Joint Scheduling and Resource Allocation in CDMA Systems

Transcription

1 TECHNICAL REPORT - JUNE Jont Schedulng and Resource Allocaton n CDMA Systems Vjay G. Subramanan, Randall A. Berry, and Rajeev Agrawal Expanded Techncal Report: A shorter verson of ths paper wll apear n IEEE Transactons on Informaton Theory. Abstract In ths paper, the schedulng and resource allocaton problem for the downlnk n a CDMA-based wreless network s consdered. The problem s to select a subset of the users for transmsson and for each of the users selected, to choose the modulaton and codng scheme, transmsson power, and number of codes used. We refer to ths combnaton as the physcal layer operatng pont PLOP). Each PLOP consumes dfferent amounts of code and power resources. The resource allocaton task s to pck the optmal PLOP takng nto account both system-wde and ndvdual user resource constrants that can arse n a practcal system. Ths problem s tackled as part of a utlty maxmzaton problem framed n earler papers that ncludes both schedulng and resource allocaton. In ths settng, the problem reduces to maxmzng the weghted throughput over the state-dependent downlnk capacty regon whle takng nto account the system-wde and ndvdual user constrants. Ths problem s studed for the downlnk of a Gaussan broadcast channel wth orthogonal CDMA transmssons. Ths results n a tractable convex optmzaton problem. A dual formulaton s used to obtan several key structural propertes. By explotng ths structure, algorthms are developed to fnd the optmal soluton wth geometrc convergence. Index Terms Cellular network, channel-aware schedulng, code dvson multple access CDMA), convex optmzaton, resource allocaton, utlty maxmzaton. I. INTRODUCTION Effcent schedulng and resource allocaton are essental components for enablng hgh-speed data access n wreless networks. In ths settng, schedulng s complcated due to the tme-varyng fadng of wreless channels. A varety of wreless schedulng approaches have been proposed that opportunstcally explot these temporal varatons to mprove the over-all system performance, e.g. [1] [20]. These approaches attempt to transmt to users durng perods when they have good channel qualty and can support hgher transmsson rates), whle mantanng some form of farness among the users. Wreless schedulng approaches can be dvded nto two classes: ) tme-dvson multplexed TDM) systems, where a sngle user s transmtted to n each tme-slot, as n the HDR system CDMA 1xEVDO) [21], [22], and ) systems n whch the transmtter can smultaneously transmt to multple users n each tme-slot, by usng a combnaton of TDM and another multplexng technque such as CDMA or OFDMA. In the latter case, n addton to decdng whch users to schedule, the avalable physcal layer resources, such as bandwdth and power, must be dvded among the users. In ths paper, we consder the V. G. Subramanan s wth the Hamlton Insttute, NUIM, Maynooth, Co. Kldare, Ireland, e-mal: vjay.subramanan@num.e. R. A. Berry s wth the Dept. of Electrcal Engneerng and Computer Scence, Northwestern Unversty, Evanston, IL 60208, USA, emal: rberry@eecs.northwestern.edu. R. Agrawal s wth the Advanced Networks and Performance Group, Motorola Inc., Arlngton Heghts, IL, USA, e-mal: rajeev.agrawal@motorola.com. The majorty of ths work was done whle V. G. Subramanan was wth the Mathematcs of Communcaton Networks Group, Motorola Inc. Hs work s also supported n part by SFI grants IN3/03/I346 and 07/IN.1/I901. The work of R. A. Berry was supported n part by the Northwestern-Motorola Center for Communcatons and NSF CAREER award CCR A prelmnary verson of ths paper was presented at the 2nd Workshop on Modelng and Optmzaton n Moble, Ad Hoc, and Wreless Networks WOpt 04), Cambrdge, UK, March 24-26, 2004.

2 2 TECHNICAL REPORT - JUNE 2009 second class of systems, where CDMA s used to multplex users wthn a tme-slot. 1 Examples of ths type of system nclude the Hgh Speed Downlnk Packet Access HSDPA) approach developed for W- CDMA [23, Chapter 11, pp ] or the 1x-EVDV approach for CDMA2000 [24]. In these systems, the physcal layer resources and nformaton rate assgned to a user are specfed by selectng the number of spreadng codes, the fracton of transmsson power, and the modulaton and codng scheme MCS). We refer to a combnaton of these as the physcal layer operatng pont PLOP). The man problem addressed n ths paper s to specfy the optmal PLOP at each schedulng nstant, whch n turn specfes the vector of user transmsson rates. Ths problem must be solved once every tme-slot e.g., 2msec n HSDPA or 1.25 msec n 1x-EVDV), and so requres a computatonally effcent soluton. We consder ths n the context of the gradent-based schedulng framework presented n [1], [2]. In ths framework, n each tme-slot the objectve s to chose the transmsson rate vector that has the largest projecton onto the gradent of the total system utlty. The utlty s a functon of each user s throughput and s used to quantfy farness. Several such gradent-based schedulng algorthms have been studed for TDM systems, ncludng the proportonally far algorthm [22], whch s based on a log utlty functon. In [1], a larger class of utlty functons s consdered that allow effcency and farness to be traded-off. The problem consdered here can be vewed as fndng the maxmum weghted sum throughput for a downlnk broadcast) channel, where the weghts are determned by the gradent of the utlty. Our soluton s general n that t also apples to other schedulng algorthms, whch may provde these weghts usng dfferent approaches. For example, these weghts could be based on queue sze nformaton as n the MaxWeght schedulng algorthms studed n [3], [4], [17], [26]. For the model studed here, the feasble rate regon s convex; hence, by varyng these weghts we can determne the boundary of ths regon. In related work, the problem of allocatng resources to maxmze the weghted sum capacty for the downlnk channel has been consdered from an nformaton theoretc perspectve n [28], [29]. Both of these works assume the use of optmal nformaton theoretc mult-user) codng/decodng. 2 The work n [29] also consders several sub-optmal transmsson strateges, such as approaches based on TDM, CDMA wthout multuser codng wth all users orthogonalzed and FDM; the focus n [29] s on dervng the long-term average throughputs over multple fadng states under a long-term average power constrant. Here, we focus on optmally allocatng resources for the specfc fadng state realzed n each schedulng tme-slot; the total power s constraned wthn each tme-slot as well. The problem wthn each tme-slot can be vewed as a specal case of the CDMA wthout multuser codng approach n [29] where the fadng s constant. However, focusng on ths case enables us to generate a much smpler optmal algorthm. We also take nto account addtonal per-user power and code constrants that are mposed by the capablty of each moble n a practcal system. 3 The algorthms n [29] make use of specfc propertes of the functon a log1 + bx) that do not generalze wth the addton of these per-user constrants. Smultaneously and ndependently of our work, 4 Kumaran and Vswanathan studed a smlar problem n [31]. They also consder the problem of maxmzng the weghted capacty wthn a tme-slot and derve several related structural characterstcs. We note that the work n [31] does not nclude per-user code constrants, but does contan an algorthm wth a per-user rate constrant. We begn wth formulatng the schedulng and resource allocaton problem n Secton II. Ths formulaton s based on a gradent-based schedulng approach from [1], [2], whch we also revew. By substtutng 1 The model n ths paper also apples to OFDMA systems when each sub-channel that may be assgned to a user has the same channel state ths may model a system n whch OFDMA sub-channels are formed by nterleavng tones from across the frequency band). A more detaled dscusson of such problems for OFDMA systems can be found n [25], [36]. 2 In the specal case of maxmzng the equal weght sum capacty n a flat fadng channel, the nformaton theoretc optmal approach s to transmt to only one user n each tme-slot [28] and hence, mult-user decodng s not requred. However, ths s not true f the users are not weghted equally or for other channel models, such a multple antenna channel. It also does not hold when addtonal per user constrants are present, as s the case here. 3 Moreover, these constrants may vary from moble to moble. For example, the ntal moble devces for HSDPA can receve up to 5 spreadng codes, whle future devces may be able to receve up to 15 spreadng codes. 4 A verson of our work was frst presented n [30].

