Analysis of Coverage and Area Spectrum Efficiency of UDN with Inter-Tier Dependence

Kaichuang Wang,Pei Li,Fei Ding,Zhiwen Pan＊,Nan Liu,Xiaohu You

National Mobile Communications Research Laboratory,Southeast University,Nanjing 210096,China

Abstract:Stochastic geometry is widely employed to model cellular network.But in most existing works,base stations (BSs) are modelled following a homogeneous Poisson point process (PPP) for one-tier network,or several independent homogeneous PPP for multi-tier network,which ignore the dependence among BSs.In this paper,a three-tier UDN (Ultra dense network) with Macrocell BSs (MBS) for basic coverage,Picocell BSs (PBSs) deployed outside the coverage area of MBSs for compensating coverage holes,and Femtocell BSs (FBSs) surrounding MBSs for capacity improvement modelled by point process with inter-tier dependence is proposed.The tier association probability,the coverage probability and area spectrum efficiency (ASE) are derived.Simulation results validate our derivation,and results show that the proposed network model has 25%-45% performance gain in ASE.

Keywords:stochastic geometry; tier association probability; Poisson hole process; area spectrum efficiency; coverage probability

I.INTRODUCTION

Ultra-dense network (UDN) is considered as a key technique to meet future network requirements [1]-[3].Stochastic geometry where BSs and users are modelled as homogeneous Poisson point process (PPP) are employed to analyze the system performance of UDN since it is possible to obtain closed-form expressions [4]-[6].

In [7],BSs are modelled as PPP for one-tier network,and closed-form expressions of the average achievable data rate are derived.The authors in [8] propose two-tier PPP model and derive coverage probability and capacity based on two different channel models.In [9],outage probability and minimum average user throughput for multi-tier independent PPP cellular network are given.BSs are modelled as a Poisson point process,optimal BS density to improve energy efficiency (EE) for one- and two-tier network is investigated in [10].However,dependence among BSs is ignored in these works.In fact,the BSs are not totally independent.

The dependence among BSs in different tiers are considered in only few works.In [11],K-tier HetNets where BSs are clustered following a Poisson Cluster Process (PCP) are proposed,and coverage probability and average achievable rate are derived.The authors in [12] consider a two-tier network where macro base stations follow a PPP and femtocells are located in the annular region of the macro-cell coverage area,the downlink coverage probability and the mean achievable rate are derived using the proposed two-tier network model.Two-tier network where the PBSs are only deployed outside the coverage area of MBSs is considered in [13],but there is no obvious improvement in the system capacity.PHP is used to model the interference field when Pico BSs are in the outer region of Macro BSs to improve the coverage,and several tight upper and lower bounds of interference are derived [14].In [15],β-Ginibre point process is used to model the distribution of base station locations,the repulsion among BSs is considered,and results show that it fits the real deployment of BSs in Paris well.However,[11-12] only consider the aggregation among the BSs which can bring about capacity improvement,while coverage holes and signal blind areas will appear.The repulsion among the BSs considered in [13-15] can reduce the interference,increase throughput and improve the basic coverage probability,but user throughput in hotspot area are not improved.Considering serious interference and poor service experience of the cell edge user,the repulsion among BSs need to be introduced to lower the interference and reduce signal blind areas.Deploying base station in hot spot,which introduces aggregation among BSs,seems to be an efficient solution to improve system capacity.However,there are no literatures that BSs are modelled by point process considering the aggregation and repulsion simultaneously.

In this paper,to reduce the blind area and improve service performance in hotspot area,(1) a model considering the aggregation and repulsion of the cellular network is proposed.Aggregation and repulsion point process is more accurate model for highly populated areas surrounded by lowly populated areas.(2) A three-tier cellular network model with inter-tier dependence where MBSs are in charge of basic coverage,PBSs deployed outside the coverage area of MBSs are used to compensate for the coverage holes and improve throughput in the outside area,and FBSs surrounding MBSs are used for the capacity improvement is given.The tier association probability,coverage probability and the area spectrum efficiency (ASE) of the proposed network model are derived.Numerical results of coverage probability and ASE are given.

The rest is organized as follows.Section II gives the three-tier cellular network with inter-tier dependence,along with channel model and user association scheme.In section III,coverage analysis and area spectrum efficiency (ASE) are given.Section IV is results on coverage probability and ASE.In section V,conclusions is given.

In this paper,we propose a three-tier cellular network model with inter-tier dependence.

Fig.1.The three-tier UDN model.The squares are MBSs and the big circles are coverage regions of MBSs with radius D.The small circles are PBSs outside the coverage regions of MBSs.The dots in each MBS are FBSs.The triangle are typical users for three kinds of BSs.

II.SYSTEM MODEL

2.1 Three-tier network

Consider a three-tier UDN with three kinds of BSs as shown in the figure 1.MBSs are overlaid by randomly distributed PBSs and FBSs,where BSs in each tier differ in transmit powers,node densities.

