Measurement-Based Opportunistic Feedback and Scheduling for Wireless Systems

Transcription

1 Measuremet-Based Opportuistic Feedbac ad Schedulig for Wireless Systems Gustavo de Veciaa ad Shailesh Patil Wireless Networig ad Commuicatios Group Dept. of Electrical ad Computer Egieerig The Uiversity of Texas at Austi Austi, TX Abstract We study opportuistic feedbac of chael state ad schedulig of users trasmissios at a wireless access poit. Our focus is o a regime where users have uow possibly slowly-varyig chael characteristics ad it is desirable to limit the resources expeded to exchagig chael state opportuistic. The paper argues that opportuistically schedulig the user whose curret rate is i the highest quatile with respect to a empirical distributio has substatial advatages. Ideed by comparig to other measuremet-based opportuistic schedulig schemes it systematically achieves high opportuism ad thus throughput i fair ad robust maer. Further the throughput pealty resultig from estimatig users distributios is limited as log as the umber of idepedet samples grows liearly with the umber of active users. Secod, a maximum quatile based schedulig scheme is cosistet the use of a simple opportuistic feedbac strategy that ca be used to substatially reduce the overheads associated with decidig which user should i fact be scheduled at ay poit i time. Usig both aalytical ad simulatio results we will show that the beefits ca be substatial for a variety of practical wireless system scearios. 1 Itroductio Motivatio. The schedulig of users data trasmissios at a wireless access poit has recetly attracted a substatial amout of attetio, see e.g., [6][15][4]. A ey feature of wireless systems relative to the traditioal wirelie systems is that, the chael capacity, or service rate, may exhibit temporal variatios. This allows oe to cosider schedulig policies that choose to sed to, or receive from, a user (or a subset of users) which at a give poit i time has (have) the best, e.g., highest, capacity. Such opportuistic schedulig ca lead to good icreases i the aggregate capacity of a wireless system, ad has thus bee adopted i various wireless stadards such as CDMA-HDR, HSDPA [2][1], ad will almost certaily play a role i future wireless systems. Wheever a access poit maes a opportuistic decisio o the user(s) to serve, it eeds to ow the curret chael capacity (or a fuctio of it) for all (or a subset) of the users. Therefore, before each decisio all users eed to trasmit their curret chael coditio to

2 the access poit. This ca be a huge overhead, compared to the gai i throughput due to opportuistic schedulig. For example, cosider a system where all users see i.i.d. (idepedet, idetically distributed) Rayleigh chael capacity fadig. Here, the gai due to opportuistic schedulig grows at most logarithmically with the umber of users i the system, while the amout of feedbac, i.e., the umber of trasmissios icreases liearly with the umber of users. This clearly uderlies the eed for reducig the amout of feedbac i opportuistic schedulig. Related Wor. Several researchers have studied the problem of reducig feedbac i opportuistic schedulig. A simple threshold based scheme was proposed i [5]. There, the idea was to allow oly users with curret chael capacity above a threshold to feedbac their curret state. This, it was show, reduced the amout of feedbac sigificatly for a small reductio i the overall throughput achieved. However, there it was assumed that the resources used to feedbac chael capacity were ot shared amog users. Whereas, the issue of reducig feedbac becomes more relevat whe the resources used for feedbac are shared amog users, for example i a TDMA system where the time used i collectig the feedbac grows liearly with the umber of users. I this paper we will focus o such a setup. I [13], a opportuistic splittig algorithm is proposed i a TDMA set up for upli schedulig (although it ca be easily modified to dowli schedulig). There, each data trasmissio was preceded by mii slots, which are used to lear the curret chael capacity of users via feedbac. Oce the user with highest chael capacity is idetified, data trasmissio esues. I opportuistic splittig, iitially a pair of thresholds depedig o the umber of users is set. At the start of the first mii slot, every user with curret chael capacity betwee the pair of thresholds coted, i.e., trasmit to the access poit. The access poit the broadcasts to all the users whether o user coteded, exactly oe user coteded, or a collisio occurred, i.e., more tha oe user coteded ad the access poit was uable to decode ay iformatio. Depedig o the broadcast message received, each user modifies his threshold accordig to a biary search lie algorithm ad users whose chael capacity is betwee the ew thresholds coted i the ext mii slot. This process cotiues util exactly oe user coteds, therefore the umber of mii slots before a trasmissio vary. This user is guarateed to be the user with the highest curret chael capacity. It was show that a average of oly 2.5 mii slots are required for the algorithm to fid the user with the highest curret chael capacity. This is sigificat as compared to liear umber of slots required for a aive feedbac scheme. Opportuistic splittig requires two way commuicatio ad coordiatio betwee the access poit ad users i every mii slot. This may be a high overhead sice the time scales ivolved are quite small. (I practical systems, a slot is of the order of millisecods, therefore each mii slot has to be smaller tha a millisecod.) To overcome this coordiatio problem, a radom access based feedbac protocol was proposed i [16], where oly users trasmit. Here, each data trasmissio is preceded by a fixed umber of mii slots (the smaller the umber of mii slots, the lesser the time used i feedbac). I each mii slot, users with curret chael capacity above a threshold coteded with some probability. If i a mii slot exactly oe user coteded, the that user s idetity is stored at the access poit, ad the access poit radomly serves oe of the idetified users. However, if o user is idetified a user is selected at radom. The threshold ad probability were optimized to maximize the overall sum capacity. However, simulatio results preseted i the paper show that opportuistic splittig performs better tha the proposed scheme. The above described wor assume that all users i the system see i.i.d. chael capacity distributio. Furthermore, to set the thresholds correctly, it is assumed that the chael capacity