3 TECHNICAL REPORT - JUNE an analytcal formula relatng the rate, power, codes, and SINR, we obtan an analytcally tractable problem wth nce convexty propertes. In Sectons III-IV, we use a dual formulaton to study ths problem. We obtan analytc formulas for many of the quanttes of nterest. For others we have to resort to a numercal search aded wth some heurstcs based on the structure of the problem). However, these numercal searches are n a sngle dmenson due to the dual formulaton) rather than over the multdmensonal PLOP space. Also, thanks to the convexty of the problem, these algorthms converge geometrcally fast. Along the way we obtan key structural propertes of the optmal soluton ncludng: 1) A tght upper bound on the number of users scheduled as a functon of the per-user code constrants; when each user can use all the codes, ths bound mples at most two users wll be scheduled. 2) Gven a code assgnment, the optmal power allocaton s gven by a water-fllng algorthm, whch s modfed to take nto account the dfferent weghts assgned to each user and any per-user power constrants. 3) For a fxed code assgnment, the optmal water-level Lagrange multpler) can be found n fnte tme. Specfcally, we gve an teratve algorthm whch wll termnate n at most M steps, where M s the number of users allocated codes. 4) For a gven water-level, the users that are scheduled are determned by smply sortng all the users based on a per-user metrc that s gven analytcally. 5) Codes are only tme-shared when tes occur n the above sort. Ths corresponds to a pont where the dual functon s not dfferentable. At these values the optmal tme-sharng can be found usng the subgradents of ths functon. We gve a complete characterzaton of these subgradents. We conclude the paper wth smulaton results comparng ths algorthm wth a base-lne heurstc n Secton V. II. GRADIENT-BASED SCHEDULING AND RESOURCE ALLOCATION PROBLEM We consder the downlnk of a wreless communcaton system wth K users. The channel condtons are tme-varyng and modeled by a stochastc channel state vector e t = e 1,t,..., e K,t ), where e,t represents the channel state of the th user at tme t. Assocated wth each channel state vector s a rate-regon Re t ) R K +, whch ndcates the set of feasble transmsson rates r t = r 1,t,..., r K,t ). Our pont of departure s the gradent-based schedulng framework n [1], [2]. In ths framework, at each schedulng nstant a rate vector r t Re t ) s selected that has the maxmum projecton onto the gradent of a system utlty functon UW t ), where K UW t ) = U W,t ), =1 and, for each user, U W,t ) s a ncreasng concave utlty functon of the user s average throughput, W,t, up to tme t. In other words, the schedulng and resource allocaton decson s the soluton to max UW t) T du x) r t = max r t Re t) r t Re t) dx r,t. 1) x=w,t For example, one class of utlty functons gven n [1], [33] s { c U W,t ) = W α,t) α, α 1, α 0, c logw,t ), α = 0, where α 1 s a farness parameter and c s a qualty of servce QoS) weght. In ths case, 1) becomes c W,t ) α 1 r,t. 3) max r t Re t) Wth equal QoS weghts, α = 1 results n a maxmum throughput rule that maxmzes the total throughput durng each slot. For α = 0, ths results n the proportonally far rule. 2)

4 4 TECHNICAL REPORT - JUNE 2009 The precedng polcy can be generalzed to allow the utlty to depend on other parameters such as a user s queue sze or delay. For example, consder the utlty U W,t, Q,t ) = c α W,t) α d p Q,t) p, where Q,t represents the queue length of user at tme t, d s a QoS weght for user s queue length and p > 1 s a farness parameter assocated wth the queue length. In ths case, 1) s replaced by 5 c W,t ) α 1 + d Q,t ) p 1) r,t. 4) max r t Re t) Specal cases of ths polcy wth c = 0 have been shown to be stablzng polces n a varety of settngs [3], [4], [17], [26]. In [27] t was shown that for specfc choces of c and d ths polcy wll maxmze the total network utlty c W α,t) α ) subject to a network stablty constrant. In general, we consder the problem w,t r,t, 5) max r t Re t) where w,t 0 s a tme-varyng weght of the th user at tme t. In the precedng examples, these weghts are gven by the gradent of the utlty; however, other methods for generatng these weghts are also possble. We note that 5) must be re-solved at each schedulng nstant because of changes n both the channel state and the weghts e.g., the gradent of the utlty). The former changes are due to the tme-varyng nature of the wreless channel, whereas the latter changes are due to new arrvals and past servce decsons. The soluton to ths problem depends on the state dependent capacty regon Re t ), whch we assume s known at tme t. 6 In ths paper, we consder a model that s approprate for a CDMA system, such as HSDPA or 1xEVDV. Ths model s parameterzed by two sets of physcal layer parameters: the number of spreadng codes, n and the transmsson power p assgned to each user. Each choce of these parameters specfes a PLOP, whch must satsfy the followng constrants: n N, 6) n N, 7) p P. 8) Here, 7) and 8) are system constrants on the total number of spreadng codes and the total system power, whle 6) s a per user constrant on the number of codes that can be assgned to user. We assume that all spreadng codes are mutually orthogonal, so that the only nterference s from other cells. Moreover, n a fully loaded system, the other cells use a constant total power and thus power allocaton per user and code does not have an mpact on the nterference. Hence, we assume that the nterference power s constant. We then let the channel state e ndcate user s receved sgnal-tonterference plus nose rato SINR) per unt power, where we have suppressed the dependence on t for convenence. 7 In ths case, the SINR per code for user s gven by SINR = p n e. We model the achevable rate per code by r n = Γζ SINR ). 5 Note that we take the negatve of the gradent of the utlty wth respect to queue length. Ths s because the queue length s decreasng n the transmsson rate assgned to a user whle the throughput s ncreasng. 6 Whle, n a practcal system, the exact channel state wll not be perfectly known at the transmtter, some estmate of t s usually avalable, for example, va channel qualty feedback. 7 In other words, f we neglect other cell nterference then e s smply the sgnal-to-nose rato SNR) of user per unt power.

