A novel channel model and optimal power control schemes for mobile mmWave two-tier networks

We present a unified system model and framework for the analytical performance study of two heterogeneous and physically-distinct, but coexisting, networks that work harmoniously at the same time, space, and frequency domains. The two-tier network model considered in this paper is an overlaying of femtocells on a macrocell. Overlaying femtocells improves the performance by offloading traffic from macrocells and providing spatial diversity. The mmWave channel model employed considers the number of clusters and rays within each cluster to vary due to the end-user mobility. This is a new and different model compared to the widely used channel models for mmWave two-tier networks. Optimal power control is formulated as a sum-rate maximization problem for downlink and uplink transmissions at two-tier networks and a power allocation scheme is proposed by following Shannon-Hartley theorem. A comprehensive and interesting performance investigation is provided, where it is shown that the upper bound on the number of admitted secondary users has a linear relationship with the outage probability threshold, logarithmic relationship with SINR and exponential relationship with channel gain factors. Simulation results show that the proposed scheme with algorithm 2 is very effective at managing the cross-tier interference and can outperform a competitive scheme from literature that is based on cognitive radio technology. The computational complexity analysis of proposed algorithms are also given, since the complexity of algorithm 2 can be a performance-complexity trade-off issue for systems with limited computation power and time requirements.


I. INTRODUCTION
The fifth-generation (5G) networks and the internet of things (IoT) promise to transform our lives. They will connect billions of devices and enable everything from driver-less cars to smart-homes, offices, cities, and the world. In order to enable these applications, much faster and more reliable communications are needed compared to the current fourth generation (4G) networks. It is envisioned that a 100x increase in area capacity compared to long-term evolution (LTE) networks will be required in 5G [1]. Multi-tier networks that overlay each other with heterogeneous radio technologies and physical properties are a promising technology that can minimize coverage gaps, provide spatial diversity gain and support the ultra-high data traffic demand of hot-spots in future mmWave Systems [2]. A popular example of multi-tier networks is the two-tier Macro-Femtocell networks [3][4][5].
It is shown that while designing such a multi-tier network the co-channel operation of macrocells and femtocells can result in inter-tier interference that may worsen the performance if appropriate interference cancellation techniques are not employed [6,7]. One of the basic solutions to mitigate interference in a two-tier network is by defining one tier to consist of macro users (MUs), also called as primary users (PUs), and the other to consist femto users (FUs), also called as secondary users (SUs), who have cognitive potential to detect unoccupied channels [8,9]. In [10], a spectrum split up method is utilized to reduce the cross-tier interference between macrocell and femtocell networks, named as fractional frequency reuse (FFR) method. FFR can be achieved by dividing up the entire available spectrum into multiple subbands and putting a constraint on access over the femtocell networks to achieve interference coordination [11][12][13][14].
A popular technique to provide better network coverage and data transmission reliability is to employ large-scale antenna arrays (LSAA) to achieve strong directional beamforming gain that can avoid interference signals between geographically distributed users in mobile mmWave massive MIMO [15]. Although, prior channel state information (CSI) at the transmitter is an important factor in the implementation, but the increased dimensionality of channel matrices due to LSAA at the transmitter causes unacceptably large beam training overhead, serious pilot contamination in massive MIMO and higher computational complexity [16]. The authors in [17][18][19][20][21][22] presented the path-loss model and MIMO channel model for line-of-sight (LoS) and non-LoS (NLoS) case studies, and characterized angular-spread models for intracluster (multipath component distribution) and intercluster (cluster distribution) evolving across a LSAA with the Laplacian and Gaussian distributions at millimetrelength electromagnetic waves.
In [23], a full-duplexing (FD) based interference alignment (IA) algorithm for small cell network operation has been proposed that can significantly improve the spectral efficiency (SE). Minimize spectrum consumption clustering (MSCC) and minimize interference leakage clustering (MILC) schemes are employed for IA to mitigate interand intra-cluster interference. A Zadoff-Chu (ZC) sequence based scheme that is feasible even for ultra-dense networks (UDN) and massive MIMO settings was presented in [24] for signal spreading in conjunction with the proposed α − η − k − µ fading model.
