Joint Scheduling of Rate-guaranteed and Best-effort Services over a Wireless Channel

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1 Jont Schedulng of Rate-guaranteed and Best-effort Servces over a Wreless Channel Murtaza Zafer and Eytan Modano Abstract We consder mult-user schedulng over the downln channel n wreless data systems. Specfcally, we consder a sngle transmtter servng multple moble users wth tmevaryng channel condtons. Wth fxed power transmsson, the channel condton of a user determnes the relable rate of communcaton. We consder a set of throughput guaranteed qualty of servce QoS users and obtan the optmal polcy that serves these users wth the least tme-slot utlzaton; thereby, maxmzng the remanng fracton of tme-slots allocated to the users wth best effort BE servce. We present a smple geometrc vsualzaton of the optmal polcy. Under a symmetrc Raylegh fadng model we obtan explct formulas that relate the varous system parameters such as the achevable throughput rate guarantee, the number of QoS users supportable and the fracton of tme-slots allocated to the BE users. Fnally, we compare the throughput results for the optmal polcy wth the random-schedulng polcy and show that gans on the order of ln can be acheved by explotng mult-user dversty. Index Terms Downln schedulng, Mult-user dversty, Optmzaton, Stochastc processes, Wreless fadng channel. I. ITRODUCTIO Rapd growth of the nternet and mult-meda applcatons has created an ever ncreasng demand for wreless data systems. Development of such systems, for example the xev-do system n [3], ntroduces numerous new challenges n provdng Qualty of Servce QoS over a wreless channel. Frst, n contrast to connecton-orented traffc, such as voce, data streams are nherently bursty and can tolerate much hgher delays. Hence, reservng resources to provde QoS s neffcent whch means that to share a common resource one needs effcent schedulng algorthms. Second, the wreless channel s tme-varyng and one can explot the varyng channel condtons among varous users to ncrease the system throughput. In the lterature, such an approach s referred to as Opportunstc schedulng [], [2], [4] or explotng Mult-user dversty [6]. In ths wor, we study optmal schedulng that explots mult-user dversty. We consder the downln scenaro wth a sngle server that represents the base staton and multple users that represent the moble handsets. The set of users are dvded nto two classes: throughput rate guaranteed QoS users and best effort BE users. The QoS users have hgh prorty servce and are guaranteed expected throughput rates f these rates are feasble; whle, the BE users have a Murtaza Zafer {murtaza@mt.edu} and Eytan Modano {modano@mt.edu} are wth the Laboratory for Informaton and Decson Systems, Department of Electrcal Engneerng and Computer Scence, Massachusetts Insttute of Technology, USA. Ths wor was supported by SF ITR grant CCR-3254, by DARPA/AFOSR through the Unversty of Illnos grant no. F and by ASA Space Communcaton Project grant number AG low prorty servce and are served when the resources are avalable. The goal of ths wor s to desgn a schedulng polcy that serves the QoS users wth the least tme-slot utlzaton so as to maxmze the remanng fracton of tmeslots avalable for the BE users. Down-ln schedulng s an actve area of research wth recent wor that ncludes [], [2], [4], [5]. The wor n [] presented varous formulatons based on utlty maxmzaton. The wor n [2] consdered the objectve of maxmzng the mnmum throughput rate, [4] maxmzed the throughput wth farness constrants whle [5] presented algorthms wth delay consderatons. Our wor dffers n presentng a smple formulaton that combnes the QoS and the BE users by abstractng the servce of BE users as the fracton of allocated tme-slots. We gve the optmalty condtons and show that a polcy