Optimal Resource Allocation in HetNetsSem Borst, Stephen Hanly, Phil Whiting

Alcatel-Lucent Bell Labs, Murray Hill, NJ 07974-0636, USAMacquarie University, North Ryde 2109 NSW, Australia

AbstractThe deployment of pico cells to cover trafc hotspots within the footprint of a macro cell provides a powerfulapproach to meet the massive growth in trafc demands fueledby smartphones and bandwidth-hungry applications. Joint opti-mization of resource allocation and user association is of criticalimportance to achieve the maximum capacity benets in suchheterogeneous network deployments (HetNets). We rst examinethe problem of minimizing the amount of resources required tosatisfy given trafc demands. We characterize the structure ofthe optimal solution, and identify a simple optimality condition interms of the physical transmission rates of the edge users betweenthe macro cell and the various pico cells. We further demonstratehow these structural properties can be leveraged in designinga distributed online algorithm for achieving a max-min fairthroughput allocation across all users. Numerical experimentsare presented to illustrate the results.

I. INTRODUCTION

Wireless cellular networks have experienced immensegrowth in trafc loads over the last few years fueled bythe rapid proliferation of smartphones and bandwidth-hungryapplications. With forecasts of a booming growth in videostreaming, the sharp rise in trafc volumes is likely to con-tinue, and put even greater strain on the capacity of cellularnetworks in the near future.

A powerful approach to expand the capacity of cellularnetworks and support further growth within the connementsof existing spectrum is to deploy pico cells so as to coverareas with high trafc density (hot spots) within the footprintsof macro cells. In LTE the deployment of such HetNetscan be enabled via eICIC mechanisms where a macro celldoes not transmit during so-called Almost Blank Subframes(ABS) so as to mitigate the interference experienced by thepico cells. In addition, so-called cell range extension canbe induced through bias values in the hand-off procedures,enticing users with weaker signal strengths into the pico cellsand facilitating greater ofoad from the macro cell [1], [8].Since the allocation of resources and association of users areevidently related, it is of critical importance that the ABSsettings and bias values are jointly optimized so as to achievethe maximum capacity benets.

Cell selection and resource allocation in HetNets are rela-tively new topics [2], [5]. Bias settings have been investigatedvia numerical experiments (e.g. [6]), but to our knowledgeno analytical approach has been developed. References [4],[7], [9] do study optimization approaches to cell selectionand interference coordination, but the models and techniquesare quite different from ours. A method for optimal resourceallocation based on Gibbs sampling is proposed in [3].

In order to gain insight in the joint optimization of resourceallocation (e.g. ABS settings) and user association (e.g. biasvalues), we focus in the present paper on a relatively simplescenario of a single macro cell with several pico cells within itscoverage area. We rst examine the problem of minimizing theamount of resources required to satisfy given trafc demands,which is key in optimizing the systems efciency and load-carrying capacity. The optimization involves determining howto share the resources between the macro cell and the variouspico cells and deciding for each of the users whether it isto be served by the macro or a pico cell. We describe thestructural properties of the optimal solution, and in particularidentify a simple optimality condition in terms of the physicaltransmission rates of the edge users between the macro cell andthe various pico cells. We also demonstrate how the optimalitycondition can be exploited in designing a distributed onlinealgorithm for obtaining a max-min fair throughput allocationacross all users. Numerical experiments are conducted forsome simple scenarios in order to illustrate the results.

II. MODEL DESCRIPTION

We consider the scenario of a single macro cell withL pico cells within its footprint (coverage area). The availabletransmission resources (e.g. frequency slices, time slots, ortime-frequency slots) are assumed to be innitely divisibleand can be allocated either for exclusive use in the macrocell or for concurrent use in the collective pico cells. Thereare some users that can only be served by the macro cell,while the rest of the users can either be assigned to the macrocell or to a (unique) pico cell through suitable adjustment ofits size (e.g. via hand-off thresholds such as bias values). Weassume that the sizes of the various pico cells can be controlledindividually, and that each pico cell is subject to a certainmaximum size. We further suppose that no interference ordirect interaction occurs among the various pico cells.

