Performance Analysis of Scheduling Policies for Delay-Tolerant Applications in Centralized Wireless Networks

Transcription

1 Performance Analyss of Schedulng Polces for Delay-Tolerant Applcatons n Centralzed Wreless Networks Mohamed Shaqfeh and Norbert Goertz Insttute for Dgtal Communcatons Jont Research Insttute for Sgnal & Image Processng School of Engneerng and Electroncs The Unversty of Ednburgh Mayfeld Rd., Ednburgh EH9 3JL, Scotland, UK Emal: {M.Shaqfeh, Norbert.Goertz}@ed.ac.uk Abstract Many schedulng schemes have been proposed n lterature to control how dfferent users access a wreless channel. Channel-aware schedulers explot the measurements of nstantaneous channel condtons of the dfferent users to obtan throughput gans by proper selecton of the users to transmt or receve n each channel state. In ths paper, the performance of channel-aware schedulng polces whch are applcable for delay-tolerant applcatons n centralzed wreless networks are analyzed n a new mathematcal framework. The framework s appled n numercal examples to compare the performance of dfferent schedulng polces n terms of ther effcency n allocatng the wreless resources. Ths effcency s measured aganst the set of all possble operatng ponts of the system, chosen, e.g., by the network operator. Index Terms centralzed wreless networks, channel-aware schedulng, power allocaton, delay-tolerant applcatons I. INTRODUCTION An mportant feature of multuser communcaton over fadng channels s multuser dversty: The ndependent fadng characterstcs of the users channels ncrease the probablty of havng one or more users wth very good channel condton at each tme nstance. The multuser dversty gan can be obtaned by explotng the ndependent fadng condtons, and schedulng only the users wth good channels. Durng the last decade, the desgn of multuser schedulng schemes for wreless networks has been extensvely studed. The algorthms whch were state-of-the-art few years ago [1], are now outperformed by a lot of new schemes. Proposals n lterature dffer n ther polces to choose the scheduled user based on channel condtons. The objectve of ths work s to systematcally compare, n a new unversal framework, known channel-aware schedulng algorthms that are applcable for delay-tolerant applcatons n centralzed wreless networks. Algorthms that consder strct delay-constrants are, however, not ncluded n our analyss. In centralzed networks, the schedulng decsons for both the uplnk and the downlnk are taken at the wreless access pont or the base staton of the network. Ths unt s provded wth a rch set of nformaton such as the traffc load and the dfferent QoS 1 requrements of the served traffc classes as well as the nstantaneous channel condtons of the wreless lnks to actve users. We assume n our analyss that the channel varatons are not too fast so that the effect of the channel measurement delay [2] s neglgble and the channel coeffcents can be estmated at the recever and be communcated to the transmtter wth suffcent accuracy at low overhead. Ths also means that we can perform coherent detecton at the recever,.e., we get no phase error and the I and the Q components are both scaled by the magntude of the channel s fadng coeffcent whch s the square root of the channel power gan. For our analyss we assume general schedulers whch may, e.g., be appled n OFDMA 2 systems wth dynamc sub-carrer allocaton DSA and adaptve modulaton and codng AMC. The OFDMA scheme enables the explotaton of multuser dversty n the frequency doman due to frequency-selectve channels n the total transmsson bandwdth and n the tme doman [3]. Therefore, OFDMA provdes a rather general settng wth other e.g. sngle carrer systems as specal cases. The schedulng decsons,.e. the decsons whch of the users are allowed to access the channel, are taken for each channel block over whch the channel fadng coeffcents can be consdered to be constant. Each channel block may consst of many slots n tme or frequency or both over whch the channel fadng coeffcents do not change as the slots belong to one channel block. Some of the schedulng algorthms we nclude n a case study to demonstrate our new analyss were orgnally proposed and appled for sngle-carrer systems e.g. the propor- 1 Qualty of Servce 2 Orthogonal Frequency Dvson Multple Access SPECTS ISBN: Authorzed lcensed use lmted to: Unverstatsbblothek der TU Wen. Downloaded on May 29, 29 at 4:19 from IEEE Xplore. Restrctons apply.