3 mii slot trasmissio slot mii slots Figure 1: Structure of a time uit. distributios are either ow at the access poit, or by the users. I practice both of the assumptios are uliely to be true. (Note that a extesio of opportuistic splittig to the case where users ca experiece oe of two possible chael capacity distributios is preseted i [14], however this is also ot a reasoable assumptio.) I practice users chael capacity variatios are heterogeous, e.g., users close to a access poit see sigificatly differet chael capacity tha those further off. Also, the chael capacity distributios are usually uow (both at the access poit ad by users) ad eed to be estimated via measuremet. Therefore, oe eeds to evaluate the pealty icurred due to estimatio. Cotributios. I this paper we propose a threshold based scheme ow as static splittig to reduce the amout of feedbac. We will cosider a TDMA setup similar to that i [13][16]. However ulie opportuistic splittig, static splittig requires oly oe way commuicatio. We will combie static splittig with a distributio based scheduler that we call maximum quatile schedulig to hadle heterogeeity i users chael capacity variatios. Maximum quatile schedulig has bee proposed by several researchers uder differet guises [8][9][3][14], the idea is to schedule the user whose curret rate is highest relative to his ow distributio, i.e., i the highest quatile. Maximum quatile schedulig allows the access poit to calculate a commo threshold idepedet of users distributio for our scheme. I this paper we also develop isight ito throughput pealty due to estimatio errors usig a distributio free boud developed i [11]. Fially, simulatio results idicate that static splittig ca perform better tha a trucated form of opportuistic splittig. Paper Orgaizatio. This paper is orgaized as follows. I Sectio 2 we give a brief itroductio to maximum quatile schedulig ad describe the proposed scheme. We discuss the throughput pealty icurred due to estimatio errors i Sectio 3. Simulatio results are preseted i Sectio 4. Sectio 5 cocludes the paper. 2 Static Splittig 2.1 System Model ad Notatio We begi by itroducig our system model ad some otatio. For simplicity, we focus o dowli schedulig from a access poit to multiple users. Suppose time is divided equal sized time uits. Each time uit cosists of equal sized mii slots (for collectig feedbac) followed by a trasmissio slot durig which at most oe user ca be served (see Figure 1). I the sequel we use the terms chael capacity ad rate iterchageably ad mae the followig assumptio o user s chael characteristics over data trasmissio time slots.

