Jont Schedulng of Rate-guaranteed and Best-effort Users over a Wreless Fadng Channel Murtaza Zafer and Eytan Modano Massachusetts Insttute of Technology Cambrdge, MA 239, USA Emal:{murtaza@mt.edu, modano@mt.edu} Abstract We address mult-user schedulng over the downln channel n wreless data systems. Specfcally, we consder a tme-slotted system wth a sngle transmtter servng multple users, where the channel condton of each user s tme varyng. Based on the throughput requrements, the user set s dvded nto two classes throughput guaranteed QoS users, and, best effort BE users. For ths system we obtan the optmal polcy that serves the QoS users wth the mnmum tme-slot utlzaton and maxmzes the total fracton of tme-slots allocated to the BE users. We show that the optmal polcy has a smple geometrc structure that can be easly vsualzed graphcally. In the specal case of Raylegh fadng, we obtan closed-form formulas that relate the achevable throughput-rate guarantee of the QoS users as a functon of other system parameters, thus, provdng closed-from relatonshps to understand the varous system tradeoffs. Analytcal comparson between the optmal and the random-schedulng polcy shows that gans on the order of ln can be acheved, where s the number of QoS users. Fnally, we present smulaton results comparng the optmal polcy under Raylegh and aagam fadng wth other heurstc polces ncludng a well nown opportunstc-schedulng polcy. Index Terms Wreless downln channel, Opportunstc schedulng, Mult-user dversty, Qualty of Servce, Raylegh fadng, aagam fadng. I. ITRODUCTIO Rapd growth of the Internet and mult-meda applcatons has created a fast ncreasng demand for data servces over wreless systems. Development of wreless data systems, such as the xev-do system n [], WMAX etc., ntroduces new challenges n provdng Qualty of Servce QoS over a wreless channel [2]. In contrast to conventonal voce traffc, data streams are nherently bursty and can tolerate much hgher delays, hence, reservng resources to provde QoS s neffcent. Therefore, n order to share a common resource, one needs effcent schedulng algorthms. Furthermore, n a wreless system the schedulng problem has an addtonal complexty assocated wth tme-varyng communcaton rates snce the channel condtons are tme-varyng. Wth multple users n the system, the transmtter can loo at the communcaton rates of the varous users and opportunstcally choose the best user to transmt to based on a requred set of objectves. In the lterature, such an approach s referred to as Opportunstc schedulng [4], [5], [7] or explotng Multuser dversty []. In ths wor, we utlze opportunstc schedulng to address the followng downln scenaro: there s a sngle server that Ths wor was supported by SF ITR grant CCR-3254, by DARPA/AFOSR through the Unversty of Illnos grant no. F4962-2-- 325 and by ASA Space Communcaton Project grant number AG3-2835. represents the base staton transmttng to multple users that represent the moble handsets. The system operates n a tmeslotted manner and n each tme-slot the base staton can serve only one user. The set of users are dvded nto two classes: throughput rate guaranteed QoS users and best effort BE users. The QoS users n the system represent sesson applcatons such as FTP, hgh data-rate web-browsng, throughput-constraned data transfers etc., whch requre the base staton to provde a certan data rate on the downln. In contrast, the BE users represent on-the-fly applcatons such as emal transfers, low prorty and latency tolerant data transfers etc. whch do not have rate requrements and are short-lved. The goal of ths wor s to desgn a schedulng polcy that provdes the requred throughput rates to the QoS users wth the least tme-slot utlzaton and maxmzes the remanng fracton of tme-slots assgned for the BE class. Down-ln schedulng and power/rate adaptaton s an actve area of research n wreless systems wth recent wor that ncludes [4] [9], [] [4]. The wor n [4] studed opportunstc schedulng under a utlty maxmzaton framewor and presented varous formulatons theren. In [5], the authors consdered the objectve of maxmzng the mnmum throughput rate among a set of users whle [6] extended the framewor to nclude a dynamc user populaton. In [7], multple smultaneous transmssons employng spread spectrum wth farness constrants was consdered and [8] presented algorthms for schedulng users wth average delay consderatons. The wors n [9], [] [4] studed transmsson power/rate adaptaton. In [9], [] the goal of the schedulng polcy was to ensure queue stablty, n [2] the am was to mnmze transmsson power subject to average delay constrants whereas [3], [4] consdered explct hard deadlne constrants over pont to pont communcaton. Our wor n ths paper dffers from the above by presentng a dfferent formulaton that combnes the QoS and the BE classes of servce. We adopt a geometrc approach to the problem and show that the optmal polcy satsfes a specal structure. The geometrc analyss s vald for a general fadng model and hence s applcable for a wde set of scenaros. In the specal case of Raylegh fadng we further obtan closed-form formulas for the varous performance metrcs. Part of the wor n ths paper has been presented earler n [3]. The rest of the paper s organzed as follows. In Secton II, we present the system model and the problem descrpton. In Secton III, we present the geometrc approach to the problem and obtan the optmal polcy. The throughput results for Raylegh fadng are presented n Secton IV; smulaton results