2 tonal far scheduler [4]. Snce these schedulng algorthms are, on a sub-carrer level, also applcable to OFDMA systems, we nclude them n our analyss. For smplcty, we assume sngle-transmt sngle-receve antenna SISO systems. The schedulng schemes we dscuss can, of course, also be used n a multple-antenna MIMO framework but the analyss s more complcated, and ths would dstract from the man ssues schedulng and resource-allocaton we lke to dscuss. In general, schedulng algorthms have two man objectves: effcent allocaton of the scarce wreless resources and achevng sutable farness crtera and QoS requrements of the dfferent applcatons. Snce the network operator may have dfferent objectves dependng on the applcatons supported by the network and the way farness s defned, one schedulng algorthm can be most sutable for a certan scenaro or defnton of farness whle t performs poorly n a dfferent scenaro. The key pont of our work s to show that t s actually possble to compare schedulng algorthms n terms of ther effcency n allocatng the wreless resources,.e. power and bandwdth, across the whole range of possble operatng ponts. Although there s a contradcton between the maxmum total throughput and farness and QoS constrants, there s no contradcton between the two objectves of effcent resource allocaton and achevng farness wth certan QoS requrements. It must, e.g., be accepted that there wll be a loss n terms of total throughput when users wth bad channels have to be served, but ths s by no means a weakness of a partcular schedulng scheme t s rather a fundamental trade-off that s descrbed by the theoretcal lmts of nformaton theory, and the key queston s how close to those lmts a schedulng scheme performs. Effcent schedulng algorthms n our sense are those whch operate on or close to the boundary of the capacty regon, whch s the set of long-term average achevable user rates for gven average power constrants. Informaton-theory [5], [6] does not only provde such characterzaton of the capacty regon, but t also provdes gudelnes for system desgns leadng to optmal performance,.e., to achevng capacty lmts. However, the optmal solutons are n most cases dffcult f not mpractcal to mplement. Thus, sub-optmal solutons whch have close-to-optmal performance and at the same tme lend themselves to an easy mplementaton are most favourable from a practcal perspectve. That s why there s a large number of schedulng algorthms suggested n the lterature. In ths paper, the channels achevable rate lmts when applyng dfferent power control polces are taken nto consderaton when comparng varous schedulng polces. In order to vsualze the performance and effcency of the schedulng algorthms, we analyze the two-user case, wth the assumpton of dfferent long-term average channel qualtes of the users. The achevable rates usng the schedulng algorthms are plotted n order to compare the performance of the algorthms. Qualtatvely, the results carry over to the M-user case. The rest of the paper s organzed as follows: frst a summary of the schedulng algorthms whch are ncluded n the comparson s provded. Then, a descrpton of the framework to evaluate the performance of the schedulng polces s gven, followed by the numercal results of the performance analyss of the schedulers. II. SCHEDULING POLICIES PROPOSED IN LITERATURE FOR USE IN A PERFORMANCE COMPARISON STUDY In lterature, there exst schedulng schemes whch use smlar polces, but wth dfferent operatng ponts 3 dependng on the control parameters of the polcy such as weghtng factors and rate offsets, whch are adjusted accordng to the farness crtera and constrants for the gven applcaton. Examples of schemes to select an operatng pont nclude proportonal user-rate ratos throughput farness [7], proportonal channelaccess ratos resource-sharng farness [8], [9], and utlty functon maxmzaton [1]. In ths secton we provde a summary, along wth a new unfed algorthmc descrpton, of schedulng polces proposed n the lterature all are applcable for both the uplnk and the downlnk that lend themselves to a comparson n our framework. Ths also ncludes some novel extensons of the orgnally proposed schemes. Of course, our choce of schedulers s far from beng exhaustve, but we beleve our selecton s sensble and provdes a useful bass for comparson of other schemes as well. Moreover, our man contrbuton s a new framework detaled n Secton III wth whch to analyze varous types of schedulers; the specfc schedulng algorthms we have pcked are actually just examples used to demonstrate the new analyss. Due to the objectves of ths work, schedulng algorthms that explctly take delay-constrants nto account such as the ones presented n [11], [12] are not ncluded n the mathematcal analyss. A. Schedulng Polces for Constant-Power Systems All polces n ths secton wll schedule exactly one user per channel block and, f scheduled, each user s power s the same. In what follows, and n partcular n Fgures 1 and 2 whch present numercal results detals are dscussed n Secton IV, the dfferent schedulng and power allocaton polces are referred to by ther equaton numbers stated below. 1 User selecton based on weghted feasble rates: Schedulng polces whch schedule user m n channel block k accordng to a weghted value of the nstantaneous feasble 4 rate R [k] of the user are gven by m = arg max μ R [k]. 1 3 An operatng pont n a delay-tolerant system s defned by the vector of long-term average rates of the users. 4 Below we wll use the well-known capacty equaton for an Addtve Whte Gaussan Nose AWGN channel to estmate the rate R [k] from a gven power P [k]. The underlyng assumptons are that we can approxmate the fadng multuser channel by a block fadng model and that the channel-codng blocksze can be made nfntely large over the fadng states of the channel model and, hence, the Gaussan channel capacty can be acheved. Moreover, we assume as dscussed n Secton I that the channel power gan h [k] s known at the transmtter, so that t can be exploted for schedulng decsons and/or transmt-power allocaton. SPECTS ISBN: Authorzed lcensed use lmted to: Unverstatsbblothek der TU Wen. Downloaded on May 29, 29 at 4:19 from IEEE Xplore. Restrctons apply.

