Delay Estimation and Fast Iterative Scheduling Policies for LTE Uplink

Transcription

1 Delay Estmaton and Fast teratve Schedulng Polces for LTE Uplnk Akash Bad WNLAB, Rutgers Unversty, Rtesh Madan Accelera MB, and Ashwn Sampath Qualcomm Abstract We desgn fast teratve polces for resource allocaton n the uplnk of LTE. We generalze recent works on teratve delay and queue based schedulng polces to more general system settngs. We model all constrants due to contguous bandwdth allocaton, peak transmt power and fractonal power control. We desgn a novel mechansm for nferrng the packet delays approxmately from the buffer status reports BSR and construct a new non-dfferentable objectve functon whch enables delay based schedulng. For frequency flat fadng, we construct an ON log L optmal resource allocaton algorthm for N users and L ponts of non-dfferentablty n the objectve functon. For a frequency dversty scheduler wth M sub-bands, the correspondng complexty s essentallyonm 2 +L 2. Through detaled system smulatons based on NGMN and 3GPP evaluaton methodology whch model H-ARQ, fnte resource grants per sub-frame, realstc traffc, power lmtatons, nterference, and channel fadng, we demonstrate the effectveness of our schemes for LTE.. NTRODUCTON Wdeband cellular systems such as LTE allow for resource allocaton wth hgh granularty of a resource block RB of 1 ms by 18 KHz [1]. Whle control sgnallng and the general framework for the physcal and medum access control MAC layers s specfed to enable effcent use of spectral resources, the exact resource allocaton algorthms for power and frequency allocaton can be desgned by an mplementor. Moreover, each cell can serve on the order of a thousand actve connectons over a bandwdth of 2 MHz. Hence, n order to take advantage of the flexblty allowed n resource allocaton, the resource allocaton algorthms have to be computatonally smple. Many schedulers n the lterature ental maxmzng the weghted sum of rates n each subframe. For example, the weghts could be based on utlty functons of average rate [2], [3], the queue length [4], [5], or head-of-lne delay [6], [7]. n the uplnk, the resource allocaton problem must consder the maxmum transmsson power of a moble and the constrants on the transmsson power mposed by fractonal power control to lmt nter-cell nterference [1], [8]. When contguous bandwdth allocaton s consdered, the problem of maxmzng the weghted sum rate n each subframe on the UL can be posed as a constraned convex optmzaton problem. For N users and M sub-bands general purpose methods can solve the problem n ONM 3. Wth peak UE power constrants, a ONM per teraton subgradent algorthm was obtaned n [9]. nteror pont methods whch have faster convergence wth an ONM 2 f N >> M Newton teraton were obtaned n [1] for uplnk resource allocaton wth addtonal fractonal power control constrants. However non-dfferentable objectve functons are not consdered under the framework n [1]. Also relevant to our paper are recent results on low complexty teratve schedulng algorthms. Many papers pror to these results had consdered schedulng to maxmze the sum of weghted rates n subframe n, where the weghts were based on the arrvals and departures n the queue of a user untl subframe n 1. The teratve polces n [11], [12] take nto account how the weghts change n subframe n to determne the resource allocaton n that subframe. The results n these papers shed a remarkable nsght that when the rate grows lnearly wth bandwdth no peak power constrants at the transmtter, as the number of users n the system grow, these rules lead to much smaller per-user queues and delays, respectvely, compared wth prevous approaches. However, the complexty of these algorthms grow wth the resource granularty even f the coherence bandwdth does not grow. n ths paper, we construct a contnuous but non-dfferentable concave reward functon based on packet delays. t can be shown that the matchng algorthm n [11] s an approxmate algorthm to maxmze ths reward functon n every subframe. Motvated by the above observaton, we consder resource allocaton to maxmze a contnuous