To make things concrete, we assume that the resources aretime allocations of arbitrary size (in seconds). Pico cell l,when expanded to its maximum size, has Nl users in it,and the rates of these users (in bits per second) when fullyserved by pico cell l are denoted by Rl,1, Rl,2, . . . , Rl,Nl .Alternatively, these users can be served by the macro cell,at rates Sl,1, Sl,2, . . . , Sl,Nl . It is possible for a user to getallocated part of the time to a pico cell and part of the time tothe macro cell. We dene l,n = Rl,n/Sl,n as the rate ratio ofuser n between pico cell l and the macro cell, and label theusers such that l,1 l,2 l,Nl . In addition, thereare N0 macro-cell-only users, with rates S0,1, S0,2, . . . , S0,N0 .

978-1-4673-3122-7/13/$31.00 2013 IEEE

IEEE ICC 2013 - Wireless Communications Symposium

5437

III. TRAFFIC LOAD MINIMIZATION

In this section we focus on the problem of minimizing thetotal amount of time required to satisfy given trafc demands.We assume that user n in pico cell l has a trafc demand ofDl,n bits, and wish to nd the time allocations to each of theusers (and in particular the association between macro and picocell) that minimize the total amount of time required. Notethat the macro-cell-only users are irrelevant to the problem,and hence we do not further consider them in this section.

Let the time allocated to the pico cells (which can servetheir users concurrently) be denoted by f seconds. Also, letxl,n and yl,n represent the amount of data (in bits) received byuser n Cl = {1, . . . , Nl} in pico cell l from that pico cell andthe macro cell, respectively. The above optimization problemmay then be formulated as the following linear program:

min f +L

l=1

nCl

yl,nSl,n

(1)

subnCl

xl,nRl,n

f l (2)

xl,n + yl,n Dl,n l, n Cl (3)f 0, xl,n 0, yl,n 0 l, n Cl (4)

While the above linear program could be solved by the sim-plex algorithm (or interior-point methods), we will show thatthe optimal solution has a specic structure, which allows it tobe obtained by lower-dimensional, or even one-dimensional,search procedures, which moreover lend themselves to dis-tributed implementation. Note that the linear program itself has1 + 2

Ll=1 Nl variables, which can be a signicant number,

particularly if there are several pico cells, and each can expandto accommodate a large number of users.

In order to examine the structure of the optimal solution of(1)-(4), dene gl(f) as the amount of time required by themacro cell to serve the residual trafc from pico cell l whenthe amount of time available to that pico cell itself is f . Thengl(f) may be determined as the optimal value of the followinglinear program (for given f ):

minnCl

yl,nSl,n

(5)

subnCl

xl,nRl,n

f (6)

xl,n + yl,n Dl,n n Cl (7)xl,n 0, yl,n 0 n Cl (8)

We now introduce some notation in order to describe theoptimal solution of (5)(7). For any nl [0, Nl], dene ml =nl, pl = mlnl, and Fl(nl) =

ml1n=1

Dl,nRl,n

+(1pl)Dl,mlRl,ml ,which may be interpreted as the amount of time required bypico cell l to serve the (fractional) nl users with the highestrate ratios. For conciseness, nl will be referred to as the sizeof pico cell l. For a given cell size nl, dene the associatedthroughput allocation xl,n(nl), yl,n(nl) by:(i) xl,n(nl) = Dl,n and yl,n(nl) = 0 for n = 1, . . . ,ml 1,

(ii) xl,n(nl) = 0 and yl,n(nl) = Dl,n for n = ml +1, . . . , Nl,(iii) xl,ml(nl) = (1 pl)Dl,ml , and yl,ml(nl) = plDl,ml , sothat xl,ml(nl) + yl,ml(nl) = Dl,ml .Also, dene nl(f) = max{nl : Fl(nl) f} as the maximumcell size that pico cell l can support when the available amountof time is f , ml(f) = nl(f), and pl(f) = ml(f) nl(f).

The next lemma provides the optimal solution to the linearprogram (5)(8), and in particular demonstrates that the opti-mal association of users between the macro and the pico cell isdictated by the ratios l,n, which will be referred to as the rateratio rule. When the macro cell rates Sl,n are approximatelyequal for all n Cl, which is a reasonable assumption whenthe pico cell is not too close to the macro base station, thisroughly corresponds to association of users based on the picocell rates Rl,n, and can be implemented via bias values.