In [25][26][27] interference mitigation scenarios to deal with various challenges such as doppler shift, blockage effect, power of the scattered waves and non-linearity of the space are studied. Doppler shifts in terms of angle of arrivals (AoAs) and angle of departures (AoDs) of the signal were compensated under beamforming network to develop a quasi time-invariant mmWave MIMO channel similar to [28][29][30][31][32][33][34][35]. Besides interference mitigation, radio resource management plays a crucial role in achieving high performance in next generation networks [36]. [37][38][39][40][41] considered Lagrangian dual decomposition technique in orthogonal frequency division multiple access (OFDMA)-based femtocell heterogeneous networks (HetNets) to design various optimization problems on power allocation, subchannel assignment, user association, load balancing and spectrum sharing subject to delay-sensitive (DS) and delay-tolerant (DT) constraints.
The work in [42] utilized mmWave FBSs to optimize the network throughput and microwave MBSs to improve energy efficiency (EE). Compared to the presented data-aided (DA) estimator for multicell decoupled two-tier femtocell networks in [43], [44] studied downlink (DL) and uplink (UL) decoupling (DUDe) in HetNets and presented significantly better performance. For the data-driven decision making in two-tier self-organizing networks (SONs), machine learning is considered in [45], where Distributed Cooperative Q-Learning (DCQL) scheme is used for power allocation in densely deployed femtocells and demonstrated much improved EE. Further [46] proposed a deep reinforcement learning (RL) based scheme to manage the DL interference and significantly improved the system capacity. In [47], authors investigated coverage probability over Rician fading channels by considering Marcum Q-function for unmanned aerial vehicles (UAVs) assisted femtocell networks and validated its energy-efficient solution. Licensed assisted access (LAA) based small cell networks have been proposed in [48] to maximize the network throughput while FBSs share licensed and unlicensed channels with MBSs and Wi-Fi respectively, where closed form expressions are established with the exchange of Lagrangian parameters through joint power and channel allocations. Other interesting research work on dragonfly, ant lion, modified firefly and ABC optimization algorithms are provided by [49][50][51].
Despite these interesting researches, the channel model considered in the aforementioned studies assumes the number of clusters and rays within each cluster to be fixed [54,55]. However, in mobile mmWave massive MIMO based communications the number of clusters and rays within each cluster varies due to user mobility [52]. In case the mmWave channel is not accurately modelled and appropriate signal processing is performed at the receiver, the received signal's quality may be significantly degraded, resulting in unreliable and slow communications. To address these open issues, in this paper we consider a novel channel model that has not been studied for mmWave two-tier networks before and propose optimal power control schemes for mobile mmWave massive MIMO based two-tier networks. The contributions of this paper are listed as follows: • A closed-form mathematical expression for a novel channel model at mmWave two-tier networks is formulized, where the variable number of clusters and rays within each cluster is modeled by well-known probability distributions. • An optimal power control scheme that employs a multichannel iterative lower-bound coefficients search algorithm is proposed to jointly maximize the sum-rate of DL and UL transmissions at mobile mmWave two-tier networks. Two lemmas are given and proved to obtain the optimal power allocation solution. Lemma 1 is used to transform the non-convex power allocation problem into its convex approximation and lemma 2 is given to prove that the transformed problem is strictly convex over given channel pairs.
The rest of the paper is organized as follows. In Section II, we present a system model considering co-existence scenario with FBSs deployment at MBS cell edge area in order to improve the network's quality of service (QoS) and formulate an optimization problem by following Shannon-Hartley theorem to maximize capacity for both UL and DL subject to the power constraints. To solve the issue of power regulation for the dense deployment of femtocells, new analytical derivations are provided along with a few properties in the form of lemmas in Section III. Then two computationally tractable algorithms focusing on a large number of subchannels and the iterative water-filling method are applied along with comparative complexity analysis. Simulation results are presented in Section IV. The paper is concluded with some remarks in Section V.