s optmal f and only f t satsfes a certan smple geometrc structure. Under symmetrc Raylegh fadng, we obtan explct formulas relatng the system parameters that nclude the achevable throughput rate guarantee, the number of QoS users supportable and the fracton of tme-slots assgned to the BE users. Fnally, we analytcally compare the optmal and the random-schedulng polcy and quantfy the gans acheved by explotng mult-user dversty. II. SYSTEM AD PROBLEM DESCRIPTIO A. System Model We consder the wreless downln scenaro,.e. communcaton from the base staton to the moble handsets, n a tme slotted system. There are multple users n the system, each user experencng tme varyng channel condtons. The channel state of a user s assumed constant n a sngle tme slot but vares over multple tme slots. We assume that the underlyng stochastc process drvng the channels states s statonary. Ths, however, does not preclude the possblty of channel correlatons over tme and among users. At the begnnng of a tme-slot, the transmtter nows the channel state of each user for that partcular slot. In a tme-slot, t serves at most one user wth full power P. Snce the users have dfferent channel condtons the relable rate of communcaton per tme slot to the users s varable. Clearly, the transmtter can explot ths varablty and select the best user for transmsson n a tme-slot based on some performance measure. The above system models a TDMA system and the recently proposed xev-do data system [3] and s a commonly used model n the lterature to study opportunstc schedulng n wreless networs [], [2], [4]. Ths s a smplfyng assumpton that models one step channel predcton

2 Let r = {r } denote the vector of relable rate of communcaton to the users n a generc tme-slot, say for example the th tme-slot. Ths means that f user s chosen to be served n tme-slot, the throughput for that user s smply r. The transmtter has nowledge of r at the begnnng of slot but does not now ths vector for future slots. Let Ω be the set comprsng of all possble rate vectors. In the th tme-slot, r s a partcular realzaton from the set Ω whch has a probablty dstrbuton nduced by the underlyng stochastc model of the channels states. A schedulng polcy, denoted as Γ r, s a rule that specfes whch user the transmtter serves n tme-slot. A statonary schedulng polcy, denoted Γ r, s one that does not depend on the tme ndex and can be represented as a map from the set Ω to the user ndex;.e. each r Ω s mapped to a unque user ndex. As the underlyng processes are statonary, t s well-nown that a statonary optmal polcy exsts, hence, t suffces to focus on statonary polces. In the rest of the paper, a schedulng polcy refers to the above map. Let X denote the throughput per tme-slot of user, then, { r, f Γ r =.e. user selected X =, otherwse The expected throughput per tme slot s E[X ]. Under ergodcty of the channel process and statonarty of the schedulng rule, t s well nown that E[X ] equals the long term throughput per slot called throughput rate of user. B. Problem Descrpton As mentoned earler, the set of users are dvded nto two prorty classes: the throughput rate guaranteed QoS users and the best effort BE users. The QoS users are guaranteed expected throughput rates whle the BE users have no such guarantees. Let there be QoS users that are guaranteed throughput rates R = R,.., R, f such a vector s feasble. By feasblty we mean that there exsts a schedulng polcy such that E[X ] R, =,..,, where X s defned as n. The objectve, now, s to serve the QoS users wth the least tme-slot utlzaton and share the remanng tme-slots among the BE users. Ths objectve provdes a smple and tractable way of ntegratng the two classes of servce. Also, typcally n most practcal systems, the populaton of BE users s large and a natural objectve whle servng such users s