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User rate and power optimization for HetNets under Poisson cluster process
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 237 (2019)
Abstract
Heterogeneous cellular networks (HetNets) consist of different tiers of base stations (BSs) to meet the everincreasing mobile traffic demand. Due to the random deployment of BSs, Poisson point process (PPP) is often used to model the BS distribution. However, low power small cells are usually clustered around the popular areas, and PPP can not reflect such a feature. To this end, we in this paper consider base station (BS) cooperation and analyze user rate and energy efficiency of HetNets based on a Poisson cluster process (PCP). A calculable formula for the average data rate (or spectral efficiency) and its approximated closed form are derived. Based on this closed form, a power minimization solution with certain spectral efficiency constraint is proposed, and the optimal cooperation radii are derived. Furthermore, we analyze spectral efficiency under a limited number of cooperative BSs in a twotier network. Finally, we propose a range expansion (RE) scheme and examine the impact of this scheme. The theoretical analyses are verified by simulation results.
Introduction
Comprising macro base stations (BSs) overlaid with dense low power BSs (hence small cells) [1], heterogeneous networks (HetNets) are being deployed to meet the rapid growth in data demand from wireless users, especially those users at the edge of macro cells. Due to the nature of dense deployment for HetNets and the power consumption of all the BSs, energy efficiency (EE) has now been considered as another key performance indicator (KPI) for future wireless networks (e.g., 5G) on top of the traditional KPI of data rate or spectrum efficiency. As a result, not only the data rate but also the EE has recently received enormous attention from the communications community [2–4].
BS locations in dense HetNets tend to be irregular and randomly placed compared with traditional cellular networks, which makes the analysis of the HetNets’ performances much more intricate. Fortunately, for the user rate and energy efficiency analysis, some tools from stochastic geometry have proved to be powerful, such as the probability generating functional (PGFL) of Poisson cluster process (PCP), the Palm characterization of PCP, and the CampbellMecke theorem. Indeed, these tools have made possible the theoretical analysis of many network metrics, such as average rate, energy consumption, as well as transmission delay.
One major challenge for HetNets is interference management. Coordinated multipoint (CoMP) transmission [5] is an effective scheme to reduce interference by transforming interfering signals to useful signals or to increase space diversity, hence enhancing the performances of HetNets. As such, CoMP is expected to play a key role in future cellular networks as an effective means of meeting higher data rates as well as expanding the indoor and celledge coverage. In [6, 7], the authors apply CoMP transmission in HetNets to enhance performances of network, and simulation results illustrate that CoMP can significantly improve the average SE and EE compared to nonCoMP. Thus, a singlestream CoMP transmission scheme (also termed macrodiversity CoMP) is exploited in this paper.
Related work and motivations
As mentioned above, there has been much effort on applying stochastic geometry to modeling and analyzing HetNets. In most works so far, cellular networks have been modeled as a tractable distribution such as Poisson point process (PPP) [8–10]. In [10], the authors derived the spectral efficiency based on the PPP model and determined the optimal received signal strength (RSS) thresholds under certain approximation. However, most small cells are usually clustered around densely populated areas, and PPP is not always suitable. Investigating a more accurate model therefore becomes imperative. To this end, the Laplace transform of the total interference in HetNets using a PCP was given in [11]. In [12], the authors compared PPP, aggregative point process, and repulsive point process to choose the optimal model for BS deployment in urban areas. In [13], it was stated that the PPP models in some case is more accurate than the hexagonal grid model, but the PCP is able to even more accurately model BS deployment. Assessing KPIs of HetNets based on the PCP model therefore becomes necessary. However, an integrated characterization for aggregate interference as well as the PGFL, based on the PCP network model, has not been reported, although such a characterization is crucial to the theoretical analysis of network performances, such as SE and EE.
