Maximizing the Capacity of Large Wireless Networks: Optimal and Distributed Solutions

Transcription

1 Maximizig the Capacity of Large Wireless Networks Optimal ad Distributed Solutios Saad G. Kiai ad David Gesbert Mobile Commuicatios Departmet, Eurecom Istitute, Sophia Atipolis, Frace {kiai, Abstract We aalyze the sum capacity of multicell wireless etworks with full resource reuse ad chael-drive opportuistic schedulig i each cell. We address the problem of fidig the co-chael (throughout the etwork) user assigmet that results i the optimal joit multicell capacity, uder a resource-fair costrait ad a stadard power cotrol strategy. This problem i priciple requires processig the complete cochael gai iformatio, ad thus, has so far bee justly cosidered upractical due to complexity ad chael gai sigalig overhead. However, we expose here the followig key result The multicell optimal user schedulig problem admits a remarkably simple ad fully distributed solutio for large etworks. This result is proved aalytically for a idealized etwork. From this costructive proof, we propose a practical algorithm that is show to achieve ear maximum capacity for realistic cases of simulated etworks of eve small sizes. I. INTRODUCTION I wireless etworks featurig multiple simultaeous trasmissio liks (cellular or ad-hoc), there exists a well kow trade-off betwee the reuse of spectral resource across these liks ad the iterferece created to oe aother by co-chael trasmissios. Power cotrol strategies were proposed to limit iterferece effects by targetig a give sigal-to-iterferece ratio (SIR) [], [] or a received sigal power-level [3], leadig to a trucated chael gai iversio. Combiig power cotrol with cell diversity was subsequetly show to icrease the umber of supported users i the uplik [4]. Recetly proposed resource allocatio techiques [5], [6], [7], [8], [9] attempt to mitigate this problem by exploitig directioal ateae, sectorizatio or clever locatio-depedet power trasmissio profiles to reduce the iterferece. However, to fully beefit from all degrees of freedom provided by the multi-user fadig chael, a more promisig solutio lies i the cocept of cochael user schedulig. I this settig, the assigmet of the users to the spectral resource is doe ot just to maximize the capacity of each idividual cell [0], but rather to maximize the capacity of all liks ad cells i a joit fashio, thus givig rise to a extesio of the cocept of multi-user diversity to a full etwork. I priciple the complexity of the above problem is high the umber of degrees of freedom is govered by the umber of cells the umber of users per cell the umber of possible schedulig slots to which a user ca be assiged. Additioally, it ca be show easily that the co-chael schedulig problem makes sese oly if some form of power cotrol is used. I this paper, we first formulate the co-chael user schedulig problem for a arbitrary etwork give the complete etworkwide co-chael gai iformatio ad a stadard power cotrol rule (gai iversio-based power cotrol). Next, we propose a idealizatio for a large etwork coied iterferece-ideal etwork, that ca be exploited to simplify the problem formulatio. We the obtai the followig strikig results ffl For iterferece-ideal etworks, maximum etwork capacity ca be reached by usig a low-complexity fully distributed schedulig protocol, based o local chael gais. This result admits a theoretical costructive proof which we further exploit to propose a multicell schedulig algorithm for realistic (o-ideal) etworks. ffl For fast-fadig, the algorithm is a geeralizatio of the sigle cell maximum capacity scheduler [0] to the multicell case. As a result, per-cell throughput maximizatio ad multicell iterferece avoidace are show to go had i had ad multi-user diversity schedulig ca also be throughput optimal i a multicellular sceario. From the aalysis above we derive a practical co-chael schedulig algorithm that ca trade-off resource fairess for system capacity. These results have applicatios i cellular/adhoc etworks with iterferece-limited trasmissio. I this paper we test the algorithms over fiite-size o-ideal cellulartype etworks ad show the throughput gais over a ocoordiated co-chael scheduler i the presece of iterferece. II. NETWORK CAPACITY MODEL Cosider a multicell system with N access