Low Complexity Grouping for Massive Scheduling in 4G Networks

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Low Complexity Groupig for Massive Schedulig i 4G Networks Qiarui Li, Lusheg Wag, Laura Cottatellucci, Navid Nikaei To cite this versio: Qiarui Li, Lusheg Wag, Laura Cottatellucci, Navid Nikaei.

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1 Low Complexity Groupig for Massive Schedulig i 4G Networks Qiarui Li, Lusheg Wag, Laura Cottatellucci, Navid Nikaei To cite this versio: Qiarui Li, Lusheg Wag, Laura Cottatellucci, Navid Nikaei. Low Complexity Groupig for Massive Schedulig i 4G Networks. WiOpt 12: Modelig ad Optimizatio i Mobile, Ad Hoc, ad Wireless Networks, May 2012, Paderbor, Germay. pp , <hal > HAL Id: hal Submitted o 18 Dec 2012 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

2 8th Iteratioal Workshop o Resource Allocatio ad Cooperatio i Wireless Networks 2012, 18 May 2012 Low Complexity Groupig for Massive Schedulig i 4G Networks Qiarui Li, Lusheg Wag, Laura Cottatellucci ad Navid Nikaei Mobile Commuicatio Dept., Eurecom, firstame.ame@eurecom.fr Abstract I this paper, we ivestigate user groupig for cooperative schedulig i a two-cell etwork. Whe the umber of trasmitters grows large, the complexity of the Hugaria algorithm optimum for user pairig becomes uaffordable i real-time systems. We cosider user groupig algorithms maximizig the etwork sum rate i cells with a massive umber of termials ad/or sesors. We provide a suboptimal user groupig algorithm which substatially reduces complexity compared to the optimum with egligible capacity degradatio. Surprisigly, the proposed algorithm outperforms the with a cosiderable lower complexity. I. INTRODUCTION Cellular etworks operatig with uiversal frequecy reuse, such as the 3GPP UTRAN Log Term Evolutio Advaced (LTE-A), ca potetially support very high throughput but suffer from cosiderable iter-cell iterferece levels. The, iterferece maagemet is fudametal i the desig of high throughput ad eergy efficiet etworks whe operatig with uiversal frequecy reuse. I the literature, cooperative schedulig schemes have bee proposed as a key approach for iter-cell iterferece coordiatio (ICIC). The beefits of joit schedulig ad power cotrol schemes for dowlik scearios with multiple cells ad sigle resource block (RB) have bee aalyzed i [1]. Importat properties, such as the upper boud ad lower boud of global system capacity have bee derived i the asymptotic coditios whe the umber of users grows large ad the system ca widely beefit from user diversity. LTE-A adopts this schedulig policy ad proposes three regular schedulig schemes: persistet, semi-persistet, ad dyamic schedulig. The latter two approaches are cooperative [2] ad their schedulig procedures aim at optimizig a give global metric, e.g. total capacity, joitly over all cells by makig use of a limited amout of exchaged iformatio amog evolved odebs (enbs). For uplik, enbs exchage received sigal stregths (RSS) ad iterferece sigal stregths (ISS) ad the supported cooperative schedulig ca achieve better global capacity, fairess, ad QoS tha o-cooperative schedulig. I [3], a multiuser multiple trasmit multiple receive ateas (MU-MIMO) sigle-cell system is cosidered ad simultaeous two-user trasmissios are optimally scheduled based o the [4]. I [5], [6], a LTE sigle-cell frequecy-selective fadig chael i uplik is cosidered. Two simultaeous SC-FDMA trasmissios are possible ad equalizatio is performed at the receiver side. Substatial gais are show (e.g. by up to 6 db) by joit user groupig ad frequecy-domai resource allocatio compared to radom user groupig ad frequecy-domai resource allocatio [6]. A user pairig algorithm satisfyig proportioal fairess quality of service costraits ad based o the Edmod s algorithm has bee proposed i [7] for siglecell system. A cetralized proportioal fair uplik schedulig scheme for a multi-cell LTE etwork with