MmWave vehicle-to-infrastructure communication: Analysis of urban microcellular networks
Yuyang Wang Kiran Venugopal Andreas F. Molisch Robert W. Heath Jr


Electrical control of the sign of the g-factor in a GaAs hole quantum point contact
    A. R. Hamilton^1
    December 30, 2023
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		Vehicle-to-infrastructure (V2I) communication may provide high data rates to vehicles via millimeter-wave (mmWave) microcellular networks. This paper uses stochastic geometry to analyze the coverage of urban mmWave microcellular networks. Prior work  used a  pathloss model with a line-of-sight probability function based on randomly oriented buildings,  to determine whether a link was line-of-sight or non-line-of-sight. In this paper, we use a pathloss model inspired by measurements, which uses a Manhattan distance pathloss model and accounts for differences in pathloss exponents and losses when turning corners. In  our model, streets are randomly located as a Manhattan Poisson line process (MPLP) and the base stations (BSs) are distributed according to a Poisson point process. Our model is well suited for urban microcellular networks where the BSs are deployed at street level. Based on this new approach, we derive the coverage probability under certain BS association rules to obtain closed-form solutions without much complexity. In addition, we draw two main conclusions from our work. First, non-line-of-sight BSs are not a major benefit for association or source of interference most of the time. Second, there is an ultra-dense regime where deploying active BSs does not enhance coverage.
	
	
	
	
	
	

§ INTRODUCTION
	
	Vehicle-to-infrastructure (V2I) communication offers the potential to enhance safety and efficiency in urban vehicular networks <cit.>. Combined with millimeter wave (mmWave) <cit.>, V2I has the potential to offer high data rates and low latency <cit.>, enabling massive data sharing among a great number and diversity of mobile devices in vehicular networks <cit.>. MmWave communication not only has access to larger bandwidths, it can also allow large yet very compact antenna arrays deployed at both the transmitter and receiver to provide high directional beamforming gains and low interference.  Compared to channels at microwave frequencies (<6 GHz), however, mmWave channels are more sensitive to blockage losses, especially in urban streets where signals are blocked by high buildings, vehicles or pedestrians <cit.>, <cit.>, and sharp transitions from line-of-sight (LOS) to non-line-of-sight (NLOS) links are more common.
	[In the current paper, LOS is defined as “optical" line of sight between the transmitter and receiver location as the papers cited here. NLOS happens when the link between the transmitter and the receiver is blocked by obstructions, specifically in our paper, is blocked by urban buildings.]This motivates the study of mmWave microcellular network performance in the context of vehicular urban areas. 
	
	

 §.§ Related Work

	
	Urban street model: Stochastic geometry has been used extensively to analyze performance in mmWave cellular  networks <cit.>. BS and cellular user locations are modeled as Poisson point processes on a two-dimensional plane, based on which the coverage probablity of a typical cellular user is derived. Also, building blockages are considered as the main source differentiating LOS and NLOS links, with a few papers analyzing different building blockage models.
	XXX REFS. NEED TO ADD REFERENCES HERE, INCLUDING TUTORIAL PAPER WITH JEFF ON MMWAVE CELLULAR, HIGHLIGHT APPROACHES AND LIMITATIONS. LOOKING FOR 3-4 MORE SENTANCES AT THE MOST
	Unfortunately, prior work analyzing mmWave cellular networks in <cit.> employed a pathloss model with a LOS probability function based on Euclidean distance <cit.>, to determine whether a link was LOS or NLOS. This works well for randomly oriented buildings <cit.>, but does not properly model V2I networks where strong LOS interference may result from infrastructure co-located on the same street. 
	
	Recent work has considered alternative topologies that may better model urban areas. 
	XXX JUST BACCELLI PAPER, ANY OTHER RELATED PAPERS??
	In <cit.>, an approach to determine LOS and NLOS BSs by approximating a LOS ball was proposed. The model was shown to be able to better approximate the LOS area than <cit.>. 
		In <cit.>, three-dimensional Poisson buildings were modeled using
		Poisson processes to characterize the correlated shadowing
		effects in urban buildings. The idea was to add one more
		dimension to the Manhattan Poisson line processes (MPLP), by modeling the floor locations as Poisson
		process. This allowed an exact characterization of coverage of
		indoor urban cellular networks.  In <cit.>, a stochastic geometry model in a Manhattan type network was analyzed, since it is a tractable yet realistic model for Manhattan type urban streets. The urban streets were modeled as one-dimensional MPLP and the coverage probability was derived considering the penetration of signal through buildings. Unfortunately, the results in <cit.> used a pathloss model mainly considering the penetration effects of signals through urban buildings, with a fixed loss for each penetration. This is not applicable for mmWave systems where penetration loss is high. In this paper, we also use the MPLP for modeling the urban street distribution, but combined with a mmWave-specific channel model. 
	
	Urban mmWave channel model: There is a vast body of literature concerning mmWave channel modeling in urban areas, see, e.g., <cit.> and references therein. One of the key characteristics of urban environment is the high density of streets and high-rise buildings. Since mmWave signals are  sensitive to blockage, which induces significant signal attenuation, LOS and NLOS links can have sharply different pathloss exponents, as was also shown in numerous measurements <cit.>, <cit.>, <cit.>, and is reflected in the standardized channel models <cit.>.  Investigations in a variety of environments showed that, in general, penetration loss increases with carrier frequency.  For modern buildings with steel concrete and energy saving windows, in particular, penetration through just one wall can incur losses in the order of 30 dB; therefore, propagation through buildings is not a relevant effect in mmWave Manhattan type urban environments <cit.>.
	
	In <cit.>,  a spatially consistent pathloss model was proposed for urban mmWave channels in microcells. Based on ray tracing, it was shown that the pathloss exponents differ from street to street and should be modeled as a function of both the street orientation and the absolute location of the BS and user equipment (UE)[Henceforth we assume a downlink so that receiver and UE can be used exchangeably.]. Hence, the signal is seen as propagating along different streets, with diffraction effects happening at the corner, instead of penetrating through the urban buildings. The pathloss is summed up by the individual pathloss on different segments of the propagation paths,  incorporating an additional loss at each corner. This shows that the  Euclidean distance might not be a good measure to characterize the pathloss effects in urban microcell networks at mmWave.  In this paper, we adopt a modified pathloss model  similar to <cit.> based on the Manhattan distance, which enables tractable analysis while still retaining the key features of the mmWave microcellular channel. 
	
	
	
	

 §.§ Contributions

	
	In this paper, we develop a tractable framework to characterize the downlink coverage performance of urban mmWave vehicular networks. Specifically, we consider snapshots of the urban microcellular network, without modeling vehicle mobility. This reduces the network to an urban mmWave microcellular networks. We model the location of urban streets by a MPLP. The width of the street is neglected, and herein the blockage effects of vehicles are not considered in the analysis. We extend our previous paper <cit.> to account for large antenna arrays and directional beamforming at mmWave. We use a modification of the sectorized antenna model for tractable analysis <cit.><cit.> and apply the new pathloss model from <cit.>. The pathloss model is characterized by the Manhattan distance of the propagation link, which, with MPLP street modeling, yields tractable results for coverage analysis. 
	
	Based on our model, we analyze coverage of randomly located UEs on the roads formed by the lines, which is different from the conventional approach where coverage is analyzed conditioned on the links being outdoors <cit.>. We adopt a new procedure in the calculation of coverage probability, compared to the previous work <cit.>.  We analyze the coverage probability by first computing the cumulative distribution function (CDF) of associated BS link gain and then the coverage probability conditioned on the associated link gain. By averaging over the conditioned received signal power, we obtain simple but accurate expression of coverage probability. 
	
Compared to <cit.>, this paper also includes the following contributions. Based on the coverage probability, we obtain insights concerning the scaling laws of coverage probability with street and BS intensities, the sensitivity of coverage to the channel conditions and the effects of LOS/NLOS interference. Also, we derive closed-form expressions of the LOS BS association probability, under different channel conditions. We then use the map data of the streets in Chicago from OpenStreetMap <cit.> and extract it using the Geographical Information System (GIS) application QGIS <cit.>. This is used to compare the ergodic achievable rate of realistic streets, MPLP street model and fixed grid models. The comparison shows that the MPLP based analysis is valid for outdoor microcell urban networks at mmWave.  

	
	
	

§ SYSTEM MODEL

	In this section, we explain the key assumptions and models adopted in this paper. First, we explain the street model in urban vehicular networks. Then, we present a tractable form of the pathloss model of mmWave microcells based on Manhattan distance from <cit.>. We introduce a modified mmWave sectorized  antenna pattern that is used for our analysis. Lastly, we formulate the signal-to-interference-plus-noise ratio (SINR) of the receiver and demonstrate the rule of the strongest propagation path.
	