3 The power allocated to the scheduled user s constant,.e. P m [k] = P. 2 Ths power allocaton rule s used by all schedulng polces wth a constant transmsson-power constrant. In the maxmum sum-throughput scheduler 5, the weghtng factors are all equal,.e., μ =1. In the Proportonal Far PF Scheduler [14], μ s nversely proportonal to the average throughput T [k] of the user n a past wndow,.e., μ = 1 T [k]. In [11] proportonal farness s suggested wth payloads dependng on the specfc applcaton. In [15] ths polcy s suggested n a generc form to maxmze throughput relatve to pre-specfed target ratos. The PF scheduler wll not be ncluded n the new analytcal framework n Secton III. The man reason s that we consder delay-tolerant applcatons: n ths case any dynamc adaptaton e.g. as above by T [k] of the scheduler parameters μ s actually counter-productve wth respect to the achevable long-term average rates we are nterested n. Further detals are dscussed n Secton II-C. 2 User selecton based on weghted channel qualty: Those polces are gven by m = arg max μ h [k], 3 where h [k] s the nstantaneous power gan of user s channel. In constant transmsson power systems, ths s equvalent to a polcy whch schedules the user wth hghest weghted receved Sgnal-to-Nose rato SNR. In [16] ths polcy s suggested n two forms: maxmum throughput μ =1 and proportonal farness μ =1/ h, where h s the long-term average channel power gan. In [9], the polcy s suggested n a generc form to acheve pre-specfed resource-sharng ratos. 3 User selecton based on feasble rates wth rate offset: In [8] t s suggested to maxmze the average total system performance whle satsfyng pre-assgned tme-fracton channel-access rate requrements of the users. The proposal s generc and applcable to any system performance measures. We nclude ths schedulng concept n the comparson wth the assumpton that throughput s the system-performance measure to maxmze. The resultng schedulng polcy s gven by m = arg max R [k]+v, 4 where R [k] s as n Secton II-A.1 the feasble rate for user n channel block k and v s a rate offset whch s adjusted such that pre-assgned resource-sharng constrants are acheved. 4 User selecton based on the cumulatve rate-densty functon: In [17] schedulng based on the cumulatve densty functon CDF of user transmsson rates R k s suggested. The concept s to schedule the user whose rate s hgh enough, but least lkely to take even larger values n other e.g. future 5 In [13] power control s used to acheve capacty. The same selecton polcy 1 maxmzes the sum-capacty under a constant transmsson-power constrant. blocks. Ths schedulng polcy s gven by m = arg max F R R [k] 1 w, 5 where F R. s the CDF of the user s feasble transmsson rates. The weghtng factors w are used to scan dfferent operatng ponts of the system based on the constrants of the applcatons. B. Schedulng Polces for Varable-Power Systems 1 User selecton based on weghted feasble rates wth waterfllg power allocaton: In [18] Proportonal Farness wth QoS provson n downlnk OFDMA s suggested. The user selecton n each sub-carrer s based on PF schedulng [14], see also Secton II-A and the power P m [k] s allocated usng the water-fllng approach [ ] P m [k] = max σ 2 1 λ m 1 h m[k],, wth user-ndvdual factors λ m = 1 λt m that depend on the average rate T m recently acheved for user m n a movng tme-wndow of lmted sze. The factor λ s adjusted such that a specfed average power constrant s met. We use a generalzed verson of ths concept: user selecton s carred out by the general selecton polcy 1 based on weghted feasble rates, and the power allocaton for user m who s assumed to be scheduled n block k s gven by [ P m [k] =σ 2 μm λ 1 ] + 6 h m [k] where [x] +. = maxx,. The factor λ s agan adjusted accordng to a long-term average power constrant, and μ m are weghtng factors used to pck a desred operatng pont. As n Secton II-A.1, the same comments apply wth repect to a dynamc adaptaton of the weghtng factors ths s further dscussed n Secton II-C. It should be noted that for constant weghts μ for all users ths s the same as the suboptmal Tme Dvson TD polcy gven n [5, Secton III-C]. 2 User selecton based on weghted channel qualty and smplfed waterfllng power allocaton: In [19] t s suggested to use a normalzed-snr-based user selecton strategy wth water-fllng power control along the sub-carrers. However, the water-fllng level λ s adjusted rrespectvely of the user selecton polcy. For comparson, we consder a smlar method n a generalzed form: user selecton s based on 3 and the power s controlled n the blocks accordng to ] +. 7 [ 1 P m [k] =σ 2 λ 1 h m [k] Agan λ has to be adjusted such that a long-term average power constrant s met. The dfference to 6 s that the weghtng factors μ used n the user selecton are not used n 7. C. Detrmental Effect of a Dynamc Adaptaton of the Scheduler In general all schedulng polces acheve ther maxmum performance when the control parameters such as weghtng SPECTS ISBN: Authorzed lcensed use lmted to: Unverstatsbblothek der TU Wen. Downloaded on May 29, 29 at 4:19 from IEEE Xplore. Restrctons apply.