possbly nondfferentable concave reward functon. We frst consder a channel model where the channel gan n the frequency doman s flat and formulate the resource allocaton problem as a non-dfferentable convex optmzaton problem. Note that n typcal cellular envronments, the channel gans can be farly correlated even for frequences 2 to 5 MHz apart [13] hence, the assumpton of frequency flat fadng s a reasonable one when the total bandwdth s up to 5 MHz 28 RBs or lower, or f the UEs are allocated to sub-bands < 5 MHz over a slower tme-scale based on nterference and channel statstcs. We use subgradent analyss to desgn algorthms wth ON log L cost per teraton wth small number of teratons for N users and L ponts of non-dfferentablty n the objectve functon. We also desgn a novel mechansm to estmate head-of-lne delays of queues at UEs wth low complexty va only queue length nformaton contaned n the buffer status reports BSR. We note our technques are equally applcable for enablng delay based schedulng n the PCF and HCF modes n WF [14]. We demonstrate the mprovement n performance due to our technques through numercal results obtaned va comprehensve numercal smulatons based on 3GPP evaluaton methodology [15]. Fnally, when frequency selectve fadng s consdered, we show how nteror pont methods wth complexty of ONM 2 + NL 2 per Newton teraton can be obtaned; note that n practce N >> L,M. A

2 longer verson of the paper contanng more detaled analyss and addtonal results s avalable at [16] for reference.. SYSTEM MODEL A. Channel Model, Power, Rate We focus on the uplnk of a sngle cell n LTE wth N UEs and the total bandwdth dvded nto M sub-bands of equal bandwdth B, wth B less than the coherence bandwdth of each user. The maxmum transmt power of each UE s P. The channel gan for UE on sub-band j s G j ; we focus on the scheduler computaton n a subframe, and don t explctly show the dependence of quanttes on tme t. The base-staton can measure the G j s va decodng the soundng reference sgnal SRS [1]. Fractonal power control n LTE lmts the amount of nterference a UE causes at base-statons n neghborng cells. A UE whch s closer to the cell edge nverts a smaller fracton of the path loss to the servng base-staton than a UE whch s closer to the servng basestaton [8]. Thus the transmt powers of a UE on dfferent sub-bands satsfy [1]: M p j γ j b j,,j, p j P, where b j s the bandwdth allocated to UE on sub-band j and γ j s a sub-band specfc constant. The nterference PSD at the servng base-staton on subband j denoted as j can be measured by the base-staton perodcally over unassgned frequency resources. The value depends on the nterference coordnaton algorthm used [17]. When a UE transmts wth power p j over bandwdth b j on sub-band j, t acheves a rate gven by treatng nterference as nose b j ψg j p j /b j j where ψ : R + R + s an ncreasng concave and dfferentable functon whch maps the SNR to spectral effcency. B. Control Sgnalng Sngle carrer frequency dvson multple access SC- FDMA s used n the LTE uplnk [1] and so a UE can be granted a number of 18 khz resource blocks n a contguous manner n frequency. The resource allocaton to the UEs s computed by the base-staton every subframe 1 ms and sgnalled to the UEs va resource grants whch nclude the contguous set of RBs allocated to the UE and the modulaton and codng scheme MCS. We assume a constant number of maxmum allowable re-transmssons for all UEs and do not adapt the re-transmsson power and resource assgnment through addtonal control sgnallng avalable n LTE. Buffer status report BSR and schedulng request SR are transmtted by the UEs to nform the base-staton about new packet arrvals at the UE. SR s one bt of nformaton used to ndcate the arrval of packets n an empty buffer at the UE and s used by the scheduler to start allocatng resources to the UE. BSRs contan a quantzed value of the number of bytes pendng transmsson at the UE 1, and are generated ether perodcally or when the queue goes from an empty 1 We gnore the effect of quantzaton n BSR, but the methods n ths paper extend easly to quantzed BSR. to non-empty