E[Q (t)] <. (7) t= Denote by P the famly of all resource allocaton whch allocate equal power to all scheduled sub-bands, and have access only to the parameters γ eff, I, and Q n order to make the resource allocaton decsons durng each slot. Let Λ be the stablty regon of the network, whch s defned as (the closure of) the set of all arrval rates λ = (λ, λ 2,..., λ K ) for whch there exsts some polcy Π P under whch the queueng system s strongly stable. We fnd a resource allocaton polcy n P whch s throughput optmal, n the sense that t keeps the queung system stable for all arrval rates n the nteror of Λ. IV. THROUGHPUT OPTIMAL RESOURCE ALLOCATION POLICY Durng each tme slot, the scheduler at the BS observes γ eff, I, and Q, and mplements the followng steps : ) Determne Î = K = I and M = Î. 2) for j = to M do () Set P j = P M. () Determne U j = { j I }. () Calculate an estmate of the outage probablty ˆP,j (r) as a functon of r for each U j. (See Secton V) (v) Calculate (v) Calculate r,j = arg max{r( ˆP,j (r))} U j. r (j) = arg max U j {Q (t)r,j( ˆP,j (r,j))}. (v) Assgn j th sub-band to (j) th UE, and transmt at rate r (j),j wth power P j. 3) end for A. Dscusson In the frst step, the scheduler determnes the set of all dstnct sub-bands reported by the UEs. Then, for each such sub-band j, the scheduler sets the power to be P j and determnes the set U j of all UEs who report that sub-band as beng one of ther best M sub-bands. In the next step,

4 the outage probablty on [, j] s computed, as explaned n Secton V. Then the scheduler computes the rate that ensures the best goodput for each UE U j. Fnally, n the last two steps, the scheduler assgns j th sub-band to the th UE that has the maxmum queue-length goodput product. Notce that the above algorthm assgns every reported subband to a unque UE. Also, no power s assgned to sub-bands that are not reported by any UE. B. Lyapunov Analyss In ths secton, we derve the optmal resource allocaton polcy as a Lyapunov drft mnmzng polcy, and prove that t s throughput optmal. We defne the quadratc Lyapunov functon L(Q(t)) = (Q (t)) 2, = and consder the condtonal Lyapunov drft (Q(t)) = E { L(Q(t + )) L(Q(t)) Q(t) }. Usng (5), straghtforward algebra gves L(Q(t + )) L(Q(t)) (A (t)) A (t)q (t) = = = = 2 a,j r,j H,j (t) 2 Q (t) a,j r,j H,j (t). Thus, takng condtonal expectatons and explotng the ndependence of A (t) and Q (t), we get (Q(t)) B + 2 Q (t)λ where 2 = = B = ( max{c,j }) 2 + Q (t)a,j r,j ( P,j (r,j )), E [ A (t) 2] <. We know from [3, Lemma 4.] that the Lyapunov drft becomng negatve for large queue backlogs s a suffcent condton for the strong stablty of the queueng system. Wth ths n mnd, we seek the polcy that maxmzes the negatve term on the rght hand sde of (9). We therefore formulate the optmal resource allocaton problem as follows. max Q (t)a,j G,j (r,j ), () subject to {a,j},{r,j} = = a,j, j, (C) = a,j,, j, (C2) r,j,, j. (C3) (8) (9) We assume that t s possble to come up wth modulaton and codng schemes for any desred rate r,j. Then, the above problem s a convex optmzaton problem and the soluton can be obtaned easly by usng Karush-Kuhn-Tucker (KKT) condtons [3]. The soluton s dscussed next. C. Mnmzng the Lyapunov Drft We now solve the convex optmzaton problem () and arrve at our resource allocaton polcy. Introducng the nonnegatve Lagrange multplers {α j },{β,j },{δ,j } for constrants (C)-(C3) respectvely, the followng condtons also must be satsfed at the optmal soluton (superscrpt ( ) denotes optmal values). Q (t)g,j (r,j) + β,j α j =,, j. () Q (t)a G,j (r,j ),j r,j + δ,j =,, j. (2) α j ( K ) a,j = =, j. (3) β,ja,j =,, j. (4) δ,jr,j =,, j. (5) Proposton. The optmal sub-band allocaton for problem () assgns a sub-band exclusvely to the UE wth the largest correspondng queue-length goodput product. Proof. It follows from (5) and (2) that f r,j >, then δ,j =,.e., =. Thus, r,j s obtaned by G,j(r,j ) r,j maxmzng goodput of th UE on j th sub-band (Step 5 of our polcy n Secton IV). Smlarly, t follows from (4) and () that f a,j >, then β,j =,.e., Q (t)g,j (r,j ) = α j. If a,j =, then β,j,.e., Q (t)g,j (r,j ) α j. Hence, the j th sub-band s assgned to the UE wth largest queuelength goodput product Q (t)g,j (r,j ). If multple UEs have the same queue-length goodput product for the same subband j, the sub-band can be shared n any arbtrary manner among these users wthout affectng optmalty n terms of the objectve functon n (). Proposton shows that the optmal sub-band allocaton assgns each reported sub-band j to the UE whch has the maxmum queue-length goodput product on the j th sub-band. Ths establshes that the polcy n Secton IV mnmzes the Lyapunov drft. Snce the proposed polcy ensures the most negatve Lyapunov drft among the class P, t seems plausble that our polcy should be able to stablze the queueng system, whenever some polcy n P can do so. The followng theorem asserts that ths s ndeed true. Theorem. The resource allocaton polcy proposed n Secton IV s asymptotcally throughput optmal, when the number of sub-bands s large. The proof s relegated to Appendx A. Our polcy s only asymptotcally throughput optmal snce the polcy uses the

