IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER

Transcription

1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER Delay Cosideratios for Opportuistic Schedulig i Broadcast Fadig Chaels Masoud Sharif ad Babak Hassibi Abstract We cosider a sigle-atea broadcast block fadig chael with users where the trasmissio is packetbased. We defie the packet) delay as the miimum umber of chael uses that guaratees all users successfully receive m packets. This is a more striget otio of delay tha average delay ad is the worst case access) delay amog the users. A delay optimal schedulig scheme, such as roud-robi, achieves the delay of m. For the opportuistic schedulig which is throughput optimal) where the trasmitter seds the packet to the user with the best chael coditios at each chael use, we derive the mea ad variace of the delay for ay m ad. For large ad i a homogeeous etwork, it is proved that the expected delay i receivig oe packet by all the receivers scales as log, as opposed to for the roud-robi schedulig. We also show that whe m grows faster tha log ) r,forsomer>, the the delay scales as m. This roughly determies the timescale required for the system to behave fairly i a homogeeous etwork. We the propose a scheme to sigificatly reduce the delay at the expese of a small throughput hit. We further look ito the advatage of multiple trasmit ateas o the delay. For a system with M ateas i the trasmitter where at each chael use packets are set to M differet users, we obtai the expected delay i receivig oe packet by all the users. Idex Terms Broadcast chael, fadig, opportuistic schedulig, packet delay, logest queue. I. INTRODUCTION RESOURCE allocatio i wireless systems aims for two coflictig goals, firstly providig quality of service such as delay ad fairess to users, ad secodly maximizig the throughput of the system. A fudametal property of wireless chaels is their time variatio due to multi-path effects ad the mobility of the users. This implies that at each chael use some users have favorable chael coditios ad other users icur deep fades. I fact, assumig a block fadig model for the chael ad havig full CSI i the trasmitter, it ca be show that sedig to the user with the best chael coditios maximizes the sum rate or throughput) of the sigle atea broadcast chael. Mauscript received Jauary 3, 26; revised July 4, 26 ad July 3, 26; accepted August 6, 26. The associate editor coordiatig the review of this paper ad approvig it for publicatio was J. Zhag. This work is preseted i part at IEEE Iter. Symp. o Ifo. Theory 24 ad IEEE INFOCOM 25. This work is supported i part by the Natioal Sciece Foudatio uder grat o. CCR-3388, by the office of Naval Research uder grat o. N , ad by Caltech s Lee Ceter for Advaced Networkig. M. Sharif is with the Departmet of Electrical ad Computer Egieerig, Bosto Uiversity, Bosto, MA sharif@bu.edu). B. Hassibi is with the Departmet of Electrical Egieerig, Califoria Istitute of Techology, Pasadea, CA hassibi@systems.caltech.edu). Digital Object Idetifier.9/TWC /7$25. c 27 IEEE I order to exploit this multiuser diversity, the base statio or the trasmitter) has to kow the chael state iformatio CSI) of all the users. I fact, this opportuistic way of trasmissio has bee proposed i Qualcomm s High Data Rate HDR) system xev-do). Other variatios of this schedulig that do ot require full CSI i the trasmitter are studied i [], [2]. However, there is a price to pay for maximizig the throughput which is fairess amog users ad delay i sedig packets. Assumig users have differet sigal to oise ratios, the throughput optimal schedulig will provide much less service to the user with the lowest sigal to oise ratio SNR) compared to that of the user with the highest SNR. Eve i a homogeeous etwork where users have equal SNRs ad so the system is log-term fair, there is o delay guaratee for trasmittig a packet to a specific user as the trasmissio is probabilistic, i.e., at each chael use each user will be chose with some probability. The other extreme would be to use a roud robi type schedulig that fairly gives service to all users ad ca guaratee a fixed delay for trasmittig a packet to each user. I applicatios with delay costraits, oe may woder how bad the worst case delay or the delay for the most ufortuate user) for the throughput optimal strategy is. I this paper, we cosider a broadcast chael with users i which users messages are idepedet. The trasmissio is packet based ad the chael is assumed to be block Rayleigh fadig ad chages idepedetly from oe block to the other. We also assume packets are dropped if outage occurs, i.e., the istataeous capacity goes below the amout of iformatio i the packet. Give the probability of outage P e, we assume packets carry a fix amout of iformatio C which oly depeds o the schedulig. For example, opportuistic schedulig is the oe that maximizes the throughput give P e. This will be further discussed i Sectio 2. We defie the delay as the miimum umber of trasmissios that guaratees all the users will receive m packets successfully. This otio of delay is clearly stroger tha the average delay i the sese that it guaratees the receptio of m packets by all users. This defiitio of the delay is specially useful for applicatios with deadlie [6]. Disregardig the throughput ad if the users are back-logged, the miimum delay of m ca be achieved by roud-robi schedulig. However, the throughput optimal strategy has to coted with delay hits. The overridig questio i this paper is to If the users are ot backlogged, there is a chace that the chose user has a empty queue. This probability must be take ito accout see Sectio 3.)