5 TECHNICAL REPORT - JUNE Here, Γ corresponds to the Shannon capacty for a Gaussan nose channel wth the gven SINR,.e., Γx) = B log1+x), where B ndcates the symbol rate.e., the chp rate/spreadng factor), and ζ 0, 1] s a scalng factor that can be used to model the gap from capacty n a practcal system. Ths s a reasonable model for systems that use sophstcated codng technques, such as Turbo codes. Redefnng e to be e ζ, the rate regon s then { Re) = r 0 : r = n B log 1 + p ) e, n } 9) n N, n N, p P. Wthout the per-user code constrants, ths s equvalent to the achevable rate-regon obtaned n [29] for TDM, CDMA wthout multuser codng and FDM, where n each case the user s subject to constant fadng over the avalable degrees of freedom. Notce that n 9), we allow the number of codes per user to take on a non-nteger value. Of course, n a practcal system these must be nteger valued. However, we wll show that, n most cases, the soluton to ths relaxed problem results n nteger values for n. We can now state the optmzaton problem n 5) as where V := max V n, p) n,p) X subject to: n N, p P, [Prmal problem] 10) V n, p) := w n ln 1 + p ) e, 11) n X := { n, p) 0 : n N }, 12) n s a vector of code allocatons, and p s a vector of power allocatons. We have normalzed the objectve by B/ ln2) to smplfy notaton. Note that the constrant set X s convex. It can also be verfed that V s concave n n, p). A. Addtonal Constrants In addton to 6)-8), there may be several other constrants on the feasble PLOPs n a practcal system. Ths ncludes the followng per user constrants:.) peak power constrant: p P,..) maxmum SINR per code) constrant: SINR = p e n S p S n e,..) maxmum rate per code 8 constrant: r = ln 1 + p ) e R/N) n n 8 As n the prevous secton, we contnue to normalze the rate, r, by B/ ln2). p e R/N) 1) n e,.

6 6 TECHNICAL REPORT - JUNE 2009 v.) mnmum rate per code constrant: r = ln 1 + p ) e n n Ř/N) v.) maxmum rate constrant: v.) mnmum rate constrant: r = n ln 1 + p ) e R n p e Ř/N) 1) n e,. p e R /n 1) n e,. r = n ln 1 + p ) e n Ř p eř/n 1) n e,. These constrants can arse due to varous mplementaton consderatons. For example, a constrant on the rate per code s mposed by the maxmum or mnmum rate of the avalable modulaton and codng schemes: a modulaton order lmtaton usually results n the former and mnmum underlyng codng rate results n the latter. On the other hand, a maxmum rate constrant arses because there s only a fnte amount of data avalable to send to each moble at any tme. A mnmum rate constrant can be used to model the case where the system s tryng to guarantee a certan level of servce to that user. 9 All of the above constrants can be vewed as specal cases of a per user power constrant wth the form: SINR = p e n [š n ), s n )],, where the functon s n ) s also dependent on the fxed for a gven optmzaton problem) parameters P, S, e, R, R/N), and the functon š n ) s dependent on the parameters Ř, Ř/N). Non-negatvty restrctons on power necessarly mply that š n ) 0. We prmarly focus on two specal cases of ths: I. s n ) s and š n ) š do not depend on n, II. s n ) s = and š n ) s = 0. We refer to these as Type I and Type II per-user power constrants, respectvely. A Type I constrant models the case where there s a maxmum and mnmum constrant on the SINR or rate per code. A Type II constrant corresponds to no per-user power constrants. Wth the per user power constrants, the constrant set X s further restrcted to X := { n, p) 0 : n N, šn )n e p s n )n },. e The set X contnues to be convex f s n )n s a concave functon of n and š n )n s a convex functon of n. Note that s n )n s ndeed concave for the two specal cases I-II) mentoned above, as well as the case of a peak power constrant, and š n )n s always convex n the prevous examples. Unless otherwse mentoned, we wll assume ths set s convex n the followng. For the maxmum rate constrant case 13), s n )n s convex n n, and so the set X wll not be convex. However, one can stll get a convex formulaton [36] for ths case by nstead vewng the rate 9 Of course, wth mnmum rate and mnmum rate per code constrants the resultng optmzaton may be nfeasble, dependng on the other constrants and the channel states. 13)

7 TECHNICAL REPORT - JUNE r as an addtonal optmzaton varable, so that the objectve s now to maxmze w r, where r s constraned to satsfy r n log 1 + p ) e, n and r [0, R ]. The fnal soluton n ths case s qute smlar to the analyss that follows n ths paper. However, to smplfy our dscusson we do not consder ths constrant here and smply focus on cases I and II above. In addton to these per user power constrants, there may also be a constrant on the maxmum number of users M scheduled n a tme-slot,.e., users wth postve code and power assgnments. 10 We wll prove later see Lemma 4.9) that such a constrant wll n most cases automatcally be satsfed by the optmal soluton assumng the selected users have enough data to send) as long as M 1 users can fully utlze the avalable code budget,.e., the sum of the N s for any subset of M 1 users s greater than or equal to N. For example, f N 5 for all and N 15, then no more than 4 users need to be scheduled n any tme-slot under the optmal scheme. III. THE DUAL PROBLEM AND CONVEX OPTIMIZATION In ths secton we begn consderng the soluton to 10), whch determnes the users to be scheduled as well as the amount of power and the number of codes to be assgned to each user. We solve the optmzaton problem by lookng at the dual formulaton. The objectve s concave and snce the constrants are lnear, there wll be no dualty gap see [34]). Ths allows us to use the soluton of the dual to compute the soluton of the prmal. A. The Dual Problem Defne a Lagrangan for the prmal problem 10) by Ln, p, λ, µ) := λ w n ln P 1 + p ) e + n p ) + µ N n ). 14) The correspondng dual functon s Lλ, µ) := max Ln, p, λ, µ). 15) n,p) X The dual problem s then gven by: Also, wth some further abuse of notaton, we defne L := mn Lλ, µ) [Dual problem]. 16) λ,µ) 0 Lλ) := mn µ 0 Lλ, µ) = mn µ 0 max n,p) X Ln, p, λ, µ). 17) 10 For example, n HSDPA such a constrant arses because the system cannot schedule more users than the number of shared control channels.