Lemma 1: (Rate ratio rule) The throughput allocationxl,n(nl(f)), yl,n(nl(f)) associated with cell size nl(f) is anoptimal solution to the linear program (5)(8).

The proof follows by considering the dual problem, deningdual variables l = l,ml(f) and zl,n = min{ lRl,n , 1Sl,n } cor-responding to the primal constraints (6) and (7), respectively.

Corollary 1: If xl,n, yl,n is an optimal solution to the linearprogram (5)(7) as specied above, then one of the followingthree scenarios must hold:

1) all users are entirely served by the macro cell:

xl =nCl

xl,n = 0, yl,n = Dl,n n Cl

2) all users are entirely served by the pico cell:

yl =nCl

yl,n = 0, xl,n = Dl,n n Cl

3) some users are served by the pico cell (xl > 0), whileothers are served by the macro cell (yl > 0), and thereexist at most one pico-cell-edge user ml Cl withxl,ml + yl,ml = Dl,ml such that:

xl,n = Dl,n, yl,n = 0 n = 1, . . . ,ml 1xl,n = 0, yl,n = Dl,n n = ml + 1, . . . , Nl

The corollary follows from Lemma 1 by distinguishing threecases: f = 0 (i.e. nl(f) = 0), f Fl(Nl) (i.e. nl(f) = Nl),or 0 < f < Fl(Nl) (i.e. 0 < nl(f) < Nl).

The statements of Lemma 1 and Corollary 1 hold for anyvalue of f , and in particular for the optimal value f ofthe linear program (1)-(4). Thus there also exists an optimalsolution xl,n, y

l,n of that linear program with the same stated

properties. Now that an optimal throughput allocation xl,n,yl,n has been obtained in terms of f , we can make two keyfurther observations:1) It only remains to determine the optimal value of f , i.e.,the value of f that minimizes g(f) = f +

Ll=1 gl(f);

2) The function gl(f) can be represented as

gl(f) = pl(f)Dl,ml(f)

Sl,ml(f)+

Nln=ml(f)+1

Dl,nSl,n

.

5438

Note that gl(f) is decreasing and piecewise linear with breakpoints bl,m = Fl(m) =

mn=1

Dl,nRl,n

, m = 0, 1, . . . , Nl, andnegative slope l,m in the interval (bl,m1, bl,m), so gl(f) isconvex since l,1 l,2 l,Nl . Further observe thatthe left and right derivatives of gl(f) may be expressed asl,nl(f) and l,nl(f+), respectively, with n

l (f) = nl(f),

n+l (f) = nl(f) + 1 for f Fl(Nl), nl (f) = n+l (f) =Nl + 1 for f > Fl(Nl), and the convention l,Nl+1 = 0.Indeed, nl (f) = n

+l (f) = m in the interval (bl,m1, bl,m),

nl (f) = m = n+l (f)1 at the break point bl,m, and nl (f) =

n+l (f) = Nl + 1 when f > bl,Nl .Since each of the functions gl(f) is piecewise linear and

convex, so is g(f), with 1+L

l=1 Nl break points bl,m and leftand right derivatives 1Ll=1 l,nl (f) and 1Ll=1 l,n+l (f),respectively. Hence the minimum of g(f) is attained either atf = 0 when

Ll=1 l,1 < 1 or at a point f

> 0 wherethe derivative of g(f) changes sign, yielding the optimalitycriterion stated in the next lemma, which will be referred toas the edge rate condition.

Lemma 2: (Edge rate condition) The time allocation f isoptimal if and only if

Ll=1

l,nl (f) 1

Ll=1

l,n+l (f)(9)

with the convention that l,0 = 1.The optimal allocation f may thus be found by a one-

dimensional search, scanning the 1 +L

l=1 Nl break pointsbl,m and checking the edge rate condition (9).

It further follows that there always exists an optimal allo-cation f such that at least one of the pico cells has no splituser (allocated time in both the macro and pico cell). Thismay also be deduced from the original linear program (1)-(4),yielding an upper bound for the number of non-basic variablesin terms of the number of binding constraints.

