H-Infinity control based scheduler for the deployment of small cell networks

Performance Evaluation 70 (2013) 513–527

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Performance Evaluation

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H-Infinity control based scheduler for the deployment ofsmall cell networksSubhash Lakshminarayana a,b,∗, Mohamad Assaad a, Merouane Debbah b

a Department of Telecommunications, SUPELEC, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, Franceb Alcatel-Lucent Chair on Flexible Radio, SUPELEC, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France

a r t i c l e i n f o

Article history:Available online 7 December 2012

Keywords:Small cell networksRandommatrix theorySchedulerBeamformingH∞ control

a b s t r a c t

In this work, we address the joint problem of traffic scheduling and interferencemanagement related to the deployment of Small Cell Networks (SCNs). The base stationsof the SCNs (which we will refer to as Micro Base Stations, MBSs) are low power deviceswith limited buffer size. They are connected to a Central Scheduler (CS) with limitedcapacity backhaul links. In this scenario, traffic flow has to be from the network to theMBS queues in such a way that the queue-length at MBS remains as close as possibleto a given target queue-length. The challenge is to design a scheduler which is obliviousto the wireless link between the MBSs and the User Terminals (UTs). For the trafficarriving at the MBS, we need to efficiently transmit it over the wireless channel to theUTs in an interference limited environment. Additionally, real time centralized interferencemanagement techniqueswill not be feasible. In this paper,wedecouple the joint schedulingand interference management into two separate parts. For the scheduling problem, wepropose aH∞ control based schedulerwhich regulates the arrival rates to the queues at theMBS. For the problem of interference management over the wireless channel, we proposea multi-cell beamforming technique and formulate a decentralized algorithm using toolsfrom the field of randommatrix theory. The beamforming vectors are designed to optimizetwo performance metrics of interest namely downlink power minimization and weightedsum rate maximization. Our simulation results show that the H∞ based queue lengthcontrol algorithm stabilizes the queue-lengths at the MBS and keeps the variation of thequeue-length around the target to a minimum.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

According to the observation made by Martin Cooper of Arraycomm, during the last century, the capacity of wirelessnetworks has doubled every 30 months. It has been observed that the biggest gains in the efficiency of spectrum utilizationhas been due to network densification, i.e., shrinking of cell sizes [1]. Hence the next paradigm shift in the field of cellularnetworks has been identified as shrinkage of cell sizes resulting in Small Cells Networks, (SCN) [2]. The reduction in the cellsizes implies reducing the distance between the transmitter and the receiver, creating the dual benefit of higher link qualitiesand more spatial reuse. It also implies reduction in the amount of power transmitted. In this document, we will refer to theSCN base stations as Micro Base Stations (MBSs). Typically the MBSs of the small cells are of the form of data access pointswith limited battery and buffer sizes installed by the users to provide better data and voice coverage over a small area.

∗ Corresponding author at: Department of Telecommunications, SUPELEC, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France.E-mail addresses: [email protected], [email protected] (S. Lakshminarayana), [email protected] (M. Assaad),

[email protected] (M. Debbah).

0166-5316/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.peva.2012.10.004

http://dx.doi.org/10.1016/j.peva.2012.10.004

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514 S. Lakshminarayana et al. / Performance Evaluation 70 (2013) 513–527

Fig. 1. System model.

However, the reduction of cell sizes imposes numerous technical challenges in the design of such SCNs. Some of thetechnical issues faced in the implementation of SCNs are the problem of interference management, traffic scheduling,optimal cell size planning, call handover issues, security, backhaul infrastructure etc. For a detailed survey on the SCNdeployment and design challenges in SCNs, please refer to [2].

In this work, we address two of the issues associated with the deployment SCNs namely interference management andtraffic flow control.

Consider the scenario in Fig. 1. The set up consists of small cells with MBS serving the UTs over the wireless channel.The MBSs are equipped with multiple antennas and the UTs have a single antenna each. Inter-cell interference is a majorbottleneck in achieving good data rates for theUTs, specially theUTs at the cell edge. Hence theMBSs perform jointmulti-cellbeamforming in order to manage interference. However, the MBSs must perform beamforming in a decentralized manner.To this end, we use joint multi-cell beamforming technique developed in [3] using tools from randommatrix theory [4]. Weconsider two performance metrics of interest namely downlink power minimization and weighted sum rate maximization.

The MBSs are in turn connected to a Central Station (CS) which forwards the traffic arriving from the backbone networkto the MBSs through wired backhaul link. We assume that the CS is an infinite reservoir of packets. Since the buffer sizesat the MBS is limited, we need an efficient flow controller which regulates the traffic from the CS to the MBS in such a waythat the queue-length at the MBS remains as close as possible to a given target queue-length (usually the capacity of thebuffer at theMBS). Themain challenge in the design of the flow controller is that it must regulate the traffic flowwhile beingoblivious to the wireless channel conditions between the MBS and the UTs. This is a very practical assumption, since the CSis typically connected to a large number of MBSs and hence the cannot keep track of the channel conditions of all the MBSs.

Wemodel the queue-length evolution as a linear dynamic systemanddevelop a robust queue-length controller/regulatorbased onH∞ control [5]. TheH∞ type robust controller controls the trajectory of a linear dynamical system under unknowndisturbances (noise) without the knowledge of the statistical distribution of the noise process. The advantage of such acontrol algorithm in our setting is that the MBS do not need to feedback the CSI of the UTs to the CS. The task of the H∞

controller is to specify an instantaneous arrival rate every time slot (from the CS to the queues present at theMBS) such thatit minimizes the variation of queue length size around the target queue length.

1.1. Comparison with cross-layer optimization in wireless networks

The problem handled in this paper is multi-disciplinary and spans the fields scheduling and power allocation in wirelessnetworks and decentralized interference management techniques in the context of cellular networks. The problem set updescribed abovehas a flavor of cross-layer design to it. Cross-layer optimizationhas been awell studied topic in the context ofsingle-hop andmulti-hopwireless networks [6–8]. In theseworks, the problem framework naturally leads to decompositionof the problem into two parts. A fairness based flow controller to regulate the traffic flow into the queues and a max-weightbased scheduler to select a set of flows whose packets can be scheduled for transmission at each time slot.

