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Performance analysis and optimization for nonuniformly deployed mmWave cellular network
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 49 (2019)
Abstract
In this paper, we propose a multitier mmWave cellular framework where sub6 GHz macro BSs (MBSs) are assumed as a Poisson point process (PPP) and smallcell BSs (SBSs), operating on either mmWave or sub6 GHz, follows nonuniform Poisson cluster point (PCP) model. This paper proposes both centralized and distributed user association algorithms. For the centralized twostep algorithm, we aim to maximize the sum rate while satisfying quality of service (QoS) and power consumption constraints based on eigenvalue analysis. Then, we derive the association probability, the coverage probability, and the average achievable rate, cosidering directivity and blockage effect, by stochastic geometry. On this basis, a distributed user association algorithm is proposed. The simulation results demonstrate the accuracy of our theoretical analysis and also reveal the effect of some parameters on the network performance. In addition, the proposed centralized algorithm can achieve nearoptimal sum rate with a low complexity.
1 Introduction
Nowadays, the fifth generation mobile communication network (5G) are undoubtedly one of the most attractive topics in both academic and industrial fields. Many researchers engage themselves to explore the new advanced methodologies and technologies in 5G, to support the booming data traffic with reduced energy consumption and improved quality of service (QoS) provision. Some enabling technologies, such as heterogeneous networks (HetNets), massive multipleinput multipleoutput (MIMO), and millimeter wave (mmWave) techniques, have been identified to bring 5G to fruition [1].
MmWave communications, which operate from 30 GHz to 300 GHz, have attracted much attention due to the shortage of microwave frequency [2]. It can provide the larger bandwidth, which can support the higher data rate transmission. Besides, mmWave communications have no effect on the traditional cellular communication in sub6 GHz, since they operate at the higher and different frequencies. Therefore, the deployment of mmWave BSs not only offloads the data traffic of existing cellular frequency, but also reduces the interference. However, a major channel characteristic of mmWave communications is its sensitivity to the blockages, which is a serious problem due to the poor performance of mmWave links at penetrating into or diffracting around solid objects [3]. Therefore, mmWave communication is commonly used in shortrange directional transmission with directional antennas instead of isotropic antennas. On the other hand, the rigorous directivity and high penetration losses lead to the fact that mmWave systems are noiselimited. Therefore, the user association algorithms designed for the interferencelimited networks cannot be directly applied in the mmWave systems [4].
Due to the two mentioned fundamental differences from conventional sub6 GHz cellular systems, the channel models of mmWave communications should be modeled carefully. Some research proposed the mmWavepattern models including the effects of blocking, directional beamforming, smallscale fading, and path loss for lineofsight (LOS) and nonlineofsight (NLOS) links [3–5]. Specifically, the blockages, resulting in the serious power loss at the high mmWave frequencies, are assumed stationary and isotropic [5, 6]. For mathematical tractability, a LOS ball model was proposed for simplifying blockage modeling where BSs inside the LOS ball were considered to be in LOS whereas any BS outside of the LOS ball was treated as NLOS [6]. In [7], this blocking model was modified by adding a LOS probability within the LOS ball, and this approach was shown to reflect several realistic blockage scenarios. Meanwhile, the pathloss exponent is considered higher in NLOS links (e.g., commonly about 4) than that in LOS links (e.g., around 2) [8]. Furthermore, some real data is obtained to verify the parameters of some blocking model (e.g., real building data in the UT Austin, downtown LA regions, NYC, and Chicago [4, 6]). It is revealed that the environment has the effect on the block behaviors, and the blocking models should be carefully selected according to the propagation environment (e.g., 3GPPlike models could be sufficient to fit the urban regions; generalized LOS ball model gives a good fit for dense random deployment of BSs, etc. [4]).
Meanwhile, the directional beamforming, for compensating the increased path loss at mmWave frequencies and for overcoming the more serious noise due to the large transmission bandwidth, is approximated by considering a sectored antenna model [9, 10]. In [9], mmWave BSs are equipped with directional antennas, and UEs perform the perfect beam alignment, whereas in [10], both the users and mmWave BSs are assumed to estimate the angles of arrival and to adjust their antenna orientations accordingly. Moreover, large antenna arrays, enabling massive MIMO and hybrid beamforming techniques, can contribute to the directive transmission. The rate of multiuser MIMO with hybrid beamforming [11] or with user pairing [12] or joint spatial division and multiplexing [13] in mmWave network are analyzed.
2 Methods/experimental
Motivated by the mathematical tractability, stochastic geometry has been accepted as a popular and useful tool to analyze performance of HetNets, and some significant results and conclusions are obtained [14–19]. Stochastic geometry is introduced into the mmWave network analysis where locations of both sub6 GHz and mmWave BSs are considered as Poisson point process (PPP), and block effects are incorporated into the system model [20]. The expressions, or upper and lower bound of some performance metrics (e.g., coverage probability, outage probability, average rate, etc.) are derived mostly under PPPbased mmWave network scenario [3, 4, 6, 9, 10]. Actually, as one of capacityincreasing techniques, mmWave BSs commonly work along with the cellular network and are distributed only on some hot spot areas. Therefore, nonuniform location distribution of mmWave BS is more appropriate to model the realistic scenario. To the best of our knowledge, few works adopt the nonuniform mmWave BS location distribution model, which will be the main difference of our paper from previous papers. In this paper, we make centralized and a distributed user association analysis for mmWave cellular system. The contributions of this paper can be summarized as

