Satisfying demands in a multicellular network: A universal power allocation algorithm

Computer Communications 36 (2013) 1373–1386

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Computer Communications

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Satisfying demands in a multicellular network: A universal powerallocation algorithm q

Veeraruna Kavitha a,b,⇑, Sreenath Ramanath a, Merouane Debbah c

a INRIA, Sophia Antipolis, 2004 Route des Lucioles, 06902 Sophia Antipolis Cedex, Franceb LIA, 339, Chemin des Meinajaries, Agroparc BP 91228, 84911 Avignon Cedex 9, Francec SUPELEC, Plateau du Moulon 3, rue Joliot-Curie, 91192 Gif-sur-Yvette cedex, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 30 September 2011Received in revised form 26 February 2012Accepted 8 April 2012Available online 27 April 2012

Keywords:Cellular networksMIMOPower allocationStochastic approximationOrdinary differential equations

0140-3664/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.comcom.2012.04.009

q This article is a journal version of the paper [13Princeton, USA.⇑ Corresponding author. Address: INRIA – Sophia

Lucioles, 06902 Sophia Antipolis Cedex, France. Tel.: +(0) 4 90 84 35 01.

E-mail addresses: [email protected] (V. Kavit(S. Ramanath), [email protected] (M. Debbah).

Power allocation to satisfy user demands, in the presence of large number of interferers (in a multicellularnetwork), is a challenging task. Further, the power to be allocated depends upon the system architecture,for example upon components like coding, modulation, transmit precoder, rate allocation algorithms,available knowledge of the interfering channels, etc. This calls for an algorithm via which each base sta-tion in the network can simultaneously allocate power to their respective users so as to meet theirdemands (whenever they are within the achievable limits), using whatever information is available ofthe other users. The goal of our research is to propose one such algorithm which in fact is universal:the proposed algorithm works from a fully co-operative setting to almost no co-operation and or forany configuration of modulation, rate allocation, etc. schemes. The algorithm asymptotically satisfiesthe user demands, running simultaneously and independently within a given total power budget at eachbase station. Further, it requires minimal information to achieve this: every base station needs to knowits own users demands, its total power constraint and the transmission rates allocated to its users inevery time slot. We formulate the power allocation problem in a system specific game theoretic setting,define system specific capacity region and analyze the proposed algorithm using ordinary differentialequation (ODE) framework. Simulations further confirm the effectiveness of the proposed algorithm.We also demonstrate the tracking abilities of the algorithm.

In heterogeneous networks, it is hard to expect the various agents to update their algorithms in a syn-chronous manner. Using two time scale stochastic approximation analysis we study the proposed algo-rithm operating in a simple example scenario, wherein the heterogeneous agents update (their powerprofiles) at different speeds.

Further, backed by numerical examples (for various generic example scenarios), we show that the algo-rithm converges to the same power profile, as long as the demands remain same, irrespective of the dis-parities in the operating speeds at different agents.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Multi-input multi-output (MIMO) combined with network den-sification promise improved network coverage and capacity formobile broadband access. But, due to an increased number oftransmit antennas and or the proximity of base stations (BS), usersat cell edges experience a higher degree of interference from neigh-boring base stations.

ll rights reserved.

] presented in WiOpt 2011,

Antipolis, 2004 Route des33 (0) 4 90 84 35 00; fax: +33

ha), [email protected]

Network MIMO or other forms of BS co-operation enable shar-ing complete or statistical knowledge of channel states (CS)amongst neighbors via back-haul links to alleviate interferenceand offer better rates to users. When back-haul is not available,each BS may estimate the local channel state information anduse the same for better performance. In some cases, a low ratefeedback from the receiver indicating the QoS of the current trans-missions is utilized, while in the worst case the transceivers are de-signed with no CS information. Thus we have a variety of systemswith varying degrees of the information about the interferingchannels. However the goal in each is the same: satisfy the de-mands of all the users. We may require higher power profiles tosatisfy the same demands when working with lesser information.Further diverse situations can arise because of the systemconfiguration like modulation, precoding, channel coding, resourceallocation etc.

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1 We show the demand meeting power profile to be a NE of a ‘leaky’ game. We callthis game ‘leaky’, because the utility of the game is upper bounded by the demands(see Definition 5, Section 3.1). In summary our aim is to provide a channel, to eachone of the users, whose (system specific) capacity is more than or equal to the user’sdemand.

1374 V. Kavitha et al. / Computer Communications 36 (2013) 1373–1386

For a given vector of power constraints at various base stations,Shannon capacity gives the maximum achievable rate, i.e., thecapacity region. This is an upper bound. We define ‘‘system specificcapacity region’’ (achievable rate region of a given system) whichdepend on coding (space–time, channel), modulation, channelstate information availability, synchronization, feedback errorsand many other things. Given a system architecture with a chosenset of parameters which define its rate allocation, modulation, etc.,the achievable rates are usually inferior to the theoretical rates andthe system specific capacity region is defined based on these rates.The system-specific capacity region for the same power constraintvaries: for example it shrinks if the number of supported discreterates reduce. Thus, the power allocated to any user to achievethe same demand rate varies with the set of system parameters.

The main contribution of this paper is a universal algorithmwhich can work with many of the systems mentioned above. It sat-isfies asymptotically the demands of all the users irrespective ofthe system in which it is operating, albeit with different powerprofiles. Each base station requires minimal information: its user’s de-mands, its total power constraint and the current transmission rates toits users. The amount of data information transmitted successfullyin a slot (per slot) basically represents the current transmissionrates. These current transmission rates are decided by the servingbase stations either using complete CSIT (algorithm can also beused as a centralized scheme in this case) or has to be estimatedcompletely blindly or using some partial information. These arealso influenced by the underlying channel.

In cellular networks, the scenarios can change with time. Forexample a base station can become active suddenly, the demandsmay change etc. We demonstrate via simulations that the pro-posed algorithms can also track the changes.

In heterogeneous networks, various agents (for example macrocells and micro cells) can operate at different speeds. However theystill can interfere with each other. We consider a simple examplescenario and demonstrate using the two time scale stochasticapproximation analysis that the proposed algorithm converges tothe same power profile irrespective of the disparities in the updaterates. We then illustrate the same for general scenarios usingnumerical simulations. The following are the contributions of thispaper:

(1) A system specific game theoretic problem formulation usingthe system specific capacity region.

(2) A Stochastic Approximation based universal power allocationalgorithm in an interference limited multi-cell network.

(3) Various properties (e.g. convergence) of the proposed algo-rithm is analyzed using an ODE framework.

(4) Simulation results demonstrate the effectiveness of the pro-posed algorithm for a variety of systems.

(5) We also establish the tracking capabilities of the algorithm.(6) We illustrate the robustness of the proposed algorithm

against the disparities in update rates at various agents.

1.1. Related work

For an excellent survey on power control in wireless networks,the reader is referred to [2] and the references there-in (e.g. [3,5–8]). In recent years, several authors have addressed distributedpower control strategies with various levels of co-operation for agiven system configuration (e.g. [3,5–7,10] etc). Typically, the de-sign objective is to maximize the total sum rate of all the users sub-ject to BS power constraints or to minimize the total transmitpower satisfying some SINR constraints of the users.

Most of the existing algorithms aim at either optimizing the to-tal power spent keeping the QoS above a required level (e.g. [5–7]etc.) and or optimize the QoS while keeping the power utilized

within a given budget (e.g. [10]). But our algorithm does not opti-mize, it only meets the demands (in the form of average transmis-sion rates) on average asymptotically.1 This relaxation helps us inproposing an algorithm that requires minimal information (hencehas minimal complexity) at the transmitters: rates at which theinformation is correctly transmitted to the user in every slot. Datais pumped out from the transmitter and hence these rates are readilyknown to the transmitter. Hence this algorithm does not require anyextra information and this can be exploited in many more ways. Forexample, one can probably use this algorithm in networks with het-erogeneous cells, i.e., when each cell has a system configuration thatcan be different from the other cells.