2 3354 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 27 characterize the delay for the throughput optimal strategy, e.g. to determie its mea ad other momets. Fially, we propose a algorithm to reduce the delay at the expese of a little hit i the throughput of the system. The results i this paper imply that opportuistic trasmissio icreases the delay by a factor of log compared to that of delay optimal strategies. Previously, the questio of the delay-throughput trade off has bee addressed by several authors i differet cotexts [5]. I sigle lik systems, the problem of how to optimally allocate the power amog chael uses such that the capacity is maximized while guarateeig the delay for sedig bits remais bouded has bee cosidered i [3], [4]. Also, the trade off betwee average power ad delay has bee addressed by Berry ad Gallager for sigle lik systems [5]. I multiuser chaels, traditioally delay ad throughput were cosidered separately ad therefore, access schemes such as ALOHA [6] were proposed to avoid collisios without exploitig multiuser diversity. As oted later i [7], [8], there has bee a large body of work to combie the physical layer ad multiple access layer see [9], [], [], [7], [8] ad refereces there i). For multiple access chaels, a decetralized variatio of ALOHA algorithm is proposed that exploits multiuser diversity []. I [9], the authors cosider the problem of characterizig the capacity regio uder a stability coditio for queues. Stability here is i the sese that the probability of the queue overflow ca be made arbitrary small by makig the buffer size sufficietly large [9]. Schedulig i broadcast chaels has bee also cosidered by several authors [2], [2], [22], [23], [2], [4], [3], [5], [6]. I [2], stabilizig parallel queues i the trasmitter is cosidered, where the coectivity of queues are radom to capture deep fades i the wireless chael. I [23], the authors icorporate the chael state iformatio i their schedulig while providig delay costraits for packets. Aalyzig the average delay over the users) ca be also doe usig the results for the geeral idepedet iput/output GI/GI/) queues ad it ca be show that the average delay is of the order of the umber users [24], [25]. However, i order to provide delay guaratee for all users, we have to study the delay for the most ufortuate user i the system. Clearly the worst case delay is a fuctio of the umber of users ad their SNRs or the probability of beig chose as the best user at each chael use). While these works give may isights ad algorithms, they leave ope the questio of how large the worst case delay is as a fuctio of the umber of users ad their SNRs for usig throughput optimal strategies. This is the mai goal of this paper. This paper is orgaized as follows. Sectio II itroduces our chael model ad our otatio. Sectio III deals with characterizig the delay for sigle atea broadcast fadig chaels. Sectio IV geeralizes the results of Sectio III to multi-atea broadcast chaels. Fially Sectio V proposes a algorithm to reduce the delay at the expese of a little reductio i the throughput ad Sectio VI cocludes the paper. II. SYSTEM MODEL AND ASSUMPTIONS I this paper we cosider a sigle atea broadcast chael with receivers. We assume a block fadig chael with m m Fig.. parallel queues i the trasmitter correspodig to users; we are iterested i the behavior of the logest queue. a coherece iterval of T, ad where the chael chages idepedetly after T secods. The trasmissio is assumed to be packet based ad the legth of each packet is T 2. For each block of legth T, the received sigal at the i th user at time t ca be writte as, y i t) ρ i h i t)st)+ i t), i,...,, ) where h i t) is the effect of chael ad i t) is additive white oise ad that both are i.i.d. circularly symmetric complex Gaussia distributed with zero mea ad variace of oe. Here ρ i is the SNR of the i th user ad St) is the trasmitted symbol at time t. We further assume idepedet memoryless chael which implies that the chael chages idepedetly to aother value after the coherece iterval of T. I the trasmitter we assume there are queues correspodig to each receiver ad receiver s messages are idepedet 3. For most of our aalysis, we will assume that there is always a packet available to be trasmitted to ay user i.e., backlogged users) 4. Fig. illustrates the arragemet of queues i the trasmitter. I fact, the mai challeges for the scheduler are to first balace the service amog all the users ad to secod exploit the multiuser diversity i the chael i order to maximize the throughput of the system. Ay schedulig strategy implies a probability for choosig each user at each chael use that may deped o the sigal to oise ratio SNR) of all users, the legth of the queue of users, ad the statistics of the chael see [4], [5], [6]). For the throughput optimal strategy, this probability oly depeds o the SNR of the user ad the chael statistics. For i.i.d chaels, it is clear that these probabilities are oly fuctios of users SNRs. Assumig that all packets have C iformatio bits for a homogeeous etwork i.e., ρ i ρ), we cosider a packet to be dropped if outage occurs, i.e., if the istataeous capacity C goes below C at the time of the trasmissio [26]. The 2 If the legth of the packet is smaller tha T, the results i this paper ca be easily geeralized. 3 Broadcast chaels, i full geerality, iclude trasmissio of commo messages betwee receivers. Here we cosider the special sceario i which the trasmitter is sedig idepedet messages to the receivers. 4 I most practical situatios, packets have fiite arrival rates ad so this assumptio may ot be valid. I sectio 3., we show how our result ca be exteded to the o-backlogged case. 2

3 SHARIF ad HASSIBI: DELAY CONSIDERATIONS FOR OPPORTUNISTIC SCHEDULING IN BROADCAST FADING CHANNELS 3355 istataeous capacity however depeds o the schedulig. For the roud-robi schedulig, C log + ρ h i 2 ) which does ot deped o. For the throughput optimal strategy 5, C however is the maximum of log + ρ h i 2 ) over i, i.e., C max i log + ρ h i 2 ). We assume if a packet is dropped, the trasmitter will be otified ad the packet will be cosidered for re-trasmissio wheever the correspodig user has the best chael coditios. If we assume that the error probability is simply the outage probability a reasoable assumptio for log packets [4]), we have P e PrC<C ). The throughput is therefore R C P e ) C PrC C ).GiveP e, ay schedulig would lead to a differet C. Note that for ay value of C, the throughput optimal strategy is to sed to the best user as this would miimize P e. Coversely, for ay fixed value of P e, sedig to the strogest user maximizes the throughput as this would allow for the largest possible C. It is also worth metioig that the maximum of i.i.d. expoetial radom variables the h i 2 ) behaves almost surely as log. Therefore for large, we do ot eed to use power cotrol to compesate for the chael variatio as the maximizatio automatically prevets havig deep fades for large umber of users with high probability. Thus, for the throughput optimal schedulig, it is