Fast Multi-Channel Gibbs-Sampling for Low-Overhead Distributed Resource Allocation in OFDMA Cellular Networks

Transcription

1 Fast Mult-Channel Gbbs-Samplng for Low-Overhead Dstrbuted Resource Allocaton n OFDMA Cellular Networks Po-Ka Huang 1, Xaojun Ln 1, Ness B. Shroff 2, and Davd J. Love 1 Abstract A major challenge n OFDMA cellular networks s to effcently allocate scarce channel resources and optmze global system performance. Specfcally, the allocaton problem across cells/base-statons s known to ncur extremely hgh computatonal/communcaton complexty. Recently, Gbbs samplng has been used to solve the downlnk nter-cell allocaton problem wth dstrbuted algorthms that ncur low computatonal complexty n each teraton. In a typcal Gbbs samplng algorthm, n order to determne whether to transt to a new state, one needs to know n advance the performance value after the transton, even before such transton takes place. For OFDMA networks wth many channels, such computaton of future performance values leads to a challengng tradeoff between convergence speed and overhead: the algorthm ether updates a very small number of channels at an teraton, whch leads to slow convergence, or ncurs hgh computaton and communcaton overhead. In ths paper, we propose a new mult-channel Gbbs samplng algorthm that resolves ths tradeoff. The key dea s to utlze perturbaton analyss so that each base-staton can accurately predct the future performance values. As a result, the proposed algorthm can quckly update many channels n every teraton wthout ncurrng excessve computaton and communcaton overhead. Smulaton results show that our algorthm converges quckly and acheves system utlty that s close to the exstng Gbbs samplng algorthm. I. INTRODUCTION In order to accommodate the exponental growth of data traffc n moble wreless networks [1], current and future cellular systems wll ncreasngly rely on hgh-densty smallcells (such as pco- and femto-cells) to sgnfcantly ncrease the traffc-carryng capacty [2]. However, the prolferaton of small-cells leads to a challengng problem of how to manage nter-cell nterference. Frst, snce the placement of small cells often faces sgnfcant constrants, the resultng network topology becomes hghly rregular. Second, the traffc densty across cells can be hghly non-unform and may exhbt tmevaryng patterns. As a result, tradtonal resource plannng strateges based on statc and regular reuse patterns are no longer adequate [3]. There s thus a pressng need to develop adaptve and even dstrbuted resource-allocaton mechansms that can dynamcally adapt to rregular topology and non-unform traffc patterns to best utlze the lmted spectrum resources. Ths work has been partally supported by the NSF through grants CNS , CNS and a MURI grant from the Army Research Offce W911NF P.-K. Huang, X. Ln, and D. J. Love are wth the School of ECE, Purdue Unversty, West Lafayette, IN USA (E-mal: huang113@purdue.edu, lnx@purdue.edu, djlove@ecn.purdue.edu) 2 N. B. Shroff s wth the Oho State Unversty, Columbus, OH USA (E-mal: shroff@ece.osu.edu) In ths paper, we focus on the downlnk of OFDMA (Orthogonal Frequency-Dvson Multple Access) systems (as n 4G LTE [4]), and our goal s to desgn such an adaptve and dstrbuted algorthm for allocatng frequency and tme resources across cells and users under rregular topology and non-unform load. Note that such a resource allocaton problem can be decomposed nto two parts. Wthn a cell, based on the set of frequency channels allocated to the cell and to ts neghborng cells, the base-staton can decde whch user to serve over each channel at each tme. Ths ntra-cell resource allocaton problem can be cast as a convex optmzaton problem, whch can be easly solved [5,6]. Then, across cells, the nter-cell resource allocaton problem decdes whch set of channels should be allocated to each cell. Due to the nherent non-convex nature of wreless nterference, ths nter-cell problem ncurs extremely hgh complexty, and exstng studes mostly focus on solutons wthout optmalty or effcency guarantees [7,8]. Recently, a class of randomzed algorthms based on Gbbs samplng has been used to solve the downlnk nter-cell resource allocaton problem n OFDMA cellular networks [8,9]. At a hgh level, n each teraton of a Gbbs-samplng algorthm, a subset of base-statons dstrbutvely decde how they wll adjust ther channel allocaton. Each base-staton n the set wll evaluate all possble adjustments of channel allocaton and learn how these adjustments wll affect the performance (whch s often modeled as utlty values [8,9]) of tself and ts neghborng cells. Then, the basestaton chooses one of the proposed adjustments wth a certan probablty. Ths probablty s carefully desgned so that the entre system follows a reversble Markov chan and has hgh probablty to move towards the globally-optmal channel allocaton. In theory, a Gbbs-samplng algorthm can be used to develop dstrbuted solutons to any nonconvex optmzaton problems wth a sutable structure (although the convergence tme to the global optmal soluton may stll be large [8,10]). However, when appled to OFDMA cellular networks wth a large number of frequency channels, the above generc verson of Gbbs samplng leads to a dffcult tradeoff between convergence speed and computaton/communcaton overhead. Note that the allocaton decson of every channel can potentally be adjusted to mprove the global performance. Thus, to expedte convergence, each base-staton should preferably evaluate a large set of proposed adjustments across multple channels. However, each proposed adjustment affects ts own cell and the neghborng cells n dfferent ways. Hence, t wll then ncur sgnfcant compu-

