Adaptive Multi-pattern Reuse in Multi-cell NetworksKyuho Son, Yung Yi and Song Chong

School of Electrical Engineering and Computer Science, KAISTE-mail: skio@netsys.kaist.ac.kr.{yiyung.song}@ee.kaist.ac.kr

Abstract-Achieving sufficient spatial capacity gain by havingsmall cells requires careful treatment of inter-cell interference(ICI) management via BS power coordination coupled with userscheduling inside cells. Optimal algorithms have been known tobe hard to implement due to high computation and signalingoverheads. We propose joint pattern-based ICI management anduser scheduling algorithms that are practically implementable.The basic idea is to decompose the original problem into two sub­problems, where we run ICI management at a slower time scalethan user scheduling. We empirically show that even with such aslow tracking of system dynamics at the ICI management part,the decom posed approach achieves high performance increase,com pared to a conventional universal reuse scheme.

I. INTRODUCTION

To achieve high spatial capacity, wireless cellular networksconsider the dense deployment of base stations (BSs) thatcover small cells. As a consequence, inter-cell interference(ICI) from neighboring BSs becomes a major source of per­formance degradation and the portion of users whose capacityis naturally limited by ICI grows. In order to fully attainthe potential gain of multi-cell networks, the coordination oftransmissions among BSs which can effectively manage ICIis essential. The key intuition of BS coordination is that theachievable rates, which depend on the amount of ICI, can beincreased by turning off some of neighboring BSs. Thus thereare cases when the increment of achievable rates preponderatesthe sacrifice of taking away transmission opportunities at theneighboring BSs. In particular, this usually happens to users atcell edges severely suffering from the ICI since the incrementof achievable rates may be sufficiently large.

A brute-force approach for mitigating ICI is the use oftraditional reuse scheme in time and/or frequency domain.However, this may waste precious radio resource since usersat different geographical locations inside cells prefer differentreuse schemes. Several schemes, e.g., fractional frequencyreuse (FFR) [1] in Mobile WiMAX, have been proposedto accommodate users in different channel conditions withdifferent reuse factors. However, these priori hand-craftedschemes are still far from optimal in the sense that they donot adapt to dynamic network environments, e.g., time-varyinguser loads/locations. In addition, opportunistic user schedulingbased on their perceived time-varying channels, needs to bejointly considered with ICI management to achieve a highperformance gain.

This research was partly supported in part by the Ministry of KnowledgeEconomy, Korea, under the ITRC (Information Technology Research Center)support program supervised by the UTA (Institute of Information TechnologyAdvancement) (IITA-2009-CI090-0902-0037). This work was also partlysupported by the IT R&D program ofMKE/IITA [2009-F-045-01, Ultra SmallCell Based Autonomic Wireless Network].

In this paper, we aim at (i) studying coupling dynamics ofinter-cell ICI management and intra-cell user scheduling, and(ii) proposing practically implementable joint algorithms thatachieve a significant performance gain. To that end, we firstpropose a pattern-based optimal algorithm that tracks time­varying channel conditions (that runs at short time scales) atboth user scheduling and ICI management, where 'pattern'corresponds to a combination of BS ON/OFF activities. Then,we show that the proposed optimal algorithm is hard toimplement due to high complexity. The key bottleneck liesin the ICI management part that requires collecting excessiveamount of feedback information from all users and also needscomplex operations to make decisions on BS coordination atevery time slot. To overcome such complexity, we decomposethe original optimization problem into two sub-problems (userscheduling and pattern-based ICI management), and solvethem with different time scales, whose complexity becomesmuch lower than that of the optimal algorithm.

The algorithm based on time-scale decomposition stemsfrom a design rationale that ICI management may not haveto track fast dynamics, e.g., fast fading channel condition.Instead, it may suffice to run the ICI management followingonly macroscopic network changes, e.g., user loads/locations,and the average channel conditions of users. In spite of suchslow tracking of system dynamics in ICI management, weempirically show that with our decomposed algorithms, theperformance increase amounts to about 6'"'-'20% (comparedto conventional universal reuse scheme), corresponding to1/2'"'-'2/3 of the optimal algorithm (that is almost impossibleto implement).

The research on mitigating ICI have recently received a lotof attentions [2]-[8]. Optimal binary power control (BPC) forsum rate maximization has been considered in [2]. In [3],[4], optimal joint ICI management (similar BPC) and userscheduling algorithms that operate slot-by-slot and requireheavy computation overheads, have been considered in slightlydifferent systems. The authors there presented an idea ofusing clustering only neighboring BSs [3] or considering onlyneighboring BSs [4] to reduce complexity. However, it stillrequires centralized coordination and complex operations per­slot basis, which hinders practical implementation.

To make algorithms practical, there have been recent ap­proaches [6]-[8], based on a slightly different time-scaleseparation approach from ours. In [6], the authors abstractusers that share similar traffic loads and channel environmentsinto classes, and perform ICI management on a very longtime scale (e.g., hours) without explicit consideration of intra­cell user scheduling. They basically design ICI managementthat tracks system dynamics at a very macroscopic level. Our

approach differs from [6] that user scheduling is explicitlyconsidered, and also our ICI management runs much faster(e.g., seconds) than that in [6]. The work with a similar time­scale separation to ours has been proposed in [7], [8] for dif­ferent systems, i.e., OFDMA systems, where they periodicallyupdates the transmit power level for different subbands forJCl management. Due to the difference in system model, weuse a different mechanism that updates patterns not powers,leading to a different sty le of algorithms and analysis. Weadditionally study the performance gap between the optimaland the decomposed algorithms.