A related concept, called satisfying equilibrium, is defined andstudied in a recent paper ([9]). Here they define the satisfying equi-librium as any profile at which the QoS of all the users is either bet-ter or the same as the specified level. Basically, the set of satisfyingequilibrium represents the domain of optimization for the prob-lems that optimize the total power utilized while maintainingthe QoS. In our paper, we propose an algorithm that satisfies thedemands for all the users at exactly the specified level via a sto-chastic approximation based zero finding method. As already dis-cussed, this zero finding method greatly simplifies the algorithm.To the best of our knowledge this is the first paper that proposesto take advantage of the relaxation obtained by avoiding theoptimization.

1.2. Organization

We introduce the system model in Section 2. In Section 3, wedescribe the system specific problem formulation. The algorithmand its analysis is presented in Section 4. Section 5 provides simu-lations. Section 6 discusses heterogeneous agents. Appendix con-tains example systems and proofs.

1.3. Notations

Boldface lower-case symbols represent vectors, capital boldfacesymbols denote matrices (IN is the N � N identity matrix). Hermi-tian transpose is denoted ð�ÞH while tr½X� represents the trace ofmatrix X. All logarithms are base-2 logarithms. Small letters repre-sent the scalars. Let ak represent the kth component of the vector a.If the vector is already indexed like for example in pj, then pk;j rep-resents its kth component. Let ðp:sÞ represent the component-wiseproduct, i.e., ðp:sÞk ¼ pksk for all k while

ffiffiffipp

represents componentwise square root. E½�� denotes expectation and Es is expectationw.r.t to s when conditioned (if any) on the other random variables.

2. System model

We consider a multi-cell MIMO system. Each base station has Mtransmit antennas and is communicating with K single-antennausers (see Fig. 1). Every user experiences both intra-cell (transmis-sions from parent BS) and inter-cell (transmissions from neighbor-ing BS) interference. Each user in a cell demands a certain rate andall these rates have to be jointly satisfied by the BS (present in thecell) while operating within a total power constraint.

Let Hj;l represent the K �M channel matrix, when the users incell j receive signals from the BS of cell l and let its elements be gi-ven by zero-mean unit-variance i.i.d. complex Gaussian entries. Letnj represent the additive white Gaussian noise at the receivers of

Fig. 1. 2D Wyner model.

Table 1System specification (I–II–III).

I CSIT II TX rate III Precoder

A Asymptotic I Ideal ZF Zero-forcingC Full CSIT D Discrete NO No precoderL Local CSIT RA Rate adaptationN No CSIT RAE RA with errors

2 One can also consider systems which have an estimate of the CS.3 We illustrate these concepts using simple rate allocation schemes. One can

extend it to other rate allocations, for e.g. schemes that incorporate fairness.

V. Kavitha et al. / Computer Communications 36 (2013) 1373–1386 1375

cell j; xj be the M length transmit vector in cell j and cl 2 ½0;1� bethe interference factor, representative of the level of interferencefrom cell l. For example, as base stations become denser, interfer-ence increases and hence cl ! 1. The signal vector (of length K)received by users in cell j is given by,

yj ¼ Hj;jxj þXN

l¼1;l–j

clHj;lxl þ nj for all j 6 N: ð1Þ

In the above the first term represents the useful signal part as wellas the intra-cell interference while the second term (summation)represents the inter-cell interference to the jth cell from itsneighbors.

If Pj represents the total power constraint in cell j, thentrðE½xjxH

j �Þ 6 Pj to satisfy the power constraint. As an example, ifthe BS in cell j uses power levels specified by pj and a precoding ma-trix Gj (of size M � K), then the transmit vector is given byxj ¼ Gjð

ffiffiffiffiffipj

p:sjÞ (sjj is a K length independent symbol vector of zero

mean and unit variance components). In this case the power con-straint leads to,

trðE xjxHj

h iÞ 6 trðE Gj

ffiffiffiffiffipj

pðGj

ffiffiffiffiffipj

pÞH

h iÞ 6 Pj for any j:

Given a precoding scheme, this constraint can equivalently be rep-resented by (for a possibly different Pj)

Pkpk;j 6 Pj: The symbol, yk;j,

received by the user k of cell j is,

yk;j ¼ hH

k;j;jxj þXK

i¼1;i–k

hH

i;j;jxj þXN

l¼1;l–j

XK

i¼1

clhH

i;j;lxl þ nk;j

¼ uk;j þ ik;j;j þXl–j

ik;j;l þ nk;j ð2Þ

where hk;j;l, the kth row of matrix Hj;l, represents the M length chan-nel vector for user k of cell j as received from the BS of cell l. In theabove, uk;j; ij;j;k and ik;j;l respectively represent the useful, intra-cellinterference and inter-cell interference signal.

2.1. System with no precoding

This paper proposes an algorithm which works for any systemin general. By system, we mean a particular multi-cell network witha given configuration like, precoding scheme, channel coding, resourceallocation etc. We will derive the exact received signal characteris-tics for one such example system. The received signal characteris-tics of the others system can be derived in a similar way. Weconsider a system with no precoding (for example, systems whichdoes not have access to channel state information). Further weconsider a system with M ¼ K and with xj ¼ ð

ffiffiffiffiffipj

p:sjÞ. The average

power in the useful, intra-cell, inter-cell interference signals of thereceived signal (after channel coding at the transmitter and chan-nel decoding at the receiver) after averaging w.r.t. to the symbolstatistics fsjg for any given channel state is:

Esj ;16j6N½juk;jj2� ¼ pk;jjhk;j;j;kj2;

Esj ;16j6N½jik;j;jj2� ¼X�k–k

p�k;jjhk;j;j;�kj2 and

Esj ;16j6N½jik;j;lj2� ¼X

�k

clp�k;ljhk;j;l;�kj2

ð3Þ

where, hk;j;l;�k is the ðk; �kÞth component of the matrix Hj;l. In the abovewe used E½sk;js�k0 ;j0 � ¼ 1fk¼k0 ;j¼j0g:

3. System specific problem formulation

Every BS has to meet its users demands, for example BS j has tomeet its users demand rates represented by rj :¼ frk;j; k 6 Kg. It hasto tune its power levels pj to achieve this. But the rates achievedwill also depend upon the powers used by the other base stations.Our goal is to find a simple universal power allocation algorithmwhich runs independently and simultaneously at all the base sta-tions and tunes the power levels to achieve the user demandsusing minimal information. The power levels depend upon the sys-tem configuration (for example channel precoding scheme, rateallocation scheme). We consider some interesting example sys-tems briefed in Table 2 and described in Appendix A. These sys-tems are referred using a three part code, I–II–III, as explainedbelow (see Table 1):

(1) The first part (I) represents the availability of channel stateinformation at transmitter2: (a) A represents an asymptotic(large number of antennas/users) system, where achievablerates for almost all CS are approximated by a constant (see[11] and references there in), (b) C for systems with completeCSIT, (c) L, systems with local CSIT, i.e., BS j knows Hj;j part ofthe CS, (d) N, systems with no CSIT.

(2) The second part (II) represents the transmission rates usedat the system3: (a) I for ideal systems which can channel codeto achieve any feasible rate, (b) D for the those systems whichcan only operate at one of the discrete rates in the setR ¼ fr1; r2; � � � rNRg (arranged in decreasing order), (c) RA for

Table 2Some example Systems. Right column gives the rate at which data is transmitted when CS is H and when system uses power profile P. More details in Appendix A.

System Description (more details in Appendix A) Rsysk;j ðP;HÞ

A-I-ZF: Large No. of antennas and users with M > K Asymptotic rates of [11] approximate the instantaneous ratesfor almost all CS and transmission at ideal rates. Zero forcing precoder log 1þ pk;j

1b�1

PN

l¼1;l–jcl

trðPl ÞK þr2

k;j

!

