techreport11-sortinghat

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SortingHat: Holisitic Resource Optimization forHeterogeneous Cellular Networks

Steven Hong, Jeffrey Mehlman, and Daniel ONeillStanford University Department of Electrical Engineering

Stanford, California 94305{ hsiying, jmehlman, dconeill } @ stanford.edu

Abstract While heterogeneous cellular networks offer the po-tential to improve network performance by augmenting coverageand enhancing trafc capacity, resource allocation is extremelycomplicated due to the heterogeneous and unplanned nature of base station deployments. In this paper, we present SortingHat , atractable optimization model which leverages the advantages of the different heterogeneous components in the network to achieve3 main objectives: user-cell pairing, power control, and channelassignment. SortingHat achieves this by casting the resourceallocation problem in a form which can be solved using iterativeoptimization algorithms such as the constrained convex-concaveprocedure (CCCP). While algorithms for user-cell pairing, powercontrol, and channel assignment have been studied extensively inthe past, this reformulation enables SortingHat to take a holisticapproach to resource allocation and jointly optimize all threeobjectives simultaneously. Numerical results show greater than30-100% improvement in median user throughput relative toprior works in which the objectives are considered separately,rather than jointly.

I. INTRODUCTION

Driven by the proliferation of smartphones and other mobile

computing devices, cellular wireless networks are strainingfrom the rapid explosion of bandwidth-intensive applications.While new protocols like LTE will provide some trafccapacity gains, service providers are increasingly favoringa heterogeneous network architecture [1] characterized bymultiple tiers of base-stations with smaller, low-power cellsdeployed within macro cells to either (1) extend coveragewhere macro cell coverage is poor, or (2) to increase totaltrafc capacity in areas with dense data usage. Currently, thereare two tiers of low power cells: operator deployed microcells and user-deployed femto cells. In moving away fromthe old-school paradigm of carefully planned, homogeneouscellular networks where every cells capabilities were roughly

equal and their geographical placements, frequency bands,and transmit powers were painstakingly chosen, operators arefaced with new and challenging multi-tier network manage-ment tasks.

The difculty of managing these networks is rooted inthe heterogeneous and chaotic nature of the base-stations.Macro, micro, and femto cells all have constrained resources,capabilities, and environmental settings which can vary alonga number of axes:

1) Backhaul Bandwidth: There exists a high degree of variability in each cells backhaul capability [2]: mostmacro cells today have copper T1 links, micro cells have

Macro Cell

Micro Cell

Femto Cell

Fig. 1. Heterogeneous Cellular Networks (HCNs) are characterizedby the co-existence of different types of cells which vary in theirallocatable power, service range, and backhaul capability.

directed microwave links back to aggregation stations,and user-deployed femto cells utilize user-owned broad-band connections. With LTE supporting up to 100Mbpsin the downlink, cell backhaul capabilities are becomingconstraints on a cells ability to meet trafc demand.

2) Total Transmit Power/Coverage Range: While macrocells will operate up to FCC spectral limits, femto and

micro cells are limited both by cost limitations and healthconcerns of the general population, leading to at least anorder of magnitude difference in total transmit power ca-pabilities of different cells. As a result of varied transmitpower abilities and geographical placement (i.e. on top of a tower, inside a building, etc.), the functional coveragearea of every cell can vary from the size of a singlehome to an entire town. Without careful power control,operators run the risk of causing excessive amounts of interference between cells and choking off the entirenetworks capacity.

3) Frequency Range: While many operators posses narrow-band spectral slices which range from 900Mhz to 2GHz,

smaller cells may be equipped only to utilize certainportions of the total available spectrum in order to lowerdeployment cost. As a result, every cell may only be ableto utilize a certain number of frequency subchannels.

4) Local User Density & Demand: The density of userscan vary wildly across time and space (i.e. cities duringthe day and at night). Additionally, every user can havevery different rate demands, ranging from voice (low rate)to streaming video (high rate).

One of the biggest challenges that dense, interference-limited, heterogeneous cellular networks (HCNs) face is inhow to effectively allocate resources. In the classical macro-


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tier setting, users were associated with the cell which providedthe best link gain, power control was done using channelinversion, and channel re-use was carefully planned in astatic fashion to mitigate interference. However, in HCNs thecoverage range of cells across tiers overlap, so it is often notclear how the pairing between users and base-station cellsshould be made. Furthermore, channel inversion at the macro-tier becomes unusable because of the excessive interference itcan cause to lower power tiers and static channel allocationresults in over/under provisioning for different cells withvarying loads. It is clear that moving forward, we should jointlyaddress user-cell pairing, power control, and subchannelselection to nd appropriate solutions.

In this work, we develop SortingHat [3], a holistic, multi-tierpolicy for heterogeneous cellular networks, which builds onprior work on globally optimal, power allocation only schemes[6][7]. In a dense network of cells and users, we unify thechoice of user-cell pairings, subchannel selections, and powerallocations in the presence of a variety of network constraints.We show that user rate demands can be reformulated into a

special convex-concave form allowing joint selection of allthree parameters. While this reformulation cedes the convexityof the original GP formulation [6], we show that this new,special form allows us to use well understood sequentialconvex programming methods to nd good solutions [20]. Wedemonstrate that this policy, despite its lack of global con-vexity, generates sensible, useful results, outperforms existingmethods, and provides a holistic framework for analysis of multi-tier heterogeneous networks.

