Kam il SENEL,Mehmet AKAR

Department of Electrical and Electronics Engineering,Bogazici University,Istanbul,34342,Turkey

1 Introduction

The next generation cellular networks will have to satisfy dem ands of higher data rates from increasing number of devices.The traditional cellular networks are not equipped to cope with these demands as providing service to indoor users with outdoor base stations is not efficient.Considering that most of the data transmission originates from indoor users,along with limitation on available spectrum,it is clear that current cellular architecture needs an overhaul in the near future.

Utilizing sm aller cells based on the idea of reducing the distance between the transmitter and receiver,received the attention of many researchers recently.Femtocell networks is an emerging technology which has potential to provide the data rates required by the services of next generation cellular networks.In theory,femtocellscan take the load from the load from the macrocell base sta tions and provide coverage to indoorusers in an efficient manner[1,2].Despite all the potential enhancement of heterogeneous networks,problem ssuch as interference management,base station association,handovers,etc.,needs to be resolved[3].In this paper,we focus on the interference m anagem ent problem.

The allocation of the available spectrum is an integral part of interference management.Re-using the spectrum between the tiers of the network introduces crosstier interference whereas a split spectrum setup needs to consider only the co-tier interference[4,5].However,due to spectrum scarcity and design difficulties,shared spectrum is preferred by the operators[6].In a shared spectrum setup,some users may end up with inferior signal quality due to interference originating from different tiers.The situation is especially arduous for macrocell users as femtocell users are usually closer to their base stations and have better channel quality.There are various approaches which focus on protecting macrocell users by utilizing the feedback from the macrocell users[7-9].Among the variety of interference management approaches in heterogeneous networks,M IMO techniques[10,11],coordinated multipoint(COMP)[12,13],successive interference cancellation[14,15]and power control are some of the techniques which have received considerable attention of researchers.

In[16],an interference management technique that protects the macrocell user’s signal quality via limiting the interference experienced by the macrocell user is introduced.A coverage coordination and power ad justment algorithm which maximizes the overall throughput of the system is given in[17].Power control problem along with base association problem is considered in[18].Another approach where each user aim s to maximize its ow n signal quality based on a non-cooperative Stackelberg gam e approach is presented in[19].An evolutionary game theoretic approach for self organization of small cells in heterogeneous networks is proposed in[20].The self optimized coverage coordination algorithm(SOCC)introduced in[21]uses the statistics of the received signals to obtain a fixed threshold value for each BS.However,the approaches mentioned above are based on the assumption that the users have access to or can estim ate the channel conditions.Furthermore,users have predefined SINR values to achieve and/or utilize threshold values for the interference they create.However,the existence of a power allocation vector such that these threshold values can be satisfied is not guaranteed.

In a related work,we have proposed a consensus based distributed power adjustment algorithm for twotier networks[22].In this paper,we extend our previous results by considering different spectrum allocation setups and two different fairness indices.

The key contributions of this paper can be summarized as follows:

.For the power control algorithm presented in[22],different spectrum allocation schemes are considered.The performance of the power control algorithm is investigated for split and shared spectrum setups.

.The convergence analysis of the power control algorithm is extended to include different spectrum allocation schemes and it is shown to converge independent of spectrum setups.

.The rate of convergence is shown to be exponential under split and shared spectrum setups.

.The fairness of the power control algorithm is investigated using two different metrics,namely Jain’s and Atkinson’s fairness indices.The algorithm is shown to achieve perfect fairness independent of the number of users or spectrum allocation setup.

.The self-organization properties of the power control algorithm are analyzed theoretically by using time varying connection matrices.The results are further verified by numerical Simulations which illustrate the performance of the power control algorithm when a new BS is deployed or disconnected from the system.

.The effect of crucial parameters on the performance of the power control algorithm is investigated.The neighbor set and connection distance parameters,which define the underlying communication matrix of the network are considered.The results suggest that rate of convergence increases with increasing flow of information.

The rest of the paper is organized as follow s.In Section 2,system setup and mathematical preliminaries are given.Theoretical analysis is carried out in Section 3,including stability analysis under time-varying underlaying communication topology and convergence rate.The verification of the results and performance evaluations are provided in Section 4.Finally concluding rem arks and possible future research directions are presented in Section 5.

2 System setup

We consider the setup given in[9]with a single macrocell of radiusRcunderlaid with femtocells each one having radiusRf.Fem tocell users are distributed inside a disc with radiusRfcentered at their associated base station.We assume that base station association is constant during adjustment process.Letpiandi∈I={1,...,N}denote BS transmission power for its useri∈I and set of users,respectively.The signal to interference and noise ratio(SINR)for useri,denoted by Γi,is

wheregijrepresents the channel gain between theithuserand thejth transmitter,videnotes the thermalnoise experienced by useri.We assume that any possible SINR gains through post processing techniques such as diversity reception,interference suppression,etc.are included in the termgii.