2 comparng the optmal and the random schedulng polcy for Raylegh and aagam fadng are presented n Secton V; and Secton VI concludes the wor. A. System Model II. SYSTEM AD PROBLEM DESCRIPTIO We consder the wreless downln scenaro, namely, communcaton from the base staton the transmtter to the moble handsets the recevers, also referred as users n a tme slotted system. There are multple users n the system, each user experencng tme-varyng channel condton. The channel state of a user remans constant for a sngle tme slot but changes over multple tme slots. We assume that the channel stochastc process s statonary and ergodc. Ths assumpton does not preclude channel correlatons over tme and among the users, thus allowng the possblty of channel states over multple tme-slots to be dependent. 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 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 [] and s a commonly used model n the lterature for the wreless downln [4], [5], [7], [8]. Let r = {r } denote the vector of communcaton rates 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 n that slot s smply r. We wll refer to r as the channel rate for user and r as the channel rate vector. The transmtter has nowledge of r at the begnnng of slot but does not now ths vector for future slots. In the th tme-slot, r s a partcular realzaton from the set comprsng all possble channel rate vectors whose probablty dstrbuton depends on the 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 gven that the channel rate vector n that slot s r. A statonary schedulng polcy, denoted Γ r, s one that depends solely on r but does not depend on the tme ndex. Clearly, such a polcy can be represented as a map from the set of channel rate vectors to the user ndex; namely, each r s mapped to a unque user ndex. As the underlyng processes are statonary, we restrct attenton n ths paper to statonary schedulng polces and such a restrcton suffces. B. Problem Descrpton The set of users n the system are dvded nto two servce classes: throughput rate guaranteed QoS users and best effort BE users. As mentoned earler, QoS users represent sesson applcatons that requre the base staton to provde a certan data rate on the downln, whereas, the BE users represent low prorty data transfer applcatons whch do Ths s a smplfyng assumpton that models one step channel predcton not have a rate requrement and are short-lved. The number of BE users s assumed large and beng short-lved t changes rapdly over tme. In such a setup, the objectve at the base staton s to provde the throughput rates to the QoS users wth the least tme-slot utlzaton so that the remanng fracton of tme-slots allocated for servng the BE class s maxmzed 2. The schedulng problem now s to obtan a rule that assgns tme-slots dynamcally over tme to meet the above objectve. Let there be QoS users n the system and denote the channel rate vector for these users as r = r,..., r. Let X r denote the throughput per tme-slot of user. We have 3, { r, f Γ r =.e. user selected X r =, otherwse The expected throughput per tme slot s E[X r]. Under statonarty of the schedulng rule, t s easy to see that X r s statonary and ergodc and that E[X r] equals the long term tme-average throughput per slot called throughput rate of user. Let R = R,.., R be the guaranteed throughput rates to the QoS users. We wll assume that R s feasble and by feasblty we mean that there exsts at least one schedulng polcy that acheves the throughput rates,.e. E[X r] R, =,.., for some polcy. Let I r be the ndcator functon for selecton of user, {, f Γ r = I r = 2, otherwse Wth ths notaton we can re-wrte X r as X r = r I r. The optmzaton problem can now be formally stated as follows, mn E[I r] subject to E[r I r] R, =,.., 3 The expectaton above s taen over the jont dstrbuton of the channel rate vector, r, for the QoS users. ote that mnmzng E[I r] s equvalent to maxmzng E[I r] 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 also assume that R s away from the boundary of the set, whch s characterzed later, comprsng all feasble 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 the paper. 2 We assume that among the BE users a greedy algorthm s used to share the slots that are allocated for the BE class. Wth a large populaton of BE users there s a hgh probablty of at least one user experencng good channel condton. Thus, maxmzng the tme-slot allocaton s then equvalent to maxmzng the sum total throughput of BE users. 3 For notatonal smplcty, explct dependence of X on Γ s not ndcated. Also, snce the servce of BE users s smply the fracton of allocated tme-slots to that class, ther channel rate vector s not requred for the optmzaton.