II. SYSTEM MODEL
We assume a cellular network comprising super high frequency (SHF) macro base station (MBS) and mmWave femto base stations (FBSs). The UL and DL characteristics of a two-tier HetNet have been investigated. The MBS operates on the sub-6 GHz frequency band and the FBSs are deployed along the cell edge area, see Fig. 1. FBS assists in improving the system capacity and also provides secondary network coverage service, i.e., as a backup coverage to MBSs, by jointly sharing sub-6GHz and mmWave bands. Femtocells underlaid macrocell network with joint operation at sub-6GHz and mm-wave bands play a key role to avoid outage and open up new opportunities to enable beyond fifthgeneration (B5G) networks for the industrial internet of things (IIoT). At the same time, this type of operation may cause severe co-and cross-tier interference, hence, there is a need for well interference management [24].
MBSs are deployed as a homogeneous poisson point process (HPPP) with intensity ϱ m ∈ R 2 . The cell edges are modelled as a PPP with intensity ϱ e ∈ [0, 2π] × R + . The deployment of FBSs along the cell edge side is assumed to be a PPP with intensity ϱ f ∈ R + . If the distance between a user and an accessible nearby LOS MBS is denoted by d ml and then this notation changes to d mn for the NLOS MBS located at the closed proximity. Likewise, d f l and d f n are the notations used to denote the distance from a user to a LOS FBS or a NLOS FBS, respectively, located at the closed proximity. Therefore, the distributions of d ml , d mn , d f l and d f n for y as the location of the user at the associated BS can be expressed by [53] f d ml (y) = 2πϱ m ye −πϱmy 2 , y < r m , where r m denotes macrocell LOS radius and r f denotes femtocell LOS radius, F d f n (y) denotes cumulative distribution function (CDF) and can be expressed by ( Let symbol x to indicate the network entity that facilitates the communication service to a given user, where x = f if the user is connected to a FBS and x = m if the user is connected to a MBS. Then, path-loss between k th BS and i th UE on n th sub-band can be expressed as below, where ξ indicates path-loss gain at a reference distance a n,i,k,t = 1; d n,i,k = a 2 n,i,k,t + b 2 n,i,k,t denotes the three dimensional (3D) distance between k th BS and i th UE on n th sub-band at t time instant, a n,i,k,t denotes the distance between k th BS and i th UE on n th sub-band at t time instant, b n,i,k,t is used to denote absolute antenna height difference between k th BS and i th UE on n th sub-band at t time instant; α is assumed for path-loss exponent which characterizes the LOS and NLOS path-loss exponents and can be expressed as follows: α ≜ ςd n,i,kα + [1 − ςd n,i,k ]α where ς ∈ {1, 0} is a Bernoulli random variable and it is becoming 1 if there is non-existence of NLOS components else 0,α andα are representing LOS and NLOS path-loss exponents, respectively, of a BS.
In this system model, suppose each transmission link consists of Z clusters regardless of MBS or FBS scattering paths. We also assume that there is Y rays between transmitter and receiver for the number of clusters z ≤ Z [52]. Therefore, the channel matrix belongs to each link can be expressed as, (11)] such properties of the mmWave channel are discussed in [54] and [55] and studied in [52]. v x Rx and v x Tx are used to denote 3D spatial response vector for the equipped antennas at the transmitter and receiver, v x ⋆ Tx represents complex conjugate of spatial response vector v x Tx , F x Rx and F x Tx denote field factors that cause antenna gain in the channel computation, θ x Rx,zy and θ x Tx,zy represent horizontal AoA and horizontal AoD related to each ray that associated to each of Z clusters, h x zy denotes multipath channel fading of Rician type and can be expressed by where m zy denotes the average channel gain at n th subband for the channel fading function related to baseband signal and m zy = E{|h x zy (n)| 2 }, K indicates Rician shape parameter, [h x zy ] d and [h x zy ] s are considered for direct-and scattered-paths of the desired transmission link and they can be expressed as follows: where ℧ x n,i,k is an arbitrary numerical value during the computation and it is dependent only on the distance between i th UE and k th BS, A Tx denotes number of equipped transmit antennas at the transmitter, the baseband signal is supposed to be modulated with the sampling period T s keeping M symbols in a single data frame, ζ is an arbitrary phase shift that uniformly distributed over [0, 2π],f d = E{f d } be the expected value of Doppler frequency shift due to velocity v in which E{.} is the statistical expectation operator and f d = fcv c cos ϕ where ϕ is uniformly distributed over [− ω 2 , ω 2 ], f c denotes carrier frequency, c stand for velocity of light.