state. BSR reports are transmtted only when resources are allocated to the UE and thus provdes only a coarse gran nformaton about the queue length at the UE.. REWARD FUNCTONS n ths secton, we defne the reward functons that we use for the optmzaton problem and relate t to the schemes used n earler works. We assume each UE to have one actve LC whch supports ether best effort or delay QoS traffc. A. Best Effort A flow,, whch s best-effort s assocated wth an average rate x t R + n subframe t whch s updated as follows: x t+1 = 1 α x t+α r, t, 1 where r s the rate at whch UE s served n the current subframe, and < α < 1 s a user specfc constant. The user experence n subframe t s modeled as a strctly concave ncreasng functon U : R + R of the average rate x t. We greedly maxmze the total utlty at each tme-step,.e., the reward functon for UE wth best effort traffc, at tme t s [18] f r = 1 α U 1 α x t+α r. 2 f we set f r = U x tr, and let α n equaton 1, the resultng scheduler s dentcal to that n [3]. Thus, our analyss offers a computatonally effcent method to mplement the schedulng polcy n [3] for the LTE uplnk wth fractonal power control. B. Delay QoS Traffc Here the user experence s a functon of the packet delays. User experence s acceptable when the packet delays are lower than a certan tolerable value. The packet arrval process s assumed to be ndependent of the tmes at whch the packets are served. Traffc for applcatons such as voce calls and lve vdeo chattng fall n ths category. At tme t, let π t be the number of packets n the queue of UE. Denote the szes and the delays of these π t packets by {s 1,...,s π t} and {d 1,...,d π t}. Then for a UE wth delay QoS traffc, we defne the reward functon as: f r = n serv r + r s jd j n serv r s j d n serv r +1 where s the length of a subframe 1 ms and n serv r s the number of packets from UE served fully f UE s scheduled at rate r,.e., { n serv r = max k : } k s j r. Lemma 3.1: f r s a contnuous concave functon. Proof: Concavty follows from the observaton that d 1 >... > d π t and contnuty s mmedate from defnton. 3

3 reward mskb slope =.76 slope =.12 slope =.3 slope = rate Kbps x 1 4 Fg. 1. Example reward functon for delay QoS flow. Example: Consder a UE wth delay QoS traffc and four packets n the queue wth delays n ms at tme t gven by d 1 = 12,d 2 = 76,d 3 = 27,d 4 = 3, and packet szes n KB are s 1 = 1.5,s 2 =.7,s 3 = 2.1,s 4 = 3. Then the correspondng reward functon f s shown n Fg. 1. C. teratve Queue and Delay Based Polces f we restrct the model n [11] to frequency flat fadng,.e., a user s ether connected to no server or all servers at any tme, the algorthm n that paper can be nterpreted as one whch approxmately maxmzes the reward functon n equaton 3. n ths work, we consder the maxmzaton of the reward functon n 3 for a much more general model wth multple rate optons, peak power constrants, and dfferent transmt PSD constrants on dfferent sub-bands. We also note that the complexty of the algorthm n [11] s ONR 2 for N users and R RBs when there are multple RBs n each sub-band of bandwdth B; the complexty of our algorthms s lower. Fnally, smlar connectons can be drawn between the scheme n [12] for frequency flat fadng and usng an objectve functon based on sums of squares of queues as n [19]; we beleve smlar connectons can be drawn for the frequency selectve fadng model through further analyss. V. ESTMATON OF PACKET DELAYS We now descrbe a method to nfer approxmate packet delays at the enb va the mechansms avalable n LTE. The man ntuton s as follows: f the base-staton estmates the queue length at tme t to be say, 1 bytes, but later decodes a BSR whch was created at tme t and has value13 bytes, the base-staton can deduce that 3 bytes arrved between tme t and the tme at whch the prevous BSR was created. Ths nformaton about the tme nterval durng whch the 3 bytes arrved can be used for makng resource allocaton decsons specfcally, schedulng polces based on packet delays can be mplemented. The man complexty s due to re-transmssons whch can lead to the BSR report arrvng out of order at the base-staton. A