C-I-ZF: Number of antennae/users not large enough. Asymptotic results not accurate. Every BS has CSIT, computestheoretical rates and transmits at ideal rates. ZF precoder log 1þ pt

k;jPN

l¼1;l–jcl

tr Htj;l

Ql Htj;l

H

� �K þr2

k;j

0BB@1CCA

C-D-ZF: Similar to C-I-ZF, but TX rate allocation from discrete set R infr2Rfr 6 RC � I� ZFk;jðP;HÞg

C-I-NO: Similar to C-I-ZF, but without Precoderlog 1þ

Esj½juk;j j2 �P

lEsl½jik;j;l j2 �þr2

k;j

� �C-D-NO: Similar to C-I-NO, but TX rate allocation from discrete set R infr2Rfr 6 RC � I� NOk;jðP;HÞg

N-RA-NO: Rate adaptation w/o CSIT. Uses blind methods to adapt to the correct rate as long as the underlyingchannel can support the same. No TX precoder

RC � D�NOk;jðP;HÞ

L-RA-ZF: Rate adaptation with local CSIT. BS has local CS, Uses blind methods to assign rates (as in N-RA-NO) andlocal CS for precodings

RCk;j � D� ZFðP;HÞ

N-RAE-NO: Similar to N-RA-NO, but with rate estimation errors, Ek;jðrÞ RN � RA�NOk;jðP;HÞ � Ek;jðRN � RA� NOk;jðP;HÞÞ

L-RAE-ZF: Similar to L-RA-ZF, but with rate estimation errors, Ek;jðrÞ RL�RA�ZFk;j ðP;HÞ � Ek;jðRL�RA�ZF

k;j ðP;HÞÞ


systems which estimate the current rate (i.e., pick the currentmaximum possible rate from the set R as explained in Appen-dix A) using rate adaptation schemes (e.g. [12]) without CSIT,(d) RAE when there are estimation errors in the rate adapta-tion algorithm (see systems N-RAE-NO and L-RAE-ZF ofAppendix A for more details).

(3) The third part (III) represents the precoder4: (a) ZF for zeroforcing precoding, (b) NO for no channel precoding.

3.1. Game theoretic formulation

As the base stations influence each other, the problem can bestbe captured using a game theoretic formulation. We begin byintroducing the components of the game. The calligraphic letters(for example P) represent the ensemble of either vectors, matricesor scalars for all the base stations.

Power profile, P :¼ fpk;jgk6K;j6N , represents the vector compris-ing of the powers used at all the base stationsand for all the users.Recall, pk;j represents the power used by the BS of cell j for user k incell j.

Channel State (CS), H :¼ fH1;1;H1;2; � � � ;HN;Ng, arranged as amatrix of dimension KN �MN, represents the channel state ofthe entire system.

Rate for a given power profile and system, Rsysk;j ðP;HÞ, repre-

sents the transmission rates allocated, to the user k by the base sta-tion j, in system represented by sys (e.g. N-RAE-NO in Table 2)when the base stations use powers P and when the CS is H. Theserates are given in the right column of the Table 2 for various exam-ple systems, whose detailed descriptions are provided in AppendixA.

Average Rate for a given system and power profile, is the ratethat is achieved on average when a given system uses the power

profile P : Rsysavg;k;jðPÞ ¼ EH Rsys

k;j ðP;HÞh i

: Let Rsysavg :¼ Rsys

avg;k;j

n ok;j: Power

constraint ðP 6 PÞ We use 6 in a special manner to facilitatedefining the power constraint. We say a power profile P is ‘‘lessthan or equal to’’ and hence satisfies the constraint defined interms of another power profile P if the two profiles satisfy theconstraints for each base station as:

Pkpk;j 6

Pk�pk;j for all j 6 N:

4 One can also consider other types of precoders (e.g. MMSE). Our analysis andproofs hold for these configurations as long as they satisfy the assumptions A.1 to 4(refer Section 4).

System Specific Capacity Region for any given power profileconstraint P and a system, sys, is defined as the collection of allpossible tuple of average rates achievable, while using powers thatsatisfy the constraints defined in terms of P, i.e.,

CsysðPÞ :¼ fRk;jg 2 RNK : for all k; jRk;j ¼ Rsysavg;k;jðPÞ

nfor some P with P 6 P

oð4Þ

This region is different for different systems. For a system with idealrates the capacity region coincides with the theoretical one. A sys-tem with discrete rates cannot always achieve the maximum possi-ble rate and hence its capacity region shrinks. It further dependsupon R, the set of supported rates. If the system has estimationerrors, the capacity region shrinks further.

Utilities and Players: Each BS j is a player and its strategyis K-dimensional power vector, pj :¼ ½p1;j; � � � ; pK;j�: Note thatP ¼ ½p1;p2; � � � ;pN�. Define the utility of player j as,5

Usysj ðpj;P�jÞ :¼

Xk

min Rsysavg;k;jðpj;P�jÞ; rk;j

n owith P�j :¼ ½p1;p2; � � � ;pj�1;pjþ1; � � � ;pN �: ð5ÞIn the above, P�j is the power vector profile excluding only the pow-ers of BS of cell j and rk;j is the demand of user k of cell j. Every sys-tem with given power constraint P and demand vectors frk;jgdefines an N-player non cooperative strategic form game:

ð½1;2 � � �N�; Usysj

n oj6NÞ: The Nash equilibrium (NE) of this game is

a power profile P� that satisfies,

p�j 2 arg maxP6P

Usysj ðpj;P��jÞ for all j: ð6Þ

From the above definitions, it is evident that,

Lemma 1. For any given system and power constraints P, if the vectorof the demands frk;jgk;j is in the corresponding capacity region CsysðPÞ,then there exists a P� 6 P, which is a NE satisfying all the demands:

Rsysavg;k;jðP

�Þ ¼ rk;j for all k; j: }

Thus, when all the base stations use the NE power profile P� ofLemma 1, all the users in each cell achieve an average rate whichequals their demand, i.e., will be able to receive the information

5 The utility of an user is the average rate at which its data is transferred. The user kof cell j requires transmission at maximum at its demand rate rk;j and hence his utilityis upper bounded by the same.


at the demand rate on average. The main aim of this paper is toobtain this NE (time) asymptotically (if required in a completely dis-tributed way) for any given system when the demands satisfy Lemma1. This NE depends on the system considered (for example higheramount of power may be required if one uses discrete rates inthe place of ideal rates) even if the power constraint and demandsare same. The proposed algorithm is a general iterative algorithmwhich works irrespective of the system considered, i.e, the proposedalgorithm converges to the system specific NE.

Remark on hypothesis of Lemma 1: It requires that the de-mands equal one of the average rates of the capacity region. Lem-ma 2 of the next section gives an easily verifiable assumptionwhich ensures this hypothesis of Lemma 1.

Set of demand meeting NE, Lsys � CsysðPÞ, is the set of NEwhich meet the demands as in Lemma 1.

We now present the Universal Power Allocation algorithm forpower constrained Multi Cell Networks (UPAMCN).

4. Universal algorithm: UPAMCN

We consider a quasi-static channel and obtain the NE of Lemma1 asymptotically by iteratively updating the power profile at thebeginning of every slot, during which the CS is assumed constant.

Basic idea6: Each BS j in every time slot knows the rates at which

data is transmitted to its users, Rsysk;j ðP;HÞ

n ok. The characterization of

these rates is provided for some examples in Table 2 (and details inAppendix A). An iterative algorithm can find the average value of it.One can then update the power vectors to force this average towardsthe demands frk;jg.

Let dtþ1k;j represent the number of bytes of data transmitted success-

fully in time slot t þ 1 by the jth base station to its user k divided bythe duration of the time slot. This ratio depends upon the power pro-file of the entire system in the previous slot (Pt) and the entire CSin the current slot (Htþ1), but (Pt , Htþ1) are only partially known atthe base stations. However dtþ1

k;j is still known at base station j as itis the source that pumps out the data. In fact, it will be preciselyequal to dtþ1

k;j ¼ Rsysk;j ðP

t ;Htþ1Þ of Table 2 by definition. Let fltg rep-resent the update step sizes.

4.1. UPAMCN algorithm

With PA representing the projection into the set A, the UP-AMCN algorithm is given by

ptþ1k;j ¼ PAj

ptk;j � lt dt

k;j � rk;j

� �h iwith Aj :

¼ p 2 RK :X

k

pk 6 Pj

( ); A :¼ A1 �A2 � � � �AN: ð7Þ

4.2. Analysis

We obtain the asymptotic analysis of the algorithm using theordinary differential equations (ODE) approach of [1]. We establishTheorem 1 given below, under the following conditions:

A.1 There exists a sequence at !1 with limtsup06i6at

ltþi=lt ¼ 0:A.2 The channel state fHtg is an independent and identically dis-

tributed (IID) sequence with finite mean and variance.