Our work is loosely connected to the cross layer framework of the above works. In our work, the scheduling aspect ofcross-layer design is replaced by beamforming on the wireless channel. Equipped with multiple antennas, the MBSs obtainspatial degrees of freedom for interference management. TheMBSs can simultaneously serve multiple UTs by beamformingon the wireless channel. The flow controller part is replaced by an H∞ based flow controller. However, unlike the fairnessbased congestion controllers in previous works, the objective of the H∞ based flow controller is to minimize the variationof the queue-length around a given target queue-length. The application of H∞ control for queue-length stabilization in thecontext of wireless network has been used in [9,10]. In particular, [10] addresses a dynamic resource allocation problem inmulti-service downlink OFDMA systems. However, to the authors’ knowledge, the joint problem of scheduling using H∞

control theory, power allocation and beamforming has not been handled before in the context of cellular networks.Throughout this work, we use boldface lowercase and uppercase letters to designate column vectors and matrices,

respectively. For a matrix X,Xi,j denotes the (i, j) entry of X.XT and XH denote the transpose and complex conjugate

S. Lakshminarayana et al. / Performance Evaluation 70 (2013) 513–527 515

transpose of matrix X. We denote an identity matrix of sizeM as IM and diag(x1, . . . , xM) is a diagonal matrix of sizeM withthe elements xi on its main diagonal. We use x ∼ CN (m,R) to state that the vector x has a complex Gaussian distributionwith mean m and covariance matrix R. We will use the notation (x)+ = max(x, 0). We will also use the notation ∥x∥2

Wto denote the weighted second order norm of the vector x given by xHWx. We use E [Y ] to denote the expected value of arandom variable Y .

The rest of the paper is organized as follows. In Section 2, we provide the system model and introduce the notations.We introduce the problem formulation in two scenarios namely downlink power minimization and weighted sum ratemaximization. In Section 3, we provide the algorithm description for the decentralized beamforming design in the twoscenarios described above. In Section 4, we provide the details of the flow control algorithm based on H∞ control. InSection 5, we compare the performance of the H∞ against standard LQG based flow control. The simulation results areprovided in Section 6.We also provide a brief description of the details ofH∞ based control and LQG control in Appendices Aand B respectively.

2. Systemmodel

The systemmodel is shown in Fig. 1. It can broken down into two parts. The frontend consisting of the MBSs and the UTsand the backend consisting of the CS and the backhaul links.

We focus on the frontend first. We consider a SCN scenario consisting of N cells and K UTs per cell where each BS isequippedwithNt antennas and eachUThas a single antenna. Each BS serves only theUTs in its cell. The notationUTi,j denotesthe j-th UT present in the i-th cell. The MBS of each cell serves only the UTs present only in its cell. Let ht

i,j,k ∈ CNt denotethe channel from the MBS i to the UTk,j during the coherence interval t . We assume that the elements of the channel vector

hti,j,k =

(σ 2

i,j,k/Nt)xti,j,k, where xti,j,k are i.i.d. complex random vectors whose elements have zero mean and unit variance.

The parameter σ 2i,j,k depends upon the path loss model between BS i and UTj,k. The channel variance has been scaled by the

factor Nt to maintain the per antenna power constraint at each base station. Additionally, we make the assumption that therandom vector xti,j,k has bounded L2 norm.1 We denote the coherence interval of the channel by the notation Tc .

We consider that a discrete time system in which the time slots are indexed by t . Before proceeding further, we willclarify the notion of time slot t in our context. We assume that arrivals and departures from the queues present at the MBShappen once every coherence interval of the channel. Hence the notation t indexes the coherence periods of the channel.

The backend part consists of the CS and the backhaul links connecting the CS to the MBS. The task of the CS is to schedulethe traffic arriving from the underlying wired backbone into the queues present at the MBS. We assume that the backhaullinks have just enough capacity to deliver all the traffic scheduled by the CS during a given coherence.

Let us denote the queue length at MBS i for the UTi,j in its cell during the time slot t by the notation qti,j.We denote thetarget buffer length at the MBS by the notation qi,j. We denote the number of packets arriving into the queue of the MBS ifor UTi,j during the time slot t by the notation ati,j. We also denote the number of packets which are transmitted out of thequeue (i, j) into the wireless link, (i.e. the service rate) during the time slot t by the notation µt

i,j.

We now describe the design parameters of the problem. Let wti,j ∈ CNt denote the transmit downlink beamforming

vector for the UTi,j during time slot t . Likewise, let Γ ti,j denote the received SINR for UTi,j and γ t

i,j the corresponding targetSINR during the time slot t. The received signal yti,j ∈ C for the UTi,j during the time slot t is given by

yti,j =

Kl=1

ht Hi,i,jw

ti,lx

ti,l +

Nm=1m=i

Kn=1

ht Hm,i,jw

tm,nx

tm,n + zti,j

where xti,j ∈ C represents the information signal for theUTi,j during the time slot t and zti,j ∼ CN (0, σ 2) is the correspondingadditive white Gaussian complex noise.

We now provide two different problem formulations depending on the metric of interest namely the downlink powerminimization and the weighted sum rate maximization.Downlink power minimization— PMIN: The downlink power minimization and scheduler design problem can be cast into thefollowing optimization problem given by

PMIN : minwt

i,j

i,j

wt Hi,jwt

i,j, ∀t = 1 . . . T (1)

s.t. Γ ti,j ≥ γ t

i,j, t = 1 . . . T , i = 1 . . .N, j = 1 . . . K

limT→∞

1T

Tt=1

qti,j = qi,j.