A power control algorithm aim s to find a power allocation vector satisfying the minimum SINR requirement,

w here γiis the minim um acceptable SINR for theith user.Com bining(1)and(2),and using vector notation we obtain

w here=d iag{γ1,...,γN},=is the norm alized noise vector;and theN×Nnormalized link gain matrixH=[hij]is given by

The target SINR γivalues are achievable with a nonnegative power vector if the spectral radius of the matrixRH(denoted as ρ(RH))is less than or equal to 1.For the case when ρ(RH)=1,γi’s are achievable if there is no noise in the system(η=0)[23].

For the case where the given SINR targets are achievable,the non-negative power vector that satisfies(2)with equality is given by

is pareto optimal in the sense that any vectorpthat satisfies(2)requires at least as much power asp?com ponent-wise.Furthermore,the maxim um achievable SINR value for the case whereR=γIis given by

Here,γ?is the m ax-m in SINR solution.A drawback is that the channel gain values and the feasible SINR values are not know n beforehand,a problem which w e can overcome by not utilizing predefined target SINR values[22].

Two different frequency allocation approaches are considered:shared and split spectrum setups.Sp litting the spectrum between the tiers of the network avoids cross-tier interference,however it reduces the available spectrum for tiers of the network.On the other hand,sharing spectrum gives every tier access to available spectrum at the cost of cross-tier interference.A detailed comparison of different approaches in spectrum allocation is given in[24].

2.1 Shared spectrum setup

The power update algorithm for the shared spectrum setup is presented in this part.Assum e that index 1 refers to the macrocell without loss of generality.Then,the power update of the macrocell BS is

whereas each femtocell BS,for alli∈I1,uses the following update

Here,N iis the neighbor set of useri,i.e.,the setofusers which exchange SINR values with useriandfij(t)denotes the connectionweights.To controltheadjustment speed of useri,a positive parameter βiis introduced.For the macrocell BS,the desired SINR value is defined as Γ1,des(t)=m in(γ(t), Γ1,ref(t)),where the reference SINR value Γi,reffor useriis given by

and the value γ(t) ∈ {γ1,γ2,...,γM}is chosen from a finite set,based on the service(e.g.,voice,data,video)the macrocell user is receiving.Γ1,des(t)provides the macrocell user two different options for power adjustment,either relying on SINR information received from femtocell users or using a predetermined threshold SINR value based on the application.

A distributed approach should try to minimize the amount of information exchange between BSs.For the algorithm given in(7),BSs only require the SINR information from the users in their neighbor set.O ther parameters,pi,Γi,required for update are know n at the BS.The required information on SINR values can be exchanged using the operator’s backbone or Internet connection of the end users.The connection weights,fij(t)’s are discussed in detail in Section 2.3.

2.2 Sp lit spectrum setup

The power control algorithm for femtocellBSs,for alli∈I,under the split spectrum setup is as

where parameters are defined in the same way as in the shared spectrum setup.Note that for the split spectrum case,a power update algorithm for macrocell BS is not included as there is no cross-tier interference in the system and two tiers of the network are decoupled.

2.3 Connection weights

The connection weights can be considered as the elements of an underlying graph.This graph is formed by the exchange of information on SINR values between BSs.LetL=[lij]be theN×Nconnection matrix defined by

wherefijare the connection weights.Note that the connection matrixLcontains all the information on the underlying graph which represents the exchange between BSs.The connection weights determine how a BS utilizes the information received from other BSs and are crucial for the performance of the power control algorithm.The connection weights are chosen with the following motivations:

i)Any exchanged information is utilized in the adjustment process.

ii)information exchange should be mutual.

iii)The adjustment should be based on the difference between a users SINR value and a scaled average of the received SINR values.

To accomplish these objectives,connection weights are assumed to satisfy the following conditions for allt.

Assumption 1i)There exists a positive constant δ such thatfij(t)≥ δ ifj∈Ni;otherwisefij(t)=0,?i,j∈ I;

ii);and

One simple choice of connection weights satisfying Assumption 1 is given by

where|Ni|denotes the cardinality of the setNiandNmaxis a bound on the maximum number of neighbors.For this set of parameters,(7)reduces to

Recall that the set of neighbors,Ni,for usericonsists of at mostNmaxusers.Nican be chosen to include the users closer to BSithan a threshold distance valueDmaxor by utilizing the user feedback reports which includes the interferer femtocell identifiers in LTE networks[4].

