Cost-Efficient Millimeter Wave Base Station Deployment in Manhattan-Type Geometry

Urban millimeter wave (mmWave) communications are limited by link outage due to frequent blockages by obstacles. One approach to this problem is to increase the density of base stations (BSs) to achieve macro diversity gains. Dense BS deployment, however, incurs the increased BS installation cost as well as power consumption. In this work, we propose a framework for connectivity-constrained minimum cost mmWave BS deployment in Manhattan-type geometry (MTG). A closed-form expression of network connectivity is characterized as a function of various factors such as obstacle sizes, BS transmit power, and the densities of obstacles and BSs. Optimization that attains the minimum cost is made possible by incorporating a tight lower bound of the analyzed connectivity expression. A low-complexity algorithm is devised to effectively ﬁnd an optimal tradeoff between the BS density and transmit power that results in the minimum BS deployment cost while guaranteeing network connectivity. Numerical simulations corroborate our analysis and quantify the best tradeoff of the BS density and transmit power. The proposed BS deployment strategies are evaluated in different network cost conﬁgurations, providing useful insights in mmWave network planning and dimensioning.


I. INTRODUCTION
Next generation cellular networks will be deployed in millimeter wave (mmWave) bands to support the data-intensive fifth generation (5G) broadband use cases in urban areas [1]- [3].In mmWave bands, large-scale antenna arrays are used at both base stations (BSs) and user equipments (UEs) to generate directional narrow beams in order to overcome the severe pathloss [4]- [6].Directional transmission enables almost interference-free communications [7], but it also imposes new challenges, as the weak penetration and diffraction of mmWave propagation make the link susceptible to physical blockages.The blockage incurs frequent link outage in highly-obstructed urban areas.This is in contrast with the conventional sub-6GHz systems, where the outage largely The associate editor coordinating the review of this manuscript and approving it for publication was Zhenyu Xiao .results from co-channel interference rather than a physical blockage.
One approach alleviating mmWave link outage is to cover each UE by multiple BSs, i.e., macro diversity techniques [8], [9].When a link from a BS is blocked, the UE can switch to another unblocked BS to restore its link.An important practical implication of imposing the macro diversity is an increased number of BSs (i.e., dense BS deployment), which will then increase the expenditures on BS installment.Another alternative is to extend the cell coverage by increasing the transmit power of each BS in order to provide sufficient cell overlap.However, this approach incurs large power consumption.Under a certain connectivity requirement, how to resolve the best tradeoff between the BS density and transmit power that minimizes the BS deployment cost is of great interest for 5G network operators.
Link connectivity is hinged upon network geometry.A Poisson point process (PPP) has been verified to be an accurate model in capturing actual urban BS distributions [10], [11].When characterizing the mmWave links, additional treatment must be given because they are susceptible to physical blockage by urban obstacles.A widely adopted approach in modeling the urban obstacles is the Boolean scheme that assumes the shapes of obstacles are either circles [12], [13] or rectangles [14], [15], and the placement of them follows PPP in stochastic geometry.Under the Boolean scheme, connectivity of urban mmWave networks was previously studied in [8], [9], [12]- [18].However, these models cannot be extended to analyzing the connectivity of a Manhattan-type geometry (MTG) in Fig. 1, which closely models the urban geometry in most of the scenarios [4], [19]- [23].Unlike PPP, the obstacles (e.g., buildings) in the MTG are located on grids partitioned by streets.
Ray tracing has been used to evaluate the mmWave link connectivity in MTG [4], [19].There were attempts to analytically model the MTG based on the random lattice process [21], [22].In the random lattice process [21], [22], the geometry is partitioned into equidistant horizontal and vertical grids.The occupation of a grid by a random obstacle is independent and identically distributed (i.i.d.) with a certain probability.Based on the lattice process, a connectivity lower bound for the mmWave MTG was analyzed in [23] by incorporating disk-shaped blockage free regions (BFRs), where BFR is an area in which a BS can transmit signals to UEs without blockage.For BS deployment, accurate MTG connectivity analysis is very important.A challenge is that analytically computing accurate connectivity in MTG is prohibitively complex and even impossible [23].As a remedy, relying on bound analysis relieves the difficulty and makes the underlying task tractable.A previously known connectivity lower bound [23] is rather loose to be used for developing the connectivity-constrained mmWave BS deployment techniques that this work is envisioning.Prior work in this topic mainly focused on conventional sub-6GHz systems [24]- [29], and their connectivity model and the environment are different from those in the mmWave MTG.
In this work, we present a connectivity-constrained minimum cost mmWave BS deployment technique, based on the application of the random lattice process in MTG.We first seek to find tight connectivity lower bounds to be adapted to the formulated optimization problem.The optimized solution resolves the optimal tradeoff between the BS density and transmit power while minimizing BS deployment cost.The resulting BS deployment greatly enhances the connectivity of mmWave networks.Contributions of the work are summarized as follows: • A new method for connectivity analysis is proposed in the mmWave MTG.The analysis is based on finding a new set of BFRs that directly expresses the network connectivity by exploiting the random lattice process.Then a tractable lower bound of the connectivity probability is provided.The new lower bound greatly improves previous known lower bounds in [23].
• We exploit the connectivity lower bound to formulate and solve a connectivity-constrained optimization problem to achieve the minimum cost BS deployment.We show that the formulated problem can be transformed into a monotonic programming problem.Because conventional monotonic optimization techniques cannot be used due to the out-of-control complexity, we devise a new low-complexity search algorithm that finds a global optimal solution of the problem.
• Numerical simulations verifying the improved performance of the devised algorithm are presented.The results illustrate that the connectivity gaps between the theoretical analysis and the proposed connectivity lower bound within 8%, showing significant improvement over the connectivity analysis in [23], whose lower bound can offset over 20%.We discuss the impacts of various factors, i.e., BS density, transmit power, obstacle density, and obstacle sizes, on the connectivity and evaluate the improved BS deployment results under different cost settings.
The rest of this paper is organized as follows.In Section II, we describe the mmWave network model and assumptions made throughout the paper.The connectivity lower bound of the mmWave MTG is analyzed in Section III.In Section IV, we formulate and solve the connectivityconstrained minimum cost BS deployment problem.Simulation results and conclusions are drawn in Section V and VI, respectively.

