Joint Scheduling and Power Allocations for. Traffic Offloading via Dual-Connectivity

Transcription

1 Jont Schedulng and Power Allocatons for 1 Traffc Offloadng va Dual-Connectvty Yuan Wu, Yanfe He, Lpng Qan, Janwe Huang, Xuemn (Sherman) Shen arxv: v1 [cs.ni] 3 Sep 215 Abstract Wth the rapd growth of moble traffc demand, a promsng approach to releve cellular network congeston s to offload users traffc to small-cell networks. In ths paper, we nvestgate how the moble users (MUs) can effectvely offload traffc by takng advantage of the capablty of dual-connectvty, whch enables an MU to smultaneously communcate wth a macro base staton (S) and a small-cell access pont (AP) va two rado-nterfaces. Offloadng traffc to the AP usually reduces the MUs moble data cost, but often at the expense of sufferng ncreased nterferences from other MUs at the same AP. We thus formulate an optmzaton problem that jontly determnes each MU s traffc schedule (between the S and AP) and power control (between two rado-nterfaces). The system objectve s to mnmze all MUs total cost, whle satsfyng each MU s transmt-power constrants through proper nterference control. In spte of the non-convexty of the problem, we desgn both a centralzed algorthm and a dstrbuted algorthm to solve the jont optmzaton problem. Numercal results show that the proposed algorthms can acheve the close-to-optmum results comparng wth the ones acheved by the LINGO (a commercal optmzaton software), but wth sgnfcantly less computatonal complexty. The results also show that the proposed adaptve offloadng can sgnfcantly reduce the MUs cost,.e., save more than 75% of the cost wthout offloadng traffc and 65% of the cost wth a fxed offloadng. I. INTRODUCTION Wth the rapd growth of smart handheld devces and moble nternet servces, traffc demand n cellular networks has been growng tremendously [2], whch causes fuent congestons and mposes a heavy burden on network operators. A cost effectve approach to releve network congeston s to offload the traffc of moble users (MUs) to spatally spread small-cell networks, e.g., femtocells and WF access Y. Wu and Y. He are wth College of Informaton Engneerng, Zhejang Unversty of Technology, Hangzhou, Chna, (emal: ewuy@zjut.edu.cn). L. Qan s wth College of Computer Scence and Technology, Zhejang Unversty of Technology, Hangzhou, Chna, (emal: lpqan@zjut.edu.cn). J. Huang s wth the Network Communcatons and Economcs Lab, Department of Informaton Engneerng, The Chnese Unversty of Hong Kong, Hong Kong (e-mal: jwhuang@e.cuhk.edu.hk). S. Shen s wth the Department of Electrcal and Computer Engneerng, Unversty of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mal: xshen@bbcr.uwaterloo.ca)

2 2 ponts (APs) [3] [4]. From the network operators perspectve, offloadng traffc can effectvely explot the addtonal network capactes provded by the mult-ters small cells and reduce the needs of costly and tme-consumng network nfrastructure upgrade. From the MUs perspectve, offloadng traffc reduces ther costs, snce small cells usually offer a low prce for moble data than cellular operators. Traffc offloadng becomes ncreasng attractve, as an ncreasng percentage of moble devces are equpped wth multple rado-nterfaces that facltate to flexbly smultaneous connectons to multple networks [5] [7]. In partcular, a new paradgm of small-cell dual-connectvty s ganng momentum n both ndustry practce [8] [9] and the 3GPP LTE-A standardzng actvtes [1] [1]. Through the dual-connectvty, each MU can smultaneously communcate wth a macro base staton (S) and a small-cell AP va two dfferent rado-nterfaces. Ths enables the MUs to flexbly schedule ther traffc between two networks to acheve effcent traffc offloadng. For nstance, an MU can schedule ts delay-senstve small-volume data traffc (e.g., voce-over-ip traffc) to the macro S, and offload ts delay-tolerant large-volume traffc (e.g., fle downloadng/uploadng) to a small-cell AP at the same tme. Nevertheless, dfferent from the centralzed resource management and orthogonal channel allocaton n cellular networks, a small-cell AP (such as WF) often allows multple MUs to share the same channels n a dstrbuted fashon, whch leads to sgnfcant mutual nterferences among the MUs. It means that f each MU aggressvely offloads ts traffc to the AP, then each MU needs to spend sgnfcant amount of transmt-power to the AP to combat the mutual nterferences, whch mght sgnfcantly reduce the beneft from offloadng traffc. Several pror studes have taken such nterferences nto consderaton when desgnng data offloadng algorthms [11] [12] [13]. However, the nterference-aware traffc offloadng desgn wth dual-connectvty s stll an open problem. Ths problem becomes especally complcated f we consder dfferent transmtpower constrants at each MU s dfferent rado-nterfaces [28] [29], whch ure the MU to carefully allocate ts transmt-powers to match the need of the scheduled traffc. To tackle the above challengng problem, we propose a jont optmzaton of the traffc schedulng and transmt-power allocatons wth the MUs dual-connectvty. Our contrbutons can be summarzed as follows. Novel Jont Optmzaton Formulaton: We formulate a cost mnmzaton problem, n whch each MU jontly determnes ts traffc schedulng to the AP and S and the transmt-powers at the two rado-nterfaces. The objectve s to mnmze the total cost of all MUs, whle meetng each MU s traffc demand and ts transmt-power constrants. To the best of our knowledge, such a formulaton targeted for the MUs traffc offloadng wth dual-connectvty has not been studed before. Centralzed Algorthm Desgn: We frst propose a centralzed algorthm for solvng the jont traffc schedulng and power control problem. In spte of non-convexty of the problem, we perform a

3 3 seres of manpulatons to transform the jont optmzaton nto an equvalent SINR-assgnment problem (here SINR denotes the sgnal-to-nterference-plus-nose rato). We then explore the hdden monotoncty of the SINR-assgnment problem and desgn a two-layered algorthm to solve t. ased on the obtaned SINRs at the AP, we derve the MUs traffc schedulng and transmt-powers to mnmze the total cost. Dstrbuted Algorthm Desgn: We next propose a dstrbuted algorthm to solve the jont optmzaton problem, by focusng on a practcally mportant case that the channel bandwdth of the AP s no smaller than the S s allocated bandwdth. In ths case, we dentfy the hdden concave mnmzaton property of the SINR-assgnment problem and propose a dstrbuted algorthm to compute each MU s SINR at the AP. ased on the obtaned SINRs at the AP, we derve the MUs traffc schedulng and transmt-powers to mnmze the total cost n a dstrbuted manner. Performance Improvement: Numercal results show that both the proposed algorthms can acheve the close-to-optmum results whch are obtaned by the LINGO (a commercal optmzaton software [4]) but wth a sgnfcant less computatonal complexty. The numercal results also valdate that the proposed traffc offloadng can sgnfcantly reduce the MUs cost, namely, savng more than 75% of the cost wthout performng any offloadng and more than 65% of the cost wth a fxed offloadng scheme. The rest of ths paper s organzed as follows. Secton II descrbes the related studes. Secton III presents the problem formulaton. Secton IV presents a seres of transformatons that facltate the followng algorthm desgns. Sectons V and VI propose the centralzed and dstrbuted algorthms, respectvely, to solve the problem. Secton VII presents the numercal results, and we conclude ths study n Sect. VIII. II. RELATED WORKS Snce the semnal studes [5] [6], there have been many studes that nvestgated traffc offloadng va nfrastructure-based small-cell networks 1. They can be roughly categorzed nto two groups as follows: Network-orented traffc offloadng. The frst group of studes manly focused on optmzng traffc offloadng from the networks perspectves. In [12], Ho et al. consdered the nter-cell nterference when accommodatng the offloaded traffc. Takng nto account the couplng effect due to the mutual nterference, the authors formulated a utlty optmzaton framework for dstrbutng traffc loads among dfferent macro-cells. In [13], Chen et al. also consdered the nterference among dfferent small cells when offloadng traffc, and proposed a framework that facltates the macro S to manage the small cells to mnmze the energy consumpton. In [15], Iosfds et al. consdered the coupled 1 esdes offloadng traffc va nfrastructure-based networks, t s also possble to offload traffc through peer-to-peer communcatons (e.g., the devce-to-devce communcatons [14]). Ths, however, s not the focus of ths study.