4 Assumptio 2.1 We assume the chael capacity (rate) for each user is a statioary ergodic process ad these processes are idepedet across users. Further we assume that the margial distributio for each user is ow a priori at the user. Some commets o this assumptio. First the chael capacities see by users might ideed be roughly statioary over a reasoable period of time particularly if users are at fixed locatios. The assumptio that users rates are idepedet is also liely to be true, though a otable exceptio is the case where mobile users move i a correlated maer, e.g., alog a highway. The assumptio that a user ows, ad i particular ca estimate, the margial distributios of the chael capacity processes may ot be completely reasoable, but simple boo eepig of the curretly achievable rate ca be used to estimate distributios. We will discuss estimatio of such distributios i Sectio 3. Note that chael capacities are ot restricted to ay specific distributio, or class of distributios, i.e., users ca udergo ay fadig process. This maes the aalysis preseted later applicable to real world scearios. I the sequel we will let x i (t) deote the realizatio of the dowli chael capacity of user i at time slot t, ad let X i be a radom variable whose distributio is that of the chael capacity of user i o a typical slot. Recall that we will be assumig X i to be idepedet across users but eed ot be idetically distributed. We deote the distributio fuctio of X i by F X i( ). Note that by assumptio F X i( ) is ow at the user. For aalysis purposes, i this paper we will oly cosider a fixed saturated regime where the umber of users i the system stay fixed with time, ad each user i the system is ifiitely baclogged. The umber of users i the system are deoted by. Before explaiig the proposed scheme, we give a brief itroductio to maximum quatile schedulig. 2.2 Maximum Quatile Schedulig As discussed earlier, maximum quatile schedulig has bee proposed idepedetly by several researchers. Specifically [8][9] proposed a CDF based scheme. While [3] proposed a so called score based scheduler ad [14] proposed a distributio fairess based scheduler. We have studied the properties of maximum quatile schedulig uder greedy user behavior i [12], i [10], we evaluate its use to achieve quality of service guaratees for real-time traffic, ad i [11] we evaluate its performace i a measuremet based set up. The mai idea of the scheme is to schedule a user who s rate is highest compared to his ow distributio, i.e., serve user l(t) durig slot t if l(t) arg max i=1,..., F X i(xi (t)). It is well ow that F X i(x i ) is uiformly distributed o [0, 1]. Let U i = F X i(x i ), the U i is also uiformly distributed o [0, 1]. Maximum quatile ca be thought of as picig the maximum amog idepedet realizatios of users (i.i.d.) U i s o every slot. Thus, it is clear that maximum quatile is equally liely to serve ay user o a typical slot, ad as a result all users get served a equal fractio of time, i.e., 1 th of time. Let U () = max[u 1,..., U ], where U j is uiformly distributed o [0, 1] j = 1,...,, the Pr(U () u) = u, u [0, 1]. (1)