5 lmtng outage probablty gven by a lmt theorem, nstead of the actual outage probablty whch s dffcult to compute. V. DERIVATION OF OUTAGE PROBABILITY In ths secton, we descrbe how the BS estmates the outage probablty on [, j] n Step 4 of our algorthm, usng only the parameters γ eff and I. We utlze a lmt theorem on the order statstcs of the s to derve an expresson for the condtonal jont dstrbuton of the s on the best M sub-bands for each UE, gven the EESM and the subband ndces. For ease of exposton, we assume that the s on the sub-bands of a gven UE are..d. exponentally dstrbuted. Ths assumpton wll hold well n the case of Raylegh fadng n a rch mult-path envronment, wth number of paths comparable to the number of sub-bands. However, we remark that the lmt theorem we are about to explot holds for a farly large class of dstrbutons namely, those whch le wthn the Gumbel doman of attracton [28]. Therefore, our polcy remans asymptotcally throughput optmal for ths class of sub-band dstrbutons. We frst state a result whch follows from [32, Theorem 5] regardng the order statstcs of M extremal values, drawn from N..d. exponental random varables. Theorem 2. Let Z, Z 2,..., Z N be a sequence of..d. unt exponental random varables, and Z (), Z (2),..., Z (N) be the correspondng order statstcs n descendng order. Then, for any fnte M, (e Z (), e Z (2),..., e Z (M) ) D (Y, Y 2,... Y M ), as N, where Z () = Z () ln N, Y = X j and X j s are..d. unt exponental random varables. Consder the th UE. Let S j = e ( γ j γ ln N) ave, for j =,..., M. Assumng that the number of sub-bands N s large, we can use Theorem 2, to approxmate the uncondtonal jont pdf of S (M) = (S, S2,..., SM ) as f (M) S (s, s 2,..., s M ) = e s M, s s 2... s M. Numercal results ndcate that ths approxmaton s good even for moderate values of N. Next, defne S eff e ( γ eff = γ ave, ln N). Choosng the parameter η n () as unty for smplcty, we have S eff = M M S j. Note that S eff s known to the BS. Next, condtoned on S eff = s and I = I, S (M) takes values only on the hyperplane M M Sj = s. Hence, we gnore the M th best and calculate the condtonal jont pdfs for the remanng sub-band s as follows. f (M ) S S eff =s,i (s =I, s 2,..., s M ) = Me (Ms f S eff M sj) (s) M, s s 2... Ms s j. Now, we can fnd the condtonal margnal densty f S j Seff =s,i (s =I j) for j =,..., M by ntegratng out the other varables n the above expresson. The outage probablty can thus be determned as ˆP,j (r) = ( N y 2 r P γ ave, M f S j Seff =Ne γ,i =I (s j)ds j. (6) ) where y = Ne. Smlarly, ˆP,M (r) can be computed from the condtonal margnal of S M. Closed form expressons for the outage probabltes (6) can be obtaned through some tedous computatons; n [33], we provde explct expressons for the cases M = 3 and M = 4. The outage probabltes computed n (6) are used n Step 4 of the resource allocaton algorthm. VI. SIMULATION RESULTS In ths secton, we present smulaton results that demonstrate the throughput gans acheved by the proposed polcy over other exstng polces. We also demonstrate that the lmtng approxmaton we use to obtan closed form outage probablty expressons s a good approxmaton. The proposed polcy (labeled as optmal n the plots) s throughput optmal among all polces that use the lmted channel feedback scheme descrbed n Secton II. Three mportant components of our polcy are: () evaluaton of the condtonal expected CQI for each sub-band from the EESM, (2) evaluaton of goodput whle accountng for outage probablty, and (3) optmal utlzaton of queue length nformaton. To llustrate the mportance of each component of our proposed polcy, we compare the proposed polcy wth the followng polces (each of the heurstc polces gnores at least one component of our proposed polcy): () a throughput optmal polcy wth perfect CQI (labelled Perfect CQI ), (2) a polcy that uses queue length nformaton but assumes that the reported EESM s the CQI for all the best M reported sub-bands (labelled Heurstc ), (3) a polcy that uses queue length nformaton and evaluates the condtonal expected CQI for the best M reported sub-bands wthout accountng for outage probablty (labelled Heurstc 2 ), and (4) a proportonally far rate allocaton polcy that uses the condtonal expected CQI and goodput evaluaton wthout usng queue length nformaton (labelled PF ). A heterogeneous sngle-cell OFDM downlnk wth K = UEs at dfferent dstances from the BS s smulated. The th UE s located at a dstance of d from the BS where d s a constant. The average for a user s assumed to be nversely proportonal to the square of ts dstance from the BS. The total number of subcarrers s 52 and there are 2 subcarrers n each sub-band. Two channel models are consdered: () IID sub-bands, and (2) Correlated sub-bands resultng from a 6-path channel wth an unform power-delay profle where each path s Raylegh fadng. The arrval traffc for the th UE s assumed to be Posson wth wth parameter λ. The channel feedback from each UE s assumed to be the best M sub-bands and EESM for these sub-bands.