reasoable to assume that all the packets have the same amout of iformatio, i.e., C roughly about log+ρ log ), idepedet of the time ad chael coditio. I this paper, we defie the packet) delay i the broadcast chael as the umber of chael uses deoted by D m, ) required to guaratee that all the users will receive m packets successfully. It is clear from the defiitio of D m, that this otio of delay refers to the worst case delay amog users or the delay for the most ufortuate user). Of course, D m, is a radom variable ad depeds o the umber of users, the umber of packets m ad also the schedulig algorithm. A delay-optimal strategy is roud-robi schedulig which clearly achieves the optimal packet) delay of m whe there is o error). However, roud-robi is ot throughput optimal which requires trasmittig to the user with the best chael coditios at each chael use. Throughput optimal strategies, o the other had, will have to coted with delay hits. The followig sectio deals with the delay for the throughput optimal schedulig. It should be also metioed that our defiitio of the delay with backlogged users suffers from the weakess that it does ot accout for the queueig delay. However, our delay is a lower boud for the overall worst case delay i a system with radom arrivals. The lower boud should be also tight whe the system is highly loaded. III. DELAY ANALYSIS FOR SINGLE-ANTENNA BROADCAST CHANNELS Opportuistic trasmissio is a probabilistic schedulig which implies that each user will be give service with some give probability. Assumig that the outage probability P e is give, the opportuistic schedulig maximizes the throughput or equivaletly C the amout iformatio bits per packet). 5 I this paper, we use the terms opportuistic schedulig ad throughput optimal strategy iterchageably. Aalyzig the average delay over all the users) ca be doe as the queue of each user ca be cosidered as a i.i.d. iput/output queue [24]. I particular, it ca be show that the average delay is of the order [25]. However aalyzig the worst case delay or the delay for the most ufortuate user i the system) requires cosiderig parallel queues of users all-together [27]. I this sectio, assumig that at each chael use the trasmitter seds to the i th user with the probability p i, which oly depeds o the SNR of all users, ad drops the packet with probability P e, we obtai the momet geeratig fuctio of the radom variable D m,. We first cosider the simple case i which the etwork is homogeeous ad P e. The we geeralize the result to the case where we have a o-zero P e ad/or a heterogeeous etwork where users are chose with differet probabilities. We obtai the mea ad variace of the delay D m, for ay m ad. We further look ito the asymptotic behavior of D m, for differet regios of m ad at the ed of this sectio. A. A Study of the Delay for Users with Poisso Arrival Before delvig ito a aalysis of D m, for the backlogged case, let us remark o the more realistic case where we have a poisso arrive for the packets with fixed rate λ.ithis case, there is a o-zero probability that the user with the best chael coditio has a empty queue. Two courses of actio ca be take: oe is to ot trasmit aythig, the other is to trasmit to the user with the best chael coditio whose chael is o-empty. The latter is a more reasoable actio, but seems very difficult to aalyze. I this sectio, we study the effect of havig radom arrivals for each queue ad fid the delay icurred by the schedulig i which o trasmissio is doe if the chose user has o packet. I order to aalyze the delay, we would eed to fid the probability of havig o packet at each queue i the steady state. Each queue has a poisso arrival process with itesity λ ad the service has a biomial distributio, i.e., with probability the queue will be served at each time slot. Therefore the characteristic fuctio for the legth of time that the queue has ot bee served ca be writte as, Sz) z i /) i z/ / z /). 2) i Therefore usig kow results for the M/G/ queue, the momet geeratig fuctio for the radom variable N deotig the umber of packets i the queue ca be writte as, λ )) z) G N z) z 3) S z)λ) where Sz) is as defied i 2). Thus, the probability of havig a empty queue is G N ) λ ). It is worth otig that i order to have all the queues i the system to be stable, λ. Now assumig that the base statio will ot trasmit ay packet if the selected queue is empty, we ca easily fid the expected delay usig the same trick as we used to aalyze the probability of droppig a packet. I particular, we may assume that there is a probability of λ ) that the packet is

4 3356 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 27 beig dropped. Therefore, the expected delay will be λ ) times more tha the delay for the case of backlogged users. However if we choose to trasmit to the strogest user with a o-empty queue, the the aalysis becomes quite formidable. The above result, however, is a simple upper boud for the delay i this case. It is also clear that the delay for the backlogged system ca be served as a lower boud for the delay i a more realistic settig where users have radom arrivals. I fact upper ad lower bouds are tight as the system becomes highly loaded. B. Homogeeous Network with No Droppig Probability Whe users are homogeeous ad assumig throughput optimal schedulig, the trasmitter chooses the i th user with probability from the pool of users sice it is equally likely for each user to have the best chael coditio. The radom variable D m, is basically the miimum umber of chael uses to guaratee all users have bee chose at least m times. This problem ca be restated as the coupo collector problem [28] which is studied by several authors i the mathematics literature see also chapter 6 of [29]). To be more precise, users ca be see as people carryig coupos ad the trasmitter is the collector that chooses radomly ad uiformly from the people ad collects his/her coupo. The questio is how may times should the collector choose to guaratee that everybody has give at least m coupos. I fact we ca state the mea value of D m, basedoaresult foud i [3]. Theorem. Newma ad Shepp [3]) Cosider a homogeeous broadcast system with users. We assume that at each chael use, the trasmitter seds to the user with the best chael coditio. The, we have, ED m, ) Sm t)e t) ) dt, 4) for ay m ad where S m t) m k tk k!. Proof: Sice the etwork is homogeeous, the probability of choosig the i th users is. Therefore, the problem is the same as the problem cosidered by Newma ad Shepp [3]. See [3] for the proof. Ispired by the proof of Theorem, we ca derive the momet geeratig fuctio of D m, defied as F z) z i Pr{D m, >i} z i b i. 5) i i Usig the geeratig fuctio F z) i 5), we ca obtai all the momets of D m, with a little effort ad by takig higher derivatives of F z) at z [3]. For example, usig the defiitio of F z) i 5), we ca write, ED m, ) F ) σ 2 D m, ) 2F ) + F ) F )) 2. 