1 The bounded L2 norm is an important assumption for the modeling of the H∞ controller which will be explained later.


The objective function of the optimization problem in (1) attempts tominimize the total power transmitted in the downlinkduring each time slot t. The first constraint equation is the target SINR constraint for each UT. The second constraint equationdenotes the scheduler design constraint which tries tomaintain the long term average of the queue-lengths (acrossmultiplecoherence intervals) as close as possible to the target queue-lengths. The received SINR in the downlink is given by thefollowing expression

Γ ti,j =

|wt Hi,jht

i,i,j|2

l=j|wt H

i,lhti,i,j|

2 +

m=i,n|wt H

m,nhtm,i,j|

2 + σ 2. (2)

The numerator term is the useful signal. The denominator terms represent the intra-cell interference, inter-cell interferenceand the thermal noise (in order as they appear in the denominator).Weighted sum rate maximization — RMAX: We now consider the problem of maximizing the weighted sum rate subject tomax-power constraint on each MBS. The optimization problem is given by

RMAX : maxwt

i,j

i,j

βi,j log(1 + Γ ti,j), ∀t = 1 . . . T (3)

s.t.

j

wtHi,j w

ti,j ≤ Pmax, t = 1 . . . T , i = 1 . . .N

limT→∞

1T

Tt=1

qti,j = qi,j

where the downlink SINR Γ ti,j is given as in Eq. (2). Here, βi,j are scalar weights for the downlink rate of UTi,j and Pmax is the

max-power constraint per MBS.The target queue-lengths in this work can be viewed as a way to model the fairness criteria between different UTs. The

target queue-lengths are a representative of the average number of packets served to each UT. The target queue-lengths canbe different for different UTs depending on the fairness objective.

3. Algorithm design

The basic optimization problem in (1) and (3) are difficult to solve due to the interdependencies of the various parametersinvolved. Hence, in this work we decouple the problem into two stages. The first stage is the traffic flow control problem forthe queues at the MBS. The design objective of the flow controller is to minimize the variance of the queue-length around agiven target queue-length. The second stage is the interference management problem.

The parameters connecting the two problems are the SINR and the service rates of the queues. The relationship betweenthe service rate of the queues at the MBS, µ and the SINR, γ , is given by the Shannon–Hartley theorem2

µ = B log(1 + γ ) (4)

where B is the total number of channel uses available during the coherence interval Tc . Accordingly, we denote the servicerate corresponding to the target SINR (γi,j) for UTi,j by µi,j and the service rate corresponding to the achieved SINR in thedownlink (Γ t

i,j) during each time slot t by µti,j. We will refer to µi,j as the target service rate for the queues.

The decoupled algorithm can be summarized as follows:

• The UTs request a target SINR during each time slot depending on their requirement.• The beamformer design algorithm computes the optimal design parameters to serve the data to the UTs.• During each time slot, the traffic flow controller regulates the amount of traffic arriving into the queues of the MBS such

that the queues are stable.

In the subsequent part of this section, we will describe the decentralized multi-cell beamforming algorithm indetail. We will defer the flow regulation part to Section 4. The problem of decentralized multicell beamforming hasreceived considerable attention in recent years [11–14]. [11] suggests beamforming design based on maximizing thevirtual signal to interference and noise ratio (VSINR) metric which is shown to attain optimal rate points. [12] proposesa distributed beamforming approach based on Kalman smoothing involving message passing between the BSs. [13,14]propose decentralized beamforming technique based on localizedmessage passing, hence eliminating the need for a centralprocessor. The decentralized algorithm proposed in this work is based on random matrix theory based approximations. Inour algorithm, the MBSs would need to exchange only the channel statistics between themselves and not the instantaneousCSI, hence reducing the information exchange. Such algorithms are asymptotically optimal (when the system dimensionsbecome very large, to be explained later in this section) and provide good approximations for practical network dimensions.We now describe the algorithm in detail.

2 We assume that the coherence interval of the channel is long enough for the transmitter to achieve the channel capacity.


3.1. Beamformer design and power allocation: the downlink power minimization case

We first provide the distributed algorithm to compute the optimal beamforming vectors and downlink powerminimization problem in (1). We take up the beamformer design problem during a given time slot t . Hence we drop thesuperscript t in subsequent equations related to the above problem. The optimization problem for the beamforming designis given by

minwi,j

i,j

wHi,jwi,j (5)

s.t. Γi,j ≥ γi,j, i = 1 . . .N, j = 1 . . . K .

The beamformer design problem is inspired from our previous work in [3].The downlink powerminimization problem in Eq. (5) ismore difficult to solve since the computation of the beamforming

vector of a given UT in turn depends on the beamforming vectors of the other UTs as well. Hence, using an approach similarto the one in [15], we convert this problem into a dual uplink problem. The uplink problem is much easier to be solved. Thedual problem in the uplink is given as

mini,j

λi,jσ2 (6)

s.t. Λi,j ≥ γi,j, i = 1, . . . ,N, j = 1, . . . , K

where the uplink SINR,Λi,j is given by

Λi,j =λi,j|wH

i,jhi,i,j|2

(m,l)=(i,j)λm,l|wH

i,jhi,m,l|2 + αi∥wi,j∥

22

where w denotes the corresponding uplink beamforming vector and λi,j represents the dual variable associated with theoptimization problem in (5). The λi,j can be viewed as the dual uplink power.

Before stating the optimal beamforming algorithm, we define the following matrices. Let Hi = [hi,1,1hi,1,2 · · ·hi,m,n · · ·

hi,N,K ] be the matrix whose columns are formed by the channel vectors from BS i to all the UTs across all the cells. Similarly,let3 = diag[λ1,1λ1,2 · · · λm,n · · · λN,K ] be a diagonal matrix with diagonal elements being the uplink power allocations. Wealso define the matrix 6i as

6i = Hi3HHi . (7)

We define the Stieltjes transform [4] of the empirical eigen value distribution of the matrix 6i by the notation m6i .Whenthe number of antennas Nt on the MBS and the number of UTs per cell K , become very large, (Nt , K → ∞)3

• The iterative function for evaluating the optimal uplink power allocation λi,j is given by

λi,j =

1 +

1γi,j

σ 2i,i,jm6i(−αi)

1 + σ 2i,i,jλi,jm6i(−αi)

−1

. (8)

• The optimal receive uplink beamformers can be evaluated as

wi,j =

m,l

λm,lhi,m,lhHi,m,l + αiI

−1

hi,i,j. (9)

• The optimal transmit downlink beamformers are computed using

wi,j =δi,jwi,j. (10)

The parameter δi,j is a scaling factor between the uplink and the downlink beamforming vectors. It can be computed asin [3].