II. SYSTEM SETUP
Suppose an urban mmWave network consisting of BSs, UEs, and obstacles (i.e., buildings) in a 2-dimensional MTG.Each UE is associated with one of the surrounding BSs that meets link quality requirement.The associated BS can be in any direction of a UE.After the association, directional link transmission is established.The link can possibly be blocked by obstacles.Outage of a UE occurs when there is no surrounding BS to communicate with.We set an arbitrary point as the origin of the R 2 space and build up the x and y axes to define a coordinate system in the MTG, as shown in Fig. 2. The subsequent analysis and derivations are based on the following assumptions and definitions.

A. OBSTACLES
We assume that obstacles in the MTG in Fig. 2 obey a random lattice process.In the random lattice process, the geometry is partitioned into equidistant horizontal and vertical grids, where the length and width of each gird are set to l > 0 and w > 0, respectively.Denoting the row grids containing the x axis and the column grids containing the y axis as the 0th row and 0th column, respectively, all rows and columns in Fig. 2 are sequentially labeled from down to up and left to right as −N LB Row , . . ., 0, . . .N UB Row and −N LB Col , . . ., 0, . . .N UB Col , respectively.The random lattice process then models the obstacles with length l and width w in urban MTG as illustrated in Fig. 2, where the occupation of each grid by an obstacle is i.i.d. with probability p o . 1  Denoting the grid at row a and column b as (a,b) , we say that occupation state of the grid (a,b) becomes z (a,b) = 1 if the grid is occupied by an obstacle and z (a,b) = 0, otherwise.Hence, the probability that a grid (a,b) is occupied by an obstacle is given by Pr Obstacles in the MTG can block mmWave links.