3 III. OPTIMAL POLICY The QoS users experence dfferent tme-varyng channel condtons, hence, ntutvely the optmal polcy must explot ths varablty gvng preference to users wth better channel condtons. Ths would ensure a hgh throughput per slot and would lead to a fewer fracton of tme-slots beng utlzed to provde the throughput guarantee. However, smply choosng the best user s not suffcent snce the throughput requrements of the QoS users and ther channel statstcs mght be very dfferent whch necesstates that these parameters must also be taen nto account. Let Ω be the set comprsng all possble channel rate vectors, r; we have Ω R +. Let the jont probablty densty functon be f r 4 so that the probablty of a subset Z Ω s gven as f rd r. We assume that f r s such that Z subsets wth zero volume n Ω or ndvdual ponts have zero probablty, thus, excludng pont mass dstrbutons. Snce a schedulng polcy maps r Ω to a unque user ndex, we wll represent a schedulng polcy as a partton of the set Ω nto + regons denoted as Z,.., Z, Z f. In a partcular tmeslot, f the channel rate vector r les wthn regon 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 5. The problem thus reduces to choosng these regons optmally to mnmze the objectve functon and satsfy the throughput rate constrant, Z r f rd r R, =,...,. In the rest of the paper, the notaton r Z r Z means that there s a neghborhood around r that s mapped s not mapped to regon Z and the probablty of ths neghborhood s non-zero. Formally, r Z mples that there exsts ɛ > such that all ˆr Ω, ˆr r < ɛ ˆr Z and fˆrdˆr > ; where the norm s the Eucldean ˆr r <ɛ dstance norm n R. The followng two lemmas gve the propertes of the optmal Z,..., Z, Z f regons. The frst lemma deals wth the regon Z f and t states that f r s mapped to Z, all rate vectors wth the 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. Proof: Appendx I A careful observaton of Lemma yelds 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 Lemma wll be volated. As ths holds for all Z, an optmal polcy has constants { } where = nf r Z r such that f r,, then r Z f. The regon Z f s shown n Fgure. 4 To avod excessve notatons, r, dependng on the context denotes both a random vector and a partcular realzaton for a generc tme-slot. 5 To elmnate unnterestng parttons the followng techncal assumptons are made. The set Ω can be parttoned nto a fnte set of components, where, each component s a connected set wth non-zero volume and every pont of ths set s arbtrarly close to an nteror pont. Such an assumpton removes the trval pont/zero volume sets. A schedulng polcy s a partton as above and each regon Z s a fnte unon of the component sets of the partton. Further, we assume that for set Ω non-zero volume sub-sets that have zero probablty have already been removed as ther mappng plays no role n the optmzaton. r 3 a 3 r 2 a 2 Z f regon Fg.. The Z f regon for = 3, threshold vector ā = a, a 2, a 3 and Ω = R +. ote that Zf = { r : r, =,..., }. Ths mplcaton s qute ntutve as t suggests that when the channel 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 requred throughput vector R for the QoS users 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, then under the optmal polcy r Z j Proof: Appendx II a r r > r j 4 The above lemma states that f the weghted comparson of th and j th component of r s n favour of the th component 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 geometrc structure for the optmal polcy. Theorem I: Optmal Structure Consder a channel rate vector r = r,..., r, then, under the optmal polcy 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
4 r 2 /a 2 = r 3 /a 3 Z 3 r 3 a 3 r 2 Z f Z 2 a r /a = r 3 /a 3 r /a = r 2 /a 2 Fg. 2. Optmal polcy structure for = 3, threshold vector ā = a, a 2, a 3 and Ω = R +. The Z regons are top truncated pyramds. Proof: Condtons and 2 follow from Lemmas and 2. Condton 3 states the obvous requrement that for optmalty the throughput constrant must be met wth equalty; snce, otherwse the excess fracton of slots that lead to a throughput above R can be assgned to the BE users. The set of r that le on the boundares for whch there s equalty n 5 and 6 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 top truncated pyramds see, for example the lght shaded Z 2 regon and t can be verfed that n ths regon, 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 functon n 3 and hence s optmal. Frst, observe that a schedulng polcy outlned n Theorem I can be rewrtten n a smplfed way as a maxmum weghted rule wth tes broen arbtrarly as follows, { Z f serve BE class, 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 III. Thus, Theorem I states that the optmal polcy must satsfy certan condtons whch mpose a weghted comparson structure on t and conversely, Theorem II completes the argument by statng that a polcy wth that structure s optmal. The results presented so far for the optmal polcy assumed that R was feasble, that s, t assumed that the optmzaton problem n 3 had a soluton and the throughput rate R could be guaranteed by some schedulng polcy. We now go bac and characterze the set of all such feasble throughput rate vectors. Let Π denote ths set; we clam that the nteror of Π s generated by consderng each