Wherexis the distance between the FBS and the center MBS.So FBSs include a parent process with densityΦmand a fixed number of daughter pointscwithin each cluster,i.e.,λ f=λmc.Denote MU,PU and FU the users served by MBS,PBS and FBS respectively.Assume users are distributed in the whole area according to a homogeneous PPP with densityλu.

2.2 Channel model and user association scheme

In this paper,fast fading and pathloss are considered.Fading coefficient between the user and its serving BS denoted ashis exponential (Rayleigh fading) with mean one.The pathloss function is given bywhereα＞2 is a path loss exponent.Transmit powers are denoted asPmfor MBSs,Ppfor PBSs,Pffor FBSs.The additive noise is ignored here for interference limited scenario [13] [20].The power received by a user located atzwith its serving BSs atyisPhl(y-z),wherePrepresentsPm,PporPf.All MBSs and PBSs operate on the same frequency,but FBSs use the orthogonal spectrum different from MBSs and PBSs [4].Assume that mobile users can only connect to MBSs and FBSs when they are located within the radiusDfrom the MBSs,and users are only allowed to connect to PBSs when they are outside the exclusion regions.Each user can associate with the BS according to maximum average received power criterion within the set of BSs they can choose.

III.COVERAGE PROBABILITY AND AREA SPECTRUM EFFICIENCY

In this section,coverage probability and area spectrum efficiency (ASE) are investigated.The conditional coverage probability of a typical user served by different kinds of BSs is given first.Then,tier association probability is derived.Based on the conditional coverage probability and tier association probability,the coverage probability and the area spectrum efficiency (ASE) are analyzed.

To calculate the coverage probability,the signal-to-interference ratio (SIR) threshold that will be used in coverage calculation for three kinds of BSs are denoted asβm,βpandβf.

3.1 Conditional coverage probability of a typical MU for fixed distance

A typical MU suffers from two kinds of interferences:interference from MBSs except its serving BSImm,and interference from PBSsIpm.

Assume the typical MU associates with the nearest MBSy0at distancerm,then

And the conditional coverage probability is

From [16],the Laplace transform ofImmis

Where F(x,y;z;w) is Hypergeometric function andδ=2/α.

Hence,the conditional coverage probability of a typical MU can be given by (4) for

Where

Since the distances between PBSs and MBSs are at leastD,Ipm⊂pm.Although the PBSs follow a Poisson hole process,independent thinning process ofwith probability exp(-λ πm D2) can be used to evaluateIpm.By replacingin (5) withλp=exp(-λmπ D2),we have

Moreover,MBSs and potential PBSs become less relevant when they are far-away from the MU [17].Hence,the conditional coverage probability of a typical MU is

3.2 Conditional coverage probability of a typical PU for fixed distance

Similar to MUs,the PUs suffer from two kinds of interferences:interference from MBSsImpand interference from PBSs except its serving BSIpp.Assume the typical PU associates with the nearest PBS at distancerp.Since the typical PU locates outside the exclusion region,there are no interfering MBSs within distanceD,thus

Where

And

To calculate the interference from other PBSs,from the potential PBSs inexcept the PBSs within the distancerpfrom the typical PU should be derived.Similar toImm,the Laplace transform ofis:

From the thinning theory [18] and replacingin (12) withλp=exp(-λmπ D2),the Laplace transform ofIppcan be written as

Similarly,the conditional coverage probability of a typical PU can be given by (13) for

3.3 Conditional coverage probability of a typical FU for fixed distance

Different from MU and PU,a typical FU only suffers from interference from other FBSsIff.From the assumption above,FBSs follows MCP.Without loss of generality,consider serving FBS at the origin and typical user at a distancerf,the Laplace transform ofIffis given by [11]:

Where

And the modified pathloss law isSince there are no interfering FBSs within radiusrf,the conditional coverage probability is the Laplace transform atso

3.4 Tier association probability

To obtain general expression of coverage probability,tier association probability should be derived.

Assume users are distributed in the whole area according to a homogeneous PPP with densityλu.Then a fractionκp=exp(-λ mπD2) of the users will be served by PBSs,and the rests are served by MBSs and FBSs.

To simplify the analysis,the overlap areas among the MBSs are ignored.As shown in Figure 2,a MBS is at the origin,cFBSs are uniformly distributed in the circle with radiusD.

The typical user follows a homogeneous PPP,and probability density function (PDF) ofRmis(r)=2r/D2,0≤r≤D.

Fig.2.Simplified model with a MBS at the origin and several FBSs around it.There are c FBSs.The small circle is MBS,the big circle is the exclusion area with radius D,the stars are FBSs,and the triangle is a typical user with distance Rm to MBS.

With above assumptions,we can derive the conditional association probability to MBS,as presented in Theorem1.

Theorem 1:The association probability to MBS is

Where

Proof:see Appendix A.