4 4 capactes of dfferent APs due to nterference, and proposed a double aucton mechansm that matches the network operators wth the most sutable APs for data offloadng. In [16], consderng the suffered nterference due to servng macro-cell MUs, Yang et al. proposed a refund mechansm for network operator to motvate the small-cell APs to admt macro-cell MUs for traffc offloadng. User-orented traffc offloadng. The second group of studes focused on optmzng the traffc offloadng from the MUs perspectves, wth the key ssues ncludng nterference management and delay-offloadng tradeoff. For nstance, n [11], Kang et al. consdered the scenaro of one S and one thrd-party WF AP, and nvestgated how dfferent MUs choose ether the S or the AP for offloadng traffc. Consderng the mutual nterference among dfferent MUs, the authors formulated the networks-selecton problem as a bnary nonlnear programmng problem for maxmzng the system-wse reward. Studes n [18] [19] proposed several dfferent schemes to motvate the MUs to delay ther traffc offloadng by leveragng the delay tolerance and better future network condtons. In [2], Im et al. proposed an MU-centrc cost-aware WF offloadng system, whch consders the MU s throughput-delay tradeoff and cost budget to decde when to offload whch type of traffc. In spte of the above studes, only few studes nvestgated the paradgm of small-cell dual-connectvty (whch has been ganng momentum n the ndustry practce [8] [9] and the 3GPP LTE-A standardzng actvtes [1] [1]) for offloadng the MUs traffc. In [22], Jha et al. dscussed several techncal challenges and potental soluton drectons regardng the small-cell dual-connectvty n cellular networks. However, to the best of our knowledge, there exsts no exstng study that nvestgated the optmal management of the MUs traffc schedulng and transmt-power allocatons for traffc offloadng va the dual-connectvty. We emphasze that although there are several studes nvestgatng how the traffc offloadng can beneft the network operators, t s less understood about how much the MUs can beneft from offloadng traffc n terms of savng the moble data cost. Our study sheds lght on ths beneft and llustrates how to optmze such beneft through both centralzed and dstrbuted approaches. A. System Model III. SYSTEM MODEL AND PROLEM FORMULATION g 1 (p1,x1) (p1a,x1a) g 2 MU1 g 2A g 1A g 3 (p2,x2) MU2 (p2a,x2a) g 3A AP S (p3,x3) MU3 (p3a,x3a) Fg. 1. Illustraton of the system model.

5 5 Fgure 1 llustrates the system model, where a macro S s servng a set of MUsI = {1,2,...,I} whch perform the uplnk transmssons 2. There also exsts a small-cell AP that can provde traffc offloadng servces to the MUs through dual-connectvty. Each MU has two rado-nterfaces, one for sendng traffc to the S and one for offloadng traffc to the AP. We use x A and x to denote MU s transmsson rates to the AP and the S, respectvely. We use p A and p to denote MU s transmt-powers to the AP and the S, respectvely 3. In the rest of ths paper, the subscrpts A and denote AP and S, respectvely. We consder that the macro S and AP operate on dfferent spectrums to provde servce 4. The small-cell AP allows multple MUs to share the same spectrum (channel) [23] [24], whch results n mutual nterference among the MUs when offloadng traffc. To make the dscussons more concrete, we adopt the throughput model under the nterference channel as [11] [13] [16],.e., gven the MUs transmt-powers {p A } I, MU s transmsson rate to the AP s p A g A x A = W log 2 (1+ j,j I p jag ja +n A ), I, (1) where W denotes the AP s channel bandwdth, and g A denotes the channel gan from MU to the AP. n A = Wn denotes the power of the background nose at the AP, wth n denotng the power densty. The S allocates orthogonal sub-channels to dfferent MUs for accommodatng the uplnk traffc (such as n OFDMA). Gven MU s transmt-power p, MU s uplnk transmsson rate to the S s ( x = log 2 1+ p ) g, I, (2) n where g s the channel power gan from MU to the S, and s the bandwdth allocated to MU by the S (we consder that the S s channel allocaton s gven n ths study). n = n denotes the power of the background nose at the S. Each MU needs to satsfy a traffc demand through the transmssons to both S and AP,.e., x A +x, I. (3) Smlar to [11] [16] [21], we consder usage-based prcng schemes by both the AP and the S. Let π A and π to denote the unt-prces announced by the AP and S, respectvely. Then, MU s total transmsson cost n one unt tme s C (x A,x ) = π A x A +π x, I. (4) 2 We focus on traffc offloadng n uplnk case, where we need to consder the MUs lmted resources. Such an ssue makes the uplnk resource allocaton more challengng than the downlnk case. Ths s also motvated by the rapd growth of user-generated contents (such as user-generated vdeos on socal networks), whch has sgnfcantly ncreased the MUs uplnk traffc volume. 3 In future work, we wll extend our model to the case of multple APs, where an MU selects whch AP to offload traffc to. 4 The small cell may operate on a separated lcensed spectrum (such as the separate carrer scheme for femtocell [25]) or on a separate unlcensed spectrum (such as the case of WF AP).

