DOWNLINK CELL ASSOCIATION OPTIMIZATION FOR HETEROGENEOUS NETWORKS VIA DUAL COORDINATE DESCENT. Kaiming Shen and Wei Yu

Transcription

1 DOWNLIN CELL ASSOCIATION OPTIMIZATION FOR HETEROGENEOUS NETWORS VIA DUAL COORDINATE DESCENT amng Shen and We Yu Electrcal and Computer Engneerng Dept. Unversty of Toronto, Toronto, Ontaro, Canada M5S 3G4 Emal: ABSTRACT Ths paper consders the optmal assocaton of remote user termnals to dfferent cells n a heterogeneous network for load balancng. Assumng fxed transmt powers at the basestatons, we adopt a network utlty maxmzaton formulaton wth a proportonal farness obectve and show that the downlnk user assocaton problem can be solved effcently usng a prcng approach where the prces are updated n the dual doman va coordnate descent. As compared to the prevously proposed subgradent method, the proposed coordnate descent algorthm does not requre the base-statons to synchronze n ther prce updates, whle stll guaranteeng convergence, whch makes t partcularly sutable for dstrbuted mplementaton. Smulatons show that the proposed method has fast convergence whle achevng near-optmal soluton. 1. INTRODUCTION Heterogeneous network s an emergng wreless cellular archtecture n whch femto or pco base-statons (BSs) are deployed throughout the geographcal area to off-load traffc from the macro BSs. By splttng the conventonal cellular structure nto small cells, the heterogeneous network archtecture allows more aggressve reuse of frequences as well as mproved coverage and hgher overall throughput for the entre network. One of the man challenges n the desgn and mplementaton of heterogenous network archtecture s the approprate settng of the transmt power levels at the femto/pco BSs and the defnton of each cell s coverage area. In most practcal deployments, the power levels at the femto/pco BSs are fxed a pror (typcally at 10-30dB below the macro BS power). The downlnk coverage areas of the small cells are then defned accordng to the receved sgnal-to-nterferenceand-nose rato (SINR). From the the user termnals perspectve, each user s smply assocated wth the BS wth the hghest SINR. Ths downlnk cell assocaton rule s often referred to as the max-sinr rule. A key problem wth the max-sinr cell assocaton s that t does not account for the vared data traffc pattern n the network, hence t does not address the load balancng problem effectvely. Load balancng s essental for a small-cell envronment, because femto/pco BSs are often deployed to allevate traffc hot-spots wth hgher-than-average user densty. In these stuatons, a heurstc that adds a bas term to the reference SINR often needs to be used n practce to address the load mbalance. Ths paper addresses the downlnk coverage and user-cell assocaton problem from a proportonally far utlty maxmzaton perspectve. We assume that the BS powers are fxed, and use a prcng approach for BS assocaton. Our man contrbuton s a dstrbuted dual prce update method based on coordnate descent whch s guaranteed to converge and s shown to acheve near-optmal performance n practce. The BS assocaton problem has been consdered extensvely n the lterature. For example, the network utlty maxmzaton approach has been consdered n [1], where heurstc prcng strateges are consdered for the ont power control and cell-ste selecton problem. The ont power control and cell assocaton problem s also consdered n [2], whch gves an optmal soluton to the problem but only under certan restrcted condtons and only for the case where the number of users and the number of BSs are the same. In [3], a dfferent specal case of only a sngle par of macro and pco BSs s nvestgated. In [4], an approxmate soluton to the ont power control, beamformng, and BS assocaton problem s gven. In [5], a game theoretcal approach s used, and the Nash Equlbrum of the problem s found. A common method for solvng the BS assocaton problem s the greedy method [6, 7, 8]. However, the performance of the greedy algorthm s not easy to control. The convergence s slow, f only a sngle user mproves ts BS assocaton at each step. But, the algorthm may exhbt oscllatory behavor f too many users swtch BSs at the same tme [7]. In ths paper, we use the dualty theory n optmzaton to tackle the BS assocaton problem. Ths approach s frst taken n [9], where the BS assocaton problem under the proportonally far utlty s solved n the dual doman usng a subgradent method. The dual varables have a prcng nterpretaton. The users determne ther BS assocaton based on /13/$ IEEE 4779 ICASSP 2013