The rest of the paper is organized as follows: In sectionII, we highlight some state-of-the-art work in HCN resourceallocation. In III, we describe the problem parameters andvariables and develop an optimization program formulation

which we will show to be non-convex. In IV, we presentSortingHat, a heuristic solver which utilizes a crisp convexapproximation to make the problem tractable. In V, wesimulate and evaluate the performance of the algorithm. InVI, we detail future directions for this work.

I I . R ELATED W ORK

Industry and academia have proposed a wide variety of resource management algorithms to deal with these multi-tiered complex networks, ranging from very straightforwardto extremely complex. These include power-control only al-gorithms, which assume subchannel and user-cell pairings

are made beforehand (simple), to power control and channelselection algorithms which use heuristic approaches like graphcoloring (complex), to cell-user pairings which are oftenconsidered separately from the other two objectives. Few, if any, of these methods address backhaul limitations of eachcell, which are becoming more and more important [2]. Webriey review the state-of-the-art for each of these approaches:

1) Power Control : If user-cell and subchannel selectionsare made beforehand, the power control problem can bereformulated into a convex problem [8] for which globalsolutions are readily found. Proposed solutions include optimalmethods based on geometric programming (GP) [6][7] as well

as heuristic methods based on power and rate adaptation [4].Most of these algorithms do not take into account how thetransmitters (cells) and receivers (users) are paired and assumethis is known a-priori. In this paper, we show that this pairingis crucial, and greatly affects the performance of any powercontrol technique. Because user demands can wildly vary andnetwork resource constraints may dictate that other user-cellpairings are better, separately considering power control, user-cell pairing, and subchannel selection is an inefcient approachand can lead to wasted spectrum. As such, these choices shouldbe made in conjunction with power allocations.

2) Power Control & Channel Selection : Complex ap-proaches which take some of these joint objectives intoconsideration have be thoroughly studied [9]-[16], but nonecover the entire gambit, and often they are based on empiricalobservations about network behaviors or unrealistic assump-tions. In fact, channel assignment alone for wireless networkshas also been studied in the past and has been shown to be NP-hard [10]. A number of heuristics have been proposed for thisproblem, including centralized, nonlinear convex optimization

[11][12], decentralized scheduling problems [16][13][14], andtheoretical graph coloring problems [15]. But as with powercontrol, these approaches [10]-[15] assume xed user-cellpairs, resulting in solutions whose performance is dependentupon the proper user-cell pairings.

3) User-Cell Pairing : Normally, user-cell pairing reliesupon the clustering of certain features such as geographicaldistance [17], local base station density [18][19], etc. Whilethese approaches can produce good results, they often donot consider multiple constraints (e.g. power, backhaul, in-terference), and this can lead to signicant sub-optimality of their solutions. In general, the entire problem, including powerallocation, user-cell association, and subchannel selection,

remains NP-hard and thus good heuristics which account forthe entire networks characteristics are needed.

III . P ROBLEM FORMULATION AND S YSTEM M ODEL

In this section we develop the system model and moti-vate the challenges for SortingHat. We consider a wirelessdownlink network with M cells serving N users. The link gains from cell i to user j , G ij , are represented by the matrixG R M N . The transmit power from cells i to user j , P ij ,are represented by the matrix P R M N .

Each cell has a maximum total transmit power constraint,specied by the vector P max R M . Additionally, each userdemands a certain rate, specied by the vector r des R N ,

where the networks ability to serve this demand is estimatedby a function of SINR. In general, the goal is to choose P such that the aforementioned constraints are satised. A cellnetwork may not be able to satisfy every users demand, sowe allow violations of this constraint via a slack rate violationvariable, s R N , to ensure that the problem is alwaysfeasible.

A. Modeling Cell-to-User Rates

We estimate the ability of cell i to serve user j s rate demandfor a particular subchannel k, r ki j via a function of the signal-to-interference-plus-noise (SINR) ratio. We utilize Foschinis


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curves for an AWGN channel [8], where the available rate (inbits/sec/Hz) for user j from cell i on subchannel k is estimatedby

r ki j log 2(1 + vSINRki j) (1)

where v provides rate curves for different modulations. Forsimplicity, we consider v = 1 for this model.

To model the SINR for each user, we consider two in-

terference models representing the dominant multiple accessapproaches that service providers employ today:

Code Division (CDMA): Each user has a reserved,orthogonal code (subchannel) across all cells, so cellsonly interfere with each other for a given user and thereis no interference between different users. The SINR fora single user-cell pairing in this model is

SINR ki j =G i, j, k P i, j, k

i = i

G i, j, k P i, j, k + N

where N R is the noise power seen by each user andk = j , i.e. each user has its own channel. We assumethat N is constant across all users.