The contribution of beamforming gain in the received signal strength (RSS) from a BS to a UE can be estimated by where B x Tx and (B x Rx ) T denote beamforming vector at the T x and transpose of beamforming vector at the R x .
[56] presents a heterogeneous network with a MBS positioned at the centre and and 4 FBSs positioned under the network coverage of MBS at the cell edge area, particularly located at (2, 2), (−2, 2), (−2, −2), (2, −2). The total number of MUs denoted by N M U is randomly positioned under the network coverage area of MBS. A co-existence scenario where a large number of femtocells implanted at cell edge area of macrocell network. But this consideration is not much useful to achieve the objective due to involving of interference effect. Figure 1 deals with a network where FBSs are deployed under the network coverage of MBS and use the same band of frequency with macrocell. This work is exclusively focus on the analysis of UL/DL transmissions alongside a bit discussion of UL/DL transmissions.
Here, the two-tier network based on OFDMA technique incorporates N M number of MBSs and N F number of FBSs in a cell. Now, the assigned BW at edge area of the network coverage of MBS is divided up into three sub-bands by employing the FFR method [57,58]. One sub-band can incorporate a number of sub-channels denoted by N SC and they are accessible to facilitate the users positioned at the area near to the cell-centre and the area belong to the cell edge by means of network service.
Thus SINR 1 , denoted by γ x n,i,k , of a typical user positioned at the middle taking the network service from the MBSs or FBSs can be written as: where I x n,i,k and I x ′ n,i,k are denoted as co-tier and cross-tier interferences respectively, where Ξ x n,i,l and [G x n,i,k ] d , used to define transmit power and beamforming gain of the desired signal respectively. κ n j ∈ {1, 0} assumes to be 1 if n th sub-channel is allocated to FBS j, else 0. The co-tier interference takes place among the network elements of the same types (e.g., between adjacent femtocells) and the cross-tier interference occurs among network elements that belong to different tiers 1 The probability density function (PDF) of SINR can be expressed as below, where µ denotes channel estimation error andγ x n,i,k denotes average SINR.  (e.g., between a macrocell and a femtocell). They can be used interchangeably based on whether x = f or x = m. Additive White Gaussian noise (AWGN) is denoted for noise power by σ 2 n,i,k .

PROBLEM FORMULATION:
Case I (For x = m): The case where a user is connected to a MBS. By following (12), the SINR of i th MU located at k th MBS on n th sub-channel can be expressed as, Likewise, the SINR at k th MBS for i th MUs on n th subchannel can be expressed as, Based on Shannon Hartley capacity formula [59], the DL and UL capacity for x = m can be written as, Case II (For x = f ): The case where a user is connected to a FBS. By following (12), the SINR of j th FU located at l th FBS on n th sub-channel can be expressed as, Likewise, the SINR at l th FBS for the FUs on n th subchannel can be expressed as, Based on Shannon Hartley capacity formula, the DL and UL capacity for x = f can be written as, Let each FBS consists, Now, we formulate the capacity maximization problem for the macro-femto heterogeneous networks as follows: where p max1 , p max2 , p max3 and p max4 are maximal powers of an MBS, an MU, an FBS and an FU respectively. Given the capacity optimization problem, next section provides the analytical derivations for power control and two search algorithms for the optimal solution, along with their complexity analysis.

III. POWER REGULATION ON CHANNEL PAIRS
While femtocells are operated in close proximity for highdensity deployment scenarios, co-tier interference becomes the main concern for the network performance. Although, the ideal channel allocation technique can ensure controlled interference irrespective of co-tier or cross-tier to satisfy the QoS requirements.