6 Most of the stochastic approximation algorithms obtain optimum of a function asthe zero of its derivative. In contrast, this algorithm obtains the profile that satisfiesthe demands, directly as the zero of the function given by the average rate minusdemand. In other words, this algorithm does not perform any optimization, but ratherjust finds a power profile that solves the equation average rate = demand for all theusers.

A.3 The instantaneous rates are bounded by the same constant,i.e., jRsys

k;j ðP;HÞj 6 B for all k; j; P and H:A.4 The average rate Rsys

avg;k;j is continuous in P for all k; j.

We will show that the UPAMCN trajectory (7) can be approxi-mated by the solution ðPðtÞÞ of the following ODE (to be precisea differential inclusion).

_pk;j ¼ rk;j � Ravg;k;jðPÞ þ zk;jðPÞ for all k; j ð8Þ

where zk;jðPÞ represents the projection term. Define the limit set ofthis ODE:

LODE :¼ limt!1[P2AfPðsÞ : s P t and Pð0Þ ¼ Pg:

The d-neighborhood of this set is defined as:

BdðLODEÞ :¼ P : jP � Pj 6 d for some P 2 LODE� �

:

Theorem 1 establishes that the trajectory ultimately spends time inthis limit set. We first establish the theorem and later study the sys-tems of previous section using this limit set.

Theorem 1. Assume A.1–4. Then for every d > 0, the fraction of timethe tail of the algorithm (for any initial power profile with eP < P)

fPsgsPt with initialization Pt ¼ ePspends in the d-neighborhood of the limit set BdðLODEÞ tends to one (inprobability) as t !1.

Proof. Refer Appendix B. h

4.3. Analyses of the specific systems

Most of the systems considered in this paper (for example, C-D-ZF) transmit at one of the rates from a discrete set R depending onthe instantaneous CS and for these one need to explicitly prove thecontinuity of the average rates. This is achieved in the following(proof in Appendix B).

Lemma 2. Assume A.1 and A.2. Then, for all the systems considered inTable 2, assumptions A.3 and A.4 are satisfied, Theorem 1 applies andhence the UPAMCN trajectory (7) asymptotically spends most of itstime in the limit set, LODE. Further, the demand meeting NE set, Lsys, isnon empty and these form the stationary points of the ODE (8),whenever for all k; j the demands satisfy

rk;j 6 supP6P

Ravg;k;jðPÞ: }

For further analysis, one needs to study the limit set of ODE (8).A limit set of an ODE usually contains limit cycles or attractors.Every demand meeting NE of Lsys is a stationary point of the ODE,whenever it is not on the boundary. The demand meeting NE ofLsys would be in the limit set, if further, we could show that theyare attractors. In that case, the algorithm spends most of its timein these attractors or in other words the UPAMCN algorithm asymp-totically meets the demands of all the users. Right now, we can onlysay that, every stationary point of ODE (8) is a demand meeting NEand any attractor of the ODE must be a stationary point. We willshow via numerical examples in the next section that the algo-rithm indeed converges to a demand meeting NE for all the sys-tems considered in this paper.

4.4. Extensions to UPAMCN

The UPAMCN algorithm works under the basic assumption thatthe BS always has sufficient data to transmit. But in reality, data

0 20000

0.2

0.4

0.6

rate

0 2000

Rate convergence(7 BS, 8 users per BS, γ = 0.5, σ 2 = 0.1)

Iterations0 2000

Fig. 2. Rate convergence for Systems S1, S2 and S3 (H1 network).

0 20000

0.1

0.2

Pow

er

0 2000

Power convergence(7 BS, 8 users per BS, γ = 0.5, σ 2 = 0.1)

Iterations0 2000

Fig. 3. Power convergence for Systems S1, S2 and S3 (H1 Network).


often arrives in real time and hence there can be situations whenthe BS can transmit at a higher rate but does not have sufficientdata. In this case we propose the following extension to UPAMCN:

btþ1k;j ¼ bt

k;j þ Btþ1k;j �min dt

k;j; btk;j

n oand

ptþ1k;j ¼ PAj

ptk;j þ lt min dt

k;j; btk;j

n o� rk;j

� �h i ð9Þ

where btk;j represents the remaining (accumulating) bytes of data to

be transmitted by BS j to the user k at the beginning of time slot tand Bt

k;j represents the fresh sample of data added to the corre-sponding buffer.

5. Numerical simulations

We consider two types of cellular networks in our simulations(Table 3). The first one is a Hexagonal network, where users in eachcell experience interference from BS transmissions of surroundingcells (typically assumed to be from the 1st tier of surrounding 6cells (see for example Fig. 1). The second one is a linear network,where users in each cell experience interference from BS transmis-sions of adjacent cells (typically two adjacent neighbors). The sys-tem configurations are summarized in Table 4. Each BS equippedwith M transmit antennas is serving K users in its cell. In all thesimulations we also compute the average rates via the following

iteration: /tþ1k;j ¼ /t

k;j þ lt dtk;j � /t

k;j

� �for all k; j: This iteration is

only a measurement procedure that is used for the purpose of cal-culating average rates of the system for the numerical examplesconsidered. How it represents the average rate can be understoodby noticing that, /t

k;j is actually a weighted average of all the

instantaneous rates fdsk;j; s 6 tg up to time t. These average rates

are used to illustrate that systems considered in these examples,asymptotically (as time progresses) satisfy the user’s demands onaverage.

The power limit on each BS is set to 1 unit. Interference factor cl

from each interfering BS, l, is set to 0.5. For the simulations consid-ered here, we choose the demand rate vector (to lie within thecapacity region and is common for all the base stations) as:r ¼ ½:065 :130 :195 :260 :325 :389 :454 :520�.

In the first set of simulations, we consider the hexagonal net-work (H1). The rate and the power convergence behavior of thealgorithm for systems S1, S2 and S3 is plotted in Figs. 2 and 3,respectively. We observe that: (1) The algorithm converges to thedemand meeting NE: we see in Fig. 2 that for all the systems, theaverage rate achieved asymptotically converges towards the de-mand rates (the dotted lines). (2) As discussed in the previous sec-

Table 3Network configurations.

Type Intf BS Tx Ant Users

L1 Linear 2 16 8L2 Linear 2 32 8H1 Hexagon 6 16 8H2 Hexagon 6 2 2

Table 4System configurations

S1 Asymptotic Ideal with ZF Precoder (A-I-ZF)S2 Rate adaptation with local CSIT and ZF precoder (L-RA-ZF)S3 Rate adaptation with local CSIT, ZF Precoder and with estimation errors

(L-RAE-ZF)S4 Rate adaptation without CSIT (N-RA-NO)S5 Rate adaptation without CSIT and estimation errors N-RAE-NO).

tions, we notice from Fig. 3, that the converged power profile(demand meeting NE) is system specific. S3 is a system with errors,the proposed algorithm still satisfies the demands asymptotically,however, the converged power profile has higher power levels incomparison with the error free systems S2 and S1. (3) Note thatS2 can also represent C-D-ZF, a complete CSIT system (see detailson Table 2 and Appendix A). From Fig. 3, we observe that the con-verged power profile of C-D-ZF (S2) is close to that of A-I-ZF (S1)system. Thus the demand meeting power profile of systems withlarge number of transmit antennas and or users and large numberof discrete levels in R is close to that of the asymptotic ideal ratesystem. Further convergence is faster with S1 system. Thus for suchsystems, UPAMCN algorithm can be used to estimate (approxi-mately) the demand meeting power profile, much faster, usingthe asymptotic rate expressions in place of instantaneous transmitrates allocated, fdt

k;jg. Note that this further avoids the need ofcomplete CSIT, as we need only local CSIT for precoding. (4) Asthe discrete levels increase, the power profile decreases and finallyconverges to that of the ideal rate. This is tabulated in Table 5.