4 factors or rate-offsets are constant. But wth constant control parameters t s mpossble to control the delay and, hence, schedulers for delay-constraned applcatons often have to be dynamcally adapted. A popular example s the Proportonal Far PF Scheduler [14], whch uses 1 to take schedulng decsons but wth the rate weghtng factors adapted accordng to μ = 1 T, where T [k] [k] s the recently acheved average rate n movng tme-wndow. Such or any other dynamc adaptaton wll decrease the achevable long-term average rate: ths mmedately follows from the convexty of the achevable rate regons. We may thnk of two ponts rate-tuples on the boundary of the scheduler s rate regon that are acheved for two dfferent parameter settngs. When the scheduler parameters dynamcally change between those two parameter sets, we can get the achevable rates pro-rata by tme-averagng the acheved rates n both cases. Hence, we obtan a pont on the straght lne connectng the two ponts on the boundary of the rate regon. As the regon s convex, any pont on ths lne wll le nsde the rate regon but not on ts boundary and, therefore, any dynamc adaptaton of the scheduler s nherently suboptmal. The detrmental effect on the acheved rate wll be the larger the larger the rate-dfferences between the two ponts are. As we consder delay-tolerant applcatons, we wll, therefore, not nclude any scheme n our theoretcal analyss that uses dynamc adaptaton of the scheduler parameters. III. MATHEMATICAL ANALYSIS OF ACHIEVABLE RATES OF DIFFERENT SCHEDULING POLICIES When a schedulng polcy allocates rate to a sngle user only n each channel block, the maxmum possble achevable rate bts/sec/hz 6 ofuser who s scheduled n block k equals R [k] = log 1+ h [k]p [k] σ 2 8 for addtve whte Gaussan recever nose wth a varance of σ 2 of course, 8 s the Shannon capacty for the AWGN channel. Wth AMC 7 a rate close to capacty can be acheved see, e.g., [2]. In practce, wreless systems support a set of dscrete rate values rather than a contnuous range. However, our objectve s not to evaluate the schedulers achevable rates for some gven set of practcal modulaton and codng schemes. Our goal s rather to evaluate the performances of schedulng schemes as such, wthout any system constrants that wll change anyway from one applcaton to another. Therefore, we use the dealsaton 8 to relate power and rate ; the relatve performances 8 of varous schedulng schemes wll carry over nto practce. 6 Ths relates to each Hz of bandwdth on the rado-frequency bandpass channel. Bandwdth s defned as the wdth of a compact set of postve bandpass frequences for whch the sgnal spectrum s allowed to be nonzero. The occuped bandwdth n each real I/Q sub-channel of an equvalent complex baseband model s half the bandpass rado-frequency bandwdth. 7 Adaptve Modulaton and Codng 8 The absolute performance of a combnaton of specfc modulaton and codng schemes can often be approxmated by 8 as well. An acceptable resdual bt or frame error rate wll often be acheved by a practcal scheme wth some farly constant power-offset aganst the theoretcal zero-error curve gven by 8. In constant power systems, P [k] n 8 s constant n all blocks n whch user s scheduled,.e., P [k] = P k. In systems applyng power control, P [k] wll be a functon of h [k] see, e.g., 6. Assumng that all the blocks have dentcal bandwdths and tme duratons, the average achevable rate bts/sec/hz of user, n the blocks n whch user s scheduled, equals R = 1 S n S log 1+ h [n]p [n] σ 2, 9 wth S the set of ndces of all channel blocks n whch user s scheduled. Of course, user does not transmt n all blocks but rather n a rato ϱ of the total number of blocks. For a very large number of consdered blocks ths rato converges aganst the probablty that user s scheduled and, hence, we set ϱ =Pr{ s the scheduled user}. 1 Thus, the actually achevable long-term average rate bts/sec/hz of user s R = ϱ R = ϱ 1+ h [n]p [n] S σ n S log The averagng n 11 over the realsatons h [n] from the set S of channel power gans can be replaced by an ntegraton over a probablty densty functon PDF of the random varable H by explotng that the random process created by a tme seres of realsatons of H from the set S s ergodc 9 : where R = ϱ f H h log 1+ h P h σ 2 dh 12 f H h. = f H h s the scheduled user 13 s the condtonal PDF of the channel power gan of user, gven that user s scheduled to transmt uplnk or to receve downlnk. The PDF 13 s dfferent from that of the actual channel power gan snce the user s transmttng wth hgher probablty when the channel gan s larger. Furthermore, note that f H h = d F H h dh 14 wth F H h the CDF 1 of the channel power gan, h, over the blocks n whch user s scheduled to transmt or to 9 The schedulng decsons depend on the channel power gans of all users and perhaps on a set of constant weghtng factors, and all those channel power gans are assumed to form ndependent and ergodc random processes. Therefore, the new random process, whch s created by consderng the power gans only when the user s scheduled, wll also be ergodc. 1 cumulated densty functon SPECTS ISBN: Authorzed lcensed use lmted to: Unverstatsbblothek der TU Wen. Downloaded on May 29, 29 at 4:19 from IEEE Xplore. Restrctons apply.