6) Next Theorem obtais F z) ad geeralizes the result of Theorem. Theorem 2. Cosiderig the settig of Theorem, we ca write the momet geeratig fuctio of D m, defied i 5) as, F z) e z t e t e t S m t)) ) dt. 7) z Proof: We evaluate F z) by the same trick as [3] i which the mea of D m, is derived. I fact, F z) ca be evaluated by otig that b i is the probability of failure i obtaiig m packets at all the users up to ad icludig the i th trial. Therefore, b i is simply the polyomial x x ) i evaluated at x... x after excludig all terms which have all x i s with expoet larger tha m. Therefore, we may write { F z) z i x x ) i} i 8) i where { } deotes the operator that removes all the terms which have all x i s with expoet less tha m. Cosiderig the followig idetities [3], z i i! i e z t t i dt, 9) z { e x +...+x } { x x ) i} i! i e x+...+x e xi S m x i )), ) where the first equality i Eq. ) is the defiitio of the expoetial fuctio ad the secod equality follows by otig that the secod term i the right had side just subtracts out the terms with all x i s larger tha m. We may the replace the itegral form for usig 9) i 8) to get, X Z i Φ F z) z e t z t i x x iψ ) dt i! z z z i Z Z Z e z t X i i Φ x x ) iψ i! e z t ψe tx +...+tx dt Y e tx i S mtx i)! i e z t e t e t S mt)) dt. ) where we replaced x i for i,..., ad we used ) to get the secod equality ad we replaced x i for i,..., to obtai the last equatio. It is ow quite straightforward to derive the variace of D m, usig F z) ad 6) as show i 2) o the ext page. C. Heterogeeous Network with Droppig Probability For the special case of a homogeeous etwork, we derived the momet geeratig fuctio of D m, i Theorem 2. I what follows, we geeralize the results to a more geeral settig i which users may have differet SNRs ad also a packet may be dropped if outage occurs. We assume the trasmitter will be otified i case a packet is dropped ad it will be cosidered for re-trasmissio wheever the correspodig user has the best SNR. Here, we assume a dt

5 SHARIF ad HASSIBI: DELAY CONSIDERATIONS FOR OPPORTUNISTIC SCHEDULING IN BROADCAST FADING CHANNELS 3357 σ 2 D m, )2 2 t S m t)e t) ) dt EDm, ) ED m, )) 2 2) ED m, )/log)) Fig. 2. m3 m2 m m umber of users) Expected delay EDm,) log for differet values of m ad. Fig. 2 shows the expected delay for m, 2, 3, 4 ad for differet umber of users for a homogeeous etwork. It is clear that whe is large ad m, the growth i the expected delay is like log. Also Fig. 2 implies that the expected delay does ot grow liearly with m for small values of m). I fact it coverges to log although the covergece seems to be quite slow. The ext subsectio deals with the asymptotic aalysis of the delay for differet regios of m ad. Remark : It is worth metioig that we ca cosider the delay i sedig m i packets to the i th user for i,...,. I particular, cosiderig the settig of Theorem 3, ad we are iterested i sedig m j packets to the j th user for j,...,iwhere i. Defiig m m,...,m i ) ad D m as the miimum umber of chael uses guaratees the receive of m j packets at the j the user for j,...,i, we ca write the momet geeratig fuctio for D m as show i 7) o the ext page. memoryless i.i.d. chael ad that the trasmitter chooses the i th user with probability p i that depeds o the SNR of all users ad their chael coditios for the throughput optimal strategy. Assumig that all the packets have the same legth, the packet for the i th user is dropped with probability of P ei. The followig Theorem states the mea ad variace of D m, for this geeral settig ad for ay m ad. The Theorem is a geeralizatio of the result of Newma ad Shepp [3] stated i Theorem. Theorem 3. Suppose we have users such that the probability of choosig the i th user is p i αi ad the probability of droppig a packet is P ei. The the momet geeratig fuctio for D m, defied i 5) is, Z F z) e z ψe P t t e i P ei t e tβ i S mtβ i)) dt, z i 3) where β i P ei )α i. I particular, assumig S m t) is as defied i Theorem, we have ED m, ) Sm β i t)e βit)) dt, 4) i ad 5) o the ext page. Proof: The proof is a geeralizatio of Theorem 2 ad we omit it for the sake of brevity. For example, as a simple cosequece of 4), we ca obtai the expected delay for the case where users are equally likely ad that the probability of droppig a packet is P e,as ED m, ) +) Sm x)e x) ) dx, P e 6) by a simple chage of variable i the itegral stated i 4). Y! D. Asymptotic Aalysis of the Momets of D m, I the previous subsectio, we obtaied the momets of D m, for a geeral settig ad for ay m ad i closed form. However, it is hard to speculate how the mea ad variace of the delay behave as fuctios of m ad. Iordertoget a better isight ito the behavior of the delay, we derive some asymptotic results for the momets of D m, ad for differet regios of m ad. Theorem 4. Assumig a homogeeous etwork ad that a packet will be dropped with probability P e, ) For m fixed ad, we have 8) ad 9) o the ext page. 6 2) For m log ad, we have ED m, )α log + O log log ). 2) P e where α 3.46 is the solutio to the equatio α log α 2. 3) For m log) r where r>isfixed ad, the ED m, ) log ) r + olog ) r ) P e m + om). 2) P e 4) For fixed ad m, ED m, ) m + om). 22) P e Proof: Here we preset the sketch of the proof for the first part ad omit the proof for the other cases for the sake of brevity. The iterested reader ca refer to [3] for the complete proofs. 6 This case has bee also proved i [3], however we preset other proof which leads to results for aother regios of m ad as well.

6 3358 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 27 σ 2 D m, )2 2 t Sm β i t)e βit)) dt ED m, ) ED m, )) 2 5) F z) z i PrD m >i) z i i e z t e t e Pet+P ) i ki+ β kt e tβp S mp tβ p )) dt 7) p ED m, ) log + m ) log log + o log log ) P e 8) σ 2 D m, ) O 2 ) 9) E{ max i x i} xf max x)dx F max x))dx F x))dx 23) Notig that the expected value of D m, is equal to 6), we first show that the itegral i 6) is i fact proportioal to the expected value of the maximum of i.i.d. χ 2 2m) radom variables. To prove that, we assume x i s for i,..., are i.i.d. radom variables with χ 2 2m) distributio. We ca the write the expected value of the maximum of x i s as show i 23) o the ext page, where f max x) ad F max x) are probability distributio ad cumulative distributio fuctios CDF) of the maximum of x i s ad F x) is the CDF of x i.we further kow that x i s are i.i.d. ad have χ 2 2m) distributio ad therefore their CDF is the icomplete gamma fuctio ad ca be writte as F x) S m x)e x. Therefore, we may write 23) as show i 24). Therefore to aalyze the mea of D m,, we ivestigate the behavior of the maximum of x i s. I [32], it is show that for m fixed, max x i behaves like i with high probability. This would the lead to the result for E{D m, } for large ad fixed m. See [3] for the precise argumet. To obtai the variace, we first ote that D m, md, which is clear from the defiitio of D m,.nowwederive the variace of D, ad, sice m is fixed, the variace of D m, has the same order. Deote by r i,fori,...,,the umber of trasmissios after trasmittig at least oe packet to i users ad before i users receive their first packet. Clearly r i s are idepedet ad have geometric distributio, i.e., Pr {r i k} ) i k ) i. The distributio of r i is obtaied by otig that r i equals