smlar approach has been ndependently proposed n [2] recently, however t gnores the effect of retransmsson falures n the analyss. Let T retx be the maxmum amount of tme between the frst transmsson of a MAC packet and the latest tme when t can be re-transmtted for H-ARQ for example, f we confgure 6 as the maxmum number of re-transmssons, T retx = 48 subframes. We estmate the number of bytes that arrved, A t n each subframet. The buffer status reports are denoted by a sequence of random three tuples: {B 1,τ 1,δ 1},{B 2,τ 2,δ 2},... where B 1 s the buffer sze reported n frst BSR, τ 1 s the tme at whch frst BSR was receved, and τ 1 δ 1 s the tme at whch the frst BSR was generated, and so on. C t denotes the number of bytes scheduled for transmsson from UE, Ĉt the number of bytes whch were successfully receved from UE, and F t the number of bytes that faled the fnal re-transmsson for UE, at tme t. We mantan the hstory of estmated queue length for each UE for duraton T retx, denoted by Q t T retx : t. Then, we update the Q matrx and the arrval vector A, at each t as follows: For every t, 1 Scheduled Bytes: Q t = Q t 1 C t. 2 Faled Bytes: Q t = Q t+f t. 3 BSR report: f a BSR report s receved at tme t,.e., there s n such that τ n = t, then update queue state as follows: f the base-staton has not receved any BSR report created after tme t δ n, then Q t δ n : t = Q t δ n : t+a t δ n where arrval A t δ n = B t Q t δ n otherwse for update arg mn [τ m δ m τ n δ n] {m: τ m<t} A t δ n = B t Q t δ n A τ m δ m = A t δ m A t δ n Q t δ n : τ m δ m 1 = Q t δ n : τ m δ m 1+A t δ n Note that Q can have negatve entres. V. FREQUENCY FLAT FADNG Here, we consder the resource allocaton to N UEs over a sngle sub-band wth bandwdth B and frequency flat fadng. We drop the dependence of quanttes n the general model on the sub-band j for example, we denote channel gan from UE to the enb as G. We allow for contguous allocaton ths s a reasonable approxmaton when B s larger than a few RBs. Roundng technques n, for example, [9] can be used to obtan ntegral solutons. The optmzaton problem to maxmze the sum of rewards for all UEs over the bandwdth allocaton vector b R N + n a subframe s: max. s.t. N G mnγ b,p f b ψ b =1 N b b max,, b B =1 4

4 whereb max s the maxmum bandwdth that UEcan use based else, f s delay QoS and b =, on the estmated queue length, Q t, for UE, and satsfes: b max G mnγ b max λ < d 1mn h,p ψ b max = Q t/ where r = h b. Proof: The lemma follows from standard arguments n, where we recall that s the length of a subframe 1 ms. for example [21], the defntons of f s, and that the subdfferental of f for delay QoS user s gven by Snce, the functon on the left s an ncreasng functon of b max, we can compute b max effcently va a bsecton search. { Problem 4 s a convex optmzaton problem wth nondfferentable objectve functon due to the lemma whch r = [d n serv,d n serv n d n serv n serv r +1, s j < r f +1], serv r s j = r follows. Lemma 5.1: The objectve functon n optmzaton problem 4 s concave n the b s for b, for all. We now evaluate the sub-dfferental of h for x, whch Proof: Consder the functon g : R + R + defned by s bounded because γ s assumed to be bounded. { } gx = xψc/x, x >,c R + s constant. Snce, ψ s ψ Gtγ, f x < P/γ assumed to be concave, t s easy to verfy va showng that the second dervatve s always negatve that g s concave as { } well. Snce, the sum of concave functons s concave, and ψ GtP x GtP x ψ GtP x, f x > P/γ h the composton of one concave functon wth another s x = [ concave, to show that the objectve functon s concave, t s ψ Gtγ GtP x ψ Gtγ, suffcent to show that the followng functon s concave ] ψ Gtγ mnc1 x,c 2, f x = P/γ hx = xψ, x,c 1,c 2 R + are constant x Note that the above functon s well defned for x. Snce, ψ s an ncreasng functon, we can wrte hx = mn{xψc 1,xψc 2 /x}, whch s the mnmum of two concave functons, and hence, concave. A. Characterzaton of Optmal Soluton We defne a functon whch maps the bandwdth allocaton b to achevable rate for user : G mnγ b,p h b = b ψ b We denote the sub-dfferental of a functon g : R R at x by gx. For