poits (AP) commuicatig with U user termials (UT) i each cell. We cosider the dowlik i which the AP seds data to the UT, but the results preseted i this paper ca be geeralized to the uplik. We assume a multiple access scheme i which spectral resource is orthogoally divided ito uits called (e.g. code-, time-, frequecy- etc.) slots. Slot assigmet occurs simultaeously i all cells ad is used to separate the trasmissios to the users of ay give cell. We eforce K- th order resource fairess, where» K» U. This meas that a schedulig frame cosists of K slots assiged to K distict users per cell. Note that this does ot ecessarily yield throughput fairess, eve with K = U, as users may ot ejoy a equal throughput due to local chael coditios. Moreover,

2 because of cocurret trasmissios i all cells i ay oe slot, a assiged user sees iterferece from all co-chael cells. A. Sigal Model To preserve light otatio we focus o the sigle atea case. For a user u i cell, the dowlik is a typical iterferece chael [], the received sigal for which is give by p p Y u = X u ix ui Z u ; where X u is the sigal from the servig AP ad Z u is additive white Gaussia oise. The sigal to iterferece-plusoise ratio (SINR), is give by, u = EjZ u j EjX u j iejx ui j If trasmit power used by a AP to serve u is P u,we have EjX u j = P u. Note that m R reflects the composite chael gai possibly icludig fast-fadig. B. Power Cotrol As is see later, power cotrol plays a key role i eablig the gais of etwork coordiatio. We assume each AP has a peak trasmissio power costrait, P MAX ad a multiplicative power cotrol factor 0 < ρ» is used to adjust the trasmitted power such that P u = ρ u P MAX. The AP trasmit power is adjusted i order to achieve a target received power at the receiver. If it is ot achievable, the AP trasmits at full power. Assumig each user ca measure ad commuicate back the power received from the servig R AP, Λ ρ u = P MAX. But, sice there is a peak power costrait P MAX, ρ is upper bouded by oe ρ u = mi ( P MAX ; ) () We poit out here that the capacity optimal schedulig policy should be joitly optimized with the power cotrol policy. Such issues are, however, beyod the scope of this paper ad will be addressed i a later paper. Power cotrol settig Depedig o the value of ad the itra-cell chael gai, a user will be receivig i full (ρ =) or reduced (ρ <) power mode. We cosider three etwork scearios. () fully power cotrolled (FPC) etwork All users achieve after power cotrol. () mixed power cotrolled (MPC) etwork Oly a fractio of users achieve. (3) o power cotrolled (NPC) etwork ρ = for all users. III. THE CO-CHANNEL USER MATCHING PROBLEM We assume that chael gais do ot vary over the schedulig frame duratio which is sized i accordace with the coherece period of the chael. Uder the K-th order resource fairess costrait, the co-chael user matchig problem cosists i selectig K users i each cell ad assigig these users to K slots so as to optimize the system utility fuctio (joit capacity). To facilitate the formulatio of the problem, we state the followig defiitios Defiitio A schedulig policy ' is a bijective mappig of the subset U, cosistig of K users chose from the set of all users i cell, oto K the set of slots, ' U 7K. Defiitio A schedulig vector I cotais the set of users scheduled i slot k across all cells (based o ') I = h u u u u N i T [;K] N ; where [I ] = u is the user scheduled durig slot k i cell. Note that because ' is a T bijectio, schedulig vectors are elemet-wise disjoit, I (a) I (b) =? 8 a 6= b. The schedulig vector is the esemble of users which iterfere with each other ad thus it determies the sum capacity for slot k. Defiitio 3 A schedulig matrix S is a K-colum matrix composed of schedulig vectors give by the schedulig policy '. i S = hi () I () I (K) This matrix describes the complete orderig of all users durig oe frame. A. System Performace The SINR for users scheduled i slot k will deped o the schedulig vector I. We ca express the SINR i cell as (I ;)= ff ρ u P MAX i ρ u P MAX i () Assumig a ideal lik adaptatio protocol, the per cell capacity i slot k ca be expressed i bits/sec/hz/cell usig the Shao capacity, C(I )= N = (I ;) (3) The etwork capacity of the system is a fuctio of the overall schedulig matrix S give by KX C(S) = C(I ) (4) K k= KX = ρ u P MAX NK k= = ff i ρ u i P MAX B. Roud Robi Schedulig A stadard approach for resource fair schedulig is roud robi (RR) i which users are give slots tur by tur i each frame. Lettig S be the set of all schedulig matrices, the etwork capacity for RR is C RR = E (SS) C(S) o (5)