In the second set of simulations, for given demand rates, wecompare the algorithm behavior for different network configura-tions L1, L2 and H1 with system S2 (see Fig. 4). We observe that:to satisfy the same demands, the base stations in L2 expend theleast power, followed by L1 and then H1. L2 performs better thanL1 due to improved transmit diversity. H1 is the worst (largernumber of interfering base stations).

In the final set of simulations, for network configuration H2, weconsider the least informed (No CSIT) systems, the rate adaptationsystems S4 and S5. We choose the common demand rate vector as½0:1694 0:1936�: Figs. 5 and 6 illustrate the average rate and powerprofile convergence. As CSIT (even local) is not available at the base

Table 5System S2.

System Demand satisfying NE (converged power)

ID .015 .032 .049 .067 .085 .105 .126 .147D1 .017 .035 .053 .072 .092 .113 .135 .158D2 .031 .049 .071 .097 .121 .146 .174 .203

ID: Ideal, D1: Discrete-100 levels, D2: Discrete-20 levels.

1 2 7 80

0.2

User num

Pow

er

Converged power(7 BS, 8 users per BS, γ = 0.5, σ2 = 0.1)

converged pwr L1converged pwr L2converged pwr H1

Fig. 4. Demand satisfying NE. System S2. (L1, L2 & H1 networks).


stations, they cannot use any precoders. Thus, it is a totally inter-ference dominated system and hence the achievable capacity re-gion is small. But the UPAMCN algorithm works even for thisleast informed system: it asymptotically satisfies the demandrates, albeit using a higher power profile. Further, the rate estima-tion errors in system S5 demand higher power levels in compari-son with the error free system S4 to achieve the same demands.

Further, we observe that the convergence to the demand meet-ing NE is quicker in those systems where base stations have moreinformation (see for e.g. Fig. 3).

5.1. Dynamic scenarios

In cellular networks, often the scenario changes. A user mightcomplete his call and depart, base station can become inactive(ex., in small cell networks some of the base stations might be

0 3000

0.16

0.2

0.24

Rat

e

Rate conve(7 BS, 2 users per

Iteratio

Fig. 5. Rate convergence for Syst

switched off to conserve energy), the demands of a user canchange, user position or user-BS associations can vary and so on.Like most of the stochastic approximation algorithms, even UP-AMCN can track the changes in such scenarios, if the update co-efficient lt converges (as t !1) to a strictly positive value. In thissubsection, we provide some simulation examples to show that theproposed algorithm effectively tracks the dynamics in the system.

In Figs. 7 and 8, we consider one such dynamic scenario. As-sume that a single cell (BS1) is active to begin with. After a while,a neighboring cell (BS2) becomes active and a while later anothercell (BS3) becomes active. We consider 8 users in each cell, withthe respective demands given by r1 ¼ ½:132 :264 :396 :528:660 :792 :924 1:056� (for users in first cell), r2 ¼ ½:099 :198 :297:396 :495 :594 :693 :792� (users in second cell) and r3 ¼ ½:066:132 :198 :264 :330 :396 :462 :528�. The UPAMCN algorithminitially converges to satisfy the demands of the users of the firstcell, then it starts running even at the second cell and the powerlevels in both the cells adjust to meet their respective users de-mands and finally when the third cell becomes active the powerlevels adjust again. We notice that at the time instances when anew BS becomes active, the demands are not met, the rates dimin-ish, but, shoots back within a small time interval (observe thenotches in Fig. 8). In all, we observe that the algorithm tracks thedynamics in the multicell system and satisfies the user demandsas long as the rates lie within the capacity region (see Fig. 8), albeitwith different power levels in different situations (see Fig. 7).

In another scenario Figs. 9 and 10, before BS2 becomes active,the demand rates of users 4 and 7 in BS1 change (observe thedemand rates changing around 3000 iterations in Fig. 10). Thenew rate vector is given by r1 ¼ ½:132 :264 :396 :554 :660:792 :878 1:056�: Again we observe that the algorithm tracks thesechanges by suitably adjusting the power levels (Fig. 9) to meet thenew demands (Fig. 10).

6. Heterogeneous agents

Often a multicell system consists of a heterogeneous network,which might support different agents with different purposes,who nevertheless interfere with each other. For example, some-times the Macrocell sites are overlaid with Small cells (Pico andFemto cells) to fill the coverage gaps and enhance the capacity ofthe network (see Fig. 11). Typically these small cells, comprisingof portable base stations are deployed in residential homes, supermarkets, office spaces and hot spots to name a few. This is just oneexample of a multi-tier network and one might find many suchscenarios. These heterogenous interfering agents still have to

rgenceBS, γ = 0.5, σ 2 = 0.1)

0 3000

ns

em S4 and S5 (H2 Network).

0 30000

0.1

0.2

0.3

0.4

0.5

0.6

Pow

er


0 3000

Iterations

Fig. 6. Power convergence for Systems S4 and S5 (H2 Network).

0 50000

0.1

0.2

Pow

er

0 5000


Iterations0 5000

BS 1 BS 2 BS 3BS3

becomesactive

BS2becomes

active

Fig. 7. Power convergence as new cells are activated.

0 50000

0.5

1

rate

0 5000


Iterations

0 5000

BS 1 BS 2 BS 3

BS2becomes

active

BS3becomes

active

Fig. 8. Rate convergence as new cells are activated.


satisfy their respective users demands. A power allocation algo-rithm, like the one proposed in the previous sections can probablybe used again to satisfy the demands. However, these heteroge-nous agents need not be synchronized, may not have the same pro-

cessing capabilities and hence their power profiles need not beupdated at the same rate.

In this section, we study the effects of some of these disparities.We will show that the UPAMCN algorithm converges to the same

0 50000

0.1

0.2

Pow

er

0 5000


Iterations0 5000

Demandchange

BS1 BS2 BS3

Fig. 9. Power convergence as new cells are activated and as demands (of User 4 and 7 of BS1) change.

0 50000

0.5

1

rate

0 5000


Iterations0 5000

BS1 BS2 BS3

Demandchange

Fig. 10. Rate convergence as new cells are activated and as demands (of User 4 and 7 of BS1) change.

Fig. 11. A heterogeneous network.


power profile in spite of ’update rate’ differences. We demonstratethis analytically via a simple example using two time scale (sto-chastic approximation) analysis, while the same is shown for morepractical scenarios via numerical examples.

via UPAMCN algorithm (Eq. (7)), all the agents (for example

base station j) update their own components (e.g. ptj ¼ pt

k;j

n ok) in

a decentralized manner, i.e., without the requirement of the otheragents updates (e.g. without Pt

�j ¼ fptj0 ; j0 – jg) and this is true for

all time t. Basically the agents (e.g. agent j) observe the estimates

required for their own iteration (e.g. dtk;j

n o) directly, avoiding the

need for the knowledge of the other’s parameter (e.g. Pt�j) thus

leading to a distributed algorithm. However, it still needs thevarious agents to be synchronized, i.e., it needs that for every basestation j the update co-efficient l is same and depends only upontime, i.e., at time t it equals lt for all j (see Eq. (7)). In a heteroge-nous network, various agents may update their components at dif-ferent rates. For example, the macro cells can operate at a muchslower rate than the small cells or vice versa. That is, (for example)there can be some index N1 such the agents with j 6 N1 update atevery time step while the agents with j > N1 update only once in jtime steps:

ptþ1k;j ¼PAj

ptk;j�

lt

dtk;j� rk;j

� �h iwhen j6N1

ptþ1k;j ¼

PAjpt

k;j� �n dt

k;j� rk;j

� �h iif j>N1; t¼jn for some n2f1;2; � � �g

ptk;j else:

8<:ð10Þ

6.1. Analysis using Two time scale algorithms

Stochastic approximation algorithms can be described in a gen-eral setting by:


ptþ1j ¼ pt

j þ ltj Wj Pt ;Vt

j

� �; Pt ¼ pt

1 � � � ; ptN

for all j 6 N; ð11Þ

where fVtjg is a random process that can for example, represent

some form of estimation error while estimating the observablesfWjg that depend upon the parameter P. These algorithms can beapproximated by the solution of the ODE (see for e.g. [1]):

_pj ¼ wjðPÞ with wjðPÞ :¼ E½WjðP;VÞ� for all j ð12Þ

and are shown to converge to a zero of the average function½w1; � � � ;wN�, under appropriate conditions whenever lt

j ¼ lt for allj.