 §.§ Network model-MPLP

	
	
	  
	
	We show in Fig. <ref> an illustrative snapshot of the Manhattan network in a Cartesian coordinate system.  Without loss of generality, a typical receiver is placed at the origin O, and the streets are assumed to be either perfectly horizontal or vertical in the coordinate system. We call the street where the receiver is located at as the typical street, i.e., the x-axis. We refer to other  horizontal and vertical streets respectively as parallel streets and cross streets. These streets are generated from two independent one-dimensional homogeneous Poisson point processes (PPP) Ψ_x and Ψ_y, with identical street intensity λ_S. Under the current coordinate system, we define the set of parallel streets as ⋃_y_i∈Ψ_y L_y(y_i), where ⋃ denotes union of sets, and L_y(y_i) denotes the parallel street with intercept (location) at y_i. The set of the cross streets is defined as ⋃_x_j∈Ψ_x L_x(x_j), with L_x(x_j) similarly defined as the cross street having intercept x_j on the x-axis. By Slivnyak's theorem [15], [33], the typical street y =0 is added to the process. BSs are deployed at the street level, and are distributed on each cross, parallel, and the typical street as independent one-dimensional homogeneous PPPs. Similar to the naming convention for the streets, we name the BSs on the typical streets as typical BSs, and cross BSs and parallel BSs on the cross and parallel streets respectively. 	
	

 §.§ Pathloss model
	
	
	We adopt a pathloss model that is based on the Manhattan distance instead of Euclidean distance. The model is similar to  <cit.>, but uses several modifications to provide tractability. Ray tracing shows that in an urban mmWave microcell,  Euclidean distance might not be a dominant parameter in pathloss modeling. Since the penetration through urban building walls is negligible at mmWave, the signal detours its way along the streets in urban canyons and changes its directions by diffractions on the buildings at intersections. Therefore, instead of the direct Euclidean distance between the BS and the receiver, the street orientation relative to the BS location, and the absolute positions of the BS and receiver are the key parameters to determine the pathloss. 
	
	It is shown by the ray tracing results that a way to model the net pathloss of a propagation link in urban mmWave microcells is to add up the pathloss on different segments of the propagation paths, with an additional loss when the waves couple into a new street canyon. The propagation path may be thought of as segmented, with the signal changing directions to find LOS paths, circumventing building blockage. We assume that there are in total of M segments along the propagation paths, i.e., M-1 corners where signal changes directions. Note that the value of M depends on the actual position of the BS and the receiver.  The individual length of the i-th segment is denoted as d_i, the pathloss exponent on the i-th segment is α_i, and the corner loss at the corner of the ith street segment and i+1-th segment is Δ (in decibel scale), where we assume corner losses at different corners are identical. 
	
	We define the LOS segment as the first segment of the propagation path from the BS and NLOS segment as the remaining segments on the propagation path. It should be noted that the LOS and NLOS defined for the segments are only indicating the order of different segments of the propagation paths, which is different from the definition of LOS/NLOS paths in traditional representations. 
	We assume that LOS segments on different streets share the same pathloss exponent α_L, while the pathloss exponent for NLOS segments is α_N. Notice that the equation is not “symmetric", i.e., the street segment that has LOS to the BS has a pathloss coefficient that is different from the one that has LOS to the UE; such a situation might occur due to the different heights of UE and BS. To clarify, the pathloss does not hold for vehicle-to-vehicle channel modeling, since the model is asymmetric. To conclude, the pathloss in the decibel scale is defined as follows 
	
    PL_dB =
    		10( α_Llog_10d_1 + α_N∑_i=2^Mlog_10d_i) + (M-1)Δ.

	With this Manhattan distance based pathloss model,  we can classify the BSs into three categories, as illustrated in Fig. <ref>:  i) BSs on the typical street (typical BSs) that have one direct propagation path to the typical receiver; ii) NLOS BSs on the cross streets (cross BSs) that have a propagation path consisting of a LOS segment (green path d_1) and NLOS segment (green path d_2) to the typical receiver, and iii) NLOS BSs on the parallel streets (parallel BSs) that have a propagation path consisting of a LOS segment (red path d_1) and two NLOS segments (red path d_2, d_3). The analysis of the strongest path of different BSs will be provided in Section <ref>. This pathloss model also bears a strong relationship to <cit.>, which considered the pathloss model in urban microcells where waves are coupled at the street corners with different angles. 
	

 §.§ Sectorized antenna model

	To leverage array gain, directional beamforming by multiple antennas is performed at mmWave BSs. For simplicity, we assume the receiver has an omni-directional antenna, and the BSs are equipped with N_t transmit antennas. We adopt a sectorized antenna model for the BS <cit.>, <cit.>, with the main lobe gain denoted as G and the side-lobe gain as g.  The beamwidth of the main lobe is θ, as shown as the red fan in Fig. <ref> and all the other directions outside the main lobe are assumed to be in the side lobe (shown in the blue circle). 
	For a uniform planar antenna array, the main lobe gain can be approximated by G = N_t, which is  the maximum power gain that can be supported with N_t-element antenna array. The side-lobe gain is evaluated by g = √(N_t)-√(3)/2πN_tsin(√(3)/2√(N_t))/√(N_t) - √(3)/2πsin(√(3)/2√(N_t)), which is calculated to satisfy the following antenna equation for constant total radiated power <cit.>, <cit.>,
	
    ∫_-π^π∫_-π/2^π/2 G(ϕ, ψ)cos(ψ)dψ dϕ = 4π,
 
	and the beamwidth is θ =   √(3)/√(N_t).
	The beamforming gain 𝒢 is therefore formulated as  
	
    𝒢 =  G,    if the receiver lies inside the main lobe,
    
    	g,    if the receiver lies inside the side lobe.

	 Since the beamforming direction of the BS is assumed to be uniformly distributed in (0, 2π), the beamforming gain 𝒢 of one typical BS with LOS visibility to the typical receiver is a Bernoulli random variable, so that
	
    𝒢 = A G + (1-A) g,

	where
	
    A    = 𝕀(p), 
    	p    = θ/2π.

	  


		
		
	

 §.§ Signal-to-interference-plus-noise ratio (SINR) 

	SINR coverage analysis is important to determine outage holes and ergodic throughput of wireless networks. While these metrics in the context of mmWave-based vehicular networks depend on both mobility and the blockage effects due to the vehicles, in this paper we simply consider snapshots of the urban microcelluar network and look at the distribution of the instantaneous SINR. This approach is taken to confirm the analytic tractability of the pathloss model described in Section <ref>, which captures the blockage and shadowing effects due to buildings and accounts for the geometry of streets in an urban environment. In this section, we will explain the key assumptions of BS association and the definition of the interference. 
	

  §.§.§ BS association

			In our model, as mentioned in Section <ref>, we assume the BSs deploy directional beamforming to exploit antenna gain, while at the receiver side, the antenna is omni-directional. During the cell discovery and BS association process, we assume all BSs do exhaustive beam search over the entire beam space by beam sweeping, each at an individual time slot. Based on the reference signal received power (RSRP) of each beam, the receiver can determine the serving BS and the associated beam by selecting the strongest RSRP. After exhaustive beam sweeping, the receiver is always aligned with the main lobe, therefore the antenna gain of the associated BS is G all the time. Therefore, the receiver is simply associated to the BS with the smallest pathloss defined in (<ref>), without including extra antenna gain.


  §.§.§ Interference

From the BS association rule, the receiver is associated to the BS with the smallest pathloss, i.e., the largest path gain, which we denote as u.  Therefore, interference arises from other BSs whose path gains are smaller than u, with an extra beamforming gain 𝒢 added on. Given the orientation of the main beam (towards the desired user), other BSs could either point the main lobe or the side lobe towards the referenced (typical) receiver, based on the sectorized antenna model. Therefore, the beamforming gain of the interference 𝒢 is random, and is represented as 
	
    𝒢 = A G + (1-A) g,

	where A = 𝕀(p) and p is the probability that the interference link from the BS has beamforming gain of G, and 𝕀(·) is the Bernoulli function. 



  §.§.§ Formulation of SINR

Based on the pathloss model in Section <ref>, there are three types of BSs to analyze: typical/cross/parallel BSs. To formulate the SINR, we first make the following assumption of the BS association rule. 	
		We assume perfect beam sweeping for the  (<ref>). 
	
We use Φ_T to denote the set of LOS link distances x_T from the  typical BSs to the receiver. The set of lengths of the horizontal and vertical links, x_C (d_1 in green, Fig. 1) and y_C (d_2 in green, Fig. 1), constituting the propagation path from the cross BSs is denoted as Φ_C. Similarly, Φ_P is used to denote the set of distances (x_P, y_P,z_P) (d_3, d_2, d_1 in red) corresponding to the propagation path from parallel BSs (see Fig. <ref>). To simplify the demonstration, we define the path gain of the LOS and NLOS segment respectively as 
	
    ℓ_L (x)  =  x^-α_L,

	and 
	
    ℓ_N (x)  = cx^-α_N,

	where x is the length of the propagation segment. The corner loss term c = 10^-Δ/10 in the total path gain expression is also captured along with the propagation loss associated with each NLOS segment in  (<ref>), with α_N denoting the NLOS pathloss exponent.
	  