3 C. Optimal Co-chael Schedulig O the other had, the schedulig policy for optimum etwork capacity (4) ca be stated as S Λ = argmaxfc(s)g (6) z } SS As S Λ gives the optimal etwork capacity, we have i geeral C(S Λ ) C RR. Iequality will be strict i most cases, thus showig the gai of coordiated etworks over ucoordiated oes. Multicell schedulig gai i NPC system It is easy to see that some scearios will result i o gai at all as show below Lemma For a o power cotrol (NPC) etwork, the etwork capacity gai associated with multicell schedulig is zero. Proof With o power cotrol ρ u =8 u, ad thus all BS trasmit at same (maximum) power. Substitutig this i () we obtai (I ;)= ff P MAX i P MAX ; (7) which is idepedet of the choice of co-chael users i other cells. It follows that the capacity will be the same o matter which users are scheduled with each other. This result idicates that the gai ca be ituitively expected to deped much o the degree of variability of chael ad power cotrol coefficiets across the etwork users, as well as o the umber of cells ad users. We ow tur to the issue of fidig the optimal S. IV. OPTIMUM SUM CAPACITY SCHEDULING As S is a discrete fiite set, (6) is a o-liear combiatorial optimizatio problem for which, fidig optimal solutios is NP-hard. jsj =(U ) N, cosiderig that a set of K schedulig vectors ca be ordered i K ways without chagig the etwork capacity. Eve for a small etwork this method remais prohibitive for N =7ad U =5, jsjß9 0. Collectig ad processig all path gai iformatio withi the coherece time will pose sigificat sigalig ad delay problems. I order to fid a distributed multicell schedulig algorithm istead, we itroduce a simplified model for etwork capacity used to later approximate the actual capacity. A. Iterferece-Ideal Networks Due mostly to the large umber of iterferece sources addig up at the receiver, we ca offer a simpler model for iterferece i large full reuse etworks. We defie the cocept of a iterferece-ideal etwork as oe i which, for ay cell user, the total received iterferece is idepedet of its locatio i the cell. Mathematically, a etwork is iterfereceideal if, for ay user u ad cell iρ ui P MAX = G ρ ui P MAX ; (8) where G does ot deped o the locatio of u. Fortuately, the iterferece-ideal etwork is a good model for a full reuse etwork with a large umber of cells We have, due to the iterferece chael gai ad the power cotrol coefficiets beig ucorrelated iρ ui P MAX ß =N i ρ ui P MAX ; ad for a large N, due to the law of large umbers, ) i ß G 8 u (for large N); N where G ca be thought of as the mea chael gai from a iterferer to ay user positio. The value of G will aturally deped o the distributio of iterferers ad the chael statistics but, as will be see, this value eed ot be kow. Moreover, variatio of the iterferece from the cell ceter to the cell boudary i a dese etwork ca be show to be quite small [] ad from a algorithmic desig poit of view, we ca cosider that all users get the average chael gai G from each AP. B. Optimum Schedulig i Iterferece-Ideal Networks We characterize the solutio to the optimal etwork schedulig problem i a iterferece-ideal etwork i a FPC sceario. Usig (8) ad () we ca rewrite () as (I ;)= ff G N X The etwork capacity will be give by C = NK KX k= = i i ff G N X i i (9) (0) We defie a vector U #, cotaiig the K users of U ordered i descedig order of itra-cell chael gais, U #=[u ; u j; u K; ] T ; where ; j; K;. We ow preset the followig result Theorem Let S #= [U # U # U N #] T ad ß(S #) be the schedulig matrix obtaied by applyig ay colum-wise permutatio o S #. The, for a iterfereceideal etwork, ß(S#) is a optimal schedulig matrix, S Λ for the problem (6). Proof See Appedix. Based o Theorem a optimal schedulig policy is for each cell to rak its users by, say, decreasig order of chael gai ad assig the best K users to the K available slots, regardless of the chael gais i other cells. As co-chael users are matched based o the rak of their chael gai, we call this schedulig policy Power Matched Schedulig (PMS).