The exam p le choice of connection weights and the resulting power update algorithm given in(12)highlights the distributive nature of the algorithm.First of all,note that the scaling term s,namely Γi(t),pi(t),N i,Nmaxare locally available at BSi.Furthermore,the values of the connection weights utilized by other users are not necessarily communicated over the network,as shown in the exam p le.The update algorithm provided in(9)is a general form and in practice an algorithm such as(12)will be used where the values of connection weights are determined beforehand.The information to be relayed between base stations is the SINR values of the users in the neighbor set.Note that,this set does not necessarily include every other user in the system.Furthermore,we introduce aNmaxparameter to limit the size of this set.The simulation results suggest that the rate of convergence increases with the size of neighbor set,however the convergence of the algorithm is not affected.

3 Theoretical analysis

In this section,the stability properties of the power update algorithms given in(6)-(7)and(9)arepresented.We show that the analysis provided in[22]is independent of the spectrum allocation setup and is valid for both shared and split spectrum allocation schemes.

3.1 Stability properties for split spectrum setup

The stability analysis for the power control algorithm given in(9)is presented in this section.The connection matrixLis modeled as a time-varying matrix,since in a wireless network a BS is not likely to exchange information with the same set of users at all times.

In order to examine the convergence properties,(9)can be represented using vector notation as

whereId=d iag{I1,...,IN}is the diagonal matrix with normalized interference values at diagonals andB=d iag{β1,...,βN}.Now,consider the quadratic Lyapunov function candidate

w hose derivative along system trajectories(13)can be computed as

Note thatis negative for all non-equilibrium points and is equal to zero for equilibrium points.Furthermore,a connected communication graph yields a Lap lacian matrixLwith all eigenvalues in the open right half of the complex plane,except for a single eigenvalue at zero.This imp lies that consensus of SINR values is achieved[22].Furthermore,

w here α andkare two positive parameters.Hence,the convergence properties of the algorithm are preserved for the split spectrum scheme and we can state the following.

Theorem 1Suppose that Assumption 1 is satisfied and the underlaying graph is connected for alltunder a split spectrum setup.Then,the algorithm described by(9)converges exponentially to a fair solution such that Γi= Γj,?i,j∈ I.

3.2 Stability properties for shared spectrum setup

The stability properties of the power update algorithm(6)for the shared spectrum case are investigated next.Similar to previous section,the connection matrixLis assumed to be time-varying in the analysis.

To this end,re-consider the candidate Lyapunov function given in(14).The case whereboils down to the split spectrum setup case provided in the previous section and is therefore om itted.

We can find an upper bound onby

whereis given by

This im p lies that the shared spectrum setup reduces to the femtocell exclusive case analysis provided in[22]which allows us to state the following.

Theorem 2Suppose that Assumption 1 is satisfied and the underlaying graph is connected for alltunder a shared spectrum setup.Then,the algorithm defined by(6)-(7)converges exponentially to a fair solution such that Γi= Γj,?i,j∈ I.

Theorem s 1 and 2 demonstrate the ability of the power update algorithm s to reach a fair solution.Furthermore,the convergence is in exponential time which is a desired property for the wireless communications sytem s due to their highly dynamic nature.Next,we illustrate the performance of the algorithm via numerical analysis.

4 Numerical results

In this section,numerical results are presented for the power control algorithm s under a simulation setup consisting of one macrocell of radiusRcwith underlaidNffemtocells of radiusRf[9].For the split spectrum setup,we assume that there is no cross-tier interference between the macrocell and femtocell BSs and the algorithm given in(9)is used to ad just the transmission powers of femtocell BSs.In the shared spectrum setup,each BS uses the algorithm given in(6).

The path loss model used is the simplified path loss model described in[26]and is as follow s:

wherePLijandDijrepresent the path loss and the distance between userjand BSi,respectively.The simulation parameters are summarized in Table 1.In the Simulations,it is assumed that the neighbor set of each femtocell BS consists of those femtocell BSs with a distance less thanDmax=100m and the neighborsethasatmostNmaxBSs.Fem tocell BSs are randomly distributed inside a disc with macrocell radiusRcand femtocell users are randomly distributed inside a disc with femtocell radiusRf.A 10 m s step size is used for all simulations(i.e.,the number of iterations is 300 for a duration of 3 seconds).

Table 1 System parameters for Simulations.

In Fig.1,a split spectrum setup example with 25 fem femtocell users is shown where self-organizing properties of the power control algorithm are depicted.The femtocell BSs are initialized at maximum power and by em ploying the power control algorithm,BSs adjust their transmission power in a way that results in equal SINR value for each user.After the initial self organization,a new femtocell BS is deployed with maxim um transmission power which creates additional interference to existing femtocell BSs.By utilizing the SINR information from its neighboring BSs,the new deployed BS adjusts its transmission power and the system reaches consensus at a new SINR value.Note that the adjustment period for a new femtocell dep loyment is shorter com pared to the initial self organization period.Finally,the change of SINR values when a femtocell BS disconnects from the network is shown.As expected,the remaining femtocell BSs achieve a higher new consensus value after the disconnection of a femtocell BS.