There are situations in which different agents can update at dif-ferent rates, i.e., lt

j need not be the same for all j. For example onemight have in (11) for some N1 < N

ltj ¼ lt

1 for all j 6 N1 and ltj ¼ lt

2 for all j > N1 with lt1 – lt

2:

Such situations are analyzed via two time scale based stochasticapproximation results (see for example [4,1]) when saylt

1 ¼ o lt2

� �. These analysis primarily assume that, for any given

fixed value of the slower component ðPs :¼ fpjgj>N1Þ, the ODE corre-

sponding to the faster components ðPf :¼ fpjgj6N1Þ

_Pf ðtÞ ¼ wf ðPf ðtÞ;PsÞ with wf :¼ ½w1; � � � ;wN1 � ð13Þ

has a unique globally stable attractor A�ðPsÞ. Under some moreassumptions, it is shown that the trajectory of the slower compo-nents is approximated by the ODE (see [1, Theorem 6.1, pp 287],[4, Theorem 1.1])

_PsðtÞ ¼ wsðPsðtÞ;A�ðPsðtÞÞÞ with ws :¼ ½wN1þ1; � � � ;wN � ð14Þ

and the slower components converge towards the limit set of theabove ODE.

So we have two sets of ODEs, joint ODE (12) for single time scalealgorithms and two coupled ODEs (13) and (14) for two time scalealgorithms. And the comparison of the limits of any algorithm (e.g.UPAMCN), with or without the same update rate by all the agents,can be done by comparing the zeros of the right hand sides of theseODEs. One can notice7 that the two sets of limit points will be‘‘same’’ in the following sense:

(a) if P�s is an attractor (hence a zero of the RHS) of the ODE (14)then P�s ;A

� P�s� ��

is an attractor of the joint ODE (12);(b) if P�s ;P�f

� �is an attractor of the joint ODE (12) then neces-

sarily P�f ¼ A� P�s� �

(because ODE (13) has a unique attractorfor any given Ps) and hence P�s is an attractor of the ODE(14).

Remark 1. This implies that the algorithm converges to the sameset of limit points, irrespective of the disparities in the updaterates. The algorithm (10), that we would like to study, is differentfrom the two time scale algorithms discussed just above. Here, weneed the result when say lt

1 > 0 for all t and lt2 > 0 only when

t ¼ jm for some integer m. We approximate this algorithm withan algorithm in which the slower component is also updated everytime slot but with a smaller update co-efficient, that is whenlt

2 ¼ lt1=jk (and in the limit jk !1). This system can now be ana-

lyzed using the two time scale ODES (13) and (14).Example Scenario, Low SNR regime: We consider Low SNR re-

gime and a single cell updating fast in comparison with the rest,i.e., as in Eq. (10) with N1 ¼ 1. Using [1, Theorem 6.1, pp 287] we

7 the precise analysis would require some extra conditions and here we are justpointing out the general outline. We obtain the exact analysis for an example in thecoming subsection.

show that the UPAMCN converges to the same demand satisfyingpower profile, as it would have done in case all the users wereupdating at the same rate (i.e., when j ¼ 1).

We consider system C-I-ZF under low SNR regime for which (ifx is small, logð1þ xÞ � x)

RC�D�ZFk;j ðP;HÞ �

ptk;jPN

l¼1;l–jcltr Ht

j;lQlHtj;l

H� �

K þ r2k;j

: ð15Þ

Let us say without loss of generality that the BS 1 updates fast, i.e., itupdates its power profile every time slot, while the rest update oncein j time slots where j is very large. We approximate it with a sys-tem in which the rest of the components are updated every timeslot, albeit with a smaller coefficient lt=jt with jt !1, say as be-low (for all k; j):

ptþ1k;j ¼ PAj

ptk;j �

lt

dtk;j � rk;j

� �h iwhen j ¼ 1

ptþ1k;j ¼ PAj

ptk;j �

ltjt

dtk;j � rk;j

� � �when j > 1:

ð16Þ

We study the above heterogenous UPAMCN using the two coupledODEs (13) and (14). The fast ODE (13) for UPAMCN, at slower com-ponents P�1, equals:

_pk;1 ¼ rk;1 � pk;1ak;1ðP�1Þ with ak;jðP�1Þ :

¼ EH1PN

l¼1;l–jcltr Hj;lQlH

Hj;lð Þ

K þ r2k;j

264375; ð17Þ

while the ODE (14) corresponding to the slower components (i.e.,BS j with j > 1)

_pk;j ¼ rk;j � pk;jwk;jðP�1Þ þ zk;j for any k and j > 1 and where

wk;jðP�1Þ :¼ ak;j p�1ðP�1Þ;P�1

�j

� �with

p�1ðP�1Þ :¼ p�1;1ðP�1Þ; . . . ; p�K;1ðP�1Þh i

and with p�k;1 :¼ rk;1

ak;1ðP�1Þ:

ð18Þ

Note in the above that p�1ðP�1Þ represents the attractor of the fasterODE when the slower components are fixed at P�1. We obtain thefollowing (Proof in Appendix C):

Theorem 3. Under the assumptions A.1–4 and for the system (16),whenever the interference level cmax :¼max cl is within a limit, thebelow result is true. For every d > 0, the fraction of time the tail of theslower components of the UPAMCN algorithm (fPs

�1gs>t with initial-ization Pt < P) spends in the d-neighborhood of the limit set, Lslow, ofthe slower ODE (18) tends to one (in probability) as t !1: Further,when Lsys the set of the demand satisfying power profiles is inside thecapacity region CsysðPÞ, it is a part of the limit set Lslow in the sense:

Lsys 2 p�1ðP�1Þ;P�1

;P�1 2 Lslow� �: }

The above theorem also characterizes the limit set of the ODEs.We show that all the demand satisfying power profiles are indeedthe attractors (for this low SNR example) and hence constitute thelimit set (see the proof in Appendix C). In fact, the result about thelimit set is correct even when we consider all agents updating atthe same rate. So the main conclusion is that the UPAMCN is noteffected by the variation of the update rates at different agents.The example considered in this section is a restrictive example,and we have partial theoretical justification in this case. However,in the next sub section, via some numerical examples, we indeedshow that UPAMCN is unaffected by disparities in the update ratesfor many general cases.

Table 6Demand Satisfying NE.

j Demand satisfying NE (converged power)

1 .0118 .0238 .0368 .0505 .0655 .0813 .0983 .11682 .0121 .0242 .0373 .0513 .0664 .0824 .0998 .11835 .0118 .0240 .0368 .0508 .0656 .0817 .0987 .1170

10 .0120 .0241 .0371 .0510 .0660 .0821 .0993 .117825 .0120 .0242 .0375 .0515 .0666 .0828 .1001 .118650 .0120 .0242 .0374 .0514 .0665 .0827 .0999 .1184

100 .0120 .0243 .0375 .0516 .0667 .0829 .1002 .1189

System S2 with heterogenous users convergence with different j


6.2. Numerical examples

We consider a network with 3 base stations as in Fig. 11. Thisnetwork has a macro BS (BS1) and two small cell base stations(BS2 and BS3), both of which update at a faster rate in comparisonwith the macro BS. Each of them support 8 users. Further, weassume system configuration S2, described in Table 4. Note thatS2 is a system which supports only finite number of transmissionrates and hence is a more practical example in comparisonwith the one studied using the two time scale analysis in theprevious subsection. To keep it simple, we assume that thedemand rates of users in each cell is the same and is given byri ¼ ½:099 :198 :297 :396 :495 :594 :693 :792� for i ¼ 1;2;3. Theinterference factor c ¼ 0:5. We show via this case study the work-ing of the algorithm and claim that due to the inherent nature ofthe algorithm, UPAMCN can converge and track in a variety ofscenarios.