Related work also includes the examination of potential ca­pacity gains (from the perspective of flow-level performance)by BS coordination [9]. Another important issue in multi­cell networks is to resolve load imbalance problem betweencells. The authors in [10], [11] explicitly balance the load bychanging user associations from the BS in hot-spot cells tothe adjacent less-crowded BS. Sang et al. [10] proposed anintegrated framework consisting of a MAC-layer cell breathingtechnique and load-aware handover/cell-site selection to dealwith load balancing. Bu et al. [11] were the first to rigorouslyconsider a formulation of network-wide proportional fairness(PF) [12] in a multi-cell network where associations betweenusers and BSs are decision variables. Although we assumethat user association is fixed, we later argue and empiricallyshow that ICI management is able to implicitly resolve theload imbalance, and the performance gain by controlling userassociation may be small.

The remainder of this paper is organized as follows. In Sec­tion II, we present our system model and problem definition.In Section III, we propose a joint pattern selection and userscheduling algorithm to solve this problem. Although this jointalgorithm is optimal, it has some implementation difficulties.In order to take into account practical concerns, we design twoalgorithms using time-scale decomposition that run at differenttime-scales in Section IV. In Section V, we demonstrate theperformance of proposed algorithms, and conclude the paperin Section VI.

II. SYSTEM MODEL AND PROBLEM DEFINITION

A. Network Model

We consider a wireless cellular network consisting of mul­tiple cells. Denote by N ~ {I, ... ,N} and JC ~ {I, ... ,K}a set of BSs and MSs (or users), respectively. A user k E JCis associated with a single BS n E N, which means that dataintended for the user k is served only by the BS n. Definea(·) : JC ~ N to be the association function, e.g., a(k) == n ifthe user k is associated with the BS n. We further denote byJCn the set of users associated with the BS n. Assume that BSstransmit data with either its given maximum power or 0, whichwe simply denote by 'ON' or 'OFF' states'. We assume thata same frequency band (or channel in short) with bandwidthW in all cells, and consider only downlink transmissions inthe time-slotted system indexed by t == 0,1, .... At eachslot, a BS can select only one user for its data transmission.

1All discussions in this paper can be readily extended to the case whereBSs can transmit data with a finite number of discrete power levels.

Channels may be time-varying, modeled by some stationary,ergodic random process with the finite state index set I andthe stationary distribution () == (() (i), i E I).

B. Network Resource and Allocation Schemes

The time-varying network resources at slot t are representedby a finite set R(t) of the K -dimensional feasible rate(bits/slot) vectors over users. A resource allocation schemethen chooses a feasible rate vector in R(t) at each slot andserves a subset of users with the chosen rate vector. A feasiblerate vector in R(t) is determined by the following two factors:(i) which BSs are activated and (ii) which users are selectedin cells for data transmission.

To formally discuss (i), we define reuse pattern (or simplypattern) p to be a combination of ON/OFF activities of BSs,which determines inter-cell interference to the correspondingscheduled users in cells. Denote by P the set of patterns. Apattern p is said to activate a BS n, if the activity of the BS nis ON under pattern p. Denote by Np c N the set of all BSsactivated by the pattern p. In parallel, we denote by P« C Pthe set of patterns that activate the BS n. Define reuse factorof a pattern p to be XP ~ I~I ~ 1, i.e., the ratio of thenumber of BSs which use a pattern p to the total number ofBSs. Denote by Xp(t) the pattern selection indicator for thepattern p, i.e., Xp(t) == 1 when the pattern p is used at slot tand 0 otherwise. Then, since only one pattern is used per oneslot, we should have:

L Xp(t) == 1. (1)pEP

In regard to (ii), define user scheduling indicator at slot tby Ik(t), i.e., Ik(t) == 1, when the user k is scheduled in itscell, and 0 otherwise. Reflecting the constraint that only oneuser can be selected in each cell, we should have:

'" Ik(t) {==~ 01" if Xp(t) = 1 and n E Np , (2)c: otherwise.kEJCn

Then, a resource allocation scheme incorporates patternselection and user scheduling that can be regarded as choosinga sequence of ((Ik(t) : k E JC), (Xp(t) : p E P)Yl~o satisfyingthe constraints (1) and (2).

We now define the transmission rates of users providedby a resource allocation scheme, depending on the choice ofpatterns. Let Gn,k(t) represent the time-varying channel gainfrom BS n to user k at slot t. The channel gain may takeinto account path loss, log-normal shadowing, fast fading andetc. The received SINR for user k at slot t when pattern pis selected and user k is served by its associated BS, can bewritten as:

{

Ga(k),k(t)P~ax .f--------'---'----------, I P E Pa(k)'

fkp(t) == NoW + ~mENp,m=F_a(k) Gm,k(t)p:;::ax .