8 8 TECHNICAL REPORT - JUNE 2009 B. Results from dualty and convex programmng From standard convex programmng see, e.g., Propostons and of [34]), we have the followng: Proposton 3.1: The dual functon Lλ, µ) s convex over the set {λ, µ) 0} and V Lλ) Lλ, µ), λ, µ 0. From the concavty of V and convexty of the doman of optmzaton, t s easy to verfy that Assumpton of [34] holds, and therefore, we have from Propostons 5.3.1, 5.1.4, and n [34] that Proposton 3.2: There exsts at least one soluton to the dual problem and there s no dualty gap. Any optmal dual soluton, λ, µ ) satsfes V = Lλ, µ ). Furthermore, n, p ), λ, µ )) s a par of optmal prmal and optmal dual solutons f and only f n, p ) X, n N, p P Prmal Feasblty λ, µ ) 0 Dual Feasblty n, p ) arg max Ln, p, λ, µ ) Lagrangan n,p) X Optmalty λ P p ) = 0, µ N n ) = 0 Complementary Slackness 18) 19) 20) 21) IV. STRUCTURE OF THE PRIMAL AND DUAL PROBLEMS In ths secton, we gve several propertes of the dual problem n 16) and the correspondng prmal problem n 10). Frst, we compute the dual functon, Lλ, µ) n 15) for a gven λ and µ. We then keep λ fxed and optmze the dual functon over µ; ths gves us Lλ) n 17). We prove that Lλ) s convex and provde bounds on the optmal λ. Usng these propertes, the optmal λ can be found wth a one-dmensonal convex search that has geometrc convergence. We fnd prmal varables n and p) that maxmze the Lagrangan for a gven λ and µ, and fndng the optmal prmal power allocaton for a gven n. A. Computng the dual functon To evaluate the dual functon, we proceed n two steps. Frst, we optmze the Lagrangan 14) over p, for a fxed λ, µ, and n. We then optmze over n to obtan the value of the dual functon. For the frst step, we defne the followng two projectons of the set X : for a gven n, let X n = {n 0 : n N, } and let X p n) = {p : n, p) X }. Then we have: Lemma 4.1: For a fxed n X n and any λ 0 and µ 0, the power allocaton p X p n) that maxmzes Ln, p, λ, µ) s gven by where p = n e s w e λ, s n ), š n ) s w e ) λ, s n ), š n ) { { w e ) := max mn λ 1, s n ) ), 22) } }, š n ). Ths lemma follows drectly from the Kuhn-Tucker condtons for the optmzaton problem. Note that the mn s not needed for Type II per user power constrants,.e., s n) =. However, the maxmum

9 TECHNICAL REPORT - JUNE p*, type I constrant p*, type II constrant λ Fg. 1. An example of the optmal power allocaton, p n 22) as a functon of λ for both a Type I and type II power constrant. s stll necessary even f š n ) = 0, to restrct attenton to non-negatve power values. The soluton can be vewed as a modfed verson of a water-fllng power allocaton across the users [32], where the waterlevel s modfed to take nto account each users weght, w, and the per-user power constrants are also taken nto account. In the case of a Type I per-user power constrant s n ) s and š n ) š ), the resultng SINR per code for a fxed λ, µ, and n s gven by p e n = s w e ) λ, s n ), š n ) = s w e λ, s, š ), 23) whch does not depend on the number of codes n. It follows that, n the Type I case, for a gven λ the total power allocated to a user scales lnearly n the number of codes. An example of p as a functon of λ s shown n Fg. 1 for both a Type I and Type II constrant. The horzontal segments of p under the Type II constrant correspond to when the maxmum and mnmum per user power constrants are actve; when these are not actve, the two curves overlap. Substtutng 22) nto the Lagrangan we have Ln, p, λ, µ) = w n ln + λ ) 1 + p e P p n ) + µ N n ) 24) where = w n hw e, s n ), š n ), λ) µn ) + λp + µn, hw e, s n ), š n ), λ) := ln1 + š n )) λ w e š n ), λ w e, 1+š n ) λ w e 1 ln λ w w e, λ [ e, w e ), 1+s n ) 1+š n ) ln1 + s n )) λ w e s n ), λ < w e. 1+s n ) 25) 26)

10 10 TECHNICAL REPORT - JUNE Type I constrant Type II constrant 3 ln1 + s ) w e w e ln1+š )w e w 1+s 1+š e š λ Fg. 2. An example of hw e, s, š, λ) as a functon of λ under a Type I and Type II power constrant. Notce that for a Type I per-user power constrant, hw e, s n ), š n ), λ) = hw e, s, š, λ) also does not depend on n. For a Type II per-user power constrant, 11 [ )] λ λ hw e, s, š, λ) = 1 ln 1 {w e w e w e >λ}. An example of hw e, s, š, λ) as a functon of λ s shown n Fg. 2 for both a Type I and Type II per-user power constrant. In both cases w e = 5. When w e 1+s λ w e 1+š the two curves overlap. For λ < w e 1+s, h grows wthout bound under a Type II constrant, whle t s lnear n ths range under a Type I constrant. For λ > w e 1+š, h decreases lnearly under a Type II constrant, whle under a Type I constrant t converges to 0 at λ = w e. For a Type II constrant, h crosses the x-axs at λ = ln1+š )w e š. In ether of these cases, snce 25) s lnear n n, t s straghtforward to optmze over n. Lemma 4.2: Wth a per-user power constrant of Type I or II, the vector of code allocatons, n, that maxmzes 25) s gven by { 0, n µ λ) < µ, = 27) N, µ λ) > µ, where µ λ) = w hw e, s, š, λ). 28) If µ = µ λ), every choce of n such that 0 n N maxmzes the Lagrangan. In other words, gven µ, the optmal code allocaton s determned for each user by checkng f µ λ) s greater than or less than µ. The last part of ths lemma follows because when µ = µ λ), 25) s not dependent on n. Usng 27) we have 12 ) w n ln 1 + p e λp n µn = [µ λ) µ] + N. Substtutng ths nto 25) yelds the followng characterzaton of the dual functon Lλ, µ). Lemma 4.3: Wth a Type I or II per-user power constrant, Lλ, µ) = [µ λ) µ] + N + µn + λp. 29) 11 The notaton 1 X denotes the ndcator functon of the event X. 12 We use the notaton [x] + = maxx, 0).