k if i the last k trials the packet is trasmitted to the i users that have already bee chose ad the i the k th chael use, oe user will be trasmitted to from the pool of i + users that have already bee chose. Usig the defiitio of D, ad r i s, it is clear that D, i r i ad therefore the variace of D, ca be writte as, σ 2 D, 2 i i 2 i. 25) i It is quite straightforward to prove that the first term i the right had side of 25) behaves like O 2 ) ad the secod term behaves like log. Therefore the variace of D m, ca be writte as σ 2 D m, m 2 σ 2 D, O 2 ). I order the prove the other cases, we eed to ivestigate the behavior of the maximum of i.i.d. χ 2 2m) radom variables whe m for large ad whe m also grows. We refer the reader to [3] for the proofs. Assumig m ad usig the result of Theorem 4, we ca state that the delay coverges to the mea almost surely usig Chebychev s iequality as show i 26) for large. This implies that the delay hit for sedig the first packet successfully to all the users is icreased from the miimum of for the roud robi schedulig to log for the opportuistic trasmissio for large. So the delay degradatio due to exploitig the chael variatio ad maximizig the throughput of the system is a multiplicative factor of log. It would be also iterestig to ivestigate the scalig law of the variace of D m, whe m also grow to ifiity; this would the imply the type of covergece to the mea for differet regios of m ad. Remark 2: For a homogeeous etwork, as opportuistic trasmissio is log term fair i.e. the probability of choosig all the users is the same), we kow that for sufficietly large m, the expected delay should behave like m. This is cofirmed by the fourth part of Theorem 4. Iterestigly, Theorem 4 further implies that if m grows faster tha log ) r where r is fixed ad greater tha oe the expected delay behaves like m. This has implicatios for the time scale after which the system behaves fairly. Moreover, if m grows logarithmically with, the expected delay is oly off by a costat factor of α 3.4, compared to the miimum delay m. Therefore, our result ca be see as the short term behavior of the delay for ay m. As metioed, the largest delay hit is whe we focus o sedig a few packets, i.e. m or m is small. The delay hit gets less whe we focus o sedig more ad more packets i.e., whe m gets larger). Therefore, i the rest of the paper, we maily focus o the delay for sedig the first packet, i.e. D,. IV. DELAY IN MULTI ANTENNA BROADCAST CHANNELS Multiple trasmit ateas have bee show to sigificatly improve the throughput of a broadcast chael. It is show that dirty-paper codig achieves the sum rate capacity

7 SHARIF ad HASSIBI: DELAY CONSIDERATIONS FOR OPPORTUNISTIC SCHEDULING IN BROADCAST FADING CHANNELS 3359 ED m, ) + P e E{ max i x i} + P e Sm x)e x ) ) dx. 24) { Pr D m, log + O log log ) } log P e log, 26) of a Gaussia broadcast chael [33], [34], [35]. However, beamformig has log bee proposed as a heuristic method to mitigate the iterferece i the trasmitter ad to sed multiple beams to differet users. Although, beamformig is ot optimal i achievig the sum rate capacity, its throughput does scale the same as that of dirty paper codig for a system with may users ad has much less complexity tha that of dirty paper codig [36], [37]. I this paper, for a system with M trasmit ateas, we assume a simple model i which the base statio trasmits to M differet receivers at each chael use. This is certaily a valid model for beamformig or chael iversio, though it does ot fit the dirty paper schedulig i which the trasmitter seds iformatio to all the users at each time. However, as far as the scalig law of the sum rate throughput is cocered, whe M is either fixed or growig logarithmically with, it ca be show that beamformig, chael iversio, ad radom beamformig all give the optimal scalig law for the sum rate throughput [32]. For a homogeeous etwork, our model for the multiple atea trasmitter implies that, at each chael use, the trasmitter seds to M differet users uiformly chose from the pool of users see [32]). I this schedulig the trasmitter seds M beams each oe is assiged to the user with the best sigal-to-oise ad iterferece ratio SINR) for the correspodig beam. As show i [32], the best SINR behaves like log with high probability for large. Therefore, we may agai assume that each packet carries a fix amout iformatio roughly about log + ρ log )). This schedulig is certaily more balaced compared to the case where we have a sigle atea system that works M times faster. This ca be justified by oticig the fact that we exclude the possibility of sedig to oe user twice or more) i each block of M trasmissios ad hece the schedulig is more balaced. I particular, assumig that there is o packet dropped as i Theorem. The, we have, D m, M) M D m, 27) where D m, M) is the delay for sedig m packets successfully to users i a M-trasmit atea system ad where D m, is the delay for a sigle atea broadcast system as i Theorem. I fact we ca compute exactly the expected delay i trasmittig the first packet successfully, i.e. E D, M)), for ay ad M. Further geeralizatio of the result to m> is o trivial ad we have ot bee able to do this; however, it is quite easy to show that D m, M) md, M). The ext theorem presets the result for m ad for ay ad M. Theorem 5. Cosider a broadcast chael with M trasmit ateas ad users. Assumig that o packet is dropped, we ca write the expected delay i sedig oe packet to all users for ay m ad as, E D,M)) X X r X k r i r ) r i M k ψ r i!ψ i M! k. 28) Proof: Similar to the proof of Theorem 3, we first ote that the mea of D, M) ca be writte as, E D, M)) Pr D, M) >k) 29) k I order to compute the probability of D, >k,wedefiethe auxiliary radom variable μ M k) as the umber of users that have received o packets after k chael uses i which the trasmitter seds to M differet users. From the defiitio of μ M, it is clear that μ M ad that D, M) >kis equivalet to μ M k) >. Therefore, Eq. 29) ca be writte as, E D, M)) Pr μ M k) > ) k k r Pr μ M k) r) 3) The probability that μ M k) r ca be computed as follows. Assumig μ M k) r implies that oly r users have received at least oe packet i k chael uses. We the defie the evet S i for i,,..., r as the evet that at least r i users have ot received ay packets amog r users that are supposed to receive a packet. This implies that there are at most i users that the trasmitter seds packets to. It is clear that for i M probability of S i is zero, sice the trasmitter certaily ca trasmit to M differet users at each chael use. For i>m, however we ca write the probability of S i as ) i k ) ) i k Pr {S i } M) r r, i k M) r i k M) M) i,,..., r. 3) where we first chose two sets of users with cardiality r ad i from the set of users ad the we distributed packets amog i of them k times by choosig M differet users at each time. Cosiderig the defiitio of μ M k) r ad the S i s, we ca use the iclusio-exclusio priciple see chapter 4 of [28]) to obtai 32). Substitutig 32) i 3), we ca write the expected delay as 33). This completes the proof for the Theorem.