6 6. Problem Formulaton We are nterested n jontly optmzng all MUs traffc schedulng {x A,x } I and the transmtpowers {p A,p } I, to mnmze the total cost. Here, CMP stand for Cost Mnmzaton Problem. (CMP): Mnmze C (x A,x ) = π A x A + π x I I I Subject to: p A P max A, I, (5) p P max, I, (6) p A +p P max, I, (7) Constrants (1),(2), and (3), Varables: (x A,x ), I and (p A,p ), I. Constrants (5)-(7) are motvated by the fact that dfferent rado-nterfaces can have dfferent restrctons on the power consumptons [28] [29]. Constrant (5) ensures that MU s transmt-power p A to the AP cannot exceed the upper-bound P max A. Constrant (6) ensures that MU s transmt-power p to the S cannot exceed the upper-bound P max. Constrant (7) ensures that MU s total power consumpton at the two nterfaces cannot exceed the upper bound P max. The key challenge to solve Problem (CMP) s due to the ntrnsc non-convexty of (1). In the rest of ths paper, we wll develop effcent algorthms to solve Problem (CMP) to acheve close-to-optmum performance. efore presentng the detals, we make the followng assumpton. Assumpton 1: Problem (CMP) s feasble. Assumpton 1 can be satsfed by mposng an admsson control polcy that selects an approprate group of MUs to serve, such that the MUs traffc demands can be satsfed wthn ther transmsson power constrants 5. Assumpton 1 enables us to focus on evaluatng the beneft of the traffc offloadng va dual-connectvty and desgnng algorthms to acheve ths beneft. efore proposng the algorthms to solve Problem (CMP), we wll frst present a seres of problem formulatons equvalent to Problem (CMP). Specfcally, we use Fg. 2 to show how these problem formulatons are related and where they are located n ths paper. IV. EQUIVALENT TRANSFORMATIONS OF PROLEM (CMP) Ths secton presents a seres of problem formulatons that facltate our later algorthm desgns. 5 A heurstc approach to perform admsson control s that the MU (let us say MU ) wth (2 R 1) n g P max s admtted.

7 7 Problem (EMP) n Secton III.: { x, p },{ x, p } A A (1)(8)(9) Problem (TPA-P) n (14)(15) Problem (SINR-P) n Secton IV.A: { } Secton IV.: { A} p A (21)(27) Centralzed Problem (SINR-M-TopP) n Secton V.C:!! F ( ) sub! Problem (SINR-M-P) n Secton IV.C: {! },! Dstrbuted (27) Problem (SINR-M-SubP) n Secton V.: {! } Problem (SINR-ME-P) n Secton VI.A: {! },! Problem (SINR-ME-TopP) n Secton VI.D:!! F ( sub! ) Problem (SINR-ME-SubP) n Secton VI.C: {! } (37)(38) Problem (SINR-MES-SubP) n Secton VI.C: { }! Fg. 2. Connectons among the problem formulatons. We also mark out the decson varables of each problem formulaton. A. Transformng Problem (CMP) nto a Transmt-Power Allocaton Problem We frst dentfy the followng property of Problem (CMP). Lemma 1: Constrant (3) s tght at any optmal soluton of Problem (CMP). Proof: We prove ths by contradcton. Suppose that (3) s not tght for some MU at an optmal soluton. Then, we can reduce x by settng x = x A, whch wll reduce the total cost (.e., the objectve functon) wthout volatng any constrant. Ths contradcts the fact that t s an optmal soluton, and hence completes the proof. y usng Lemma 1 and eq. (2), we express x and p as functons of p A for each MU as follows: ) x = W log 2 (1+, I, (8) p = n 2 R g (1+ p A g A j,j I p jag ja +n A 1 p Ag A j,j I pjagja+na)w n g, I. (9) Smlar to [16] [17], we consder the practcal scenaro that the prce of the AP s lower than that of the S (.e., π A < π ), whch motvates the MUs to offload traffc to the AP. y usng (1), (8), and (9), we transform Problem (CMP) nto an equvalent Transmt-Power Allocaton Problem (TPA-P): (TPA-P): Maxmze p A g ) A (π π A )W log 2 (1+ I j,j I p jag ja +n A p A g ) A Subject to: W log 2 (1+ j,j I p, I, (1) jag ja +n A p A PA max, I, (11) n g 2 R n g 2 R Varables: p A, I. (1+ (1+ 1 p Ag A j,j I pjagja+na)w 1 p Ag A j,j I pjagja+na)w P max + n g, I, (12) +p A P max + n g, I, (13)

8 8 Problem (TPA-P) only nvolves {p A } I as varables. Here, (1) comes from Lemma 1,.e., each MU s x A cannot exceed. Constrants (11), (12), and (13) come from (5), (6), and (7), respectvely. Problem (TPA-P) ndcates a tradeoff n offloadng traffc as follows. To mnmze the MUs total cost, all MUs should offload ther traffc to the AP as much as possble. However, aggressve traffc offloadng causes a heavy nterference at the AP, whch ncreases the MUs power consumptons (but satsfyng (11), (12), and (13)). Notce that Problem (TPA-P) s more complcated than the well-studed power control problems over nterference channels (e.g. [31] [32]), because t takes nto account each MU s transmt-powers at two dfferent rado-nterfaces. Ths consequently yelds the non-convex constrant (13) whch couples each MU s p A and p together. We also recall that all MUs {p A } I are also coupled due to the nterference at the AP. Hence, Problem (TPA-P) s very dffcult to solve.. Transformng Problem (TPA-P) nto an SINR-Assgnment Problem To solve Problem (TPA-P), we need to make some further equvalent transformatons. We use θ to denote MU s acheved SINR at the AP as follows: θ = p A g A j,j I p jag ja +n A, I. (14) ased on (14), we have the followng result that connects {θ } I wth {p A } I. Proposton 1: Gven any profle of transmt-powers {p A } I whch s feasble for Problem (TPA-P), the correspondng profle of {θ } I gven by (14) ensures that the followng result holds: and we always have I p A = n A g A θ 1+θ < 1. θ 1 1+θ 1 I Proof: Please refer to Appendx I for the detals. θ, I, (15) 1+θ ased on Proposton 1, we can use {θ } I to substtute {p A } I and transform Problem (TPA-P) nto an equvalent SINR-assgnment problem as follows: (SINR-P): Maxmze I(π π A )W log 2 (1+θ ) Subject to: θ 2 R W 1, I, (16) n A g A n g 2 R θ 1 1+θ 1 I 1 (1+θ ) W n A θ 1 g A 1+θ I Varables: θ, I. 1 I θ PA max, I, (17) 1+θ P max + n g, I, (18) θ + n 2 R g 1+θ 1 (1+θ ) W P max + n g, I,(19) θ < 1, (2) θ +1