11 TECHNICAL REPORT - JUNE µ 1 λ) µ 2 λ) µ 3 λ) µ 4 λ) λ Fg. 3. An example of µ λ) for a system wth K = 4 users and a Type I per-user power constrant. B. Optmzng over µ We now turn to optmzng the dual functon over µ. We restrct our attenton to ether a Type I or Type II per-user power constrant, so that the dual functon s gven by 29). To begn, we sort the users n decreasng order of µ λ) n 28), where tes are broken arbtrarly. Assume that the users are numbered correspondng to ther poston n ths orderng,.e. so that µ λ) µ +1 λ) for all. 13 Let j 1 be the largest nteger such that µ j 1λ) 0 and j 1 =1 N < N. If no such user can be found, set j = 1. Note that f š = 0 for all, then µ λ) 0 for all, n whch case j wll be the frst user that would fll up the total code budget f all users receved ther maxmum per-user code allocaton. By conventon set µ K+1 λ) = 1 [µ K λ)], where [x] = [ x] +. Let N j := N j 1 =1 N. Lemma 4.4: Wth a Type I or Type II per-user power constrant, Lλ) := mn Lλ, µ) µ 0 = j 1 =1 and the mnmzng µ s gven by µ λ) := [µ j λ)] +. µ λ)n + [µ j λ)] + N j + λp, 30) Proof: For µ λ) < µ < µ 1 λ), from 29) t can be seen that the dervatve of Lλ, µ) n µ s gven by N 1 j=1 N. Hence, j s the largest nteger for whch Lλ, µ) wll be ncreasng n the correspondng nterval,.e., Lλ, µ) wll be ncreasng f and only f µ > µ j λ). The lemma then follows. From Lemma 4.2, µ s a threshold separatng the users that get ther full code allocaton from the users that get allocated no codes. As µ s decreased, more users wll be allocated ther full code allocaton. Lemma 4.4 shows that the threshold µ λ) that mnmzes the dual functon s such that the full code budget s utlzed. Fgure 3 shows an example of the curves µ λ) as a functon of λ for a system wth K = 4 users, under a Type I per-user power constrant. Also ndcated on the fgure are the values of λ for whch each curve µ λ) crosses the x-axs. Consder the case where N = N for all. In ths case, j = 1.e. the user wth the maxmum value of µ λ) for the gven value of λ. Therefore, for λ < ln1+š 2)w 2 e 2 š 2, µ λ) wll be the upper envelope of the curves shown n the fgure. For λ > ln1+š 2)w 2 e 2 š 2 all of the µ λ) wll be less than 0 and so µ λ) = Of course, as λ changes ths orderng wll change, n whch case we must re-number the users.

12 12 TECHNICAL REPORT - JUNE 2009 Remark: When w w j, e > e j, and s s j then t can be shown that µ λ) µ j λ), for all λ. It follows that n ths case, user wll be always be gven a full code allocaton before allocatng any codes to user j. Furthermore, assume the schedulng rule s the maxmum throughput verson of 3),.e. the case where α = 1 and the class weghts are all equal, so that the w s are constant and dentcal across users. In ths case, stll assumng that f e > e j then s s j ) packng users nto the code budget n order of decreasng e s s optmal. C. Fndng a Lagrangan Optmal Prmal Soluton. We next consder fndng prmal values n, p ) such that n, p ) = arg max Ln, p, λ, µ λ)) 31) n,p) X for a gven λ 0. Here, µ λ) s the optmal µ gven by Lemma 4.4. Gven the optmal λ = λ, then from Proposton 3.2, such an n, p ) wll be an optmal soluton for the prmal problem f t also satsfes prmal feasblty 18) and complmentary slackness 21). We gve a procedure for selectng such a par n the followng. If the λ λ, ths procedure can also be used to fnd a canddate feasble ñ. In the next secton, we construct a feasble p correspondng to ñ. From Proposton 3.1, we have 14 V V ñ, p) Lλ) V ñ, p). We contnue restrctng our attenton to Type I or II per-user power constrants. From the results n Sectons IV-A and IV-B, t can be seen that a soluton to 31) s equvalent to fndng n = arg max {n X } µ λ) µ λ)) + n, 32) and settng p as n Lemma 4.1. As n the prevous secton, we agan assume that the users are ordered n decreasng order of µ λ) so that µ λ) = µ j λ). When 15 µ j 1λ) > µ j λ) > µ j +1λ) and µ j λ) 0, then there s a unque feasble n that optmzes 32) and satsfes µ λ)n n ) = 0. Ths s gven by N, < j, n N j =, = j and µ λ) 0, 0, = j and µ 33) λ) = 0, 0, > j. Note that ths soluton wll always satsfy n N, wth equalty f µ λ) > 0. Also note that n 33) s always an nteger code allocaton. Defnton 4.1: A scalar d R s a subgradent of Lλ) at λ f n L λ) Lλ) + λ λ)d, λ 0. Proposton 4.1: Let ˆn, ˆp) be a soluton to 31) for a gven λ whch satsfes ˆn N, and µ λ)n ˆn ) = 0. Then P ˆp s a subgradent of Lλ) at λ. 14 Ths can be used as a stoppng crteron n a practcal teratve algorthm. 15 Recall that by conventon µ K+1λ) = 1 [µ K].

13 TECHNICAL REPORT - JUNE Proof: Usng the defnton of µ λ) we have L λ) = L λ, µ λ)) = max Ln, p, λ, µ λ)) n,p) X Lˆn, ˆp, λ, µ λ)) = V ˆn, ˆp) + λp ˆp ) + µ λ)n V ˆn, ˆp) + λp = V ˆn, ˆp) + λp + λ λ)p ˆn ) ˆp ) 34) ˆp ) ˆp ) = Lλ) + λ λ)p ˆp ). 35) The nequalty n 34) follows because N ˆn 0 and µ λ) 0; equalty n 35) holds because µ λ)n ˆn ) = 0. Note that the code allocaton gven by 33) and the correspondng power allocaton n Lemma 4.1 satsfy the assumptons of Proposton 4.1 and so provde a subgradent of Lλ). Later n Corollary 4.1, we show that all subgradents of Lλ) can be found n ths way. When there s a te and more than one µ j λ) = µ λ), then there may be multple n that optmze 32) and satsfy µ λ)n n ) = 0 and n N. There wll also be multple canddates for n f there s no te, but µ j = However, for the optmal λ, every such n may not result n a power allocaton that s feasble and satsfes complmentary slackness. For an arbtrary λ, dfferent choces of n wll result n dfferent subgradents for Lλ). Next, we examne resolvng such tes. Frst, we show how to resolve these tes to fnd the maxmum and mnmum subgradents of Lλ). 17 Let there be l 0 users wth < j and k 1 users wth j whose µ λ) are ted wth µ j λ), where l + k 1,.e., 18 µ j l 1λ) > µ j lλ) = µ j λ) = µ j +k 1λ) > µ j +kλ). Let I λ = [j l, j + k 1] denote the set of these users. The objectve n 32) wll not depend on n, for I λ. Note that the orderng of these users based on µ λ) s arbtrary. Frst we consder resolvng ths te to fnd the maxmum subgradent of Lλ) at λ. It follows from 16 It can be seen that f š = 0, then the case of µ j λ) = 0 s trval because user j wll not receve any power regardless of ts code allocaton. 17 That these are ndeed the maxmum and mnmum follows from Corollary The case where l + k = 1 captures the stuaton where there are no tes and µ j = 0.