Suppose that ℧ n1 and ℧ n2 be the set of links between the FBS and the FU can perform the operation of communication on the channel pair n 1 and n 2 . Similarly, φ n1 and φ n2 assume to be the set of links between MBS and MU on the same channel pair. Now, achievable sum-rate on the given channel pair can be expressed as follows, R(p n,k , p n,i , p n,l , p n,j ) = As sum-rate maximization maximizes the capacity in the cognitive femtocell networks, hence sum-rate maximization problem employing power control can be expressed as follows:  (13), (14), (17) and (18), respectively. The sets ℧ n1 , ℧ n2 , φ n1 and φ n2 are decided by Ψ n,i , Ψ n,k , Φ n,j , Φ n,l , respectively. Only the power constraints (21k) − (21n) parts are taken into account by assuming the fulfilment of QoS needs with proper channel allocation.
The problem of selecting a set of cochannel users and allocating power among them to maximize the weighted system sum rate subject to a given power constraint is shown to be a non-convex combinatorial problem [60][61][62]. Such problems can be transformed into a convex problem by deriving the tightest lower-bound of the logarithmic function of weighted system sum rate [60][61][62]. Thus the problem P 1 given in (23) is not a strictly convex problem due to the fact that (22) is a non-concave function. Therefore, in order to achieve universal optimum of P 1, it is the necessary condition that P 1 should be a convex optimization problem. In the following sections, we propose flexible power control scheme to obtain the solution over the channel pair.
R 1 (p n,k , p n,i , p n,l , p n,j ) ≜ R 2 (p n,k ,p n,i ,p n,l ,p n,j ) = n,i + ψ A. FLEXIBLE POWER CONTROL SCHEME: For the proposed flexible power control scheme, we derive and employ the concave property, i.e., tightest lower-bound of log 2 (1 + u) for u ≥ 0, to get the optimal solution of nonconvex problem P 1. In this context, the below lemma [60] is a useful lower-bound solution to transform the P 1 problem into a convex approximation. Lemma 1: A lower bound inequality of the logarithmic function is written by, where ψ and ϕ denote lower bound co-efficients and are given by, However, ψ log 2 u + ϕ is very close to log 2 (1 + u) and the lower bound is tight and equal to log 2 (1 + u) at u = u 0 ■ By Lemma 1, P 1 can be extended to a new problem as given below: where R 1 (p k , p i , p l , p j ) is given by (25), in which ψ n,l represent lower bound co-efficients and DL/UL SINR corresponding to n th sub-channel at t th iteration.
With the assumption ofp n,k = log 2 (p n,k ),p n,i = log 2 (p n,i ),p n,l = log 2 (p n,l ), andp n,j = log 2 (p n,j ), R 1 (p n,k , p n,i , p n,l , p n,j ) is re-expressed as a concave function in (26). Then we transform P 2 into a new problem P 3 as follows: Subject to: By applying the Sylvesters criterion [63], the strict concavity of s(x 1 , x 2 ) can achieve on the space spanned by n,j,l ) is also a strictly concave function. Hence, (26) can be said to be a strictly concave/convex by obeying the law given in [64] which is any non-negatively weighted sum of concave functions remains concave. Therefore Lemma 2 follows. ■ An optimization problem P 3 with constraints given in (27) that describe the boundary of the region can solve by applying the method of Lagrange multipliers and the Lagrangian is given by arg max wherep = [p n,k ,p n,i ,p n,l ,p n,j ] denotes approximation vector; λ n , and λ

4[t]
n are Lagrange multipliers 2 . The Lagrange function is (29) and the Lagrange problem with fixed ψ [t] and ϕ [t] is n ,λ subject to: λ n,l ] denotes approximation vectors. By following the 1 st -order necessary conditions of equality and inequality constrained problem, we derive the near optimal stationary power points of (28) with respect top n,k keeping λ n,i ± λ 1[t] n p n,k log e 2, where (a) follows the property given in (32).
∂ ∂p n,k 2p n,k = 2p n,k log e 2 = p n,k log e 2, where,p n,k = log 2 (p n,k ). Based on the expressions in (31), the following equations of the near optimal stationary power points at iteration (t + 1) hold, 2 The Lagrange multiplier is a real number, so it doesn't matter if we use ±λ for exact equality constraints but '+' is the preferred sign for an inequality constraint, r(p 1 , p 2 ) ≤ 0.   Figure 2: The power plane illustration for two-users.