8 336 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 27 Pr μ M k) r) PrS r ) PrS r )+...+PrS ) r r ) ) k ) r i PrS i ) r) r i k ) M) r i 32) i M i im E D, M)) Pr μ M k) > ) k r r) k M) k r im ) ) k r i ) r i 33) i M Remark 3: It is worth metioig that we ca also obtai the geeratig fuctio F z) that would lead to the momets of D, M) for ay M ad. I fact, F z) is equal to F z) X k X z k PrD,M) >k) X r X k r i M k zk ) r i r ψ r i!ψ! k i 34) M Usig 6) ad 34), we ca easily obtai the variace ad other momets) of D, M). Although Theorem 5 gives us the exact value of the expected delay for ay umber of users, it does ot make clear how much improvemet o the delay we ca get i usig multi-atea trasmitter over that of the sigle atea system. We ca i fact asymptotically aalyze the expected delay derived i Theorem 5 for large umber of users to get a better ituitio about this result. Theorem 6. Cosider the settig of Theorem 5. The the expected delay i sedig at least oe packet to all users usig a M-atea trasmitter derived i 28) behaves like E D, M)) k k M r r + O). 35) for large ad whe M grows o faster tha log. Proof: The proof is quite ivolved ad we omit it due to lack of space. The iterested reader is referred to [3] for the proof. For the special case of M, the problem reduces to the coupo collector problem whe m oe packet). It ca be easily show that the expected delay is equal to i i log. Clearly the result of Theorem 5 cofirms this result for oe trasmit atea, i.e. M. Remark 4: As metioed i 27), usig multiple trasmit ateas i the trasmitter should improve the delay. We may write the improvemet o the expected delay by usig M trasmit ateas over that of sigle atea case as show i 36). Eq. 36) implies that whe M is ot growig faster tha log, the gai i delay is a factor of M which comes from the fact that we are trasmittig packets M times faster. Therefore, multiple trasmit atea systems icur pretty much the same delay as that of a sigle atea trasmitter that operates M times faster whe there is o chael correlatio. Although the gai o delay i usig multiple trasmit ateas is ot that much, multiple trasmit ateas ca sigificatly improve the log term fairess i a heterogeeous etwork. More precisely, i [32], it is proves that if M grows logarithmically with the umber of users, the probability of choosig each user become idepedet of its SNR ad approaches to. Moreover, whe there is chael correlatio, multiple atea systems ca sigificatly reduce the delay by decorrelatig i time the effective chael through meas such as radom beamformig [32], [38]. V. TRADING DELAY WITH THE THROUGHPUT: d-algorithm Previously, we showed the delay hit i usig the optimal throughput schedulig is a log fold icrease compared to the miimum achievable delay. I this sectio, we propose a algorithm that ca reduce the expected delay for sedig the first packet at the price of a little throughput degradatio. The goal is to improve the log fold degradatio i the delay without too much reducig the throughput of the system. I order to improve the delay, we have to itroduce more optios to the scheduler at each chael use. For sigle atea systems, this ca be doe by lookig at the d best users i terms of capacity ad trasmit to the user amog those d users that has received the least umber of packets. We call this schedulig the d-algorithm. For a large umber of users ad fixed d, it is quite easy to show that the capacity of the best user ad that of the d th best user is quite close almost surely. This i fact guaratees that the throughput degradatio usig our algorithm is ot that much. The ext Theorem quatifies the performace of the d algorithm precisely. Theorem 7. Cosider the settig of Theorem ad suppose the trasmitter uses the d algorithm. We deote the expected delay i sedig the first packet by ED,). d The, for ay d, ED d, ) d dx + O) 37) xd Asymptotically, we ca further prove that if d is fixed, ED, d)) ED, d lim lim ) ED, ) log d. 38) Proof: I order to compute the expected delay, we agai defie the variable r i as the umber of chael uses after

9 SHARIF ad HASSIBI: DELAY CONSIDERATIONS FOR OPPORTUNISTIC SCHEDULING IN BROADCAST FADING CHANNELS 336 Gai o the expected delay with M atea trasmitter M r r M 2 M + O ). 36) sedig at least oe packet to i users ad before completig the trasmissio of at least oe packet to i users. Clearly r i has a Geometric distributio as, Prr i k) p i ) k p i k, 2,... 39) d where p i is the probability that all the d best users have bee chose before, therefore p i i d i p i d) d i 4) d), Notig that D, i r i, ad also usig the fact that the mea value of r i is p i, we ca obtai the expected value of D, as ED d,) id p i id ii )...i d+) )... d+) id i d+ where we used a simple upper boud for i d) / d).toevaluate the summatio i the right had side of 4), we may take itegrals from x to x d +from both sides of x/) d x /) d x )/) d, 42) to obtai d/ ED, d ) dx + O), 43) xd which completes the proof for the first part of the Theorem. To prove the secod part, we defie the itegral i the right had side of 43) as G). The it is quite easy to show that whe d is fixed, we have ) d 4) G) lim log lim d d )d ) d. 44) where we used the L Hopital s rule i 44). Cosiderig that ED, ) scales like log as proved i Theorem 4, the secod part of the theorem immediately follows from 44). Fig. 3 shows the delay improvemet for differet values of d ad for differet umber of users. As d icreases the delay improves though with less pace. Clearly, we ca get most of the improvemet by just checkig the the best two users d 2) ad further icreasig d will ot improve the expected delay as much as before. There is of course a price to pay o the rate for the delay improvemet. I order to see the throughput hit, we look ito the ergodic throughput of the chael deoted by Rd)) usig the d algorithm defied as ) Rd) E log +ρ max k h i 2 45) i ED, d)) umber of users) d2 d3 d4 d5 Roud Robi Fig. 3. Expected delay ED, d ) for differet values of d ad. where max k deotes the k th maximum ad k is a