The effectiveness of our algorithm lies in the fact that the computation of the uplink power parameter λ depends only onthe channel statistics and not upon the actual channel realization. Hence in order to compute the beamforming vectors,MBSs only need to exchange the channel statistics (of their UTs) between themselves. This tremendously reduces thecomputational complexity and the information to be exchanged between between the BSs especially in a fast fading scenarioin which the channel realization changes rapidly where as the channel statistics remain constant over fairly long periods oftime (assuming low mobility scenario).

3 The details of the algorithm can be found in [3].


Note however, that the algorithm is optimal in the asymptotic setting implying that the SINR constraints are perfectlysatisfied only when the number of antennas on the MBS and the number of UTs tend to infinity. However, in a practicalscenario, when the network dimensions are finite, the authors in [3] show with the help of simulation results that theactual achieved SINR fluctuates around the target SINR. The fluctuations around the target SINR depends upon the channelrealizations across the multiple coherence intervals.

3.1.1. The problem of feasibilityWe now include a practical constraint in the beamformer design problem namely the max-power constraint on each

MBS. We rewrite the optimization problem of Eq. (5) incorporating the max-power constraint.

minwi,j

i,j

wHi,jwi,j (11)

s.t. Γi,j ≥ γi,j, i = 1 . . .N, j = 1 . . . Kj

wHi,jwi,j ≤ Pmax i = 1 . . .N.

The optimization problem in (5) has a solution only if the max-power constraints for each MBS is satisfied. In other words,if the problem is infeasible, it implies that given the max-power constraints on the MBSs, the targeted SINR lie outside thecapacity region of the system. In this work, we address this problem by selectively reducing the target SINR of a few UTs.Weprovide a heuristic approach to select the UTs whose target SINRs can be reduced. For the cells in which

j w

Hi,jwi,j ≥ Pmax,

do the following steps.

1. Denote Pi,j as the downlink power for the UTi,j given by Pi,j = wHi,jwi,j. Among the UTs belonging to the cells for

which power constraint is violated, select the set of UT(s) which has the maximum ratio δi,j =Pi,jqi,j. We define the

UTi,j′ = argmaxjδi,j.2. If the target SINR γi,j′ > τ , decrease the target SINR for UTi,j′ by a small amount ∆. Else, choose the UT(s) for which the

parameter δi,j′ is the next highest.3. Solve the optimization problem in Eq. (5) with the new set of target SINR constraints.4. Repeat the steps 1–4 until the problem becomes feasible.

The intuition behind the algorithm is as follows. During each time slot t , if the max-power constraints for the BSs is not met,we start by choosing the set of UTs for which δi,j is maximum. The numerator term is the power consumed by the UT. If theUT is consuming a lot of power, then the UT probably has a bad channel realization and hence we reduce its target SINR. Thedenominator term is the queue-length at time slot t . Minimum queue-length implies that the UT(s) has a smaller amountof information bits to be served and hence we can afford to reduce the data rate of that UT. The parameter ∆ is a designparameter which can be chosen suitably depending on the simulation scenario. The parameter τ is a small quantity whichensures that each UT gets at least a minimum rate. No UT goes unserved.

Therefore, the actual achieved SINR in the downlink fluctuates around the target SINR both due to the asymptotic natureof the decentralized beamformer design algorithm and also the feasibility issue. Additionally, note that the L2 norm of thefluctuations is bounded (because of the boundedness assumption of the L2 normof the channel). The fluctuations of the SINRresult in a variation of queue-length around a given target queue-length. Moreover, the fluctuations depend on the actualchannel realization during each time slot of which the CS is oblivious. Hence, we need an effective queue-length controlalgorithm which accounts for the fluctuations of the SINR. We will defer the description of the H∞ based flow controller toSection 4.

3.2. Beamformer design and power allocation: the weighted sum-rate maximization case

Wenow take up theweighted sum ratemaximization case in (3). As in the case of downlink powerminimization problemdiscussed in the previous subsection, we decompose the problem in (3) into two parts namely the sum-rate maximizationand queue-length stabilization. The problem of weighted sum rate maximization is given by

maxwt

i,j

i,j

βi,j log(1 + Γ ti,j), ∀t = 1 . . . T (12)

s.t.

j

wtHj wt

i,j ≤ Pmax, t = 1 . . . T , i = 1 . . .N.

Optimizing over the beamforming vector in the case of MIMO downlink channel is a particularly challenging problem evenin the centralized scenario (MIMO broadcast channel) which has been addressed in the past literature [16,17]. Noting thatthe main objective of the paper is to bring out the effectiveness of the H-Infinity queue length controller, we avoid thecomplications associated with the above optimization problem by fixing the direction of the beamforming vector and


optimize only over the power allocation associated with the UTs. We choose the MMSE beamforming vector to fix thedirection of beamforming vector (di,j) given by

di,j =

m,l

hi,m,lhHi,m,l + I

−1

hi,i,j

m,lhi,m,lhH

i,m,l + αiI

−1

hi,i,j

2

(13)

and the beamforming vectorwi,j is given by

wi,j = Pi,jdi,j

where Pi,j is the power allocation to theUTi,j.As shown in the iterative algorithm in [16,17], theMMSEbeamformingdirectionis a reasonable choice for optimizing the weighted sum rate. Hence the modified optimization problem is given by

maxP ti,j

i,j

βi,j log

1 +P ti,j|d

t Hi,jht

i,i,j|2

l=jP ti,l|dt H

i,lhti,i,j|

2 +

m=i,nP tm,n|dt H

m,nhtm,i,j|

2 + σ 2

, ∀t (14)

s.t.

j

P ti,j ≤ Pmax, t = 1 . . . T , i = 1 . . .N.

Once again, as in the previous case, we assume that the BSs are only allowed exchange the channel statistics and not theinstantaneous CSI values. Hence, the MBSs optimize the power allocation based on the asymptotic approximations of theassociated quantities which are given by

|dt Hi,jht

i,i,j|2−

σ 2i,i,jm6i

1 + σ 2i,i,jm6i

2a.s.