threshold vector ā > Z r and obtanng the correspondng R that can be acheved for the polcy n 9 for that partcular ā. To see why ths s true consder the followng. Gven any ā >, we frst construct a polcy as gven n 9. Snce ths s a vald schedulng polcy the correspondng R wth R = E[r I ] s feasble; hence, Π must at least nclude all such R. ow, conversely, pc a feasble R n the nteror of Π, then, from Theorem I we see that a schedulng polcy can be re-mapped to have the optmal geometrc structure or equvalently there exsts ā > for whch the polcy n 9 s optmal. For a gven R, we now from 8 that the threshold vector ā for the optmal polcy s chosen such that Z r f rd r = R, =,..,. Ths can be solved usng numerous technques of fndng the postve root of a non-lnear vector equaton. In practce, however, the densty functon f r may not be nown apror n whch case the vector ā can be adjusted n real tme usng stochastc approxmaton algorthms smlar to those outlned n [4], [5]. For a comprehensve and thorough treatment of stochastc approxmaton algorthms see [7]. We now consder the specal case of Raylegh fadng n the next secton and obtan explct expressons for varous system metrcs. IV. DIMESIOIG In ths secton, we apply the general results obtaned n the last secton to a Raylegh fadng scenaro. From a practcal perspectve whle such a fadng model mght be restrctve, nevertheless, from a systems vewpont the closed form formulas obtaned provde mportant tradeoff lmts between the allocaton of resources to the QoS and the BE users and can be used as a frst cut calculaton n system desgn. For other fadng dstrbutons a smlar analyss can be carred out, albet, closed form expressons may not always be possble and certan quanttes would need to be evaluated numercally, as done n Secton V for an llustratve aagam fadng scenaro. To proceed, we consder the followng specalzatons to the earler model. The users experence ndependent dentcally dstrbuted..d flat Raylegh fadng, hence, h 2 s Exponentally dstrbuted, where h s the magntude of the channel gan/fade state. 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 and P s the transmsson power. A lnear power-rate relatonshp s a good model n varous scenaros such as the low SR regme n whch most CDMA systems operate, ultra-wdeband transmsson and fxed modulaton schemes and has been studed earler n the lterature [5]. 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. Lastly, we tae the guaranteed throughput rate the same for all QoS users, namely, R = R,..., R. A. Throughput Characterzaton Let γ denote the fracton of tme-slots allocated to the BE users. We frst obtan the threshold vector n terms of γ as follows. Due to symmetry n f r and R, clearly, the regons
5 Z, =,.., are dentcal, hence, the { } s for the optmal polcy are equal and the threshold vector s gven as ā = a,.., a. ow, the threshold value n terms of γ s as follows. Lemma 3: Let γ be the fracton of tme-slots allocated to the BE users, then 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 = whch agrees wth the fact that γ = no slot for the BE users mples Z f s null and smlarly, γ = a whch agrees wth the fact that γ = all slots for the BE users mples Z f = R +. ow, usng the optmal structure of regon Z we can obtan an expresson for the requred throughput rate R n terms of the threshold value a. 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 3 + + = 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 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 get 3. = ote from 3 that R s monotoncally decreasng n a, hence there s a one to one relatonshp between R and a. Stated equvalently, gven a certan R value, there s a unque threshold a that acheves t. 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 umber of QoS users,, and v The average channel condton, µ, of the users. j Fg. 3. R/mu.9.8.7.6.5.4.3.2. ncreasng gamma gamma = gamma =.2 gamma =.4 gamma =.6 gamma =.8 2 4 6 8 2 4 umber of users, Plot of R/µ versus for the optmal polcy for varous γ values. 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 by, = ln γ / + + R µ = + 2 γ + 6 Proof: The result follows from Lemmas 3 and 4. From 6, we see that R depends lnearly on µ, thus as expected, for a gven, γ, the throughput guarantee s hgher f µ s ncreased. ow, re-phrasng 6, theoretcal lmts for varous performance measures can be deduced as follows. Maxmum Throughput Rate: By settng γ =, we can obtan the maxmum throughput rate R max for each QoS user when no slots are allocated for the BE users. Ths s gven as, R max = µ = + 2 7 Fgure 3 s a plot of R/µ versus for dfferent γ values. The functon R max /µ s the topmost curve correspondng to γ =. As R max s monotoncally decreasng n, ts maxmum value s at = and equals R max /µ =. Ths s expected as the maxmum rate achevable when all the slots are assgned to just one QoS user equals E[r] = µ. Maxmum umber of QoS Users: Fx a value of R and γ, the maxmum number of QoS users such that throughput of each s at least R s gven by, max R, γ = max R R 8 Obvously f the values of R, γ are such that there s no nteger that acheves t, the system values n ths case are nfeasble. Fgure 4 s a plot of R/µ versus γ for varous values of. Infeasblty arses when γ, R /µ pont les above the = curve n Fg. 4.