We propose a centralized twostage user association and power control algorithm to maximize the network sum rate while satisfying SINR and power consumption constraints. The optimization problem is divided into user association and power control subproblems. Specifically, for user association, the eigenvalue analysis is performed to drop communication links successively that cause the maximum sum of the interference power in the network until the feasibility condition is satisfied. In addition, binary power (BP) allocation is applied to solve the optimal power allocation.

We derive analytical expressions on the association probability, the coverage probabilities, and the average achievable rate, considering directivity and blockage effect, using stochastic geometry. On this basis, a distributed distancebased user association method is proposed to decide the pattern of BSs (conducting in either sub6 GHz or mmWave).
The remainder of this paper is organized as follows. In Section 3, the system model for mmWave cellular networks is described. The proposed centralized algorithms are presented in Section 4. In Section 5, the analytical expressions for the association probability, coverage probability, and the average achievable rate are derived. Further, a distributed distancebased user association method is proposed to decide the patterns of BSs. Simulation results are provided in Section 6 to compare the performance of the proposed algorithms, which are followed by our conclusions in Section 7.
3 System model
We consider a downlink mmWave cellular network. Specifically, macro base stations (MBSs) are uniformly distributed in \(\mathbb {R}^{2}\) according to Poisson point process (PPP) with density λ_{mu}; smallcell base stations (SBSs) are generated by Poisson cluster process (PCP) Φ_{s} where the parent points following PPP with density λ_{p} forms the centers of these clusters and the daughter points (i.e., SBSs) are uniformly distributed around the cluster center within the radius R [19]. The number of daughter points within each cluster is c. In a hybrid deployment, MBSs, under sub6 GHz, are essential to provide wide coverage to guarantee a consistent service for UEs whereas mmWave may be used in SBSs mainly to deliver high rate to individual UEs. Specifically, all MBSs and some SBSs share sub6 GHz, whereas the rest of SBSs operate at the mmWave bands. It is worth noting that SBSs in one cluster will choose the same pattern, i.e., either sub6 GHz or mmWave. Without loss of generality, we assume that a SBS cluster is under sub6 GHz with probability p_{su}, i.e, the density of sub6 GHz SBSs λ_{su}=cp_{su}λ_{p}; otherwise, it conducts under mmWave, i.e, the density of mmWave SBSs λ_{sm}=cp_{sm}λ_{p} where p_{sm}=1−p_{su}. Thus, we regard BS location distribution as three independent tiers, i.e., sub6 GHz MBS tier, sub6 GHz SBS tier, and mmWave SBS tier, as is shown in Fig. 1.
The two main characteristics of mmWave transmission are blockage effect and directivity at transmitter and/or receiver.