9 9 Notce that we ntroduce constrant (2) n Problem (SINR-P) for the convenence of later dscussons, and Proposton 1 shows that ntroducng such a constrant does not reduce the feasble set of Problem (TPA-P). Once we solve Problem (SINR-P), then the optmal soluton {p A } I of Problem (TPA-P) can be computed based on eq. (15). In the specal case where all MUs traffc demands are small enough (see Fg. 6 n Sec. VII), t s optmal for all MUs to offload ther entre traffc to the AP for mnmzng the cost. The partcular structure of Problem (SINR-P) enables us to derve a suffcent condton for ths specal case to happen. 2 W Proposton 2: (Complete-Offloadng Stuaton) The optmal soluton of Problem (SINR-P) s θ = 1, I when both of the followng two condtons (.e., (C1) and (C2)) hold: (C1): ( ) 1 > I 1, and (C2): n A g 1 I W A W I I +1 mn{pmax 2 R 2 R 2 R W,PA max }, I. Proof: We frst derve θ = 2R W 1 by settng constrant (16) tght, whch corresponds to that each MU offloads ts entre traffc to the AP. We then derve Condtons (C1) and (C2) by ensurng that {θ } I s compatble wth all constrants. We obtan Condton (C1) by substtutng θ = 2 R and obtan Condton (C2) by substtutng θ = 2R W W 1 nto (2), 1 nto (17) and (19). Notce that (18) s always satsfed, snce no traffc s sent to the S when each MU s θ = 2R W 1. Therefore, Condtons (C1) and (C2) together guarantee that the optmal soluton of Problem (SINR-P) s θ = 2R W 1, I. C. Equvalent Form of the SINR-assgnment Problem wth {ρ } I and ρ Problem (SINR-P), however, s stll dffcult to solve due ts non-convexty. To solve t effcently, we further ntroduce a one-to-one mappng as follows ρ = θ 1+θ, θ = ρ 1 ρ, I. (21) Snce I ρ < 1 always holds (.e., Proposton 1), we ntroduce another postve varable ρ such that ρ + I ρ = 1. Such a condton wll help smplfy the complcated denomnators n (17) and (19). y usng {ρ } I and ρ, we equvalently transform Problem (SINR-P) nto Problem (SINR-M-P) as

10 1 follows (the letter M means Medum, and Appendx II presents the detaled procedures). (SINR-M-P): Mnmze π A )W log 2 ρ + I(π ρ j (22) Subject to: ρ 1 1 n A ρ PA max g A ρ ρ + n g 2 R j,j I 2 W ρ j j,j I, I, (23), I, (24) ρ + Pmax + n g j,j I n g 2 R ρ j W W, I, (25) + n A ρ P max + n, I, (26) g A ρ g ρ + ρ = 1, (27) I Varables: ρ and ρ, I. Notce that constrants (23)-(27) here correspond to (16)-(2) n Problem (SINR-P), respectvely. Proposton 3: Let (ρ,{ρ } I) denote an optmal soluton of Problem (SINR-M-P). Then, θ = ρ 1 ρ, I corresponds to an optmal soluton of Problem (SINR-P) 6. Proof: The key dea s to show that the feasble regon of Problem (SINR-P) and that of (SINR-M-P) form a one-to-one mappng, f Problem (SINR-P) s feasble. Please see Appendx II for the detals. Proposton 3 allows us to solve Problem (SINR-P) by solvng Problem (SINR-M-P). In partcular, Problem (SINR-M-P) can be solved through a two-layered structure,.e., we frst solve the subproblem n whch the value of ρ s gven, and then we fnd the best ρ (,1] to mnmze the objectve functon. V. CENTRALIZED ALGORITHM TO SOLVE PROLEM (SINR-M-P) We propose a centralzed algorthm to solve Problem (SINR-M-P). Sec. V-A shows the two-layered structure. Wth the layered structure, we desgn an algorthm to solve a subproblem of Problem (SINR- M-P) n Sec. V-, based on whch we desgn an algorthm to solve the top problem n Sec. V-C. A. Two-Layered Structure of Problem (SINR-M-P) Problem (SINR-M-P) s stll a non-convex optmzaton. Nevertheless, solvng Problem (SINR-M-P) can be vertcally separated nto two steps, namely, solvng a seres of subproblems n whch the value of ρ s gven n advance, and then solvng a top-problem that fnds the best ρ (,1] for mnmzng the objectve functon. The detals are as follows. 6 It s worth emphaszng that Proposton 3 does not hold f Problem (CMP) s nfeasble (.e., Assumpton 1 fals to hold).

11 11 1) (Subproblem under a gven ρ ): We consder a subproblem under a gven ρ (,1] as follows: (SINR-M-SubP): F sub (ρ ) = Mnmze π A )W log 2 ρ + I(π ρ j (28) j I,j Subject to: {ρ } I G (ρ) H (ρ), (29) Varables: ρ, I. Constrant (29) means that the profle {ρ } I for all MUs should belong to the ntersecton of sets G (ρ) and H (ρ). Here, set G (ρ) can be characterzed by (23), (24), (25), and (26) under a gven ρ as follows: { G (ρ) = {ρ } I ρ 1 1 n A ρ 2, I; PA max, I; g W A ρ ρ j Pmax + n W g ρ, I; j,j I n g 2 R n g 2 R ρ + j,j I ρ j W + n A ρ P max + n }, I. (3) g A ρ g Meanwhle, set H (ρ) s characterzed by constrant (27) under a gven ρ as follows: { H (ρ) = {ρ } I ρ 1 ρ }. (31) I Note that we change the equalty n (27) nto the nequalty n H (ρ), and ths does not change the optmal soluton due to Proposton 4. Let {ρ,sub,(ρ ) } I denote an optmal soluton of Problem (SINR-M-SubP). Proposton 4: If Problem (SINR-M-SubP) s feasble under a gven value of ρ, then I ρ,sub,(ρ ) = 1 ρ always holds,.e., the optmal soluton of Problem (SINR-M-SubP) s consstent wth (27). Proof: Please refer to Appendx III for the detals. 2) (Top-problem to fnd ρ ): y solvng the bottom problem and obtanng F sub(ρ ) as a functon of a gven ρ, we next solve the top-problem to fnd the best ρ for mnmzng the objectve functon: (SINR-M-TopP): Mnmze F sub (ρ ), Subject to: < ρ 1, Varable: ρ.. Monotoncty of Problem (SINR-M-SubP) and Algorthm (Cen-Sub) for Soluton We focus on solvng Problem (SINR-M-SubP) n ths subsecton. 1) (Monotoncty of Problem (SINR-M-SubP)): We solve Problem (SINR-M-SubP) by usng technques from monotonc optmzaton, whch tackles a class of optmzaton problems wth monotonc objectve functons and monotonc constrants [33] [34]. The feasble regon of a monotonc optmzaton problem can be represented by the ntersecton of a normal set (.e., Defnton 1 below) and a reversed normal set (Defnton 2). Thanks to the monotonc property of the objectve functon and the constrants, the feasble regon can be well approxmated by a set of poly-blocks (wth an arbtrary desrable precson),