−−−−−→Nt ,K→∞

0 (15)

|dt Hi,lht

i,i,j|2−

(σ 2i,i,jσ

2i,i,l/Nt)m′

6i1 + σ 2

i,i,jm6i

2 1 + σ 2i,i,lm6i

2 a.s.−−−−−→Nt ,K→∞

0 (16)

|dt Hm,nht

m,i,j|2−

(σ 2m,i,jσ

2m,m,n/Nt)m′

6m1 + σ 2

m,i,jm6m

2 1 + σ 2m,m,nm6m

2 a.s.−−−−−→Nt ,K→∞

0. (17)

Here m6i is the Stieltjes transform of the matrix 6i =

m,l hi,m,lhHi,m,l + I

. The reader can refer to [3] for the derivations

using randommatrix theory. The details have been omitted in this paper. In order to simplify notations, let us introduce

Gi,i,j ,σ 2i,i,j

1 + σ 2i,i,jm6i

2 Gi,i,l ,σ 2i,i,l

1 + σ 2i,i,lm6i

2Gm,i,j ,

σ 2m,i,j

1 + σ 2m,i,jm6m

2 Gm,m,n ,σ 2m,m,n

1 + σ 2m,m,nm6m

2 .Let us denote the downlink SINR obtained by replacing the numerator and denominator terms by their asymptoticequivalents by Γi,j,

Γi,j =Pi,jσ 2

i,i,jGi,i,jm6il=j

Pi,lGi,i,jGi,i,lm′6i

+

m=i,nPm,nGm,i,jGm,m,nm′

6m+ σ 2

. (18)

Note thatwith the asymptotic approximation, the downlink SINR Γi,j is the same across the time slots (since Γi,j only dependsupon the channel statistics which is constant across time slots). We re-frame the problem with asymptotic approximationsgiven by

RasympMAX : max

Pi,j

i,j

βi,j log1 + Γi,j

(19)

s.t.

j

Pi,j ≤ Pmax, i = 1 . . .N.


TheMBSs can now solve the optimization problem RasympMAX by exchanging only the channel statistics between themselves.We

make the high SINR approximation and use log(1+x) ≈ log(x), for large x. However, the optimization problem in log(Γi,j) isnot concave in terms of P. Hence, we proceed by using geometric programming approach [18] andmake the variable changeto Pi,j = log(Pi,j).

RasympMAX : max

Pi,jlog

exp(Pi,j)σ 2

i,i,jGi,i,jm6i

− log

l=j

exp(Pi,l)Gi,i,jGi,i,lm′

6i+

m=i,n

exp(Pm,n)Gm,i,jGm,m,nm′

6m+ σ 2

(20)

s.t.

j

exp(Pi,j) ≤ Pmax, i = 1 . . .N.

The problem is now a convex optimization problem in terms of the variable Pi,j (refer to [18] for details). We Introduce theLagrange variable δi for the max-power constraint and formulate the Lagrangian as

L(P, δ) =

i,j

βi,j logexp(Pi,j)σ 2

i,i,jGi,i,jm6i

(21)

− log

l=j

exp(Pi,l)Gi,i,jGi,i,lm′

6i+

m=i,n

exp(Pm,n)Gm,i,jGm,m,nm′

6m+ σ 2

(22)

−

i

δi

i,j

exp(Pi,j)− Pmax

. (23)

Let us also denote the terms inside log in Eq. (22) by the notation Ii,j.We also define P =

P1,1, P1,2, . . . , P1,K , . . . , PN,K

.

The solution to the optimization problem can now be given by

minδ

maxP

L(P, δ). (24)

We will use the gradient based method to solve the dual problem [19]. At the iteration count τ for the gradient basedalgorithm, for a given δ = δ(τ ), let us define

P(τ ) = argmaxP

L(P, δ(τ )). (25)

The update equation for the dual variable is given by

δi(τ + 1) = δi(τ )− α

j

exp(Pi,j(τ ))− Pmax

∀i = 1, . . . ,N. (26)

In order to solve the optimization problem in (25), we once again use the gradient basedmethod. The gradient ofL(P, δ(τ ))with respect to Pi,j at iteration ζ is given by

∆i,j(ζ , τ ) =∂L(P, δ(τ ))

∂ Pi,j

Pi,j=Pi,j(ζ ,τ )

.

In particular, we have

∆i,j(ζ , τ ) = βi,j −l=j

βi,l exp(Pi,l(ζ , τ ))Gi,i,lGi,i,jm′6i

Ii,l

−

m=i,n

βm,n exp(Pm,n(ζ , τ ))Gi,m,nGi,i,jm′6m

Im,n− δi(τ ) exp(Pi,j(ζ , τ )).

We iteratively update the power allocation until convergence

Pi,j(ζ + 1, τ ) = Pi,j(ζ , τ )+ κ∆i,j(ζ , τ ) i = 1, . . . ,N, j = 1, . . . , K

where κ is a small step size. The value Pi,j(τ ) is the value of the gradient algorithm at the convergence point given by,

Pi,j(τ ) = limζ→∞

Pi,j(ζ , τ ).


Now, let use denote the solution of the optimization problem RasympMAX as

µi,j = maxPi,j

RasympMAX

Pi,j = argmaxPi,j

RasympMAX .

The MBSs precompute the power allocation based on the channel statistics to Pi,j and assume a precomputed service rateµi,j. The precomputed service rate can be viewed as an estimate of the service rate offered to the queues at the CS. However,in reality, the actual service rate offered to the queues depends on the channel realization and fluctuates around the targetservice rate. The service rate obtained per time slot is given by

µti,j = log

1 +Pi,j|dt H

i,jhti,i,j|

2l=j

Pi,l|dt Hi,lht

i,i,j|2 +

m=i,n

Pm,n|dt Hm,nht

m,i,j|2 + σ 2

.The CS has only an estimate of the service rate provided to the UTs given by µi,j. However, the CS being oblivious to theactual channel realization does not have the knowledge of the actual service rates in the downlink.