sigle RB has bee proposed i [8]. Two frequecy ad power sub-optimal allocatio schemes for the sceario with multiple cells ad multiple RBs i LTE uplik are proposed i [9]. Optimizatio problems for cooperative schedulig are i geeral discrete ad ofte have o-polyomial (NP) complexity. This makes cooperative schedulig ot feasible o a real system whe the umber of active UEs becomes very large as i the case of sesors ad machie-type commuicatios [10], shortly referred as user equipmet (UE) commuicatios. I this cotributio, we desig algorithms for cooperative schedulig i the uplik of a two-cell LTE-A system whe a massive umber of UEs populates the cells ad the UEs are uiformly distributed i the cell. We utilize the kowledge of the statistical distributio of the RSS ad ISS at both enbs to reduce the complexity of the schedulig algorithm. We propose two schedulig algorithms that achieve ear-optimum performace with a lower complexity tha the Hugaria algorithm. The key idea of the proposed algorithms cosists i partitioig the UEs of each cell i subsets ad establishig a bijective mappig betwee subsets of differet cells. The, a assigmet algorithm is applied to pair UEs belogig to the subsystems geerated by partitioig ad bijectio. The iitial partitio step allows to keep the dimesio of the assigmet problem reasoably low ad reduce drastically the schedulig complexity. The algorithm dubbed partitioig ad greedy groupig, surprisigly, achieves higher performace tha the with a sigificatly lower complexity. Our aalysis shows that, for the proposed system model, coordiated schedulig does ot beefit from user diversity, the average performace does ot icrease with the umber of UEs i the system, ad the does ot appear to be asymptotically optimum. Techically, this is due to the fact that the radom variables that defie our problem are strogly correlated. The remider of the paper is orgaized as follows. Sectio II presets the system model ad problem statemet. I sectio III, schedulig statistical patters based o Hugaria algorithms is derived ad two heuristic cooperative schedulig algorithms are proposed. Sectio VI presets the umerical results of the proposed algorithms ad compares them with / Copyright is with IFIP 460

3 exitig algorithms. Sectio V provides cocludig remarks ad future directios. II. SYSTEM MODEL AND PROBLEM DESCRIPTION I this paper, we cosider two adjacet enbs with a massive umber of UEs i uplik commuicatios. Each UE commuicates oly with the earest enb, formig a star topology withi hexagoal cells. Whe a UE trasmits to its enb, the overhearig adjacet enb receives this sigal as iter-cell iterferece. We assume that each cell is populated by uiformly distributed UEs. The chael states for all UEs to both enbs are static. The two enbs are deoted by enb K ad enb L, while the UEs i the two cells are deoted by UEK i, for i =1,..., N K, ad UE j L, for j =1,..., N L, respectively. The chael atteuatio from UEK i to enb K ad enb L are deoted by gk i ad hi K. Similarly, chael atteuatio from UE j L to enb L ad enb K are deoted by g j L ad hj L. The atteuatio coefficiets gx,h i i x, with x {K, L}, follow a log distace pathloss model with o fadig, i.e., if d is the distace betwee a UE ad a enb, the correspodig atteuatio coefficiet is give by d α, where α is the pathloss expoet ragig typically from 2 to 4. Two UEs, oe i each cell, may trasmit simultaeously to the respective enbs. If UEK i ad UEj L trasmit simultaeously, their cotributed spectral efficiecy is give by rij K = log(1 + Pgi K ) Ph j L +σ2 ad rji L = log(1 + Pgj L ), respectively, where P is the Ph i K +σ2 trasmittig power ad σ 2 is the variace of the additive white Gaussia oise (AWGN). Let N = mi(n K,N L ). We divide the available time resource i max(n K,N L ) equal time slots ad cosider a cooperative schedulig that assigs a slot to each of N K N L UEs i the cell with higher umber of trasmitters. For the remaiig UEs, the