Frequency Division (OFDMA): Each user utilizes anarrowband subchannel with a cell, and experiences co-channel interference from other users. The SINR for asingle user-cell pairing and subchannel in this model is:

SINR ki j =G i, j, k P i, j, k

j = j iGi, j, k P i,j, k +

i = i

Gi, j, k P i, j, k + N

where the rst term in the denominator represents theinterference from other cells servicing other users onthe same subchannel and the second term represents theinterference from other cells attempting to service user jon the same subchannel.

B. Optimization Problem: Joint BS-user selection, power al-location, and channel assignment

Mathematically, we can express this optimization problemwith variables P R M N K , s R N as follows:

minimize f (P, s ) (2)subject to (Backhaul bandwidth constraints: )

N

j =1

(log2(maxi, k

(1 + SINR ki j))) i , i (3)

(Cell transmit power limits: )K

k =1

N

j =1

P i,j,k P maxi , i = 1 ,...,M (4)

(Transmit power is nonnegative: )P i,j,k 0, i , j , k (5)(Satisfy rate demand minus slack: )log 2(maxi,k (1 + SINR i j)) r

desj s j , j (6)

(Slack variable must be nonnegative: )s j 0, j = 1 ,...,N (7)

This formulation captures all of the different components of an HCN as discussed in I. Variation in backhaul capabilitycan be modeled through i in constraint (3). The differencesin transmit power/coverage range are accounted for by P maxiin constraint (4). Smaller cells can be restricted to certainsubchannels by specifying that P i,j,k = 0 for the restrictedsubchannels k in constraint (5). And lastly, different user ratedemands are modeled through varying r desj in constraint (6).

Depending on the situation, there are several convex objectives which might be of interest such as minimizing totalpower consumption,

f 0(P, s ) = 1 T P 1 (8)

minimizing a weighted sum of user slacks to represent QoSimportance on users demands,

f 1(P, s ) = wT s (9)

or any weighted combination of these two objective functions

f 2(P, s ) = 0f 0(P, s ) + 1f 1(P, s ) (10)

The max i,k () in constraints (3) and (6) can be motivated asfollows: the users demand should always be served by the bestavailable subchannel and cell SINR, and total user demand ona given cell should not exceed its backhaul ability, respectively.If we pre-allocate a cell i and subchannel k for each userand ignore the backhaul constraints entirely , the problemreduces to a convex, geometric program. But pre-allocatinguser-cell pairs and subchannels precludes the optimizationfrom encompassing all network constraints, which can lead tovery suboptimal results. While our formulation unies user-cell pairing, channel assignment and power allocation, ascurrently formulated it is not convex. In IV we address thisnon-convexity and develop a tractable heuristic.

IV. A LGORITHM D ESIGN

In this section, we present SortingHat, an HCN resourceallocation algorithm which utilizes an iterative, sequential con-vex programming method, the Constrained Convex-ConcaveProcedure (CCCP), to nd solutions for user subchannelselection, user-cell pairings, and transmit power allocations.

CCCP approximates objectives or constraints which are adifference of two convex functions by taking the negativepiece (i.e., the non-convex one), linearizing it locally around acurrent, feasible point via Taylor approximation, and usingthis linear approximation to pose a convex function. The

approximate convex program is then globally solved, theresulting optimal variable is plugged back into the linearizedproblem, and this process is repeated until some stoppingcriterion is met. Because the approximation is local, thealgorithm iteratively solves the program, updating the local,Taylor approximations along the way. The effectiveness of thealgorithm depends on the specic problem formulation andthe chosen starting point, but in many circumstances it worksvery well. See [20] for more detail on this procedure.

To express our optimization problem as a convex-concaveprogram and use CCCP, we begin by reformulating constraints(3) and (6). Specically, we re-formulate the log 2(max i, k (1+


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log 2 max i, k (1 + SINR i j ) = log2 maxi, k

1 +G i, j, k P i, j, k

j = j i G i, j, k P i,j, k + i = i Gi, j, k P i, j, k + N (11)

= log2 maxi, k

1 +G i, j, k P i, j, k

j,i Gi, j, k P i,j, k G i, j, k P i, j, k + N (12)

= log2 maxk

1 +max i (G i, j, k P i, j, k )

j,iG

i, j, kP

i,j, k max

i(G

i, j, kP

i, j, k) + N

(13)

= maxk

log2j,i Gi, j, k P i,j, k + N

j,i G i, j, k P i,j, k max i (G i, j, k P i, j, k ) + N (14)

= maxk

log 2j,i

G i, j, k P i,j, k + N log 2j,i

Gi, j, k P i,j, k maxi

(G i, j, k P i, j, k ) + N (15)

= maxk

a j (P ) bj (P ) (16)

SINR ki j )) term which appears in both affected constraintsas almost a fully convex-concave equation, giving both con-straints almost convex-concave forms. At the top of this page

in equations (11)-(16), we step through the algebra requiredto reform this function and review the reasoning here:

From eq. (11) to (12), the co-channel interference is re-written as the summation of the total power emitted byeach cell multiplied by the link gains minus the powerfrom the selected cell, G i, j, k P i, j, k .

From eq. (12) to (13), maximizing the ratio over the icells can be brought into the numerator and denominator,since the same G i, j, k P i, j, k term is affected in both.