5 The the rate distributio see by user i o a slot that it gets served is the same as F 1 X i (U () ). Therefore, the average throughput see by user i is G i mq() = E[F 1 X i (U () )] = E[Xi,() ]. (2) where X i,() is maximum of i.i.d. copies of X i, i.e., X i,() := max[x i 1,..., X i ], where X i j X i, j = 1,...,. Compared to other schemes proposed i literature, maximum quatile schedulig is foud to be very useful i a realistic sceario, i.e., a sceario where users have heterogeeous chael capacity distributios, ad parameters/weights associated with a schedulig scheme have to be estimated via measuremet. Maximum quatile ot oly maximizes the amout of opportuism exploited while beig fair, but is also quite robust to measuremet errors (this will be discussed later also). I fact maximum quatile schedulig does better tha the sum throughput optimal strategy proposed i [7], we refer the reader to [11] for details. Therefore i our proposed scheme we try to serve a user with high quatile, i.e., high F X i(x i (t)). 2.3 Proposed Scheme Recall that a slot cotaied mii slots used to collect feedbac. Usig these mii slots we try to idetify a user whose curret rate is i a high quatile, i.e., high F X i(x i (t)), so as to maximize the total quatile served. (Note that we do ot ecessarily try to idetify the user with the highest quatile, we will revisit this poit later.) Apart from the several advatages of selectig a user with high quatile discussed above, we will show i sequel, it also simplifies threshold calculatio. For simplicity assume that the umber of users is such that a itegral value. I the proposed scheme, a user is associated with exactly oe mii slot, with users associated with each mii slot. A user ca coted for the slot (i.e., sed its feedbac) oly o the mii slot with which it is associated. I other words, users are split ito static groups, with users belogig to the same group allowed to coted oly i the same mii slot. This reduces the chaces of collisio i a mii slot. Oce a user is associated with a mii slot, it calculates a quatile threshold deoted by q i, i = 1,...,. We will give the details of calculatig q i later. Cosider slot t, recall that the rate supported by user i durig slot t is deoted by x i (t). At the start of a mii slot, durig the first mii slot all users associated with it ad havig curret rate above F 1 X i (q i ) coted, i.e., if x i (t) > F 1 X i (q i ) they trasmit the quatile of their curret rate F X i(x i (t)) to the access poit. If exactly oe user coteds, we assume that the access poit is able to idetity the user that coteded, ad the value of its quatile, ad store this iformatio. If more tha oe user coteds, i.e., a collisio occurs, the the access poit stores the fact that there was a collisio o the first mii slot. If o user coteds, the o actio is tae by the access poit. This process of cotedig is repeated i all the subsequet mii slots. Oce the cotedig for mii slots is fiished, if the access poit was able to idetify at least oe user, it serves the user with the highest quatile amog the idetified users. However, if it fails to idetify such a set, it chooses to serve a radomly selected user. This ca occur i two ways, if collisios happeed i oe of the mii slots, the a user is radomly selected from all the users. However, if a collisio occurred o at least oe of the mii slot, the a user is radomly selected amog the groups of users that could have coteded o the mii slots o which collisios occurred. This icreases the chace of fidig a user with a high quatile. We ow discuss calculatig the threshold q i. Let S i be the evet deotig the selectio of user i for service, ad 1 S i be the idicator fuctio for S i. Recall that we would lie to

6 adjust the thresholds q i s so as to maximize the overall sum quatile of the users served, i.e., E[ i=1 F X i(xi )1 S i]. Recall that F X i(x i ) are i.i.d. across users, therefore each user is equally liely to achieve a high quatile. Therefore to maximize E[ i=1 F X i(xi )1 S i], the scheduler is equally liely to select a user, i.e., i, Pr(S i ) = 1. Also, the maximum umber of users that ca coted i ay mii slot is. These symmetries allows us to have a commo threshold q across users, i.e., q i = q, i = 1,...,. Now cosider E[ F X i(x i )1 S i] = i=1 E[F X i(x i ) S i ] Pr(S i ) = 1 i=1 E[F X i(x i ) S i ]. i=1 Agai by symmetry, E[F X i(x i ) S i ] is equal across users, therefore it is sufficiet to choose a value of q that maximize E[F X i(x i ) S i ] for ay user i. For simplicity, from hereo we will drop the super script i otatio deotig the user i this subsectio. Note that i our scheme, eve if a user is correctly idetified by the access poit ad selected for service, it may ot have the highest quatile. Let S j, j =,..., deote the correct idetificatio ad selectio of user whe it has the j th highest quatile. (Note that a user has to be at least the highest i its group to to be idetified, therefore the miimum value of j is.) Let S r deote radom selectio of a user (whe o user has bee idetified by the access poit). The E[F X (X) S] = E[F X (X) S j ] Pr(S j ) + E[F X (X) S r ] Pr(S r ). (3) j= Now ad Pr(S ) = 1 q u 1 ( q u ) 1 du = E[F X (X) S ] E[U () ] = (1 1 ) + 1(q 1 q ), (4) + 1, (5) where U () is as defied i (1). Furthermore sice we will be taig maximum amog quatiles of successfully idetified users with quatiles greater tha q, j =,...,, E[F X (X) S j ] 1 + q 2. (6) Note that if the value of q is high, the the above iequality ca be treated as a approximate equality. The from (5) ad (6), oe ca rewrite (3) as E[F X (X) S] (1 1) + 1(q 1 q ) Pr(S ) q 1 Pr(S j ) + E[F X (X) S r ] Pr(S r ). 2 (7) Now 1 j= Pr(S j ) is the probability that the access poit is uable to idetify the user with the highest quatile, but is able to idetify at least oe user i the remaiig 1 mii slots that do ot cotai the user with the highest quatile. Let p s = (1 q)q 1 j=