Indeed, as both EE and SE are now viewed as crucial performance metrics in 5G networks, analysis of EE and SE has gained much attention recently. Reference [14] proposed a new cooperative multipleinput multipleoutput (MIMO) scheme to improve the spectrum efficiency while maintaining the same spatial multiplexing and diversity gains as traditional MIMO schemes. Reference [15] studied the EE and area SE with respect to the number and size of microcells. References [16, 17] studied the performances of crosstier cooperation scheme based on the PPP network model, and the numerical results illustrate that such a cooperation strategy can significantly boost ergodic capacity and reduce outage probability. Crosstier cooperation scheme is expected to be a major technique to provide higher data rates. However, none of the above works has applied this cooperation schemes to a PCP network. As a result, Reference [18] derived new analytical expressions for the coverage probability and area spectral efficiency on clustered devicetodevice millimeter wave networks. In [19], the authors investigated the outage probability, the coverage probability, and the average achievable rate based on a tractable lower bound of the PGFL when the nodes follow a PCP. Reference [20] found that the coverage probability in [19] is not always accurate for analyzing PCP network performances. However, the EE and SE of PCP modeled HetNets, based on crosstier cooperation, have not been analyzed so far.
Over the above situation, this paper targets at the singlestream CoMP over a PCP HetNet, where crosstier cooperation is based on the RSS threshold. From the perspectives of EE and SE, we derived a tractable result for SE based on PCP and then proposed an energy optimization solution. Moreover, we also analyzed spectral efficiency in case of the limited number of cooperative BSs in a twotier HetNet and proposed a range expansion (RE) scheme based on CoMP, which balances the load of macro BS.
Contributions
The main contributions of this paper are therefore as follows: (1) a computationally efficient formula for spectral efficiency is derived under the PCP model, and then approximated as a closed form; (2) a comprehensive comparison with the corresponding PPP results is carried out; (3) a power optimization scheme is proposed by optimizing the cooperative radius of each tier; and (4) a scheme based on a limited number of cooperative BSs and a range expansion (RE) strategy are proposed, which can ease the load of macro BS and enhance the network EE.
Paper organization
The remainder of this paper is structured as follows. In Section 3, we give a brief description on a particular PCP model and present the downlink system model. The formula of spectral efficiency is derived based on PCP and an optimal solution for each tier’s cooperative radius is determined in Section 4. In Section 5, we propose an effective BS cooperation and RE strategy based on CoMP. Section 6 presents the numerical results and related illustrations. A list of major symbols and their meanings are shown in Table 1.
PCP and downlink system model
Poisson cluster process (PCP)
A Poisson cluster process (PCP) consists of two kinds of point processes: the parent points in each tier following homogeneous Poisson point process (PPP) with the density λ_{p,k} form the centers of the clusters, while the daughter nodes are scattered around cluster centers. Matern cluster process (MCP) is a special case of PCP, in which the daughter nodes are of uniform distribution within a circle of radius R_{a} with the corresponding parent point being the cluster center. The parent points indicate the centers of hotspots, and the daughter nodes represent the locations of BSs. In this paper, we mean MCP by PCP. This special PCP model and the PPP model are illustrated in Fig. 1. In each cluster of the kth tier, the number of daughter nodes is fixed to be c_{k}, the mean density of the kth tier is therefore c_{k}λ_{p,k}. The probability density function of such PCP daughter nodes is given by
where d is a daughter node’s position relative to the cluster center, and d is the distance between the daughter node and the center of the relevant cluster.
System model