14 14 TECHNICAL REPORT - JUNE 2009 Lemma 4.1 and Corollary 4.1 that ths s the soluton to the followng lnear program LP): max {n I λ } P res I λ s w e λ, s, š ) n e subject to: 0 n N, I λ n N res, I λ µ λ)n res I λ n ) = 0. [LPmax] Here, P res := P <j l s w e, s ) λ, š N e and N res := N <j l N are the resdual power and codes avalable for the users n the te. The mnmum subgradent can also be found va a LP gven by mn {n I λ } P res I λ s w e λ, s, š ) n e. [LPmn] subject to the same constrants as n LPmax. The structure of these lnear programs permts a smple greedy soluton. For LPmax, f µ λ) = 0, then the soluton to LPmax s clearly to assgn ˆn = 0 for all I λ. Otherwse, f µ λ) > 0, order the users n I λ n ncreasng order of s w e, s ) λ, š 1 e. Let ˆΘ : I λ I λ be a permutaton of I λ accordng to ths orderng, so that f s w e, s ) λ, š 1 e < s w j e j, s ) λ j, š 1 j e j, then ˆΘ) < ˆΘj). For LPmn, we nstead order the users n decreasng order of s w e λ, s, š ) 1 e and denote ths orderng by the permutaton ˇΘ. Let ĵ be the smallest nteger such that ĵ =j l N ˆΘ 1 ) N res; f no such nteger exsts, set ĵ = j + k 1. Let ǰ denote the correspondng nteger usng the ˇΘ orderng. For I λ, set N, ˆΘ) < ĵ, ˆn = N, ˆΘ) = ĵ, 0, ˆΘ) > ĵ, where N ˆΘ 1 = mn{n ĵ) res ĵ 1 =j l N ˆΘ 1 ), N ˆΘ 1 ĵ) }. Let ň denote the correspondng code allocaton usng the ˇΘ orderng. Lemma 4.5: The code allocaton ˆn n 36) solves LPmax for µ λ) > 0; the correspondng code allocaton ň solves LPmn, for all values of µ λ). When µ λ) = 0, the soluton to LPmax s ˆn = 0 for all I λ. The proof of ths lemma follows from a smple nterchange argument. Fndng both of these solutons nvolves a sort over the users nvolved n a te, and thus each have a complexty of O I λ log I λ )). Typcally, f a te occurs, only a small number of users wll be nvolved. To gan some ntuton as to why ths s the case, note that tes occur whenever two or more of µ λ) curves n Fg. 3 cross for a gven value of λ. Moreover, t can be shown that any two such curves wll only cross at one pont. Hence, t follows that f the parameters w and e are ndependently chosen accordng to an absolutely contnuous dstrbuton, then wth probablty one a te wll not nvolve more than two users. Gven the soluton to LPmax n 36), let n = N, < j l, ˆn, j l ˆΘ) j + k 1, 0, j + k. denote the correspondng complete code allocaton. In two specal cases, ths wll be a prmal optmal code allocaton. Lemma 4.6: The par n, p ) gven by 37) and 22) are a prmal optmal soluton f ether 36) 37)

15 TECHNICAL REPORT - JUNE ) λ = 0 and LPmax has a non-negatve soluton, 2) The soluton to LPmax s zero. Ths lemma follows drectly from notng that n both of these cases, the soluton wll satsfy both the complmentary slackness and prmal feasblty condtons n Prop Note that when λ = 0, s w e, s λ, š ) = s for all, 19 and thus the ˆΘ-orderng corresponds to sortng the users based on s e. A correspondng code allocaton can be defned based on ˇΘ and ň ; f ths results n a soluton to LPmn of zero, then t wll also be prmal optmal. If the soluton to LPmax s negatve, then all the subgradents of Lλ) at λ wll be negatve. Lkewse, f the soluton to LPmn s postve, then all the subgradents wll be postve. However, f LPmax has a postve soluton and LPmn has a negatve one, then Lλ) wll have a zero subgradent at λ; a feasble code allocaton correspondng to ths zero subgradent wll be prmal optmal. In ths case, there must exst an α [0, 1] such that ) w e ) P res = α λ, s ˆn, š e Solvng for α above, set I t s + 1 α) I t s w e λ, s, š ) ň e ) ñ = αˆn + 1 α)ň 38) for all I t and let n denote the correspondng complete code allocaton as n 37). Lemma 4.7: If the soluton to LPmax s postve and the soluton to LPmn s negatve, then n constructed usng 38) and the correspondng p are a prmal optmal soluton. Once agan, ths follows from notng that by constructon the code and power allocatons satsfy the assumptons n Prop Ths gves a prmal optmal soluton; but dependng on the number of users nvolved n the te, t may not be the prmal soluton wth the mnmum number of users scheduled. As dscussed n Sect. II-A, n practce there may be constrants on ths number. The next lemma gves an upper bound on the mnmum number of users scheduled n an optmal soluton. Usng typcal parameter values for a HSDPA system, ths bound wll be no greater than 4. Lemma 4.8: For a Type I or II power constrant, an optmal code allocaton can always be found such that at most N/N mn + 1 users wll be scheduled, where N mn := mn N. Proof: At the optmal λ, f the condtons n Lemma 4.6 are satsfed then the code assgnment n 37) s optmal and wll result n no more than N/N mn + 1 users scheduled. Therefore, we need only consder the case where these condtons are not satsfed,.e., λ > 0 and the soluton to LPmax s strctly greater than 0. When λ > 0, from complementary slackness and Prop. 4.1, a prmal optmal code allocaton must result n a zero subgradent of Lλ). Such a code allocaton s a soluton to the followng feasblty problem: maxmze n 1 19 Ths wll arse only wth a Type I power constrant. subject to: P 1 n s w e ) e λ, s, š = 0 n = N 0 n N,..