2 taton/communcaton overhead f each base-staton has to calculate the potental ncrease/decrease of the utlty values at the affected cells for every possble adjustment. In ths paper, we propose a fast mult-channel Gbbs samplng algorthm that addresses ths dffculty. In our proposed algorthm, each base-staton can evaluate the proposed adjustments coverng multple channels wthout ncurrng excessve computaton and communcaton overhead. If we vew the utlty value under each adjustment as the soluton to an optmzaton problem, then the key dea behnd our proposed algorthm s to vew each adjustment as a perturbaton to a common optmzaton problem. We can then quckly and accurately estmate the perturbed utlty values for each base-staton wthout actually solvng addtonal optmzaton problems. We derve condtons under whch the transton probabltes of the resultng Markov chan approach those of the orgnal Markov chan under standard Gbbs samplng. Our smulaton results confrm that the proposed algorthm acheves both fast convergence and low computaton/communcaton overhead. Further, t can dynamcally adjust the global channel allocatons to adapt to dfferent network condtons and load patterns. There have been a number of recent studes that use Gbbs samplng to desgn dstrbuted algorthms for the complex optmzaton problems n wreless networks [8,9,11] [16]. Under a graph-based nterference model, Gbbs samplng has been used to study lnk schedulng [12] [14], channel selecton and user assocaton [11]. However, the nterference model used n these studes does not account for SINR and s mpractcal for cellular systems. Wth a more accurate SINR-based nterference model, recent studes use Gbbs samplng to study schedulng and power control [15,17,18], jont power control and user assocaton [16], channel allocaton [9], and jont power-control/user-assocaton/channelallocaton [8]. Our work dffers from these studes by utlzng a new method to estmate the perturbed utlty values across multple channels and hence can acheve fast convergence wth low computaton/communcaton overhead. Our work s also closely related to [7]. However, the latter uses a greedy approach for nter-cell channel allocaton and power control, and thus the resultng soluton does not possess global optmalty. Fnally, for a comprehensve revew of other exstng algorthms for resource allocaton problem n wreless networks and ther relatonshp to Gbbs samplng, we refer the readers to [8]. The rest of the paper s organzed as follows. The system model s presented n Secton II. In Secton III, we provde an overvew of the Gbbs samplng and explan the dffculty n exstng Gbbs-samplng algorthms. We propose and analyze the fast Gbbs samplng algorthm n Secton IV and present the smulaton results n Secton V. Then, we conclude. II. SYSTEM MODEL Consder a cellular system usng OFDMA (as n LTE [4]). There are J base-statons sharng K channels. Each channel has a bandwdth W c = W T /K, where W T s the total avalable bandwdth n the system. Assume that tme s slotted 1. Let P j be the maxmum power that a basestaton j can apply over all channels. We do not consder power control n ths paper. Hence, we assume that the power that a base-staton j can apply on a channel s P j /K. We assume that each moble user s assocated wth a unque base-staton. For each base-staton j, let S j be the set of S j users assocated wth t. Correspondngly, for a user, let A() be the base-staton that user s assocated wth,.e., S A(). Let G j be the channel gan from base-staton j to user, whch captures channel attenuaton due to path loss and slowly-varyng shadow fadng. In ths paper, we wll focus on the resource allocaton problem for downlnk transmsson. Ths problem can be vewed n two parts: the ntra-cell control problem and the nter-cell control problem. The ntra-cell control determnes user schedulng wthn a base-staton, whle the ntercell control determnes the channel allocaton across basestatons. Snce nter-cell control ncurs a hgher overhead, our goal s to fnd one fxed nter-cell channel-allocaton pattern based on a gven system settng. Then, based on the fxed channel allocaton, each base-staton can schedule dfferent users across channels and tme-slots. In practce, ths decomposton means that the nter-cell channel allocaton s updated at a slower tme-scale than ntra-cell user schedulng. Specfcally, defne the ndcator varable I jk = 1 f basestaton j uses channel k, and I jk = 0, otherwse. Let I = [I jk ] represent the global channel-allocaton vector across all base-statons and