0, otherwise,

where No is the noise spectral density. Here, the noise spectraldensity is assumed to be equivalent over all users for simplepresentation. Following the Shannon's formula, the data ratefor user k on reuse pattern p at slot t is given by:

rkp(t) == W log2 (1 + fkp(t)) .

TABLE ISUMMARY OF NOTATIONS III. OPTIMAL ALGORITHM

c. Problem Definition

In this paper, we aim at proposing the joint pattern selectionand user scheduling that maximizes the long-term network­wide utility whenever possible, i.e., solves the followingoptimization problem Q:

(7)

(4)

(3)

(5)

(6)

if Q == 1,otherwise,

U(l) == L Uk(Rk)kEJC 1

L L 7fkp < 1,pEPl kEJC 1 XP

'!rkp 2: 0, Vk E JCI,Vp E PI,

u; == L '!rkprkp, Vk E JC I ·

pEPl

subject to

max( 7r kp :k E JC ,p EP )

where Q is nonnegative and Wk is positive.

For general networks, it is quite difficult to characterize the

optimal fractions of time for user-patterns ('!rk~ : k E JC, p E

P, i E I). However, we will show that it is indeed possibleto explicitly characterize them for symmetric networks withstatic channels. Here, a network is said to be symmetricif all BSs have the same number of users whose channelcharacteristics are equivalent each other. Fig. 1 depicts anillustrative example of a linear two-cell network having threepatterns where (Xl, X2, X3) == (1,0.5,0.5). Recall that Xp, thereuse factor of pattern p, is equal to the ratio of the numberof BSs which use a pattern p to the total number of BSs.Since the network is symmetric, it is enough to analyze thefollowing optimization problem Q-symmetric for a referenceBS only:

Q-symmetric:

We have found that the problem Q-symmetric has aninteresting structure of optimal solution described by theLemmas 3.1 and 3.2. Let Xprkp be the effective rate on patternp for user k, which is the normalized data rate w.r.t. Xv- Notethat there is a trade-off between the reuse factor X» and thedata rate r kp- If the user k chooses the pattern p with thelower value Xt» then the less BSs are active in the network,and accordingly the higher data rate rkp is expected, and viceversa. And we adopt the generalized (w, Q) -fair utility functionin [13] where Uk(Rk) is given by:

In this section, we first study the structure of optimalsolutions analytically for simple scenarios to gain insights. andthen describe a optimal pattern selection and user schedulingalgorithm that converges to the optimal solution of Q.

Here, we derive the constraint (4) for the above two-cellexample. However, this can be readily extended to generalsymmetric networks.

A. Structure of optimal solution for symmetric networks withstatic channels

max U == L u(n) == L Uk(Rk)nEN kEJC

subject to R E R,

fraction of time for pattern pfraction of time that user k is served with pattern p

set of BSs, N == {I, ,N}set of users, JC == {I, , K}association function from JC to Nset of users associated with BS n

system (network-wide) utilityutility of BS nutility function of user k's average throughputachievable rate regionvector of long-term user throughputs, R == (HI, ... ,HK)pattern selection indicator at slot tuser scheduling indicator at slot t

channel gain between BS n and user k at slot ttransmit power of BS nnoise spectral densitysystem bandwidthreceived SINR of user k on pattern p at slot tinstantaneous data rate of user k on pattern p at slot t

set of patterns, P ~ {I, ... ,P}set of patterns that can be used by BS nset of BSs allowed to use the pattern preuse factor (PF) of pattern p

Q:

NJC

a(·)JC n

Uu(n)

Uk(·)RR

X(t)I(t)

Gn,k(t)p~ax

NoW

rkp(t)rkv(t)

Note that rkp(t) == 0, Vp ~ PaCk), i.e., user k cannot receiveany data rate if its associated BS a (k) is not activated bythe pattern p. Also notice that rkp (t) is the potential datarate when the user k is scheduled, i.e., its actual data ratebecomes 0, when other user, say k', associated with the BSa(k), is scheduled for service. We assume that each BS nknows instantaneous achievable data rates for all its associatedusers through channel feedbacks. We further assume that BSshave infinite amount of data to be destined to users.

where R == (Rk, k E JC) is the vector of long-term userthroughputs; the network-wide utility U is just the summationof utilities of all BSs iu'», n EN), or of utilities of all users(Uk, k E JC). Assume the standard condition of differentiabilityand strictly increasing concavity of Ui: The set R E JR.f, theset of all achievable rate vectors over long-term, is shown tobe a closed bounded convex set. Denote by '!r~i) the portionofyattem p. for the i-th channel state. By further denoting by'!rk~ and r k~ the fraction of time that is served by user k andthe data rate of user k (if scheduled) on the pattern p and fori-th channel state, respectively, we can characterize R by:

R = {R = (Rk : k E K) IRk = L L 8(i)7fk~rk~ ,iET pEP

'" '!rei) < '!rei) Vi Vp Vn '" '!rei) < 1 Vi }.L....t kp - p' , , 'L....t p - ,kEJC n pEP

Fig. I. Example of a linear two-cell symmetric network

~Pattern l

~P:;;:::;---'" n; .- .!.llie~r~C~ - - - - ~(i)~ /\ no \Dle~r~CEL.* - -

Pattern 3 ----. R .--~ k2 _--

~ .--------~O 0 I 0 0)----

BS 1 user k BS 2

(reference BS) ········ ::::~9:~[y.~!i;.i·::::. .... . . . . . . . . . .. . . .