B. BASE STATIONS AND TYPICAL USER EQUIPMENT
The placement of mmWave BSs in MTG obeys a homogeneous PPP BS = {B} with density λ, where B represents the location of a particular BS in R 2 .We focus on the problem of outdoor mmWave BS deployment, where the outdoor BSs serve the outdoor UEs.Both outdoor BSs and UEs are located in grids that are not occupied by obstacles.All outdoor BSs are assumed to have the same transmit power value P Tx in 1 For ease of presentation, all grids in the MTG are assumed to have the same size and occupation probability p o .Extension of the model to the cases where grids have different sizes and occupation probabilities is straightforward.One can also emulate more practical scenarios.For example, setting occupation probability to p o = 0 of the grids in a same row or column represent a street.A practical scenario is actually a realization of the random lattice process in Fig. 2.
Watt, where 0 < P Tx ≤ P Max Tx and P Max Tx is the maximum allowed transmit power value.
Connectivity probability of a typical outdoor UE U o located at the origin in Fig. 2 is a function of parameters l, w, p o , λ and P Tx , which is denoted as p U o (l, w, p o , λ, P Tx ).Note that moving the UE U o (i.e., the origin in R 2 ) to a different location inside (0,0) results in a different connectivity probability.We define p c (l, w, p o , λ, P Tx ) as the minimum connectivity probability among all U o ∈ (0,0) , Due to the i.i.d grid occupation statistics and PPP-obeying BS distribution, the value of p c (l, w, p o , λ, P Tx ) is the same over different grids.This being said, without loss of generality, we focus on analyzing connectivity of UEs located at the grid (0,0) to achieve the p c (l, w, p o , λ, P Tx ).We shall refer to the minimum connectivity probability p c (l, w, p o , λ, P Tx ) as the network connectivity.

C. LINK BUDGET
An mmWave link is mainly characterized by the line-ofsight (LoS) and a few low-order reflection paths [11], [30].However, in MTG, the low-order reflection paths have a negligible contribution to the network connectivity because they suffer from much severer attenuation than the LoS path due to additional reflection losses, longer propagation distance, and higher blockage probability [23], [31].Hence, our focus is given to the connectivity contributed by the major LoS components, while neglecting minor reflection components.
For an LoS link with length r meters, we adopt the pathloss model at 28 GHz [31] PL(r) = 61.4+ 21 log 10 (r) in dB.
The received power at the UE U o can be given by where G Tx and G Rx are directional beam gains at the BS and UE, respectively.We let P Th be the minimum power required at the receiver for successful signal detection.Setting P Rx = P Th in (3) allows us to get the BS coverage radius whose maximum value R Max = f −1 (P Max Tx , P Th ).

D. BLOCKAGE FREE REGION (BFR)
If there exists an outdoor BS B ∈ BS that can serve the UE U o , we say that B is in the BFR of the U o , namely, B ∈ .This means any B ∈ simultaneously satisfies two conditions: (i) B is located in the disk with radius R in (4), that is, where Therefore, the BFR can be formally defined by Since ⊂ O (R), only BSs located in O (R) can contribute to the connectivity of UE U o .Hence, in Fig. 2, we focus on the region consisting of grids (a,b) for where , and where x represents the minimum integer that is greater than or equal to x, and l and ŵ, as shown in Fig. 2, are the length and width of the sub-rectangle of (0,0) in orthant 1. (5).

III. CONNECTIVITY ANALYSIS
The outdoor UE U o in Fig. 2 is in connectivity when there are BSs in the BFR in (6), i.e., ∩ BS = ∅.Hence the p U o (l, w, p o , λ, P Tx ) in (2) can be expressed as The goal of this section is to derive a tight lower bound of the p U o (l, w, p o , λ, P Tx ) that admits a computationally feasible closed-form expression.Throughout this section, we shall use p U o to denote p U o (l, w, p o , λ, P Tx ) for simplicity.Because connectivity probability of the outdoor UE U o ∈ (0,0) is the focus, we hold z (0,0) = 0.