2 ther achevable utltes mnus the BS prce, and the BSs update ther prces based on a subgradent. Ths paper uses the same problem formulaton as n [9], but nstead of usng subgradent update, we propose a prce update based on coordnate descent n the dual doman. As compared to the subgradent method, the advantages of the proposed method are two fold. Frst, the performance of the subgradent method s often senstve to parameter settng, whle the proposed method s not. Second and more mportantly, the mplementaton of subgradent update requres synchronzaton among the BSs n the sense that all BSs must update ther prces usng the same step sze at the same tme n order to ensure convergence. In contrast, the proposed method requres only mnmal coordnaton among the BSs and s therefore more suted for dstrbuted mplementaton. 2. PROBLEM FORMULATION Consder the downlnk of a heterogeneous cellular network wth L BSs wth fxed (but possbly dfferent) transmsson power spectrum densty (PSD) levels, and a total of user termnals across a geographc area (possbly nonunformly). A maxmum frequency reuse factor of 1 s assumed; the total avalable bandwdth s W. Let SINR be the SINR of user f t s assocated wth BS. To smplfy the problem, we assume frequency-flat channels and a flat transmt PSD level so that the SINR s constant across the frequences. Further, we assume that f a total of users are assocated wth the BS, then each user smply gets 1/ of the total bandwdth 1. In ths case, the data rate of the user assocated wth BS can be calculated as R = W log(1 + SINR ). (1) Ths paper adopts a proportonally far (or log-utlty) obectve functon, and seeks user-cell assocatons that maxmze the sum of each user s utlty, log (R ). We follow the problem formulaton of [9]. Let x be bnary varables (1 or 0) denotng whether or not user s assocated wth BS, and let be the number of users assgned to BS. The user assocaton problem can be wrtten as: maxmze x, subect to ( ) W x log log(1 + SINR ) (2) x = 1, = 1,..., (3) x =, = 1,..., L (4) = (5) x {0, 1} (6) 1 Ths s ustsfed n [9] as Round-robn schedulng can be shown to maxmze the log-utlty. where (3) states that each user can only assocate wth one BS, and (5) states that all users n the network must be served. To smplfy the notaton, we ntroduce a as a = log (W log(1 + SINR )). (7) The obectve functon (2) can now be wrtten as f(x, ) = A. Lagrangan Dual Analyss a x 3. ALGORITHM log( ) (8) The BS assocaton problem (2) s a dscrete optmzaton problem, whch s typcally dffcult to solve. However, t turns out that ts dual problem can be explctly derved. Ths forms the bass of the prcng update algorthm of ths paper. Introduce dual varables µ = (µ 1,..., µ L ) for the constrant (4) and ν for (5). The Lagrangan functon of the optmzaton problem wth respect to these two constrants s L(x,, µ, ν) = a x log( ) µ ( ) x ν The dual functon g( ) can now be stated as { max L(x, g(µ, ν) =, µ, ν) s.t. x = 1; x {0, 1}. (9) (10) The above optmzaton problem has the followng explct analytc soluton: { 1, f = () x = 0, f () where () = arg max (a µ ) (11) and = e µ ν 1. (12) Substtutng (11) and (12) back nto (10), we obtan the dual obectve g(µ, ν) = max (a µ )+ ( e µ ) ν 1 +ν. (13) The dual problem s now ust the mnmzaton of the above g(µ, ν) over µ and ν. Note that f () n (11) s not unque, x can be assgned value 1 for any of the BSs wth maxmum (a µ ) wthout affectng the value of g(µ, ν) n (13). The man pont of ths paper s that the optmzaton of g(µ, ν) can be done va coordnate descent. Ths approach s nspred by the development of aucton algorthm by Bertsekas [10]. The BS assocaton problem consdered n ths paper can be thought of as a generalzaton of the 1-to-1 assgnment problem solved by the aucton algorthm [10] to the -to-1 case. 4780