We consider a general model of downlink HetNets which includes K network tiers, and the BS positions of each tier form a PCP denoted by Φ on \(\mathbb {R}^{2}\). BSs across tiers are distinguished by their deployment mean density c_{k}λ_{p,k}, transmit power p_{k}, and path loss exponents α_{k}. User density is denoted by λ_{u}. Both BSs and users are equipped with a single antenna. Without any loss of generality, the general path loss is given by \(PL(x)=\left \x \right \^{\alpha _{k}}\), where ∥x∥ indicates the distance between a typical user at the origin and BS at x. Inspired by [10], we also consider a usercentric crosstier BS collaboration scenario. Figure 2 illustrates a twotier HetNet involving a mix of macro and pico BSs, where the typical user can be served by cooperative BSs from both tiers.
We employ the Palm measure [21] to characterize the total signal and the aggregate interference at the typical user located at the origin and served by crosstier cooperative BSs. For convenience, the set of cooperative BSs is denoted by C_{C}. BSs in the C_{C} transmit the same data to the typical user (hence a noncoherent macrodiversity CoMP strategy) and the RSS at the user exceeds some RSS threshold. Namely, for the kth tier, the BS located at x_{k} belongs to the cooperative group C_{C} if \({p_{k}}{h_{k}}{\left \ {{x_{k}}} \right \^{ {\alpha _{k}}}}\geq \theta _{k}\), where θ_{k} is the RSS threshold for the kth tier, and h_{k} denotes the power fading coefficient of the kth tier.
Assuming that the cooperative region in each tier is circular, the cooperative radius R_{k} can then be given by
Denoting by C_{I}=Φ∖C_{C} the set of the interfering BSs, the signaltointerferenceplusnoise ratio (SINR) at the typical user is given by
where σ^{2} is the noise power.
Spectral efficiency and power optimization
In this section, we derive and optimize two key metrics, i.e., the spectral efficiency (SE) and energy efficiency (EE) based on the probability generating functional (PGFL) of the Poisson cluster process in a Ktier HetNet. Both metrics depend on the SINR in [3]. As in [10], the Laplace transform is used to characterize the aggregate interference (the denominator) and the total signal power (the numerator plus the denominator). For comparison, the same denotations have also been employed as in [10].
Laplace transform of the total signal power
For the proposed PCP model, the number of daughter nodes in each cluster is assumed to be fixed, and the parent points are not included. All cooperative BSs are assumed to transmit the same signal to the “typical user" at the origin. The total signal power (the signal plus the interference) received at this user is given by
where ϕ_{k} denotes the set of BS at the kth tier.
From the above formula, the Laplace transform of total signal power P is
where \({{\mathcal {L}}_{h}}(s{p_{k}}{\left \ {{x_{k}}} \right \^{ {\alpha _{k}}}})\) denotes the Laplace transform of received signal power at x_{k}, and (a) is due to the fact that h_{k} is exponentially distributed with mean one.
On the other hand, the above expression is directly related to the PGFL of PCP as given in [22]
wher
and c denotes the number of daughter nodes in each cluster.
The lower bounded PGFL and the upper bounded conditional PGFL of the PCP are given by [19]. In order to evaluate accurately the spectral efficiency, we derive below the precise expression by algebraic operations.
Lemma 1
The Laplace transform (i.e., (5)) of total signal power P is estimated as
where
Proof
Let \(v(x) = \frac {1}{{1 + s{p_{k}}{{\left \ x \right \}^{ \alpha _{k} }}}}\). By applying (6) to (5), we have □
where (a) uses the change of variables ∥r∥=∥x−y∥, and (b) follows from the polar representation.
Then by substituting (8) into (6) and then (5), we obtain (7) in Lemma 1. Computing the integral in (6) numerically is a very arduous task, but (7) in Lemma 1 is now much more computationally efficient.
Spectral efficiency
From the information theory, we can achieve Shannon bound ln(1+SINR) for an instantaneous SINR. Thus, the spectral efficiency in units of nats/s/Hz is calculated as
where P and I denote the total signal power and aggregate interference power, respectively.
Theorem 1
In a Ktier HetNet, the spectral efficiency of a typical user at the origin under the proposed model can be evaluated as
Proof
Please see Appendix 1. □
Although it is not a closed form, the above integral can be analysed numerically. As \(B_{k}{{(y)}^{{c_{k}}}}1<0\), the spectral efficiency is an increasing function of the cooperative radii R_{k}. As such, we can design a larger cooperative region of BS in HetNets to meet higher data rates (subject to the overhead incurred).
In order to compare the performances (i.e., SE and EE) under PPP and PCP, we adopt the same setup for a typical user as in a PPP model. Assuming fading coefficient h∼exp(1), the spectral efficiency under PPP [10] with the density λ_{ppp,k}=c_{k}λ_{p,k}=λ_{k} can be express as
where
and Γ(·) denotes the gamma function.