all channels. For a user assocated wth base-staton A(), the rate r k that user wll receve on channel k (assumng I A(),k = 1) s a functon of the sgnal-to-nterference-plus-nose rato (SINR). Wthout loss of generalty, we use the Shannon formula [19],.e., r k = W c log 2 (1+ = W c log 2 (1+ P A() I A(),k G,A() /K W TN 0/K+ j A() PjI jkg ) j/k P A() I A(),k G,A() W TN 0+ j A() PjI jkg j ), where N 0 s the thermal nose densty. Note that Equaton (1) accounts for the nterference from all base-statons. In realty, the nterference between two dstant base-statons wll be small. Hence, n practce t s reasonable to approxmate (1) by only consderng the nterference from neghborng base-statons. Specfcally, let N j denote the set of neghborng base-statons 2 for base-statonj. Letη,A() max = {z A(),z/ N A() } P zg z. We can then approxmate (1) by r k = W c log 2 (1+ P A() I A(),k G,A() W TN 0+η max,a() + {j N A() } PjI jkg j ). (2) Note that r k r k. Further, snce ηmax j s ndependent of I, the value of r k only depends on the channel allocaton of the neghborng base-statons of A(). Later, we wll see that ths property s crtcal for developng a dstrbuted algorthm. Hence, n the rest of the paper, we wll use (2) to model the 1 In LTE, each channel n a tme slot represents a resource block that can be assgned by each base-staton to one assocated user. [4] 2 For example, N j could be the base-statons wthn a certan dstance from j. However, our result does not depend on a specfc defnton of N j. (1)

3 rate of each user. Further, we assume that the neghborng relatonshp s symmetrc,.e., f h N j, then j N h. For ntra-cell control, we assume that at any gven tmeslot and for a gven channel, the base-staton can only serve one assocated user. However, a base-staton can use the same channel to serve dfferent users across tme slots,.e, tmedvson multplexng 3. Let φ k be the fracton of tme that base-staton A() serves user on channel k. Note that we must have S j φ k 1 for each base-staton j and each channel k. The average rate R receved by user s gven by R = K k=1 φ kr k. We assocate a utlty functon U (R ) for user, whch captures the satsfacton level of user when ts servce rate s R. Recall from (2) that the rate r k of each user s determned by the channel-allocaton vector I. Wth a fxed vector I, the ntra-cell control for base-staton j can then be modeled as the followng optmzaton problem: max [φ k ] 0 subject to U (R ), S j R = K k=1 φ kr k, and S j φ k = 1. (3) Problem (3) s a convex optmzaton problem and can be solved by standard technque [5]. Let V j ( I) denote the optmal objectve value of problem (3) for base-staton j. Then, the nter-cell channel-allocaton problem can be wrtten as: max I=[I jk ] I J V j ( I), (4) j=1 where I s the set of all channel-allocaton vectors. Unfortunately, Problem (4) s a combnatoral problem that s n general very dffcult [8]. In partcular, snce the number of all channel-allocaton vectors (2 JK ) grows exponentally wth the number of base-statons and the number of channels, even a centralzed soluton wll ncur extremely hgh complexty 4. In the next secton, we wll ntroduce dstrbuted and low-complexty algorthms based on Gbbs samplng. III. GIBBS SAMPLING In ths secton, we provde an overvew of dstrbuted algorthms based on Gbbs samplng, whch can be used to solvng Problem (4). We then revew several types of exstng Gbbs-samplng algorthms and explan ther performance tradeoffs n terms of the convergence speed and the computaton/communcaton overhead. Ths dscusson motvates the new algorthm that we wll propose n Secton IV. A. Overvew of Gbbs Samplng Suppose that the channel-allocaton vector I s updated n an teratve manner, and let I t be the vector at the t-th teraton. In Gbbs samplng (and many other randomzed algorthms), one forms a Markov chan wth I t as the state 3 Ths s standard n LTE system. 4 In a LTE system wth 20MHz of total bandwdth, the number of channels s around 100. If we consder a 19 cells (two rngs) layout, the number of channel-allocaton vectors s ! such that the statonary dstrbuton of the chan has the form of the Gbbs dstrbuton: P( I t = I) π( I) = 1 e 1 J T j=1 Vj( I), (5) Z T where T s called the temperature of the Gbbs dstrbuton [20, Chapter 7], and Z T s a normalzaton constant. Clearly, when T s small, the channel-allocaton vector(s) I wth the optmal utlty wll be reached wth probablty close to 1. Hence, such a Markov chan can be used to teratvely fnd the optmal I. To form such a Markov chan, Gbbs samplng sets the transton probabltes n a partcular way [20, Chapter 7, Secton 6.1]. We start wth a few notatons. Let A = {(j,k) 1 j J,1 k K} denote the set of all basestaton/channel pars, and let H A be a subset for whch the network consders to update the channel allocaton n a gven teraton. Recall that I s a vector of ndcator varables, each of whch corresponds to an element n A. We further use I(H) and I(A\H) to denote those components of I that correspond to elements n H and A\H, respectvely. Smlar to the vector I, let x and y be the ndcator vectors before and after a gven teraton, respectvely. Snce