5 receve. We can wrte F H h equvalently as follows: F H h = F H h s the scheduled user 15 Thus, we obtan = Pr{H h s the scheduled user} 16 = Pr{H h, s the scheduled user} 17 Pr{ s the scheduled user} = Pr{H h, s the scheduled user} 18 ϱ f H h = d Pr{H h, s the scheduled user} ϱ dh. 19 Hence, we can rewrte 12 as where R = v H h. = v H h log 1+ h P h σ 2 dh 2 d dh Pr{H h, s the scheduled user}. 21 For adaptve power allocaton systems, the power P h n 2 contans λ whch needs to be adjusted to mantan the average power constrant P,.e., λ s selected such that P = v H h P h dh = P. 22 For systems wth a constant-power constrant the stuaton s much smpler, as, f user s scheduled n block k, P h [k] = P and, hence, P [k] = P. The next step n order to evaluate 2 and 22 s to derve the relaton of v H h n terms of the known weghtng factors μ m and the known uncondtonal channel PDFs f Hm h m of all users. We consder polces that schedule no more than one user n each channel block.e., there s no tme-sharng between any two users wthn a channel block wth constant channel gan. In order to fnd the probablty that a user s scheduled we defne, as a novelty, contnuous non-decreasng auxlary functons g j h [k] whch can be used as follows to take schedulng decsons: user s scheduled n block k f and only f h j [k] <g j h [k] j 23 By 23, the functons g j h [k] are as yet only mplctly defned. We wll descrbe below how to obtan them; of course, they wll depend on the specfc schedulng polcy chosen. The functons g j h [k] descrbe the borders of the regons wthn the channel-gan vector-space h. = {h 1,h 2,...} over whch the dfferent users are scheduled. In what follows we drop the block ndex k, as the functons g j. actually descrbe how decsons are taken by some polcy for a gven set of channel coeffcents 11 channel state. 11 Of course the coeffcents depend on the block ndex k and therefore the value of g j wll also depend on k. The schedulng polcy tself that s descrbed by g j s, however, not dependent on the block ndex k. TABLE I g j h FOR THE SCHEDULING POLICIES UNDER CONSIDERATION. EACH SCHEDULING POLICY IS CHARACTERIZED BY A SELECTION POLICY TO SELECT THE USER m IN BLOCK k, AND A POWER-ALLOCATION POLICY P m FOR THE SCHEDULED USER. IN THE TABLE WE REFER TO THE EQUATION NUMBERS OF THESE POLICIES. THE BLOCK INDEX k IS OMITTED FOR BREVITY. m P m g j h μ μ 1+h P j 1 P μ μ j h + expv v j[1+h P] 1 P 5 2 F 1 h j [F h h ] w j w μ λ 1 6 μh μ j μ j λ : h > λ μ and :h λ μ In general, the schedulng polces we consder have the format m = arg max y h 24 where y s an ncreasng functon of h. Then the only possblty that user s scheduled s ff 12 the channel power gans h j j are below certan values whch are specfed by the g j h functons. For example, f the scheduler s applyng polcy 3,.e. y h =μ h, then user s scheduled ff for every other user j μ j h j <μ h h j < μ μ j h. = gj h 25 Ths defnes the functon g j h for ths schedulng polcy. As another example, the scheduler may be applyng polcy 1,.e. y h =μ R = μ log1 + h P. Then user s scheduled ff for every other user j μ j log 1+h j P < μ log 1+h P 26 [ μ 1+h j P < exp log ] 1+h P 27 μ j 1+h P μ μ j 1. h j < = g j h 28 P Usng the appled procedures n these examples, we can also obtan g j h for all other schedulng polces under consderaton. We summarze the results for g j h n Table I. Havng defned the functons g j h, we can now go back 12 f and only f SPECTS ISBN: Authorzed lcensed use lmted to: Unverstatsbblothek der TU Wen. Downloaded on May 29, 29 at 4:19 from IEEE Xplore. Restrctons apply.