(1)
The actual mmWave transmission directivity is approximated by a sectored model [3, 4, 6, 9, 10], where a UE receives a signal with directivity gain G_{max} if the UE’s angle θ with respect to the best beam alignment is within the main beamwidth 2ω of the serving mmWave cell and with directivity gain G_{min} otherwise. This is formulated by
$$ {G}\left(\theta \right) = \left\{\begin{array}{ll} {{G_{\max}}}&{\left \theta \right \le \omega}\\ {{G_{\min}}}&{{\text{otherwise}}} \end{array}\right. $$(1)whereas G(θ)=1 under sub6 GHz cell. Due to the fact that θ is independent of all other variables, we simplify the G(θ) as G_{j} for the jtier directivity gain.

(2)
We approximate the blockage effect by a modified LOS ball model under mmWave link [4], i.e., a UE within a distance D to an mmWave SBS is LOS link, otherwise is considered NLOS link.
Thus, the received SINR at the typical user (at origin o) is
where x is the location of the associated BS, Φ is the set of all BSs’ locations, h_{x} is the smallscale fading coefficient between UE at origin and BS at x and is assumed to be independent and identically distributed (i.i.d.) Rayleigh fading [9]. The path loss exponent of an mmWave link depends on the link distance whereas the path loss exponent of a sub6 GHz link is independent of the link distance. Specifically, the path loss exponent a_{k} equals a_{s} if the link is under sub6 GHz, equals a_{l} if the link is LOS under mmWave and equals a_{n} otherwise. The additive noise is complex Gaussian distributed [19]. The transmit power of MBSs, SBSs operating at sub6 GHz, and SBSs operating at mmWave frequency are P_{mu}, P_{su}, P_{sm}, respectively. The aggregated interference for UE at x can be expressed as [4, 21]
where Φ_{j} represents the set of jtier BS locations and

k=mu means that k tier is sub6 GHz macro BS tier.

k=s means that k tier is sub6 GHz SBS tier.