Hence, it is clear that in both the optimization problems (downlink power minimization and weighted sum ratemaximization) there is a mismatch between the estimated service rate and actual service rate obtained to the queues. Wemodel the mismatch between the targeted service rate and the actual service rate as unknown noise and apply H∞ basedflow controller described in Section 4 to regulate the arrival rates into the queues.

In what follows, we provide a flow control algorithm based on H∞ control.

4. Flow controller design and queue-length stability at the MBS

We now describe the flow controller design. The flow controller has to regulate the flow into the queues without theknowledge of the wireless channel conditions (which implies that the scheduler does not have the knowledge of the actualreceived SINR in the downlink, µt

i,j).Let us now focus on the queue length dynamics at the MBS. The equation for the queue-length evolution at the MBS is

given by

qt+1i,j = qti,j + ati,j − µt

i,j + U ti,j i = 1 . . .N, j = 1 . . . K . (27)

Itmust be recalled thatwe assumed the CS to be an infinite reservoir of packets. The flow controller at the CS (to be describedlater in this section) specifies ati,j, the number of packets that should flow into the queue at theMBS from the CS at each timeslot t andµt

i,j is the number of packets departing the queue (based on the downlink SINR corresponding to the beamformingvector design). U t

i,j is a constant which is added in order to keep the queue-length positive at all instants of time. Let ussubtract the target queue-length qi,j from the LHS and the RHS of (27). We also add and subtract the target service rate ofeach queue, µi,j on the RHS of the Eq. (27). Hence Eq. (27) after rearranging becomes

qt+1i,j − qi,j = [qti,j − qi,j] + [ati,j − µi,j] + [µi,j − µt

i,j] + U ti,j. (28)

We perform a change of variables and define ψ ti,j = qti,j − qi,j and the vectorized version by the notation ψt

= [ψ t1,1, ψ

t1,2,

. . . , ψ t1,K , . . . , ψ

tN,K ]

T .We also define uti,j = ati,j − µi,j and ζ t

= (µi,j − µti,j). Its vectorized notations are given by

ut= [ut

1,1, ut1,2, . . . , u

t1,K , . . . , u

tN,K ]

T

ζt = [ζ t1,1, ζ

t1,2, . . . , ζ

t1,K , . . . , ζ

tN,K ]

T .

Therefore, Eq. (28) in its vectorized form becomes

ψt+1= ψt

+ ut+ ζt . (29)

Note that we have also included theU ti,j term in the noise process (ζt). We assume that theMBSs feed back the queue-length

information during each time slot to the CS over the backhaul links (which are assumed to be perfect). Hence, the CS hasperfect state variable information (queue-length). However, the CS does is oblivious to the achieved SINR in the downlink.Hence, the CS does not have the knowledge of the noise process. Additionally, the CS does not have the knowledge of thestatistical distribution of the noise process.

The objective of the flow control design problem is to minimize the cost function given by

J =1T

T

t=1

∥ψt

∥2+ ∥ut

∥2 . (30)


Fig. 2. Example of a network with 3 Cells. The crosses represent the location of the BSs and the dots represent the location of the UTs randomly scatteredinside the cells. The distances of a UT from the three BSs are also provided.

The first term of the cost function in Eq. (30) denotes the penalty for deviating from the target queue length. The secondterm tries tomaintain the arrival rate at each queue as close as possible to the target service rate γi,j. Note that the schedulerhas the knowledge of the error term ζt at each instant of time t but does not have the knowledge of statistics of the noiseterm.

Eq. (29) represents a linear dynamic system with the state variable denoted by the notation ψt , a control parameterdenoted by the notation ut and an unknown process noise denoted by ζt . In particular, Eq. (29) denotes the evolution of thestate variable and (30) the quadratic cost to be minimized. We use a controller based on H∞ control algorithm with closedloop perfect state information pattern and finite observation window [5, Theorem 3.2] to solve the above problem. The taskof the controller is to specify an arrival rate into the queue by calculating the control parameter ut at each time slot t .

The reason why the H∞ filter makes controlling decisions without making any assumptions of the noise processes isbecause H∞ control assumes the worst case noise. That is, it tries to minimize the cost function assuming the worst casenoise. For this reason, the H∞ control is called the mini-max control. Additionally, the scheduler is trying to minimize thelong term cost function.

Once again, we would also like to clarify the main idea behind maintaining the actual queue-length as close as possibleto the target queue-length. In the context of the flow regulator (can be also viewed as a congestion controller), the targetqueue-lengths can be a representative of the fairness objective.We target a specific queue-length value as away tomaintainfairness between different UTs. However, this target queue-length can be different for different UTs. This way of modelingthe fairness enables to cast the problem as a quadratic cost minimization problem and use the control theoretic approach.

For details of the H∞ control design please refer to Appendix A.We can also model the imperfections introduced at the backhaul by adding an observation noise process. In this case the

CS makes an observation of yt given as

yt = ψt+ ηt (31)

where ηt is a noise parameter. Additionally, we can model delays introduced in the backhaul using H-Infinity control withclosed loop delayed information pattern [5, Theorem 3.4].

However, in order to keep the presentation simple, we do not consider the observation noise process and assume thatthe CS has a perfect observation of the queue-lengths.

We now provide numerical results in order to illustrate the working of the H∞ based scheduler.

5. Numerical results

We consider a hexagonal cellular system with a cluster of 3 cells as shown in Fig. 2. Each cell has a MBS which servesonly the UTs in its cell. The number of transmit antennas on each MBS scales with the number of UTs in the cell such thattheir ratio is a finite constant. In particular, we assume Nt = K . The MBS are connected to the CS. We assume that the targetqueue-length of the buffers at all the UTs to be 20 kBs. We also assume the number of channel uses per coherence intervalof the channel (B in Eq. (4)) as one-third the target queue-length. The channel realizations are assumed to be i.i.d. across thecoherence intervals.

We consider a distance dependent path loss model in which the UTs are assumed to be arbitrarily scattered inside each

cell. In this case, the path loss factor fromMBS i to UTj,k is given by σ 2i,j,k =

1

di,j,k

βwhere di,j,k is the distance between MBS

i to UTj,k, normalized to the maximum distance within a cell, and β is the path loss exponent which lies usually in the rangefrom 2 to 5 dependent on the radio environment. We normalize the variance of the total received noise to σ 2

= 1. We alsoassume that no user terminal is within a normalized distance of 0.1 from the closest BS.