If (p ( ′ ,w,n) , R ) denote optimum solution for total powers and data rates corresponding to a particular λ n and w, then optimum solution of the problem with Lagrange Multipliers (λ ′ n , λ ′′ n ) and weights (w 1 , w 2 ) implies that λ We need to adjust λ n and w in order to satisfy the following constraints: , so that weighted power sum and weighted data rate sum become higher than any other tentative set of consideration either in the colored triangle or rectangle region.
In the consideration of two optimum solutions such as (p ( ′ ,wa,λ a n ) , R ( ′ ,wa,λ a n ) 2 , p ( ′′ ,wa,λ a n ) , R ( ′′ ,wa,λ a n ) 2 ) and ) for (w a , λ a n ) and (w b , λ b n ) respectively, the optimum solution (p ( ′ ,wa,λ a n ) , R ( ′ ,wa,λ a n ) 2 , p ( ′′ ,wa,λ a n ) , R ( ′′ ,wa,λ a n ) 2 ) for (w a , λ a n ) turn into the following expression, as the part of (35) is on the right certainly utilize the condition that optimize the Lagrangian for (w ) turn into a higher value for the Lagrangian.
Based on the same statement, the optimum solution n ) can lead into the following expression, By adding (35) and (36), we get Further, (37) for a pair of users can be re-expressed for N -user as where and ∆p = [∆p ′ , ∆p ′′ · · · ∆p N ] T denote vectors for the corresponding weights, Lagrange multipliers, data rates and powers.
Based on the condition of w and λ n , (38) can be classified into the following cases: (38) is a useful expression for establishing a operation to determine w and λ n that carry out the constraints. Due to negative product of ∆λ n and ∆p, ∆λ n for the required ∆p should be somewhere around the colored triangle plane on the opposite side of the ∆p vector. With the regulation of λ n by varying the ∆λ n vector, (39) promises with certainty to achieve very near to (p ′ ,max , p ′′ ,max ) provided that ∆λ n is not very high. It is illustrated in Figure 2 that the ∆λ n drives (p ′ ,w,λn , p ′′ ,w,λn ) to the next point on the inner side of the circle. Hence, the Lagrange multipliers' solutions can then iteratively express by applying the gradient descent method as below, where ϖ [t] stands for a sequence of scalar step size at t th iteration and [Z] + = max(0, Z) 4 .
Since MC-ILBCS algorithm can only achieve a lowerbound solution that is suboptimal, we propose another algorithm based on sub-channel iterative Lagrange multipliers search. This algorithm achieves an optimal solution, but it comes with additional computational complexity.

IV. NUMERICAL RESULTS AND DISCUSSION
In this section, numerical results are provided for the proposed power control schemes. Since the channel model     studied is novel, only a limited performance comparison with state of the art research can be done, where the CDF of SE for the scheme by [57] and this paper's proposed schemes: algorithms 1 and 2 are compared. However, a    based upon the corresponding transmission distance have been computed according to the following assumptions. The path-loss in dB corresponding to d n,i,k is calculated as [65]: L(d n,i,k ) = [44.9−6.55 log 10 (h BS )] log 10 (d n,i,k )+34.46+ 5.83 log 10 (h BS ) + 23 log 10 ( fc 5 ) + e n,i,k φ n,i,k , where h BS is the BS height that is selected as 30m and 10m for MBS and FBS, respectively; carrier frequency f c = 2.5GHz; additionally, e n,i,k indicates the number of walls and φ n,i,k indicates the wall penetration loss. It is assumed that φ n,i,k =5dB for e n,i,k =1 and φ n,i,k =12dB for e n,i,k =2. Figure 3 shows the macrocell traffic offload ratio versus number of FBS deployed in the system. It can be seen that as the number of FBS increases the offload ratio increases and with a very dense deployment of FBS it is possible to completely offload macrocell traffic to FBSs. This can be used for ultra-high speed and reliable communications in future wireless networks. When the number of FBSs is between 1 to 100, the increase at offload ratio is very high and this increase slows down when number of FBSs are more than 100. This shows that the addition of even small number of FBSs initially provides significant gain. Figure 4(a)-(c) shows the upper bound on the number of admitted secondary users versus the outage probability threshold, SINR and channel gain for different values of spectral efficiency. Figure 4(a) shows that there is a linear relationship between the upper bound on the number of admitted secondary users and the outage probability threshold. Also, as the spectral efficiency increases the number of admitted secondary users increases. In figure 4(b) the relationship