Due to the complexity of the spectral efficiency expression in (10), we now derive a closed form by ignoring the noise (interference is the dominated issue in a dense HetNet).
Lemma 2
Ignoring the noise in dense HetNets, Eq. (10) can be simplified as follows
where
Proof
Please see Appendix 2. □
Note that q is not related to the cooperative radii, and it is a constant for a given deployment density and transmit power. As shown in Appendix 2, q−ln(f) is in fact the lower bound of τ in (10). When R_{k} becomes larger, however, the gap between (10) and q−ln(f) rapidly tapers off, hence the close approximation of (12) for τ. Equation (12) is a closed formula, which makes the following optimization problem easier to solve.
Minimizing energy consumption via optimizing cooperative radii
For the proposed model, the BSs inside the cooperative clusters can communicate with the typical user located at the origin, and the average power consumption when serving such a user is given by [10]
where P_{bh,k} is the backhaul power dissipation when serving one user (i.e., the typical user) in the kth tier, P_{sp,k} denotes the corresponding signal processing power consumption, and a_{k} denotes the slope of the kth tier power consumption with respect to transmit power p_{k}. The second term of P_{u} in (13) denotes the average static power for serving one user, and it is independent of the load of BSs.
In the following, we will provide a solution for optimizing the average power consumption of the HetNet for the typical user in (13). In reality, it is very important to determine the network parameters which can minimize the network power consumption. Since the average energy consumption P_{u} per user and the average consumed energy P_{av} are related with P_{av}=λ_{u}×P_{u}, minimizing P_{u} is critical to designing an energysaving network.
For a given cooperative radius R_{1} of the first tier, we want to find out the optimal cooperative radius R_{2} of the second tier in a twotier HetNet, which minimizes the energy consumption while ensuring a certain rate to the typical user. Note that the energy consumption in (13) increases with the cooperative radii, which suggests that we should determine the minimum cooperative radii while guaranteeing the minimum user rate. Intuitively, the user rate should increase with cooperative areas, and the rate in (10) is indeed an increasing function of the cooperative radii. Assuming that the firsttier cooperative radius R_{1} is known, we can determine the optimal value of the secondtier cooperative radius R_{2} through dichotomy. The problem will be transformed into a simple problem with a single variable (with the constraint from (10)):
The above problem formulation can be solved by the wellknown dichotomy algorithm^{Footnote 1}.
Now, we want to form the general problem to determine the optimal values for R_{1},R_{2},···,R_{K} in a Ktier network, which minimize the power consumption in (13) while satisfying the minimum spectral efficiency requirement. Note that the second term of P_{u} in (13) is independent of R_{1},R_{2},···,R_{K}, and can be ignored. Hence, by applying (12) and (13), the optimization problem can be formulated as follows:
Clearly, both the constraint condition and the optimization objective function are of closed form. The problem can then be solved easily by a linear programming method as follows.
Theorem 2
The optimal cooperative radii approximately satisfy
where R_{o,k} denotes the optimal value of the kth tier cooperative radius, and P_{a,i}=P_{bh,i}+P_{sp,i}+a_{i}p_{i} denotes the total power dissipation when serving a typical user in the ith tier.
Proof
Please refer to Appendix 3. □
Combining (16) and (17), the optimal value of each tier can be determined by solving the resultant system of nonlinear equations for a given data rate τ_{0}, which optimizes the PCP networks’ energy consumption.
Once the power P_{u} has been minimized, the network energy efficiency η_{ee}, defined as the ratio of the average spectral efficiency to the average power consumption, can be calculated as
Limited BS number and range expansion
In this section, we examine two implementation variations of the above collaboration strategy: (a) limited number of cooperative BSs and (b) range expansion (RE).
Limited number of cooperative BSs
In order to accommodate the everincreasing traffic demand, low power BSs are densely deployed in HetNets. The number of crosstier cooperative BSs for each user may be too high in popular areas, due to the overhead incurred and the marginal rate increase.