6 to calculate v H h. We obtan from the defnton 21 of v H h and from 23: v H h = = d Pr{H h, s scheduled} dh 29 d Pr{H h,h j <g j H j} dh 3 = d h f H x j Pr{H j <g j x}dx dh 31 wth 31 followng from the ndependence of the channel power gans of the users. From the dfferentaton of the ntegral n 31 we obtan v H h = f H h j Pr{H j <g j h } 32 = f H h j F Hj g j h 33 wth f H the uncondtonal and statonary PDF of user s channel gans and F Hj h j =. h j f Hj xdx the uncondtonal CDF of the channel power gans for the users j. Equatons 2 and 22 can now be evaluated by usng v H h accordng to 33. Note that 33 nvolves the known smple channel models uncondtonal PDFs and CDFs for the users channel coeffcents. The structural propertes of the schedulng polcy and the power allocaton scheme are completely captured by the newly defned g j h -functons: by use n 2, these g j h -functons allow for a smple evaluaton of the achevable rate of any schedulng polcy wth sngle-user selecton n each block k see Table I. By f necessary numercal evaluaton of the ntegral 22, the g j h -functons also allow to adjust the control factor λ whch s used, e.g., n power control polcy 6. Note that the evaluaton of 2 and 22 by means of the g j h - functons s a great smplfcaton n comparson to a tmesmulaton of the scheduler wth ts assocated power control strategy and tme-averagng of the rates over many channel realsatons to obtan a statstcally sgnfcant values for the rate-averages. Moreover, the g j h -functons provde a useful tool for an analytcal charactersaton of schedulng polces; we provde results n Secton IV that would, due to the extensve smulaton tme requred, be very hard to obtan by smulaton and tme-averagng. IV. NUMERICAL RESULTS FOR THE ACHIEVABLE RATES OF DIFFERENT SCHEDULING POLICIES We now apply 33 to evaluate 2 and 22: we assume the magntudes a of the users channel coeffcents to have Raylegh or Rce dstrbutons [21, pp ], [2, pp ]. As the channel power gan, h, s the square of the channel-coeffcent s magntude, we have to use the varablesubsttuton h = a 2 n the orgnal Raylegh/Rce PDFs. As qa =a 2 s monotoncally ncreasng for a >, weusethe standard rule f H h = fa a q a wth q a =2a a=q 1 h and q 1 h = h to obtan the PDF of the channel power gan h from the PDF of the channel coeffcent s magntude a. For use n 33 we obtan for the Raylegh-case: f H h = 1 h exp h 34 h F H h =1 exp h 35 h wth h the long-term average channel power gan. For the Rce-case we fnd f H h = κ +1 exp κ κ +1 h I 2 h h F H h = κ+1 h h κ +1 κh h 36 e κ+x I 2 κx dx 37 wth I. the zero-th order modfed Bessel functon 13 of the frst knd, and κ s the fadng parameter that s defned as the rato κ. = h,los /h,nlos of the average power gans h LOS on the lne-of-sght path and h NLOS on the non-lne-of-sght path: κ =means we get Raylegh fadng, and κ means no fadng. The long-term average channel power gan s agan denoted by h. We wll consder the two-user case n two examples. Ths s for clarty only, as t s dffcult to vsualze and compare hgher-dmensonal rate regons. Of course, the mathematcal concepts and the results from Secton III can also be appled to the general M-user case. Moreover, the numercal results for the two-user case presented below provde nterestng nsghts that wll carry over to the M-user case. Fgures 1 and 2 show numercal results. Smlar to Table I, we ndcate by ther equaton numbers the user selecton polcy and power allocaton polcy of each schedulng scheme shown n the fgures. In Fgure 1 the achevable rates n bts/sec/hz, over all possble operatng ponts of the system, are depcted usng the polces whch schedule a sngle-user per channel block wth constant transmt power. We consder a Rce fadng channel for the frst user κ =1, wth a long-term average channel power gan that s 1dB hgher than that of the second user, who has a Raylegh fadng channel. Fgure 1 shows that the weghted feasble-rates polcy 1 s the best. However, the weghted channel-gans polcy 3 works almost as good for all operatng ponts and, thus, s attractve for generc schedulers, as 3 does not need to assume a partcular relaton such as the AWGN capacty equaton between power and rate. Another advantage of ths polcy s that, unlke the weghted feasble-rate polcy, t has a contnuous probablty dstrbuton and thus, wth probablty of one, a sngle user s maxmzng the scheduler metrc. 13 Ths functon can be represented [21, p. 44] by the nfnte seres I x = 1 k= 2 k k! 2 x2k, x. As,fork suffcently large, the denomnator wll domnate the result, a lmted number of summands wll suffce to get accurate results. Ths means that 36 can be evaluated wthout explct use of any Bessel functon and that 37 can be evaluated wthout any numercal ntegraton. SPECTS ISBN: Authorzed lcensed use lmted to: Unverstatsbblothek der TU Wen. Downloaded on May 29, 29 at 4:19 from IEEE Xplore. Restrctons apply.