k=sm means that k tier is mmWave SBS tier.
The conditions that UE associates with MBS tier, sub6 GHz SBS tier and mmWave SBS tier are illustrated in Figs. 2, 3, and 4, respectively. Unlike sub6 GHz networks, mmWave cellular systems will be noiselimited due to the directivity and blocking effects discussed earlier, in conjunction with the large bandwidth which brings in much more noise power [4, 9]. In such cases, the SNR distribution can be used as an approximation of the SINR. This is in contrast to sub6 GHz cellular networks, which are often interferencelimited, meaning SIR ≈ SINR instead.
In this paper, we consider an mmWave cellular network in which users intend to communicate with the BS in either mmWave or sub6 GHz band. In this mmWave cellular system, we propose a centralized and a distributed user association algorithm, respectively. In Section 4, we propose a centralized user association and power control algorithm. The main idea of the centralized algorithm is to design the user association and transmit power of users to maximize the network sum rate while satisfying the individual target SINR constraints for all links. Note that the centralized algorithm depends on the global channel state information (CSI) possibly at a centralized controller, which is significant to the power design. Note that the centralized algorithm requires global channel state information (CSI) possibly at a centralized controller, which may incur high CSI feedback overhead. To resolve this issue, we propose a distributed user association method in Section 5, which requires distance about the link between the transmitter and its corresponding receiver only. The main goal of the distributed algorithm is to maximize the coverage probability, thereby the basis of which is to calculate the coverage probability. Therefore, we derive the coverage probability as well as average achievable rate in Section 5. Moreover, the probability distribution function (pdf) of associated distance and the association probability are also derived in Section 5, which are the necessary components of the expression of coverage probability. Thus, we analyze the mmWave cellular network performance from the perspectives of centralized and distributed control.
4 A centralized twostage user association and power control algorithm
A main feature of mmWave cellular networks is that the mmWave links are managed by MBSs in a centralized manner. Commonly, the centralized algorithm can provide the upper performance bound for its distributed counterpart. In this section, we suppose that MBS is able to acquire global channel state information (CSI). Based on the system model in Section 2, we generate a mmWave cellular network of N sub6 GHz MBSs and M mmWave SBSs. On this basis, we would like to maximize the network sum rate under SINR and power consumption constraints. The optimization problem can be formulated as
where (4) corresponds to the SINR requirement to be satisfied at all M+N receivers. All M+N inequalities are put together in a matrixform inequality, as is shown in (4). The element in the normalized gain matrix F is \({F_{ki}} = \left \{\begin {array}{*{20}{c}} {\frac {{\gamma _{T}}{\beta _{ki}}}{\beta _{kk}},}&{k \ne i,k \le N}\\ {0,}&{\text {otherwise}} \end {array}\right.\), where \({\beta _{ki}}={G_{ki}}{h_{ki}}{\left \ {x_{ki}} \right \^{ {\alpha _{k}}}}\) consists of the antenna gain, channel gain, and path loss from transmitter k to receiver i. The elements in b are auxiliary variables which can be considered as the normalized SINR threshold for every receiver and \(\mathbf {b}=\left [\frac {{\gamma _{T}}{N_{0}}}{\beta _{11}},\frac {{\gamma _{T}}{N_{0}}}{\beta _{22}},...,\frac {{\gamma _{T}}{N_{0}}}{\beta _{(N+M)(N + M)}}\right ]\). The power control vector P=[P_{1},P_{2},...,P_{N},P_{N+1},...,P_{N+M}]^{T}. \(P_{\text {mu}}^{\max }\) and \(P_{\text {sm}}^{\max }\) are the maximum power of MBS and mmWave SBS, respectively. I is the identity matrix. In this section, we focus on the condition p_{sm}=1, which means that all SBSs operate at the mmWave frequency. Without loss of generality, the results can be extended to the condition 0≤p_{sm}<1, i.e., the sub6 GHz SBSs are included in the optimization.