contnuous concave functons over the set of reals, the subdfferental at x s the set of slopes of lnes tangent to f at x. Let b R + denote the soluton to the resource allocaton problem 4. The followng lemma shows that an optmal allocaton n a gven subframe s one for whch the followng quanttes are equal for all users wth non-zero bandwdth allocaton: for best effort user, the margnal utlty tmes the ncremental rate when more bandwdth s allocated to t, and for delay QoS user, the delay of the oldest packet whch s not served completely tmes the ncremental rate when more bandwdth s allocated to t. Lemma 5.2: There exsts a λ > such that f s best effort, then λ U 1 αx t+α r h b, f b > λ < U 1 αx tmn h, f b = else, f s delay QoS and b >, f n serv r s j < r, λ d n serv r +1 h b else f n serv r s j = r λ [d n serv mn h b,d n serv +1max h b ] subgradent of objectve 1 3 λ 1 2 pkt 1 pkt 1 pkt 2 pkt 3 power lmted pkt 4 pkt 5 user 1 user 2 pkt 2 power 1 1 lmted number of RBs Fg. 2. Optmalty condton We llustrate the optmalty condton va a two user example. The total bandwdth to be shared s 1 RBs, or 18 KHz. All packets are of sze 5 bts. The packet delays of the two users n the gven subframe are User 1: [45,33,135,8,2] User 2: [17,15,14,11,8,2] The rate at whch the users can be served as a functon of the RBs are gven by: { b1 log b khz h 1 b 1 = b 1 log b 1 b 1 > 5 18khz { b2 log b khz h 2 b 2 = b 2 log b 2 b 2 > 8 18khz where the 5 and 8 RB thresholds and correspondng SNRs of.5 db and 4 db are derved from fractonal power control constrants n Secton -A. The subgradent of the rewards for both the users as a functon of bandwdth allocaton, and

5 the optmal bandwdth allocaton are shown n Fg 2 the optmal resource allocaton s 5 RBs to each user, and the optmal dual varable λ s shown n the fgure. For each user, the fgure also shows the number of RBs requred to fully serve a gven number of packets and the number of RBs at whch the user becomes power lmted,.e., the maxmum peak power constrant lmts the transmsson power rather than the fractonal power control whch lmts the transmt PSD. B. Computaton of Optmal Soluton The optmzaton problem 4 entals the maxmzaton of the sum of concave functons subject to a lnear nequalty constrant. Whle, n prncple, the optmal resource allocaton scheme can be computed va a bsecton search on the dual varable λ, two dffcultes arse: There may be multple values of b for whch the subgradent of f h s equal to λ. See, for example, the frst packet for user 1 n Fg. 2. As a result the dual functon s non-dfferentable and the bsecton search may not converge [22]. f λ belongs to the subdfferental at a pont b of non-dfferentablty of ether f or h, the values of the gradent of f h may be arbtrarly dfferent at b +ǫ and b ǫ for an arbtrarly small ǫ. Ths can also be seen n Fg 2. We use Algorthm 1 to compute the optmal soluton of problem 4. The convergence analyss s almost dentcal to that n Sec. 6 n [22]. An accurate soluton can typcally be computed n about 1 teratons. Algorthm 1: Bsecton search for optmal λ Gven startng value of λ, λ, b, b and tolerance ǫ. repeat Bsect: λ = λ+λ/2. Allocate bandwdth for all : f λ > max f max h then set b =. else b s such that λ [mn f r mn h b, 5 max f r max h b ] where G tmnγ b,p r = b ψ b end Update: f N =1 b B >, λ = λ, b = b, else λ = λ, b = b. untl λ λ < ǫ Feasble Soluton: f b b > then set α = B b b b. else set α =. end b = αb+1 αb The startng values of λ and λ can be generated usng the followng smple lemma proof s straghtforward and omtted; the values of b and b are obtaned by repeatng the Allocate Bandwdth step n Algorthm 1 for dual varables λ and λ, respectvely. Lemma 5.3: The optmal dual varable λ satsfes λ λ λ where λ = max =1,...,N [ ψ G tγ [ G tp λ = ψ B G tp max f Bψ B G tp ] max f B ψ ] G tp B, for some The man computatonal step n each teraton of Algorthm 1 entals solvng 5 N tmes we now show ths can be done n OlogL tme when the reward functon f for user s non-dfferentable at at most L ponts. The composton of functon f wth h s a concave functon as shown n Lemma 5.1. Hence, to compute the bandwdth