radom variable uiformly distributed betwee ad d. Usig results o the extreme value theory, it is quite straightforward to show that, lim Rd) R), 46) whe d is fixed. The proof is based o the fact that is d is fixed, the first ad the d best user both have SNR of about logsee [39], [32]. Eq. 46) implies that i the limit of large, the differece of the throughput of the d algorithm ad the maximum throughput coverges to zero. Remark 5: It is worth metioig that the trasmitter may use a roud-robi type schedulig ad also exploits the chael. This ca be doe by sedig to the best user amog users at the first chael use, ad the sedig to the best user amog users that have ot bee chose ad so o. This method ca make sure that the worst case delay is equal to. The ergodic throughput of this scheme ca be writte as, R RR E { k ) } log +ρ max i 2 i k 47) Assumig that the chael is Rayleigh fadig, we ca show that i the limit the ratio of R RR over R) is oe. Of course, the covergece i 46) for d-algorithm holds i a stroger sese. Moreover, it is worth metioig that this schedulig may require packets with differet amout of iformatio. Remark 6: Aother approach to trade the delay with throughput is to cosider a threshold for the capacity ad to sed to the user that has received the least umber of packets amog the users with istataeous capacity above the threshold value C Th ). I this case, we basically have a radom d that has a biomial distributio where the biomial parameter q depeds o the threshold value C Th. We ca i

10 3362 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 9, SEPTEMBER 27 fact boud the delay for sedig oe packet to all users usig d algorithm as, { d+ } ED, )E d {E{D, d}} E d ) d. l l 48) where d has biomial distributio with parameter q Pr{log + ρ i h i 2 ) C Th }. VI. CONCLUSION Providig quality of service QoS) ad also maximizig the throughput i a cellular system are the mai challeges that require desigig the physical layer ad multiple access layer together. I this paper, we cosider the dowlik of a cellular system i.e., a broadcast chael) ad we also cosider a otio of worst case delay which is defied as the delay D m, icurred i receivig m packets by all the users i the system. Clearly this defiitio of the delay is stroger tha the average delay ad represets the worst case delay amog the users. I order to maximize the throughput, the trasmitter has to sed a packet to the user with the best chael coditio which icreases the delay. The mai goal of this paper is to aalyze this delay icrease. Assumig a block fadig i.i.d. chael ad a sigle atea broadcast system with backlogged users, we derive the momet geeratig fuctio of the delay for ay m ad ad for a geeral hetereogeous etwork where a packet ca be dropped if outage capacity occurs. We further discuss how our results ca be exteded to the o-backlogged case. Asymptotically, for a homogeeous etwork where the throughput optimal schedulig is log-term fair i.e., the probability of choosig users are equal), the result implies that the average delay i sedig oe packet to all users behaves like log as opposed to for a roud robi schedulig. We also prove that whe m grows like log ) r,forsomer>, the to the first order the delay scales as m. This roughly determies the time-scale required for the system to behave fairly. We also look ito the delay aalysis for a system equipped with multiple trasmit ateas. Fially we propose a algorithm that without sacrificig too much o the throughput ca sigificatly improve the delay. The algorithm always cosiders the first d user with the best chael coditios ad trasmits to the oe that has received the least umber of packets. There are still questios remai to be aswered. For example, i the model we cosidered, all the users always have packets of equal size for trasmissio, it would be quite iterestig to geeralize the results to the case where each user have a radom rate of arrival or differet trasmissio rates ad aalyze the behavior of the legth of the logest queue amog users. REFERENCES [] X. Qi ad R. Berry, Exploitig multiuser diversity for medium access cotrol i wireless etworks, i Proc. of INFOCOM 23, pp [2] S. Shamai ad E. Telatar, Some iformatio theoretic aspects of decetralized power cotrol i multiple access fadig chaels, i Proc. Iformatio Theory ad Networkig Workshop 999. [3] I. Bettesh ad S. Shamai, Optimal power ad rate cotrol for fadig chaels, i Proc. Veh. Tech. Cof. 2, pp [4] G. Caire, G. Taricco, ad E. Biglieri, Optimum power allocatio over fadig chaels, IEEE Tras. If. Theory, vol. 45, o. 5, pp , July 999. [5] R. A. Berry ad R. G. Gallager, Commuicatio over fadig chaels with delay costraits, IEEE Tras. If. Theory, vol. 48, o. 5, pp , May 22. [6] N. Abramso, The ALOHA systems-aother alterative for computer commuicatios, i Proc. Fall Joit Comput. Cof. 97, pp [7] R. Gallager, A perspective o multiaccess chaels, IEEE Tras. If. Theory, vol. 3, o. 3, pp , Mar [8] A. Ephremides ad B. Hajek, Iformatio theory ad commuicatio etworks: a ucosummated uio, IEEE Tras. If. Theory, vol. 44, o., pp , Oct [9] D. N. Tse ad S. V. Haly, Multiaccess fadig chaels. I. polymatroid structure, optimal resource allocatio ad throughput capacities, IEEE Tras. If. Theory, vol. 44, o. 7, pp , Nov [] L. Tog, V. Naware, ad P. Vekitasubramaiam, Sigal processig i radom access: a cross layer perspective, IEEE Sigal Processig Mag., July 24. [] M. J. Neely ad E. Modiao, Dyamic power allocatio ad routig of time-varyig wireless etworks, IEEE J. Sel. Areas Commu., vol. 23, o., Ja. 25. [2] A. Gati, E. Modiao, ad J. Tsitsiklis, Optimal trasmissio schedulig i symmetric commuicatio models with itermittet coectivity, available at [3] A. Eryilmaz ad R. Srikat, Schedulig with Quality of Service Costrait over Rayleigh Fadig Chaels, i Proc. IEEE Coferece o Decisio ad Cotrol 23, pp [4] A. Stoylar ad K. Ramaa, Largest weighted delay first schedulig: large deviatios ad optimality, Aals Applied Probability, o., pp. 48, Nov. 2. [5] S. Borst, User level performace of chael aware schedulig