(11)t:-.U(t) = L Uk(Rk(t - 1))r k(t )kEIC

maxX (t ) ,I ( t )

~M 0.4

~N

iO 0.35-g.. 0.3N

E~ 0.25c.o 0.2co~ 0.15c.

~ 0.1

B. Joint pattern selection and user scheduling algorithm

We now present an optimal joint pattern selection and userscheduling algorithm. To that end, we use a standard gradient­based algorithm, e.g. , Stolyar's gradient algorithm [15], thatselects the achievable rate vector maximizing the sum ofweighted rates where the weights are marginal utilities at eachslot. Then , it suffices to solve the following problem at eachslot, which jointly determines the pattern selection X(t) =(Xp(t) : p EP) and user scheduling I(t) = (h( t) : k E K):

Q-joint:

0.45

0.05

Fig. 2. Numerical example of the linear two-cell symmetric network whereeach BS has two users; user 1 is in the inner region of the cell and user 2is in the edge of the cell, whose instantaneous data rate vectors are given by(r ll , r12) = (10,11 ) and (r21 , r22) = (3, 8) .

Fig. 2 depicts the optimal portion of patterns with respectto the fairness criterion a. We fix the number of users asshown in the Fig. 1, that is, each BS has are two users:one is in the center and the other in the edge of the cell,IK 11 = 2, IKll l = 1, IKd = 1. When a = 1, the optimalportion of patterns can be given by (10): (7r1' 7r2 , 7r3) =(1/2 ,1 /4 ,1 /4). Accordingly, user throughputs can be easilycalculated: (R 1 , R2 ) = (7rllrll, 7r22r22) = (7r1rll, 7r2r 22) =(5,2). When we decrease a , the portion of pattern 1 increasesas expected. In the extreme case , throughput maximization(a = 0), only user 1 having a better channel is alwaysserved with pattern 1, and user 2 cannot be served at all, i.e.,(7r1 , 7r2) = (1,0). On the other extreme case (a ---+ (0), max­min fairness is achieved such that the throughputs of user 1and user 2 become identical.

(9)

(10)

7r1 = L 7rkp*(k) = IK lll/I K 11 andk ElC ll

7r2 = 7r3 = L 7rkp*(k) = IK 121 / (2IK 11) .kElC 12

{IK11-1, ifk EKll ,

7rkp*(k) = (2 1V"1 1) -1 , 'fk V"l'v I E 1'v12 ,

where IK lll is the set of users such that Rk1 ~ 2Rk2, i.e.,the set of center users, and IK d is the set of users such thatRkl < 2Rk2, i.e., the set of edge users. Thus , the optimalportion for each pattern (7r1 ' 7r2 , 7r3) is given by

Lemma 3.1: For symmetric networks with static channels,the objective (3) is maximized if and only if

{ ~ O , ifp =p*(k) , *7rkp = 0 otherwise where p (k) = arg max Xprkp.2

" p

This implies that each user, if served, only utilizes its patternhaving the largest effective rate.

Lemma 3.2: For the generalized (w,a)-fair utility function,the optimal fractions of time for user-patterns is given by:

( 1-", / ' )1/ '"7rkp* (k) = WkXp* (k)r kp*(k) AO ,

(1/ 1-0< 1-0< )'" (8)

where Ao = L pEP1 LkElC1Pwk "'Xp:'(k{ k;* (k) ,

where KIp is the set of users whose most effective pattern isp, i.e., p*(k) = p if k E KIp.

Please refer to our technical report [14] for proofs. Now,we give a numerical example to illustrates the property of theoptimal solution.

Example I: Consider the example of the linear two-cell sym­metric network in the Fig. 1. In this example, we have threepatterns p EP = {I , 2, 3} where N 1 = {1,2} ,N2 = {I},N.., = {2} and (Xl , X2, X3) = (1,0.5,0.5). Suppose that allusers have the same utility function with (w ,a) = (1,1). By(8), we can obtain Ao = IK 11 and the optimal time fractionsof user-patterns is given by

Note that in the case of proportional fair (a = 1) the optimalportion of each pattern depends only on and is proportionalto the number of users in the sets of center and edge users.However, for general cases (a i=- 1), its closed form is verycomplex because the optimal portion of each pattern dependson the data rate rkp* (k) for all users due to (8). Thus, we relyon numerical computations for a ~ 0, a i=- 1.

2For simplicity, we ignore the ease when more than two patterns achievethe same largest value.

subject to L X p(t) = 1, (12)pEP

'" I k(t){ :::; 1, if Xp(~) = 1 and n E Np, (13)6 = 0, otherwise,

kElC n

where r k(t) = LpEP Xp(t)Ik(t) rkp(t) is the data rate

assi~ned to user k at slot _t and Rk(t) = t L~=l r k(r )= Rk(t - 1) + lOt h (t ) - Rk(t - 1)] (by letting lOt = l /t)is the long-term throughput for user k up to slot t.