only those components correspondng to H wll be changed, we must have x(a\h) = y(a\h). Further, for any x and y, we use ( y(h), x(a\h)) to denote another ndcator vector whose components correspondng to H and A\H are taken from y and x, respectvely. At each teraton t, a set H s chosen wth probablty p H, and only the elements n H wll be updated. The requrement on the probablty dstrbuton [p H,H A] s that: Every base-staton/channel par (j, k) must belong to at least one H A such that p H > 0. (6) Hence, every (j, k) has a non-zero probablty of beng chosen for update. Let I(H) be the set of all possble values of I(H). Based on the channel-allocaton vector I t = x, Gbbs samplng uses the followng transton probablty: where P( I t+1 = y I t = x) = {H: y(a\h)= x(a\h)} p H π( y(h) x(a\h)), π( y(h) x(a\h)) = e 1 J T j=1 Vj( y(h), x(a\h)). Jj=1 V j( z, x(a\h)) { z I(H)} e 1 T (8) In other words, f H s chosen, then update x(h) to y(h) wth probablty gven by (8). Note that (8) s smply the condtonal probablty for I(H) = y(h), gven that I(A\H) = x(a\h), when I follows the dstrbuton n (5) 5. It can be easly verfed that the resultng Markov chan s aperodc and rreducble. Further, t s reversble (satsfyng the detaled balanced equatons), and ts statonary dstrbuton s exactly (5). 5 Note that the normalzaton constant Z T does not appear n (8). Ths s very useful because calculatng Z T would have ncurred exponental complexty. (7)

4 However, n general the transton probablty n (7) depends on the update at all base-statons. Hence, ths procedure requres centralzed mplementaton. In order to derve a dstrbuted algorthm, we explot the fact that the value of V j ( I) only depends on the channel allocaton at base-staton j and the base-statons n N j (see (2)). In other words, let A j = {(h,k) h = j or h N j,k = 1,...,K}. Then, we can wrte V j ( I) = V j ( I(A j )). We then use the followng dstrbuted procedure. At each teraton, frst choose a set of base-statons D 6 wthout common neghbors,.e., For any j,j D, N j N j =. (9) Then, each base-staton j D chooses a subset of channels H j {(j,k) k = 1,...,K} to update. For j / D, we let H j =. Let H = J j=1 H j. Note that by ths procedure, the value of V j ( ) n (8) can only change ether because base-staton j tself updates ts own channels H j (f j D) or because at most one neghborng base-staton h N j updates ts channels (f j N h and h D). We can then show that the expresson (8) can be decoupled as follows: π( y(h) x(a\h)) = j Dπ( y(h j ) x(a\h j )), (10) where π( y(h j ) x(a\h j )) s equal to e 1 T [ V ] j( y(h j), x(a j\h j))+ {h N j } V h( y(h j), x(a h \H j)) { z j I(H j)} e 1 T [ V j( z j, x(a j\h j))+ {h N j } V h( z j, x(a h \H j)) (11) (Detals are avalable n our onlne techncal report [21].) Note that for a base-staton j D, expresson (11) does not depend on the updates at other base-statons n D. Hence, each base-staton j can do the update ndependently,.e., base-staton j uses the transton probablty (11) to decde whch new channel-allocaton decson y(h j ) t wll pck. Fnally, f each base-staton j only updates one channel,.e., H j = {(j,k)} for some k 1,...,K, then z j only contans one element z jk that can take ether 0 or 1. Hence, the expresson (11) can be further smplfed to: π( y(h j ) x(a\h j )) = exp(y jk jk ) 1+exp( jk ), (12) and the value of jk s gven by jk = 1 T [V j(1, x(a j \H j )) V j (0, x(a j \H j ))] {h N [V j} h(1, x(a h \H j )) V h (0, x(a h \H j ))], + 1 T where the frst argument of V j (, ) and V h (, ) corresponds to the element H j = {(j,k)}. B. Exstng Mult-Channel Gbbs Samplng Algorthms The approach descrbed n Secton III-A mmedately leads to dstrbuted mplementaton. However, when the number of channels to be updated s large, ths procedure can stll lead to hgh computaton/communcaton overhead. To see ths, let I(H j ) denote the set of possble values of I(H j ) that base-staton j may transt to. Then, n order to compute the 6 Ths set D s smlar to the decson schedule n [13,14]. ]. transton probablty n (11), each base-statonh {j N j } needs to compute the value of V h ( z j, x(a h \H j ) for every possble value of z j I(H j ). Further, f h N j, the result also needs to be sent to base-staton j. Recall that each computaton of V j ( ) requres solvng the optmzaton problem (3). Hence, dependng on the dfferent mplementaton optons outlned below, there wll be a challengng tradeoff between computaton/communcaton overhead and convergence speed. Opton 1 (Sngle-channel update): In each teraton, each base-staton j D only updates one channel,.e., H j = 1. Hence, I(H j ) = 2, and the computaton/communcaton overhead s the lowest. However, f K s large, updatng one channel at a tme may lead to slow convergence, as can be observed from our smulaton results n Secton V and n [8]. Opton 2 (Mult-channel update): In each teraton, each base-statonj D can update multple channels,.e., H j > 1. The convergence speed wll become faster as we ncrease H j. However, I(H j ) = 2 Hj. Hence, the computaton overhead ncreases exponentally wth H j. Opton 3 (Sequental update): Ths opton