allocaton for UE as gven n equaton 5, we can use a bsecton on b. Frst we obtan how many packets should be served fully such that the correspondng bandwdth requred, b, satsfes equaton 5 n OlogL tme. Then, we compute b. We compute the range of subgradents for packet η as b = h 1 η 1 k=1 s, b = h 1 η k=1 s SGη = d η[mn h b,mn h b] where we recall d η and s η are the delay and sze for ηth packet queued at UE. Note that the nverse of h s smple when b < P/γ ; otherwse t can be computed va bsecton. The number of packets to be served completely s η = η 1. Note that h has at most one pont of dscontnuty, say ˆb. f b ˆb b for η = η 1 n 6, then b = ˆb f λ/d η h ˆb ; else update b or b approprately. A smlar method can be used for best effort traffc and the analyss s omtted here due to lack of space. A. Smulaton Framework V. SMULATON RESULTS The algorthms n the prevous secton were smulated usng a detaled system smulator where the MAC layer sgnallng was modeled fathfully, and the PHY layer performance was abstracted va modelng of fadng channels, transmsson power, and capacty computatons as n [15]. A hexagonal regular cell layout wth three sectors per ste was smulated wth the parameters as noted n Table. For fractonal power control parameter values P = 6 dbm, α =.6 smlar to those n [8], a 19 cell 57 sector smulaton wth wrap around was frst performed to determne the nterference over thermal ot at the base-staton of a cell to be 6 db on an average. n subsequent smulatons, only one cell was smulated wth the ot assumed to be constant n tme and frequency. Ths drastcally reduces the smulaton tme whle stll accountng for the nter-cell nterference. 6

6 Parameter Value Channel Profle TU-T PedA Moble Speed 3 km/hr Log-Normal Shadowng σ =8.9 dbm ntra-ste Shadowng Correlaton 1. nter-ste Shadowng Correlaton.5 Cell Radus 1 km No. of UEs/cell 2 No. of RBs 11 Max UE Tx Power 23 dbm No. of Tx & Rx Antenna 1 enb & UE Antenna Gans db Thermal Nose Densty -174 dbm/hz BSR perodcty 5 ms max. number of retransmsson 6 TABLE SMULATON PARAMETERS The tme varyng channel gans, G s, were assumed to be measured perfectly at the base-staton n each subframe. The MCS was pcked on the bass of the channel gan from the UE and a rate adaptaton algorthm to target an average of two H-ARQ transmssons for successful decodng was used. We use the mutual nformaton effectve SNR metrc MESM [23]; we frst obtan the effectve SNR accordng to the modulaton alphabet sze and then use that value to smulate an event of packet loss accordng to the packet error rate for the effectve SNR. We model the tmelnes for Schedulng Request SR, resource grants, Hybrd-ARQ, ACK/NACKs, and BSR as descrbed n Sec.. We assume error free transmsson of control messages n our smulatons. Two types of traffc, lve vdeo and streamng vdeo were modelled as per the descrpton n [19]. B. Results We consder two topologes for smulaton: a macro-cell wth the path loss between the base staton and UEs randomly selected between 1 db and 135 db [24], mcro-cell wth path loss n the range 17 db to 115 db. We smulate three schedulng algorthms: teratve Delay whch maxmzes the reward functon n Sec. -B, teratve Queue whch mnmzes sum-of-squares of queue lengths as n [19] and smlar to [12], non-teratve maxmum weght where a UE wth the hghest queue length tmes spectral effcency for frst RB s allocated bandwdth untl the queue s draned or the UE becomes power lmted before allocaton to the next UE. We note that the computatonal algorthms n ths paper are applcable to computng resource allocaton for schedulng polces and, and that polces smlar to do not consder the change n reward functon of the UE n a gven subframe. 