Stochastic Geometry Analysis of MultiAntenna TwoTier Cellular Networks
Abstract
In this paper, we study the key properties of multiantenna twotier networks under different system configurations. Based on stochastic geometry, we derive the expressions and approximations for the users’ average data rate. Through the more tractable approximations, the theoretical analysis can be greatly simplified. We find that the differences in density and transmit power between two tiers, together with range expansion bias significantly affect the users’ data rate. Besides, for the purpose of area spectral efficiency (ASE) maximization, we find that the optimal number of active users for each tier is approximately fixed portion of the sum of the number of antennas plus one. Interestingly, the optimal settings are insensitive to different configurations between two tiers. Last but not the least, if the number of antennas of macro base stations (MBSs) is sufficiently larger than that of small cell base stations (SBSs), we find that range expansion will improve ASE.
I Introduction
Heterogeneous networks (HetNets), which is composed of MBSs overlaid with SBSs, has been recognized as a promising approach to cope with the 1000x traffic demand [1]. However, the differences in transmit power and BS density between MBSs and SBSs make the properties of HetNets different from the traditional singletier networks.
To characterize the performance of HetNets, stochastic geometry has been identified as a powerful tool because of its accuracy and tractability. By modeling MBSs and SBSs as independent Poisson point processes (PPPs), the performance of HetNets has been extensively studied in [2][3][4]. Especially, the technique of cell range expansion for loadbalancing was considered in [4]. It is found that, although range expansion improves the fairness between users, the overall throughput will be degraded. However, [2][3][4] only considered singleantenna BSs, the key properties of multiantenna HetNets cannot be revealed. In multiantenna singletier networks, the ASE and energy efficiency have been studied in [5][6]. [7] analyzed the error probability of singletier Poisson distributed networks. However, the interplay between different tiers in HetNets cannot be understood in [5][6][7]. To take a step further, the coverage probability of multiantenna HetNets was investigated in [8][10][9]. [11] studied the coverage probability of multiantenna HetNets with transmitreceive diversity. However, [8][10][9][11] focused more on the analysis of signaltointerferenceplusnoiseratio (SINR), which is usually in complex form. It is difficult to analyze the properties of multiantenna HetNets under different system parameters theoretically based on the previous works. As a results, the impact of differences between two tiers on system performance have not been well investigated.
In this paper, we derive the expressions and tight approximations for users’ average data rate in multiantenna twotier networks. Through the more tractable approximations, we study how different system parameter configurations affect the users’ average data rate theoretically. Furthermore, also based on the approximations, we obtain the optimal number of active users of each tier to maximize ASE. We study the optimal settings under different system parameters. We find that the ratio of BS density and transmit power and the range expansion bias mainly affect the users’ data rate and the optimal settings.^{1}^{1}1 are the ratios of BS density, transmit power between MBSs and SBSs, which will be defined in the main body of this paper. Moreover, the results based on the approximations are validated through numerical results. The novel and insightful findings of this paper as follows:

The average data rate of macro or small cell users is insensitive to the number of active users of the other tier. If range expansion is considered, the users’ data rates of both tier increase with . However, if range expansion is not considered, the users’ data rate is insensitive to .

For each tier, the ratio between the optimal number of active users and the sum of the number of antennas plus one nearly remains fixed, which is insensitive to different system configurations. With the number of active users set as the optimal number, ASE increases linearly with the number of antennas of both tier.