can be conceptually vewed as a hybrd between Opton 1 and Opton 2. In each teraton, lke Opton 2, each base-staton j D can update multple channels,.e., H j > 1. However, basestaton j updates these channels sequentally. Specfcally, base-staton j chooses unformly at random a permutaton (v 1,v 2,,v Hj ) of the elements n H j,.e., each permutaton s chosen wth probablty 1/ H j!. Then, the teraton s dvded nto H j rounds, and only channel v l s updated n each round l = 1,..., H j as n Opton 1. It can be shown that the resultng Markov chan stll has the same statonary dstrbuton as (5) [14]. Note that the computaton overhead now ncreases lnearly wth H j. However, after each round, the updated channel allocaton must be communcated to the neghborng base-statons so that future computaton of Problem (3) can use the most-recently updated channel allocaton. As a result, more rounds lead to more control messages exchanged between base-statons and slower speed. In sum, when K s large, all optons ether suffer slow convergence or hgh computaton/communcaton overhead. IV. FAST GIBBS SAMPLING ALGORITHMS In ths secton, we propose a fast Gbbs samplng algorthm wth only one round of communcaton, low computaton overhead, and mult-channel update n each teraton. We frst sketch the basc skeleton of ths algorthm below. Fast Mult-Channel Gbbs Samplng: In each teraton, 1) Each base-staton j = 1,..., J solves Problem (3) based on the exstng channel-allocaton decsons. 2) Randomly choose a subset D of base-statons and a subset H j of channels for each base-statons j D such that the condtons n (6) and (9) are satsfed. 3) Base-staton j chooses unformly at random a permutaton (v 1,...,v Hj ) of the H j channels. 4) The teraton s dvded nto H j rounds. In the l-th round, the base-staton j updates channel v l usng the

5 transton probablty n (19) and (20) shown later. The above procedure s smlar to Opton 3. However, the key dfference s n Step 4, where we wll propose a new method n (19) and (20) to compute the transton probablty wth low overhead. We explan the thought process as follows. Let x and y agan denote the channel-allocaton vectors before and after an teraton, respectvely. At the l-th round, we can use z l = ( y( l 1 n=1 v n), x(a\ l 1 n=1 v n)) to express the channel allocaton to update from 7. Recall that n Opton 3 (and smlarly n Opton 1), the transton probablty at round l s gven by (12), whch s rewrtten below after takng nto account the changes n prevous rounds: π( y(h j,vl ) z l (A\H j,vl )) = exp(y j,v l j,vl ) 1+exp( j,vl ), (13) where H j,vl = {(j,v l )}, and j,vl can be wrtten as [ j,vl = 1 T Vj (1, z l (A j \H j,vl )) V j (0, z l (A j \H j,vl )) ] [ Vh (1, z l (A h \H j,vl )) V h (0, z l (A h \H j,vl )) ]. + 1 T {h N j} (14) Note that the frst argument n V j (, ) and V h (, ) corresponds to the channel allocaton for v l. Now, when the number of channels s large, and the sze of H j s not too large, the channel-allocaton vector z l dffers from x by only a few elements. LetVj 0 andv0 h be the optmal objectve values of Problem (3) for base-staton j and basestaton h N j. Then, we can expect that V j and V h should be close to Vj 0 and Vh 0. The key dea s then to use the solutons to Problem (3) forvj 0 andv0 h to predct the value n (14). In ths way, we can approxmate the transton probablty wthout ncurrng addtonal computaton/communcaton overhead n each round. To llustrate the dea, we start wth some basc propertes of the solutons to Problem (3). Let R = [R ] and φ = [φ k ]. Assocate dual varables λ to the frst equalty constrant of (3). LetΦbe the doman ofφ gven n (3). The dual objectve functon of Problem (3) s then [5]: g( K λ) = max U (R ) λ (R φ k r k ). (15) R, φ Φ S j S j k=1 Let us frst focus on the change of utlty at base-staton j. Recall that the value of j,vl s determned by the utlty dfference wth or wthout a channel v l allocated to basestaton j. Suppose that Problem (3) s solved twce for a base-staton j D, frstly wth a channel v l assgned to base-staton j and secondly wthout the channel v l assgned to t. Let ( R, φ ) and λ be a par of optmal prmal and dual varables for Problem (3) when channel v l s assgned to base-staton j, and let V j be the correspondng optmal objectve value. Smlarly, let R (wo), φ (wo), λ (wo), and be the correspondng values for Problem (3) when channelv l s not assgned to base-statonj. Note thatr (wo),v l = V (wo) j 7 Of course, elements outsde A j and h Nj A h may also change. However, ther changes do not affect the transton probablty at base-staton j as can be seen n (13) and (14). 