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Modeling Co-location in Multi-Operator mmWave Networks with Spectrum Sharing

Competing cellular operators aggressively share infrastructure in many major US markets. If operators also were to share spectrum in next-generation mmWave networks, intra-cellular interference will become correlated with inter-cellular interference. We propose a mathematical framework to model a multi-operator mmWave cellular network with co-located base-stations. We then characterize the SINR distribution for an arbitrary network and derive its coverage probability. To understand how varying the spatial correlation between different networks affects coverage probability, we derive special results for the two-operator scenario, where we construct the operators' individual networks from a single network via probabilistic coupling. For external validation, we devise a method to quantify and estimate spatial correlation from actual base-station deployments. We compare our two-operator model against an actual macro-cell-dominated network and an actual DAS-node-dominated network of different scales. Using the actual deployment data to set the parameters of our model, we observe that coverage probabilities for the model and actual deployments not only compare very well to each other, but also match nearly perfectly for the case of the DAS-node-dominated deployment. Another interesting observation is that spectrum and infrastructure sharing has a lower rate coverage probability for lower thresholds, which would make it less suitable for low-rate applications.

between network operators [8]- [10]. While spectrum sharing is a future possibility, infrastructure sharing is already a reality, and there has been a progression in the cellular operator industry towards sharing network infrastructure such as the network core, backhaul, and cell towers as a means of expanding coverage at a reduced cost [11], [12]. When multiple closed-access cellular networks share spectrum, inter-network interference adds to intra-network interference. When cellular networks also share infrastructure, inter-network interference becomes coupled to intra-network interference because many of the base-stations (BSs) of the different networks are stationed at the exact same location. In this paper, we propose a mathematical framework that accurately models the co-location of BSs of multiple operators that share spectrum licenses, and suggest how to estimate model parameters from actual deployments.

A. Background and Related Work
Cellular operators have already been sharing network infrastructure through a variety of business models to increase their coverage and capacity while reducing capital and operational expenditures [13]. We surveyed three geographically diverse US cellular market areas and tallied the sites that are occupied by a single operator and those that are shared by two or more operators. These sites included macro-cellular towers, rooftops, and DAS nodes. Figure 1 shows the percentage of sites shared with one or more competing operators. For every market, the bar plot shows the sharing ratio per operator, i.e. the percentage of the operator's BSs that are co-located with those of at least one other. Sharing ratios range from 11% to 78%, with some markets displaying more aggressive site sharing than other markets. This trend is expected to continue in next generation cellular networks through a dense overlay of multi-operator and virtual-host small cells targeting enterprise and entertainment venues, on top of the existing layer of macro towers housing BSs of competing operators [14]- [17].
Spectrum is a valuable asset that can also be shared by competing cellular networks [18]- [20].
Studies have suggested that cognitive radios are able to efficiently utilize existing sparse, sporadically used spectrum and facilitate spectrum sharing between different networks [10], [21]- [23]. Other studies have called for Licensed Shared Access (LSA) to enable 5G small cell to access mmWave spectrum bands that are already in use [24]. Combining spectrum and infrastructure sharing between operators couples inter-network interference with intra-network interference. It is unclear how this affects the coverage of the involved networks. The objective of this paper is to model infrastructure sharing between networks that have access to the same spectrum, and to study the effect of varying the extent of sharing on the downlink rate achieved by network subscribers.
In recent work, the performance of a number of mmWave cellular systems spanning different combinations of spectrum and access sharing methods was analyzed in a stochastic geometry framework [20]. Two particular systems were studied: a two-operator system with closed access and full spectrum sharing, and a two-operator system where all the BSs of the two operators are housed on the same towers. The work in [20], however, did not consider a system with partial base station co-location.
The same authors also established the feasibility of secondary licensing in mmWave licensed, yet the model they used to represent the locations of the primary and secondary BSs did not generalize to scenarios where the primary and secondary networks share infrastructure [25]. In other work related to infrastructure sharing, a statistical approach to model multi-operator networks with shared deployment patterns was presented in [26], but performance of the models therein was evaluated only through simulation. In a subsequent paper by the same authors [27], the impact of spatial clustering, network density, and spectrum access coordination on network coverage in a multi-operator system was studied analytically. Finally, in [28], different configurations of infrastructure and spectrum sharing were considered, and corresponding SINR and rate coverage probabilities were compared and evaluated against different channel, antenna, and BS patterns. Prior work such as [20], [25]- [28] lacks a model that reproduces any extent of co-location between the BSs of any set of operators, and that allows straightforward analysis of key performance metrics like SINR and rate coverage probabilities.