smply maxmzng the sum-throughput. Clearly, under a large populaton of BE users, maxmzng the tme-slot allocaton s equvalent to maxmzng the total throughput of such users 2. Let I r be the ndcator functon for selecton of user, {, f Γ r = I = 2, otherwse Wth ths notaton we can re-wrte X as X = r I. The 2 Tme slots allocated for BE users can be shared n a greedy fashon, thus, maxmzng the sum throughput of these users. optmzaton problem can now be stated as follows, mn E[I ] subject to E[r I ] R, =,.., 3 where the expectaton s taen over the jont dstrbuton of r for the QoS users. ote that mnmzng E[I ] s equvalent to maxmzng E[I ] whch equals the fracton of tme-slots avalable for the BE users. We assume that R >,.e. R >,.., R >. If some R =, we can neglect that user and the problem reduces to dmensons. We assume that R s feasble and away from the boundary of the set comprsng all achevable throughput rate vectors. Ths assumpton s solely to smplfy the mathematcal exposton by avodng the lmtng condtons at the boundary and does not affect the results presented throughout ths paper. III. OPTIMAL POLICY The QoS users experence dfferent tme-varyng channel condtons, hence, ntutvely the optmal polcy must explot the varable communcaton rates to the users by selectng the best user to have a hgh throughput per tme slot. The choce of whch user to serve must also account for the dfferent throughput rate guarantees among users and ther varyng channel statstcs. Clearly, for optmalty the nequalty n 3 must also be met wth equalty. Let r = r,.., r be the rate vector n a generc tme-slot for the QoS users 3 ; ths vector les n the set Ω R +. Let the jont probablty densty functon be f r such that the probablty of some regon Z Ω s gven as Z f rd r. The restrcton on f r s that subsets wth zero volume n Ω or ndvdual ponts have zero probablty. Snce a schedulng polcy maps r Ω to a unque user ndex, we can represent t as a partton of the set Ω nto + regons denoted as Z,.., Z, Z f. In a partcular tme-slot say slot, f the transmsson rate vector r Z, user s selected for servce whereas f r Z f, no QoS user s selected and the slot s used to serve the BE users. The problem thus reduces to choosng these regons optmally to mnmze the objectve functon and satsfy the throughput rate constrant,.e. Z r f rd r R, =,..,. As ndvdual ponts n Ω have zero probablty, we wll refer to regons wthn Ω 4. The notaton r Z r Z means that there s a neghborhood around r that les does not le n Z. Formally, there exsts ɛ > such that ˆr Ω, ˆr r < ɛ ˆr Z. The followng lemma gves the necessary condton for the optmalty of regon Z f. It states that for optmalty f r s mapped to Z, all rate vectors wth th component larger than r cannot be mapped to Z f. Lemma : Under the optmal polcy, suppose r = r,.., r Z then ˆr = ˆr,.., ˆr > r,.., ˆr Z f. 3 To mae the notatons smple, r, dependng on the context denotes a random vector and also a partcular realzaton for a generc tme-slot. 4 Regons wth zero probablty densty can be removed from Ω as ther mappng does not affect optmalty.

3 r 2 r 2 Z f regon r 2 /a 2 = r 3 /a 3 Z 2 r /a = r 2 /a 2 a 2 Z Z 3 a r a a 3 r r 3 r 3 a 3 Zf r /a = r 3 /a 3 Fg.. The Z f regon for = 3, threshold vector ā = a, a 2, a 3 and Ω = R +. ote Zf = { r : r, =,..., }. Fg. 2. Optmal polcy structure for = 3, threshold vector ā = a, a 2, a 3 and Ω = R +. The Z regons are bottom truncated cones. Proof: We omt a rgorous proof for brevty but the man des that f there s a ˆr Z f wth ˆr > r then we can re-map the regons such that the objectve functon n 3 decreases. Ths s acheved by mappng a small neghborhood of ˆr to Z and mappng a neghborhood of r to Z f whle ensurng that the throughput constrants stll hold. As ˆr > r one can show that objectve functon under the new map s strctly lower than the earler