6 Rato, R/mu.8.6.4.2 ncreasng = = 2 = 4 = 8 = 4.2.4.6.8 Gamma Fg. 4. Plot of R/µ versus γ for values of =, 2, 4, 8, 4. Maxmum Value of γ: Gven R and, the value of γ that solves the equaton n 6 gves the maxmum fracton of slots that can be allocated to the BE users. Fgure 4 wth ts axes nverted gves a plot of γ versus R/µ for dfferent. B. Comparson wth Random-schedulng Polcy To understand how much gan can be acheved, we present an analytcal comparson of the optmal polcy wth the random schedulng polcy. The random polcy assgns a tmeslot 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 /. Clearly, ths polcy does not explot the varyng channel condtons for schedulng the users. Due to the random nature of the assgnment each QoS user gets γ/ fracton of tme-slots and snce the users have statstcally dentcal channel condtons, the throughput rate of each QoS user, denoted R r, s gven as, γ R r = µ 9 Let us now fx a value of γ for both the optmal and the random polces,.e. under both polces, γ fracton of slots are assgned to the BE class. Let R opt, R r denote the correspondng throughput rate provded to each QoS user. Then, as shown below, the gan defned as R opt /R r s on the order of ln. To show ths result, we need the followng lemma. Lemma 5: For any γ,, we have the followng relatonshp 6 Proof: Appendx IV ln = Θln 2 γ Theorem IV: The throughput gan of the optmal polcy as compared to the random polcy, defned as R opt /R r, for γ, satsfes the relatonshp, R opt R r = Θln 2 6 The followng notaton s followed: f = Og means that there exsts a constant c and nteger such that f cg for >, f = Θg means that f = Og and g = Of Proof: Appendx V Observe that as the throughput rate per QoS user for both the optmal and the random polcy tends to zero. Equaton 9 states that R r decreases as / whereas 45 n Appendx V states that by usng the optmal polcy R opt decreases more slowly as ln/. Hence, we get a gan on the order of ln. The above logarthmc behavor can be attrbuted to the exponental dstrbuton of the rate under Raylegh fadng and whle such channel statstcs are smplfed models, n practce one could expect gans along these orders for moderate QoS user populaton. V. SIMULATIO RESULTS To valdate the theoretcal results derved n the earler sectons, we present smulaton results obtaned for two fadng dstrbutons, Raylegh and aagam. The setup for the smulatons s as follows: we consder a tme duraton of seconds and dvde t nto, slots, thus, each tme-slot s of length mllsecond. For the sae of smplcty, the QoS users all experence..d channel fadng. We assume a lnear relatonshp between the channel rate and the fade state squared magntude;.e. r h 2. Thus, for Raylegh fadng the rate, r, at whch data can be transmtted n a slot s Exponentally dstrbuted wth densty fr = e r/µ µ, r ; whle for aagam fadng, r has a Gamma dstrbuton gven m m as fr = r m µ Γm e mr/µ, r, where m s the fadng parameter [6]. The mean channel rate, µ, for each user s taen as, µ = 8 Kbts/sec for both the dstrbutons. At each tme-slot, a random vector of channel rates for the QoS users s drawn from the respectve dstrbuton. Gven ths channel rate vector, the partcular schedulng polcy decdes whch QoS user to serve or to allocate the slot to the BE class. In the former case, the chosen QoS user, say user, receves a throughput rate of r whle for the others the throughput rate s n that slot. In the latter case, all QoS users get a throughput n that slot. We smulate the optmal, the random, the greedy Tme Dvson Mult-Access TDMA and an opportunstc schedulng polcy studed n [] whch we refer to as Opportunstc Proportonal Far OPF polcy. In case of the optmal polcy, the schedulng decson s taen as gven n 9 where the threshold vector ā s computed usng the formulas n Secton IV. The random polcy maes a schedulng decson as descrbed n Secton IV-B. For the greedy TDMA and the OPF polcy the schedulng decson s taen as follows. Let T denote the runnng tme-average of the throughput rate for the th QoS user. At the begnnng of each tme-slot, consder all QoS users for whch T < R where R s the requred throughput guarantee. In the greedy TDMA polcy the user wth the maxmum channel rate s selected whereas for the OPF polcy the user that maxmzes the metrc r /T s selected. If for all QoS users T R, the slot s allocated to the BE class. We frst numercally valdate the theoretcal results obtaned n Secton IV. We consder Raylegh fadng wth 3 QoS users each havng a throughput rate guarantee of R = 2 Kbts/sec.