Denoteκ m,κp,κfthe tier association probability a user is associated with MBSs,PBSs,and FBSs respectively.WithAm,the tier association probabilities can be easily written as,

3.5 General coverage probability

Based on the conditional coverage probabilities in (8),(14) and (17),the general coverage probability for MU,PU,and FU can be written as

The PDF ofrpis given by Weibull distribution [19]Wherekandvare variables related toD.

However,the accurate expressions for(r) and(r) are difficult to obtain,and approximation is needed.Ignoring the overlap area between exclusion regions of MBSs,the average coverage area of a MBS isSm=Am πD2,whereAmis the probability a user is associated with MBS,and the average coverage area of a FBS isSf=(1-Am)πD2/c.Hence,the coverage regions of MBSs and FBSs are assumed to be circles with radiusandrespectively.

Assume the users are uniformly distributed in the whole plane,then

And

Hence,the general coverage probability of the UDN is

3.6 Area spectrum efficiency of the network

Assume each BS hasNbresource blocks (RBs),and one RB can be allocated to only one user at the same time.Under this assumption and a homogeneous PPP distribution of users,the average number of users served by each MBS,PBS,FBS,denoted asr espectively,can be given.

Since mean number of users in the Voronoi region of a MBS in a PPP scenario isλ u/λm,thenis given by

Similarly,

And the ASE as defined in [4] is

IV.NUMERICAL RESULTS

Numerical results for the coverage probability and area spectrum efficiency of PHP and the proposed model are given in this section.The main parameters for numerical evaluation are listed in table 1.

Fig.3.Coverage probability of the system versus βp.

Fig.4.Coverage probability vs βm.

Fig.5.CDF of user SIR.

4.1 Coverage probability

Figure 3 shows the simulation and numerical (theoretical) results of coverage probability of the network versusβp(the SIR threshold of PBSs).It can be seen that the Monte Carlo simulation and the theoretical analysis agree well,thus validating our derivation.Hence,in the following,only numerical results are given.

Figure 4 shows coverage probability versusβmfor PHP and proposed model.It can be seen that both of them decrease withβmand the proposed model has the same coverage probability performance as PHP model.

4.2 Area spectrum efficiency

Figure 5 shows the cumulative distribution function (CDF) of user SIR.It can be seen that there is a 5dBgain in SIR in the proposed model compared with PHP model due to the deployment of FBSs.

Figure 6 shows the ASE gain (defined as the ratio of proposed model to PHP (c=0) versus the number of FBSs in each MBS under different radius of MBSs.For smallc,ASE of the proposed network increases with the number of FBSs since FBSs are closer to the users thus providing larger SIR.ASE gain increases slowly for relative largecsince the demand of users in exclusion area are already fulfilled and more FBSs will have little help in ASE gain.For largec,ASE gain decreases because the serious interference among FBSs results in decrease of coverage probability of FUs.Hence,there exists a maximum ASE gain,and results show that the max gain increases withD.

Figure 7 shows the change of the ASE gain with the D.Maximum performance gain in ASE increases monotonously with D because more users are offloaded to FBSs which are closer to the users thus providing larger SIR.

Figure 8 shows the coverage probability decreases withDwhencis fixed due to large distance to the connected MBS,soDis set as 50-75m to provide coverage in our analysis.Results shows whenDis set as 50-75m,there will be 25%-45% maximum performance gain in ASE compared with existing schemes(c=0).

The impact of user densityλuon ASE is shown in Fig.9.It can be seen that with the increase ofλu,ASE increases accordingly.

V.CONCLUSION

In this paper,we propose a three-tier cellular network model with inter-tier dependence.Tier association probability of users and the coverage probability is obtained,and ASE of the network is analyzed.Results shows with the deployment of FBSs,there will be 25%-45% performance gain in ASE with nearly the same coverage probability compared with existing schemes.

APPENDIX A

PROOF OF THEOREM 1

From figure 2,the typical user associates with the MBS only when the RSRP (Reference Signal Receiving Power) from MBS is larger than all the FBSs in this circle.Hence,

Where (a) can be obtained from the independence of FBSs.

As shown in figure 10,sinceη=(Pf/Pm)1/α,obviouslyη＜1,Amdepends on the overlapping areaS(shaded area as shown in Fig 10).To calculateS,two situations need to be considered.

For situation 1 shown in figure 10 (left),S=π(ηRm)2.For situation 2 shown in figure 10 (right),

Fig.6.Performance gain of ASE versus the number of FBS in each MBS.

Fig.7.Maximum Performance gain of ASE versus the radius D of MBS coverage area.

Fig.8.Coverage probability versus radius of MBS coverage area.

Fig.9.ASE versus the number of FBS.

Fig.10.Two situations for the calculation of overlapping area S.

Fig.11.S1 and S2 for the calculation of S in situation 2.

Then,

And

Finally,in situation 2:

So the overlapping area is:

The PDF ofRmis(r)=2r/D2,0≤r≤Dand the proof is complete.

ACKNOWLEDGEMENTS

This work is partially supported by National 863 Program (2014AA01A702),national major project (2016ZX03001011-005) and national natural science foundation project (61521061).