map. Interestngly, Lemma mples a specal structure on Z f as follows. Let a be the nfmum value of the frst component among all vectors r Z ;.e. a = nf r Z r. ow, any ˆr Z f must be such that ˆr a ; otherwse the above lemma wll be volated. As ths holds for all Z, the optmal polcy s such that there exsts constants { } such that f r, then r Z f. The regon Z f s shown n Fgure. Ths mplcaton s qute ntutve as t suggests that when the rate vector of the QoS users s below some threshold vector bad channel condtons, the QoS users must not be scheduled and the slot must be used to serve the BE users. The vector ā depends on the throughput guarantees, R and the densty functon f r. Gven that R does not le on the boundary of feasble throughput rates, t follows that ā s at least a postve vector a >,.., a > and the regon Z f = { r : r Ω, r } s not null non-zero probablty. We now proceed to obtan the structure of the regons Z, =,..,. Lemma 2: Consder regons Z, Z j, j and the correspondng thresholds,. Suppose r Z f and satsfes, r > r j 4 then under the optmal polcy r Z j Proof: Appendx I The above lemma states that f the weghted comparson of the th and the j th component of r s n favour of user, t s not optmal to serve user j. The weghts are the nverse values of the correspondng components of the threshold vector ā. The above mplcaton s ntutve as condton 4 means that n some sense user has a better channel condton than user j and hence servng user j s not optmal. Combnng the above two lemmas, we obtan the followng necessary condtons for the optmal polcy. Theorem I: ecessary Condtons Consder r = r,.., r then the optmal polcy s such that there exsts a threshold vector ā wth the followng structure, r Z f f t satsfes, r <, =,.., 5 2 r Z, =,.., f t satsfes, r 3 > r j, j =,..,, j 6 r > 7 Z r f rd r = R, =,.., 8 Proof: Condtons and 2 follow from Lemmas and 2. Clearly, as stated n Condton 3, for optmalty the throughput constrant must be met wth equalty. The set of r such that there s equalty n 5 and 6 have zero probablty and can be mapped to any Z wthout affectng optmalty. It can also be observed that the set of condtons n Theorem I are exhaustve and map every r Ω to a unque user ndex. Thus, gven ā, we have a unque partton of Ω nto regons Z,..., Z, Z f. In Fgure 2, we present a geometrc pcture of these regons for = 3. As seen from the fgure the Z regons are bottom truncated cones and t can be verfed say, for example Z 2 regon that 6 s satsfed. ext, we present the suffcency argument by provng that a schedulng polcy of the form as n Theorem I mnmzes the objectve n 3 and hence s optmal. Frst, observe that a schedulng polcy outlned n Theorem I can be re-wrtten n a smplfed way as a maxmum weghted rule as follows, { Z f no QoS user, f r, =,.., Γ r = r argmax, otherwse 9 where { } are such that E[r I ] = R,. Theorem II: Suffcency Consder the optmzaton problem n 3 and let R be feasble, then polcy Γ defned n 9 s optmal. Proof: Appendx II.

4 Thus, Theorem I states that the optmal polcy must satsfy certan condtons whch mpose a weghted comparson structure on the polcy and conversely, Theorem II completes the argument by statng that any polcy wth that structure s optmal. ow, vector ā s chosen such that Z r f rd r = R, =,..,. Ths can be solved usng technques of fndng the postve root of a non-lnear vector equaton. For general densty functons, t s dffcult to obtan analytcal expressons for ā. In practce, however, vector ā can be adjusted n real tme usng standard stochastc approxmaton algorthms smlar to those outlned n [], [2], [8], [9]. Interestngly, as dscussed next n Secton IV, one can solve for ā n closed form under a symmetrc Raylegh fadng model. From a system perspectve, ths analytcal study helps us obtan explct results for varous mportant performance measures such as the achevable throughput