7 Throughput rate n Kbts/sec 8 6 4 2 qos user qos user 2 qos user 3 Throughput rate n Kbts/sec 5 5 qos user qos user 2 qos user 3 2 4 6 8 tme n seconds 2 4 6 8 tme n seconds Fg. 5. Runnng tme-average of throughput rate for Raylegh fadng wth 3 QoS users, R = 2 Kbts/sec. Throughput gan, R opt /R r 5 4 3 2 thpt gan log +.8 2 3 4 umber of QoS users, Fg. 6. Throughput gan, R opt /R r, for Raylegh fadng wth γ =.3. Fg. 7. Runnng tme-average of throughput rate for aagam fadng wth fade parameter m =.6, γ =.3 and 3 QoS users. Throughput gan, R opt /R r 8 7 6 5 4 3 2 thpt gan 2 3 4 umber of QoS users, Fg. 8. Throughput gan, R opt /R r, for aagam fadng wth fade parameter m =.6 and γ =.3. Fgure 5 gves a plot of the runnng tme-average of throughput rate under the optmal polcy. As can be seen from the plot, the long-term requred rate s acheved very qucly n tme wthn almost a second and s mantaned thereafter wthn a close range. Thus, wthn a very short tme nterval the requred throughput rate can be provded to the QoS users. A smlar trend s observed when the parameter values are vared. In Fgure 6, we fx γ =.3,.e. the BE class s assgned 3% of the slots. The fgure gves a plot of the smulated throughput gan R opt /R r as a functon of ; where R opt, R r s the throughput rate of each QoS user under the optmal and the random polcy respectvely. In conformaton wth the result R n 2, we see from the plot that opt R r grows logarthmc n. We next consder aagam fadng wth the fadng parameter m =.6. In Fgure 7, we fx γ =.3 and plot the runnng tme-average of the throughput rate for the optmal polcy wth 3 QoS users. For the case of aagam fadng, becomes, ma µ t m e t dt = γ Γm from whch the optmal threshold s evaluated numercally by fndng the root of the above non-lnear equaton. The long-term rate provded to each QoS user n ths case s R = 494 Kbts/sec. Agan as before, the throughput rate s acheved very qucly n tme and s mantaned thereafter wthn a close range. In Fgure 8 we compare the throughput gan of the optmal polcy versus the random polcy. As seen from the plot the optmal polcy acheves a substantal gan n throughput even wth aagam dstrbuton. In fact, the gan s hgher now because the Gamma dstrbuton wth m =.6 has a larger varance than the Exponental wth the same mean. As a result, the optmal polcy whch opportunstcally explots rate varatons gves a hgher gan n comparson to random assgnment. We now present smulaton results that compare the performance of the optmal, random, TDMA and OPF polces. We consder 3 QoS users wth Raylegh fadng and the mean channel rate of each QoS user, µ = 8 Kbts/sec. Fgure 9 plots the total fracton of slots utlzed by the QoS users under each polcy versus the throughput rate requrement of each QoS user. The quantty, total fracton of slots used by QoS users, s the tme-slot allocaton to the BE class. Frst, as expected the random polcy has the worst performance and utlzes the maxmum tme-slots to provde the throughput rate guarantees. Snce the OPF, TDMA and optmal polcy explot the channel varatons and opportunstcally schedule the users, the tme-slot utlzaton s lower as compared to the random polcy. The OPF polcy performs worse than the TDMA polcy whch s expected snce the TDMA polcy by beng greedy has a hgh throughput per slot and hence utlzes fewer tme-slots. Fnally, as expected the optmal polcy uses a substantally lower fracton of tme-slots than all the polces. VI. COCLUSIO We addressed the ssue of downln schedulng over a wre-
8 Total fracton of slots used by QoS users.8.6.4.2 random OPF TDMA optmal 5 5 2 25 Throughput rate of each QoS user n Kbts/sec [3] M. Zafer, E. Modano, A Calculus Approach to Mnmum Energy Transmsson Polces wth Qualty of Servce Guarantees, Proceedngs of the IEEE IFOCOM 25, vol., pp. 548-559, March 25. [4] M. Zafer, E. Modano, Contnuous-tme Optmal Rate Control for Delay Constraned Data Transmsson, 43rd Annual Allerton Conference on Communcaton, Control and Computng, Montcello, Sept. 25. [5] P. Lu, R. Berry, M. Hong, Delay-Senstve Pacet Schedulng n Wreless etwors, IEEE WCC, ew Orleans, 23. [6] M. aagam, The m-dstrbuton - A general formula of ntensty dstrbuton of fadng, Statstcal Methods n Rado Wave Propagaton, W. C. Hoffman, Ed. London, England: Pergamon, 96 [7] H. Kushner, G. Yn, Stochastc approxmaton algorthms and applcatons, Sprnger, ew Yor, 997. [8] Gradshteyn I.S., Ryzh I.M., Table of Integrals, Seres and Products, Academc Press, Fg. 9. Comparson of the fracton of slots utlzed by the random, OPF, TDMA and optmal polces. less channel ncorporatng the QoS and best effort servces. We consdered a set of rate guaranteed users and obtaned the optmal polcy that serves these users wth the least tmeslot utlzaton, thereby, maxmzng the tme-slot allocaton to the BE users. Equvalently, the optmal polcy also solves the problem of maxmzng the rate guarantee for the QoS users gven that a certan fracton of tme-slots must be allocated to the BE users. We presented a geometrc vsualzaton of the optmal polcy and under Raylegh fadng we derved analytcal expressons quantfyng the varous system metrcs. Analytcal comparson wth the random-schedulng polcy showed that throughput gans on the order of ln can be acheved by explotng mult-user dversty. Fnally smulaton results show substantal gans acheved by the optmal polcy as compared to other