scheduler assigs a sigle slot to two UEs, oe i each cell. We assume that the selectio of the N K N L UEs trasmittig aloe is arbitrarily doe by the enb accordig to priority or quality of service criteria. Therefore, i the followig, we focus o the user groupig problem of two sets of N UEs over N time slots. Let us deote by π a permutatio of the set N = {1,..., N} ad by π i its ith elemets. Without loss of geerality, a schedulig ca be represeted by a permutatio π ad the time slot i is assiged to UEi K ad UEπ L i, simultaeously. The average spectral efficiecy of the two cell etwork correspodig to permutatio π is γ(π) = 1 ri,π K N i + rπ L i,i. (1) i=1,...,n Let Π be the set of all possible permutatios, a optimal cooperative schedulig allocatio maximizes the average spectral efficiecy of the system, i.e., it is the permutatio π that maximizes γ(π). This optimizatio problem reduces to the classical problem of assigmet that ca be solved optimally by the Hugaria algorithm [4] with polyomial complexity i the case of two-cell system. More specifically, it ca be solved with a complexity order O(N 3 ). A lower complexity algorithm is provided by the, which has a complexity i the order of O(N 2 log N). Although approaches with polyomial complexities are available, their applicatio to cells with massive umber of active UEs becomes rapidly uaffordable. This motivates a desig of low complexity algorithms tailored to the peculiarities of the system at had. III. CELL PARTITIONING AND GROUPING SCHEME To uveil statistical properties of optimum schedulig i systems with a massive umber of UEs we applied the Hugaria method [4] to a two-cell system with a massive umber of UEs radomly geerated. Fig. 1 shows the obtaied pairig for two-cell system, each of them populated by 1000 UEs, uder the assumptios of a log distace pathloss model without fadig for the chael atteuatios, ad equal trasmit powers for all the UEs. A UE of a certai color i cell L is paired with some user of the same color i cell K. Fig. 1 is plotted such that UEs i cell L have gradually varyig colors from ceter to edge. It ca be see that users with higher RSS i cell K, i.e. closer to enb K ad with greater gk i, ted to be paired with users with small ISS i cell L, i.e. farther from enb L ad with lower h j L. A mirrorig figure ca be obtaied whe the colorig is drive by users i cell K. This shows that the statistical patter is symmetric betwee the two cells. Additioally, if we oly cosider the capacity of UEs i cell L, the statistic patter is close to the oe show i Fig. 1, except some blur for edge UEs. y/km x/km Fig. 1: Statistical patter usig. These observatios suggest a low complexity algorithm based o the orderig of RSS ad ISS withi a cell. Before detailig our cell partitioig ad groupig schemes, we show a iterestig property of user groupig based o the orderig of RSS ad ISS withi a cell. Let us focus o cell K. We call quick pairig i cell K the algorithm that sorts UEs i cell K i descedig order of gk i ad UEs i cell L i ascedig order of h j L ad, the, pairs UEs with idetical orderig idex. This algorithm provides the optimum pairig whe the objective fuctio to be maximized is the spectral efficiecy of cell K as show i the followig theorem

4 Theorem 1. The quick pairig algorithm maximizes the objective fuctio 1 N i=1 N rk i,π i over all possible permutatios π Π. Proof: Let us observe that, for ay two slots i ad j, if gk i g j K ad hi L hj L, it results (Ph i L + σ 2 + Pg i K)(Ph j L + σ2 + Pg j K ) =(Ph i L + σ 2 + Pg j K + Pgi K Pg j K ) (Ph j L + σ2 + Pg i K + Pg j K Pgi K) =(Ph i L + σ2 + Pg j K )(Phj L + σ2 + Pg i K ) +(Ph i L + σ2 + Pg j K )(Pgj K Pgi K ) +(Ph j L + σ2 + Pg i K )(Pgi K Pgj K ) +(Pg j K Pgi K)(Pg i K Pg j K ) =(Ph i L + σ 2 + Pg j K )(Phj L + σ2 + Pg i K) + P 2 (g i K g j K )(hj L hi L) (Ph i L + σ2 + Pg j K )(Phj L + σ2 + PgK i ). (2) Let us assume that the UEs i cell K are ordered