If the number of antennas of MBSs is sufficiently larger than that of SBSs, range expansion will improve ASE. Besides, the region of the number of antennas where range expansion improves ASE is insensitive to . However, larger will expand the improvement region.
Ii System Model
We consider a downlink twotier network where the macro tier is overlaid with cochannel deployed SBSs. The positions of MBSs and SBSs are modeled as PPPs with density . The MBSs and SBSs have transmit power and are equipped with antennas. The singleantenna users are located according to some stationary point process, which is independent of . We adopt standard path loss propagation model with path loss exponent . User association is based on the longterm average biased received power. The range expansion bias for SBSs is .
Each MBS and SBS serves active users at each slot. We consider zeroforcing (ZF) precoding and assume perfect channel state information at each BS. Thus, we have . We follow the infinite user density assumption in [8][10]. Therefore, there are at least users covered by each MBS and SBS. The transmit power is allocated equally for the active users in each cell. The small scale fading on each link is i.i.d. Rayleigh fading. The thermal noise is ignored in this paper. Without loss of generality, we will focus on the analysis of a typical user located at the origin [12]. The signaltonoiseratio (SIR) of the typical user located at the origin is
(1) 
where indicates the BS the typical user associated with, and depends on which type BS is. According to [8][10], we know the equivalent channel gains , and .^{2}^{2}2Actually, the proposition are Gamma distributed is an accurate approximation, which is widely adopted in previous works [6][8][9][10].
Iii Analysis of the Typical user
In this section, we will study the average data rate of the typical user, especially, the impact of different configurations between two tiers will be discussed.
Iiia Theoretical Analysis
Taking average over spatial distribution and channel power distribution, we can obtain the average data rate of the typical user, which is provided in the following theorem.
Theorem 1
The average data rate of the typical user in macro tier and small cell tier are given by
(2) 
where .
Proof:
The proof is given in the appendix. \qed
Theorem 1 provide tractable expressions for the average data rate of users in multiantenna twotier networks. However, have complicated relationships with other system parameters. To further study how the differences between two tiers impact on the network performance, we provide more tractable approximations for , which are demonstrated to be quite tight in our numerical results.
Theorem 2
The approximations for are
(3) 
where .
Proof:
The proof is given in the appendix. \qed
From Theorem 1, are unrelated to the number of antennas of the other tier. Interestingly, besides the number of antennas, do not depend on the number of active users of the other tier either. In fact, numerical results show that the original expressions are also insensitive to the number of active users of the other tier, i.e., behave similarly to . Moreover, based on Theorem 1, we find that the ratios and mainly affect .
Lemma 1
When , we have 1) increases with , , and ; 2) increases with and , but decreases with . However, when , are unrelated to , and can be simplified as .
Proof:
The proof is given in the appendix. \qed
It is a wellknown fact that the average data rate increases with for macro users, but decreases with for small cell users. However, instead of numerical results, this property is demonstrated theoretically through the tight approximations in this paper. Furthermore, it is interesting to point out that both increase with and when , which has never been reported in previous works. That is to say, the more small cells or the higher transmit power of small cells, the average data rate of both tiers will be larger. Another interesting result is that the approximations of users’ data rate do not depend on BS density nor BS transmit power when , which is similar to the singleantenna scenario [4]. The properties in Lemma 1 will be validated for the original expressions through numerical results.
Fig. 1 depicts with different system parameters. We can find that the approximations in Theorem 2 is quite accurate. Moreover, all the mentioned properties of are validated for . Firstly, we find are insensitive to the number of active users of the other tier. The reason is that, instead of , it is the total powers which dominate the crosstier interference. Secondly, when , we observe that both and increase with , . By increasing , we know that the intercell interference from the small cell tier will be stronger. However, the users will be closer to the associated BSs, which makes the desired signal power stronger. From Fig. 1, we know that the increase of desired signal power is larger than the increase of the interference power, which leads to the fact that increases with . Similarly, for increasing , we also know that the increase of desired power is larger than the interference power. Different from the case , when , are insensitive to and . That is to say, when , the increase of desired signal power due to large is counterbalanced by the increase of intercell interference. Comparing the results for and , we find range expansion will improve but degrade . This property is similar to the singleantenna HetNets [4]. However, we consider a more general case where multiantenna BSs are considered. Above all, all theoretical results based on the approximations are validated for . The impact of differences between two tiers on users’ data rate has been revealed.
Iv Area Spectral Efficiency Optimization
In this section, we will study the optimal to maximize ASE. The key properties of under different system parameters will be revealed. How range expansion affects ASE will be discussed.
Iva The Optimal Number of Active Users
Based on Theorem 1, the expression of ASE can be expressed as . We find that are crucial system design parameters, which affect ASE of both tiers. Thus, we attempt to obtain the optimal to maximize ASE. It is difficult to maximize directly because of the complex expressions of . Therefore, we obtain that maximize the approximations instead. Fortunately, due to the tightness of the approximations, the numerical results demonstrate that the results based on the are consistent with .
In detail, we formulate the following problem,
(4) 
Substituting into the above problem and relaxing to , we arrive at the following problem,
(5) 
where . For the optimization problem , we have the following observations.

Firstly, can be optimized separately to maximize .

Secondly, the optimal do not depend , i.e., for arbitrary , the optimal numbers of active users are . ^{3}^{3}3In fact, the number of active users should be rounded.