Remark 3.3: If we fix the user scheduling I(t) and chooseutility function as Uk(Rk) == Rk in Q-joint, then this problemis reduced to binary power control (BPC) problem for sum-ratemaximization in [2].

The optimization problem Q-joint is an integer program­ming. A naive approach is the exhaustive search of all possiblecombinations of pattern selections and user schedulings. Withthe help of Lemma 3.4 telling us the nice property of theproblem, we can develop the joint optimal pattern selectionand user scheduling algorithm that requires lower complexitythan the exhaustive search.

Lemma 3.4: For afixed pattern p, then the problem Q-jointcan be decomposed into the following INp I independent intra­cell user scheduling problems:

IV. TIME-SCALE DECOMPOSED ALGORITHM

A. Algorithm Description

In contrast to the centralized joint pattern selection and userscheduling algorithm in Section III, user scheduling in prac­tice is typically undertaken by individual BSs independentlywithout any coordination and information exchange with otherBSs. In this section, in order to take into account such au­tonomous feature in user scheduling as well as overcome highcomputation and feedback overheads in the optimal algorithm,we run user scheduling at every slot, but pattern portion changeless frequently, say, every Tp >> 1 slots. We first describeour algorithm (see Fig. 3 for a pictorial description), and thenexplain the rationale behind it.

and updates the following variables for all users k E lCn withsome constants 0 < 131, 132, 133 < 1:

User scheduling algorithm

At each slot t, each BS n E Np(t) activated by pattern p(t)selects the user k~(t), i.e., Ik':tp(t) == 1,

k~(t) == arg max U~(Rk(t - l))rkp(t),kEJC n

Two algorithms with different time scales interact with eachother as follows: The pattern portion change algorithm adjuststhe portion of reuse patterns tt for every Tp slots, usingthe variables Rk (t), 7fkp (t), Tkp (t). These variables essentiallycorrespond to the long-term averages of Ik(t )rkp(t), 7rkp (t),and r kp (t) which are progressively updated at every slot bythe user scheduling algorithm. This time-scale decompositionand the way of interaction between two algorithms impliesthat we design and operate the pattern portion algorithm to letit tract just average interference levels and channel conditions,not fast time-varying ones like the joint optimal algorithmin Section III. Remarking that user scheduling algorithm canbe carried out autonomously, we can significantly reducethe actual (amortized) complexity per slot, which makes our

(1 - (31)Rk(t - 1) + (31Ik(t)rkp(t) ,(1 - (32)1rkp(t - 1) + (32Ik(t) ,

{(I - (33)rkp(t - 1) + (33 rkp(t) , if Ik(t) == 1 ,rkp(t - 1), otherwise,

Rk(t)1rkp(t)

rkp(t)

Pattern portion change algorithm

For every T slots, each BS n E N computes the partialderivative Drn ) and sends it to the central coordinator,

D~n) = L U£(Rk)·(~PfkP)' P E r;kEJC n p

Then, the central coordinator calculates the gradient vectorD == (D 1 , D2 , ... ,Dp ) by collecting D~n) from all BSs,

o; == L D~n), pEP,nEN

and updates the pattern portion vector 7r as follows,

7r ~ Proj", tt =1, (7r + ,D),L... pEP p

where ProjA (.) denotes an orthogonal projection on a set A.

Joint pattern selection and user scheduling algorithm

Remark 3.5: A similar argument has been made in a differ­ence setting [4], but we present this Lemma for completeness.Please refer to our technical report [14] for a proof.

Note that the total number of combinations for our jointalgorithm is polynomial O(P·K) while the that of the naiveexhaustive search is O(P·KN ) . For each pattern p, we selectthe best user having the largest value of U~ (R k(t - 1))rkp(t)from (14) and then the value of the selected user is used inthe pattern selection algorithm. We then find the best patternp*(t) that maximizes the sum of weighted rate U~(Rk(t ­1))rkp* (t) of the scheduled users. The proof of convergenceto the optimal solution is a slight extension to [15], [16] thatstudied only user scheduling for a fixed pattern. We skip theproof.

This joint algorithm requires instantaneous channel feed­back from all users in the network. We assume that at eachslot t, user k estimates its own SINR for all patterns p E PaCk)

upon listening to pilot signals, calculates the instantaneousdata rate r kp (t) and then reports this information to the centalcoordinator through its associated BS.