rate guarantee, the number of QoS users supportable and the fracton of tme-slots allocated to the BE users. IV. DIMESIOIG We have shown that an optmal polcy has a weghted structure as represented n 9 for some threshold vector ā. Here, we consder a symmetrc Raylegh fadng scenaro under whch closed form expressons can be obtaned for varous performance measures. To proceed, we mae the followng specalzatons to the earler model. The rate per tme slot of a user s assumed proportonal to the fade state square magntude;.e. r = h 2 P, where s a constant, h s the magntude of the fade state and P s the transmsson power. Ths lnear relatonshp s a good approxmaton of the Shannon capacty formuln the low SR regme and n ultra-wdeband transmsson and has been studed earler n the lterature [7]. The users experence ndependent dentcally dstrbuted..d flat Raylegh fadng, hence, h 2 s exponentally dstrbuted. As r s proportonal to h 2, the dstrbuton of r s also exponental and s gven as fr = e r/µ /µ, r where µ = E[r] s the average throughput rate of a user f t s served n all the tme-slots. Fnally, the guaranteed throughput rate s the same for all QoS users,.e. R = R,..., R. A. Throughput Characterzaton Intutvely, the fracton of tme-slots remanng for the BE users, denoted as γ, wll depend on the parameters R,, µ. As R, ncreases, γ should decrease whereas f µ ncreases hgher communcaton rates to the QoS users, the throughput guarantee can be acheved n fewer slots and γ should ncrease. Equvalently, gven γ,, µ, one can also as for the maxmum throughput-rate guarantee achevable for the QoS users. Our goal n the subsequent analyss s to obtan expressons for all these performance measures. It s clear that due to symmetry n f r and R, the regons Z, =,.., are dentcal Ω = R +. Hence, the { } s are equal and the threshold vector s gven as ā = a,.., a. The followng lemma relates the threshold value a wth γ. Lemma 3: Let γ be the fracton of tme-slots allocated to the BE users, the threshold value a for the optmal polcy s gven by, a = µ ln γ / Proof: From Theorem I, the regon Z f s gven as Z f = { r : r a, =,..., }. By ergodcty, the probablty of ths regon equals γ and by the..d channel assumpton, f r = f r = fr. Thus we get, a... a fr dr = γ Evaluatng the ntegrals for the exponental dstrbuton gves, γ = e a/µ 2 Re-wrtng the above expresson gves the result n. Observe from that γ = a = and γ = a whch corroborates the ntuton that γ = mples Z f s null and γ = all slots for BE users mples Z f = R +. Lemma 4: Under the optmal polcy, the throughput rate guarantee R for a gven threshold value s gven by, R = = a + µ e +a/µ Proof: Gven a threshold vector ā = a,..., a, the regon Z s gven as, Z = { r : a r, r j r, j }. As R = E[r I ] we get, r r R =... r fr dr fr j dr j 4 a j where f r = f r = fr by the..d assumpton. For the exponental dstrbuton, 4 smplfes to, r e r/µ R = e r/µ dr 5 µ a Usng the bnomal expanson, e r/µ = e r/µ, 5 can be solved to gve 3. = Conversely, one can also solve 3 to obtan the value of a that would acheve rate R. As R s monotoncally decreasng n a, the value of a that acheves R n 3 s unque. Elmnatng a from and 3 we obtan a unfed relatonshp among the system quanttes: Throughput rate R, Fracton of tme-slots, γ, allocated to the BE users and umber of QoS users,, n the system. Theorem III: Under the model assumptons stated earler wth QoS users n the system and γ [, ] fracton of tme-slots allocated to the BE users, the maxmum throughput rate R for each QoS user s gven as, = ln γ / + + R µ = + 2 γ + Proof: The result follows from Lemmas 3 and 4. 6