Furthermore, with the optimal , the approximations of ASE , which will increase linearly with .
Through the second order derivatives, it is not difficult to show that both and are concave function of . Therefore, the optimal can be obtained by setting first order derivative as zero.
Theorem 3
is the solution of . is the solution of . Both of the two equations have a unique solution located in , which can be obtained through the bisection method.
Proof:
We take as an example. Take the first order derivative , we arrive at the mentioned equation. is a decreasing function of . Besides, for the boundaries , it is not difficult to show that and . Therefore, there is a unique solution located in , which can be solved through bisection method. \qed
From Theorem 3, after rounding and , we can obtain the optimal that maximizes . Interestingly, although it is obvious that depends on , and , we find that are insensitive to these parameters in numerical results. The optimality of for the original expression will be also validated through numerical results.
IvB Numerical Illustrations
By exhaustive search, we obtain the optimal that maximizes . Fig. 2 illustrates under different system parameters. It is obvious that is either or , is either or . That is to say, the solution based can be applied to directly. Interestingly, we find is always located in even under different system parameters. Therefore, the optimal are insensitive to different configurations between the two tiers. Specifically, and for different , , .
When the number of active users set as optimal, Fig. 3 depicts with respect to . We find the mismatch between and is quite small, which demonstrates the tightness of Theorem 2. As we mentioned in subsection A, with optimal and , will increase linearly with . Consistent with , we can see that also increases linearly with . The reasons for this phenomenon are as follows. From Theorem 3, we know that are approximately fixed portion of , respectively. In such situation, from Theorem 2, we know that nearly remain fixed for different . However, the number of active users will increase linearly with . In summary, will increase linearly with .
With the optimal , Fig. 4 depicts with considering different range expansion bias . In singleantenna HetNets, it is found that range expansion will degrade the overall ASE [4]. However, it is not true for multiantenna HetNets. From Fig. 4, if is sufficiently larger than , range expansion will improve the ASE. The reasons are as follows. As we mentioned in Theorem 2, range expansion will improve but degrade . If is sufficiently larger than , the value of will be sufficiently larger than . Hence, will benefit more from range expansion. Besides, as , the larger is, the more macro users will benefit from the range expansion. Thus, the improvement of due to range expansion will make up for the degradation of . Fig. 5 illustrate the regions where range expansion improves or degrades ASE under different system parameters. It is interesting to point out that, the improvement/degradation region is insensitive to . Instead, the difference in transmit power significantly affects the improvement/degradation region. Specifically, the large is, the improvement region will be larger. Based on Theorem 2, we know that increasing improves both . However, from Fig. 5, we know that the improvement of is larger than that of , which leads to the fact that the improvement region will be expanded with the increase of .
V Conclusions
In this paper, we have studied the users’ average data rate and the optimal settings in multiantenna twotier networks. With the help of the tractable approximations, the key properties of system performance have been revealed. We find that the users’ data rate of each tier is insensitive to the number of active users of the other tier. If , we find users’ data rate increases with . However, if , the users’ data rate is insensitive to . For the purpose of ASE maximization, we find the optimal number of active users of each tier is fixed portion of the sum of the number of antennas plus one. The optimal settings are insensitive to different system parameters. With optimal settings, ASE will increase linearly with the number of antennas. Moreover, we find that, if the number of antennas of MBSs is sufficiently larger than SBSs, range expansion will improve ASE. The improvement region will expand with larger .
This work has revealed the key properties of multiantenna twotier networks. Further extension of this work is to consider more sophisticated multiantenna techniques, such as coordinated beamforming and transmitreceive diversity.
Proof:
We follow the similar approach with [13]. However, we obtain much more simple expressions for multiantenna HetNets. Take as an example. , where is the distance distribution between the macro user and the serving MBS [4]. Following the lemma in [14], , where and . As is Gamma distributed, we have . From the properties of PPP, we have , . Above all, substituting into the primary integral, we can derive after some algebraic manipulations. \qed
Proof:
The main procedures are similar to Theorem 1, except that we derive an approximation for by averaging the equivalent channel gains and . Specifically, we have
(6) 
Proof:
First consider the case . We first show increases with and . Let . Based on Leibniz integral rule, we only need to prove the integral item increases with . Take the first order derivative, we have . We find increases with , i.e., . Thus, . Therefore, increases with and . Similarly, we can prove increases with and . Then, we will prove increase with . Indeed, we have . It is obvious that and . Thus, , i.e., increases with . Following similar procedures, we can prove decreases with . For the case , the results can be obtained easily by substituting . \qed
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