However, this joint pattern selection and user schedulingalgorithm still has implementation difficulties. Apart from thecomputational complexity of this algorithm, the central coor­dinator running the algorithm needs to collect the followinginformation from each BS n EN: instantaneous data raterkp(t) of all its associated users k E lCn on its availablepatterns pEPn.- The total amount of feedbacks is quitelarge, i.e, (LnEN IlCn \IPn I), though they may be deliveredalong with high speed wired links. Furthermore, a series oftasks, including information feedback from BSs to the centralcoordinator as well as the computation and the distributionof central coordinator's decision, should be performed in oneslot.

p*(t) == argmax L [max U~(Rk(t - l))rkP(t)] ,pEP «r kEJC n

nEJv p

k~(t) == arg max U~(Rk(t -l))rkp*(t), Vn E N p*.kEJC n

Central coordinator

max(h(t),kEKn )

subject to

L U~(Rk(t - l ))I k (t )rkp(t )kEKn

L h(t) < 1.kEKn

The user scheduling algorithm solving Q-scheduling isstraightforward. Each BS n E Np allowed to use the pattern pindependently chooses the best user k~ (t) among it associateduser set K n , i.e., h ;,(t) = 1:

k~(t) = arg max U~(Rk(t - l ))rkp(t ), \:In E Np, (19)«ex;

and updates the portion of reuse patterns following the in­creasing direction of network utility.

7r +- Proj L pE"P 7l"p= l , (7r + 'YD ) . (18)

Based on the updated portion of patterns, the central co­ordinator predetermines the sequence of patterns for next Tp

slots that satisfies:

and updates the following variables for the future purpose ofthe pattern portion change algorithm:

(1 - (31)Rk(t - 1) + !3d k(t )rkp (t ) ,(1 - (32 )itkp(t - 1) + !32h (t ) ,

{(I - (33 )Tkp(t - 1) + !33rkp (t), if h(t) = 1 ,Tkp (t - 1), otherwise,

where /31, /32, /33 > 0 are small averaging parameters; Rk (t),itkp(t ) and i'kp(t ) are the average throughput of user k, theaverage fraction of time that user k is served with pattern p,and the average instantaneous data rate when the user k isserved with pattern p, respectively.

Remark 4.1: There are two key differences between thealgorithm in [8] and ours. First, they additionally introducea virtual scheduler to obtain the fraction of time that thescheduler chooses user i for transmission in sub-band j (theirnotation: cPi j ). In our algorithm, however, we just obtain thefraction of time that user k is served with pattern p (ournotation: itkp) using the actual scheduler without any extraalgorithm. Second, they do not reflect time-varying nature ofthe data rate available to user i in sub-band j (their notation:Rij ) by assuming this rate does not change with time. Inour algorithm, the long-term average of data rate of user kon pattern p (our notation: i'kp) is not just the average ofinstantaneous data rata. We take the average of instantaneousdata rata only if the user k is really served by the scheduler.

(the total number of pattern p) / t; ~ 7rp , \:Ip E P.

While there may be many strategies, a nice candidate is a ran­dom strategy. The central coordinator sequentially determinesthe sequence of patterns by rolling a P-dimensional die Tp

times with probability of the pattern p being 7rv-

Now we develop the user scheduling algorithm under thefixed pattern given by the pattern portion change algorithm.From the Lemma 3.4, for the given pattern the network-wideuser scheduling problem can be decomposed into independentintra-cell user scheduling problems. Therefore, each BS needsto solve the following problem Q-scheduling:

Q-scheduling:

(15)

(17)

(16)

D =" D(n)p L.-J p

neN

2

D ~ au = '" D (n) pp a7r ~ p , p E ,

p n EN

BSNrr== = =;-;= = =;---;==== i1

algorithms much more implementable. We will discuss theprice of such complexity reduction, i.e., utility gperformancegap with the optimal algorithm in the Subsection IV-C.

3While we differentiate Rkp on 7[p , we assume that <Pkp is constant.Please refer to [8] and its technical report for rigorous proof.

where.'

Fig. 3. Proposed time-scale decomposed algorithms

aRk itkA, - p -'l-'kprkp = - rkp'a7rp 7rp

Note that three parameters (Rk, itkp and i'kp) required torun this pattern portion update algorithm can be attained byattained by the user scheduling algorithm over long time.And then the central coordinator gathers information from allBSs and calculates the partial derivative of the network utilityD p ~ au/a7rp by aggregating these partial derivatives of thelocal utility,

B. Rationale of Time-scale Decomposed Algorithms

The pattern portion change algorithm is a standard gradientprojection algorithm for the following problem Q-pattern:

Q-pattern:

m;x L Uk(Rk) = L u,(L cPkP7rPi'kP )kEK kEK pEP

subje ct to L 7rp = 1,pEP

where cPkp E [0,1] is the probability that the user k isscheduled when pattern p is selected, i.e., cPkp . 7rp = itkp

For each of the pattern portion update epoch, i.e., everyTp slots , each BS n needs to calculate the partial derivativeD~n) ~ au (n)/ a7rp of per-cell utility u(n) with respect to theportion of pattern p and send these information to the centralcoordinator.

TABLE IICOMPARISO N BETWEEN JOI NT O PTIMA L ALGORI T HM (JOA) AND TI ME-SCALE DECOMPOS ED AL GORI T HM (TDA)

I Joint optimal algorithm I Time-scale decomposed algor ithm

Time-scale of algor ithms every slotevery slot (user scheduling)

every 1" slot (pattern portion change)Amount of feedbacks to each BS n at each slot liCnllPnl liCnlAmount of feedbacks to the central coord inator L ,n",M liCnllPnl L n",M IPnl

Period of feedback to the central coordinator 1 Tp

Convergence speed fast reasonable speed (depending on Tp )

Recall that the opportunistic scheduler likely to serve the userwhose current channel quality is high relative to his ownrate statistics. In other words, our rkp reflects the multi-userdiversity gain from exploiting the channel fluctuation.