As a result, the number of associated BSs at a typical user should be limited to an acceptable level. Therefore, we now propose a cooperation tactic where the number of cooperative BSs is restricted to N (N≤4). In other words, the typical user at the origin communicates with N BSs based on maximum RSS, and all other BSs become noncooperative. Of course, the SE always increases when a new BS is added to the cooperative BS set, but the SE gain becomes more and more marginal and the overhead still increases linearly. Hence, there exists a tradeoff between SE and the overhead (including designing costs of BS and energy consumption). It is therefore meaningful to put a limit on the number of cooperative BSs, which can optimize EE and the overhead cost efficiency. As an example, we consider a twotier HetNet, and the cooperative BSs number in this case is given by
where U_{k} is the number of cooperative BSs in the kth tier.
The effect of this BS constraint will be shown via simulation.
Range expansion
In HetNets, deployment of macro BSs are for large coverage ranges, which may lead to a heavy load for macro BSs in crowded areas, and the energy consumption also heavily depends on the loading of macro BSs due to macro BSs’ high power consumption. Therefore, we now propose a range expansion (RE) scheme based on CoMP to alleviate the loading of macro BSs and improve the EE.
Generally, the transmission power of macro BSs is much larger than that of pico BSs, and as a result, many users will still receive the strongest signal from the macro BS even in hotspots. The RE scheme is to favor the selection of low power BSs by introducing a bias to the association threshold for low power BSs [23]. Due to the smaller coverage, lower power BSs can provide more resource blocks to each user than macro BSs. References [23–25] have shown that RE can achieve load balancing and enhance performances of networks without a loss of average user throughput. For example, in a twotier HetNet, due to the bias, users are more likely to connect to the pico BSs than the macro BSs even if \({p_{1}}{h_{1}}r_{1}^{ {\alpha _{1}}} > {p_{2}}{h_{2}}r_{2}^{ {\alpha _{2}}}\). Figure 3 illustrates how the typical user is served based on the RE scheme in such an example. Here, we consider a twotier HetNet comprising macro BSs and pico BSs, which follow a PCP. Consequently, the macro BSs can collaborate to serve the typical user at the origin only when \({p_{1}}{h_{1}}{\left \ {x} \right \^{ {\alpha _{1}}}} \ge \varepsilon _{1} {\theta ^{*}}\) and the pico BSs can serve this user if \(\varepsilon _{2}{p_{2}}{h_{2}}{\left \ {x} \right \^{ {\alpha _{2}}}} \ge {\theta ^{*}}\), where θ^{∗} denotes normalized RSS threshold (each tier then still has the same RSS threshold), ε_{1} and ε_{2} are the first power bias and the second power bias, respectively. Therefore, the cooperative radii after biasing are given by
where \(R_{1}^{*}\) and \(R_{2}^{*}\) denote the modified macrotier radius and modified picotier radius after utilizing a power bias, respectively.
From above expressions, the macro BSs cooperation region shrinks but the pico BSs cooperation region expands when the power bias coefficient is greater than 1. The energy efficiency under various bias values will be examined via simulation in the next section.
Numerical results and discussions
In this section, we present some numerical results of the user rate and energy efficiency for a twotier HetNet consisting of macro and pico BSs. The PCP cluster radii of this twotier HetNet are R_{a,1}=200m and R_{a,2}=100m, respectively. Unless otherwise specified, the parameters are those from [26]: the transmit power p_{1}=20W,p_{2}=0.13W, and the slope of the power consumption: a_{1}=4.7,a_{2}=4. We now compare the results under the same parameters for two different point processes: PCP and PPP.
Validation of Laplace transform on PCP model
Figure 4 shows simulation and numerical results of the Laplace transform for a twotier HetNet computed (i.e., (7)) under the PCP model. The PCP parameters are λ_{p,1}=1/(π×300^{2}),λ_{p,2}=1/(π×200^{2}),p_{1}=40W,p_{2}=0.13W,c_{1}=1,c_{2}=10, and the cooperative radii R_{1}=400m, R_{2}=100m. The simulation results match well with the analytical integrations in the whole range of s and different path loss exponents (assuming α_{k}=α here), which verifies the validity of the derived Laplace transform of the aggregate interference and total signal powers. The reason for the differences among the α curves is that for larger a α value, the total interference attenuates more, while the Laplace transform is an exponentially decaying function.
Comparison between PPP and PCP