12 12 and an optmal soluton can be attaned at one of the vertexes of the poly-blocks. Ths yelds an effcent approach, referred to as the polyblock-approxmaton, to solve the monotonc problems [34]. The advantage of the monotonc optmzaton s that t does not ures the problem to be convex, hence s wdely applcable for solvng a wde range of practcal engneerng problems (see [33] for more detals). Defnton 1: (Normal Set) A set G R n + s normal, f for any two ponts x and x R n + wth x x and x G, we always have x G. Defnton 2: (Reversed Normal Set) A set H R n + s a reversed normal set, f for two ponts x and x R n + wth x x and x H, we always have x H. Wth the above termnologes, we can show the followng result. Proposton 5: Gven a fxed ρ, Problem (SINR-M-SubP) s a monotonc optmzaton problem. Proof: We frst note that the objectve functon (28) s ncreasng n {ρ } I. Moreover, the left hand sdes of (23)-(26) are ncreasng n {ρ } I,.e., G (ρ) s a normal set and H (ρ) s a reversed normal set. Hence, Problem (SINR-M-SubP) nvolves the mnmzaton of an ncreasng functon, subject to a feasble regon gven by G (ρ) H (ρ). Thus, accordng to [33], (SINR-M-SubP) s a monotonc optmzaton. 2) (Algorthm (Cen-Sub) for Solvng Problem (SINR-M-SubP)): ased on Proposton 5, we propose Algorthm (Cen-Sub) based on the dea of poly-block approxmaton to solve Problem (SINR-M-SubP). We should pont out that, snce Problem (SINR-M-SubP) mnmzes an ncreasng functon, Algorthm (Cen-Sub) s desgned to construct the seres of the poly-blocks that approxmate the lower boundary of the feasble regon as close as possble. Ths s new n the monotonc optmzaton lterature ! ( )! ( )! ( ) k z ( ) k x The removed regon k z ( ) k x ( ) (a) Illustratons of Steps (b) Illustraton of Step 12 1 (c) Illustraton of " k! 1 1 Fg. 3. The poly-block approxmaton used n Algorthm (Cen-Sub). Each sold node denotes a vertex. The shaded area denotes the feasble regon, and the red-lne constructed by the vertces denotes the approxmated lower-boundary of the feasble regon. The key component of Algorthm (Cen-Sub) s the Whle-Loop (Lnes 2-18), whose purpose s to teratvely construct the poly-blocks that approxmate the lower-boundary of G (ρ) H (ρ) wth an ncreasng precson. Fgure 3 provdes a sketch to llustrate the procedures. Specfcally, n the k-th 7 The pror applcatons of monotonc optmzaton often focus on maxmzng an ncreasng functon [35].

13 13 Algorthm (Cen-Sub): to solve Problem (SINR-M-SubP) under a gven ρ (,1] 1: Intalzaton: Set the current best soluton CS =, and the current best value CV =. Set ndex k = 1 and ǫ as a small postve number. Set the flag for stoppng as f stop =. Intalze set T 1 = {}. We use V({ρ } I ) = I (π π A )W log 2 (ρ + ) j,j I ρ j for easy presentaton. 2: whle f stop = do 3: Select vertex z k argmn {V({ρ } I ) {ρ } I T k }. 4: Construct a lne between z k and pont o whose element o = mn ρ }, I. { R W,ρ P max A ga n A 5: Fnd the ntersecton pont x k between the above constructed lne and the lower boundary gven n H (ρ) by bsecton search. 6: f V(x k ) < CV then 7: Update CV = V(x k ) and set CS=x k. 8: end f 9: f x k z k < ǫ then 1: Set f stop = 1. 11: end f 12: Update the set of vertexes as T k+1 = (T k \{z k }) 13: Remove all vertexes z T k+1 \G (ρ). 14: f T k+1 s empty then 15: Set f stop = 1. 16: end f 17: Set k = k+1. 18: end whle 19: Output: Set {ρ,sub,(ρ ) } I s equal to CS, and F sub (ρ ) = CV. { } z k +(x k zk )e, I.,1 teraton, set T k denotes the current set of vertexes. In T k, we fnd a vertex z k that yelds the smallest objectve value (Lne 3) 8. Then we perform the followng two tasks: Task ): to update the current best soluton (CS) and the current best value (CV). We frst con- { 1 1 struct a lne fromz k to a specal upper-boundary pontowth ts elemento = mn 2 R W,ρ P max A ga n A,1 ρ }, I (Lne 4) (each element o of pont o represents the maxmum possble value of ρ, based 8 We use the vector x (n bold letter) to denote the profle {x } I,.e., x and {x } I are nterchangeable.

14 14 on (23),(24), and (27)). We then fnd the correspondng ntersecton pont (denoted by x k ) between the constructed lne and the lower boundary of the feasble regon (Lne 5) 9. We use V(x k ) and x k to update the CV and the CS n the k-th teraton (Lnes 6-8). Task ): to construct poly-blocks T k+1 for the next round teraton. We use vertex z k and the ntersecton pont x k to construct the new poly-blocks that can approxmate G (ρ) H (ρ) closer (Lne 12) 1. The essence of Lne 12 s to remove the regon n whch the optmum cannot exst. Algorthm (Cen-Sub) termnates f z k and x k are close enough (Lnes 9-11), or f we cannot expect to fnd a better soluton (Lnes 14-16). C. Algorthm (Cen) for Solvng Problem (SINR-M-TopP) and Soluton of Problem (CMP) 1) Algorthm (Cen) to solve Problem (SINR-M-TopP): Next we propose Algorthm (Cen) to solve Problem (SINR-M-TopP), by usng Algorthm (Cen-Sub) as a subroutne to solve the subproblem. Algorthm (Cen) performs a one-dmensonal lnear search over ρ (,1] wth the step-sze top (the Whle-Loop on Lnes 2-11). For each gven value of ρ, we use Algorthm (Cen-Sub) to evaluate F sub (ρ ) (Lne 5) and update the currently best soluton (Lne 7). We perform the feasblty-test for Problem (SINR-M-SubP) under each ρ (Lne 3), such that we avod nvokng Algorthm (Cen-Sub) when subproblem (SINR-M-SubP) s nfeasble and hence save the computatonal tme. Remark 1: The dffculty n solvng Problem (SINR-M-TopP) s that we cannot obtan F sub (ρ ) n a closed-form expresson. Fortunately, n Problem (SINR-M-TopP) the sngle varable ρ s restrcted wthn a fxed nterval (,1] that s ndependent on the other parameters. Ths property allows us to use the one-dmensonal lnear search wth a fxed step-sze top to fnd ρ. A smaller step-sze top wll lead to a more accurate soluton n the search. In Sec. VII, we adopt top =.5 (.e., 2 samples wthn (,1]) for most of the smulatons, snce we fnd that a top smaller than.5 yelds a very lmted performance mprovement wth a sgnfcant ncrease n computatonal tme (see Table III for detals). We emphasze that although the proposed Algorthm (Cen) provdes an effectve approach to solve Problem (SINR-M-P), we cannot clam the obtaned {ρ } I as the global optmum due to the use of lnear search wth a fxed step-sze top. Nevertheless, we have the followng asymptotcal result. Proposton 6: Algorthm (Cen) can yeld the asymptotcally optmal soluton (.e., ρ and {ρ } I) for Problem (SINR-M-P), as top approaches to zero. Proof: Usng an extremely small step-sze top gong to zero enables us to enumerate all possble values of ρ (,1]. For each enumerated ρ, the subroutne (Cen-Sub) can fnd the correspondng 9 Thanks to the monotoncty of the constrants, we can use the bsecton search to fnd x k effcently. 1 In Lne 12, scalar x k (or z k ) denotes the -th element of vector x k (or vector z k ), and vector e denotes the vector wth the -th element equal to 1, and all other elements equal to. All vectors n ths paper are of dmenson 1 I.