3 B. Subgradent Method We frst revew the subgradent method for optmzng g(µ, ν) as proposed n [9]. Observe frst that f µ s are fxed, g( ) s a dfferentable convex functon of ν, and the optmal ν can be found as ν (t+1) = log eµ(t) 1. (14) However, g( ) s not a dfferentable functon of µ, so nstead of takng the gradent wth respect to µ, the subgradent method updates µ n each step accordng to µ (t+1) = µ (t) α t ( e µ(t) ν(t) 1 (t) ), = 1,..., L (15) where α t s the step sze and (t) = x, where x s as defned n (11). Note that agan x may not be unque because () n (11) may not be unque, but any vald set of x s gves a vald subgradent. One problem wth the subgradent method s that ts convergence speed depends heavly on the choce of step szes α t. Possble choces of α t nclude constant step sze (but the constant s dffcult to choose) or dmnshng step szes (whch guarantee convergence but can be qute slow n practce). As a baselne for comparson, ths paper adopts the self-adaptve scheme of [11] as suggested n [9]. We refer the detaled algorthm descrpton to [11], and only menton that the scheme nvolves qute a few parameters, namely γ t, ρ 1, β < 1, as well as δ 1 and δ. Note that because all the µ s need to be updated at the same tme usng the same step sze (n order to ensure convergence), the dstrbuted mplementaton of the subgradent method requres synchronzed prce updates across the BSs. The dual coordnate descent algorthm proposed n the next secton removes such a requrement. C. Dual Coordnate Descent The basc dea of the proposed approach s to recognze that the dual functon (13) s n a closed form, and t can be optmzed n a coordnate descent fashon n closed form. Fxng all the µ s, we see that ν can be updated accordng to (14). Fxng ν and all µ s except one of them, we see that g(µ, ν) s n fact the sum of a contnuous pece-wse lnear functon and an exponental functon. So we can take ts left and rght dervatves and choose µ to be such that the left-dervatve at µ s less than or equal to zero, and the rght dervatve s greater than or equal to zero. Mathematcally, defne functons { } f 1 (µ ) = C, where C = a µ = max {a s µ s } s (16) and f 2 (µ ) = e µ ν 1. (17) µ * f 1 f 2 (a) f 1 and f 2 have ntersecton µ * f 1 f 2 (b) no ntersecton Fg. 1. Two cases of updatng µ n dual coordnate descent It s easy to see that the left partal dervatve of g(µ, ν) wth respect to µ s ust f 2 (µ ) f 1 (µ ). So, fxng all other dual varables, the µ that mnmzes g( ) s ust µ (t+1) = sup {µ f 2 (µ ) f 1 (µ ) 0} (18) The dual coordnate descent algorthm s descrbed below: Algorthm 1 Dual Coordnate Descent eµ 1 ; Intalzaton: Set µ = 0,. Set ν = log repeat 1) For = 1,..., L, update µ accordng to (18); 2) Update ν accordng to (14); untl Convergence; Set user-bs assocaton accordng to (11). Resolve tes f necessary. The dual coordnate descent algorthm s qute ntutve. The dual varable µ s the prce of the BS, whle a s the utlty of the user f t s assocated wth BS. Each user maxmzes ts utlty mnus the prce among all possble BSs, whle the BSs choose ther prces n an teratve fashon to balance ther loads. Fg. 1 llustrates the prce update condton, whch seeks µ to match f 1(µ ) and f 2(µ ). Here, f 1( ) s a step functon. The functons f 1 ( ) and f 2 ( ) may not ntersect, but the optmal µ can always be determned unquely. As mentoned earler, the man advantage of the dual coordnate descent algorthm as compared to the subgradent method s that the BSs do not need to synchronze n ther prce updates. In fact, the order of prce updates n Algorthm 1 can be arbtrary. Snce each dual update step always decreases the dual obectve, the teratve algorthm s always guaranteed to converge. To completely specfy the algorthm, we also need to descrbe how to recover the prmal varables x from a set of dual solutons. Ths s done usng (11), but a user may have several BSs wth equal (a µ ). To resolve such tes, one may use greedy search or relaxaton methods. In our smulatons, only a very small number of users are typcally nvolved n tes, so te-breakng by exhaustve search s also feasble. 4781

4 Dual current Dual best Dual Coordnate Descent Subgradent Method: δ 1.02 Subgradent Method: δ 1 Subgradent Method: δ 1 =90, γ k Iteratons Cumulatve Dstrbuton Max SINR Dual Coordnate Descent Subgradent Method: δ 1.02 Subgradent Method: δ 1 Subgradent Method: δ 1 =90, γ k Data Rate (Mbps) Fg. 2. Convergence: Dual coordnate descent vs Subgradent Fg. 3. Data rate CDF after 56 teratons D. Dualty Gap Because of the fact that g(µ, ν) s not dfferentable everywhere, the coordnate descent algorthm s not guaranteed to converge to the optmal dual varables. Further, because of the nteger constrants, there may be a non-zero optmal dualty gap between the prmal and the dual problems. For the BS assocaton problem consdered n ths paper, the dualty gap can actually be characterzed n closed form. Let (µ, ν) be a dual feasble soluton, and (x, ) be the prmal soluton recovered by (11) and te-breakng. It s possble to show that f(x, ) = g(µ, ν) ( ) log e µ ν 1 (19) By weak dualty, g( ) s greater than or equal to the prmal optmum f ( ). Thus, the dfference between the current prmal obectve f(x, ) and the true optmum s bounded by ( ) log. Note that whenever e µ ν 1 = e µ ν 1 for a BS, as n Fg. 1(a), t does not contrbute to the dualty gap. When a BS s nvolved n tes, the dualty gap s mnmzed when s made as close to e µ ν 1 as possble. 