From eq. (13) to (14), log() is non decreasing on R ++ sothe maximum over k channels can be pulled out behindthe log function.

In its nal form (16), the users optimal rate over all subchan-nels, cell pairings, and power allocations is a convex-concavefunction nested inside of a maximization over all k channels.

In order to provide a purely convex-concave form for bothof the affected constraints and utilize CCCP, we must rstremove the max k () wrapping the function. Thus, within eachiteration (iter ) of the SortingHat algorithm, we

1) Select a subchannel kj for each user j , through asimple, greedy selection scheme in which the users withthe largest slack violations are assigned in succession tochannels with the minimum co-channel interference. Thisis shown as steps (3)-(10) in the Pseudo Code.

2) Optimize with CCCP , by linearizing the two constraints(rate and backhaul) for the selected subchannel around acurrent suboptimal power allocation matrix P (iter-1) . Withthis approximate, convex form in hand, we utilize off-the-shelve solver convex program solver CVX [21]. Thisis shown as steps (11)-(16) in the Pseudo Code.

After each iteration, power allocation P (iter) and slack vio-lation s (iter) are used to seed the next iteration of SortingHat.

A. Subchannel Selection

Channel sharing among multiple users is a critical issue inmobile cellular systems. In this paper, we focus on channel

SortingHat Pseudo Code:IV-A lines (3)-(10); IV-B lines (11)-(16)

(1) initialize P (0)R

M N K(to feasible starting point)

(2) for iter := 1 to T (3) Sort (kj , s j ) pairs by descending values of s j(4) for index := 1 to min( users K 1)(5) kindex = argmin k = K ( j,i G i, j,k P

( iter 1)i,j,k

(6) Gi, j,k P ( iter 1)i, j,k )

(7) end

(8) if ( obj ( iter ) obj ( iter 1) < 0)(9) P ( iter ) := P ( iter 1)

(10) end

(11) Find Taylor expansions around P ( iter 1) = (20) , (22)(12) [P ( iter )

i, j, k, s j ] = CVX.solve( P

( iter 1)i, j, k

) {(13) variables: P ( iter ) R M N K , s R N (14) minimize (23)(15) subject to: (5) , (4) , (7) , (20) , (22)(16) };(17) end

assignments in OFDMA systems. In this context, the channelswe will refer to are equally spaced in the frequency domainand are numerically ordered ( k = 1 , 2, 3 etc.) from low-frequencies to high frequencies. Multiple channels cannotbe simultaneously assigned to the same user. To meet therequirements of users with higher rate demands, more power

must be allocated to those users on their assigned channel.Aside from this rule, we place no spatial restrictions onchannel re-use; adjacent cells are free to re-use the samesubchannel if SortingHat decides that is the right choice.

The subchannel selection algorithm works as follows: Werst sort the users in descending order by the amount of slack violation they have following the prior iteration of SortingHat.We pick the user with the largest slack and assign it to thechannel in which it will experience the smallest amount of co-channel interference (i.e. if no one nearby is using thatchannel, its probably a good choice for that user). Then, theuser with the 2nd most amount of slack is selected and the


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procedure is repeated.The problem with this approach is that oscillations can

occur, for instance consider the case when there are N userswhich are initially assigned to the same channel and thusexperience signicant co-channel interference. Now assumethat there are a total of N available channels, but that N 1of them are already being used by neighboring cells andhence users assigned to these channel will experience a smallamount of co-channel interference. If all of the users followthis reassignment algorithm, they will all go to the remainingchannel which is free of interference, but in doing so will endup in the situation they were originally in, i.e. all of theminterfere with each other. In the next iteration, the channelwhich they have just vacated will end up being the optimalchannel and thus these users will end up oscillating betweenthe 2 channels, and the algorithm would never converge.

Ideally, each subsequent user would be able to accountfor the interference of the earlier users on new channelsbefore making their new subchannel selection. However, anew power allocation will not be determined until the nextiteration of SortingHat, so this is not possible. Our solutionfor this oscillation problem is to deny subsequent users fromchoosing channels which have already been selected by userswith more slack in this iteration . Thus, the maximum numberof users that can be assigned to a different channel withineach iteration is equal to the total number of channels lessone (K 1). This approach also has the added benet that notall users should change, because as some users depart frombusy, interference-limited subchannels, these subchannels willhave less interference and the performance of the users whichhave stayed in those subchannels may improve as well.

B. CCCP Formulation

The constraints (3) and (6) of the optimization problemhave been reformulated as the difference of two convexproblems as shown in eq. (11)-(16) (within a max k () term).Within each iteration of SortingHat, once new subchannelselections for users with the most slack are determined, wecan remove the max k () term around both constraints and solvethe optimization problem using CCCP [20]. In the nal form(16), the rst term a j (P ) is a concave composition and thesecond term bj (P ) is the negation of a concave composition(i.e a convex function). Since the rate demand is a concaveconstraint, we must linearize bj (P ) in order to utilize CCCP,and vice-versa for the backhaul constraint. We review thelinearization procedure for both constraints below:

1) Rate Constraint : To linearize bj (P ), we rst calculatea subdifferential bj (P ). Because our space is limited, weshow only the subdifferential formulation for the CDMA SINRmodel, but the calculation is similar for the OFDMA SINRmodel. To simplify notation, we drop the k index (since thechannel is already selected) and utilize vector notation, where pj and gj are the j th column vectors of P and G, respectively:

bj (P ) =

j,i G i, j max i diag (gj ) pj

ln(2) j,i Gi, j P i,j max i diag (gj ) pj + N (17)

The subdifferential of max i diag (gj ) pj is a vector withzeros in all positions except for the i th index of gj whichprovides the maximal G i, j P i, j pair. For example, if the i

th

index of diag (gj ) pj is the maximal entry of the vector, then

maxi

diag (gj ) pj = 0 0 G i, j 0 0T

(18)

Finally, this results in the subdifferential formulation

bj (P ) =G1 , j . . . G i 1, j 0 G i +1 , j . . . G m, j

T

ln(2) gT j pj max i diag (gj ) pj + N (19)

and we have the CCCP approximation for the non-convex rateconstraint (6),

a j (P ) bj (P (iter 1) ) bj (P

(iter 1) )T (P P (iter 1) ) r desj s j(20)

where P (iter 1) is the value of P generated in the last iteration.2) Backhaul Constraint Reformulation : A similar ap-

proach is taken to satisfy the backhaul constraint (3) butbecause it is convex, we must linearize the concave componenta j (P ) in order to use CCCP. The subgradient of a j (P ) is

a j (P ) =j,i G i, j

ln(2) j,i G i, j P i,j + N (21)

and the CCCP approximation for the non-concave backhaulconstraint(3) becomes

j

a j (P (iter 1) ) bj (P )+ a j (P (iter 1) )T (P P (iter 1) ) j

(22)Since the CCCP approximations for equations (20) and (22)

are calculated independently and linearize different halves (i.e.a j (P ) for one and bj (P ) for the other), the procedure onlyproduces a meaningful result if the opposing linearizationsconverge to the same result. To ensure this outcome, weaugment the objective function such that it minimizes theabsolute difference between the two estimates, i.e.

f 3(P, s ) = 1 T P 1 + 1 T s +j

| f j (P (iter 1) )

+ f j (P (iter 1) )T (P P (iter 1) ) bj (P ) (f j (P ) bj (P (iter 1) ) bj (P (iter 1) )T (P P (iter 1) ) | (23)

Exchanging the original backhaul constraint (3) and originalrate constraint (6) for the CCCP constraints (20) and (22)yields a fully convex optimization program which SortingHatcan sub-optimally solve using CCCP (since the rest of theproblem is convex). This procedure is outlined in steps (11)-(16) of SortingHats pseudo code.

V. E VALUATION

To the best of our knowledge, no current approaches solvethe cell-user assignment, power allocation, and channel al-location problem simultaneously. To make fair comparisonswith previous work, we analyze the separate components of


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User Number

D e s

i r e

d R a

t e

( B i t s

/ s / H z

)

Desired RateUnsatisfied Rate after Iteration 1Unsatisfied Rate after Iteration 2

Fig. 2. Top: Cell-User Pairing for (A) Initial allocation to Cell 7 (B) After 1 CCCP Iterations, (C) After 2 CCCP Iterations. The colors denote which celleach user is assigned to and the numbers correspond to the user number. Bottom: Demanded and Slack Rate per User over 2 CCCP Iterations

SortingHat individually against two prior approaches. First, wecompare against an algorithm which performs optimal power

allocation using GP [6][7] but uses a clustering approach toassign users to cells rst [17]. To obviate the need for channelallocation, we utilize the CDMA interference model for thisanalysis. Second, we compare our algorithm against an ap-proach [9] where each cell randomly and without replacementassigns a channel to each paired user. Here, we utilize theOFDMA interference model to analyze the performance of SortingHats channel allocation.

For SortingHat and the compared approaches, the keyperformance metric in all simulations is the slack violationvariable, s , which captures the amount of unsatised ratedemand per user. For each user, it ranges from s j = 0 , i.e.the rate demand is entirely satised, to s j = r desj , i.e. the ratedemand is completely unsatised. The goal, of course, is tohave s j = 0 for all j .

Test Scenario: Weve built a compact simulator to comparethe different approaches. To emulate the unplanned deploy-ment patterns of typical HCNs, the simulator randomly placem cells on a two-dimensional square grid of xed size, D D .There are n users are scattered throughout the grid, each witha unique rate requirement to mimic different end-user applica-tions. The upper plots in Figure (2) show one such realizationwith a D = 100 service area. The numbered, colored starsdenote m = 9 cells. There are 3 tiers of cells: Macro, Micro,and Femto. Macro cells (cell 3,7) have total allocatable power

of 35 units. Micro cells (cell 2,5,6) have total allocatable powerof 25 units and Femto cells (cell 1,4,8,9) can allocate 10units units of power. All cells are capable of serving everysubchannel. The numbered, colored circles denote n = 25users, randomly scattered throughout the service area, andtheir coloring denotes the user-cell pairing. Using the randompositions of the cells and users within the grid, we calculatea corresponding static gain matrix G using basic, line-of-sightRF propagation models. If there are multiple subchannels percell in the simulation, there is no frequency selectivity (i.e.the gain is at across subchannels). The blue circles centeredaround every cell represent the effective service range of each

one (based on their total available power).