15 15 Algorthm (Cen): to solve Problem (SINR-M-TopP) 1: Intalzaton: Set a small step-sze top. Set ρ = top. Set the CV as a very large number. 2: whle ρ < 1 do 3: Check the feasblty of Problem (SINR-M-SubP) wth Algorthm (Cen-Sub-FC) n Appendx IV. 4: f Problem (SINR-M-SubP) s feasble then 5: Use Algorthm (Cen-Sub) to obtan {ρ,sub,(ρ ) } I and F sub (ρ ). 6: f F sub (ρ ) < CV then 7: Set CV = F sub (ρ ). Set ρ = ρ, and ρ = ρ,sub,(ρ ), I. 8: end f 9: end f 1: Update ρ = ρ + top. 11: end whle 12: Output: ρ and {ρ } I. optmal soluton based on the monotonc optmzaton theory [33] [34]. Hence, Algorthm (Cen) can yeld the asymptotcally optmal soluton for Problem (SINR-M-P), as top approaches to zero. However, Algorthm (Cen) wth a very small step-sze top wll consume a very long computatonal tme. We show n Sec. VII that by usng an approprately chosen small step-sze top =.5, Algorthm (Cen) s able to yeld the close-to-optmum soluton whle consumng an affordable computatonal complexty. The detaled complexty of Algorthm (Cen) depends on both the step-sze top and the complexty of Algorthm (Cen-Sub). However, accordng to [33] [34], t s very dffcult to quantfy the complexty of Algorthm (Cen-Sub) drectly (because t strongly depends on the geometrc structure of the feasble regon). In Sec. VII, we evaluate the complexty of Algorthm (Cen) va numercal examples. 2) Optmal Soluton of Problem (CMP): Algorthm (Cen) outputs ρ and {ρ } I for Problem (SINR- M-P). ased on Proposton 3, each MU s SINR at the AP (.e., the soluton of Problem (SINR-P)) s θ = ρ 1 ρ, I. Correspondngly, the soluton of Problem (CMP) can be derved as follows. ased on Proposton 1 and eq. (1), MU s transmt-power and transmsson rate to the AP are p A = n A g A ρ 1 I ρ and x A = W log 2 ( 1 1 ρ ), I, respectvely. Moreover, based on (8) and (9), MU s rate and transmt-power at the S are p = n ( ) 2 (1 ρ g ) W 1 and x = +W log 2 (1 ρ ), I, respectvely. Thus, we solve Problem (CMP) completely.

16 16 To perform Algorthm (Cen), the S needs to collect each MU s prvate nformaton, e.g., the channel gans {g A,g }, the transmt-power capactes {P max A,Pmax,Pmax }, and the demand. Ths may not be always feasble n practce, due to prvacy concerns and consderaton of communcatons overhead. Ths motvates us to study whether t s possble to solve Problem (SINR-M-P) n a dstrbuted fashon. VI. DISTRIUTED ALGORITHM FOR SOLVING PROLEM (SINR-M-P) We frst consder mplementng Algorthm (Cen) n a dstrbuted manner, whch turns out to be challengng due to the centralzed nature of the subroutne (Cen-Sub) to solve the monotonc optmzaton. Hence, we wll focus on a practcal network scenaro as follows where a dstrbuted algorthm s possble. Assumpton 2: The channel bandwdth of the small-cell AP s no smaller than the allocated bandwdth by the S,.e., W. Assumpton 2 s motvated by the fact that a small-cell AP often provdes a larger bandwdth than the cellular S. For example, IEEE 82.11ac enables a total channel bandwdth up to 16MHz [27], whch s larger than that of 6MHz n the 3GPP LTE-A standard [8] [9]. We wll later show that Assumpton 2 helps convexfy (26) such that Problem (SINR-M-P) becomes a concave mnmzaton problem. Ths nvolves some addtonal equvalent problem transformatons as follows. A. Equvalent Form of Problem (SINR-M-P) and Its Two-Layered Structure To fnd a structure sutable for a dstrbuted soluton, we re-express Problem (SINR-M-P) as follows. (SINR-ME-P): Mnmze I(π π A )W log 2 (1 ρ ) (32) Subject to: constrants (23),(24), and (27) 1 Pmax + n W g ρ, I, (33) n n g 2 R 2 R n (1 ρ ) W A ρ + P max g g A ρ Varables: ρ and ρ, I. + n g, I, (34) ased on (27), the above (32), (33), and (34) are equvalent to (22), (25), and (26), respectvely (that s why we add addtonal letter E after M to label the problem). Problem (SINR-ME-P) also has a two-layered structure, namely, we can frst solve the subproblem wth a gven ρ, and then solve the top-problem to determne the best ρ (,1]. The subproblem and the top-problem are gven as follows.

17 17 1) Subproblem under gven ρ : The subproblem under a gven ρ can be gven as follows. (SINR-ME-SubP): F sub (ρ ) = Mnmze I(π π A )W log 2 (1 ρ ) Subject to: constrants (23),(24),(27),(33), and (34), Varables: ρ, I. 2) Top problem for fndng ρ : ased on the soluton of subproblem, we next solve a top-problem: (SINR-ME-TopP): Mnmze F sub (ρ ), Subject to: < ρ 1, Varable: ρ. We emphasze that dfferent from solvng Problem (SINR-M-P) n a centralzed manner as n Secton V, we wll solve the above Problems (SINR-ME-SubP) and (SINR-ME-TopP) n a dstrbuted manner.. Subproblem (SINR-ME-SubP): A Concave Mnmzaton Problem To solve Problem (SINR-ME-SubP) n a dstrbuted manner, we frst prove the followng. Proposton 7: Under Assumpton 2, Problem (SINR-ME-SubP) s a concave mnmzaton problem. Proof: Gven ρ, (23), (24), (27), and (33) are all lnear n {ρ } I. Meanwhle, under Assumpton 2, (34) s convex and thus defnes a convex feasble regon for each ρ. Thus, the feasble regon of Problem (SINR-ME-SubP) s a convex set. We can further verfy that the objectve functon s a concave functon. Thus, Problem (SINR-ME-SubP) nvolves the mnmzaton of a concave functon over a convex feasble set,.e., s a concave mnmzaton problem [37] [38]. The concave mnmzaton problem s known as a mult-extremal global optmzaton. For the general concave mnmzaton optmzatons, t s very dffcult to characterze an optmalty condton, and we only know that the optmal soluton(s) exst at one or multple vertces of the feasble regon [37]. Motvated by ths observaton, several numercal schemes (see [38] for more detals) have been proposed for enumeratng the vertexes of feasble regon. However, these schemes suffer from two common drawbacks: the hgh computatonal complexty and the centralzed mplementaton. For nstance, n [36], the authors proposed a parameterzed branch-and-bound algorthm to solve the spectrum balancng problem (whch was proved as a concave mnmzaton problem). However, the proposed algorthm s centralzed wthout an upper-bound characterzaton of the computatonal complexty. Therefore, to explot the concave mnmzaton property to solve Problem (SINR-ME-SubP), we need to dentfy some addtonal structural propertes of the feasble regon n our problem. Fortunately, n Problem (SINR-ME-SubP), only constrant (27) couples all MUs decsons. Now let us tentatvely gnore (27) (whch wll be taken nto account later on). Then, under a gven ρ, (23),(24),(33), and (34) yeld a [ ] decoupled feasble nterval for each MU n the form of ρ M,M,(ρ),(ρ ), whch s much smpler to deal wth. Ths property sgnfcantly smplfes the feasble regon. To analytcally characterze M,(ρ)