Fig.1 An example of SINR change under split spectrum setup.

In Fig.2,a shared spectrum example and comparison with the SOCC algorithm is presented[21].The convergence of the average SINR for fem to BSs and m acro BS are depicted for both methods.It is seen that the power control algorithm leads to a fair equilibrium SINR level that is higher than the SINR levels for fem to BSs using the SOCC algorithm.A predetermined threshold value Γth=5.41 dB defined in[21]is used for the SOCC algorithm.An important drawback of using a predetermined threshold is that if the threshold value is not feasible,the algorithm diverges and the test for feasibility of a specific value requires the know ledge of channel gains.As show n in Section 3,the proposed power update algorithm s converge independent of channel gains or initial conditions.

Fig.2 Comparison of the consensus based power control algorithm with the SOCC algorithm.

4.1 Fairness analysis

In this part,the fairness performance of the power control algorithm s is investigated.For this purpose,two different fairness indices are utilized,namely Jain’s fairness index and Atkinson index.The performance analysis is carried out via com paring with SOCC algorithm under different spectrum allocation schemes as well as different number of users.

The Jain’s fairness index(FI)[27]is a fairness criterion that has been w idely used for resource allocation[4,28].fiis given by

The value of fichanges between 1 and 1/Ncorresponding to best and worst case of fairness,respectively.The simulation results for the change in fairness index are shown in Fig.3.The results are in agreement with the analytical results obtained in Section 3.Every simulation converges to a final state where fiisequalto 1,for both shared and shared and split spectrum setups and independent of the number of users.However,the speed of convergence is reduced with increasing number of users and as seen in Fig.3,split spectrum setup converges faster com pared to split spectrum setup;this is to be expected due to the initial inferior signal quality of the macrocell user.

The Atkinson index(AI)introduced by[29],is another fairness index which has been used for fairness performance of resource allocation algorithm s[30].AI is defined by

Here,μ denotes the average SINR value achieved by users andNis the number of users.? is a positive parameter which determines the sensitivity of the measure to higher or lower values for the given distribution.For values of ? around 0,the distribution of higher values have more weight in the AI value and as ? increases the weight of lower values increases. The best value of the AI is 0 corresponding to perfect fairness and the worst case is represented by an AI value of 1.The simulation results with two different ? values are illustrated in Figs.4 and 5 with ? =1 and ? =0.5,respectively.The fairness performance of the consensus based power control algorithm is com pared with the SOCC algorithm.Similar to Jain’s Fairness measure,perfect fairness is achieved by the power control algorithm and the SOCC algorithm results show com parable fairness values.Furthermore,the difference between two algorithm s lessens as ? decreases.This is a result of giving less weight to lower SINR values on AI value.

Fig.3 Fairness analysis.

Fig.4 Evaluation of fairness based on Atkinson Index(?=1).

Fig.5 Evaluation of fairness based on Atkinson Index(?=0.5).

4.2 Convergence speed evaluation

The effect of two important parameters,DmaxandNmax,on convergence speed is investigated in this part.Nmaxparameter is an upperlimit on the maxim um number of users that can be in the neighbor set of a user..This parameter effectively determines the maximum number of SINR values that a user can receive from other users and its scaling term.The change on SINR values with differentNmaxvalues is depicted in Fig.6.An important point is that the convergence of the algorithm is not effected by theNmaxvalue,however the speed of convergence is.Using a larger value ofNmaxresults in a slower convergence rate as for the fixedDmaxvalue the number of neighbors that can send information is fixed,and with a largeNmaxthese values are utilized with a smaller scaling term which results in a slower convergence rate.

Fig.7 show s the effect of changingDmaxvalues under a fixedNmaxvalue of 8.Similar to the previous case,the value ofDmaxdoes not change the convergence properties of the algorithm,but it effects the convergence rate.The convergence speed of the algorithm increases with increasingDmax,which indicates the significance of utilizing the available SINR values.As users are able to communicate with more users and obtain more information on SINR values,the convergence rate increases.

Fig.6 The effect of N m ax.

5 Conclusions

In this paper, we have investigated the performance of a power adjustment algorithm under different spectrum allocation schemes for two-tier femtocell networks.The convergence properties of the power control algorithm are shown to be independent of the spectrum allocation.Furthermore,the convergence rate analysis reveals that the algorithm converges exponentially regard less of spectrum allocation.In Simulations,the self-organizing and perfect fairness properties of the algorithm are illustrated under new femtocell deployments or disconnection of femtocells.Furthermore,it is observed that the rate of convergence increases with increasing neighbor femtocell SINR information;however,this is to be justified by future theoretical work.

Fig.7 The effect of D m ax.

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