We can see that the problem in (P1) is nonconvex, which is difficult to obtain the optimal solution. Therefore, we decouple this problem into two subproblems, i.e., user association subproblem and power control subproblem. On this basis, we propose a centralized twostage user association and power control algorithm, denoted by TS algorithm to get the closely local optimum. In the following, we provide the solution to the two subproblems separately.
4.1 The first stage: user association
The first stage is to find the feasible user association set to satisfy (4) that can be considered as the prerequisite of the next power control. The matrix F is comprised of nonnegative elements and is irreducible because all the links interfere with each other. By the PerronFrobenious theorem [22, 23], the SINR constraint set in (4) is nonempty if and only if the maximum modulus eigenvalue of F is smaller than 1, i.e., ρ(F)<1. Otherwise, the power control solution is infeasible. The explanation for this conclusion is that the network prefer to support more transmissions, but sometimes it has to drop some links to satisfy SINR constraints. The key idea to solve this problem is to drop communication links successively that cause the maximum sum of the interference in the network until the feasibility conditions are satisfied. It is worthy noting that since the mmWave BSs will not cause the interference due to the high attenuation loss, none of mmWave BSs will be dropped in the first stage.
4.2 The second stage: power allocation
After the user association, the original optimization problem (P1) can be reformulated as P2, i.e., a tractable powerconstrained sum rate maximization problem.
where \(\phantom {\dot {i}\!}N^{\prime }\) is the number of the updated sub6 GHz BSs set after the user association stage.
Geometric programming (GP) is a widely adopted method to obtain the optimal solution to the powerconstrained sum rate maximization problem [24] and binary power (BP) can provide the near optimal performance but with low complexity [25]. Therefore, we tackle the power control subproblem by BP. In addition, the optimal mmWave BS transmit power is P_{i}=Psmmax,i=N^{′}+1,N^{′}+2,...,N^{′}+M due to the fact that mmWave link is interferencefree.
The details of TS algorithm are describe below.
Centralized twostage user association and power control algorithm
Initialization: Set gain matrix F^{t} for t=0 according to the global CSI. All cellular and mmWave links are active.
Step 1: Calculate ρ(F^{t}). If ρ(F^{t})<1, solve the sum rate maximization by BP to obtain the power allocation method. Then, the algorithm stops. Otherwise, go to step 2.
Step 2: Calculate \(q = \arg \;\underset {q \in \{ 1,2,...,N + M\}}{\max } {\left \ {\mathbf {f}_{q}} \right \_{2}}\), where f_{q} is the column of the matrix F^{t}, i.e., \({\mathbf {F}^{t}} = \left [ {\mathbf {f}_{1}^{t},\mathbf {f}_{2}^{t},...,\mathbf {f}}_{N + M}^{t} \right ]\).
Step 3: Remove the qth column and row in F^{t}. t←t+1. The updated matrix F^{t} is the reduced matrix. Return to step 1.
5 Distributed user association analysis of mmWave cellular network
In the previous section, we propose a centralized twostage user association and power control algorithm. Although centralized algorithm can achieve a good performance, the main challenge is the frequent coordination between transmitters and the high signaling overheads for sharing CSI. In order to solve this problem, we conduct a distributed user association analysis for mmWave cellular network. Specifically, we derive some key performance metrics considering the directivity and blockage effect. On this basis, we propose a distributed distancebased user association method to decide the pattern of BSs (either sub6 GHz or mmWave pattern).
In order to enable the derivations and the theorems in this section easier to understand. Table 1 lists the symbols and parameters for distributed user association.
For the parameters k, i, j, and l, they represent ktier, itier, jtier, and ltier. The following theorem provides the per tier association probability, which is essential for deriving the main results in the sequel.
5.1 Performance analysis
In this subsection, we derive the association probability, coverage probability, and the average achievable rate under mmWave cellular network.
The following theorem provides the per tier association probability, which is essential for deriving the main results in the sequel.