The above-described one-dimensional search can either beperformed as a centralized operation, by gathering the trafcdemands Dl,n and rates Rl,n and Sl,n of all the users, orimplemented in a distributed fashion by having the picocells compute the gl(f) functions and passing these to themacro cell. In either case however, the search entails anofine computation, and hence we now proceed to describean iterative (L + 1)-dimensional method which lends itself toan online implementation.

For given cell sizes n1, . . . , nl, dene Fmax(n) =maxl=1,...,L Fl(nl), representing the maximum amount of timerequired by any of the pico cells, and the set B(n) ={l : Fl(nl) = Fmax(n)} containing the pico cells with themaximum time requirement. The method searches for thevalues of the nl(f)s in an iterative way, and exploits the factthat the edge rate condition in a certain sense is also sufcientfor optimality, as stated in the next lemma.

Lemma 3: If nl = Nl for all l B(n) and(B(n)) =

lB(n)

l,nl 1 +(B(n)) =

lB(n)

l,n+l,

(10)

with nl = nl and n+l = nl+ 1, then the associated timeallocation f = Fmax(n) is optimal.

The proof follows by noting that f > Fl(Nl), thus nl (f) =n+l (f) = Nl +1 and l,nl (f) = l,n+l (f) = 0 for all l B(n),so that the edge rate conditions (9) and (10) coincide.

We now describe an iterative scheme for nding cell sizesn1, . . . , nL that satisfy the optimality conditions stated in theabove lemma. For given cell sizes n1, . . . , nL, calculate thetime requirements Fl(nl), l = 1, . . . , L, and then determineFmax(n) = maxl=1,...,L Fl(nl) and B(n) = {l : Fl(nl) =Fmax(n)} as above.(1) For every pico cell l B(n) with nl < Nl, increment thesize nl by a small amount in order to expand.(2a) Furthermore, if (B(n)) < 1, then decrement the size nlof each pico cell l B(n) by a small amount l = Rnl inorder to shrink.(2b) On the other hand, if +(B(n)) > 1, then increment thesize nl of each pico cell l B(n) with nl < Nl by a smallamount l = Rn+l in order to expand.Choosing the changes in the pico cell sizes proportional to theedge rates ensures that the time requirements of the pico cellsin B(n) all remain equal.

It may be shown that for suitable step sizes , the cellsizes converge to values that satisfy the conditions stated inLemma 3, and are hence optimal.

IV. MAX-MIN FAIRNESS

In the previous section we focused on the problem ofminimizing the total amount of time required to satisfy giventrafc demands. A closely related problem is to maximize thethroughput utility of the various users for a given amount oftime. Specically, interpreting the optimal throughput valuesas trafc demands, we see that any Pareto-optimal throughputallocation must satisfy the structural properties established inthe previous section. Since the optimal throughput values arenot known a priori, this does not generally provide an actualway of solving the utility maximization problem. However,in the particular case of maximizing the minimum weightedthroughput across all users, we do know beforehand that thethroughput values may be set proportional to the target ratios.In that case, these targets may be taken as trafc demands, andextensions of the search algorithms described in the previoussection can be applied as we will show.

Denoting by wl,n the relative throughput target of user n inpico cell l, and adopting the convention x0,n 0, the problemmay be formulated as the following linear program:

max minl=0,...,L,n=1,...,Nl

(xl,n + yl,n)/wl,n

subNl

n=1

xl,nRl,n

+L

l=0

Nln=1

yl,nSl,n

1 l = 1, . . . , L,

5439

or equivalently,

max z

subNl

n=1

xl,nRl,n

+L

l=0

Nln=1

yl,nSl,n

1 l = 1, . . . , L,

z (xl,n + yl,n)/wl,n l = 0, . . . , L, n Cl,For given cell sizes nl [0, Nl], dene ml = nl, pl =

ml nl,

Fl(nl) =ml1m=1

wl,mRl,m

+ (1 pl)wl,mlRl,ml

,

and

F0(n) =N0n=1

w0,nS0,n

+L

l=1

(pl

wl,mlSl,ml

+Nl

m=ml+1

wl,mSl,m

),

with n (n1, . . . , nL). The value of Fl(nl) represents theamount of time required by pico cell l to provide wl,m bits touser m, m = 1, . . . ,ml1, and (1pl)wl,ml bits to user ml.Note that the value of Hl(nl) = 1/Fl(nl) is the weightedharmonic mean of the pico cell rates of the users in pico cell lwhen the size is nl, and may be interpreted as the weightedmax-min throughput in pico cell l per unit of time. The valuesof F0(n) and H0(n) = 1/F0(n) have similar interpretations.Also, dene Fmax(n) and B(n) as before. Note that the cellsin B(n) have the maximum time requirement for the giventarget throughputs, and thus determine the minimum weightedmax-min fair throughput.