18 18 and M,(ρ), we frst defne the followng functon: wth J () = n g 2 R J (ρ ) = n 2 n (1 ρ ) W A ρ +, I, (35) g g A ρ and J (1) = na g A 1 ρ. We have the followng results regardng functon J (ρ ). Lemma 2: Under Assumpton 2 and a gven ρ, each MU s functon J (ρ ) s monotoncally ncreasng wthn (,1], f Condton (C3): na g A 1 ρ > n not hold, then there exsts a specal pont χ,(ρ) gven by ( n A g 1 χ,(ρ) = 1 n g A ρ g 2 R W, I holds. If Condton (C3) does ) W, (36) 2 W such that functon J (ρ ) decreases wthn ρ (,χ,(ρ)] and ncreases wthn ρ [χ,(ρ),1]. Proof: Please refer to Appendx V for the detals. ased on Lemma 2, we obtan the followng results regardng M,(ρ) and M,(ρ ). Proposton 8: Under Assumpton 2 and a gven ρ, constrants (23),(24),(33), and (34) together yeld a decoupled feasble nterval for each MU as ρ [M,M,(ρ),(ρ )], where M,(ρ) = max 1 Pmax + { n W g,,µ,(ρ), and M,(ρ ) = mn 1 1 n g 2 R 2 W,PA max g A ρ,1 ρ,µ n,(ρ) A ( ) The tuple of µ,µ,(ρ),(ρ ) depends on whether Condton (C3) (n Lemma 2) holds or not as follows: ( ) ) When Condton (C3) holds, the tuple of µ,µ,(ρ),(ρ ) s gven by (1,), f J () > P max + n g ; ( ) ( { }) µ,µ,(ρ),(ρ ) =,argmax v J (v) = P max + n g, v 1, f J () P max (,1), f J (1) < P max ( ) ) When Condton (C3) does not hold, the tuple of µ,µ,(ρ),(ρ ) s gven by ( ) µ,µ,(ρ),(ρ ) = (1,), f J (χ,(ρ)) > P max + n g ; + n g J (1); + n g. }. (,1), f J (χ,(ρ)) P max (,argmn { v χ,(ρ) v 1,J (v) = P max + n g }), f J (χ,(ρ)) P max + n g, and J () P max ( { } ) arg max v v χ,(ρ),j (v) = P max + n g,1, f J (χ,(ρ)) P max + n g, and J () P max + n g, and J (1) P max + n g ; + n g, and J () P max ( { } arg max v v χ,(ρ),j (v) = P max + n g,argmn f J (χ,(ρ)) P max + n g, and J (1) P max + n g ; + n g, and J (1) P max + n g { v χ,(ρ) v 1,J (v) = P max + n g }), + n g, and J () P max + n g, and J (1) P max + n g.

19 19 Notce that the value of χ,(ρ) s gven n (36). Proof: Please refer to Appendx VI for the detals. Note that n the thrd case when Condton { (C3) does not hold, we need to calculate argmn v χ,(ρ) v 1,J (v) = P max + n g }. Ths can be effcently computed by a bsecton search, as J (ρ ) s ncreasng for ρ [χ,(ρ),1] (Lemma 2) n ths case. For the other smlar cases n Proposton 8, the bsecton search s also applcable. Usng Proposton 8, we can re-wrte Problem (SINR-ME-SubP) n the followng equvalent form. (SINR-MES-SubP): F sub (ρ ) = Mnmze I(π π A )W log 2 (1 ρ ) Subject to: M,(ρ) ρ M,(ρ), I, (37) constrant (27), Varables: ρ, I. In Problem (SINR-MES-SubP), each MU can calculate ts M,(ρ) and M,(ρ ) usng only ts prvate nformaton wthout communcatng wth the S. Remark 2: We can nterpret Problem (SINR-MES-SubP) as a resource allocaton problem. Specfcally, gven ρ, the S allocates a total budget 1 ρ of vrtual currency to all MUs (.e., (27)). Varable ρ corresponds to the amount of vrtual currency allocated to MU, and each MU has a ds-utlty functon (π π A )W log 2 (1 ρ ) (accordng to the objectve functon) dependng on ts ρ. Each MU s ρ should fall wthn [M,(ρ),M,(ρ )] (.e., (37)). The S s objectve s to allocate ts budget to the MUs to mnmze the total ds-utlty. C. Dstrbuted Algorthm (Ds-Sub) for Solvng Problem (SINR-MES-SubP) The concave mnmzaton property of Problem (SINR-MES-SubP) together wth ts smple form (followed by Proposton 8) lead to two mportant gudelnes for the S to allocate vrtual currency to the MUs: Gudelne-I, each MU prefers to obtanng a larger ρ (fallng wthn [M,(ρ),M,(ρ )]), snce ts ds-utlty functon s decreasng ρ ; and Gudelne-II, recevng a larger ρ can yeld a larger margnal decrease n MU s ds-utlty, snce ts ds-utlty functon s concave. Wth the two gudelnes, we desgn a dstrbuted algorthm (Ds-Sub) to solve Problem (SINR-MES-SubP), whch works as follows. MU ntalzes ts current ρ current = M,(ρ), and reports ρ current effectve budget as Θ effectve = 1 ρ I ρcurrent and sets Θ current = Θ effectve. The S quantzes Θ effectve wth a small step-sze sub, whch yelds Θeffectve sub to the S. The S ntalzes ts unts of currency. The S performs an teratve process to allocate these unts to the MUs (the Whle-Loop on Lnes 2-8). In each teraton, the S allocates one unt of currency (n the amount sub ) to an approprate MU based on the MUs bds. To compete for ths unt, MU calculates ts bd b (Lne 3) representng