A. CONNECTIVITY PROBABILITY
Seen from Fig. 2, the MTG is divided into four orthants by the x and y axes.We name anticlockwise the orthants as orthant 1, 2, 3, and 4, where orthant 1 is outlined by the semipositive x axis and y axis.The UE U o is in connectivity when it can be served by an outdoor BS in any of the four orthants.The first step of our method is to analyze the counterparts of the and BS in each orthant i, i.e., (i) and (i) BS , respectively.The connectivity p U o in ( 9) is obtained as We note in Fig. 3 that the occupation state of a grid at row 0 or column 0 can simultaneously affect the BFR (i) of two orthants.This implies that the connectivity of an orthant is correlated with the connectivity of other orthants, which inflicts difficulty in computing (10).
To decouple the correlation, we denote points of the BFR at the row 0 or column 0 as which varies with the occupation states of grids at row 0 and column 0, as exemplified in Fig. 3. Conditioned on a realization of C, the (i) for i ∈ [1,4] become into mutually independent.The independence leads to the p U o in (10) conditioned on the jth realization of C, denoted by C j , as where p The connectivity probability p U o in (10) over random C is thus given by where E C [p U o |C ] is the expectation of p U o |C in terms of C, and J is the number of realizations of the C in (11).In each realization of the obstacle distribution, we compute four variables based on the obstacle occupation states of grids on the four semi-x and -y axes, where Pr(η i ), ∀i are computed using the same approach.For instance, we have where (1 − p o ) η +y is the probability that grids (a,0) for a ∈ [1, η +y ] are not occupied by obstacles.
Finding the p U o in ( 13) needs the closed-from expression of p In what follows, we focus on the computation of p keeping in mind that the exactly same approach can be applied to the calculations of p

B. CONNECTIVITY PROBABILITY IN ORTHANT 1
Without loss of generality, we focus on an arbitrary realization C j ∈ C in (11).The subscript j of p (1) 12) is omitted whenever there is no ambiguity.
In Fig. 2, conditioned on a realization of C, the connectivity in orthant 1 is characterized by the occupation state set Under different realization of D, we can obtain different (1) .This allows us to express the p where (a) is because, for PPP-distributed BSs, the probability that there are BSs in a (1) (i.e., (1)  ∩ (1) BS = ∅) is given by 1 − exp − λS( (1) ) , in which S( (1) ) is the area of (1) .The (1) m|C in (21) is the mth instance of (1) conditioned on C. In (21), one should find the S( (1) m|C ) for every m.According to definition of the BFR in (6), each (1) m|C has an irregular shape, as shown in Fig. 4, making computation of its area challenging.To circumvent the difficulty, we alternatively derive a tractable lower bound of the p t|C is the maximum inscribed BFR of a Referring to [23, (8)], a simple method finding the t|C } T t=1 is to set each (1) t|C to be the subregion of a disk O(r) in orthant 1 (i.e., sector-shaped (1) t|C ), where r is a discrete random variable.The sector-shaped { (1) t|C } T t=1 will lead to a loose lower bound in (23) because the shape of each (1) m|C is usually far from its sector-shaped approximation.For instance in Fig. 4, the maximum inscribed sector in a (1) m|C is much smaller than the (1) m|C in area.To obtain a relatively tight lower bound in (23), we find a new BFR set { As observed from Fig. 3, the (1) m|C in (23) for all m have a common region, denoted by C (1) , which is the subregion of the C overlapped with orthant 1. Inside the C (1) , a tractable subregion C (1) is derived to be the common region of the targeted { (1) t|C } T t=1 .As in Fig. 3, the C (1) is L-shaped and is determined by the η +y in ( 14) and η +x in (15).When η +y < N UB Row and η +x < N UB Col , the C (1) is limited by occupation states of grids and has regular shape.The C (1) can be directly set to C (1) = C (1) whose area S(C (1) ) = l ŵ + wη +y + ŵ l + lη +x . ( When η +y = N UB Row and/or η +x = N UB Col in Fig. 3, part of the (η +y ,0) and/or (0,η +x ) is in the C (1) , leading to irregular shape of C (1) .In this case, the C (1) is set to be the C (1)   excluding the part of the (η +y ,0) and/or (0,η +x ) .We can compute the S(C (1) ) using ( 24) by updating η +y = η +y − 1 and/or η +x = η +x − 1.
Finding the BFR set { (1) t|C } T t=1 is now of interest.For ease of computing S (1) t|C − S(C (1) ), we consider each where n (1) b−1 , where n In (26), the n b guarantees the obtained (1) b−1 , as illustrated in Fig. 5, ensures the (1) t|C ⊂ F in ( 6) if there is no obstacle in the (1) t|C .Hence, the (1) t|C in ( 25) can be a BFR defined in (6).