 We denote h_o as the small scale fading of the typical receiver o from the associated BS and h_i as the small scale fading of the ith BS in the Poisson point processes. 𝒢_i is a beamforming gain associated with each interfering BS, defined in (<ref>), and N_0 is defined as the noise variance. Φ_T', Φ_C' and Φ_P' are the set of segment lengths of the interfering BSs. 	Conditioning on the associated BS link gain u (which includes both the path gain PL in (<ref>) and the antenna beamforming gain, which is always G), the SINR can be formulated as follows, in terms of interference components, respectively from the typical BSs I_ϕ_T, cross BSs I_ϕ_C and parallel BSs I_ϕ_P,
	
    SINR   = h_o u/N_0 + I_ϕ_T(o) + I_ϕ_C(o) + I_ϕ_P(o), 
    with 
    		I_ϕ_T(o)    = ∑_x^i_T∈Φ_T'𝒢_ih_i ℓ_L (x^i_T),
    
    		I_ϕ_C(o)    = ∑_(x^i_C,y^i_C)∈Φ_C'𝒢_ih_i ℓ_N(x^i_C)ℓ_L(y^i_C), 
    and 
    		I_ϕ_P(o)    = ∑_(x^i_P, y^i_P, z^i_P)∈Φ_P'𝒢_i h_iℓ_N(x^i_P)
    		ℓ_N(y^i_P)ℓ_L(z^i_P).
	
	Based on the assumption in Section <ref>, and conditioning on the associated BS path gain as u, we have the following constraints for the sets of interfering BSs' segment lengths Φ_T', Φ_C' and Φ_P' in (<ref>) – (<ref>) as
	
    Φ_T'    = {x_T∈Φ_T| ℓ_L(x_T)  <  u}, 
    Φ_C'    = {(x_C,y_C)∈Φ_C| ℓ_N(x_C)ℓ_L(y_C) <  u}, 
    and Φ_P'    = {(x_P,y_P, z_P)∈Φ_P| ℓ_N(x_P)
    		ℓ_N(y_P)ℓ_L(z_P) <  u}.

	The above constraints are based on the assumption that perfect beam sweeping is done for each surrounding BS in the initial access, which leads to (<ref>) – (<ref>). 
	

 §.§ Analysis of strongest path

Given a BS at fixed location (either a typical, cross or parallel BS), the received power from the BS is still not clear, even though we have already defined our pathloss model in (<ref>). This is because, first, the Manhattan pathloss model bears huge differences with the Euclidean distance based pathloss model. Secondly, given a BS location, there could be multiple paths for the signal to reach the receiver within the grid-type Manhattan city. Since we assume the antenna pattern at the BS is sectorized, there exists radiated power to all directions, with different antenna gains With different paths routed for a signal radiated from all directions, the received power comes from different paths. To make it tractable, we make the following assumption for the analysis herein.  

	
		For analysis, we only consider the path from BS to the typical receiver with the largest received power. 
	 
	To be the strongest path, the path should have i) shorter individual path segment lengths, ii) fewer individual segments, hence fewer corners and smaller corner loss, since pathloss is calculated by multiplying individual segment pathloss and one extra multiplication might reduce the path gain by orders of magnitude, iii) larger beamforming gain.
	(only for the analysis of the interfering link.  Because for the BS association case, the receiver is always associated with the main lobe, with identical beamforming gain of G). 
	
	
	

The strongest path for the BS association is simply the path with the smallest pathloss, since the receiver is always associated with the main lobe of the beam, with an identical beamforming gain as G. The strongest path for the interfering link analysis, however, is not necessarily the path with the smallest pathloss, due to the existence of  beamforming gain. Next, we demonstrate some different cases of the relative location of the receiver to the BS, in terms of the strongest receiver path. Fig. <ref> and Fig. <ref> illustrate the potential strongest paths of a typical and a cross BS. In each of the cases, there is one direct path which has fewer corners and one detoured path which detours its way before reaching the receiver. For typical BSs, the detoured path has four more corners than the direct path; while for the cross BSs, there are two more corners. Each corner introduces an approximately extra 20dB loss, which is much more significant than the effects compensated by the beamforming gain difference. Therefore, even if the departure direction of the direct path lies inside the side lobe, the strongest path should still be the direct path.
	
	For the parallel BSs, both the detoured and direct paths have two corners, which makes it hard to identify the strongest path (see Fig. <ref>). In addition, the BSs are categorized to two types. It could either be in the same block as the receiver (e.g., BS 1) or different block as the receiver (e.g., BS 2), as shown in Fig. <ref>. For the same block BS,  the strongest path could either be the green dashed line or the green solid line. For the different block BS, however, the strongest path could traverse any of the cross streets and could point either left for right. 
	Section <ref> assumes that the strongest path travels the LOS segment towards the receiver, the thinning rule follows from the typical and cross BSs. The assumption is that only when the main beam points towards the corner in the direction of the receiver of the LOS segment, the beamforming gain can be main lobe gain G. 
	To make the analysis tractable, we make the following assumption. 
	
	
		For the strongest path of the parallel BSs, the signal travels along the LOS segment (first segment of the path) in the direction towards the receiver, rather than away from it. 
	
	With this assumption, to find the strongest path for the parallel BS at different blocks as the receiver, we provide the following proposition. 
	
	
		The strongest propagation path from a parallel BS is via either the cross street Θ_R closest to the receiver or Θ_B closest to the BS. 
		
			Conditioning on the location of the parallel BS, the segment y_P and the corner loss 2Δ of all propagation paths are the same, hence, the pathloss on the vertical link and the two corner losses can be taken out while formulating the following optimization problem. 
			
			For the interfering BS, since 𝒢 is a random variable taking values of G or g, we have 𝒢≤ G. Hence, the  maximum path gain of the parallel BS G_p can be upper bounded by 
			
    G_P   ≤ G - 2Δ - 10 α_Nlog_10y_P + 10G_M
       ≤ G - 2Δ - 10 α_Nlog_10y_P+ 10max{G_M}.

			where 
			
    G_M =  -α_Nlog_10x_P - α_Llog_10z_P,

			We then formulate the optimization problem of G_M as 
			
    x_P, z_P∈(0,W)maximize      -α_Nlog_10 x_P - α_Llog_10 z_P
       subject to       x_P + z_P = W.

			The objective function can be expressed as P(x) = -α_Nlog x -α_Llog (W-x)  , x∈(0,W),  whose second order derivative is 
			
    P”(x) = α_N/x^2 + α_L/(W-x)^2.
 
			The second order derivative of P(x) is positive for all α_L, α_N, and W, which means P(x) is convex. Denoting the distance from Θ_R to the receiver as x_1 and the distance from Θ_B to the receiver as x_2, and using the convexity of P(x), we have 
			
    P(λ x_1+ (1-λ) x_2) <λ P(x_1) + (1-λ)P(x_2)
        <max{P(x_1), P(x_2)} ∀λ∈(0,1)  and  x_1, x_2∈(0,W).

			In (<ref>),  P(λ x_1+ (1-λ)x_2)  parameterizes all path gains of the propagation paths via any cross street lying between Θ_R and Θ_B, with different values of λ selected. From the second inequality in (<ref>), all these propagation paths have smaller path gain than that going through the streets specified in this proposition, which concludes the proof. 
		
		
	
	Since the pathloss exponent of the segment z_P is α_L and that of the segment x_P is α_N, with α_L<α_N, it is intuitive that the strongest path is more likely to be via the street closest to the receiver, i.e., Θ_R. 
	
	To conclude the discussion on the uniqueness of the propagation path in the system model considered in this paper, we demonstrated that for both the typical and cross BSs, the propagation path is unique and also easy to identify based on the strongest path analysis above. For the parallel BS,  irrespective of whether the BS is located in the same block as the receiver, there are only two potential paths to be the strongest, and for analysis, we choose the path which traverses the cross street that is closest to the receiver.  
	  

§ COVERAGE ANALYSIS

	The coverage probability serves as an important metric in evaluating system performance, since it is closely related to ergodic rate and throughput outage. In this section, we compute the coverage probability of a typical receiver in the MPLP microcellular network. First, we explain the independent thinning of the BSs considering the sectorized beam pattern of the mmWave BSs. Then, we analyze the CDF of the associated BS link gain based on the assumption that the receiver is associated to the closest BS (with smallest pathloss). In addition, we derive an accurate and concise  expression of the coverage probability. Finally, we examine the effects of the various components that contribute to interference mmWave microcellular networks. 
	