B. Contributions
In this paper, we propose an analytical framework to model a multi-operator mmWave system with any desired co-location extent of any set of operators. We use independent Poisson point processes (PPPs) of different densities to describe the co-location pattern of any group of operators. This pattern denotes locations of sites that undergo dual BS co-location between every pair of operators, triple BS co-location between every group of three operators, and so on. Hence, BS locations of any operator can be represented by superposing the appropriate point processes. To evaluate the performance of a participating cellular network, we derive the SINR probability of coverage of a typical user of this network.
We then focus on the more tractable case of two operators to understand how varying the spatial correlation between the two networks affects coverage probability. We begin with a mother point processes representing the BS locations of the two operators combined, and then derive two child point processes while forcing a desired degree of spatial correlation by probabilistically coupling the resultant processes on the same probability space (here, spatial correlation denotes the linear correlation of the number of BSs of the two operators in an area of space). We use the notion of second moment measure to quantify spatial correlation which we capture by the overlap coefficient. The two-operator system allows us to analyze the impact of correlation on the performance of the individual networks.
To evaluate and compare the how the SINR probability of coverage varies as a function of overlap, we consider two perspectives of infrastructure sharing: fixed individual densities (FID), and fixed combined density (FCD). In FID, the densities of the individual networks' BSs are fixed with varying overlap. In FCD, these densities increase as overlap increases, but the density of the total BSs is fixed. In a real deployment, FID corresponds to relocating BSs to sites that are already occupied by a competitor, while FCD corresponds to expanding into such sites.
Finally, we compare the probability of rate coverage and median rate for different systems under FID and FCD. To measure how accurately our model reflects the performance of an actual two-operator system, we compare the SINR probability of coverage obtained for those two systems. Since there are no mmWave networks currently deployed, we suggest ways of extracting model parameters, namely, overlap and individual densities, from actual deployments. We consider both macro-tower-dominated deployments and deployments predominantly comprised of distributed antenna system nodes (DAS nodes) in major US cellular markets. Our results show that coverage probabilities for the PPP model and actual deployments compare very well, and they are even almost identical in the case of the DAS-node-dominated deployment.
The rest of the paper is organized as follows. Section II describes the channel and multi-operator system model. Section III gives the expressions of the probability of SINR and rate coverage. In Section IV, we consider the two-operator model from a different angle which allows us to quantify the spatial correlation between two BS deployments, and estimate its value from actual deployments.
Section V presents numerical results and provides some insights. Finally, we conclude in Section VI.

II. SYSTEM MODEL
We consider a mmWave cellular network of M cellular operators that share spectrum licenses as well as infrastructure. We combine independent point processes representing BS locations to produce any amount of co-location of any subset of the M operators. Our model reduces to the two extremes of co-location described in [20]  PPPs is another PPP [29]. Moreover, given the Poisson nature of these point processes, any collection thereof shares no points in common almost surely; hence "double counting" of sites is not a concern.
These blocks, independent and disjoint, constitute the most elementary components of the model.
Additionally, we consider the collection { Φ m }, m ∈ O, where Φ m describes the locations of all sites housing BSs of operator m, both shared and non-shared, and is given by It is important to note the difference between Φ S , S ∈ P(O), and Φ m , m ∈ O. The first denotes a block and is the main building block of the model. The second characterizes the BS locations of an operator and is the result of combining different blocks. Consider, for example, the two-operator scenario (M = 2). The locations of BSs are represented by Φ { 1 } , Φ { 2 } , and Φ { 1,2 } , which in turn are subsets of Φ 1 or Φ 2 . We make the simplifying assumption that all operators own licenses of an equal amount of spectrum, and leave the generalization to future work.
We make the following assumptions about the blockage and channel models.
Blocking model: We assume the independent blocking model where the link established between the typical user and a BS located at a distance r away can either be LOS (denoted by L) with probability p L (r) or NLOS (denoted by N) with a probability p N (r) = 1 − p L (r). We adopt the exponential blocking model introduced in [7], where p L (r) = exp(−βr). Hence, conditioned on the typical user, each system block Φ S of density λ S is divided into two independent non-homogeneous PPP as a direct result of the independent thinning theorem [29], and we obtain the two sub-blocks: • L S containing all BSs with LOS links to the user. It has density λ S,L (r) = p L (r)λ S and measure Λ S,L .
• N S containing all BS with NLOS links. It has density λ S,N (r) = p N (r)λ S and measure Λ S,N .
Then, it follows that the average number of BSs in the sub-blocks L S and N S in the Euclidean ball B 0 (r) centered at the origin and of radius r is where γ(., .) is the lower incomplete gamma function.
Transmit and noise power: We assume that all BSs transmit at a fixed power P t . We consider a noise power spectral density N 0 and a total bandwidth B.
Path loss: We consider the power-law path loss functions for LOS and NLOS links: where c L and c N correspond to the power attenuation at r = 1 for LOS and NLOS links.
Directionality gain: Similar to [20], base stations are equipped with steerable antennas characterized by a main-lobe gain G and side-lobe gain g, while user mobile devices have a single omni-directional antenna. Hence, all points { X k } representing BS locations are endowed by marks { G k } which are where θ b is the half beamwidth and assumed to be identical accross all BSs. Since signals received from co-located BSs are transmitted from antenna arrays pointed in different directions to serve different users, we can assume that directionality gains are independent. In reality, actual array patterns can be different from those produced by this model because of scattering and dispersion [30]. Nevertheless, we use this model for analytical tractability. Small-scale fading: We assume that the channel undergoes flat Rayleigh fading. Equivalently, the fading power H k of the signal received from the BS at X k is exponentially distributed with unit mean.
We verify in Section V that the relative performance remains unchanged when Nakagami fading and lognormal shadowing are used. Despite the fact that the large-scale propagation losses of co-located transmitters of opposite networks are equal at any distance, we assume that signals received from these transmitters at any point undergo independent fades. This is reasonable given the different locations of BS antennas on the tower are typically further than the (vertical) coherence distance of the channel.