Finding a Feasible Initialization: The CCCP algorithm

requires a feasible starting point for iteration 0. SortingHatis initialized by pre-assigning all users to one particular celland allocating its total power equally among all of the users.The other cells powers are initialized to zero.

A. CDMA Interference Model

We begin by analyzing the user-cell pairing and powercontrol component of SortingHat under the CDMA Interfer-ence Model (where there is one, orthogonal channel per user).The top of Figure 2 shows the progressive user-cell pairingswith an initial allocation made to cell 7 over two iterations(Figure2TopA). The bottom of Figure 2 shows the desired

rates per user and the evolution of slack (unsatised rate) peruser over two iterations. After the rst iteration many of theusers have met their rate demands (i.e., have no slack), whilecell-boundary users tend to have signicant amounts of slack.

These cell-boundary users are interesting cases because theirmatchings can dictate the performance of other users. Takeuser 24 for example, who is serviceable by cell 2 and to acertain extent by cell 8. After the rst iteration (Figure 2TopB)it has been assigned to cell 2, but because of the distancebetween them and the heavy load cell 2 is already supporting, 50% of user 24s desired rate has not been satised. Afterthe second iteration, it is interesting to note that user 24 hasswitched to cell 8. This switch improves the performance of

users 14 and 25 because cell 2 has more power after ofoadinguser 24, while user 24s demand is almost entirely satised bycell 8 (Figure 2TopC). Similarly, user 15 also switches to cell8 despite having all of its rate satised in the second iteration.This enables cell 6 to service user 5 and further reduce thetotal slack rate. By folding the user-cell pairing into the convexoptimization constraints, SortingHat is better able to balancethe load against the network constraints and improve overallperformance.1. What effect does the initial starting point have? Sinceany CCCP-based algorithm is not guaranteed to nd the globaloptimum, an important consideration is the effect of the initial


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Fig. 4. The median users using SortingHat have almost 80% of their desiredthroughput satised. The same median users using GP with Cell Clusteringhave less than 45% of their throughput satised.

starting point. To measure the impact of different startingpower allocations, P (0) , on the convergence rate and nalallocation, Figure 3 shows the total slack violation versus theiteration number for ve different initializations. Regardless of starting point, the algorithm converges to the nal allocationwithin 2 to 3 iterations. Interestingly, while some of theseinitial allocations did provide different P nal allocations,they all managed to converge to within approximately 1Bits/s/Hz of the same total slack violation objective.

Compared Approach: Power allocation for a predeterminedset of user-cell pairings can be re-expressed as a GeometricProgram (GP) [6][7] and solved to nd globally optimalsolutions. While the power allocations are globally optimalfor a given set of user-cell pairings, the pairing is xed andthus does not allow load balancing via cell-user re-pairing. Wemake the user-cell pairing by clustering users with the cell thatprovides the maximum link gain G i, j . We run the simulatorten times with randomly generated user and cell locations and

generate allocations with both approaches.Figure 4 plots the CDF of the percentage of desired rate

satised per user, i.e. r j /r desj , with the blue line denotingSortingHats performance and the dotted red line denotingthe GP with cell clustering. As you can see, SortingHatsignicantly outperforms GP with cell clustering because it si-multaneously optimizes user-cell pairing and power allocationto load-balance the network. While SortingHat outperformsthe GP with cell clustering throughout, the disparity is mostobvious in the lower portion of the CDF. With SortingHat,the lower 10% of users have more than 60% of their desiredthroughput satised while with the GP, the same segment of

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Fig. 5. User-Cell pairs found by nearest cell clustering - Femtos 1 and 4are oversubscribed while femtos 8 and 9 are under-subscribed.

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Fig. 6. SortingHats reduced slack violation corresponds to a 32% increase innetwork throughput compared to GP with nearest cell clustering and convergesto the global optimum found by a GP with SortingHats user-cell pairing.

users have less than 25% of their throughput satised.To shed some light on this disparity, Figures 5-7 plot the

results for the GP and cell clustering approach, applied to thesame scenario as shown in Figure 2. In Figure 5, we can seethat assigning users to the nearest cell produces different user-cell pairs than SortingHat did, because name accounts for cellcapabilities in its procedure. The cell clustering over-assignsusers to femto cells 1 and 4, which were subsequently unableto support the rate demands.

2. How far from optimal is SortingHat? The performanceof SortingHat and the GP differs because pairing users to thenearest cell can result in either over provisioning cells, orpairing with suboptimal types (e.g. pairing with low-powerfemtos which are far away). Total slack violation over veiterations of SortingHat is plotted in Figure 6 and showsthe performance gap between SortingHat, a GP with cellclustering, and a GP with the user-cell pairing generated bySortingHat. Interestingly, SortingHat converges to the same

global optimum found by the GP using the user-cell pairingsgenerated by SortingHat. Thus for this scenario, the power-allocation of SortingHat converges to the optimal power allocation for a given set of user-cell pairings.