D. CONNECTIVITY LOWER BOUND
To use the obtained { (1) t|C } T t=1 to compute the lower bound in (23), we first show that the { (1) t|C } T t=1 satisfies the three conditions specified above (23).According to (25), the area

S
(1) t|C = S(C (1) )+ b .Since a (1) 1) in (25) obtained by = n (1) 0 = η +y , n (1) 2 = 0, . . ., n (1)  η +x = 0 is the common region of all instances (1) m|C in (21), each m|C can find a maximum inscribed BFR in { (1) t|C } T t=1 by using (22).Seen from Fig. 5, setting occupation states of grids crossed and overlapped with a (1) t|C to be 0s and occupation states of the remaining grids in the MTG to be 1s, we obtain a (1) m|C whose maximum inscribed BFR must be the (1) t|C .We can then compute the lower bound in (23) as where Pr N (1) t follows Lemma 1.Using the same approach, we define N (i) for i ∈ [2,4] and find p (i) U o |C for each i ∈ [1,4] to compute the lower bound of p U o in (13) as Lemma 1: Probability of a realization of N (1) is computed as Pr N (1)  = where Pr n b−1 is the probability that n (1) b is selected from its domain (28) determined by n b−1 .
Remark 1: Different realizations of N (1) in ( 27) yield BFR (1) t|C s in (25) with different shapes.Compared to the sectorshaped { (1) t|C } T t=1 , as illustrated in Fig. 4 and 5, the derived { (1) t|C } T t=1 can approximate instances of (1) in ( 23) in a more accurate way.Hence, we can obtain a relatively tight connectivity lower bound, as can be seen in Section V.
Remark 2: When computing the p U o in (30), we first use the BS coverage radius R to determine the N LB Row , N UB Row , N LB Col , and N UB Col in (8), and the α b s in (26) for different b and i ∈ [1,4].These variables are all integers and used to determine the p U o .Hence, the p U o is an implicit function of R.

IV. BASE STATION DEPLOYMENT PROBLEM
We find the location of U o ∈ (0,0) in Fig. 2 that has the minimum p U o (l, w, p o , λ, R) in (30), where R = f −1 (P Tx , P Th ) in (4).According to (2), the p U o (l, w, p o , λ, R) at this U o location is a lower bound of the network connectivity p c (l, w, p o , λ, R), denoted by p c (l, w, p o , λ, R).In this section, we use p c (l, w, p o , λ, R) to construct the network connectivity constraint and then formulate and solve the minimum cost BS deployment problem in the mmWave MTG.