7 deote the probability of a successful trasmissio i a typical mii slot. The oe ca approximate 1 j= Pr(S j ) as 1 Pr(S j ) (1 Pr(S ))(1 (1 p s ) 1 ). (8) j= Fially Pr(S r ) = 1 Pr(S j ), (9) j= ad we ca approximate E[F X (X) S r ] as E[F X (X) S r ] 1 2, (10) i.e., the average quatile of the selected user whe it is selected completely radomly (ad ot coditioed o the access poit beig uable to idetify ay user). The (7) ca be further rewritte as E[F X (X) S] Pr(S ) (1 1) + 1(q 1 q ) q 1 Pr(S j ) Pr(S r), (11) where the probabilities are defied i (4), (8) ad (9). Recall that we wated to fid the value of threshold q that maximized the value of sum quatile of users served. This ca be obtaied umerically by searchig over [0, 1] ad fidig the value of q that maximized (11). I Figure 2 we plot the variatio of optimum threshold q for a icreasig umber of users for = 5, 7, 9. The threshold icreases with for a give, ad with for a give, this is expected. Note that (11) is idepedet of users chael capacity distributios. The expressio derived i (11) ca also be used to fid a approximate value of q eve whe is ot a itegral value. Some fial commets, oe observes that opportuistic splittig ca also be combied with maximum quatile schedulig to hadle heterogeeity i users chael capacity. However, opportuistic splittig was desiged to oly fid the user with the highest quatile/rate. I a practical system the umber of mii slots may be limited, ad if the scheme is uable to fid the user with the highest quatile i those may mii slots, a user has to be chose at radom. This is ot desirable. Whereas if static splittig is usuccessful i fidig the user with the highest quatile, it tries to serve a user with high quatile. The possibility of servig a high quatile user (ad ot the highest) is captured i (11), i fact the expressio also captures the performace of the scheme eve whe a user is selected at radom. Therefore maximizig (11), ca lead to better performace as compared to opportuistic splittig. We will verify this i Sectio 4. 3 Pealty due to Estimatio Errors As discussed earlier, opportuistic splittig ad the scheme proposed i [16] assume that the chael capacity distributios of users are either ow to the users or are ow at the access poit. This assumptio is also required for static splittig (rate distributio has to be ow by j=

8 Optimal threshold q = 5 = 7 = Number of users associated with a mii slot / Figure 2: Variatio of optimal threshold q with ad. the user). However this is uliely, ad oe has to estimate the distributio via measuremet. Therefore, oe eeds to evaluate the pealty icurred due to errors i estimatio. Suppose the quatile of the curret rate of a user is estimated usig the previous m samples of the user s rate. The empirical distributio of user i durig slot t based o m previous samples m,t is deoted by F ( ) ad is give by X i F m,t (x) = 1 X i m m 1{X i (t j) x}. (12) j=1 Note that the above way of estimatig is similar to the score fuctio described i [3]. Thus maximum quatile schedulig of users based o estimated distributios, would choose user l(t) for service durig slot t if l(t) arg max i=1,..., F m,t X i (x i (t)), (13) with ties beig broe arbitrarily. It ca be show that eve with estimated distributios as give i (12), maximum quatile schedulig will serve each user a equal fractio of time. Recall that the G i mq() (2) is the throughput experieced by user i uder maximum quatile schedulig with whe chael capacity distributio is perfectly ow. Let Gi mq (, m) be the throughput experieced by user i uder maximum quatile schedulig whe users rate distributio is estimated accordig to (12). We ow state a Theorem from [11] that gives a distributio idepedet boud o the relative error betwee G i mq() ad G i mq(, m).