Theorem 1
The association probability in the kth tier is
where \({{\mathcal {E}}_{{Y_{l}}}}\left (y \right)\) is expressed as
where \({F_{R_{l}}}\left ({r_{l}}\right) = 1  \exp \left ({  \pi {\lambda _{l}}r_{l}^{2}} \right)\) is the cumulative distribution function (cdf) of the distance of the nearest ltier BS.
Proof
See Appendix A. □
From Theorem 1, we further derive the the distance between a user and its serving BS, of which the premise is that the user is associated with the kth tier.
Corollary 1
The probability distribution function (pdf) of ktier associated distance
where \(\Pr \left [{{r_{k}} > {x_{k}},{P_{r,k}} > \underset {j,j \ne k}{\max }\ {P_{r,j}}}\right ]\) is obtained through the method to calculate A_{k}, in which the only difference is the integral interval over r_{k} is x_{k} to infinity instead of zero to infinity.
Depending on the association probability and the pdf of the associated distance, we further obtain the coverage probability and average achievable rate, which are the two important indicators to evaluate the network performance.
In this paper, a UE is said to be in coverage if it is able to connect to at least one BS with SINR above its threshold.
Theorem 2
The coverage probability is
where the first term can be calculated by averaging over x_{k}, and the pdf of x_{k} is given by Corollary 1. \({\mathbb {E}_{x_{k}}}\!\left [\!\!{\exp \!\! \left ({\!\! \frac {{\gamma _{k}}\sum \limits _{z \in {\Phi _{j}}} {{P_{j}}{h_{z}}{z^{ {\alpha _{s}}}}}}{{P_{k}}x_{k}^{ {\alpha _{s}}}}}\!\right)}\!\right ]\) and \(\mathbb {E}_{{x_{k}}}^{!}\!\!\left [\!\!{\exp \! \left ({ \frac {{\gamma _{k}}\sum \limits _{z \in {\Phi _{j}}} {{h_{z}}{z^{ {\alpha _{s}}}}}}{x_{k}^{ {\alpha _{s}}}}} \right)} \right ]\) have been derived in [19].
Proof
See Appendix B. □
Last, we derive the average achievable rate of mmWave cellular network. The average achievable rate \(\mathfrak {R} \) can be derived by the method to analyze the coverage probability in Appendix B.
Theorem 3
The average achievable rate is
where \({\mathfrak {R}_{k}}\) is the average achievable rate on the condition that user is associated with ktier and Pr(SINR_{k}(x_{k})>e^{t}−1) can be easily deduced according to (25) and (26) in Appendix B.
5.2 Distributed distancebased user association method
On the basis of the expression of coverage probability, together with the known associated distance, we propose a simple method to judge which pattern of BS (sub6 GHz or mmWave) are preferred by comparing the conditional coverage probabilities of sub6 GHz BS and mmWave BSs. In order to make the problem mathematically trackable, we set p_{sm} = 1 in the following analysis.
The smtier coverage probability conditioned on the associated distance is
The mutier coverage probability conditioned on the associated distance is [16]
where B(x;y,z) is the incomplete beta function.
The user association decision is determined by the coverage probability conditioned on the associated distance, that is, the user will associate to the nearest mmWave BS if φ_{sm}≥φ_{mu}, otherwise associates to the nearest MBS.
Remark 1
According to Jensen inequality, [26] \({a_{1}}\exp \left ({{b_{1}}{x^{\alpha _{\text {sm}}}}}\right) +{a_{2}}\exp \left ({  {b_{2}}{x^{{\alpha _{\text {sm}}}}}}\right) \ge \exp \left ( \left ({a_{1}}{b_{1}} + \right.\right.\)\(\left. \left. {a_{2}}{b_{2}} \right){x^{{\alpha _{\text {sm}}}}} \right)\), the sufficient condition for the inequality φ_{sm}≥φ_{mu} is \(\exp \left ({ \left ({{a_{1}}{b_{1}} + {a_{2}}{b_{2}}} \right){x^{{\alpha _{{\text {sm}}}}}}} \right) \ge \exp \left ({  c{x^{2}}} \right)\). Therefore, the user will associate with nearest mmWave BS if \(D\! \le \! x\! \le \! {\!\left (\!{\frac {c}{{{a_{1}}{b_{1}} + {a_{2}}{b_{2}}}}} \right)^{\!\frac {1}{{\alpha _{n}}  2}}} \) or \( x <\! \min \! \left \{ {D,{{\left ({\frac {c}{{{a_{1}}{b_{1}} + {a_{2}}{b_{2}}}}} \right)}^{\frac {1}{{\alpha _{l}}  2}}}} \right \}\).
6 Results and discussion
In this section, we present numerical and simulation results on association probability, coverage probability, and average rate for mmWave cellular network and then evaluates the performance of the proposed centralized TS algorithm. The simulation parameters are listed in Table 2.