 §.§ Independent thinning of BSs

	Based on the sectorized antenna model in Section <ref> and the properties of PPP, the BSs are independently thinned to generate two independent PPPs of BSs with antenna gain of G and g <cit.>. We define p_T as the thinning probability, and λ_B is the density of all active BSs deployed on the road side. After independent thinning, the densities of thinned BSs with antenna gain of G and g are respectively λ_B p_T and λ_B(1-p_T). For the typical BSs, the thinning probability is p_T which equals to the probability that the receiver lies inside the main lobe, as defined in (<ref>). For the cross BSs and the parallel BSs, we assume that only the BSs pointing towards the corner where the diffraction happens have beamforming gain as G. Hence, cross and parallel BSs have identical thinning probability as that of typical BSs. 
	For the parallel BSs, even though the uniqueness analysis of strongest path in Section <ref> is not as straightforward as the typical and cross street BSs, the thinning probability remains the same. Section <ref> assumes that the strongest path travels the LOS segment towards the receiver, and the thinning rule follows that of the typical and cross BSs. The assumption is that only when the main beam points towards the corner in the direction of the receiver, the beamforming gain can be G.
	To conclude, the thinning probabilities for three types of BSs (typical, cross and parallel) are identical, which are equal to the probability that the interfering BS has a beamforming gain of G

    p_T  =  p,
	
where p is given in  (<ref>).  The value of p is hard to evaluate from a physical point of view, because propagation is dominantly down a street canyon, and it is dependent on the distribution of the interfering BS beam direction and multiple reflections along the street canyon. We can actually use any value for “p" that occurs in practice. To make the exposition more clear, we pick the value as p_T = p = θ/2π, where θ is the beamwidth, under the assumption that the main lobe of the interfering BSs is uniformly distributed in the angular domain of (0, 2π). 
	

 §.§ Distribution of associated BS link gain

	To simplify SINR coverage analysis, we assume all links (association/interfering) experience independent and identically distributed (I.I.D.) Rayleigh fading with mean 1, h∼exp(1). We denote the normalized transmit power P_B = 1 and represent the noise variance by N_0. Since the SINR expression in (<ref>) is conditioned on the associated BS link gain u, we first analyze the distribution of u. Based on strongest BS association law, the receiver can be associated to either a typical/cross or parallel BS. The following lemma provides the cumulative density function (CDF) of the associated BS link gain of the typical/cross/parallel BS respectively. 
	
	
		The CDFs of the associated BS link gain of the typical BSs u_1 = max_(x_T∈Φ_T){ℓ_L(x_T)}, cross BSs u_2= max_(x_C, y_C)∈Φ_C{ℓ_N (x_C)ℓ_L(y_C)} and parallel BSs u_3   =max_(x_P, y_P, z_P)∈Φ_P{ℓ_N(x_P)ℓ_N(y_P)ℓ_L(x_P)} are approximated as 	
    F_u_T(u)    = exp(-γ_Tλ_Bu^-1/α_L), 
    
    			F_u_C(u)      =exp(-γ_Cλ_B^α_L/α_T u^-1/α_N),
    
    			F_u_P(u)   ≈  2√(2γ_Pλ_Sλ_B^α_L/α_N u^-1/α_N)K_1(2√(2γ_Pλ_Sλ_B^α_L/α_Nu^-1/α_N)),

		where 
		
    γ_T   = 2G^1/α_L
    			,
    γ_C    = 2^1+α_L/α_Nλ_S(cG)^1/α_NΓ(1-α_L/α_N),
    γ_P   = √(16 λ_S^2 Γ(1-α_L/α_N)(2λ_B)^α_L/α_N (Gc^2)^1/α_L
    				u^-1/α_N)γ_P   = γ_C c^1/α_N,

		and K_1(·) is the 1-st order  modified Bessel's function of the second kind <cit.>. 
	
	
		See Appendix <ref>. 
	
	
	
	Based on properties of the modified Bessel function, when the argument μ of K_1(μ) becomes small, it can be approximated as <cit.> 
	
    K_1(μ)∼μ^-1.
	
	Since the argument of the modified Bessel function in (<ref>) scales with λ_S^2 λ_B^α_L/α_N, and the corner loss term c further reduces the value, so that the approximation in (<ref>) applies. Consequently, we can approximate (<ref>) as
	
    F_u_P(u)   ≈ 
    2√(2γ_Pλ_Sλ_B^α_L/α_N u^-1/α_N)(2√(2γ_Pλ_Sλ_B^α_L/α_N u^-1/α_N))^-1
       = 1 ,
	
	which implies that, generally, the largest gain from a parallel BS is small, i.e., the probability of being associated with a parallel BS is negligible.
	
	Using Lemma <ref>, the CDF of the associated BS link gain U = max{u_T, u_C, u_P} can be evaluated as
	
    F_U(u) = ℙ(max{u_T, u_C, u_P}<u) 
       =^(a)ℙ(max{u_T}<u) ℙ(max{u_C}<u) ℙ(max{u_P}<u)
       ≈^(b)exp(-γ_Tλ_B u^-1/α_L)exp(-γ_Cλ_B^α_L/α_N u^-1/α_N),

	where (a) is based on the fact that the locations of the typical/cross/parallel BSs are mutually independent, (b) follows the results of Lemma <ref> and the observation that the association with parallel BSs is negligible.
	

	
	Fig. <ref> compares the numerically evaluated CDF of the associated BS link gain for the following cases: i) association only with typical BSs, ii) with typical/cross BSs, and iii) considering all association cases, against the theoretical result given in (<ref>). 	
	The simulation parameters we use are summarized in Table <ref>. The parameters are applicable to all of the following simulation results, unless stated otherwise. 
	
It is seen that the analytic result matches well with the numerical result. It can also be seen that the empirical CDF curves obtained with and without the association with the parallel BSs coincide. This verifies the analysis in Lemma <ref> and the subsequent approximation for largest gain seen by parallel BSs. Also, the curves show that the cross BSs association is also small compared to the typical BSs association, with the given simulation parameters.
	The simulation shows that LOS association with the typical BSs is dominant in the urban mmWave microcellular networks. 

	 

	

 §.§ Coverage probability

	In this section, we derive a closed-form expression for the coverage probability p_c(u, T) conditioned on the associated BS link gain as u. The coverage probability conditioned on u is defined as 
	
    p_c(u,  T) = ℙ(SINR >T| u).
	
	Using (<ref>) – (<ref>), (<ref>) can be expanded in terms of the Laplace transforms of interference conditioned on u and noise as follows.
	
    p_c( u, T) 
    		=ℙ(h>Tu^-1(N_0+I_ϕ_T(o) + I_ϕ_C(o)+I_ϕ_P(o)))
       =^(a)exp(-Tu^-1N_0)ℒ_I_ϕ_T(Tu^-1)ℒ_I_ϕ_C+I_ϕ_P(Tu^-1),
 
	where (a) is based on the assumption of I.I.D. Rayleigh fading channels, and ℒ_(·)
	is the Laplace transform (LT) of random variable (·). Note that we cannot completely decouple the interference terms since the propagation links from the cross and parallel BSs could potentially share the same path segments, thus making their individual interference not independent. To analyze the problem, we start by examining the parallel BS interference. 
	
	
		The LT of the interference from the parallel BSs I_ϕ_
			P is upper bounded by
		
    ℒ_I_ϕ_P(T, u)⪆ 2√(2γ_Pλ_Sλ_B^α_L/α_Nϱ(T)^α_L/α_N u^-1/α_N)
       ×
    			 K_1(2√(2γ_Pλ_Sλ_B^α_L/α_Nϱ(T)^α_L/α_Nu^-1/α_N))≈ 1,

		where γ_P is defined in (<ref>), 
		and 
		
    ϱ(t)    = ∫_1^∞1/1+t^-1μ^α_Ldμ.

		
			The proof follows from the proof of Lemma <ref> given in Appendix <ref>, and is provided in Appendix <ref>.
		
	
	
	Since the lower bound of LT of the parallel interference evaluates to 1 , which indicates that the interference from parallel BSs is small enough to be neglected, i.e., I_ϕ_P≈ 0.  Hence, the correlation of cross and parallel interference can be neglected and the coverage probability in (<ref>) can be reformulated as 
	
    p_c( u, T) 
    		≈exp(-Tu^-1N_0)ℒ_I_ϕ_T(Tu^-1)ℒ_I_ϕ_C(Tu^-1),
 
	which is derived in the following Theorem. 
	
	
		The coverage probability conditioned on the channel gain u of the associated link is
		
    p_c(u, T)    ≈exp(-β_1 u^-1)exp(-β_2λ_B u^-1/α_L)
       ×exp(-β_3λ_B^α_L/α_Nu^-1/α_N),

		where
		
    β_1 = TN_0, 
       β_2  =γ_T(p_Tϱ(T)+(1-p_T)  ϱ(Tg/G)),  
       β_3  = γ_C(p_T^α_L/α_Nϱ(T)^α_L/α_N+(1-p_T)^α_L/α_Nϱ(Tg/G)^α_L/α_N),

		and ϱ(·) is defined in (<ref>). 
		
			See Appendix <ref>.
		