7 Second User Average Rate bts/sec/hz Round Robn Second USer Average Rate bts/sec/hz Round Robn Polcy Frst User Average Rate bts/sec/hz Frst User Average Rate bts/sec/hz Fg. 1. Achevable long-term average rates of the constant-power-per-block schedulng polces for the two-user case. The schedulng polces are ndcated by ther equaton numbers n the legends; 2 s the equaton number of the constant-power allocaton polcy. The channel coeffcent of the frst user has a Rce-dstrbuton κ = 1, whle Raylegh fadng apples to the second user. The average channel power gan of user 1 s 1 db hgher than that of user 2. The Round-Robn polcy schedules user 1 n odd-numbered channel blocks and user 2 n even-numbered blocks, regardless of ther channel coeffcents; power allocaton s also by 2. Polcy 4 concdes wth the lmts acheved by polcy 1 whch has the best possble performance for constant power at the maxmum sum-throughput pont 14. The latter s acheved by polcy 1 when ts weghtng factors are all equal, and t s acheved by polcy 4 when the rate offsets are all zero. For all other ponts polcy 4 has degraded performance n comparson to polcy 1, and ths degradaton s larger when the system operates at low spectral effcency low sum-rate; the degradaton at hgh spectral effcency s however very small. Polcy 5 s smlar to polcy 4 n that t has degraded performance compared to 1, but unlke 4 ts best performance agan concdng wth polcy 1 s not at the maxmum sum-throughput but rather on another operatng pont whch depends on the fadng channel models; that s why we observe dfferent results n Fgures 1, 2 for both polces. In Fgure 2 we nvestgate schedulng polces nvolvng power control. We assume Raylegh fadng channels for both users, wth 1dB hgher average channel power gan for the frst user. For comparson, we also nclude the constant-power polces nvestgated n Fgure 1 already 15. Fgure 2 demonstrates that power control s helpful for the users wth low-spectral effcency low rate, whle usng constant power s justfed when operatng at hgh spectral effcency. As a compromse, we may use a lmted number 14 Polces 4 and 1 also concde for the the trval case that ether user 1 or user 2 are scheduled all the tme, but ths s no longer a multuser case. 15 We do not duplcate results, as the channels are dfferent from those n Fgure 1. Fg. 2. The performance of varous schedulng polces secton II n a two-user case. The channels are assumed to be Raylegh faded wth 1 db dfference of the average channel power gans. of dfferent power levels as long as ths does not cause nterference problems to be better able to support users wth bad channels. The power allocaton usng waterfllng n sngle-user systems means allocatng more power when the channel s better. In multuser communcaton, t also means allocatng more power to the user wth hgher rate-reward μ. Thus, polcy 7 s a bad choce. The selecton polcy 1, combned wth power allocaton polcy 6, gves a better performance. V. CONCLUSIONS We have presented a new framework to evaluate the performance of schedulng polces whch select at most one user per channel block. The mathematcal framework was appled n numercal examples to compare the performances of schedulng polces known from lterature. The performance s dependent on the user selecton strategy as well as the power control polcy. Although the exemplary analyss of the twouser case does not provde an exhaustve numercal evaluaton of the schedulng polces consdered, t hghlghts mportant results whch, qualtatvely, carry over to the general case. As a summary, we obtan the followng results: There s no contradcton between effcent resource allocaton and maxmzng any network performance metrc for delay-tolerant applcatons. A good scheduler uses a polcy that closely acheves capacty for the operatng pont selected. Maxmzng any utlty functon or mantanng farness crtera should be done such that a sutable operatng pont s chosen on the capacty regon s boundary. Wth known channel coeffcents at both ends, the optmal schedulng polcy does not depend on the statstcs of the channel fadng process. SPECTS ISBN: Authorzed lcensed use lmted to: Unverstatsbblothek der TU Wen. Downloaded on May 29, 29 at 4:19 from IEEE Xplore. Restrctons apply.