20 2 how eager MU s to get the unt 11 and submts the bd to the S. MU also sends the S a sgnalng f ndcatng whether t can further ncrease ts ρ current or not. After recevng all MUs {b,f } I, the S selects the MU (let us say MU k) whch submts the largest bd wth f = 1, and grants MU k the unt (Lne 4). Correspondngly, MU k ncreases ts ρ current k by sub (Lne 5). The teraton process wll termnate f ether the S uses up ts effectve budget or the S fnds that every MU s f =,.e., every MU has reached the maxmum value of ρ current. In a nutshell, Algorthm (Ds-Sub) provdes an effcent approach to determne the best vertex of the feasble regon that can mnmze all MUs total ds-utlty (under a gven ρ ). Algorthm (Ds-Sub): dstrbuted algorthm to solve Problem (SINR-MES-SubP) 1: Intalzaton: Each MU sets ts ρ current = M,(ρ) and f = 1, and reports (ρ current,f ) to the S. The S sets the effectve budget of the currency asθ effectve = 1 ρ I ρcurrent andθ current = Θ effectve. 2: whle Θ current sub and I (1 f ) = do 3: MU calculates ts bd as b = 1 (1 ρ current and submts the tuple of (b,f ) to the S. 4: The S collects each MU s tuple of (b,f ). )ln2 and the sgnallng f = I(ρ current 5: The S selects MU k = argmax I b f, and notces MU k for allocatng sub. 6: MU k, whch s notced by the S, updates ts ρ current 7: The S updates ts budget as Θ current = Θ current sub. 8: end whle = ρ current + sub. + sub M,(ρ)), 9: Output: The S sets F sub (ρ ) = (π π A )W I log 1 2( b ln2 ). Each MU sets ρ,sub,(ρ = ) ρcurrent. D. Dstrbuted Algorthm (Ds) for Solvng Problem (SINR-ME-TopP) We next propose a dstrbuted Algorthm (Ds) to solve Problem (SINR-ME-TopP). In Algorthm (Ds), the S performs a one-dmensonal lnear search over ρ (,1]. For each enumerated ρ, the S and the MUs nvoke Algorthm (Ds-Sub) as a subroutne. Algorthm (Ds) works as follows. In Lne 3, each MU determnes ts feasble nterval ρ [M,(ρ),M,(ρ)] based on Proposton 8. In Lne 5, the S determnes the feasblty of Subproblem (SINR-MES-SubP). 11 Each MU s bd corresponds to ts margnal decrease n the ds-utlty when obtanng one more unt of the currency. Here, we assume that MUs wll truthfully report ther bds, and we wll leave the study about the ncentve ssues n a future study.

21 21 If Subproblem (SINR-MES-SubP) s feasble, the S and the MUs perform Algorthm (Ds-Sub) to evaluate F sub (ρ ), based on whch the S and the MUs update ther respectve currently best solutons (Lnes 7-9). Otherwse, the S contnues to evaluate the next value of ρ (Lne 11). Algorthm (Ds): to solve Problem (SINR-ME-TopP) 1: Intalzaton: The S sets a small step-sze top for updatng ρ, and ntalzes ρ = top. The S sets the current best value (CV) as a very large number. 2: whle ρ 1 do 3: Gven ρ, each MU uses Proposton 8 to determne ts M,(ρ) and M,(ρ ). Each MU reports ts ndvdual feasble nterval [M,(ρ),M,(ρ)] to the S. 4: The S collects each MU s tuple of (M,(ρ),M,(ρ)). 5: f M,(ρ) M,(ρ ), I and I M,(ρ ) 1 ρ I M,(ρ ) are met then 6: The S and the MUs perform Algorthm (Ds-Sub). As a result, the S obtans F sub (ρ ), and each MU obtans ρ,sub,(ρ ). 7: f F sub (ρ ) < CV then 8: The S sets CV = F sub (ρ ), and each MU sets ts ρ = ρ,sub,(ρ ). 9: end f 1: end f 11: The S updates ρ = ρ + top. 12: end whle 13: Output: Each MU outputs ρ as the soluton of Problem (SINR-ME-TopP). Lne 5 checks the feasblty of Problem (SINR-MES-SubP) based on (27) and (37), and avods nvokng Algorthm (Ds-Sub) when Problem (SINR-MES-SubP) s nfeasble, thus saves the computatonal tme. Fgure 5 n Sec. VII wll verfy the gan of ths operaton. Consderng that Algorthm (Ds-Sub) ures no more than 1 ρ sub teratons ured by Algorthm (Ds) wll be no more than sub top. teratons, the overall number of As a comparson, we emphasze that the centralzed Algorthm (Cen) proposed n Sec. V does not ure Assumpton 2 and can be appled to more general settngs than the dstrbuted Algorthm (Ds). However, the downsde of Algorthm (Cen) s that t only explots the monotonc structure of Problem (SINR-M-P), and thus ures a longer computatonal tme than Algorthm (Ds). The numercal results n Table I and Fg. 5 wll verfy ths pont.

22 22 E. Dstrbuted Power Control Algorthm for Achevng {θ } I After obtanng ρ, each MU can derve the soluton for Problem (CMP). MU s SINR at the AP s θ = ρ 1 ρ (accordng to (21)), and the transmsson rate to the AP s x A = W log 1 2( 1 ρ ) (accordng to (1)). Meanwhle, MU s transmsson rate to the ( S s x = R +W ) log 2 (1 ρ ) (accordng to (8)), and the transmt-power to the S s p = n g 2 (1 ρ )W 1 (accordng to (9)). However, as shown n (15), all MUs {p A } I are coupled together. Fortunately, by settng θ as MU s targeted SINR, the MUs can adopt the mnmum power control scheme [39] to reach {p A } I n a dstrbuted manner. In the mnmum power control scheme, each MU ntalzes ts p A = and then keeps on updatng p A based on ts measured SINR for achevng θ targeted. Due to the space lmtaton, we skp the detaled descrpton of ths power control scheme, and nterested readers please refer to [39] for the detals. We emphasze that, as the soluton for Problem (SINR-P), {θ } I s guaranteed to be feasble to all MUs transmt-power constrants, whch guarantees that the resultng {p A } I (obtaned by the mnmum power control scheme) s also feasble to the orgnal Problem (CMP). VII. NUMERICAL RESULTS We consder a network scenaro that the S s located at the orgn, and the small-cell AP s located at (35m,m). The MUs are randomly located wthn a crcle, whose center s (32m,m) and the radus s 2m. Ths means that the MUs are closer to the AP than to the S (otherwse, there s lttle beneft of consderng traffc offloadng). We use the smlar method as [26] to model the channel power gan,.e., g A = A l, where l A κ A denotes the dstance between MU and the AP, and κ denotes the power-scalng factor for the path-loss (we set κ = 4). We further assume that A follows an exponental dstrbuton wth unt mean due to channel fadng. Fgure 4 plots two examples of the network scenaros (namely, an 8-MU scenaro and a 12-MU scenaro), whch wll be used n the followng smulatons. We set the total channel bandwdth of the AP shared by all MUs as W = 2MHz (82.11a/b/g/n standard [27]) and the channel bandwdth of the S to a sngle MU as = 5MHz (close to a WCDMA channel [28]). For each MU, we set P max.2w [28], P max =.25W (.e., Power Class-3 of moble devces), Pmax A = =.35W, and n = W/Hz. To account for the economc cost, we set π = $1/G [3] and π A = $2/G. As stated earler, we use top =.5 n Algorthms (Cen) and (Ds), and we valdate the choce of top =.5 at the end of ths secton by comparng dfferent top. A. Accuracy and Effcency of Algorthm (Cen) and Algorthm (Ds) Table I presents the accuracy and computatonal effcency of Algorthm (Cen) and Algorthm (Ds). We consder an 8-MU scenaro wth the randomly generated channel gans{g A } I = [.1256,2.818,.221,.381,.591,.2528,1.4989,.681] 1 4 and {g } I = [ ,1.264,.5815,2.5812, ,2.628,2.3551] 1 8. We vary each MU s from 2Mbps to 8Mbps (as Problem (CMP) s