We start by looking into the association probabilities and the different factors that affect the probability. Figure 5 illustrates the association probability of sub6 GHz MBS, sub6 GHz SBS, and mmWave SBS against the mmWave probability p_{sm}. Figure 5 shows that the simulation results match the analytic results well, which validates our analysis. It is also seen in Fig. 1 that more UEs associate with mmWave BS with the growth of p_{sm}, which is in agreement with our intuition. Moreover, it can be seen that MBSs still undertake more UEs due to higher transmission power.
Figure 6 shows that coverage probability decreases with the larger c value whereas the gap between different c diminishes with the increasing mmWave probability p_{sm}. That is because larger c value under low p_{sm} results in more sub6 GHz SBSs that cause more interference. However, the effect of c on coverage probability becomes weaker with the increasing p_{sm} due to the fact that more SBSs conduct in interferencefree mmWave pattern with the increase of p_{sm}. The extreme case is that the gap disappears for p_{sm} = 1, as is shown in Fig. 2. Thus, the results in Fig. 2 demonstrates that mmWave links can contribute to coverage probability.
Similarly, the higher achievable average rate is along with the larger p_{sm} in Fig. 7. It is worth noting that a cross point appears with the increasing p_{sm} under different c. That means less sub6 GHz SBSs (i.e., lower interference) or more mmWave SBSs (i.e., better channel state) can achieve higher achievable average rate.
From Fig. 8, we can see that the cluster radius R has no strong effect on UE association when SBS density is small, which further verifies that a PCP can be approximated as a PPP under small value c. Moreover, it is noticeable that mmWave support highest data rate even under the least number of associated UEs by comparing Figs. 8 and 9. Therefore, mmWave communications outperforms sub6 GHz communications on rate delivery for high directivity and interferencefree transmission environment. It can be seen in Fig. 10 that the coverage probability grows in accordance with the increasing cluster radius whereas it tends to keep stable for small values of c. Higher c value results in stronger nonuniform distribution, which can implicitly confirm that nonuniformity can reduce coverage area.
This finding obtained in Fig. 11 is in agreement with the published paper [6] that the coverage probability decreases with the growing SINR threshold. Also, it illustrates that more mmWave links are able to enhance the coverage probability. This is mainly due to interferencefree transmission environment under mmWave communications.
As expected, the sum rate decreases with the increasing SINR threshold in Fig. 8. Meanwhile, we compare the performance of the proposed centralized TS algorithm (using BP) and geometric programming (GP) algorithm where the performance of GP is commonly adopted as a benchmark. The observation in Fig. 12 is that TS algorithm is always approximate to the GP for c=1. Although the gap between GP and BP becomes obvious with the larger c, BP is near to the optimal results under low SINR. The results demonstrate nearoptimality of BP under low SINR constraint. Moreover, the complexity of the BPbased transmitter design is reduced since only a twolevel power control is required, which is a potential in the design and analysis of wireless networks.
7 Conclusion
This paper proposes a realistic multitier mmWave cellular framework where sub6 GHz MBS deployment is assumed as a PPP, and SBS operating on either mmWave or sub6 GHz follows PCP model. In this system, we propose both centralized and distributed user association algorithms. For the centralized twostep algorithm, we aim to maximize the sum rate while satisfying QoS and power consumption constraints based on eigenvalue analysis. Then, we derive the association probability, the coverage probability, and the average achievable rate, incorporating directivity and blockage effect, by stochastic geometry. On this basis, the distributed user association algorithm is proposed. The simulation results demonstrate the accuracy of our theoretical analysis and also reveal that the effect of some parameters on the network performance. In addition, the proposed centralized algorithm can achieve nearoptimal sum rate with a low complexity under nondense scenario.