Fig. 3. Scatter Plot showing the variation of queue-length around the target, Target SINR = 9 dB, Target Buffer Length = 20 kBs, β = 3.6, T = 100.

Fig. 4. NMSE of Queue-Length Vs The No. of UTs. Target SINR = 9 dB, Target Buffer Length = 20 kBs, β = 3.6, T = 100.

We perform our simulations for the case of downlink power minimization (Section 3.1) and assume a target SINR of 9 dBfor all the UTs during each time slot. First let us focus on the case with no max-power constraint on the MBSs (Pmax = ∞).We run our simulations for T = 100 time slots. We first plot the variation of queue-length values (averaged across the UTs)against the number of UTs per cell in Fig. 3. The bubbles represent queue-length values at each time slot for a given numberof UTs (overlapped on the same point) and the horizontal line represents the target queue-length.

In order to quantify the variation in the queue-lengths around the target queue size, we define the Normalized MeanSquare Error (NMSE) of the queue-lengths given by

NMSE =1

NKT

Tt=1

i,j

(qti,j − qi,j)2

qi,j.

The NMSE for the arrival rates also follows a similar definition. We plot the NMSE of the queue-lengths against the numberof UTs in Fig. 4.

It can be seen that in both the Figs. 3 and 4 (in this case the curve corresponding to Pmax = ∞), as the number of UTsincreases, the variance of the queue-length becomes lesser. This is in accordance with our beamforming algorithm. Recallthat as the number of UTs become large, the achieved SINR in the downlink gets very close to the target SINR for every givenchannel realization. Hence, the variation of the queue-length around the target also reduces.

In fact, when the number of UTs goes to infinity, effectively, we do not require queue-length control anymore. The arrivalsinto the queue is then deterministic equal the the target queue-length at all the time slots.


Fig. 5. Mean Achieved Rate per UT (left y-axis) and Mean Power Consumed (Right y-axis) per UT Vs Max-Power of the BSs. Target SINR = 9 dB, TargetBuffer Length = 20 kBs, β = 3.6, T = 100, 10 UTs per cell.

We now introduce the max-power constraint on each MBS. When the max power constraint of a particular MBS is notsatisfied, we use the approach developed in Section 3.1.1 to redesign the system parameters. In Fig. 4, (curves correspondingto Pmax = 30, 40), we plot theNMSE of the queue-lengthswith power constraints. Please note that Pmax indicated in the plotsis normalized by K , the number of UTs. It can be seen that as the Pmax per MBS reduces, the NMSE parameter increases. Thisgoes well with intuition since with the max-power constraint, the error between the target SINR and actual achieved SINRin the downlink increases because of redesigning of system parameters. The max-power plots will be useful in determiningthe excess buffer capacity beyond the target queue-length that the system designer will have to provide.

In Fig. 5, we plot the average downlink transmitted power per UT and the average rate achieved per UT against the givenmax-power constraint (Pmax) in the same plot with 10 UTs per cell and 100 channel realizations. It can be seen that morethe power consumed, more is the average rate achieved per UT. This implies that with more power, the MBSs can meet thetargeted SINR without having to reset them to lower values. Hence, the error between requested SINR (target) and actualachieved SINR is lower. Also, for low levels of Pmax, it can be seen that the increase in total downlink transmitted power isalmost linear as a function of Pmax. This is due to the fact that at low Pmax, all the MBSs are transmitting nearly at their peakin order to provide the best possible data rates for the UTs (with the power constraint). However, as Pmax increases beyondcertain value, the system performance is close to the case of Pmax = ∞. This is due to the fact that with higher powers, thetargeted SINRs lie inside the capacity region.

6. LQG control vs H-Infinity

In this section, we compare the effectiveness of our H-Infinity controller in minimizing the fluctuations of the queue-length as opposed to a standard LQG controller.4 It must be recalled that the LQG control is optimal in terms of minimizingthe expected cost when the noise process in Gaussian in nature. We apply the LQG control to our system and compare theresult with the H-Infinity based flow controller. Note that if the fluctuations of the downlink SINR around the target SINRwere to be indeedGaussian, the cost incurred by the LQG controllerwould have been lesser than that of H-Infinity controller.

We simulate the evolution of the queue-length for the casewithH-Infinity control and LQG controller and plot of NMSE ofthe queue-length for the two cases (Fig. 6). The simulation results show that the H-Infinity flow controller has a lower NMSEas compared to the LQG controller in terms of and hence keeps the fluctuations of the queue-length to minimum. The aboveobservation shows that the noise process (resulting from the fluctuations of the downlink SINR) is not Gaussian. It justifiesour model of assuming a linear dynamical system in which the statistical distribution of the noise process is unknown. Wethus apply a robust controller such as H∞ controller that keeps the queue-length fluctuations to a low value and ensuresstability. (See Fig. 6.)

7. Conclusions

In this work, we addressed the joint problem of traffic scheduling using H∞ control theory and decentralizedbeamforming design using tools from random matrix theory in the context of cellular networks. The task of the H∞ basedflow controller is to regulate the arrival process to the queues at the MBS from the CS. The H∞ based flow controllerstabilizes the queue-lengths at the MBS without the knowledge of the wireless channel between the MBS and the UTs.The H∞ controller keeps the fluctuations around the target queue-length to a minimum as compared to LQG based control.

In this work we decoupled the two problems of traffic scheduling and beamforming design. A natural extension of thisworkwould be to jointly address the two problems of flow control and beamforming design and develop a cross-layermodelto address the same.

4 We provide a brief description of a LQG control in Appendix B.


Fig. 6. Comparison of H∞ control Vs LQG control, Target SINR = 9 dB, Target Buffer Length = 20 kBs, β = 3.6, T = 100.