5 Rato, R/mu Plot of R/mu for varous values of gamma gamma = gamma =.2 gamma =.4 gamma =.6 gamma = umber of users,. Fg. 3. Plot of R/µ versus for values of γ =,.2,.4,.6,.8. Rato, Throughput/mu Fg. 4. gamma =.2 optmal random 5 5 umber of users,. Rato, Throughput/mu gamma =.4 optmal random 5 5 umber of users,. Plot comparng R/µ for the optmal and the random polcy. An nterestng observaton s that R vares lnearly wth µ where µ s the average channel condton of the QoS users. Re-phrasng 6 we see that gven R and γ, max = max R R s the maxmum number of supportable QoS users wth rate guarantee R, f a soluton exsts. Fnally, gven R and, the value of γ n 6 s the maxmum fracton of slots that can be allocated to the BE users. Fgure 3 s a plot of R/µ versus for dfferent γ values. B. Comparson wth Random-schedulng We, now, compare the performance of the optmal polcy wth a random schedulng polcy that s very smple to mplement and does not explot the varyng channel condtons among the users. Specfcally, the random polcy assgns a tme-slot to the BE users wth probablty γ and to the QoS users wth probablty γ. Among the QoS users the slot s then randomly assgned to one of the users wth equal probablty /. Due to the random nature of assgnment each QoS user gets γ/ fracton of tme-slots and the users have statstcally dentcal channel condtons. Thus the throughput rate of each QoS user, denoted R r, s gven as, γ R r = µ 7 Fgure 4 plots R opt /µ and R r /µ versus for γ =.2,.4, where R opt s the throughput for the optmal polcy as gven n 6. We, next, quantfy the gan, defned as R opt /R r, for large and show that t s on the order of ln. Proposton : The throughput gan, defned as R opt /R r, of the optmal polcy as compared to the random polcy s, R opt R r = Θln 8 Proof: Startng wth 6, the summaton over the frst terms can be evaluated as follows. Let α = γ, then, tang γ, we have α,. = α+ + = α x dx = α x dx = = α = γ 9 We can now re-wrte 6 as, R µ = γ ln + α + α γ + 2 = 2 Snce 9 holds for all α, we get the dentty, = x + + = x. Dvdng both sdes of ths equaton by x and ntegratng from to α, we get, α + α x + 2 = dx x = α dx = α = γ 2 x The nequalty above follows by notng that x s postve, monotoncally decreasng for x [, ], and has a maxmum value equal to at x =. Usng 2 we can bound the summaton term n 2 as, α + γ = + 2 γ γ lnγ γ whch s fnte for γ >. Consderng the log term n 2 we see that, ln α = ln γ = γ / + γ2/ 2 + γ3/ = Θln. Thus, for any < γ < and large, the log term n 2 domnates and we can express R opt as, R opt µ = γ Θln 22 From 7 and 22 we get the result n 8, Observe that as the throughput for both the optmal and the random polcy tends to zero. Equaton 22 smply states that R opt decreases as ln/ whle 7 states that R r decreases as /. Hence, we get a gan on the order of ln. The above logarthmc behavor arses due to the nfnte support and the exponental dstrbuton of the rate under Raylegh fadng. Whle such channel statstcs are smplfed models, n practce one could expect gans along these orders for moderate QoS user populaton. V. COCLUSIO We addressed the ssue of downln schedulng over a wreless channel ncorporatng the QoS and best effort servces. We consdered a set of rate guaranteed users and obtaned an optmal polcy that serves these users wth the

6 least tme-slot utlzaton, thereby, maxmzng the tme-slot allocaton to the BE users. Ths wor opens up nterestng questons about QoS guarantees over wreless channels. Whle we consdered long-term rate guarantee as a QoS measure, future wor sees to address schedulng over a wreless channel wth more general QoS requrements, for example, strct delay constrants on the data such as those that arse n vdeo streamng and multmeda applcatons. REFERECES [] X. Lu, E. Chong,. Shroff, A framewor for opportunstc schedulng n wreless networs Computer etwors, 4, pp , 23. [2] S. Borst, P. Whtng, Dynamc rate control algorthms for HDR throughput optmzaton, IEEE IFOCOM, Alasa, Aprl 2. [3] A. Jalal, R. Padovan, R. Panaj, Data throughput of CDMA-HDR a hgh effcency hgh data rate personal communcaton