Figure 5 compares the spectral efficiency (SE) versus the firsttier and the secondtier cooperative radii for PPP and PCP models. In order to compare PCP and PPP, we have chosen some parameters as in [10] : λ_{1}=1/(π×250^{2}),λ_{2}=1/(π×50^{2}), α_{1}=4.3,α_{2}=3.8, and set c_{1}=1,c_{2}=4. As can be seen from Fig. 5, the average user rate of PCP is worse than that of PPP, and the reason is that the main interferers are more likely to be closer to the typical user for PCP network (hence the higher interference level) than in the case of PPP. Furthermore, the spectral efficiency increases with the macrotier cooperative radius and the pico cooperative radius, and this is due to the number of cooperative BSs increasing for larger cooperative regions. We note that the picotier BSs after some sufficiently large distance contribute very marginally to the user rate. Therefore, the RSS threshold of the second tier should not be too low (to limit the number of the secondtier cooperative BSs), considering the corresponding collaboration overhead.
Figure 6 shows the energy efficiency (EE) versus the mean network cooperative radii R_{1} and R_{2} for PCP and PPP. The network model parameters are λ_{u}=1/(π×50^{2}),c_{1}=1,c_{2}=4,P_{bh,1}=P_{bh,2}=11W,P_{sp,1}=55W,P_{sp,2}=2.5W,P_{0,1}=75W, and P_{0,2}=4.3W. We can see that the picotier cooperative radius R_{2} has an optimal value for a given R_{1} which leads to a maximum EE. The case is similar to that of PPP, but the EE of PCP is poorer compared with that of PPP, since the average rate of PPP is better than that of PCP, and energy consumption of the two cases are the same due to the same mean network density. Note that the EE is a decreasing function of the macrotier cooperative radius, and this is because a large cooperative area of a macrotier results in an increase in the number of the cooperative macro BSs, while the main energy consumption is heavily dependent on the number of cooperative macro BSs.
Validity of Lemma 2
Figure 7 shows that the optimal and the upper bound cooperative radius values of the second tier versus the minimum user rate for two different cooperative radii of the first tier. By utilizing (12), we can acquire the upper bound curves, and the optimal radius can be calculated approximately by applying the binary search algorithm to (14). It can be seen that the optimal radius increases with the minimum rate τ_{0}, since the increase of the minimum rate τ_{0} means that more BSs need to be connected to the user, which results in an increase of cooperative radius. Furthermore, the gap between the optimal radius in the case of R_{1}=300m and that in the case of R_{1}=600m increases with the minimum rate τ_{0}. This is because the macro BSs in a specific cooperative area contribute limited impact on user rate, and more pico BSs are therefore needed to satisfy the increment of the τ_{0}. As τ_{0} increases, the upper bound increasingly approaches the optimal value, which testifies that the derived closed form is effective, especially for high data rates.
Limited number of cooperative BSs and range extension
Figure 8 shows the average achievable rate versus the secondtier cooperative radius for a different number of cooperative BS based on PPP and PCP. We can see that the achievable rate of PCP is still worse than that of PPP for a fixed number of cooperative BSs, and the reason is the same as for Fig. 5. It can be seen that the achievable rate does not change when R_{2} is greater than a specific value, which reveals that the achieved rate has an upper bound for a given number of cooperative BSs, under either PPP or PCP. The reason is that the number of cooperative BS reaches the set limit N in the cooperative areas. Moreover, the increase in the number of associated BSs can improve the achievable rate, but the rate gain increment is a decreasing function of N. In the meantime, adding one additional BS means more energy consumption. As a result, BSs located at the edge of cooperative areas is likely to decrease the EE, which also justifies limiting the number of cooperative BSs.
Figure 9 presents the simulation results of the EE based on RE with respect to power bias. Here, we assume that the twotiers’ power biases are both equal to ε. It can be seen that the EE is a concave function of the bias. No matter PCP or PPP, there exists an optimal bias (approximately ε=7 dB) and it seems to be nearly independent of the normalized RSS threshold. This figure reveals the RE can improve EE as a consequence of decreasing the macro BS loading in comparison with the nonRE scheme (i.e., ε=0 dB). Equivalently, it is obvious that the EE decreases with the normalized RSS threshold θ^{∗} decreasing because of more cluster edge BSs. It is interesting to observe that for θ^{∗}=− 65 dBm, the bias ε has a relatively weaker effect on the EE (due to the large preexpansion collaboration range). These curves also reveal that the EE of PCP is always lower than that of PPP, as the major interfering BSs in a PCP network are more likely to be closer to the user, which worsens the performances.