	
	
	Using Theorem <ref> and the distribution of the associated BS link gain in (<ref>), the SINR coverage probability can be evaluated as
	
    P_c(T) = ∫_0^∞ p_c(u,T) f_U(u)du,

	where p_c(u, T) is provided in (<ref>), and the probability density function (PDF) f_U(u) can be obtained from the derivative of the CDF derived in  (<ref>).
	Though the coverage probability serves as an important metric in evaluating performance, it is not sufficient to characterize how much data rate the system can support. We next use a simplified definition of throughput 𝒯(β) to quantify the data rate, 
	
    𝒯 (η)= log_2(1+η)P_c(η),

	where η is the SINR threshold and P_c(η) is the coverage probability given in (<ref>). 
	

 §.§ The effect of LOS and NLOS interferers

	
	In Proposition <ref>, we showed that the parallel BSs interference can be neglected in the analysis. In this section, we further compare the effects of typical interference I_ϕ_T and cross interference I_ϕ_C.
	For tractable analysis, we assume the receiver is associated to the typical BS, so that we can have simpler associated BS link gain distribution. The analysis is based on the application of Jensen's inequality to the individual LT of I_ϕ_T and I_ϕ_C. 
	
	From Theorem 1, by unconditioning the associated BS link gain u, the LT of the interference of BSs on the typical street is 
	ℒ_I_ϕ_T(T) = 𝔼_u[exp(-β_2λ_Bu^-	1/α_L)]
	and the LT of the interference due to the NLOS BSs on the cross streets is 
	ℒ_I_ϕ_C(T) = 𝔼_u[exp(-β_3(λ_Bu^-1/α_L)^α_L/α_N)]. Define two convex functions φ_1(u) = exp(-u) and φ_2(u) = exp(-u^α_L/α_N). 		
	Since we assume the BS is associated to the typical BS in this case, the CDF of the associated BS link gain u becomes  
	
    F(u) = exp(-γ_Tλ_B u^-1/α_L).

Based on the CDF of u given above in (<ref>), we can derive the expectation of u^-1/α_L as 
	
    𝔼_u[u^-1/α_L] = 1/γ_Tλ_B,

	hence, by Jensen's inequality, the lower bound of ℒ_I_ϕ_T(T) becomes 
	
    ℒ_I_ϕ_T(T)≥ℒ^LB_I_ϕ_T(T) = exp(-β_2/γ_T) 
       = exp(-p_Tϱ(T)-(1-p_T) ϱ(Tg/G)).
 Similarly, we have derive the expectation of (λ_B u^-1/α_L)^α_L/α_N = λ_B^α_L/α_N u^-1/α_N as
	
    𝔼_u[(λ_B u^-1/α_L)^α_L/α_N] = (1/γ_T)^α_L/α_NΓ(1+α_L/α_N),

	with the lower bound of ℒ_I_ϕ_C(T) evaluated as 
	
    ℒ^LB_I_ϕ_C(T) = exp(-(1/γ_T)^α_L/α_Nβ_3 Γ(1+α_L/α_N))
        = exp(-2λ_Sc^1/α_NΓ(1-α_L/α_N)Γ(1+α_L/α_N) ε).

	where we denote ε as 
	
    ε = (p_T^α_L/α_Nϱ(T)^α_L/α_N+(1-p_T)^α_L/α_Nϱ(Tg/G)^α_L/α_N).

The argument inside the exponential function of (<ref>) scales with λ_S and c^1/α_N, where we have λ_S≪1 and c≪ 1. The inside argument is therefore effectively small. Another factor that might influence the cross street interference is the ratio between the pathloss exponents of the LOS/NLOS segments r = α_L/α_N. 

Generally, when α_N is much larger than α_L, Γ(1-α_L/α_N)Γ(1+α_L/α_N) is not that big, which leads to
	
    ℒ^LB_I_ϕ_T(T)≪ℒ^LB_I_ϕ_C(T)≈ 1.

	The lower bound is shown to be fairly close to 1 but much larger than the lower bound of the LT of the typical interference. This indicates that in this case, the cross interference is much smaller than the typical interference, and also is negligible. 
	
	When α_N is very close to α_L, however, Γ(1-α_L/α_N)Γ(1+α_L/α_N) can be very large, which averages out the effects of small λ_S and c^1/α_N. In this case, cross interference can also contribute significantly in some certain urban canyons, where α_N→α_L. This scaling law also leads to an intuitive insight that when the street intensity increases, the effects by cross BS interference grow larger. 
	
	From (<ref>), it can be seen that the lower bound of LT ℒ^LB_I_ϕ_C(T) scales exponentially with  -λ_S c^1/α_N, and
 Γ(1-α_L/α_N)Γ(1+α_L/α_N). When α_N is large, 
	Fig. <ref> gives a comparison between the analytic and simulation results of the coverage probability with different selections of α_N, and the cases when considering no interference (noise only), considering interference from only typical BSs, from both typical and cross BS interference, and from all interference. When α_N = 7, it is shown from the first five curves in the legend that the coverage probability curves with different interference components almost overlap. This verifies the corresponding proof in Proposition <ref> that the parallel interference can be neglected, and the validity of Jensen's inequality lower bound analysis in (<ref>)  and (<ref>). For  Fig. <ref> and Fig. <ref>, we set the corner loss as Δ = 20dB. It will also be shown in Section <ref> that with the corner loss ranging from 30 dB to 0 dB (no corner loss case), the coverage probability does not vary significantly. For the black curve pair, where α_N = 2.51→α_L = 2.5, and the green curve pair α_N= 2.52,  there do exist certain differences between the coverage probability considering only the typical interference and considering also the cross interference, even though when α_N = 2.52, the difference becomes small already. We then choose the value α_N = 3, and it is shown that the coverage probability curves almost coincide with that of α_N = 7. Hence, we can conclude that under the Manhattan distance based pathloss model, α_N does influence the contribution of cross street BS interference to the coverage probability. In most of the cases, the NLOS interference (from both cross and parallel BSs) is negligible; when and only when α_N→α_L, the cross interference becomes significant enough to have an impact on the coverage probability. The effect of different selection of α_N can also be observed from the parameter γ_C in (<ref>), which scales with Γ(1 - α_L/α_N), too. In the case when α_N→α_L, the absolute value of the argument inside the exponential function representing the CDF of maximum cross BS power becomes large, which makes the CDF grow smaller, hence making it easier to be associated with a cross BS. For the following analysis, for ease of explanation, we adopt the pathloss exponent as α_N = 7, which is the value recorded the measurement results <cit.>, <cit.>.
	
	

§ SCALING LAWS WITH NETWORK DENSITIES 

	In this section, we analyze the scaling laws of the coverage probability and the association probability with the network densities, i.e., street intensity λ_S and BS intensity λ_B. We apply tight approximations to the coverage probability and reveal interesting interplay between the performance and network deployment. 
	

 §.§ Scaling laws for coverage probability

	In this section, we focus on answering the following questions: i) how densely should BSs be deployed in  urban streets to maximize coverage at a minimum cost? ii) how does the coverage change for different densities in different cities?
	

  §.§.§ Scaling law with BS intensity

	The interference limited scenario targets an asymptotic case, where  the noise can be neglected and thus focus fully on the interplay between network intensities. This scenario can either be achieved by  high BS intensity (per street) or by dense streets deployment. 
	Based on the coverage probability given in (<ref>) and (<ref>), after neglecting the noise term and changing variables x =λ_B u^-1/α_L, the expression for the coverage probability becomes
	
    P_c(T)   = ∫_0^∞exp(-(β_2+ γ_T)x)exp(-(β_3+ γ_C)x^α_L/α_N)
       ×(γ_T+γ_Cα_L/α_Nx^α_L/α_N-1)dx,

	where the parameters β_2, β_3, γ_T and γ_C are provided in Section <ref>.
	
	
	
	
	Under the Poisson models for BSs and the Manhattan distance pathloss model, one  observation from (<ref>) is that the coverage probability is independent of the BS intensity.  On one hand, when both street and BSs intensities grow large, it is intuitive that with ultra dense deployment of BSs, i.e., λ_B→∞, both the associated link gain and interference become large. And therefore  their effects on the coverage probability cancel out, which leads to an asymptotic value of the coverage probability. On the other hand, when only the street intensity itself grows large, the scenario also becomes interference limited. In this case, the coverage probability is still a constant, however densely the BSs are deployed. This reveals an important insight that when street intensity grows large, the increase of coverage probability by deploying denser BSs is less significant. We plot Fig. <ref> to demonstrate the above two observations in an ultra-dense network where intensity of BS grows large. First, it is shown that from  approximately λ_B = 0.05 (average BS spacing of 20m) for different street intensities, the coverage probability starts to converge to the asymptotic value. Second, with denser street distribution (e.g.,  λ_S= 0.1, red curve), the increase of coverage probability is less prominent. Also, denser street distribution leads to lower asymptotic coverage probability. 
	
	
	
	
	
	
	

  §.§.§ Scaling law with street intensity

	In the last section, we demonstrated the impact of different city streets (with different intensities) on the coverage probability enhancement. Next, we reveal the scaling laws between the coverage probability and the urban street intensity. One important thing to note is that in the dense street case, the street intensity λ_S is not arbitrarily large, where the most dense streets might have at least 20m average spacing between them, with λ_S = 0.05. We provide the following proposition to quantify how the coverage probability changes under different street intensities and prove it herein.  
	