9 Theorem 3.1 Cosider a fixed saturated system with users whose chael capacity variatios satisfy Assumptio 2.1. Suppose the chael capacity distributios i such a system are estimated via (12) base o m idepedet samples of a user s chael ad users are served usig maximum quatile schedulig, the G i mq() G i mq(, m), m, ad the relative throughput pealty is bouded by G i mq() G i mq(, m) G i mq() 1 m + 1 (1 ( m m + 1 ) ). Opportuistic splittig attempts to idetify the user with the highest quatile. If the distributios are estimated, the scheme will mae the same mistaes as made by the access poit uder (13). Therefore Theorem 3.1 also applies to opportuistic slittig. Oe ca show that for reasoably large, the above stated boud icreases with. Now i static splittig, we may ot select the highest user, but we will be selectig the highest amog, 2,..., users (i.e., the highest i a mii slot, or the highest amog two mii slots ad so o). Also, because there is o relative pealty icurred whe users are selected at radom, the overall relative pealty icurred by static splittig is liely to be lower tha that idicated by the boud, or lower tha that icurred by opportuistic splittig. (Note that static splittig selects a user oly if its curret quatile is above a threshold, therefore Theorem 3.1 does ot directly apply to it.) We verify this cojecture i Sectio 4. 4 Simulatio Results We simulated the proposed scheme to observe its performace. We first describe the setup used for simulatios. For simplicity, we assume that all users udergo i.i.d. Rayleigh fadig with a mea SNR of 2. The badwidth associated with each user is 500 KHz ad we assume that codig achieves the Shao rate. We set = 5, ad icrease the umber of users from 5 to 35 i steps of 5, i.e., the maximum umber of users that ca coted i a mii slot are equal across slots ad icrease from 1 to 7. The threshold q is set as discussed i Sectio 2. We also compared our scheme s performace with that of opportuistic splittig. Note that a mii slot i opportuistic splittig cosists of two trasmissios, whereas a mii slot i our scheme cosists of oly oe trasmissio. Also ote that i our scheme, at the ed of mii slots a extra trasmissio is required from the access poit iformig the user it has selected for service, i.e., i total + 1 trasmissios are required. Therefore to be fair, we compare our scheme with a trucated form of opportuistic splittig where at most (+1) = 3 mii slots 2 are used. At the ed of 3 mii slots if the algorithm is uable to fid the user with the highest quatile, the it selects a user at radom. Note that 3 is greater tha the average of 2.5 slots eeded for the scheme to coverge. We performed two experimets, i the first experimet we observed the throughput performace of the schemes whe the chael capacity distributios are perfectly ow. We plot the loss i throughput (due to lower feedbac) relative to the case whe the user with the highest quatile is ow i Figure 3. It is clear from the figure that static splittig outperforms opportuistic splittig. Whe there are 10 users i the system, static splittig has a 40% lower loss compared to opportuistic splittig.

10 0.12 static splittig opportuistic splittig 0.1 Percetage loss i throughput Total umber of users Figure 3: Relative throughput loss due to low feedbac uder static splittig ad opportuistic splittig with icreasig umber of users. I the secod experimet we study the pealty suffered due to estimatig chael capacity distributios. The distributios were estimated usig m = 100 idepedet previous samples. We plot the throughput pealty compared to perfectly ow distributios case for both static splittig ad the trucated opportuistic splittig i Figure 4. As expected static splittig performs better. 5 Coclusio I this paper we preseted a simple scheme for reducig feedbac i opportuistic schedulig etwors. The scheme is ovel i the sese that oe ca calculate the thresholds idepedet of users distributio maig it very much applicable to real world scearios. We also develop isights ito the loss icurred whe rate distributios are estimated. Ulie previous wor our approach is focussed o fidig a user with high quatile, ot ecessarily the user with the highest quatile. The advatage of such a approach is that it ca lead to better overall performace. This is verified by simulatio results. Refereces [1] 3GPP TS UTRA high speed dowli pacet access (HSDPA): Overall descriptio, Release

11 Relative throughput pealty icurred due to estimatig distributios static splittig opportuistic splittig Total umber of users Figure 4: Relative throughput pealty due to estimatio error uder static splittig ad opportuistic splittig.

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