well-nown polces n the lterature. REFERECES [] 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. [2] M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, P. Whtng, and R. Vjayumar, Provdng qualty of servce over a shared wreless ln, IEEE Communcatons Magazne, pp. 5-54, Feb. 2. [3] M. Zafer and E. Modano, Jont Schedulng of Rate-guaranteed and Besteffort Servces over a Wreless Channel, IEEE CDC-ECC 25, Sevlle, Span, Dec. 25. [4] X. Lu, E. Chong,. Shroff, A framewor for opportunstc schedulng n wreless networs Computer etwors, 4, pp. 45-474, 23. [5] S. Borst, P. Whtng, Dynamc rate control algorthms for HDR throughput optmzaton, IEEE IFOCOM, Alasa, Aprl 2. [6] S. Borst, User-level performance of channel-aware schedulng algorthms n wreless data networs, IEEE/ACM Transactons on etworng, vol. 3, no. 3, pp. 636-647, 25. [7] Y. Lu, E. Knghtly, Opportunstc far schedulng over multple wreless channels, IEEE IFOCOM, San Francsco, 23. [8] 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. [9] L. Tassulas and A. Ephremdes, Dynamc server allocaton to parallel queues wth randomly varyng connectvty, IEEE Transactons on Informaton Theory, vol. 39, pp. 466478, Mar. 993. [] P. Vswanath, D. Tse and R. Laroa, Opportunstc Beamformng usng Dumb Antennas, IEEE Trans. on Informaton Theory, 486, June, 22. [] M. eely, E. Modano, and C. Rohrs, Power Allocaton and Routng n Mult-Beam Satelltes wth Tme Varyng Channels, IEEE Transactons on etworng, vol., no., pp. 38-52, Feb. 23. [2] D. Rajan, A. Sabharwal, B. Aazhang, Delay bounded pacet schedulng of bursty traffc over wreless channels, IEEE Transactons on Informaton Theory,, vol. 5,, pp. 25-44, Jan. 24. APPEDIX I PROOF OF LEMMA The proof s based on a contradcton argument where we begn by supposng that for the optmal polcy there s a ˆr Z f wth ˆr > r. By re-mappng the regons we wll show that the objectve functon n 3 decreases, thus, contradctng the optmalty clam and provng ˆr Z f. We are gven that r Z, hence, there s a neghborhood of r, whch we denote as S, that s mapped to Z,.e. S Z and S = { x x Ω, x r < δ } for some δ >. Further, by assumpton ˆr Z f, there s a neghborhood of ˆr gven as, S 2 = { x x Ω, x ˆr < δ 2 } for some δ 2 >, such that S 2 Z f. ow re-map the regons as follows. Map S Z f and S 2 Z. To ensure the new mappng s feasble we must satsfy the QoS rate constrant for user whch entals the followng equalty. x f xd x = x f xd x 22 S 2 S The left sde above s the throughput acheved over regon S 2 under the new map and the rght sde s the throughput lost by re-mappng S to Z f. A set of δ, δ 2 > exst that satsfy 22; to see ths note that the ntegral over any regon {S } 2 = s a postve, contnuous functon wth respect to δ, non-ncreasng as δ decreases and tends to zero as δ. Hence, startng wth the largest δ, δ 2 values that satsfy the S, S 2 defnton and then decreasng these values one can obtan {δ, δ 2 > } such that each ntegral above s postve and the two are equal. ow, vewng δ 2 as a functon of δ, t s clear that f a soluton exsts for some δ then for all δ δ a soluton exsts by the contnuty and decreasng property of the ntegrals. We now proceed by choosng δ δ. Usng the Frst Mean Value theorem, [8], we can tae the x outsde the ntegrals as follows, S x f xd x = r + ɛ S f xd x and S 2 x f xd x = ˆr + ɛ 2 S 2 f xd x, where the {ɛ } 2 = depend on {δ } 2 = or equvalently on δ as δ 2 depends on δ through 22. Wth ths, we can re-wrte 22 as, ˆr + ɛ 2 f xd x = r + ɛ f xd x 23 S 2 S ow, loong at the objectve functon n 3, the change n ts value due to the re-map equals the probablty of regon
9 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.. Fgure showng the mappngs for the proof of Lemma 2. S 2 added from Z f to Z mnus the probablty of regon S removed from Z. Thus, J = f xd x + f xd x S S 2 ˆr + ɛ 2 = f xd x 24 r + ɛ S 2 Let c = ˆr r, then, c > snce by assumpton ˆr > r. Usng the Frst Mean Value theorem, we also have ɛ as δ. Thus, for any c we can scale δ to be small enough ˆr such that +ɛ 2 r +ɛ >. Further, snce the ntegral n 24 s the probablty of S 2 whch s strctly postve regons wth zero probablty are unnterestng and have been removed from Ω, we fnally get, J <. Ths completes the contradcton argument. APPEDIX II PROOF OF LEMMA 2 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 > r j 25 We now gve a re-mappng of the regons such that the objectve functon n 3 decreases or equvalently the probablty of Z f regon ncreases, thus, provng 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 x Ω denote a generc rate vector. Snce by assumpton r Z j, there s a neghborhood around r gven as S = { x x Ω, x r < δ } for some δ >, such that S Z j. ext, snce the optmal polcy satsfes Lemma ts volaton would mae the polcy non-optmal to start wth 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 ;.e. S 2 Z and S 2 = { x x Ω, < x m < δ 2 } for some δ 2 >. Fnally, snce R does not le on the boundary