III. COVERAGE ANALYSIS
We use the SINR probability of coverage as the system performance metric, which is defined as the value of the SINR CCDF at a threshold T P c (T ) = P (SINR > T ) .
Suppose that the typical user associates with b th BS of the n th network at a distance R via a link of type τ (b, n) which can be LOS or NLOS. We define c τ (b,n) and α τ (b,n) to be the path loss constant and exponent corresponding to τ (b, n), and σ 2 to be the thermal noise power normalized by the transmit power, i.e. σ 2 = N 0 B/P t . We also define I to be the interference from all blocks and is expressed as Therefore, the SINR of the typical user is The first term of the sum accounts for the interference from BSs of the same operator, while the second term describes the interference from all BSs of different operators. Note that X j,m and X j ,m ,  In the remainder of this section, we analyze the coverage probability of a typical user of Network 1 since the coverage analysis of all networks is mathematically identical. The networks could have different coverage due to the various parameter values. However, this does change the analysis. We first investigate the association of the typical user of Network 1 to any of its BSs. We then compute its SINR probability of coverage and derive the rate probability of coverage which is a tangible metric in quantifying user experience.

A. Association Criterion
The probability that a typical user of Network 1 is covered depends on what block they are associated with. Association could take place through any of the 2 M −1 blocks of { Φ S }, 1 ∈ S, and any of their sub-blocks. Since these blocks and sub-blocks are independent, the events of associating to distinct blocks are disjoint. Hence, we can compute the total probability of coverage by adding the joint probabilities of coverage and association.
We first define some notation. Let ν S,τ = Λ S,τ •B 0 , where Λ S,τ is the measure for the appropriate subblock of T S , and the subscript τ denotes an arbitrary link type. The operator • denotes composition, i.e., ν S,τ (r) = Λ S,τ (B 0 (r)). Additionally, let D L be the exclusion function of LOS transmitters of Network 1 when the user is associated with a NLOS transmitter of the same network. Similarly, let D N be the exclusion function of NLOS transmitters when the user is associated with a LOS transmitter.
An exclusion function gives the radius of the region around the tagged BS within which no other BSs in the same or different blocks exist. These functions are given in [7] as Moreover, let A T S be the event of association with sub-block T S . Define P c (T ; Φ S ) and P c (T ; T S ) to be the probabilities that the user is in coverage for a threshold T and that the user is associated to a BS in any block Φ S and sub-block T S thereof: These probabilities can be obtained by integrating the ccdf of the appropriate SINR given by (8), (6), and weighted by the PDF f R (·; T S ) of the length R of the established link with a BS of T S as What remains to be derived is the PDF f R (.; T S ) for an arbitrary sub-block T S . To accomplish this, we follow the derivation in [20]. We draw an analogy between the sub-blocks that a typical user of Network 1 can associate with, defined in this paper, and the tiers as defined in [20]. In [20], a typical user (of Network 1, let's say) is permitted to access Network 1, their home network, and every other network that is in the access class of Network 1. In a closed-access system, the access class is Network 1, and in an open-access system, the access class is all networks. The key to computing f R (r; T S ) is computing the probability f o R (r; T S ) that all BSs of every other sub-block T T , T ∈ P(O) \ S, 1 ∈ S that are accessible by the typical user are outside the exclusion radius r.
For an arbitrary sub-block T, this is given by the void probability µ T (r) = P (T(B 0 (r)) = 0). Since all sub-blocks are mutually independent, f o R (r; L S ) is given as the product of void probabilities The density f R (·; L) is obtained according to [31, Section V-C] and [20, Equation (9)] as Since the events { A T } of association with different sub-blocks { T } are disjoint [20], the SINR coverage probability P c (T ) for the typical user is obtained by adding these individual block coverage probabilities over all accessible block of Network 1: Next, we derive the expression for P c (T ; T S ) for an arbitrary sub-block.