16 16 TECHNICAL REPORT - JUNE 2009 Ths s a LP and the feasble set s a K dmensonal bounded polyhedron. 20 By Lemma 4.7, ths polyhedron s non-empty,.e. the LP has a soluton. However, the soluton gven n Lemma 4.7 may result n more than N/N mn + 1 users scheduled. In ths case, we show that ths LP must have another soluton wth the desred property. In partcular, t must have an extreme pont soluton; we consder such an extreme pont code allocaton. At an extreme pont, at least K constrants must be bndng, two of whch are the two equalty constrants. Ths means that at least K 2 users must have n set equal to ether 0 or N and so at most 2 users wll have a fractonal code assgnment. Frst, assume N/N mn s an nteger. If N/N mn users have n = N, then clearly to satsfy the second constrant, no other users can have postve code allocatons. Lkewse, f no more than N/N mn 1 users have n = N, then from the above argument at most N/N mn = N/N mn + 1 users wll have a postve code allocaton. Smlarly, f N/N mn s not an nteger, then at most N/N mn 1 users can have n = N to satsfy the second equalty, and so at most N/N mn + 1 users wll have a postve code allocaton. Though n general 37) may result n more than N/N mn + 1 users beng scheduled, n several key specal cases ths soluton wll also nvolve no more N/N mn + 1 users. Ths s useful n practce, snce determnng the soluton n 37) s less complex than solvng the LP n the proof of Lemma Lemma 4.9: For a Type I or II power constrant, the code allocaton n 37) results n no more than N/N mn + 1 users beng scheduled n ether of the followng cases: 1) At most two users are nvolved n a te; 2) For all users I λ, N N res. The second condton n ths lemma mples that the per-user code constrants wll be nactve for any soluton to LPmax or LPmn. 22 In ths case, the soluton to LPmax and LPmn wll nvolve one user each and the combnaton n 38) wll nvolve only these two users. 23 Note that when N = N, ths condton wll always be satsfed. Based on the above dscusson, we outlne a procedure for fndng a prmal feasble n gven an arbtrary λ. Ths can be used to construct a feasble soluton n a sub-optmal algorthm, whch does not fnd the optmal λ. Te breakng rule: 1) Solve LPmax, f the soluton s non-postve, or λ = 0, resolve the te usng ˆn. 2) Otherwse, solve LPmn, a) If the soluton s negatve use ñ n 38) to resolve the te, b) otherwse use ň. For a gven λ, we denote by n λ) the code allocaton gven by usng ths te breakng rule. If the optmal choce of λ s used, n λ) wll be an optmal code allocaton. Otherwse, t s the allocaton that corresponds to the mnmum postve subgradent f all subgradents are postve) or the maxmum negatve subgradent f all subgradents are negatve). D. Optmzng the power allocaton In ths secton, we consder the optmal prmal power allocaton, p, gven a fxed non-negatve code allocaton n,.e., we want to solve V n) := max V n, p) p X pn) subject to: p P. 20 Note, for convenence we formulate ths LP as a functon of all K users nstead of just the I λ users nvolved n the te. 21 Solvng ths nvolves lstng all the extreme ponts and determnng the one that works. 22 In practcal systems, ths condton wll often be satsfed. For example, n a HSDPA system wth N = 15 and N = 15 or 10, then ths condton wll always be satsfed. 23 If µ λ) = 0, then the soluton of LPmax wll nvolve zero users, and the combnaton n 38) wll nvolve only one user. 39)

17 TECHNICAL REPORT - JUNE Ths can be solved by fndng λ n) usng the dual formulaton and then computng the optmal p n) as n Lemma 4.1. We note that the results n ths secton are not restrcted to Type I or Type II per user power constrants but wll hold for any reasonable per-user constrants. 24 not just those dscussed n Secton II-A. Wthout loss of generalty, we remove any users wth zero code allocatons. Let M be the number of remanng users wth postve code allocaton, and assume these are numbered = 1,..., M. We frst need to check f the problem s nfeasble,.e., f M =1 p mn := n e š n ) P. If ths s the case, then 39) wll have no feasble solutons. We also check f the sum power constrant s nactve,.e., M =1 p max := n e s n ) P. If ths s the case, the optmal power allocaton s smply p = n e s n ). Henceforth, we assume the problem s feasble and the power constrant s actve. In ths case, the sum power constrant must be satsfed wth equalty for the optmal powers, otherwse at least one of the powers can be ncreased resultng n a larger value of the objectve functon. We can now construct a Lagrangan for 39) as M L n p, λ) := w n ln 1 + p ) e =1 + λ n P p ). 40) Notce that f µn n ) = 0, L n p, λ) wll be equal to the orgnal Lagrangan n 14). The dual functon correspondng to 40) s gven by L n λ) := max L n p, λ). 41) p X pn) Also, note that when optmzng over powers, the constrant set s always convex regardless of the functon s n )n. Maxmzng L n p, λ) over p s essentally the same as the problem for Lp, n, λ, µ) covered n Secton IV-A. The optmal p s gven by 22) as before. Substtutng ths nto 41) yelds L n λ) = M w n hw e, s n ), š n ), λ) + λp. =1 From basc convex optmzaton theory, we know that L n λ) s convex n λ. Furthermore, t can be shown that L n λ) s contnuously dfferentable n λ. To see ths note that from 26), for each, d hw e, s n ), λ) = d λ šn ) w e, 1 w e 1 λ, w e s n ) w e, λ < w e λ, 1+š n ) w e λ < w e, 1+s n ) 1+š n ) 1+s n ), 24 By reasonable constrants we refer to constrants such that 0 š n ) s n ). 42)

18 18 TECHNICAL REPORT - JUNE 2009 whch s contnuous n the three ntervals as well as at the two break ponts. Ths allows us to conclude that L n λ) s mnmzed by the set ponts at whch the dervatve s zero. Note that for each user, 42) s constant n two of the three ntervals; hence, t s possble that there are multple ponts at whch the dervatve s zero. The followng lemma gves an alternatve characterzaton of the λ whch mnmzes L n λ). Let a and b be the two break ponts for each user = 1,..., M,.e., a := w e, and 1+s n ) w e 1+š n ). b = Lemma 4.10: A λ > 0 s the soluton to the dual problem mn λ 0 L n λ) f and only f λ = P P n P n w 1 [a,b )λ) e s n )1 [0,a )λ) š n )1 [b, )λ)+1 [a,b )λ)), 43) where, by conventon, f numerator and denomnator of the rght-hand sde are both zero, then we set ths equal to λ. Proof: Note that whle the optmal λ may not be unque, the set of optmzers must form an nterval by the convexty of L n λ). Snce for any gven λ, the p that maxmzes the Lagrangan s unque, t follows from complementary slackness that λ > 0 s optmal f and only f the correspondng p satsfes p = P. Substtutng n p from 22) we have that λ > 0 s optmal f and only f n n š n )1 [b, )λ) e P = + s n )1 [0,a )λ) + e n w e e λ 1)1 [a,b )λ). The desred result then follows from smple algebra. Note that f the rght-hand sde of 43) s 0, then the 0 frst term on the left-hand sde of 44) must be zero. Ths corresponds to all users ether beng assgned ther maxmum or mnmum ndvdual power, n such a way that the total power constrant s exactly met. Such a power allocaton, wll not depend on small varatons n λ, provded that λ does not enter a new nterval n 42) for some user. 25 Let λ n) denote an optmal value of λ for a gven code allocaton, and let p n) denote the correspondng optmal power allocaton gven by 22). Ths lemma says that f λ n) > 0, t must satsfy 43). Next we show that a soluton to ths equaton can be found n fnte-tme. Sort the set {a, b = 1,..., M} nto a decreasng set of numbers {x[l]; l = 1,..., 2M}, where tes are resolved arbtrarly. For l = 1,..., 2M, let P sum [l] denote the total power p where p s gven by 22) wth λ = x[l]. Let l be the smallest value of l such that P sum [l] P. Assumng that λ n) > 0 such an l must exst.) Lemma 4.11: For a gven n, f the sum power constrant s actve, 26 an optmal λ n) can be found n fnte-tme and s gven by the rght-hand sde of 43) wth λ = x[l ]. Proof: Note that as λ decreases, the rght-hand sde of 43) s rght-contnuous and only changes values when λ = x[l], l = 1,..., 2M. Durng any nterval when the rght-hand sde s 0, by our conventon, 0 the value changes contnuously n λ; but ths does not effect the followng argument.) Hence, an optmal λ must be gven by evaluatng the rght-hand sde of 43) wth λ = x[l] for some l = 1,..., 2M. Also, note that as λ decreases, the total power, p s ncreasng. By assumpton the sum power constrant s actve at the optmal soluton. Thus, we have x[l 1] > λ n) x[l ]. Combnng these observatons, the lemma follows. The dea behnd ths lemma s llustrated n Fg. 4, whch shows an example where only two users have postve code allocatons. The optmal power allocaton for each user, p from 22) s shown as a functon of λ, as well as the total power p 1 + p 2. In ths example, for a total power of P, x[l ] = a 1, and the optmal λ can then be calculated usng Lemma Indeed, t follows that ths s the only case n whch the optmal λ s not unque. 26 We make ths assumpton for smplcty of exposton. The algorthm can easly be modfed to take nto account the case where ths constrant s not actve and wll stll complete n fnte tme. 44)