wreless system, IEEE Vehcular Technology Conf., vol. 3, 2. [4] Y. Lu, E. Knghtly, Opportunstc far schedulng over multple wreless channels, IEEE IFOCOM 23, San Francsco, 23. [5] S. Shaotta, A. Stolyar, Schedulng Algorthms for a Mxture of Real-Tme and on-real-tme Datn HDR, Proc. Internatonal Teletraffc Congress ITC-7, Brazl, Sept. 2. [6] P. Vswanath, D. Tse and R. Laroa, Opportunstc Beamformng usng Dumb Antennas, IEEE Tran. on Informaton Theory, 486, June, 22. [7] P. Lu, R. Berry, M. Hong, Delay-Senstve Pacet Schedulng n Wreless etwors, IEEE WCC 23, ew Orleans, LA. [8] H. Kushner, G. Yn, Stochastc approxmaton algorthms and applcatons, Sprnger, ew Yor, 997. [9] I. Wang, E. Chong, S. Kularn, Weghted averagng and stochastc approxmaton, Math. of Control, Sgnals and Systems, 997. r APPEDIX I PROOF OF LEMMA 2 For brevty, we smply outlne the steps nvolved n the proof and omt the techncal steps. The proof s based on a contradcton argument. To begn, consder r Z f and suppose that for the optmal polcy, r Z j such that > r j. We now gve a re-mappng of the regons such that the objectve functon decreases or equvalently the probablty of Z f regon ncreases, thereby, showng that the earler mappng cannot be optmal. As the lemmnvolves only the th and j th component, we wll focus only on these components. Let the neghborhood around r that s mapped to Z j be denoted as S. We can represent S as S = { x : x Ω, x r < δ } for some < δ δ m where δ m s the largest δ such that S Z j. By the assumpton r Z j, there exsts δ m >. ow, snce the optmal polcy satsfes Lemma we now that s the nfmum value of the th component among x Z. Thus, there exsts a pont m wth m = and a regon around m, denoted S 2, that maps to Z. The regon S 2 can be represented as S 2 = { x : x Z, < x m < δ 2 } for δ 2 >. Fnally, snce R does not le on the boundary of feasble throughput vectors there exsts n wth n j = > and a regon around n, denoted S 3, that maps to Z f. The regon S 3 s S 3 = { x : x Z f, < n j x j < δ 3 } for δ 3 >. Thus, we have regons S, S 2, S 3 that are not null and as defned above. ow re-map these regons as follows. Map S Z, S 2 Z f and S 3 Z j as shown n Fgure 5b. By approprately choosng the δ s, one can ensure that the throughput constrants are satsfed and also show that the objectve functon s smaller under the new mappng. x j S 3 ε Z f x j = x S ε Z 2 r. S ε Z j x Fg. a: Orgnal mappng x j S 3 Z j S 2 r. Z f Fg. b: ew mappng Fg. 5. Fgure showng the mappngs for the proof of Lemma 2. APPEDIX II PROOF OF THEOREM II We wll prove optmalty of polcy Γ, defned n 9, by showng that for any other feasble polcy Γ we have E[I ] E[Ĩ] where I r and Ĩ r are the ndcator functons for the respectve polces. We now that polcy Γ satsfes the throughput-rate constrants wth equalty,.e. E[r I ] = R. If Γ does not, ts trval to prove that Γ cannot be optmal. ow, suppose Γ also satsfes the rate constrants wth equalty,.e. E[r Ĩ ] = R, then, the objectve functon for polcy Γ can be re-wrtten as, E[Ĩ] = E[Ĩ] E[r Ĩ ] R 23 where { } s the threshold vector for polcy Γ. ote that the second term n 23 s zero. Re-arrangng 23 we get, [ E[Ĩ] = E r ] R Ĩ + 24 For any vector r we have the followng two cases. Case : Suppose r,, then, polcy Γ does not choose any QoS user Equaton 9 and I =, =,...,. ow, snce r, we have r,. Ths mples that whether Γ chooses or does not choose a QoS user we have the followng nequalty, r Ĩ = r I 25 Case 2: Suppose r > for some ndex. Let j be the chosen ndex for polcy Γ, then, from 9 we see that r j / has the maxmum value. Thus, r j r, and also rj <. Agan rrespectve of what Γ chooses, r Ĩ r j = From 24, 25 and 26 we get, [ E[Ĩ] E r ] I + r R = S x Z I 26 E[I ] where the last equalty follows from 23 replacng Ĩ wth I. Ths completes the proof.

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