		
		1) When the BS intensity λ_B is large, the coverage probability decreases linearly with the street intensity λ_S. 
		2) When λ_B is small, the coverage probability increases linearly with λ_S. 
		
			In terms of the linear scaling law and its dependence on the BS intensity, we provide the following steps of the proof:  
			

  
Linear scaling law
			First, from (<ref>) – (<ref>), the coverage probability can be rewritten as 
			
    P_c(T) =P_1 + P_2,

			where 
			
    P_1= ∫_0^∞exp(-β_1 u^-1)exp(-(β_2+γ_T)λ_B u^-1/α_L) 
       ×exp(-(β_3  + γ_C)λ_B^α_L/α_Nu^-1/α_N)(λ_Bγ_T/α_Lu^-1/α_L-1)du,

			and 
			
    P_2 = ∫_0^∞exp(-β_1 u^-1)exp(-(β_2+γ_T)λ_B u^-1/α_L)
    ×exp(-(β_3  + γ_C)λ_B^α_L/α_Nu^-1/α_N)(γ_C/α_Nλ_B^α_L/α_Nu^-1/α_N-1)du.

			We then rewrite the second part in (<ref>), using integration by parts, as
			
    P_2 = γ_C/γ_C+β_3∫_0^∞exp(-β_1 u^-1)exp(-(β_2+γ_T)λ_B u^-1/α_L)
       ×∂[exp(-(β_3  + γ_C)λ_B^α_L/α_Nu^-1/α_N)]/∂ u
        =γ_C/γ_C+β_3 - γ_C/γ_C+β_3∫_0^∞exp(-(β_3  + γ_C)λ_B^α_L/α_Nu^-1/α_N)
       ×∂[exp(-β_1 u^-1)exp(-(β_2+γ_1)λ_B u^-1/α_L)]/∂ u.

			In both (<ref>) and (<ref>), only β_3 = ζ_1 λ_S, and  γ_C = ζ_2 λ_S depend on λ_S. Further, β_3 scales linearly with  γ_C, which itself is small due to the terms λ_S and c^1/α_N. Then, by applying a first-order Taylor approximation exp(-x)≈ 1-x to 
			exp(-(β_3  + γ_C)λ_B^α_L/α_Nu^-1/α_N) ≈ 1  -λ_s(ζ_1+ ζ_2)λ_B^α_L/α_Nu^-1/α_N in (<ref>) and (<ref>), we can see P_1 and P_2 scale linearly with λ_S, hence proving the linear scaling law of coverage probability with λ_S. 	Fig. <ref> compares the exact coverage probability in (<ref>) and that with the Taylor  approximation. It is shown that under different street intensities λ_S = 0.001, 0.01, 0.02, the exact results match well with the Taylor approximations. This verifies the accuracy of using Taylor approximation to prove the linear scaling law. Another observation here is that when the street density is relatively small, e.g., λ_S= 0.001, the coverage probability is insensitive to the NLOS pathloss exponent α_N, since the coverage almost remains a constant with α_N ranging from 3 to 10. When streets become dense, the coverage probability decreases faster with growing α_N.  This is consistent with the fact that α_N only affects pathloss of the NLOS links, and NLOS BS is negligible in either association or interference. 
			

  
Dependence on BS intensity
			To demonstrate the different scaling laws of coverage probability with BS intensities, we segregate the components in (<ref>) which are dependent on λ_S in the integral, and define it as Υ(λ_S), which is 
			
    Υ(λ_S)= exp(-λ_S(ζ_1 + ζ_2)λ_B^α_L/α_Nu^-1/α_N)
       ×(λ_Bγ_T/α_Lu^-1/α_L-1+λ_sζ_2λ_B^α_L/α_N/α_Nu^-1/α_N-1),

			the derivative of which is 
			
    Υ'(λ_S) = λ_B^α_L/α_N/α_Nu^-1/α_N-1exp(-λ_S(ζ_1 + ζ_2)λ_B^α_L/α_Nu^-1/α_N)
       ×(ζ_2 - (ζ_1 + ζ_2)α_N[γ_Tλ_B/α_Lu^-1/α_L + λ_Sζ_2 λ_B^α_L/α_N/α_Nu^-1/α_N]).

			Since the exponential part from (<ref>) is always positive, and ζ_2 and ζ_1 are independent of λ_B, there exists a threshold λ_B^*, which satisfies
			
    γ_Tλ^*_B/α_Lu^-1/α_L + λ_Sζ_2 λ^*_B^α_L/α_N/α_Nu^-1/α_N = ζ_2/(ζ_1+ζ_2)α_N.
 	Hence, when λ_B>λ_B^*, Υ'(λ_S)<0, which indicates that when BS intensity is larger, coverage probability decreases with λ_S. Further, when λ_B<λ_B^*, denser streets lead to a higher coverage probability. 		
	


		
	
	
	
	Fig. <ref> illustrates the linear scaling of the coverage probability with the intensity of streets λ_S. It first can be observed that the coverage probability scales linearly with the intensity of streets, and the coverage probability increases with λ_S while it decreases with  corner loss Δ, when the BS intensity is relatively small λ_B = 0.005. Also, the coverage probability decreases with λ_S with large BS intensity λ_B = 0.01, while it increases with corner loss in the meantime. This implies that when the BS deployment is dense, interference becomes dominant and larger corner loss reduces the interference; when BSs are relatively sparse, small corner loss strengthens the signal from the cross BSs, thus making the associated link gain  stronger and enhancing the coverage probability. Also, it can be observed that when the corner loss becomes small (e.g., the no shadowing loss case Δ = 0dB), the coverage probability becomes more sensitive to the change of street intensities, which is shown by a larger slope of the curve of coverage probability. This is because the smaller corner loss makes the cross BS interference more prominent, thus increasing the sensitivity of coverage probability to the street intensities. 
	
	
	From the above analysis, the microcellular network does not work efficiently in a scenario where both BS and street intensities are large. When the BSs are sparsely deployed in an urban landscape with increasing street intensities (i.e., where blocks are small), then a typical UE is more likely to be associated with a BS on cross streets, and also can have a larger associated BS link gain. When λ_B grows large, however, the system becomes interference-limited, thus dense BS deployments in dense streets only contribute to more interference and lower the coverage probability. This sheds light on an import conclusion that ultra-dense BS deployment should be avoided in an urban canyon with dense street densities.
	
	
	

 §.§ Scaling law for BS association probability 

		 I don't need to change this because this is all using the parameters of γ and β. 
	

 §.§ LOS probability
	
	In this section, we analyze the probability the link is LOS under the MPLP model.  This result will also be applied to the calculation of pathloss in <cit.> in Section <ref> for comparison of the new Manhattan distance based model and previous Euclidean distance based pathloss models. We compare the result of the closed-form LOS probability to that in 3GPP microcellular LOS model, which shows a good match between the MPLP LOS probability and the realistic microcellular scenario in 3GPP. This justifies the accuracy of MPLP in urban street modeling. 
	
	
	
	
	
	
	
		In the MPLP, the LOS probability of a propagation link from a BS at Euclidean distance d is 
		
    p_LOS^MPLP(d) =  1-exp(-4dλ_S)/4dλ_S.

		
		
			We illustrate the LOS probability analysis in Fig. <ref>. 
			


			According to Slivnyak's theorem <cit.>, <cit.>, <cit.>, we first add a horizontal street S_o crossing the receiver at o. Conditioning on the distance between the BS and the receiver as d, there are in total of N_h+2 potential locations of BSs on the horizontal street and a total number of N_v potential BSs on the cross streets, at an Euclidean distance of d (which are the intersections of the streets with the circle centered at the receiver O of radius d). Because the streets are modeled as a Poisson line process in the MPLP model, N_h and N_v are Poisson random variables with N_h/2∼Poisson(2λ_Sd) and N_v/2∼Poisson(2λ_Sd). Hence, the probability of LOS link is
			
    p_LOS^MPLP(d) = 𝔼[2/2+N_h+N_v],

			which simplifies to 
			
    p_LOS^MPLP(d)   =^(a)∑_k=0^∞(4λ_Sd)^k/(1+k)k!exp(-4λ_Sd)
        =  exp(-4λ_S)/4λ_Sd∑_k=1^∞(4λ_Sd)^k/k!
       =^(b)1-exp(-4dλ_S)/4dλ_S,

			where (a) is derived based upon the distribution of Poisson process and (b) utilizes the power series of the exponential function:  exp(x) = ∑_k=0^∞x^k/k!. 
		