1 Macro cell Topology: We consder 2 UEs wth a mx of lve vdeo and streamng vdeo traffc. Snce lve vdeo has a tghter requrement for packet delays, we bas the scheduler to assgn lve vdeo users 5x prorty compared to streamng vdeo users for same packet delay. Smulatons were performed for low load and hgh load cases: 1 Hgh Load: 5 UEs have lve vdeo traffc, each wth a mean rate of 3 kbps. For the other 15 UEs wth streamng vdeo traffc, we mmc an adaptve-rate streamng mechansm n whch the data rate for each user depends on the qualty of ts Head of Lne Delay mllseconds Actual Head of Lne Delay Estmated Head of Lne Delay Tme mllseconds Fg. 3. HoL delay estmaton performance channel to the base-staton,.e. a user close to the base-staton transmts a better qualty vdeo compared to a cell-edge user. For smulatng hgh-load, the traffc parameters are vared for each UE such that they generate traffc at 8% of the average data rate they receved wth full buffer traffc. 2 Low Load: 5 UEs have lve vdeo traffc wth a mean rate of 2 kbps. The UEs wth streamng vdeo traffc are now set to operate at 4% of ther full buffer average data rate. We frst study the performance of the delay estmaton mechansm descrbed n Secton V. Fgure 3 shows the estmated head of lne HoL delay and the actual HoL delay at a UE over a perod of 1 second. The estmated values can be seen to follow the actual delays but the accuracy s lmted by the granularty of BSR messages,.e., f there are multple arrvng packets between two successve BSR messages, the packets are bundled as one n our mechansm resultng n relatvely small errors n HoL estmaton. Next we show the performance of the head of lne delay based schedulng scheme computed as the soluton to the optmzaton problem n 4 wth the reward functon n 3. Fgure 4 shows the medan and 95th percentle delays of the lve vdeo UEs for the two baselne and the head of lne delay based schedulers for low and hgh loads. The delays experenced by the lve vdeo users are consstently less n the case of HoL delay based schedulng wth the non-teratve scheme resultng n an average 95th percentle delay 1.6x hgher than wth the HoL delay schedulng. The queue based scheme also results n slghtly hgher delays, on an average 1.1x compared to 95th percentle delays for HoL schedulng. A more pronounced mprovement s observed for the streamng vdeo users, as shown n the delay plots n Fgure 5. n ths case, the non-teratve and queue based schemes result n 6.2x and 5x more delays compared to HoL delay schedulng n terms of 95th percentle latences. Fnally, Fgure 6 shows the combned delay numbers for uplnk packets from all the UEs n the hgh load smulaton. As can be seen from the fgure, the teratve queue based and delay based schemes result n smlar delays for lve vdeo users due to preferental assgnment. However ths results n large delays for the streamng vdeo users for both non-teratve and queue based schemes: close to 11x and 8x respectvely compared to HoL delay based schedulng n terms of 95th percentle delays. Thus, leveragng the approxmate packet delays obtaned va our method leads to sgnfcant performance mprovement over queue based

7 Delay mllseconds Non teratve Queue Based HoL Delay Based Medan 95 %le Low Load Smulaton Medan 95 %le Hgh Load Smulaton Delay mllseconds per UE Cell wde Fg. 4. Lve vdeo users: delay performance Medan 95th %le Medan 95th %le Non teratve Queue Based HoL Delay Based Delay mllseconds Non teratve Queue Based HoL Delay Based Medan 95 %le Low Load Smulaton Medan Fg. 5. Streamng vdeo users: delay performance Delay mllseconds Lve Vdeo UEs %le Hgh Load Smulaton Streamng Vdeo UEs Medan 95th %le Medan 95th %le Non teratve Queue Based HoL Delay Based Fg. 6. Cell-wde delay performance of all packets n macro cell smulaton schedulng. Moreover, even for the queue based scheduler, the computatonal methods n ths paper are very useful. 2 Mcro cell Topology: n order to compare these schedulng schemes n a smaller cell topology, we ran a second smulaton wth 2 UEs located wthn a regon wth path loss db from the base staton. Each UE, n ths smulaton, carres streamng vdeo traffc wth the mean data rate randomly selected between 3-2 Kbts/sec. Decouplng the mean traffc rate wth the path loss hghlghts the relatve performance of the schedulng algorthms n real deployments where pror knowledge of user demand s rarely known. ndvdual and cell wde delay numbers are shown n Fgure 7, whch shows that 95th percentle delays for nonteratve and queue based schemes are 1.8x and 1.4x more than those for the HoL delay based schedulng. Fg. 