B. Interference Characterization
Computing the probability of coverage under Rayleigh fading can be readily reduced to finding the Laplace transform of interference. To illustrate this, (5) can be expanded as where F H is the CCDF of the fading encountered by the signal emitted from the tagged BS which can be expressed as a single exponential. Noting that F H (u) = e −u , (14) becomes Since the BSs of Network 1 are spatially co-located with those of other networks, inter-network interference is no longer independent and L I|R is not the product of Laplace transforms of internetwork and intra-network interference. To resolve this, we reformulate the interference expression in (6), which is given with respect to the correlated point processes of { Φ m } m∈O , as a sum over the . We give the resulting expression in the following proposition.
Proposition 1. Given that the typical user associates to a BS at a distance r in the LOS sub-block L T , T ∈ P(O), then the Laplace transform of the interference random variable is given as where u τ (s, t) = E G L H|G (sc τ Gt −aτ ) . Moreover, if fading power is exponentially distributed with unit mean and antenna gain follows a Bernoulli distribution, L I L (s) is given as The Laplace transform is the product of Laplace transforms of independent random variables corresponding to different classes of BSs, grouped according to the network they belong to as well as the type of potential link they can establish with the user. The first term of (16) represents the contribution of the LOS BSs of different operators co-located with the typical user's tagged BS at a distance r away, since there are a total of |T | co-located BSs in sub-block L T . The second term gives the LOS interference from all the sites where no BS of Network 1 is deployed. The third term gives the LOS interference from all the sites housing BSs of Network 1 averaged outside the exclusion ball associated with the tagged BS. The remaining terms are almost identical to the second and third terms, with the only difference being that they account for NLOS interference.
The probability of coverage is expressed in terms of the interference Laplace transform (see (10) and (15)). Now that we have determined the Laplace transform, we give the coverage probability in the next proposition. The ultimate metric for evaluating the performance of a cellular network is the per-user downlink rate distribution since it reflects an aspect of service quality experienced by the user. We can transform the SINR coverage probability into a rate coverage probability with a few assumptions. The amount of bandwidth resources allotted to a typical user is a function of the total number of users served by the associated BS, as well as the total available bandwidth B. We assume a fair resource allocation algorithm where the BS scheduler divides bandwidth resources equally among each of the N U users of spatial density λ U . Due to the closed-access nature of our multi-operator system, users of a particular operator can only connect to their operator's home network. Hence, the mean number of connected users in a cell can be given based on the approximate load model in [20], [32], [33] as N U = 1 + 1.28 λ U λ . Finally, the probability that a typical user of Network 1 experiences a rate of at least R bps is P (Rate > R) = P c 2 RN U /B − 1 .

IV. THE TWO-OPERATOR CASE
In this section, we analyze the probability of coverage for the two-operator case. This scenario is an important special case because it allows to parameterize the system using only three quantities: the densities of the two networks and the extent of overlap between them. In addition, it is difficult to simplify the general case because one has to sweep the densities { λ S } of all the underlying blocks { Φ S }, and consider all possible ways these blocks could be combined to construct the point processes { Φ m } describing the BS locations of every operator.
In this section, we describe how to construct the operator point processes when there are only two operators, and analyze the probability of coverage for Network 1. Next, we describe how to estimate the parameters of our model from actual two-operator deployments. Then, we consider two perspectives to study how the coverage probability changes as a function of the overlap between the two networks.