i decreasig order of gk i. Let π be ay permutatio of UEs i cell L. We focus o ay arbitrary pair i, j of time slots with i < j ad h i L hj L. The, the total capacity correspodig to the schedulig iduced by permutatio π ad the total capacity correspodig to the permutatio π + obtaied from π switchig UE i L ad UEj L are related by r π = N =1 r K,π N PgK = log(1 + Ph π =1 L + ) σ2 = PgK log(1 + )+ Ph π L + σ2 =1,...,N i,j log [ L + σ2 + Pg i K )(Phπj =1,...,N i,j log =r π +, log(1 + L + σ2 )(Ph πj Ph π L Pg K + σ2 )+ [ L + σ2 + Pg j K )(Phπj ] L + σ2 + Pg j K ) L + σ2 ) ] L + σ2 + PgK i ) L + σ2 )(Ph πj L + σ2 ) where r π ad r π + deote the total capacities of cell K correspodig to the schedulig π ad π + respectively, ad the iequality is a straightforward cosequece of (2). The, by applyig repeatedly this swappig with the above metioed criterio, we obtai the quick pairig as optimum schedulig policy. Although the quick pairig is optimum whe the objective fuctio to be optimized is the spectral efficiecy of a sigle cell, it is ot a optimal solutio for maximizig the total capacity of the two cells. Therefore, based o Theorem 1, we propose the followig heuristic scheme to partitio each cell ito subsets of equal size ad pair them: Sort the UEs i cell K i a descedig order of RSS ad the UEs i cell L i a ascedig order of ISS; Partitio the N users i each cell ito subsets such that the i th subset cotais users i the order positios from (i 1)N +1 to (i 1)N + N ; Cosider the subsystem cosistig of the ith subsets of cell K ad cell L, ad apply ay stadard pairig algorithm to it. The proposed partitioig allows to keep the dimesio of the assigmet problem reasoably low hece reduce drastically the schedulig complexity. We propose the applicatio of the Hugaria or the to each subsystem. Hece, we obtai two algorithms that we dub partitioig ad Hugaria groupig ad partitioig ad greedy groupig, respectively. The prelimiary partitioig ad subsequet groupig decreases the time complexity of the algorithms from O(N 3 ) to O( N 3 ) whe Hugaria groupig is performed, ad 2 from O(N 2 log N) to O( N 2 log N ) whe greedy groupig is applied. Other groupig algorithms ca be applied withi a subsystem as log as their objective fuctio is the total capacity of the two-cell system. IV. NUMERICAL PERFORMANCE ANALYSIS I this sectio we assess the performace of the proposed algorithms i terms of both attaied average total capacity ad complexity by umerical simulatios. The heuristic partitioig ad groupig algorithms are compared to cooperative schedulig based o ad. Sice the class of greedy/myopic algorithms is a large group of suboptimum approaches aimig at solvig combiatioal optimizatio problems based o the idea to select the locally best choice i each decisio stage without cosiderig overall global optimality we briefly describe the adopted i our simulatios. Let us itroduce a matrix R (N) with compoets R (N) ij = rij K + rl ji. The cosists of N steps. At step l, it selects the maximum etry from the matrix R (N l+1) ad costructs a matrix R (N l) obtaied from R (N l+1) by removig the colum ad the row correspodig to the selected etries. A first group of simulatios is performed assumig α =2, geeratig radomly, accordig to a uiform distributio, a umber of users N varyig from 100 to 500, ad fixig the umber of subsets i the partitio equal to 10. The results are averaged over 100 experimets. Fig. 2 shows the average capacity 1 of the two-cell system as a fuctio of UEs i each cell for the Hugaria ad s, for the proposed algorithms ad for the quick pairig. Radom pairig 1 Note that the optimizatio does ot chage if we cosider average capacity istead of total capacity. However, the former metrics eables a isightful graphical compariso of systems with differet umber of UEs. 462