7. ndvdual and Cell-wde Delay performance for mcro cell smulaton V. FREQUENCY SELECTVE RESOURCE ALLOCATON We extend the analyss n [1] for frequency selectve fadng to concave functons f such as the delay based reward functon whch are thrce contnuously dfferentable everywhere except at L ponts where they are only contnuous. We can re-wrte such a functon as L f r = f l mn ρl ρ l 1,[r ρ l 1 ] + l=1 where ρ 1 <... < ρ L are the ponts of nondfferentablty and f l : R + R are thrce contnuously dfferentable concave functons defned as f l x = f ρ l 1 +x f ρ l 1, x [,ρ l ρ l 1 ] wth l 1,ρ =, and satsfy f lx < f,l 1y,x [,ρ l ρ l 1 ], y [,ρ l 1 ρ l 2 ]. We also assume xψ 1 y/x s concave for all x,y > ; ths s true for example, when ψ s the Shannon capacty formula, and for practcal M-QAM schemes. Consder the followng convex optmzaton problem over r l s, r j s rate for user on sub-bandj, andb j s bandwdth for user on sub-band j: max. s.t. N =1 l=1 l=1 L f l r l, L M r l r j,, r l ρ l ρ l 1,,l N b j = B, j, =1 M b j N + j ψ 1 r j /b j 1 P,, G j Gj γ j r j b j ψ N + j, r j,b j,,j. The frst constrant mples that the total rate for a user s the sum of rates over sub-bands, the second constrant s on total bandwdth allocaton n a sub-band, thrd constrant s on peak power at the UE n a subframe, and the fourth constrant 7

8 models fractonal power control. The followng lemma follows easly from the constructon of the f l s: Lemma 7.1: f rj,b j s a soluton to the optmzaton problem 7, then f j r j s the maxmum sum reward for any feasble resource allocaton. General purpose nteror ponts methods to solve the above optmzaton problem have a complexty of ONM +NL 3 per teraton we explot the structure to reduce t to ONL 2 + M 2. Note that n practce L and M are much smaller than N. n order to construct a soluton for whch the bandwdth allocaton s contguous n frequency to satsfy the SC-FDMA requrements, we can use the heurstc n [1]. The man computaton to solve 7 s to determne the Newton step at each teraton whch entals solvng a set of lnear equatons of the form we omt the detals due to lack of space, the exact expressons can be obtaned followng the steps n [25]: H 1 A T... x 1 [ ] HN. a x N = b y A where H R L+M L+M, A R M NL+M, x R L+M, a R NL+M, y,b R M. We frst elmnate the x s as x = H 1 a A T L+M 1+1:L+M y where A T k:m s the submatrx of AT gven by rows k to m. We nvert H 1 n OL 2 +M 2 tme, solve for y n OM 3 tme M lnear equatons n M varables, and back-substtute y to obtan x. To nvert H, we note that t decomposes as K 1... H = [ h h + T K M g 1 ] [ + c c T... gl ] +g g T where g R L+M, h R L, c R M. Usng the matrx nverson lemma we can nvert H n OL 2 +M 2 tme. V. CONCLUSONS We desgned a general computatonal framework n ths paper to enable a wde array of onlne schedulng polces n a computatonally effcent manner. We modeled the constrants due to fractonal power control, and formulated an optmzaton problem wth non-dfferentable objectve functon. We showed how to estmate the packet delays on the uplnk va the BSR reports, and proposed a novel schedulng polcy based on packet delays. Numercal results demonstrated that usng packet delay estmates for the uplnk can lead to sgnfcant reducton n packet delays as compared wth a queue length based scheduler. There are many nterestng drectons for future work. For example, we can further study the connectons wth the work n [11], [12]. n terms of mplementaton, an nterestng queston s whether we can desgn approxmaton algorthms for the uplnk bandwdth packng problem whch are optmal accordng to some metrc. REFERENCES [1] E. Dahlman, S. Parkvall, J. Skold, and P. Bemng, 3G Evoluton, Second Edton: HSPA and LTE for Moble Broadband. Academc Press, 28. [2] H. J. Kushner and P. Whtng, Convergence of proportonal-far sharng algorthms under general condtons, EEE Trans. Wreless Commun., 24. [3] A. Stolyar, On the asymptotc optmalty of the gradent schedulng algorthm for mult-user throughput allocaton, Oper. Res., 25. [4] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, R. Vjayakumar, and P. 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