3. How do backhaul constraints effect SortingHat? Becausenearest cell clustering does not take into account backhaulconstraints, SortingHat outperforms GP with clustering evenmore when backhaul resources are constrained. In Figure 5,the nearest neighbor pairing leaves femto cells 8 and 9 unused,increasing the backhaul and power requirements of cells 2,5,and 6. Because femto cells are not subject to the backhaulconstraints of the network, such an allocation is particularly


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Fig. 7. SortingHat and GP with nearest cell clustering utilize identicalbackhaul bandwidths but SortingHat reduces slack violations, providing 56%more network throughput by intentionally ofoading trafc to femtocells

ineffective when the backhaul is limited. To demonstrate this,Figure 7 shows that when the backhaul resources are nite(represented by the dashed lines), SortingHat utilizes theseresources in a much more efcient manner by ofoading asmuch trafc onto the femto cells as possible. As a result,while the backhaul usage is more or less the same (shown bythe convergence of the 2 dotted lines in Figure 7, SortingHatprovides nearly twice the network throughput that GP can.

B. OFDMA Interference Model - Dynamic Channel Allocation

We now consider an OFDMA model in which simulta-neous transmissions between different users interfere witheach other if they are assigned to the same channel. Underthis interference model, we evaluate SortingHats subchannelallocation component described in IV-A. Because the generalsolution to the channel assignment problem is known to be NP-hard, we compare SortingHat against another heuristic channelallocation algorithm.

Compared Approaches: Given a set of user-cell pairs foundwith either nearest cell clustering (strawman1) or SortingHatsCCCP approach (strawman2), each cell randomly assigns achannel to each of its paired users. The cell uniformly selectsfrom the set of channels without replacement until the set isexhausted, and then repeats this process with the original setuntil all users have been assigned [9]. The total number of channels is the same for SortingHat and both of the strawmanapproaches. Since the strawman algorithms do not considerinter-cell interference or user-cell pairing, we expect that theywill perform poorly in comparison to SortingHat. Again, werun the simulator ten times with randomly generated userand cell locations and generate allocations using all three

approaches.Figure 8 plots the CDF of the percentage of satised

rate per user, with the blue line denoting the performanceof SortingHat, the dotted red line denoting strawman1, andthe dot-dash black line denoting strawman2. As expected,SortingHat signicantly outperforms both approaches. WithSortingHat, the median 50% of users having more than 80%of their desired rate satised, while strawman1 and strawman2satisfy only 65% and 40% of user rate demands, respectively.

Because each approach differs in only a single aspect, wecan determine exactly where SortingHats performance gainscome from. First, strawman1 and strawman2 only differ in

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Fig. 8. SortingHat outperforms strawman1 and 2, with the the medianusers having more than 80% of their desired rate satised. The same mediansegment of users have less than 65% and 40% of their throughput satisedin strawman1 and 2, respectively.

the user-cell pairing method, and as such the performanceadvantage can be attributed to SortingHats iterative pairingwhich considers important factors like the remaining numberof unused channels on each cell. Second, strawman2 andSortingHat differ in only how channels are allocated, thus theperformance gap exists because SortingHat considers the inter-ference and distance between different users in its assignmentprocess, whereas strawman2 does not.

To assess these performance results in more depth, Figure9 shows the channel allocation results for SortingHat, straw-man1, and strawman 2 for a particular instance in which thereare ve channels total. In these gures, the coloring denotesthe user-cell pairing, and the numbers next to each user denotewhich channel it is using. If a user is denoted by an x, thismeans that its chosen cell has too many users and cant servethat demand.

5. Why does SortingHat outperform strawman1? Sort-ingHat outperforms strawman1 because it takes co-channelinterference into account when assigning user-cell pairs and

as such, ensures that no cell is assigned more users than thetotal number of channels available (which in this case is ve).In strawman1 in Figure 9B, we see that cell 7 is heavily over-assigned users and as a result is unable to service two of them. Whereas in SortingHat, cell 7 only uses four of theavailable ve channels. This shows SortingHats ability to takeinto account important factors, such as the remaining numberof unused channels on each cell, when determining user-cellpairs.6. Why does SortingHat outperform strawman2? Sorting-Hat outperforms strawman2 because it tries to minimize theco-channel interference by assigning repeated channels as farapart as possible. In Figure 8, the performance gap between

between SortingHat and strawman2 is much less pronouncedthan the gap between SortingHat and strawman1 because usersare not over assigned to cells. In fact, for low-density usageareas such as the top right corner with cell 2, we found thatSortingHat only slightly outperformed strawman2 because thecells total set of channels is not exhausted. The differencebetween the channel allocations of SortingHat and strawman2is subtle. In particular, notice that the user which is closestto cell 7 is actually assigned to cell 4 on channel 1, andthis channel is the only one that cell 7 does not use. All of cell 7s neighboring cells, however, do use channel 1 to serveusers. Since cell 7s service region overlaps almost every other


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cell, it creates signicant interference on every neighboringcells channels. This shows SortingHats natural generationof smart channel re-use; by minimizing the number of usersassigned to cell 7, SortingHat is able to minimize the amountof interference caused by cell 7 and effectively re-use the rstchannel 3 times (cell 6,4,3 all use channel 1) as opposed to

once with cell 7.7. How do different initial starting points affect perfor-mance? As we saw in the CDMA model, different initialstarting points impact the performance of the CCCP. To mea-sure the impact of the initialization, Figure 10 plots the totalslack violation from different starting points in comparison tostrawman1 and 2. We see that despite the simplicity of ourchannel selection algorithm, SortingHat converges to similarsolutions, all of which outperform strawman1 and 2.