5 average capacity partitioig ad Hugaria partitioig ad greedy quick pairig radom pairig umber of UE odes Fig. 2: Compariso of optimal ad suboptimal algorithms: average system capacity versus umber of UEs per cell. schedulig time/ s partitioig ad Hugaria partitioig ad greedy quick pairig umber of UE odes Fig. 3: Compariso of optimal ad suboptimal algorithms: processig time versus umber of UEs per cell. Let us observe that the partitioig ad greedy groupig algorithm reduces to the ad to the quick pairig as the subsets i the partitio is equal to 1 ad to N, respectively. Similarly, the partitioig ad Hugaria groupig boils dow to the Hugaria ad quick pairig algorithms i aalogous situatios. The, it is iterestig to ivestigate performace ad complexity of the proposed algorithms as the umber of partitio subsets varies. The umerical aalysis is preseted i Figures 4 ad 5 i terms of average capacity ad processig time, respectively. I cotrast to the fact that performace ad complexity of the partitioig ad Hugaria groupig algorithm decreases as the umber of subsets icreases, the performace of the partitioig ad greedy groupig algorithm has a optimum umber of partitio subsets both for the average capacity ad the time processig. The two optima are very close each other ad quite high: betwee 30 ad 50 subsets. Surprisigly, there is a regio where the time processig decreases while the performace icreases. The, for practical implemetatio the umber of partitio subsets eeds to be optimized. A aalytical performace aalysis exceeds the scope of this cotributio. I fact, the state of art o the theoretical aalysis of radom assigmet problems is still at its ifacy to be applied to the complex settig of the problem at had. However, some qualitative isights o the system are still possible. Let us cosider the radom variables ri,π K i + rπ L i,i. Their margial distributio matches very accurately a gamma distributio. Figure 6 shows this matchig for a probability desity fuctio (p.d.f) obtaied i a regular hexagoal cell with edge of legth 1 ad α =2. The margial radom variable fits a Gamma distributio with Γ(k, θ) = Γ( , ). If all the radom variables had bee idepedet ad idetically distributed, the performace of the system could have bee aalytically determied by applyig the results i [11] with the followig coclusios: is adopted as bechmark to assess the beefits of pairig. Iterestigly, the average capacity for a sufficietly large umber of UEs stays costat ad the user diversity does ot eable ay capacity gai. We will further discuss this aspect i some coclusive remarks i this sectio. Fig. 3 shows the complexity of the ivestigated algorithms i terms of processig time for icreasig umber of users. The compariso amog the aalyzed algorithms shows that ay pairig algorithm outperforms sigificatly a radom pairig. Additioally, the two proposed algorithms attai almost the same average system capacity. They outperform the ad the quick pairig ad achieve almost the same performace as the optimum. The compariso i terms of complexity chages substatially: the partitioig ad greedy groupig algorithm has a complexity sigificatly lower tha the greedy algorithm ad the partitioig ad Hugaria groupig algorithm while it attais substatially better performace tha the former ad the same performace as the latter. As expected, it has higher complexity tha the sigificatly suboptimal quick pairig. The asymptotic performace of optimal groupig are give by Nθ log N + Nkθ log log N + O(N), i.e. the average utility scales as log N ad icreases with the system size. As a example, the average performace of a radom assigmet problem with Γ( , ) is aroud 13.7 ad for N = 100 ad N = 500. This implies that the optimizatio eables pairig of users with performace i the tail of the distributio. The is asymptotically ear-optimal, i.e. it coverges asymptotically to the optimal algorithm. The same could have bee show for the proposed algorithms. Due to a strog correlatio of the radom variables that defie the assigmet problem, the average performace of the system at had does ot scale with the umber of UEs ad, also for large umber of users, the average performace is close to the mea of the margial distributio with a cosiderable performace loss compared with a hypothetical system defied by i.i.d. radom variables. Additioally, the is suboptimal also i asymptotic coditios. 463