0 for all users. Snce Problem (3) s convex, the KKT condton [5] must hold, and we have: K λ = U (R ), R = φ k r k. (16) k=1 If user s served at channel k,.e., φ k > 0, we also have: λ r k = max u S j λ u r uk. (17) A smlar set of equatons wll also hold for Problem (3) wthout channel v l. Intutvely, ths set of equatons suggest that f a channel v l s taken away from a base-staton j, t wll be taken away from user (see (17)). For ths user, the correspondng utlty decrease wll be approxmately U (R )r,v l = λ r,v l = max u Sj λ u r u,v l. Hence, t suggests that the value of max u Sj λ ur u,vl would be a good estmate of the utlty dfference between V j and V (wo) j. The followng lemma makes ths ntuton more precse. Lemma 1: Suppose that there exsts a constant D such that λ (wo) λ D. Then, 1) V j V (wo) j 2) V j V (wo) j +max S j λ r,v l. +max S j λ r,vl +D S j φ,v l r,vl. Proof: Detaled proof can be found n [21] When the number of channels K s large, and only a few channels H j are updated n one teraton, we would expect that the dual varables wll not change much across rounds,.e., the value of D wll be small. Lemma 1 then states that max Sj λ r,v l becomes a good estmate of the utlty dfference when a channel v l s removed from basestaton j. Note that we can replace λ by λ (wo) and use max Sj λ (wo) r,vl to estmate the utlty dfference. The latter wll add another error term Dmax Sj r,vl, whch agan wll be small f D s small. Further, for any round l, we can replace λ or λ (wo) by the optmal dual varables before round 1. In that case, the error terms wll accumulate lnearly n l. However, as long as l s bounded, the error terms can stll be bounded. We have estmated the utlty dfference for a base-staton j D. For ts neghborng base-staton h N j, the stuaton s slghtly dfferent because base-staton h does not change ts own channel allocaton. Instead, the user rates r,vl, S h, change due to the change of channel allocaton by basestaton j. If the base-staton h does not use channel v l, then ths change wll not affect ts utlty. If the base-staton h does use channel v l, suppose that the user rates wthout and wth channel v l assgned to base-staton j are r,v 1 l to r,v 2 l, respectvely, for all S h. We can then vew the change at base-staton h as the concatenaton of two steps: basestaton h frst removes channel v l wth user rates r,v 1 l and then adds back channel v l wth user rates r,v 2 l. In order to dfferentate the base-statons, we use λ j and λ h to denote the optmal dual varables for Problem (3) at base-staton j and h, respectvely. By Lemma 1, we can then approxmate the total utlty dfference by V h V (wo) h max S h λ hr 1,v l +max S h λ hr 2,v l, (18)

6 where V h and V (wo) h are the optmal utlty for base-staton h when base-staton j uses or does not use channel v l, respectvely. Takng all the above nto account, the transton probablty at round l at base-staton j can be approxmated as: π( y(h j,vl ) z l (A\H j,vl )) = exp(y j,v l j,vl ) 1+exp( j,vl ), (19) where H j,vl = {(j,v l )} and j,vl can be wrtten as j,vl = 1 T max + 1 T {h N j} [ max S h λ h r1,v l +max S h λ h r2,v l S j λ jr,vl ] I h,vl. (20) where all the optmal dual varablesλ j and λ h can be taken as those before round 1. Our proposed fast mult-channel Gbbs samplng algorthm then uses (19) and (20) n Step 4. Remark: Despte the smlarty between (13) and (19), we emphasze that they lead to sgnfcantly dfferent speed and overhead. In (19) and (20), snce the values of the optmal dual varables before round 1, whch are calculated n Step 1 of the proposed algorthm, are used, base-staton j can use one control message to send nformaton about H j to base-staton h and ask base-staton h to return the correspondng values n j,vl for all channelsv l nh j, agan usng one message. Then, base-staton j can carry out the H j rounds of transtons wthout addtonal communcaton/computaton nvolvng neghborng base-statons. Thus, the computaton/communcaton overhead s sgnfcantly reduced, and the channel updates can be carred out wth much faster speed than Opton 3 (usng (13)). A. Analyss Next, we wll analyze the performance. Ideally, we would lke to show that as the number of channels K ncreases, the statonary dstrbuton of the Markov chan usng the approxmate (19) and (20) wll approach the statonary dstrbuton of the orgnal Markov chan usng (13) and (14). However, snce the statonary dstrbuton of the orgnal Markov chan also changes wth K, t appears to be dffcult to drectly analyze the convergence n terms of the statonary probablty. Instead, n ths subsecton, we wll focus on showng how the approxmate transton probablty n (19) approaches (13). As we have dscussed, t s suffcent to focus on one base-staton j D and study the utlty dfference wth or wthout a sngle channel v l (see Lemma 1). Intutvely, as the number of channels K ncreases, the contrbuton of any one channel,.e., the term max Sj λ r,v l n Lemma 1, decreases as Θ(1/K). Hence, f we can show that the error term D φ,v l r,vl n Lemma 1 decreases faster than Θ(1/K), then by settng the temperature T = α/k, the effect of the error term wll approach 0 as K ncreases. However, the bound D n general depends on the partcular channel-allocaton patterns. In the sequel, we wll study condtons under whch D can be bounded unformly over all transtons. Due to space constrants, we wll present the results wthout proofs. The nterested readers are referred to our techncal report n [21] for detaled proofs. Agan, we focus on the settng n Lemma 1, where Problem (3) s solved twce, frstly wth channel v l allocated to base-staton j and secondly