algorithms i wireless data etworks, i Proc. INFOCOM 23. [6] M. Agrawal ad A. Puri, Base statio schedulig of requests with fixed deadlies, i Proc. INFOCOM 22. [7] S. Kumar ad P. R. Kumar, Performace bouds for queueig etworks ad schedulig policies, IEEE Tras. Auto. Cotrol, vol. 39, o. 9, Aug [8] P. R. Kumar ad S. Mey, Stability of queueig etworks ad schedulig policies, IEEE Tras. Auto. Cotrol, vol. 4, o. 2, Feb [9] E. Yeh ad A. S. Cohe, Throughput ad delay optimal resource allocatio i multiaccess fadig chaels, i Proc. IEEE ISIT 23, pp [2] J. I. Capetaakis, Tree algorithms for packet broadcast chaels, IEEE Tras. If. Theory, vol. 25, o. 9, pp , Sept [2] L. Tassiulas ad A. Ephremides, Dyamic server allocatio to parallel queues with radomly varyig coectivity, IEEE Tras. If. Theory, vol. 39, o. 2, Mar [22] A. Eryilmaz, R. Srikat, ad J. Perkis, Stable schedulig policies for broadcast chaels, i Proc. IEEE Iter. Symp. Ifo., July 22, p [23] M. Adrew, K. Kumara, K. Ramaa, A. Stoylar, P. Whitig, ad R. Vijaykumar, Providig quality of service over a shared wireless lik, IEEE Commu. Mag., vol. 39, o. 2, pp , Feb. 2. [24] J. F. Kigma, Iequalities i the theory of queues, J. Royal Statistical Society: Series B, vol. 32, o., pp. 2, Ja. 97. [25] M. J. Ferguso, O the cotrol. stability, ad waitig time i a slotted ALOHA radom access system, IEEE Tras. Commu., vol. 23, o., Oct [26] L. H. Ozarow, S. Shamai, ad A. D. Wyer, Iformatio theoretic cosideratios for cellular mobile radio, IEEE Tras. Veh. Techol., vol. 43, o. 2, pp , May 994. [27] A. Ephremides ad R. Zhu, Delay aalysis of iteractig queues with a approximate model, IEEE Tras. Commu., vol. 35, o. 2, Feb [28] W. Feller, A Itroductio to Probability Theory ad its Applicatios. Joh Wiley ad Sos, Ic., 967. [29] N. L. Johso ad S. Kotz, Ur Models ad Their Applicatio. Joh Wiley ad Sos, Ic., 977. [3] D. J. Newma ad L. Shepp, The double dixie cup problem, Amer. Math. Mothly, vol. 67, o., pp. 58 6, Ja. 96. [3] M. Sharif ad B. Hassibi, Delay aalysis of throughput optimal schedulig i broadcast fadig chaels, Techical Report, Califoria Istitute of Techology, available at masoud/delaybc.pdf, 24. [32] M. Sharif ad B. Hassibi, O the capacity of MIMO BC chael with partial side iformatio, IEEE Tras. If. Theory, o. 2, pp , Feb. 25.

11 SHARIF ad HASSIBI: DELAY CONSIDERATIONS FOR OPPORTUNISTIC SCHEDULING IN BROADCAST FADING CHANNELS 3363 [33] P. Viswaath ad D. N. Tse, Sum capacity of the vector Gaussia broadcast chael ad dowlik-uplik duality, IEEE Tras. If. Theory, vol. 49, o. 8, pp , Aug. 23. [34] G. Caire ad S. Shamai, O the achievable throughput of a multiatea Gaussia broadcast chael, IEEE Tras. If. Theory, vol. 49, o. 7, pp , July 23. [35] S. Vishwaath, N. Jidal, ad A. Goldsmith, Duality, achievable rates ad sum rate capacity of Gaussia MIMO broadcast chale, submitted to IEEE Tras. If. Theory, 22. [36] M. Sharif ad B. Hassibi, A compariso of time-sharig, DPC, ad beamformig for MIMO broadcast chaels with may users, i Proc. Iteratioal Symp. o Iformatio Theory 24. [37] Y. Xie ad C. Georghiades, Some results o the sum rate capacity of MIMO fadig broadcast chael, i Proc. Iter. Symp. i Advaces i Wireless Commu. 22. [38] P. Viswaath, D. N. Tse, ad R. Laroia, Opportuistic beamformig usig dump ateas, IEEE Tras. If. Theory, vol. 48, o. 6, pp , Jue 22. [39] M. R. Leadbetter, Extreme value theory uder weak mixig coditios, Studies i Probability Theory, MAA Studies i MAthematics, pp. 46, 978. Masoud Sharif received his Ph.D. i Electrical Egieerig 25) from Califoria Istitute of Techology. I 25, he was a post-doctoral scholar i the EE departmet at Caltech. Sice Jauary 26, he has bee a assistat Professor at Bosto Uiversity. Dr. Sharif was awarded the C.H. Wilts Prize i 26 for best doctoral thesis i Electrical Egieerig at Caltech. He is a member of the Ceter for Iformatio ad Systems Egieerig at Bosto Uiversity. His research iterests iclude adhoc ad sesor etworks, multiple-user multipleatea commuicatio chaels, cross-layer desig for wireless etworks, ad multi-user iformatio theory. His recet research has focused o collaborative commuicatio scheme i ad-hoc ad sesor etworks ad the capacity of multiple atea broadcast chaels. Babak Hassibi was bor i Tehra, Ira, i 967. He received the B.S. degree from the Uiversity of Tehra i 989, ad the M.S. ad Ph.D. degrees from Staford Uiversity i 993 ad 996, respectively, all i electrical egieerig. From October 996 to October 998 he was a research associate at the Iformatio Systems Laboratory, Staford Uiversity, ad from November 998 to December 2 he was a Member of the Techical Staff i the Mathematical Scieces Research Ceter at Bell Laboratories, Murray Hill, NJ. Sice Jauary 2 he has bee with the departmet of electrical egieerig at the Califoria Istitute of Techology, Pasadea, CA., where he is curretly a associate professor. He has also held short-tem appoitmets at Ricoh Califoria Research Ceter, the Idia Istitute of Sciece, ad Likopig Uiversity, Swede. His research iterests iclude wireless commuicatios, robust estimatio ad cotrol, adaptive sigal processig ad liear algebra. He is the coauthor of the books Idefiite Quadratic Estimatio ad Cotrol: A Uified Approach to H2 ad H Theories New York: SIAM, 999) ad Liear Estimatio Eglewood Cliffs, NJ: Pretice Hall, 2). He is a recipiet of a Alborz Foudatio Fellowship, the 999 O. Hugo Schuck best paper award of the America Automatic Cotrol Coucil, the 22 Natioal Sciece Foudatio Career Award, the 22 Okawa Foudatio Research Grat for Iformatio ad Telecommuicatios, the 23 David ad Lucille Packard Fellowship for Sciece ad Egieerig ad the 23 Presidetial Early Career Award for Scietists ad Egieers PECASE). He has bee a Guest Editor for the IEEE Trasactios o Iformatio Theory special issue o space-time trasmissio, receptio, codig ad sigal processig, was a Associate Editor for Commuicatios of the IEEE Trasactios o Iformatio Theory durig 24-26, ad is curretly a Editor for the joural Foudatios ad Treds i Iformatio ad Commuicatio.