8 For constant power and sngle user selecton per block 1 s the best polcy. If power control s appled, 6 should be used to adjust the power. Some schedulng polces are good for some operatng ponts only, but they are not generc. There are polces whch can be used n generc schedulers such as 3 because they have close-to-optmal performance for all set of operatng ponts. Dynamc on-lne varaton of the weghtng factors degrades the performance and wll be avoded by a good scheduler f the applcatons are delay tolerant. VI. ACKNOWLEDGEMENTS The work reported n ths paper has formed part of the Core 4 Research Program of the Vrtual Centre of Excellence n Moble and Personal Communcatons, Moble VCE, whose fundng support, ncludng that of EPSRC, s gratefully acknowledged. Fully detaled techncal reports on ths research are avalable to Industral Members of Moble VCE. The authors would also lke to thank for the support from the Scottsh Fundng Councl for the Jont Research Insttute wth the Herot-Watt Unversty whch s a part of the Ednburgh Research Partnershp. [12] S. Ryu, B. Ryu, H. Seo, and M. Shn, Urgency and effcency based packet schedulng algorthm for OFDMA wreless system, n Proceedngs IEEE Internatonal Conference on Communcatons ICC, May 25, vol. 4, pp [13] R. Knopp and P. A. Humblet, Informaton capacty and power control n sngle-cell multuser communcatons, n Proceedngs IEEE Internatonal Conference on Communcatons ICC, Seattle, WA, USA, June 1995, pp [14] P. Vswanath, D. Tse, and R. Laroa, Opportunstc beamformng usng dumb antennas, IEEE Transactons on Informaton Theory, vol. 48, no. 6, pp , June 22. [15] B. Borst and P. Whtng, Dynamc rate control algorthms for HDR throughput optmzaton, n INFOCOM, Apr. 21, vol. 2, pp [16] L. Yang and M. Aloun, Performance analyss of multuser selecton dversty, IEEE Transactons on Vehcular Technology, vol. 55, no. 6, pp , Nov. 26. [17] D. Park, H. Seo, H. Kwon, and B. Lee, Wreless packet scheulng based on the cumulatve dstrbuton functon of user transmsson rates, IEEE Transactons on Communcatons, vol. 53, no. 11, pp , Nov. 25. [18] T. Nguyen and Y. Han, A proportonal farness algorthm wth QoS provson n downlnk OFDMA systems, IEEE Communcatons Letters, vol. 1, no. 11, pp , Nov. 26. [19] Y. Ma, Proportonal far schedulng for downlnk OFDMA, n Proceedngs IEEE Internatonal Conference on Communcatons ICC, June 27, pp [2] A. Goldsmth, Wreless Communcatons, Cambrdge Unversty Press, 25. [21] J. G. Proaks, Dgtal Communcatons, McGraw-Hll Internatonal Edtons, thrd edton, REFERENCES [1] Y. Cao and V. L, Schedulng algorthms n broad-band wreless networks, Proceedngs of the IEEE, vol. 89, no. 1, pp , Jan. 21. [2] D. Avdor, S. Mukherjee, J. Lng, and C. Papadas, On asymptotcally far transmsson schedulng over fadng channels wth measurement delay, IEEE Transactons on Wreless Communcatons, vol. 5, no. 7, pp , July 26. [3] P. Svedman, S. Wlson, L. Cmn, and B. Ottersten, Opportunstc beamformng and schedulng for OFDMA systems, IEEE Transactons on Communcatons, vol. 55, no. 5, pp , May 27. [4] P. Bender, P. Black, M. Grob, R. Padovan, N. Sndhushayana, and A. Vterb, CDMA/HDR: A bandwdth-effcent hgh-speed wreless data servce for nomadc users, IEEE Communcatons Magazne, vol. 38, no. 7, pp. 7 77, July 2. [5] L. L and A. Goldsmth, Capacty and optmal resource allocaton for fadng broadcast channels Part 1: Ergodc capacty, IEEE Transactons on Informaton Theory, vol. 47, no. 3, pp , Mar. 21. [6] D. Tse and S. Hanly, Multaccess fadng channels Part 1: Polymatrod structure, optmal resource allocaton and throughput capactes, IEEE Transactons on Informaton Theory, vol. 44, no. 7, pp , Nov [7] Z. Shen, J. Andrews, and B. Evans, Adaptve resource allocaton n multuser OFDM systems wth proportonal rate constrants, IEEE Transactons on Wreless Communcatons, vol. 4, no. 6, pp , Nov. 25. [8] X. Lu, E. Chong, and N. Shroff, Opportunstc transmsson schedulng wth resource-sharng constrants n wreless networks, IEEE Journal on Selected Areas n Communcatons, vol. 19, no. 1, pp , Oct. 21. [9] M. Shaqfeh and N. Goertz, Channel-aware schedulng wth resourcesharng constrants n wreless networks, accepted for publcaton n Proceedngs IEEE Internatonal Conference on Communcatons ICC, 28. [1] G. Song and Y. L, Utlty-based resource allocaton and schedulng n OFDM-based wreless broadband networks, IEEE Communcatons Magazne, vol. 43, no. 12, pp , Dec. 25. [11] Z. Han, X. Lu, Z. Wang, and K. Lu, Delay senstve schedulng schemes for heterogeneous QoS over wreless networks, IEEE Transactons on Wreless Communcatons, vol. 6, no. 2, pp , Feb. 27. SPECTS ISBN: Authorzed lcensed use lmted to: Unverstatsbblothek der TU Wen. Downloaded on May 29, 29 at 4:19 from IEEE Xplore. Restrctons apply.