The power profile at the macro cell is updated once in j timeslots while the small cells update every time slot. We plot the UP-AMCN power updates for two values of j (1 and 10) in Figs. 12 and13. We see from the figures that the demands are met, power pro-file converges to the same point irrespective of j, albeit with differ-ent speeds. We further observe from the two figures that theconvergence speed of the slower components is proportional toj, which is quite intuitive (the effective number of updates before

0 50000

0.05

0.1

Pow

er

0

Power conv(3 BS, 8 users pe

Iterati

κ =1κ =10

BS 1 BS

Fig. 12. Power convergence w

0 50000

0.5

1

rate

0

Rate conve(3 BS, 8 users pe

Iterati

κ =10κ=1

BS 1 B

Fig. 13. Rate convergence w

convergence remain the same). More surprisingly the convergencespeed of the faster components does not depend much upon j(though both the systems are coupled). The reason for this beingthe following: the attractors of the faster components P��1ðp1Þare (Lipschitz) continuous in the slower component p1 and hencewill vary little with small changes in the slower component andhence it appears to have converged faster.

We further illustrate the robustness of UPAMCN algorithmagainst update rate disparities in Table 6. Here, we tabulate theconverged power profile of all the eight users of the macro cellwith different j. We do not tabulate the quantities correspondingto the small cells as they do not change much with j. We observe

5000

ergencer BS, γ = 0.5, σ 2 = 0.1)

ons0 5000

2 BS 3

ith heterogenous users.

5000

rgencer BS, γ = 0.5, σ 2 = 0.1)

ons0 5000

S 2 BS 3

ith heterogenous users.

Table 7Time to converge.

j User 8 Small cell (BS 2) User 8 Macro cell (BS1)

1 506 5072 505 9575 501 2321

10 506 454025 507 1126950 506 22907

100 506 45014


that UPAMCN converges to the same power profile for all the val-ues of j. In Table 7, we also tabulate the number of time stepsneeded to converge to a ball which is within 5 % of the demandrates for all the users of the macro cell. We see that (roughly) thenumber of time steps for the convergence is proportional to j.

7. Conclusions

Mobile broadband users demand certain rates depending on theend application and QoS requirements. The base station servingthese users has to allocate power to satisfy user demands operatingwithin its own total power budget. Intra-cell and inter-cell interfer-ence diminish the available rates in multicell networks. Neighboringbase stations can co-operate to exchange some form of channel stateinformation depending on backhaul capacity and processing powerto alleviate interference and thus enhance achievable rates. Further,system specific components like modulation, coding, rate allocation,channel estimation and synchronization impact the achievablerates and hence the power allocation. We observe that the powerprofile satisfying the demands depends upon the exact configura-tion of the system and we propose an universal power allocationalgorithm which works in a multitude of such settings. The stochas-tic approximation based universal power allocation algorithm runsat each BS, independently and simultaneously to meet the user de-mands as long as the demands are achievable. The power allocationis formulated as a game problem. A system specific capacity regionis defined and the proposed algorithm is analyzed using an ODEframework. The proposed algorithm works well in a multitude ofsystem configurations as demonstrated via simulations and analy-sis: it converges to the system specific power profile, which satisfiesthe demands asymptotically.

The proposed algorithm assumes that the serving BS always hassufficient amount of data to transmit. However, in many applica-tions, the data is available in real time. We mention a possibleextension of the same in the paper.

Cellular networks with heterogenous users can have differentprocessing powers and may update their power profiles at differ-ent rates. We show that the proposed algorithm works irrespectiveof the disparities in the update rates using two time scale analysisand numerical simulations: the algorithm converges to the samepower profile as long as the demands and the system configurationremain the same, irrespective of the disparities in update rates.

Acknowledgments

This research is carried out in the framework of the INRIA andAlcatel-Lucent Bell Labs Joint Research Lab on Self Organized Net-works and the Alcatel-Lucent chair on Flexible Radio. The work ofV. Kavitha is supported by CEFIPRA.

Appendix A. Example systems

(1) Asymptotic Ideal Rate system: In a multicellular system withlarge number of antennas at the BS and large number ofusers, the rate for a given CS can be obtained using random

matrix theory. For example, in [11] the asymptotic rates arederived for a zero forcing (ZF) precoder. It is shown that foralmost all realizations of CS, the rate can be approximated bythe expression given below in Eq. (19). Further, we considera system in which, the base stations use channel codingschemes to transmit very close to the theoretical rates.When this system (which we call as asym-ideal-zeroforcingor in short A-I-ZF according to our notations) uses powerprofile P and when the channel state (CS) isH, the BS j trans-mits to the user k at rate ([11]) (when M > K):

RA�I�ZFk;j ðP;HÞ � log 1þ

pk;j

1b�1

PNl¼1;l–jcl

trðPlÞK þ r2

k;j

!ð19Þ

where, b ¼ M=K is the ratio of number of transmit antennason the BS to the number of users and cl 2 ð0;1Þ representsthe interference from cell l. This rate is same for almost allCS H as it is an asymptotic rate. Similar expression is avail-able for the case with M ¼ K in [11].

(2) Ideal rates using complete CSIT: If the number of antennae/number of users is not large enough, the asymptotic resultsare not accurate. If BS has access to CSIT (and if each BS couldchannel code to obtain rates closer to the ideal rate) thenwith ZF precoder it transmits at rate:

RC�I�ZFk;j ðP;HÞ ¼ log 1þ

ptk;jPN

l¼1;l–jcltr Ht

j;lQ lHtj;l

H� �

K þ r2k;j

0B@1CA

Q l :¼ HtlH

Htl H

tlH

� ��1Pt

l Htl H

tlH

� ��1Ht

l

For the same configuration, but without transmitter precod-ing, the instantaneous transmission rate (as obtained usingShannon’s capacity expression), from Eq. (2) is:

RC�I�NOk;j ðP;HÞ ¼ logð1þ gk;jÞ where

SINR;gk;j :¼ E½juk;jj2�qk;j

with

noiseþ interference;qk;j :¼X

l

E½jik;j;lj2� þ r2k;j

ð20Þ

(3) Finite number of Rates: Ideal rate systems are not realistic,they can’t be implemented in practice. We consider a sys-tem, in which the BS can transmit at one of the available dis-crete rates from the set R. When transmitter has CSIT, itknows the exact theoretical rate and hence will pick thelargest rate from set R that is smaller than the current the-oretical rate:

RC�D�ZFk;j ðP;HÞ ¼ inf

r2Rfr 6 RC�I�ZF

k;j ðP;HÞg; ð21Þ

RC�D�NOk;j ðP;HÞ ¼ inf

r2Rfr 6 RC�I�NO

k;j ðP;HÞg: ð22Þ

(4) Rate adaptation Without CSIT: It is once again not realistic toassume the knowledge of complete CSIT. There are manyschemes that estimate the rate blindly or using some partialCSIT (e.g. [12]). The UPAMCN algorithm is a very generalalgorithm and works with all those systems which satisfyassumptions A.1–4. These are quite simple assumptionsand most of the systems can satisfy these and hence thealgorithm works for majority of the blind/partial CSIT sys-tems.We explain one such blind system wherein, the BS estimatesthe transmission rates without knowledge of CSIT. Eachtime, the BS begins by attempting at the highest availablerate r1. If the data is not received correctly (information


obtained via a feedback from the receiver), the BS sendssome more information about the same data packet so thatthe overall rate now is the second highest r2. This procedurerepeats until the two agree upon the correct rate. Weassume that this rate adaptation system is always successful,i.e, it can estimate the actual rates without errors. Such asystem does not require CSIT, however the final rate atwhich the transmission takes place depends upon the cur-rent channel state in exactly the same way as in the caseof C-D (or A-D for large antenna and user case and note thereis no channel coding in this case as there is no CSIT) andhence,

RN�RA�NOk;j ðP;HÞ ¼ RC�D�NO

k;j ðP;HÞ ð23Þ

(5) Rate Adaptation with local CSIT: All the base stations havelocal CSIT, i.e., BS j knows the Hj;j part of CS. However theycan’t estimate the current rates just based on local CSIT.So, they once again use rate adaptation technique as inthe system (4). They can however design a better systemby using for example a zero forcing precoder. In thiscase, as in system (4) the rate will be adapted to theactual underlying rate and hence will be same as that inC-D-ZF:

RL�RA�ZFðP;HÞ ¼ RC�D�ZFk;j ðP;HÞ ð24Þ

(6) Rate Adaptation with errors: There can be some errors in rateadaptation algorithm of system (4) or (5). In this case

RN�RAE�NOk;j ðP;HÞ ¼ RN�RA�NO

k;j ðP;HÞ � Ek;j RN�RA�NOk;j ðP;HÞ

� �ð25Þ

RL�RAE�ZFk;j ðP;HÞ ¼ RL�RA�ZF

k;j ðP;HÞ � Ek;j RL�RA�ZFk;j ðP;HÞ

� �ð26Þ

where (assuming independent errors) Ek;jð�rÞ can take valuesin the subset R \ fr 6 �rg with a given probabilitydistribution.