Appendix A. H-Infinity control

In this section,we provide a brief overviewon the solution to the discrete-time systemdynamicwith unpredictable noise.(Please note that in order to maintain consistency with the previous literature in this field, we deviate from the guidelinesset for notations in the introduction section this paper.) We consider the following state space equation:

xk+1 = Akxk + Bkuk + Dkwk (32)

where state vector xk, control vector uk and noise vector wk take values respectively in Rn,Rp and Rm. The objective is todesign a controller that minimizes the following cost function:

L(u,w) = |xk+1|2Qf

+

Kk=1

|xk|2Qk

+ |uk|2 (33)

where Qf and Qk are the norm matrices. The problem to solve is then a linear control problem with quadratic cost andunpredictable noise. Please note that when the noise is Gaussian, the solution to this problem is to use the LQG (LinearQuadratic Gaussian) technique. When the noise is unpredictable (we do not have information on the distribution of thenoise), an efficient way to solve the aforementioned problem is to use the H∞ technique. The H∞ controller adopted in thissection is the robust controller discussed in [5] and based on zero sum game theory. To solve our problem using H∞, weintroduce the following cost function:

Lγ (u,w) = |xk+1|2Qf

+

Kk=1

|xk|2Qk

+ |uk|2− γ 2

|wk|2 (34)

where, γ 2 is the level of attenuation. This is a mini-max optimization problem where the cost function is minimized overmaximum unknown disturbance. This problem is a zero sum game with two players. The cost Lγ is minimized by player 1and maximized by player 2 using vectors uk and wk. One can refer to [5] for more general class of discrete-time zero-sumgames, with various information patterns, where sufficient conditions for the existence of a saddle point are provided whenthe information pattern is perfect and imperfect state.

The H∞ controller can be obtained according to the following proposition [5, Theorem 3.2];There exists a unique feedback saddle point solution if

γ 2I − DTkMk+1Dk > 0, k ∈ [1, K ],

where the sequence of nonnegative definite matricesMk+1, k ∈ [1, K ], are generated according to

Mk = Qk + ATkMk+13

−1k Ak MK+1 = Qf (35)

where,

3k = I +BkBT

k − γ−2DkDTk

Mk+1. (36)


Under the conditions above, there exists unique feedback saddle point policies given by,

u∗

k = −BTkMk+13

−1k Akxk (37)

w∗

k = γ−2DTkMk+13

−1k Akxk (38)

with the corresponding unique state trajectory generated by the difference equation

x∗

k+1 = 3−1k Akx∗

k , x∗

1 = x1.

Appendix B. LQG control

We now provide a brief description of standard LQG control algorithm. Consider the following state space equation andthe observation equations given by

xk+1 = Akxk + Bkuk + Dkwk

where xk is the state variable, uk the control andwkt being the process noise. The sequencewt is Gaussian distributed withzero mean and covariance matrix given by

EwtTws

= R1δ

t,s

where δt,s are Dirac functions. The cost function to be minimized is given by

L(u,w) = |xk+1|2Qf

+

Kk=1

|xk|2Qk

+ |uk|2 .

The LQG control with perfect state observations is summarized as follows [20]. Starting with the initial state xkk=0 = 0,

solve the following Ricatti equations given by

Mk = ATMk+1A − ATMk+1B(BTMk+1B + I)−1BTMk+1A + Qk MK+1 = Qf .

The optimal control strategy which is a linear control law in the state estimate is given by

uk = −Lkxkwhere the matrix Lk is given by

Lk = (BTMk+1B + I)−1BTMk+1A.

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Subhash Lakshminarayana received his B.E. degree in Electronics and Communication engineering from VisvesvarayaTechnological University, India, in 2007 and hisM.S. degree in Electrical and Computer Engineering from TheOhio State Universityin 2009. He otained his Ph.D. degree in the field of wireless communication at The Alcatel Lucent Chair on Flexible Radio and theDepartment of Telecommunications at SUPELEC, France in 2012. His research interests include cross layer design in wirelessnetworks, MIMO systems, stochastic network optimization and randommatrix theory. He has been a student intern at the IndianInstitute of Science (IISc), Bangalore, India where he has worked on modeling and performance analysis of optical burst switchednetworks. He is currently working as a research associate at the Singapore University of Technology and Design (SUTD).

Mohamad Assaad received the B.E. degree in electrical engineering with high honors from Lebanese University, Beirut, in 2001.He received the M.Sc. and Ph.D. degrees (with high honors), both in telecommunications, from Ecole Nationale Superieuredes Telecommunications (ENST), Paris, France, in 2002 and 2006, respectively. While pursuing the Ph.D. degree, he was aResearch Assistant with the Wireless Networks and Multimedia Services Department, INT, France, working on cross-layer designin UMTS/HSDPA system and interaction of TCP with MAC/RLC and physical layers. Since March 2006, he has been with theTelecommunications Department, SUPELEC, where he is currently an Associate Professor. His research interests include cross-layer design, resource optimization in wireless networks and multi antenna systems.

Merouane Debbah entered the Ecole Normale Supérieure de Cachan (France) in 1996 where he received his M.Sc and Ph.D.degrees respectively. He worked for Motorola Labs (Saclay, France) from 1999 to 2002 and the Vienna Research Center forTelecommunications (Vienna, Austria) until 2003. He then joined theMobile Communications department of the Institut Eurecom(Sophia Antipolis, France) as an Assistant Professor until 2007. He is now a Full Professor at Supelec (Gif-sur-Yvette, France), holderof the Alcatel-Lucent Chair on Flexible Radio and a recipient of the ERC starting grant MORE (Advanced Mathematical Tools forComplex Network Engineering). His research interests are in information theory, signal processing and wireless communications.He is a senior area editor for IEEE Transactions on Signal Processing. Merouane Debbah is the recipient of the ‘‘Mario Boella’’award in 2005, the 2007 General Symposium IEEE GLOBECOM best paper award, the Wi-Opt 2009 best paper award, the 2010Newcom++ best paper award as well as the Valuetools 2007, Valuetools 2008, Valuetools 2012 and CrownCom2009 best studentpaper awards. He is a WWRF fellow. In 2011, he received the IEEE Glavieux Prize Award.

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H-Infinity control based scheduler for the deployment of small cell networks