19 TECHNICAL REPORT - JUNE p* 1 λ) p* 2 λ) p* 1 λ)+p* 2 λ) P a a b b 2 λ Fg. 4. Example llustratng Lemma Lemma 4.11 provdes an algorthm for solvng 43) by calculatng P sum [l] startng wth l = 1 and stoppng when the total power constrant s volated. Also, note that wth the above orderng, the rghthand sde of 43) can be recursvely calculated as l ncreases. The algorthm complexty s OM log M) due to the sort of {x[l]}. Recall, M s the number of users wth postve code allocatons. As dscussed after Lemma 4.9, ths wll typcally be on the order of 1-4. Also, note that under a type II per-user power constrant, a = 0. Thus wth no per-user power constrants, only the M values of x[] correspondng to the b s need to be consdered n the above search, and a smpler algorthm results. E. Optmzng the dual over λ Recall, Lλ) s the mnmum of the dual functon over µ 0. The soluton to the dual problem, L s thus gven by L = mn λ 0 Lλ). We consder ths problem and several characterstcs of Lλ) n the followng. Frst we show that Lλ) s convex n λ. 27 Lemma 4.12: Wth a Type I or Type II per-user power constrant, Lλ) s convex n λ. Proof: From Lemma 4.4, Lλ) = j 1 =1 µ λ)n + [µ j λ)] + N j + λp, where the users are re-ordered accordng to µ λ) for each λ. Ths can be re-wrtten as: Lλ) = max µ λ)n + λp n N where, N = { n : = max n N L nλ), 45) n N, 0 n N, }. 27 Ths lemma also follows from Prop. 4.1, snce a functon wll only have a subgradent at every pont f t s convex. Here we gve an alternatve proof that does not rely on subgradents.

20 20 TECHNICAL REPORT - JUNE 2009 We have already establshed n Sect. IV-D that for each n, L n λ) s convex n λ. Snce the maxmum of a set of convex functons s also convex, t follows that Lλ) s convex. In 45), Lλ) s expressed as the maxmum of an nfnte number of the functons L n λ). Next we show that n fact only a fnte number of such functons are needed to characterze Lλ), e.g. Lλ) = max L n λ) 46) n N Π where N Π s a fnte subset of N. Specfcally, from Lemma 4.4, t follows that for each permutaton of the users, we only need to consder a sngle greedy code allocatons whch uses all the codes,.e. a code allocaton as n 33) that sequentally assgns each user the maxmum feasble number of codes untl the code budget s full. We can then set N Π to be the set of such code allocatons, one for each permutaton. Now we turn to fndng the optmal λ. From Lemma 4.12, ths s the mnmum of an unvarate convex functon, and so t can be found by usng a one-dmensonal convex search technque, such as the bsecton method or a Fbonacc search [34]. Also note that, from 22) f λ > ln1+š ) š w e, then user wll be allocated zero power. Therefore the optmal λ, must satsfy 0 λ ln1 + š ) max w e max w e. 47) š These bounds provde a startng pont for the algorthms consdered n the next secton. As noted n Secton IV-D, L n λ) s contnuously dfferentable. From 46), we then have: Lemma 4.13: Wth a Type I or II per user power constrant, Lλ) s dfferentable for all λ for whch there exsts a unque n N Π, wth L n λ) = Lλ). When there s not a unque n N Π, ths s exactly the te case dscussed n Secton IV-C. Ths s llustrated n Fg. 5. Shown are three curves L n λ) correspondng to dfferent code allocatons; Lλ) s the upper envelope of these curves whch s shown n bold. Lλ) s dfferentable, except for at the two ndcated places where a te occurs. At the te values, the dervatves of the L n λ) curves nvolved n the te wll be the correspondng subgradents dscussed n Secton IV-C. Indeed, as the next corollary shows, any subgradent of Lλ) can be found n ths way. Corollary 4.1: Gven any subgradent d of Lλ) at λ, there exsts prmal values ˆn, ˆp) that satsfy the assumptons of Proposton 4.1 so that P ˆp = d. Proof: At any λ, f L n λ) = Lλ) for some n N Π, then the prmal values n, p) whch defne L n λ) wll satsfy the assumptons of Proposton 4.1 and gve a subgradent of Lλ) that corresponds to the dervatve of L n λ) at λ. If there s a unque n N Π, wth L n λ) = Lλ), then from Lemma 4.13, Lλ) s dfferentable and so has only one subgradent, whch s gven by the above. Next consder the case where there are multple n N Π such that L n λ) = Lλ). Snce each L n λ) s contnuously dfferentable and convex and Lλ) s the maxmum of these, t follows that the maxmum subgradent of Lλ) must be gven by the dervatve of L n +λ), where n + s one of the n nvolved n the te that satsfes Lλ + ɛ) = L n +λ + ɛ) for small enough ɛ. Lkewse, the mnmum subgradent must be gven by the dervatve of L n λ), where n s one of the n nvolved n the te that satsfes Lλ ɛ) = L n λ ɛ) for small enough ɛ. Any other subgradent can be found by consderng a code allocaton that s an approprate convex combnaton of the maxmum and mnmum. As λ decreases from the upper bound n 47), users receve a postve code allocaton based on the orderng of ln1+š ) š w e. For large enough λ ths orderng can determne the optmal code allocaton. To be precse, for the remander of ths secton, consder the case where š = 0 for all. In ths case, ln1+š ) š w e = w e by takng a lmt as š 0). Assume the users are ordered n decreasng order of w e, n the case of a te, order the users n decreasng order of w. If the w s are also ted, then order the users arbtrarly. Let Φ be a permutaton of the users correspondng to ths orderng. Usng ths permutaton, let j denote the smallest value j such that j 1 =1 N Φ 1 ) < N j N Φ 1 ). =1