Conclusions
In this paper, we have derived computationally efficient new formulas for Laplace transforms of the aggregate interference and total signal power for PCP HetNets. By using these formulas, we have derived the spectral efficiency and the energy efficiency expressions for a typical user. We have demonstrated that the performances of PCP networks are worse than those of PPP networks. The optimal cooperative radii are also determined under the condition of a minimum rate, which minimizes the energy consumption. Additionally, we examined the case of limited number of cooperative BSs and the RE scheme (to balance the load between macro BSs and pico BSs). In terms of energy efficiency, we observed that there exists an optimal solution for the bias, which maximizes the energy efficiency.
Appendix 1. Proof of Theorem 1
From [27], we have the following useful lemma
By applying (22) to (9), the rate is given by
where (a) follows by substituting t=s(I+σ^{2}), and \({{\mathcal {L}}_{I}}(s)\) denotes the Laplace transform of the aggregate interference power of HetNets.
The group of interfering BSs is denoted by C_{I}=Φ∖C_{C}, and the aggregate interference range is outside the circle with radius R_{k}. Therefore, following a similar process to Lemma 1, we can express approximately the Laplace transform of aggregate interference as
where (a) follows from the fact that BS position can be approximated by the relevant cluster center.
Finally, by substituting (7) and (24) into (23), we can obtain (10) in Theorem 1.
Appendix 2. Proof of Lemma 2
Proof
A useful result from [28] is given by □
where a real part of μ and v is greater than zero.
The first integral in (10) is
where (a) is from substituting \(\left \ u \right \ = {\left ({s{p_{k}}} \right)^{ \frac {2}{{{\alpha _{k}}}}}}{\left \ y \right \^{2}}\), and (b) follows from the fact that 1−(1+x)^{−n}≤nx.
Then, by substituting (26) into (10), we ignore the noise and focus on the interference, and the spectral efficiency is approximately calculated as
where (a) follows from fact that v=f, μ=1 and p=1 in (25).
Appendix 3. Proof of Theorem 2
By applying the Lagrangian multiplier method [29], we can establish the objective function as follows:
Differentiating (28) with respect to R_{1},R_{2},···,R_{K}, and ζ, respectively, we can obtain
where L_{i}=2πλ_{p,i}c_{i}p_{i}.
Obviously, we can select one equation arbitrarily from (29) to determine the coefficient ζ:
R_{o,1},R_{o,2},···,R_{o,k},···,R_{o,K} satisfy these equations. Then, we can obtain (16) and (17) in Theorem 2.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Notes
 1.
The dichotomy algorithm (also commonly known as binary search algorithm) means, in our case, the following operation: select a very large R_{2} value at the beginning and then test the constraint in (14) at the middle point of R_{2} (i.e., 0.5 R_{2}). If (14) is satisfied, take the middle point of the lower half of R_{2} and test the constraint in (14) again; if not, take the middle point of the upper half of R_{2} and test (14). Repeat the same steps until the process converges to a certain value (which is the optimal vale for R_{2}).
Abbreviations
 BS:

Base station
 CoMP:

Coordinated multipoint
 RSS:

Received signal strength
 EE:

Energy efficiency
 HetNets:

Heterogeneous cellular networks
 KPI:

Key performance indicator
 MCP:

Matern cluster process
 MIMO:

Multipleinput multipleoutput
 PCP:

Poisson cluster process
 PGFL:

Probability generating functional
 PPP:

Poisson point process
 RE:

Range expansion
 SE:

Spectral efficiency
 SINR:

Signaltointerferenceplusnoise ratio
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FCZ proposed the main ideas and directions, supervised the research, and modified and rewrote some sections of the paper. XJ performed simulations and wrote a draft of the paper. Both authors read and approved the final manuscript.
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Jiang, X., Zheng, F. User rate and power optimization for HetNets under Poisson cluster process. J Wireless Com Network 2019, 237 (2019) doi:10.1186/s1363801915582
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Keywords
 Heterogeneous networks
 Poisson cluster process
 Energy efficiency
 User rate
 Stochastic geometry
 Range expansion