4 PMS is completely distributed as local chael gai is the oly schedulig criteria. Note that a side-effect of the policy is to group users with similar chael quality levels, possibly creatig a ufair service. V. MULTI-USER DIVERSITY AND FAIRNESS Iterestigly, whe we choose K =(o resource fairess), Theorem leads to schedulig the user with the best chael gais i each cell. This ca be iterpreted as a geeralizatio of multi-user diversity schedulig to the multicell case. Clearly, for K > there is a ufair repartitio of iterferece amog the differet slots with best users gettig also the least iterferece. Thus, etwork capacity is optimized at the expese of throughput fairess which is reasoable from a iformatio theoretic poit of view ad exteds the well-kow capacity/fairess trade-off kow i sigle cell scearios [0], [3]. Notice that K ca be selected (betwee ad U) by the service provider to vary the resource fairess/throughput trade-off. For K =oly multi-user diversity gai is obtaied without regard for resource fairess, while K = U provides full resource fairess at the cost of capacity. VI. NUMERICAL RESULTS The performace of PMS is compared with RR through Mote Carlo simulatios uder a full resource fairess costrait (K = U). A 800 MHz hexagoal cellular system with km. radius cells is cosidered, with 30 users/cell radomly spread accordig to a uiform distributio. Both iter-cell ad itra-cell AP-UT liks are based o the COST- 3 [4] path loss model icludig zero mea ormal shadowig with a stadard deviatio of 0 db ad fast-fadig οcn(0; ). correspods to a SNR target of 30 db ad P MAX = W. These parameters result i a MPC etwork which serves to compare the schemes i a realistic settig. For PMS, the schedulig matrix is give by Theorem ad, i accordace with (5), RR is modeled by selectig a radom permutatio of the schedulig matrix for each frame. The compariso of the two schedulig policies is represeted by the Network Capacity Gai fi, give by fi = C(SΛ ) C RR.We show traces of etwork capacity obtaied with N = 3 ad N =9i figs. & respectively. As the umber of cells icreases, iterferece averagig reduces variatio i etwork capacity yieldig a icrease i gai. As expected, PMS outperforms RR i all cases ad moreover, the gai icreases with system size (fig. 3). Moreover, the gai is greater i the presece of both shadowig ad fast-fadig. This leads to the coclusio that icrease i system size, as well as greater chael variatio, improves performace. VII. CONCLUSION We address the problem of multicell schedulig for wireless etworks. We show that large gais are obtaied from itercell coordiatio due to the iter-cell iterferece variability that stems from power cotrol ad fadig. We show that the optimal scheduler ca be efficietly approximated by a fully distributed multicell scheduler. I the optimal scheduler each cell raks its users accordig to decreasig chael gais. The multi-cell scheduler is also cosistet with maximizig the capacity of each cell idepedetly through multi-user diversity. REFERENCES [] J. Zader, Distributed cochael iterferece cotrol i cellular radio systems, IEEE Tras. Veh. Techol., vol. 4, pp , Aug. 99. [] G. J. Foschii ad Z. Miljaic, A simple distributed autoomous power cotrol algorithm ad its covergece, IEEE Tras. Veh. Techol., vol. 4, 993. [3] J. F. Whitehead, Sigal-level-based dyamic power cotrol for cochael iterferece maagemet, i Proc. IEEE Vehicular Techoy Coferece, May 993, pp [4] S. V. Haly, A algorithm for combied cell-site selectio ad power cotrol to maximize cellular spread spectrum capacity, IEEE J. Select. Areas Commu., vol. 3, pp , 995. [5] K. K. Leug ad A. Srivastava, Dyamic allocatio of dowlik ad uplik resource for broadbad services i fixed wireless etworks, IEEE J. Select. Areas Commu., vol. 7, pp , May 999. [6] K. Chawla ad X. Qiu, Quasi-static resource allocatio with iterferece avoidace for fixed wireless systems, IEEE J. Select. Areas Commu., vol. 7, pp , Mar [7] I. Koutsopoulos ad L. Tassiulas, Chael state-adaptive techiques for throughput ehacemet i wireless broadbad etworks, i Proc. IEEE INFOCOM, Alaska, Apr. 00. [8] V. Tralli, R. Veroesi, ad M. Zorzi, Power-shaped advaced resource assigmet for fixed broadbad wireless access systems, IEEE Tras. Wireless Commu., vol. 3, pp. 07 0, Nov [9] T. Boald, S. Borst, ad A. Proutière, Iter-cell schedulig i wireless data etworks, i Proc. Europea Wireless, Cyprus, Apr [0] R. Kopp ad P. Humblet, Iformatio capacity ad power cotrol i sigle-cell multiuser commuicatios, i Proc. IEEE ICC, Seattle, Jue 995, pp [] T. M. Cover ad J. A. Thomas, Elemets of Iformatio Theory. New York Joh Wiley & Sos, 99. [] S. G. Kiai ad D. Gesbert, Optimal ad distributed schedulig for multicell capacity maximizatio, Joural versio submitted. [3] P. Viswaath, D. Tse, ad R. Laroia, Opportuistic beamformig usig dumb ateas, IEEE Tras. Iform. Theory, vol. 48, pp , Jue 00. [4] Urba Trasmissio Loss Models for Mobile Radio i the 900 ad 800 MHz Bads, EURO-COST Std. 3, 99. Network Capacity (bits/sec/hz/cell/slot) RR Network Capacity Per Frame Mea RR Network Capacity PMS Network Capacity Per Frame Mea PMS Network Capacity Frames Fig.. Trace of etwork capacity values for 3 cells ad 30 users per cell. Idepedet chael realizatios based o shadowig ad fast-fadig are geerated o a frame by frame basis. APPENDIX PROOF OF THEOREM We prove the optimality of S# by first showig that it is valid for N cells ad two slots. This is the exteded to K slots.