	
	The LOS probablity given in the 3GPP microcellular model is <cit.>
	
    p_LOS^3GPP (d) = min{1, 18/x}(1-exp(-x/36))+ e^-x/36.
First, it should be noted that the expression of LOS blockage probability under MPLP model has a similar form as that in the 3GPP microcellular model. By fitting the result to 3GPP microcellular LOS probability in (<ref>) with a minimum mean squared error regression, the street intensity is λ_S  = 0.0092 (we use λ_S = 0.01 in simulations). The comparison in Fig. <ref> also shows that the LOS probability obtained in (<ref>) has a close match to that in the 3GPP model. It should be noted that even though our pathloss model is based on Manhattan distance, Euclidean distance is still a widely adopted metric in understanding urban cellular system, since it is generally easier to measure and manipulate. We use the Euclidean distance based LOS probability to evaluate the Euclidean distance based pathloss model in <cit.> and also to verify MPLP street models by comparing to 3GPP Euclidean distance based LOS models. 
	In this section, we analyze the BS association under the Manhattan distance based pathloss model in MPLP. We start with the analysis of association probability. Given the CDF of the associated BS link gain in Section <ref>, we derive the probability the receiver is associated with a LOS BS on the typical street.
	
		The probability χ_T that the receiver is associated with a typical BS is 
		
    χ_T=^(a)𝔼_u{ℙ(u_C<u  | u_T = u )} = 𝔼_u_T{ℙ(u_C<u_T)}
       =^(b)∫_0^∞exp(-γ_Cλ_B^α_L/α_Nu^-1/α_N - γ_Tλ_B u^-1/α_L)
       ×γ_Tλ_B/α_Lu^-1/α_L-1 du
       =^(c)γ_T∫_0^∞exp(-γ_C x^α_L/α_N-γ_T x)dx,

		where (a) is conditioned of maximum path gain of typical BSs is u, (b) is based on the CDF of the maximum path gain of typical/cross BSs, (c) follows by change of variables x = λ_B u^-1/α_L. 
	
	Since the argument of the second exponential function in (<ref>) is the multiplication of λ_S and an additional attenuation of corner loss, the argument inside tends to be small. Similar to the approximation in Section <ref>, we approximate the association probability by 
	
    χ_T^Approx =  ∫_0^∞exp(-μ)(1 - ζ_2/γ_T^α_L/α_Nλ_Sμ^α_L/α_N)dμ
        = 1- 2^α_L/α_N+1γ_C/γ_T^α_L/α_N[sinc(α_L/α_N)]^-1λ_S,

	where sinc(x) = sinπ x/π x. Because the sinc function monotonously decreases with x (0<x<1), the association probability with a typical BS decreases with α_L, while it  increases with α_N. Fig. <ref> provides the comparison of the exact association probability in (51) and the approximation result in (52). The approximation in (52) is tight when there exists corner loss Δ = 20 dB, while the gap increases when the corner loss increases. There exists a linear scaling law for the association probability with the street intensity in the scenarios with significant shadowing loss at corner, which is shown in Fig. <ref>. Also, different from α_N, which only impacts on the NLOS BS pathloss, the LOS pathloss exponent α_L is involved in both the calculation of typical/cross BS pathloss. The decrease of typical association probability with larger α_L implies that the LOS link pathloss is more sensitive to the changing pathloss exponents. Also, it is intuitive that the increase of α_N enhances the association probability since it further attenuates the transmit signal from cross street BSs. It should be noted that it is meaningful to examine the interplay between the coverage probability and these exponents values, since the pathloss exponent in reality is not fixed (we extract two reasonable parameters for the ease of analysis in this paper), but is a random variable varying from streets to streets <cit.>. The interplay of pathloss exponents and typical BS association probability provides insight into BS association behaviors under various channel conditions of different urban canyons. 
	
	
	In addition, from (<ref>) there is a linear scaling law of the typical BS association probability with the intensity of cross streets in Fig. <ref>. Also, it should be noted that with the corner shadowing loss, even in an extremely dense street network, e.g., λ_S= 0.1, the association probability with typical BSs χ_T is still greater than 0.7. Only when in the case with no street corner loss, the association probability χ_T decreases significantly with the street intensity λ_S.  The above association probability analysis illuminates another important observation that considering shadowing loss at a reasonable value, cross BSs play a minor role in BS association under the Manhattan distance based microcellular pathloss model. Similar effects on coverage probability have been demonstrated in Section <ref>. 
	
	
	
	
	
	
	Hence, we can make  the following conclusions about the BS association. First, the BS association probability with the typical BSs is independent of the BS intensities.  Second, the association probability decreases linearly with the intensity of the cross streets. Third, typical BS association is less likely when the LOS pathloss exponent α_L increases. 
	
	

§ COMPARISON OF DIFFERENT STREET MODELS

	In this section, first, we fit the  pathloss data of the Manhattan distance based pathloss model to different Euclidean distance based models and get the parameters for Euclidean distance pathloss models.  Then we compare the coverage probability and the ergodic rate of the Manhattan distance based pathloss model adopted in this paper and the Euclidean distance pathloss models. Second, we compare the ergodic rate  ℛ = 𝔼{log_2(1+SINR)} under different street modelings, respectively MPLP model in this paper, the fixed grid model and realistic street data of
	

 §.§ Pathloss models comparison

	We compare our model with two different Euclidean distance based pathloss models. The first model computes the pathloss directly by Euclidean distance d, where PL_dB(d) = 10α̃log_10d + Δ_1, α̃ is the pathloss exponent, and Δ_1 is the offset for straight-line linear regression of the Euclidean pathloss model. In <cit.>, a key parameter for characterizing coverage in mmWave wireless networks is the distance dependent blockage probability,  which has been analyzed in Section <ref>. In the urban microcell downlink scenario, we define it as p_B(d) = 1 - p_LOS(d) and apply different pathloss exponents α̃_L and α̃_N for unblocked (LOS) and blocked (NLOS) links. The pathloss is calculated by PL_dB(d) = (1-𝕀(p_B(d))) (10α̃_Llog_10d +  Δ_2^L)+ 𝕀(p_B(d))(10α̃_Nlog_10d + Δ_2^N), where 𝕀(x) is the Bernoulli function with parameter x, Δ_2^L and Δ_2^N are respectively the offsets for LOS and NLOS pathloss formulas. 
	
	The fitting parameters are given in Table <ref>. Based on the linear regression results in Table <ref>, we compare the coverage probability and ergodic rate of these three models in Fig. <ref> and Fig. <ref>. It is shown that the three models show significant difference in coverage probability and ergodic rate. This motivates
	us to do further theoretical analysis in our proposed model.
	
	
	
		
	
	
	

 §.§ Street models comparison
	
	In this section, we compare the ergodic rate under three different urban street models, the MPLP street modeling in this paper, fixed grid model (fixed spacing between streets) and realistic street deployments in Chicago, using the Manhattan distance pathloss model.  The ergodic rate is defined as ℛ = 𝔼[1 + SINR]. The raw street data is obtained by OpenStreetMap powered by open source software and  <cit.>, <cit.>. We extract the map data by using GIS tool QGIS <cit.>. The simulated area is a region in Chicago given in  Fig. <ref>, and the extracted map which includes street and node information is plotted in Fig. <ref>.
	
	The parameters of the simulation scenario under the three street models are obtained based on the map we extracted from Chicago city. We assume all the street models have the same size (1.659× 2.002 km^2). It can be counted from Fig. <ref> that the number of the vertical and horizontal streets are respectively 15 and 8 (we only count main streets which are shown explicitly in the map). Also, we assume the three models have the same (mean) street numbers. These leads to the derivation of the horizontal street density as λ_sh≈ 4.8 /km, and the vertical street density as λ_sv≈ 7.5 /km. The densities are then applied to generate two independent PPPs for horizontal and vertical streets in the MPLP model, and set the spacing between two adjacent street respectively as S_h = 133.5 m and S_v =  207.4 m. The comparison of the ergodic rate under the three models is given in Fig. <ref>. From this figure, the ergodic rates are close under these different street models, which nearly coincide. The major reason for the observation is the negligible contribution of NLOS interference on the performance of Manhattan type mmWave microcellular networks from the analysis. The result, however, not only substantiates the negligibility of NLOS interference in MPLP networks, but  shows that the conclusion is also applicable to fixed grid and realistic urban canyons. Therefore, MPLP is an appropriate street model in understanding Manhattan type networks, which can yield simple yet accurate results and also provide interesting insights on the scaling of performance metrics. 
	 
	
	
	

§ CONCLUSION

	In this paper, we proposed a mathematical framework to model a Manhattan-type microcellular network under the urban mmWave communication system by stochastic geometry.  We first analyze the distribution of the path gain to the BS. We then derive an exact yet concise expression of the coverage probability. The LOS interference from the BSs on the same street as the serving BS is the dominating factor in determining the coverage probability, while BSs on cross and parallel streets have insignificant effects in most of the cases. We showed that in the ultra-dense network where intensity of BSs grows large, 
		
	
	the network is interference-limited and the coverage probability approaches an asymptotic value.  Also, the coverage probability scales linearly with the intensity of streets, and displays an interesting interplay with the BS intensity: i) when BS deployment is dense, coverage probability decreases with street intensity; ii) when BS intensity is small, the coverage probability increases with street intensity. This implies that the system does not work efficiently when both BS and street intensities are large. Therefore, there is no need to deploy many BSs in an already dense urban street environment. In addition, we showed that in most of the cases,  the LOS BSs still dominate, from the perspective of both BS association, as well as coverage, unless in the case when α_N→α_L. Finally, we numerically compared the ergodic rates under MPLP, fixed spacing and a realistic street deployment in Chicago city. The ergodic rates under these street models match well, reinforcing the validity of MPLP as a realistic yet accurate urban street model in mmWave microcellular anlaysis. 
	