6 average capacity partitioiog ad Hugaria partitioig ad greedy quick pairig probability umber of groups Fig. 4: Compariso of optimal ad suboptimal algorithms: average system capacity versus umber of partitio subsets. schedulig time/ s partitioig ad Hugaria partitioig ad greedy quick pairig umber of groups Fig. 5: Compariso of optimal ad suboptimal algorithms: processig time versus umber of partitio subsets. V. CONCLUSION AND FUTURE EXPERIMENTS Two heuristic cooperative schedulig algorithms for dese cells are provided based o the statistical schedulig patter for cells with a massive umber of UEs. The partitioig ad greedy groupig algorithm ca sigificatly decrease the processig time with egligible performace degradatio i terms of capacity. Surprisigly, the proposed algorithm has complexity sigificatly lower tha the with sigificatly higher performace. I future studies, the optimal umber of partitio subsets for a fixed umber of UEs i the cell should also be ivestigated. I this cotributio, the aalysis is based o the assumptio of log distace pathloss model for the chael. More complex models should be cosidered to determie if partitioig ad greedy groupig is always efficiet ad applicable. Fially, a theoretical performace aalysis of the radom assigmet problem for correlated utilities is highly desirable for a deep uderstadig of the problem at had joit capacity Fig. 6: histogram ad fitted probability desity fuctio of joit capacity ACKNOWLEDGEMENT This work was developed i the cotext of the CONECT project - Cooperative Networkig for High Capacity Trasport Architectures ( ad partially fuded by the Europea Commuity s Seveth Framework Programme uder grat agreemet o It was also partially supported by Agece Natioale de la Recherche, with referece ANR-09- VERS-001. REFERENCES [1] D. Gesbert, S. G. Kiai, A. Gjedemsjo ad G. Oie, Adaptatio, coordiatio, ad distributed resource allocatio i iterferece-limited wireless etworks, Proceedigs of the IEEE, vol. 95, o. 12, pp , Dec [2] K. Igema, Peterse et al., A overview of dowlik radio resource maagemet for UTRAN log-term evolutio, IEEE Commuicatio Magazie, vol. 47, o. 7, pp, 86 93, July 2009 [3] Emauele Viterbo ad Ari Hottie, Optimal user pairig for multiuser MIMO, i Proc. ISSSTA, 2008, pp [4] D. Jugickel, The, Chapter 14, Graphs, Networks ad Algorithms, Spriger-Verlag Berli Heidelberg, [5] M. Ruder, U.L. Dag, ad W. Gerstacker User Pairig for Multiuser SC- FDMA Trasmissio over Virtual MIMO ISI Chaels I Proceedigs of IEEE Global Commuicatios Coferece (Globecom 2009), Hoolulu, HI, November/December 2009 [6] M.A. Ruder, Daiyog Dig, Uye Ly Dag, W.H. Gerstacker, Combied User Pairig ad Spectrum Allocatio for Multiuser SC-FDMA Trasmissio, 2011 IEEE Iteratioal Coferece o Commuicatios (ICC), 5-9 Jue 2011 pp [7] Nikuj Aggarwal, R. Saravaa Maickam, ad C. Siva Ram Murthy, Cross-Layer User Pairig for CSM i IEEE Networks, IEEE Commuicatios Letters, Vol. 15,.5, pp , May [8] P. Frak, A. Muller, H. Droste ad J. Speidel, Cooperative iterfereceaware joit schedulig for the 3GPP LTE uplik, i Proc. PIMRC, Sept. 2010, pp [9] M. Jalloul, A. M. El-Hajj ad Z. Dawy, Uplik iterferece coordiatio/avoidace i LTE systems, i Proc. NGMAST, July 2010, pp [10] 3rd Geeratio Partership Project; 3GPP TR v1.6.1, System improvemets for Machie-Type Commuicatios (MTC), Techical Specificatio Group Radio Access Network, Evolved Uiversal Terrestrial Radio Access Network (E-UTRAN), Release 11, [11] W. Szpakowski, Combiatorial optimizatio through order statistics, Secod Aual. Iteratioal Symposium o Algorithms,Taiwa,

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