5 Network Capacity (bits/sec/hz/cell/slot) RR Network Capacity Per Frame Mea RR Network Capacity PMS Network Capacity Per Frame Mea PMS Network Capacity Frames Fig.. Trace of etwork capacity values for 9 cells ad 30 users per cell. Idepedet chael realizatios based o shadowig ad fast-fadig are geerated o a frame by frame basis. Network Capacity Gai.5.5 Fast fadig ad Shadowig Shadowig oly Number of Cells Fig. 3. Network capacity gai versus umber of cells for differet propagatio scearios. Gai icreases with system size as optimizatio space icreases. Greater chael variatio icreases performace gap betwee the two schedulig policies thereby icreasig gai. Lemma For a arbitrary umber of cells N ad two slots, let S# N = 0 u ; u ; u ; u ;.. u ;N u ;N C A The optimal schedulig matrix for (6), S Λ = S# N. Proof We show that iterchagig users i M<Ncells will result i either o chage or a decrease i etwork capacity (M = N will result i same capacity). Without loss of geerality let these be the first M cells. We employ lighter otatio by lettig G k represet the chael gai betwee user scheduled i slot k =; ad it s servig AP. Capacity before the swappig is give by C Λ = X k= = ff G " MX i= G ki j=m # ; G kj ad after the swap C 0 = = = ff G " MX i= ff G " MX i= As G G8, we declare» G i Lettig fi ; = M X i= ff ; = N X we eed to show C Λ C 0 = j=m» G j g (x) = = G i G i fi ; = M X i= ff ; = N X j=m j=m j=m ; G i ff G (x fi ;) # G j # G j G j ff G ; (x fi ;) g (ff;) g(ff;) 0 8 ff ;» ff; Fortuately, it ca be show that the differetial of g (x) is egative [], makig it a decreasig fuctio. Thus C Λ C 0 0. This proves that S Λ = S# N. Next, we defie a operator Ql;k(S) which orders the users i colums (slots) l ad k of the schedulig matrix i decreasig order of chael gai. h Ql;k(S) = I () I () I (l ) (I (l) ; I ) ; I (l) I (k ) (I (l) ; I ) ; I (k) I (K)i ; where (u; v) N N obtaied through (u; v) i; = max (i i;g vi i) ; (u; v) i; = mi (i i;g vi i) Lemma 3 For a arbitrary schedulig matrix S, C(Ql;k(S)) C(S) Proof As oly colums l ad k are maipulated, the capacity due to other colums remais uchaged. From Lemma, the capacity of two slots arraged i decreasig order of chael gais will be more tha whe they are arraged i ay other fashio. Thus, C(Ql;k(S)) C(S). Lemma 4 For a arbitrary schedulig matrix S QK ;K Q ;K Q;3Q ;K Q;3(Q ;(S)) = S# Proof From Lemma 3, the capacity of the schedulig matrix after each Q operatio will be greater tha the previous. The successive K(K ) Q operatios will result i the perfectly ordered matrix S#. Sice there is a icrease i capacity at every step, C(S#) C(S). This cocludes the proof.