BS1@

~P, = {I, 2]

Pattern 1: (B81 , B82) = (ON,ON)Pattern 2: (B81 , B82) = (ON,OFF)Pattern 3: (B81, B82) = (OFF,ON)

I

Ja) Linear two-cell network

BS2@~

P2 = {I, 3]

C. Complexity Reduction and Its Price

Our time-scale decomposed algorithm still involves signal­ings from BSs to the central coordinator. However, we cansignificantly reduce feedback overheads because the period­icity of the feedback is stretched from every slot to everyTp slots. Moreover, the amount of feedbacks is reduced from(LnEN IJCnIIPnl) to (LnEN IPnl), i.e., requires only theBS-Ievel feedback, not the user-level channel feedback. Theamount of feedbacks to each BS n from its associated users ateach slot is also reduced from IJCnllPnl and IJCnl because usersneed to send channel information only for the predeterminedpattern. Table II compares the joint pattern selection and userscheduling algorithm with the proposed algorithms based ontime-scale decomposition.

This complexity reduction for implementability comes atthe cost of performance gap with the joint optimal algorithm.This is because the ICI management part in the decomposedalgorithm cannot fully exploit instantaneous inter-cell channelvariations, and only intra-cell channel variations are oppor­tunistically utilized. Note that in the joint optimal algorithm,both pattern selection and user scheduling fully exploit bothinter-cell and intra-cell time-varying channel conditions at afast time scale.

As an example, consider a two cell network where two usersare located at the edge of each cell. Their achievable rates arelimited by severe ICI. The decomposed algorithm will find thefollowing TDMA-like solution: BS I and BS 2 are exclusivelyactive in order to mitigate the ICI, i.e., the portion for thepattern in which both BSs are active is nearly zero. However,suppose that both (time-varying) inter-cell channel gains fromBS I (or 2) to the user in BS 2 (or I) are in deep fading atsome time slot. You can imagine this case as if there were abig wall between two cells. Then the user in cell I (or 2) is notinterfered by the BS transmission in cell 2 (or I). Therefore,serving two users simultaneously is transiently optimal in thisinter-cell deep fading case, whereas the pattern that only oneBS is active is the solution of the average ICI mitigation. Jointoptimal algorithm can find this optimal solution by trackingthis fast fading while the decomposed algorithm cannot. Wefinally comment that as we will see in the Section V, in absenceof fast fading, the performance gap becomes negligible.

,(~) Two~t ierl11 l1 lt i ~cell l1etworkcoll~J)osed ~ofl9 ~ce l ls

Fig. 4. Network configurations.

V. SIMUL ATION RE SULT S

A. Simulation Setup

We consider two cases of network configuration: one is thelinear two-cell network and the other is the two-tier multi­cell network composed of 19 cells. In both cases, the distancebetween BSs is 2km. In the linear two-cell network, there arethree patterns P = {l , 2, 3}. Under pattern I, both BSs areON, and under pattern 2 (resp. 3), only BS I (resp. 2) is ON.In the two-tier multi-cell network, we consider I I patterns.Under pattern I, all BSs are ON so that each BS receives all theICI from all over the network. However, under well-designedpatterns 2,,-,4 or 5"-'11 (see Fig. 4 for the pattern design), theBS using these patterns can expect the first-tier ICI mitigationor the mitigation of ICI from one of its neighbors, respectively.

To evaluate the performance under various user distributionscenarios, we introduce a variable, so-called, 'user distributionoffset' p E [0, 1] , which adjusts the minimum distance betweenthe BS and the user to p x (cell radius). Basically, we randomly

-+- GAT: JO--GAT: TO--GAT: UNI

Geometric average of user throughputs -. - AET: JOA(GAT) -. - AET: TOA

-e-AET: UNI

Geometric average of user throughputs -+- GAT: JO(GAT) --GAT: TO

--GAT: UNI-+-AET: JOA-.-AET: TOA-e-AET: UNI

7

(a) Without last lad ing

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9User distribution offset, p

D. Imbalance load: BS coordination vs. association change

We also test the performance in the linear two-cell networkhaving imbalanced loads. We located 2 users 900m away fromBS 1 and 2 x LI users 900m away from BS 2, respectively,where LI quantifies the load imbalance. In the network withimbalanced load, we have performed simulation to investigatethe amount of additional gain that BS association change canprovide. We compare the following two different approaches:(i) Association change: load-aware handover in [fO} and (ii)BS coordination: our TDA. As a baseline, we also plot theperformance of UN\.

Originally users are associated with the closest BS offeringthe best signal strength. In the case of association changealgorithm, however, if the expected throughput measure in [10]from the other BS is greater than that from the current BS,

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9User distribution offset , p

(b) With fast fading

Fig. 5. Throughput performances of three algorithms in two-cell network:joint optimal algorithm (lOA), time-scale decomposed algorithm (TDA) anduniversal reuse (UNI).