wthout. Our frst lemma s: Lemma 2: If the utlty functon s U (R ) = log(r ), then λ λ (wo) D f and only f R R (wo) D. R R (wo) Proof: The result can be easly shown by (16). Ths lemma shows that to bound the dfference n λ and λ (wo), we can nstead focus on the dfference n R and R (wo). Hence, n the followng, we wll let U (R ) = log(r ) and focus on the dfference between R and R (wo). Lemma 3: R R (wo), for all S j. Lemma 3 s qute ntutve. It states that after an addtonal channel v l s allocated to base-staton j, the optmal rate for all assocated users of base-staton j cannot decrease. Next, we would lke to bound the ncrease R R (wo). Here, we need a condton on the heterogenety of the user rates. Let and r max = W T max S j log 2 (1+ P j G j W T N 0 +ηj max ) (21) P j G j r mn = W T mnlog 2 (1+ S j W T N 0 +ηj max + ). h N j P h G h (22) Then, from (2), we have max { Sj,k}r k r max /K and mn { Sj,k}r k r mn /K. Let B = r max /r mn. Lemma 4: R R (wo) +Br max /K for all S j. Lemma 4 mples that the ncrease (R R (wo) ) must be bounded by Br max /K, whch decreases as Θ(1/K). The bound can also be shown to be tght n the order sense. The nterested readers can refer to [21] for detals. Combnng Lemmas 3 and 4, we then have R R (wo) R R (wo) Br max KmnR mnr (wo) S j S j. (23) It remans to bound the denomnator. Lemma 5: Let c be a fxed constant n (0,1). Suppose that at least a fracton c of the K channels s allocated to each base-staton. Then, mn[mnr,mn R (wo) ] cr mn / S j. S j S j Note that by Lemma 5 we can conclude that the rghthand-sde of (23) does decrease asθ(1/k). Combnng wth Lemmas 1 and 2, we then obtan the followng man result. Let Ĩ denote the set of all possble I such that the condton n Lemma 5 holds. Proposton 6: Suppose that U (R ) = log(r ), T = α/k, and at least a fracton c of the K channels s allocated to each base-staton. Further, suppose that S j and H j are upper-bounded by some constants. Then, we have lm sup j,vl j,vl = 0. (24) K x, y Ĩ Proposton 6 confrms that whenk s large, and each update nvolves a relatvely small number H j of channels, then the

7 transton probablty of the proposed fast Gbbs samplng wll approach that of the orgnal Gbbs samplng. Remark: Note that the condton n Lemma 5 suggests the followng modfcaton to the proposed algorthm. Suppose that the channel-allocaton vector x before an teraton satsfes x Ĩ. Then, we only need to ensure that the vector y that the algorthm transts to s stll n Ĩ. Each base-staton j can ensure ths property dstrbutvely by settng the transton probablty (19) to zero f the state after round l s not n Ĩ. Remark: Our analyss may be conservatve snce we focus on the worst-case scenaro. However, t gves us some confdence that the proposed fast Gbbs samplng algorthm wll lkely follow the trajectory of the orgnal Gbbs samplng. In fact, our smulaton results n Secton V show that such convergence occurs even for a moderate number of channels. A detaled dscusson s provded n [21]. Fnally, n our smulatons, we found that the standard gradent algorthms [5] are stll qute slow for solvng Problem (3) n each teraton. To resolve ths problem, we develop a second-order algorthm [5, Chapter 10] [22] to expedte the convergence. For detals, please refer to [21]. V. SIMULATION In ths secton, we use smulaton to evaluate the fast Gbbs samplng algorthms proposed n Secton IV. We frst smulate a LTE network wth 19 same-szed hexagon cells arranged n a three-rng structure. (There are a center cell, an nner rng of 6 cells, and an outer rng of 12 cells.) Each basestaton s at the center of a cell, and the nter-ste dstance (ISD) s 500m. A total bandwdth of 20Mhz s dvded nto 50 channels. The power of each base-staton s 47dbm and s equally shared over the 50 channel. Each cell has 10 users randomly placed nsde ts coverage area. The utlty functon of each user s log( /10 6 ). The channel gans G j are modeled by a path-loss component wth exponent n = 2.2 and log-normal shadow fadng wth standard devaton σ = 8dm. The neghborng set N j for each base-staton j conssts of any base-statons wthn 500m range. We frst compare the convergence of the proposed fast Gbbs-samplng algorthm wth the sequental-update verson of standard Gbbs samplng,.e., Opton3n Secton III- B. 8 For all versons of Gbbs samplng, the temperature T of the Gbbs dstrbuton s We also compare wth the Metropols-Hastngs algorthm [8]. The Metropols-Hastngs algorthm s smlar to Gbbs samplng, but t chooses the transton probablty dfferently. Specfcally, after each base-staton chooses a subset of channels H j to update, the Metropols-Hastngs algorthm randomly chooses a feasble proposed state y(h j ) and compares π( y(h j ), x(a\h j )) wth π( x) accordng to (5), where x s the orgnal state. If the former s larger, the base-staton accepts state y(h j ) wth probablty one. Otherwse, the base-staton accepts state y(h j ) wth probablty π( y(h j ), x(a\h j ))/π( x). 