The next lemma provides a sufcient condition for a set ofcell sizes (n1, . . . , nL) to produce a weighted max-min fairallocation. A similar iterative algorithm as described in theprevious section can be adopted for obtaining cell sizes thatsatisfy this optimality condition.

Lemma 4: If the cell sizes n1, . . . , nL satisfy

(B(n)) =

lB(n)l,nl

1 +(B(n)) =

lB(n)l,n+l

,

with nl = nl and n+l = nl+1, then the throughput allo-cation fxl,n(nl), (1f)yl,n(nl), with f = F

max(n)F0(n)+Fmax(n)

is weighted max-min fair.The proof follows by considering the subnetwork consisting

of the macro cell and the pico cells in B(n). Suppose we wishto provide target throughput values wl,m to each of the users inthis subnetwork. Viewing the wl,m values as trafc demands,Lemma 3 implies that F0(n)+Fmax(n) is the minimum timerequirement to meet these targets.

Now observe that the throughput allocation fxl,n(nl),(1 f)yl,n(nl) yields a minimum weighted throughput off/Fmax(n) = (1 f)/F0(n) = 1/(F0(n) + Fmax(n)),and suppose that this is not weighted max-min fair. Thenthere must exist a different allocation, which only uses aunit total amount of time, and provides a throughput wl,mto each of the users in the original network, with >1/(F0(n)+Fmax(n)). Reducing all the allocations by a factor yields an allocation which provides a throughput wl,m to

Fig. 1. f + g1(f) + g2(f) as a function of f for N = 30 per pico cell

each of the users, and only requires a total amount of time1/ < F0(n)+Fmax(n). Since the latter allocation is feasiblein the subnetwork as well, this contradicts the earlier statementthat the minimum time requirement is F0(n) + Fmax(n).

V. NUMERICAL RESULTS

We now present numerical results for some simple illustra-tive scenarios. We rst examine the problem of determiningthe minimum toal amount of time required to satisfy giventrafc demands, and then turn to the problem of nding themax-min fair throughput allocation.

We start with a scenario with L = 2 pico cells, each withN = 30 users, placed inside a macro cell at random positionswithin an annulus with inner radius 100 m and outer radius500 m. The path loss from the pico cell is 0 dB at 100 m andthe path loss exponent is 4. The users in pico cell l have acommon macro cell rate Sl, which is determined by the pathloss. The trafc demand is 100 Kbps per user.

The plot in Figure 1 shows the total amount of time f +g1(f)+ g2(f) required to satisfy the given trafc demands asa function of the amount of time f allocated to the pico cells.Note that f + g1(f) + g2(f) is a convex function as observedbefore, and that the minimum is achieved before either picocell is exhausted, i.e., it has been allocated a sufcient amountof time to carry its full trafc demand. (Exhaustion for one ofthe pico cells occurs around f 0.92.) The rate ratios aroundthe optimum are found to be

R1(f)S1

= 0.4041,R2(f)

S2= 0.5986,

so the sum is roughly 1 (more precisely 1.0027), which isin close agreement with the edge rate condition (9). In afurther experiment the number of users in each pico cellwas reduced to N = 5 in order to bring out the effect ofdiscreteness more clearly. As the graph in Figure 2 indicates,the function f + g1(f)+ g2(f) is indeed piece-wise linear, asobserved before. The minimum is therefore attained where thedirectional derivative changes sign, as it does around f 0.55.The rate ratios around the optimum are found to be

R1(f)S1

= 0.4221,R2(f)

S2= 0.6057,

yielding a slightly negative overall derivative of 0.0278. Atthe optimum and just above, the rate ratio for pico cell 1drops to R1(f)/S1 = 0.3937, and the overall derivative turnsslightly positive, 0.0006.

5440