between the upper bound on the number of admitted secondary users and SINR is shown to be logarithmic, where for a given spectral efficiency most of the increase is obtained at the low SINR regime, between 1-5 dB. Figure 4(c) shows that there is an exponential relationship between channel gain and the upper bound on the number of admitted secondary users. Also the number of admitted secondary users increases as the spectral efficiency increases. It is evident from above discussion that channel gain has the most effect on increasing the number of admitted secondary users, accordingly engineers that aim to increase this value shall focus on improving channel gain. Figure 5 shows the CDF of SE for various FFR values, where UL T x power per user = {0.1, 0.5} W and the number of FBSs = 16. It can be seen from the figure that for both UL T x power scenarios the performance increases as the FFR value increases from 1 to 3 and then it degrades as the FFR = 4. Thus, the optimal reuse factor is shown to be 3, this is in accordance with the literature. Performance improvement due to varying the FFR value demonstrates the effectiveness of FFR technique for interference management for cell-edge users. Further, the increase of UL T x power per user significantly improves the performance where the performance gain stays constant for considered FFR values. Figure 6 shows the CDF of SE for the scheme by [57] and this paper's proposed schemes: algorithms 1 and 2. The scheme by [57] is based on cognitive radio technology and comprised of two steps that is channel sensing with fractional frequency reuse and resource scheduling to manage heterogeneous interference problem. Proposed algorithm 1 uses multi-channel iterative lower-bound coefficients search and algorithm 2 employs sub-channel iterative Lagrange multipliers search, where the details are given in section III-A. It can be seen from the figure that the scheme by [57] is very effective at mitigating severe interference since it achieves the lowest percentage of users achieving SE between 0 to 0.5 bits/s/Hz compared to algorithms 1 and 2. However for the rest of the SE region considered in the simulations the proposed scheme with algorithm 2 outperforms the other scheme by [57] and the algorithm 1. Thus, algorithm 2 is a good candidate for next generation ultra-high speed communications where high SE regions are interested. Further, it should be noted from table II that algorithm 2 has a higher computational complexity compared to algorithm 1. For systems with limited computation power and time requirements there can be a performance-complexity trade-off. Figure 7 shows the cross-channel gain estimation versus number of FUs. It can be seen from the figure that as the number of FUs decreases the cross-channel gain decreases. This is due to a smaller number of users distributed over the geographical area and accordingly a reduced multi-user gain. Cross-channel gain is especially important for mobile applications that may require frequent vertical handovers. Having high cross-channel gain improves reliability for these applications, thus high user-density systems are favorable for their deployment.

V. CONCLUSION
This paper proposes a channel model for mobile mmWave massive MIMO based two-tier networks. The user mobility introduces a variable number of clusters and rays within each cluster at the mmWave channel. These properties of the mmWave channel have not been studied for two-tier networks. Studying this new channel model and power control schemes have been proposed. A flexible power control scheme is given that maximizes the total system capacity. It uses a multi-channel iterative lower-bound coefficients search algorithm to find the optimum solution. Number of SUs admitted to the system, SE, cross-channel gain estimation and macrocell traffic offload ratio metrics are investigated and discussed. Finally, it should be noted that the proposed channel model and schemes can be extended to multitier networks where the number of tiers is larger than two. As the first study of the channel model and proposed schemes in a multi-tier network architecture, this paper considers a twotier case for the algorithms and mathematical derivations to be easy to follow. Lack of available data and priori research studies on the proposed channel model are limitations of this study. Authors encourage researchers to use this model at their studies to increase the availability of data and provide opportunity for performance comparison between different proposals.