4. NUMERICAL RESULTS We valdate the proposed algorthm n a 7-cell wrap-around topology wth 7 macro BSs and 3 femto/pco BSs per cell, for a total of 28 BSs. The macro-bss are 1.4km apart. The femto BSs are located 0.5km from the macro BSs. A total of 840 users are dstrbuted unformly across the network. The transmt power spectrum denstes (PSDs) of macro-bss and femto-bss are set to be -27dBm/Hz and -47dBm/Hz, respectvely, over a 10MHz bandwdth. PSD of background nose s assumed to be -174dBm/Hz. The channels are modeled wth a dstance-dependent path-loss gven by log 10 (d)db (where d s n km) and a log-normal shadowng wth standard devaton of 8dB. An antenna gan of 15 db s assumed. Fg. 2 compares the convergence behavor of the dual coordnate descent algorthm wth that of the adaptve subgradent method. Each teraton here refers to ether a sngle update of µ or a subgradent update of all µ s. We see that the dual coordnate descent converges to wthn 10 2 of the optmum wth only three rounds of teratons per BS, whle the convergence of the subgradent method s very senstve to parameters. Here, we set ρ = 1.2, β = 0.9, and δ = n adaptve subgradent [11] and see that dfferent settngs of δ 1 and γ k can result n qute dfferent convergence behavors. We notce n Fg. 2 that the dual coordnate descent algorthm does not converge to the optmum. Ths s due to the fact that t s possble for coordnate descent to get stuck n a suboptmal pont. Ths gap s qute small, however. Fg. 3 shows the cumulatve dstrbuton of data rates after 56 teratons for the varous algorthms. We see that both the dual coordnate descent and the subgradent algorthms, when converged, offer substantal rate mprovement as compared to the max-sinr BS assocaton rule. The 50th-percentle rate s ncreased by about 30%. Ths s a consequence of off-loadng traffc from the macro-bss to the femto-bss. 5. CONCLUSIONS Ths paper proposes a new cell assocaton algorthm for the heterogenous network based on the use of the prcng approach for BS assocaton and dual coordnate descent for prce updates. Comparng to prevous approaches, the proposed algorthm has faster convergence and leads to a soluton structure more sutable for dstrbuted mplementaton. It also leads to more balanced traffc loads and proportonally farer allocaton of resources than the SINR-based cell assocaton. 4782

5 6. REFERENCES [1] J.-W. Lee, R. R. Mazumdar, and N. B. Shroff, Jont resource allocaton and base-staton assgnment for the downlnk n CDMA networks, IEEE/ACM Trans. Netw., vol. 14, no. 1, pp. 1 14, Jan [2] R. Sun, M. Hong, and Z.-Q. Luo, Optmal ont base staton assgnment and power allocaton n a cellular network, n IEEE Workshop on Sgnal Processng Advances n Wreless Communcatons (SPAWC), June 2012, pp [3] S. Corroy, L. Falconett, and R. Mathar, Cell assocaton n small heterogeneous networks: Downlnk sum rate and mn rate maxmzaton, n IEEE Wreless Commun. Networkng Conf. (WCNC), 2012, pp [4] M. Sanab, M. Razavyayn, and Z.-Q. Luo, Optmal ont base staton assgnment and downlnk beamformng for heterogeneous networks, n IEEE Internatonal Conference on Acoustcs, Speech, and Sgnal Processng (ICASSP), Mar. 2012, pp [5] L. Jang, S. Parekh, and J. Walrand, Base staton assocaton game n mult-cell wreless networks, n IEEE Wreless Commun. Networkng Conf. (WCNC), 2008, pp [6] R. Madan, J. Borran, A. Sampath, N. Bhushan, A. handekar, and T. J, Cell assocaton and nterference coordnaton n heterogeneous LTE-A cellular networks, IEEE J. Sel. Areas Commun., vol. 28, no. 12, pp , Dec [7] T. Bu, L. L, and R. Ramee, Generalzed proportonal far schedulng n thrd generaton wreless data networks, n INFOCOM, Apr. 2006, pp [8]. Son, S. Chong, and G. de Vecana, Dynamc assocaton for load balancng and nterference avodance n mult-cell networks, IEEE Trans. Wreless Commun., vol. 8, no. 7, pp , July [9] Q. Ye, B. Rong, Y. Chen, M. Al-Shalash, C. Caramans, and J. G. Andrews, User assocaton for load balancng n heterogeneous cellular networks, submtted for publcaton [Onlne]. Avalable: May [10] D. P. Bertsekas, The aucton algorthm: A dstrbuted relaxaton method for the assgnment problem, Annals of Operatons Research, vol. 14, pp , Dec [11] D. P. Bertsekas, Convex Optmzaton Theory, Athena Scentfc,