A. PROBLEM FORMULATION
We first define the objective function.Based on the properties of PPP, the average number of outdoor BSs per square meter and the power consumption at these BSs are given by λ(1 − p o ) and (P Tx + P C )λ(1 − p o ), respectively, where P Tx is the transmit power and P C is the constant circuit power at each BS [32].Let c BS be the cost of installing an outdoor BS and c Pw be the cost of unit power in a certain period. 2The average BS deployment cost per square meter is given by With the minimum required network connectivity ζ Con , the connectivity-constrained minimum cost BS deployment problem is formulated as where λ and P Tx are decision variables, while geometry parameters l, w, and p o are fixed at this stage.It is verified from the definition of the objective function in (33) and derivations of the network connectivity p c (l, w, p o , λ, R) (i.e., p U o (l, w, p o , λ, R) in (30)) that the objective function and p c (l, w, p o , λ, R) in problem (33) are both monotonically increasing with respect to the BS density λ and transmit power P Tx .These can be intuitively known since high BS density and large BS coverage radius contribute to the the range of a subinterval, the increment of R at step 3 may skip the subinterval, and the minimum R in this subinterval is not included in the R. Besides, a large t will lead to non-negligible bias between the obtained values in R and true values of the minimum R in each subinterval.To avoid the above issues, the t should be small enough.However, a too small value of t may negatively affect the algorithm efficiency.
We now quantitatively show an appropriately value for t.According to (8) and (26), integers in ϒ in (37) are used to describe the number of grids inside the boundary of the disk O(R).Since R in different subinterval have different ϒ, the range of a subinterval should have the magnitude order similar to sizes of grids.We intuitively set the increment t = 0.02 min(l, w). ( Since t in (38) can be large when both l and w are large, we further limit t ≤ 0.
Compute ϒ(i) in (37).With the error tolerance , the number of iterations in the Bisection method for each R ∈ R is given by log (λ UB − λ LB )/ [33].In each iteration, we compute the p c (l, w, p o , λ, R) once.In total, it is computed times, where |R| is the cardinality of set R. While computing the p c (l, w, p o , λ, R), we need to compute the p (i) U o |C j (l, w, p o , λ, R) , ∀i ∈ [1,4] for J in (18) times.Seen from (29), complexity of p where ϑ > 0 is a newly introduced small integer.The Pr N (1) in ( 29) can be directly computed in (31) by replacing the min α b−1 , ϑ .In this way, the complexity of p The ϑ should be properly selected.If ϑ is critically small, there are a few elements in { U o |C (l, w, p o , λ, R) in ( 29) will become to be a loose lower bound.On the other hand, large ϑ will nullify the computational reduction efforts because of min α In this work, Algorithm 2 is designed to determine the ϑ that achieves a tradeoff between the lower bound tightness and complexity.In the algorithm, we initialize ϑ = 1 and then gradually increase it.The increment of ϑ terminates when: (i) The computational time of the overall connectivity lower bound p c (l, w, p o , λ, R) exceeds a predefined threshold Max , or (ii) The increment of p c (l, w, p o , λ, R) from the previous iteration is smaller than a tolerance threshold > 0. By integrating the Algorithm 2 into the above Bisection method, we finally obtain a low-complexity search algorithm.

V. SIMULATION STUDIES
Numerical results are presented to verify the connectivity analysis and evaluate the BS deployment strategy.In the connectivity comparison, lower bounds in [23] are taken as benchmarks.Throughout this section, the length and width of each grid are set to be l = 15 m and w = 10 m, respectively unless stated otherwise.

A. NETWORK CONNECTIVITY
In Algorithm 2, we know larger ϑ can yield tighter connectivity lower bound p c (l, w, p o , λ, R).We now quantitatively evaluate the impact of ϑ on the p c (l, w, p o , λ, R). t m |C is large.This is to say, the lower bound in ( 23) is tight even though each (1) m|C is not accurately approximated by its maximum inscribed BFR.When ϑ is already large, increasing ϑ contributes little to tightness of the derived lower bound.This explains why gaps between the curves with ϑ = 4 and ϑ = 6 are minor.Seen from Fig. 6, ϑ = 5 is enough to obtain a tight connectivity lower bound.Since ϑ = 5 is not significantly large, the computational complexity of p c (l, w, p o , λ, R) is relatively low.
Fixing ϑ = 5 and R = 120 m, we evaluate the connectivity lower bound p c (l, w, p o , λ, R) versus BS density λ with different p o .As shown in Fig. 7, the connectivity lower bound p c (l, w, p o , λ, R) becomes critically tight when p o is relatively large.This is because, in large obstacle occupation probability p o , the connectivity p  21) is dominated by the instances of (1) with small areas.These small-area (1) m|C s are usually close to their common region C (1) in (25) and thus can be well approximated by our derived BFR set in (25).In practical urban scenarios, the obstacle density is usually high, leading to relatively large p o .Hence, our derived connectivity lower bound can be tight in mmWave MTG.Setting ϑ = 5 and p o = 0.6, we show the connectivity lower bound p c (l, w, p o , λ, R) versus the BS density λ for  different coverage radius R in Fig. 8.It is observed that, when R is relatively large, e.g., R = 50 m, increasing R (increasing transmit power) contributes little to the network connectivity.It means connectivity of an outdoor UE is mainly contributed by nearby BSs.This is due to the physical blockage, which is severe when path length is large.Compared to Monte-Carlo simulations, our derived connectivity lower bounds show great tightness, and the maximum bias is around 8%.This significantly outperforms the benchmark, whose offset can be over 20%.Seen from Fig. 8, the connectivity lower bound is monotonically increasing with respect to the BS density λ and coverage radius R.
The effect of different obstacle sizes on the network connectivity lower bound p c (l, w, p o , λ, R) is investigated in Fig. 9 under p o = 0.6, R = 100, and ϑ = 5.It is observed that increasing sizes of grids will dramatically improve the p c (l, w, p o , λ, R).It is due to the fact that larger l and w will result in fewer number of grids in O(R).Given the obstacle occupation probability p o , it is easier to obtain BFR (1) in (16) with large area, which benefits the connectivity.