8 We do not report the mult-channel update verson,.e., Opton 2, due to the exponental computaton/communcaton overhead. However, the convergence speed of Opton 2 should be comparable to Opton 3. The smulaton result s shown n Fg. 1(a), where we plot the total system utlty as a functon of the teratons. As readers can see, for all versons of Gbbs samplng algorthms, updatng fewer channels n each teraton leads to a slower convergence speed. Further, although the Metropols- Hastngs algorthm can update a larger number of channels wth low computaton overhead, the proposed change n each teraton may not always lead to larger utlty. Thus, ts convergence speed s stll slow. In contrast, the proposed fast Gbbs-samplng algorthm enjoys much faster convergence when updatng multple channels at a tme. By comparng the curves sequental-5 and fast-5, we can observe that the utlty evoluton of the fast algorthm s close to that of the sequental update verson. Fnally, the fast-15 curve converges even more quckly than other curves. These results confrm that our proposed algorthm s suffcently accurate for fndng the optmal channel-allocaton. Further, t leads to fast convergence and low overhead. Next, we smulate a settng wth non-unform load and evaluate whether the proposed algorthm can adapt to load patterns. Specfcally, we compare fast Gbbs-samplng algorthm wth (1) unversal 1-reuse, where each base-staton uses all the channels, and (2) strct FFR (Fractonal Frequency Reuse), where 1/4 of the channels are shared by all basestatons, and the rest 3/4 of the channels are allocated accordng to a 3-reuse pattern [3]. Note that accordng to the 3-reuse pattern, we can dvde the cells nto three groups. We then let the cells n group 1, 2, and 3 have 20 users, 10 users, and 1 user, respectvely. For all schemes, once the nter-cell channel allocaton s determned, each cell solves Problem (3) for ntra-cell user schedulng. The CDF of the resultng user rates under dfferent schemes are shown n Fg. 1(b). As we can see, the user rates under unversal 1-reuse are much poorer due to the strong nter-cell nterference. Strct FFR performs better. However, snce t does not react to the non-unform load, ts performance s stll sgnfcantly worse than our proposed algorthm. A deeper analyss of the results reveals that under our algorthm, roughly 27 channels, 20 channels, and 3 channels are allocated to the base-statons n group 1, 2, and 3, respectvely. As a result, the acheved user-rates are consstently better. We next smulate our algorthm for a heterogenous network wth both macro- and pco-cells (see Fg. 1(d)). Specfcally, there are three macro-cells wth the ISD of 500m. On the boundary of every two macro-cells, there s a pco-cell. Further, one addtonal pco-cell s placed n the nteror of the coverage area of each macro-cell and s placed away from the boundary pco-cells. The power and coverage radus of each pco base-staton s 27dbm and 75m, respectvely. Each macro- or pco-cell has 10 users randomly placed n the cell. The neghborng set N j for each macro and pco base-staton conssts of those base-statons that are mmedate neghbors (refer to [21] for detals). We compare fast Gbbs-samplng wth (1) unversal 1- reuse, where each base-staton uses all the channels, and (2) strct FFR, where macro base-statons use strct FFR descrbed earler [3], whle pco base-statons use 1-reuse.

8 (a) (b) (c) (d) Fg. 1. (a): The utlty evoluton of dfferent Gbbs-samplng algorthms. Sequental represents Opton 3 (sequental update). Fast represents fast Gbbs-samplng. Metro represents the Metropols-Hastngs algorthm. The number behnd the descrpton of the algorthm s the number of channels updated by the chosen base-staton n each teraton. (b): The cumulatve dstrbuton functon (CDF) of the user rates under non-unform load. (c): The cumulatve dstrbuton functon (CDF) of the user rates under heterogeneous topology. (d): The topology of a heterogeneous network wth three macro-cells and sx pco-cells. The smulaton result s shown n Fg. 1(c), where our proposed algorthm demonstrates even more sgnfcant performance gans. As we can see, the user rates under unversal 1-reuse s agan very poor due to the strong nter-cell nterference. The performance of pco-cells under strct FFR s also poor because pco-cells stll receve strong ntercell nterference. In contrast, wth our fast Gbbs-samplng algorthm, the channel allocaton s automatcally adjusted to manage nterference, and thus the user rates are mproved sgnfcantly. Specfcally, the upper-rght-hand sde of the CDF represents nteror pco-cells whose user rates are nearly doubled. The lower-left-hand sde of the CDF represents the macro-cells and boundary pco-cells whose user rates are also sgnfcantly mproved. We select the macro-cell #1, the pco-cell #4, and the pco-cell #7 n Fg. 1(d) and observe the number of channels allocated to each of them to be 7, 12 and 43, respectvely. Moreover, only two channels allocated to the boundary pco-cell #4 are shared by the two neghborng pco-cells or the three neghborng macro-cells. Such an adaptve channel allocaton not only mproves the overall system utlty, but also helps to mprove the CDF of the user rates consstently across all users. VI. 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