Appendix B. Proofs

Proof of Theorem 1. As a first step, we rewrite the algorithm as in[1]:

Ytk;j :¼ rk;j � dt

k;j; ptþ1k;j ¼ PAj

ptk;j þ ltYt

k;j

h i:

Define, F t :¼ r Ps; Ys�1k;j

n ok;j; for alls 6 t

� �and let Et represent the

expectation w.r.t. F t , the filtration. Under the assumptions A.2 andA.3 clearly the condition expectation

Et Ytk;j

h i¼ gp

k;jðPtÞ :¼ rk;j � Ravg;k;jðPtÞ for all k; j and t:

For every j, the constraint set Aj satisfies the assumption (A3.2),

page 107 of [1]. By assumption A.3 Ytk;j; t

n ois uniformly integrable

and hence satisfies assumption A.2.1, pp. 258 [1]. They also satisfythe assumptions A.2.3 to A.2.7 of pages 258, 259 [1] withgt ¼ �g ¼ gp and with bt ¼ 0 nt ¼ 0 for all time t. Assumption A.2.2,pp 258, [1] is satisfied because of our Assumption A.4. Let zj repre-sent the projection or constraint term, the minimum force neededto keep the vector pj in Aj. Then by Theorem 2.3, pp. 259, [1] the

UPAMCN algorithm trajectory Ptconverges weakly to the trajectoryof the solution of the ODE (8) (in the sense as explained in [1]). Fur-ther by the same theorem of [1], for any d > 0, the fraction of time

that the tail sequence fPsgsPt , with initializations ptk;j ¼ ~pk;j for

every ðk; jÞ, spends in the d-neighborhood of the limit of set of theabove ODE (8), BdðLODEÞ, goes to one (in probability) as t !1. }

Proof of Lemma 2. The boundedness assumption A.3 is direct fordiscrete rate systems and is also true for ideal rate systems as seenfrom the formulas. The ideal rates are point wise continuous andare bounded and hence by bounded convergence theorem satisfythe continuous assumption A.4. The same for the discrete rates isgiven by Lemma 3.

The continuity assumption A.4 now also holds for the rate adap-tation system with errors, L-RAE-NO, whenever the statistics of theerrors fEk;jg are independent of the power profile or when they arecontinuous in P. Thus for all the systems considered in this paper The-orem 1 applies.

Conditions for existence of demand meeting NE: For all thesystems considered so far, the hypothesis of Lemma 1 is satisfied,i.e., LNE is non empty whenever the power constraints are sufficientto cater to the demand rates. This fact is established by the conti-nuity of the average rates w.r.t. the power profile, i.e., the estab-lishment of the assumption A.4. To be precise Lemma 1 issatisfied, i.e., LNE is non empty whenever for all k; j rk;j 6

supP 6 PRavg;k;jðPÞ: }

Lemma 3. The average rates Rsysavg;k;j for systems C-D-ZF and C-D-NO

are continuous w.r.t. power profile P for all k; j.

Proof. Let Rk;jðP;HÞ represent the corresponding ideal rate (therate before discretization) for the given CS H. From all the rate for-mulas in this paper, we can see that these rates bounded and arecontinuous in P, for all H. For the discretized systems, the averagerates can be written as,

Ravg;k;jðPÞ ¼Xi6N

qði;PÞri where

qði;PÞ :¼Z

1fri�16Rk;jðP;HÞ6rigdCðHÞð27Þ

with dC representing the Gaussian measure. For a given P, theprobability of the sets of the type (the boundaries of the sets usedwhile defining the indicators in (27))

CðfH : Rk;jðP;HÞ ¼ rigÞ ¼ 0;

because of the continuity of the Gaussian measure. Hence, the pointwise functions of integral (27) are continuous w.r.t. toP for almost allH. Thus the lemma follows by bounded convergence theorem. }

Appendix C. Proofs related to two time scale algorithms

Proof of Theorem 3. We obtain this proof via the [1, Theorem 6.1,pp 287]. The [1, Assumptions A6.0, A6.1, A6.2, A6.3 and A63.5, pp287] hold for this example as is shown in the proof of Theorem 1.Thus it remains to prove the [1, Assumption A6.4], in order to apply[1, Theorem 6.1, pp 287]. The ODE corresponding to the fastercomponent (BS 1) for a given value of the other user’s parametersP�1 is given by (17) (see [1] for details). It is easy to see that thisODE has unique solution,

pk;1ðtÞ ¼ p�k;1ðP�1Þ � e�ak;1ðP�1Þt with

p�k;1ðP�1Þ :¼ rk;1

ak;1ðP�1Þfor all k

and that p�1ðP�1Þ is its unique globally stable attractor. Further,there exists a constant m <1 such that,


jak;1ðeP�1Þ � ak;1ðP�1Þj 6mcmax

r4k;1

jeP�1 � P�1j with

cmax :¼max cl for all k; ð28Þ

and thus the function p�1 is locally Lipschitz, satisfying [1, Assump-tion A6.4]. By [1, Theorem 6.1, pp. 287], the tail of the trajectory ofUPAMCN spends its time mainly in the limit set of the mean ODE(see [1] and with wk;jðP�1Þ :¼ ak;j p�1ðP�1Þ;P�1

�j

� �),

_pk;j ¼ rk;j � pk;jwk;jðP�1Þ þ zk;j for any k and j > 1:

We now characterize this limit set, denoted by Lslow. We will showthat every internal (which is not on the boundary of the constraintset) demand satisfying power profile is a part of Lslow: i) it is easy tosee that p�1 P��1

� �;P��1

is an internal demand satisfying power pro-

file if and only if P��1 is an internal zero of the RHS of the above ODE;ii) below, we will show that every internal zero of the RHS of theabove mean ODE, will be an asymptotically stable attractor andhence is in Lslow.

Let P��1 be any internal zero of this ODE and let ek;j :¼ pk;j � p�k;jfor every k and j > 1. Consider a neighborhood of P��1 which iscontained inside the constraint set (hence the projection termwould be zero) and in this neighborhood we have (notep�k;jwk;j P��1

� �¼ rk;j)

_ek;j ¼ p�k;jwk;j P��1

� �� pk;jwk;jðP�1Þ

¼ �ek;jwk;j P��1

� �þ pk;j wk;j P��1

� �� wk;jðP�1Þ

� �:

Let E represent the vector fek;jgfk;j>1g. There exists a constant cdepending upon the radius r and P��1 such that for all E withjEj 6 r (using the upper bound (28)),

< _E; E ><Xk;j>1

�wk;j P��1

� �þ cmmax

� �jEj2:

Thus by [14, Global existence theorem, pp 169-170],

jEjðtÞ6e

Xk;j>1

�wk;jðP��1Þþcmmaxð Þtwhen initial condition Eð0Þ2fE : jEj< rg:

Hence as the interference reduces, i.e., as mmax ! 0 the exponentterm becomes negative (in which case, the error tends to zeroasymptotically). For these small values of interference, every inter-nal zero is an asymptotically stable attractor. }

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Satisfying demands in a multicellular network: A universal power allocation algorithm