6 Fgures and 2 show the average queue length (averaged across UEs and tme slots) versus the aggregate arrval rate (.e., sum of λ s) for the IID and correlated sub-band cases respectvely. λ s chosen as (K + )λ,.e., each UE has a dfferent arrval traffc rate, and λ s vared to change the arrval traffc load. Also, M = 3 and the number of sub-bands N = 43. It s clear that the proposed polcy can support sgnfcantly hgher arrval traffc for the same average queue length than the heurstc polces. The Perfect CQI polcy s also shown to quantfy the loss due to lmted feedback. It s also clear that the proposed polcy provdes smlar performance gans even n the correlated sub-band case. Smlar results have been observed for M = 4 (not shown due to space lmtatons). N = N = N = 43 N = Weak lmt..d sub bands 2 2 Average backlog per user per slot.9 Average backlog per user per slot Optmal Heurstc Heurstc 2 PF Perfect CQI 5 5 Aggregate arrval rate (bts per slot).4.2 Fg.. IID sub-bands case: M = 3, N = 43, η =. Optmal Heurstc Heurstc 2 PF Perfect CQI 5 5 Aggregate arrval rate (bts per slot) Fg. 2. Correlated sub-bands case: M = 3, N = 43, η =. Fgure 3 shows the condtonal CDF of the CQI of the best sub-band gven a partcular EESM for the best M sub-bands. Four cases of N (the total number of sub-bands) are shown. Note that the number of subcarrers s 2N. It can be observed that the weak lmt approxmaton s very good for N = 22 and N = 43 for the IID case. Fg. 3. Condtonal CDF of best sub-band gven γ eff =.5 + log N, M = 3, η =. VII. CONCLUSION We proposed a queue-aware polcy for allocatng sub-bands n the LTE downlnk when each UE reports the best M subband ndces and a sngle effectve CQI for these bands. The throughput optmalty of the proposed polcy was shown usng the Lyapunov stablty framework. The polcy assgns each sub-band to the UE wth the best queue-length goodput product for that sub-band. The goodput was obtaned by dervng analytcal expressons for the condtonal outage probablty of each sub-band gven the effectve CQI. The condtonal outage probablty was derved by explotng a lmt theorem on the jont dstrbuton of the of the best sub-bands. The proposed polcy supports sgnfcantly hgher arrval traffc than exstng polces lke: () proportonal far allocaton based on CQI that does not consder queue nformaton, (2) queueaware polces that use the effectve CQI as the CQI of each sub-band, and (3) queue-aware polces that do not account for outage n the estmaton of goodput. APPENDIX A PROOF OF THEOREM Proof. If the arrval rate vector λ s stablzable by some polcy Π P then ɛ = (ɛ, ɛ 2,..., ɛ K ) wth ɛ > such that λ b,j r,j ( P,j (r,j )) ɛ,, where b,j s the fracton of j th sub-band allocated to th UE and r,j s the rate assgned to [, j] by polcy Π. Next, we nvoke Scheffé s Lemma [34], whch asserts the unform convergence of P,j (r) ˆP,j (r) to zero. Thus, for large N, P,j (r) ˆP,j (r) δ N,j, r,, j, where δ,j N s a small postve number ndependent of r. Snce for every sub-band, our polcy assgns a,j and r,j such that

7 K = Q (t)a,j r,j ( ˆP,j (r,j )) s maxmzed, the followng nequalty holds good j, {b,j }, {r,j }. Q (t)b,j r,j ( P,j (r,j ) δ,j) N = Q (t)a,jr,j( P,j (r,j) + δ,j). N = For our polcy, the Lyapunov drft can be upper bounded as (Q(t)) B Q (t) ɛ (a,jr,j + b,j r,j )δ,j N. = Note that only M of the a,j and b,j are non-zero for each user. Thus, for any ɛ, there exsts a large enough N for whch ɛ (a,jr,j + b,j r,j )δ,j N >,, whch ensures that the Lyapunov drft becomes negatve as queues grow..e., the proposed polcy stablzes all the arrval rates whch can be stablzed by any other polcy for large enough N. Hence t s asymptotcally throughput optmal. ACKNOWLEDGMENT The authors thank Dr. Sheetal Kalyan (IIT Madras) for her useful comments. The second author acknowledges support from the IU-ATC project, funded by the Department of Scence and Technology, Government of Inda, and the UK EPSRC Dgtal Economy programme. REFERENCES [] C. 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