8 Appendix
8.1 A. Proof of Theorem 1
The associated probability in the ktier is given by
where a is given from the independence of PPP. The pdf of r_{k} is given by [16]. G is a vector involving the directivity gains of all links.
First we calculate \(\Pr \left ({{P_{k}}{G_{k}}r_{k}^{ {\alpha _{k}}} > {P_{j}}{G_{j}}r_{j}^{ {\alpha _{j}}}} \right)\)
1. If k,j∈{mu,su} and j≠k
2. If k∈{mu,su} and j = sm
where the variable \({Y_{j}}=r_{j}^{\alpha _{j}}\). \({\ell _{Y_{j}}}\left (y \right)\) and \({\mathcal {E}}_{Y_{j}}\left (y\right)\) are the pdf and cdf of variable Y_{j}, respectively. The cdf of variable Y_{j} is given by
where the last term in (19) can be calculated based on the pdf of distance between user and the nearest jtier BS. Therefore, \({\mathcal {E}}_{Y_{j}}\left (y \right)\) is expressed as
where \({F_{R_{j}}}\left ({r_{j}}\right) = 1  \exp \left ({  \pi {\lambda _{j}}r_{j}^{2}} \right)\) is the cdf of the distance of the nearest jtier BS.
3. If k = sm and j∈{mu,su}
Thus, the ktier associated probability is
where a averages the directivity gain which is a trivial work due to the fact that the directivity gain of each link is independent of other variables. Additionally, averaging over r_{k} is also a simple work once we know the pdf of r_{k} [10].
8.2 B. Proof of Theorem 2
The coverage probability can be expressed as
where Pr(SINR_{k}(x_{k})>γ_{k}) is given by
where the interference can be ignored when UE is in the coverage of mmWave SBS, i.e., k=sm and the interference cannot be neglected when UE is in the coverage of sub6 GHz SBS or MBS, i.e., k∈{mu,su}. The following is the discussion about the associated distancebased conditional ktier coverage probability,
(1) If k=sm,
(2) If k=mu,su,
where \({\mathbb {E}_{x_{k}}}\! \!\begin {aligned}\left [\!{\exp \! \left ({ \frac {{\gamma _{k}}\sum \limits _{z \in {\Phi _{j}}} {{P_{j}}{h_{z}}{z^{ {\alpha _{s}}}}}}{{{P_{k}}x_{k}^{ {\alpha _{s}}}}}} \right)} \right ]\end {aligned}\) and \(\mathbb {E}_{{x_{k}}}^{!}\! \!\begin {aligned}\left [{\!\exp \! \left ({  \frac {{\gamma _{k}}\sum \limits _{z \in {\Phi _{j}}} {{h_{z}}{z^{ {\alpha _{s}}}}}}{{x_{k}^{ {\alpha _{s}}}}}}\right)} \right ]\end {aligned}\) of PPP and PCP have been derived in [19], respectively.
Thus, the coverage probability is given by
where the first term can be calculated by averaging over x_{k}. The pdf of x_{k} is given by Corollary 1. \({\mathbb {E}_{x_{k}}}\begin {aligned}\left [{\exp \left ({  \frac {{\gamma _{k}}\sum \limits _{z \in {\Phi _{j}}} {{P_{j}}{h_{z}}{z^{ {\alpha _{s}}}}}}{{{P_{k}}x_{k}^{ {\alpha _{s}}}}}} \right)} \right ]\end {aligned}\) and \(\mathbb {E}_{{x_{k}}}^{!}\begin {aligned}\left [{\exp \left ({  \frac {{{\gamma _{k}}\sum \limits _{z \in {\Phi _{j}}} {{h_{z}}{z^{ {\alpha _{s}}}}}}}{{x_{k}^{ {\alpha _{s}}}}}}\right)} \right ]\end {aligned}\) have been derived in [19].
Abbreviations
 5G:

the fifth generation mobile communication network
 BP:

Binary power
 CSI:

Channel state information
 GP:

Geometric programming
 HetNets:

Heterogeneous networks
 LOS:

Lineofsight
 MBS:

Macro base station
 MIMO:

Multipleinput multipleoutput
 mmWave:

Millimeter wave
 NLOS:

Nonlineofsight
 PCP:

Poisson cluster point
 PGF:

Probability generating function
 PPP:

Poisson point process
 QoS:

Quality of service
 SBS:

Smallcell base station
 SINR:

Signal to interference plus noise ratio
 UE:

User
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Funding
This work is supported by the National Natural Science Foundation of China (No. 61701037 and 61601503), Beijing Natural Science Foundation (No. L172033) and the 111 Project of China (B16006).
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The idea of this work was proposed by XZ and YL. YW performed the simulation. XZ wrote the manuscript. JB has given critical revision and helped revise the manuscript. All authors read and approved the final manuscript.
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Zhang, X., Liu, Y., Wang, Y. et al. Performance analysis and optimization for nonuniformly deployed mmWave cellular network. J Wireless Com Network 2019, 49 (2019). https://doi.org/10.1186/s136380191370z
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DOI: https://doi.org/10.1186/s136380191370z