C. Two-tier 19-cell network case

B. Linear two-cell network case

Fig. 5 shows the GAT and AET performances of threealgorithms in the linear two-cell network. In the case withoutfading, the performances of both JOA and TDA are almostsame, where they increase the GAT and the AET by 20",85%(depending on user distribution) and 85% compared to UN\.We observe a higher performance gain when user distributionoffset is larger (i.e., more users are located at cell edges).This is because the lCI management is mainly targeted forthe performance improvement of edge users. With fading,however, as discussed in subsection IV-C, there is a perfor­mance gap between JOA and TDA due to loss in opportunismof time-scale decomposition. Still, the TDA outperforms theUNI in terms of both the GAT (6",20% depending on userdistribution) and AET (33%). Note that TDA can attain morethan 1/2 (at p = 0) and up to 2/3 (at p = 0.9) of the GATperformance gain that is achieved by JOA.

In Fig. 6 shows the GAT and AET performances in thetwo-tier multi-cell network composed of 19 cells. Although theperformance gain is a little small compared to the simple lineartwo-cell network case, trends are similar to those in Fig. 5 asa whole. With fading, the TDA outperforms the UNI in termsof both the GAT (5",25% depending on user distribution) andAET (25%). Similar to the two-cell case, TDA can still attain1/2",2/3 of GAT performance gain that is achieved by JOA.

distributed users in each cell with this mmimum distancerestriction. For example, if p = 0, users are uniformlygenerated over the cell. On the other hand, if p goes to 1,users are only located in the edge of the cell.

The maximum powers of BSs are all the same with 20W.Channel models are implemented following ITU PED-B pathloss model [17] and Jakes' Rayleigh fading model. Thechannel bandwidth is 10MHz, and the time-slot length is 5ms,as specified in the IEEE 802.16e standard. The pattern updateperiod Tp = 500, and the step size is chosen to be a typicallysmall value, i.e., (31 = (32 = (33 = "( = 0.001. For each givenparameter set, we ran simulations over 50000 slots.

We consider the performance of the (i) conventional uni­versal reuse scheme (UN!), in which all BSs in the networkare always active without any lCl management, as a baselineand compare the performance of the following two algorithmsnormalized by UNIVERSAL: (i) the joint optimal algorithm(lOA) and (ii) the algorithm based on time-scale decomposi­tion (TDA). As performance metrics, the geometric average ofuser throughputs (GAT) and the average of edge user through­puts (AET) are used. We use GAT since maximizing this isequivalent to the system objective (sum of log throughputs).The AET is the measure of cell edge performance defined asthe average throughput of users located at cell edges. In oursimulation, we treat 'edge users' as those who are more than800m away from their associated BSs in our setup.

10

--- BS coordiat ion - TDA- + - As s ocia tion change - [1OJ-- Universal reuse - UNI

46 8Load imbalance

(jj" 140..0:2:';;;" 12"50.

o§, 10::::le£Qj(/)::::l

'0 6(l)Ol

~(l) 4>coo

.<:;

Q3 2E0(l)

<.9 02

-+-GAT: JO--GAT: TD--GAT: UNI- + - AET: JOA-e-AET: TDA-e-AET: UNIGeometric average of user

throughputs(GAT)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9User distribution offset , p

7

(a) Without fast fading

Fig. 7. Assoc iation change vs. BS coordination under imbalanced loadscenario.

VI. CON CL USION

In this paper, we have focused on the problem of joint ICImanagement and user scheduling in multi-cell networks. Wehave shown that the joint optimal algorithm is too complex (interms of computational and signaling overhead) to be imple­mented in practical systems. To overcome this complexity andmake the algorithm practical, we have decomposed the originaloptimization problem into two sub-problems, where we runICI management at a slower time scale than user scheduling.This time-scal e decomposition stems from a design rationalethat ICI management may not have to track fast changingdynamics, and it may suffice to attain much gain just byrunning it based only on macroscopic network changes. Weempirically show that even with such a slow tracking ofsystemdynamics at the ICI management, our algorithm achieves highperformance gain compared to a conventional universal reusescheme, as well as is practically implementable compared tothe very complex optimal algorithm.

R EFER ENC ES

-+-GAT: JO--GAT: TD

Geometric average of user throughputs __GAT : UNI(GAT) - + - AET: JOA

-e-AET: TDA-e-AET: UNI

7

(b) With fast fading

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9User distribution offset, p

Fig. 6. Throughput performances of three algorithms in 19-cell network:joint opt imal algorithm (lOA), time-scale decomposed algorithm ("fDA) anduniversal reuse (UNI) .

then the user changes its association. When the LI is small,users do not change their associations. When we increase LImore than 6, the association change from the hot-spot cell (BS2) and the under-loaded cell (BS I) happens (moving one, twoand three users at LI=6, 8 and 10, respectively) by the load­aware handover in [10]. As can be seen in Fig. 7, however,the gain from the association change is marginal.

On the other hand, using the BS coordination, we canimplicitly resolve the load imbalance by preventing the hot­spot cell (BS 2) from being turned off, i.e., provide moretransmission chances compare to the BS I. In brief, the BScoordination originally developed for ICI mitigation can alsoresolve the load imbalance, and the improvement of than BScoordination is better than that of the association change.

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