of feasble throughput vectors the regon Z f s not null. Hence, there exsts n wth n j = > and a regon around n, denoted S 3, that maps to Z f ; namely, S 3 Z f and S 3 = { x x Ω, < n j x j < δ 3 } for some δ 3 >. The regons S, S 2, S 3 are depcted n Fgure a. S x Z ow re-map these regons as follows. Map S Z, S 2 Z f and S 3 Z j as shown n Fgure b. To ensure the new mappng s feasble we must satsfy the QoS rate constrants for user and user j, whch entals the followng equaltes. x f xd x = x f xd x 26 S 2 S x j f xd x = x j f xd x 27 S 3 S Equaton 26 matches the throughput lost for user due to the re-map of S 2 Z f and the throughput ganed by S Z, whle 27 gves a smlar equalty for user j. To see why a set of {δ } 3 = exst that solve the above equatons, note that the ntegral over any regon S s a contnuous, postve functon of δ, decreasng or non-ncreasng as δ decreases and tends to zero as δ. Hence, startng wth the largest δ that satsfes the S defnton, decrease t untl a δ 2 s obtaned that solves 26. By the non-nullty of S, S 2 and the above property of the ntegrals such a soluton δ, δ 2 > exsts. Smlarly obtan a δ, δ 3 that solves 27. Fnally, tang δ as the mnmum of the two solutons, re-obtan δ 2, δ 3 such that both 26 and 27 are satsfed. ow, vewng δ 2, δ 3 as functons of δ, t s clear that f a soluton exsts for some δ, then, for all δ δ a soluton exsts by the contnuty and decreasng property of the ntegrals. We now proceed by choosng δ δ. Usng the Frst Mean Value theorem, [8], we can re-wrte the above ntegrals as, + ɛ 2 f xd x = r + ɛ f xd x 28 S 2 S + ɛ 3 f xd x = r j + ɛ 4 f xd x 29 S 3 S where the {ɛ } above depend on the {δ } or equvalently on δ as δ 2, δ 3 depend on δ through 26 and 27. ext, loong at the objectve functon n 3, the change n ts value due to the re-map equals the probablty of regon S 3 added from Z f to Z j mnus the probablty of regon S 2 removed from Z. Thus, J = f xd x + f xd x S 2 S 3 r + ɛ = r j + ɛ 4 f xd x 3 + ɛ 2 + ɛ 3 S Let c = r rj, then, from 25 we have c >. From the Frst Mean Value theorem we also have ɛ as δ. Thus, for any gven c we can scale δ to be small enough such r that +ɛ +ɛ 2 rj+ɛ4 +ɛ 3 >. Further, snce the ntegral n 3 s the probablty of S whch s strctly postve, we fnally get J <. Ths completes the proof. APPEDIX III 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, t s 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 3 where { } s the threshold vector for polcy Γ. ote that the second term n 3 s zero. Re-arrangng 3 we get, [ E[Ĩ] = E r ] R Ĩ + 32 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 33 Case 2: Suppose r > for some ndex. Let j be the chosen user for polcy Γ, then, from 9 we see that r j / has the maxmum value. Thus, rj r, and also rj <. Agan rrespectve of what Γ chooses, r Ĩ r j = From 32, 33 and 34 we get, [ E[Ĩ] E r ] I + r I 34 R = E[I ] where the last equalty follows from 3 replacng Ĩ wth I. Ths completes the proof. APPEDIX IV PROOF OF LEMMA 5 To prove the lemma we need to show the followng two relatonshps, ln/ γ = Oln and ln = Oln/ γ. We begn by provng the frst relatonshp. Snce γ, and s a postve nteger, we have < γ <. Tang a power seres expanson of we get, ln γ = ln +γ / +... + γ / γ +γ + γ / +... + γ 2... 35 + γ / +... + γ / = ln 36 γ ln = ln ln γ 37 γ The nequalty above follows, snce γ < + γ / +... + γ / ; thus we get ln/ γ = Oln. To prove the reverse relatonshp,.e. ln = Oln/ γ, proceed as follows. Usng the standard nequalty, ln + 2 +... +, we get, γ ln γ + γ 2 +... + γ γ / + γ2/ +... + γ/ 2 ln γ snce < γ < 38 where the last nequalty above follows by truncatng the power seres expanson of ln γ. Thus, ln γ ln/ γ whch gves ln = Oln/ γ. APPEDIX V PROOF OF THEOREM IV Startng wth 6 we can wrte t as, R µ = ln γ + γ + = γ + + + 2 39 = Consder the frst term n 39 above; t can be evaluated as follows. Let α = γ, then, snce γ, we have α,. = α+ + = α a = = = α x dx x dx 4 α = γ 4 Equalty a above follows by nterchangng the summaton and the ntegral and usng the Bnomal expanson. Thus, we get, ln α = α + + = ln γ α. ow, consder the second term n 39 and proceed as follows. Frst, snce 4 holds for all α, we get the dentty, = x + + = x. Dvdng both sdes by x and ntegratng from to α, gves, α α x x + = dx = x α + α x + 2 = dx x = α dx = α = γ 42 The nequalty above follows by notng that x s postve, monotoncally non-ncreasng over x [, ], for fxed, and has a maxmum value equal to at x =. Equaton 42 further gves, α + γ = + 2 x
γ γ lnγ γ whch s fnte for < γ < and snce γ γ s monotoncally ncreasng n wth a fnte lmtng value, t s bounded for all. Thus, we get, α + γ + 2 ln/γ 43 γ = ow, usng the above smplfcatons we can re-wrte 39 as, R µ = γ ln + α + α γ + 2 = 44 For γ,, the frst term wthn bracets above, grows as ln α = Θln usng Lemma 5 whereas the second term s bounded from 43. Hence, for large, R opt can be expressed as, R opt µ = γ Θln 45 From 9 and 45 we get the result n 2,