VI . C ONCLUSION AND FUTURE W ORK

As cellular providers transition from single-tier networks

to more heterogeneous deployments, resource allocation willbecome more challenging. In this paper, weve presented Sort-ingHat, a resource allocation algorithm for HCNs, and showhow it can be used to analyze important network performancemetrics such as per user throughput, backhaul constraints,and co-channel interference. The benets of SortingHat wereillustrated through several numerical examples, which showeda 30-100% improvement in median user throughput resultingfrom optimizing user-cell association, power control, andchannel assignment jointly rather than separately.

There are numerous possible extensions to this work, whichinclude nding a systematic way to distribute the algorithm to

each cell in order to implement SortingHat practically. Wealso plan to consider real deployments and investigate howan online version of SortingHat adapts to statistically varyinggains, mobile users, and changing rate demands.

R EFERENCES

[1] V. Chandrasekhar, J. G. Andrews, and A. Gatherer, Femtocell networks:a survey, In IEEE Commun. Mag , 2008.

[2] S. Chia, M. Gasparroni, P. Brick, The Next Challenge for CellularNetworks: Backhaul, In IEEE Microwave Mag. , 2009.

[3] J.K. Rowling, Harry Potter and the Sorcerers Stone, Scholastic Pub-lishing , 1998.

[4] A. Akella, G. Judd, S. Seshan, P. Steenkiste, Self-Management inChaotic Wireless Deployments, In ACM MOBICOM , 2005.

[5] V. Chandrasekhar, J. Andrews, T. Muharemovic, Z. Shen, A. Gatherer,Power Control in Two-Tier Femtocell networks, In IEEE Trans. W.Comm. , 2009.

[6] M. Chiang, C. Tan, D. Palomar, D. ONeill, D. Julian, Power ControlBy Geometric Programming, In IEEE Trans. W. Comm. , 2007.

[7] D. Julian, M. Chiang, D. ONeill, S. Boyd, QOS and Fairness Con-strained Convex Optimization of Resource Allocation for Wireless Cel-lular and Ad Hoc Networks, In IEEE INFOCOM , 2002.

[8] G. Foschini, Z. Miljanic, A simple distributed autonomous power controlalgorithm and its convergence, In IEEE Trans. Veh. Tech. , 1993.

[9] V. Bhandari, N. Vaidya, Capacity of Multi-Channel Wireless Networkswith Random (c,f) Assignment, In ACM MOBIHOC , 2007.

[10] C. Sung, W. Wong, Sequential Packing Algorithm for Channel Assign-ment Under Cochannel and Adjacent Channel Interference Constraint,In IEEE Trans. Veh. Tech. , 1997.

[11] R. Madan, S. Boyd, S. Lall, Fast Algorithms for Resource Allocationin Wireless Cellular Networks, In IEEE Trans. Net. , 2010.

[12] V. Chandrasekhar, J. Andrews, Spectrum Allocation in Tiered CellularNetworks, In IEEE Trans. Comm. , 2009.

[13] K. Sundaresan, S. Rangarajan, Efcient Resource Management inOFDMA Femto Cells, In ACM MOBIHOC , 2009.

[14] C. Ko, H. Wei, On-Demand Resource-Sharing Mechanism Design inTwo-Tier OFDMA Femtocell Networks, In IEEE Trans. on Veh. Tech. ,2011.

[15] K. Ramachandran, E. Belding, K. Almeroth, M. Buddhikot,Interference-Aware Channel Assignment in Multi-Radio WirelessMesh Networks, In IEEE INFOCOM , 2006.

[16] Y. Yuan, P. Bahl, R. Chandra, T. Moscibroda, Y. Wu, AllocatingDynamic Time-Spectrum Blocks for Cognitive Radio Networks, In ACM MOBICOM , 2007.

[17] A. Ganti, T. Klein, M. Haner, Base Station Assignment and PowerControl Algorithms for Data Users in a Wireless Multiaccess Framework,In IEEE Trans. W. Comm. , 2006.

[18] R. Yates, Integrated Power Control and Base Station Assignment, In IEEE Trans. Veh. Tech. , 1995.

[19] J. Lee, R. Mazumdar, N. Shroff, Joint Resource Allocation and Base-Station Assignment for the Downlink in CDMA Networks, In IEEE/ACM Trans. Net. , 2006.

[20] B. Sriperumbudur, G. Lanckriet, On the Convergence of the Concave-Convex Procedure, In Neural Information Processing Systems , 2009.

[21] M. Grant, S. Boyd, CVX: Matlab Software for Disciplined ConvexProgramming, v. 1.21, http://cvxr.com/cvx , 2011.

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