	
	
	

§ PROOF OF LEMMA <REF>

	
Since the receiver is always associated with the main lobe of the BS, which provides the smallest pathloss, the beamforming gain is always G. Hence, the CDF of the largest received power from the typical BSs is 
	
    F_u_T(u)  = ℙ(max_x∈Φ_T Gx^-α_L<u)
        = ℙ(min_x∈Φ_T x>G^1/α_Lu^-1/α_L)
       =^(a)exp(-2λ_B  G^1/α_Lu^-1/α_L) = exp(-γ_Tλ_B u^-1/α_L),

	where (a) is based on the distribution of closest distance x to one fixed point of one-dimensional PPP with intensity λ, and min{x} follows an exponential distribution, with parameter, min{x}∼exp(2λ), and also follows the independent thinning rule of BSs on the typical street of BSs with main lobe pointing to the receiver. 
	
	Similarly, the CDF of the largest received power from the BSs on the cross streets can be derived as follows 
	[1]
	
    F_u_C(u)= 𝔼_Φ_C[∏^(x_C, y_C∈Φ_C)ℙ(x_C^-α_Ny_C^-α_LcG<u)]
       =𝔼_x_C[𝔼_y_C[∏^(x_C, y_C)∈Φ_Cℙ(x_C^-α_Nmin(y_C)^-α_LcG<u) | x_C]]
       =^(a)𝔼_x_C[∏^x_Cexp(-2λ_B 
    		x_C^-α_N/α_L(cG)^1/α_Lu^-1/α_L)]
       =^(b)exp(-2λ_S∫_0^∞ 1- exp(-2λ_B 
    		x_C^-α_N/α_L(cG)^1/α_Lu^-1/α_L) dx)
        = exp(-2λ_S(2λ_B)^α_L/α_N(cG)^1/α_NΓ(1-α_L/α_N)u^-1/α_N)
        = exp(-γ_Cλ_B^α_L/α_Nu^-1/α_N),

	where (a) is derived by first conditioning on x_C,  and (b) is based on the probability generating functional (PGFL) of PPP. 
	
	Here, we provide an approximation of the CDF result of the associated BS link gain, based on the assumption that the strongest path is always via the cross street closest to the receiver, in Section <ref>. The CDF can be derived as 
	
    F_u_P(u)≈ℙ(⋂^(x_P,y_P, z_P )∈Φ_P x_P^-α_Ny_P^-α_Nz_P^-α_Lc^2G<u) 
       =^(a)𝔼_x_P, y_P[∏^x_P∏^y_Pexp(-2λ_B G^1/α_L u^-1/α_Lc^2/α_Lx_P^-α_N/α_Ly_P^-α_N/α_L)]
       =^(b)𝔼_x_P[∏^x_P𝔼_y_P[∏^y_Pexp(-2λ_B( Gc^2/ux_P^α_Ny_P^α_N)d^1/α_L)| x_P]]
       =^(c)𝔼_x_P[exp(-2λ_S(Gc^2(2λ_B)^α_L/u)^1/α_NΓ(1-α_L/α_N)/x_P)]
        = ∫_0^∞ 2λ_Sexp(-γ_Pλ_B^α_L/α_Nu^-1/α_N x^-1 - 2λ_Sx)dx
       =^(c) 2√(2γ_Pλ_Sλ_B^α_L/α_N u^-1/α_N)K_1(2√(2γ_Pλ_Sλ_B^α_L/α_Nu^-1/α_N)),

	where ⋃ denotes the intersection of all of the events defined in the set (x_P,y_P, z_P )∈Φ_P, (a) is derived conditioned on x_P, y_P and (b) is derived conditioned on x_P, (c) is based on the PGFL function, (d) follows the equation <cit.>, 
	
    ∫_0^∞exp(-β/4x - γ x)dx = √(β/γ)K_1(√(βγ)).

	By simple calculations, we can conclude the proof. 
	

§ PROOF OF PROPOSITION <REF>

		The derivation of the LT of the interference coming from the BSs on the parallel streets is similar. Based on the proof in Proposition <ref>, the LT of the interference can be lower bounded by the LT assuming all the interfering beamforming gain is G, which is formulated as  
	
    ℒ_I_ϕ_P (Tu^-1)⪆
       𝔼_ϕ_P[exp(-∑_(x_P,y_P, z_P )∈Φ_P Tu^-1hx_P^-α_Ny_P^-α_Nz_P^-α_Lc^2G)] 
       =^(a)𝔼_x_P{∏^y_Pexp(-2λ_B  G^1/α_Lϱ(T)u^-1/α_Lc^2/α_L(xy)^-α_N/α_L)}
       =^(b)𝔼_x_P{exp(-γ_Pϱ(T)^α_L/α_Nλ_B^α_L/α_Nu^-1/α_N x_P^-1)}
        = ∫_0^∞ 2λ_Sexp(-γ_Pϱ(T)^α_L/α_Nλ_B^α_L/α_Nu^-1/α_N x^-1 - 2λ_S x)dx
        =2√(2γ_Pλ_Sλ_B^α_L/α_N(ϱ(T))^α_L/α_N u^-1/α_N)
       ×
    K_1(2√(2γ_Pλ_Sλ_B^α_L/α_N(ϱ(T))^α_L/α_Nu^-1/α_N)),

	where (a) and (b) follow the standard procedures in analysis of stochastic geometry and are similar to the proof of Laplace transform of I_ϕ_T and I_ϕ_C above in Appendix B. 
	
	

§ PROOF OF THEOREM 1

	
	We respectively give the LT of the three kinds of interferers ϕ_T, ϕ_C and ϕ_P. The LT of the typical BS interference ℒ^G_I_ϕ_T(s) with beamforming gain as G can be  given by 
	
    ℒ^G_I_ϕ_T(s) = 𝔼[exp(-sG∑_x_T∈Φ_T hx_T^-α_L)]
        =exp(-2λ_Bp_T∫_(u/G)^-1/α_L^∞𝔼(1-exp(-sGhx_T^-α_L)))
        = exp(-2λ_Bp_T∫_(u/G)^-1/α_L^∞1/1+s^-1G^-1x_T^α_Ldx_T).

	For the interference with beamforming gain as g, the LT ℒ^g_I_ϕ_T(s)  can be derived as 
	
    ℒ^g_I_ϕ_T(s) =𝔼[exp(-sg∑_x_T∈Φ_T hx_T^-α_L)]
        = exp(-2λ_B(1-p_T) ∫_(u/G)^-1/α_L^∞𝔼(1-exp(-sghx_T^-α_L)))
        = exp(∫_(u/G)^-1/α_L^∞-2λ_B(1-p_T)/1+(Tg/G)^-1uG^-1x_T^α_Ldx_T).

By applying change of variables to (<ref>) and (<ref>), and combining the results above, the LT of the interference on the typical street can be formulated as 
	
    ℒ_I_ϕ_T(s) =	ℒ^G_I_ϕ_T(s)	ℒ^g_I_ϕ_T(s)
       =exp(-γ_Tλ_Bu^-1/α_L∫_1^∞1/1+T^-1μ^α_Ldμ)
        = exp(-β_2 λ_Bu^-1/α_L).

	Similarly, the LT of the cross interfering with beamforming gain G follows the the proof in Appendix A and proof of ℒ^G_I_ϕ_T(s), which can be given by 
	
    ℒ^G_I_ϕ_C(s) =𝔼[exp(-∑_(x_C, y_C)∈Φ_Csh x_C^-α_N y_C^-α_L c G )]
       =𝔼[∏_x_C𝔼[∏_y_cexp(-shx_C^-α_Ny_C^-α_LcG)|x_C]]
        = 𝔼[∏_x_Cexp(-2λ_Bp_T (cG)^1/α_Lx^-α_N/α_Lϱ(T))] 
       =		exp(-2λ_S (2λ_B p_T)^α_L/α_N(cGϱ(T)^α_L/u)^1/α_NΓ(1-α_L/α_N)).

Combining the LT of the cross interference with beamforming gain g, the LT of the cross interference ℒ_I_ϕ_C(u) derived accordingly. 
    ℒ_I_ϕ_C(u) 
       = exp(-2^1+α_L/α_Nλ_S (cG)^1/α_NΓ(1-α_L/α_N)ελ_B^α_L/α_Nu^-1/α_N)
        = exp(-γ_Cελ_B^α_L/α_Nu^-1/α_N) = exp(-β_2 λ_B^α_L/α_Nu^-1/α_N),

where ε was defined in (<ref>).
ieeetr
	