B. BASE STATION DEPLOYMENT
We evaluate the BS deployment strategy under ϑ = 5 and p 0 = 0.6.For link budget in (3), we set the beam gain G Tx = G Rx = 13 dB, and power successful detection threshold VOLUME 7, 2019

FIGURE 1 .
FIGURE 1.An example of the urban MTG.

FIGURE 2 .
FIGURE 2. Urban mmWave MTG modeled by a random lattice process.
represents the length of the LoS segment U o X connecting points U o and X, and (ii) U o B is not blocked by VOLUME 7, 2019 any obstacle.Defining the region occupied by obstacles as A = (a,b) |z (a,b) = 1, ∀a, b , the condition (ii) is equivalent to

FIGURE 3 .
FIGURE 3. Illustration of the set C in (11) in a random lattice modeled MTG.
o |C in (21) by finding a tractable BFR set { (1) t|C } T t=1 with T elements satisfying three conditions: (i) S (1)t|C for every t ∈ [1, T ] is known, (ii) each(1) m|C can find a maximum inscribed BFR from {(1)

1 )
m|C in which M t is the index set of (1) m|C s who have the same maximum inscribed BFR (1) t|C .Obviously, tightness of the lower bound in (23) is guaranteed if each (m|C is accurately approximated by its maximum inscribed BFR.This requires the BFR set { (1) t|C } T t=1 contains large number of elements with various shapes.
NEW TRACTABLE BFR SET IN ORTHANT 1

1 )
t|C consists of a rectangular region at each column b ∈ [1, η +x ] and tractably express it in the form(
Overhead of Algorithm 1 is low due to the limited iteration number R Max /t and low-complexity step 5. To implement the Bisection method for each R ∈ R, one should first find λ UB and λ LB satisfying p c l, w, p o , λ UB , R > ζ Con , p c l, w, p o , λ LB , R < ζ Con .
o |C (l, w, p o , λ, R) is dominated by η +x b=1 min n (1) 0 , α (i) b , which is huge when n (1) 0 = η +y , η +x , and α (i) b are large.In what follows, we focus on reducing the computational overhead of the p (i) U o |C (l, w, p o , λ, R) in the case of huge o |C (l, w, p o , λ, R) in (29) as the example, we reduce its complexity by shrinking the min α , ϑ , ∀b ∈ [1, η +x ],

FIGURE 6 .
FIGURE 6. Connectivity lower bound p c l , w , p o , λ, R versus BS density λ under different ϑ with p o = 0.4 and R = 120.
Fig. 6 displays the p c (l, w, p o , λ, R) versus BS density λ under different ϑ for p o = 0.4 and R = 120 m.For large λ, the analytical connectivity is close to that of Monte-Carlo simulation in even small ϑ.This is because large λ guarantees 1 − e −λS (1) m|C ≈ 1 − e −λS (1)

FIGURE
FIGUREConnectivity lower bound p c l , w , p o , λ, R versus BS density λ under different p o with ϑ = 5 and R = 120.

FIGURE 8 .
FIGURE 8. Connectivity probability p c l , w , p o , λ, R versus BS density λ under different R for p o = 0.6 and ϑ = 5.