A. Two-Operator Model
The two-operator cases allows for a more natural construction of the operator point processes that allows to describe the model using a few parameters: the densities of the two networks and the extent of overlap between them. We start with a mother point process Φ of density λ, and extract the two child point processes from it. We identify Φ as sites managed by a network infrastructure provider, and Φ 1 and Φ 2 as sites leased to two independent operators. We build our model by capturing the density of the first network (Network 1) and the density of the second (Network 2), denoted by λ 1 and λ 2 .
We introduce the overlap coefficient ρ(·, ·) as a measure of spatial correlation between Network 1 and Network 2 over two sets. For any two given sets A and B, ρ(A, B) is a function of the covariance between the number of Network 1 sites in A and that of Network 2 in B as Notice that the numerator of the overlap coefficient is a function of the cross-moment which describes the interaction between two point processes Φ 1 and Φ 2 . The normalization by the total density λ is necessary so that one can compare the spatial correlation of two networks across different markets of distinct sizes. Since we are interested in the correlation of these two point processes over the entire geographical window W , we use ρ as a shorthand notation to ρ(W W ). Proposition 2 shows that ρ is in fact directly proportional to λ, which is a direct result of extracting the child processes from a mother process that is Poisson. If the mother process is not Poisson, the overlap coefficient will not necessarily be proportional to the total density λ.
We now explain how to mathematically construct Φ 1 and Φ 2 from Φ. The key to construct Φ 1 and Φ 2 from Φ is the coupling technique, where we enforce that the derived point processes have some points { X k } of Φ in common by coupling them on the same probability space. We begin by marking the points of Φ with independent random variables { U k } k≥0 uniformly distributed between 0 and 1. We next consider two parameters a and b, where 0 ≤ b ≤ a ≤ 1, and the retention functions q 1 (X k ) = a and q 2 (X k ) = 1 − b. A retention function assigns to every point of a point process a probability of being retained, or alternatively, discarded [29]. Here the probability that a point X k in Φ is retained by Φ 1 and Φ 2 is a and 1 − b, respectively. As a result of thinning Φ separately with q 1 and q 2 , the following two child processes can be obtained where ω ∈ Ω, and Ω is the common sample space. Let Φ 12 = Φ { 1,2 } = Φ 1 ∩ Φ 2 be the point process describing the locations of shared sites, and let λ 12 be its density. We now give a proposition that validates the above construction of the individual networks from a greater one.
Proof: See Appendix B.
Since ρ turns out to be the fraction of co-located BSs, and 0 ≤ ρ ≤ 1, we use it as a proxy for λ 12 to obtain λ 1 + λ 2 − ρλ = λ. In the next proposition, we give the expression for the interference Laplace transform, which is the stepping stone towards the coverage probability expression.
Proposition 3. Given that the typical user establishes a LOS link of length r with a BS of Network 1 that is not co-located with a BS of Network 2, the Laplace transform of the interference random variable is If, otherwise, the tagged BS of Network 1 is co-located with a BS of Network 2, then the Laplace transform of interference is given as Proof: This follows directly from Proposition 1 with M = 2, S ∈ { { 1 } , { 1, 2 } }, and S = { 2 }.
We examine the extreme cases of infrastructure sharing, namely full spatial independence and full spatial co-location, through (20). These two cases mimic the closed access with full spectrum sharing and co-located BSs with closed access and full spectrum license sharing systems studied in [20]. In particular, values of ρ = 1 (and a = 1) fold the Laplace transform expression in (20) into a product of the second and the fourth terms. These two factors account for the interference contribution from all BSs of the two networks that are located in the LOS and NLOS exclusion balls B 0 (r) and B 0 (D L (r)) centered at the typical user. Note that the Laplace transform expression for NLOS interference follows the same derivation, so it is excluded.
Before giving the final proposition, we introduce some notation. Let Φ 1 = Φ 1 \ Φ 2 and Φ 2 = Φ 2 \ Φ 1 be two point processes with respective densities λ 1 and λ 2 . Now, we give a corollary that relates the probability of coverage of a typical subscriber of Network 1 in a two-operator system to the different system parameters.
Corollary 2. The SINR probability of coverage of a typical user of Network 1 in a two-operator system is given by Proof: This is a special case of Proposition 1 with M = 2 and S ∈ { { 1 } , { 1, 2 } }.
As in Corollary 1, the inner summation in (22) is across all collections S of operators in which operator 1 is present. Every block Φ S is divided into two sub-blocks L S and N S , and their contributions to the total probability of coverage are weighted by the pdfs f R (.; L S ) and f R (.; N S ).

B. Statistics of Actual Deployments
To assess how well our model matches with an actual two-operator deployment of comparable network densities and overlap, we first need to estimate these parameters from the actual deployment.
Unfortunately, there is one realization of the point process we seek to model per market, so the first order (the densities) and second order statistics (the overlap) need to be estimated from this single realization. Hence, we conduct our statistical analysis on a single observation and assume that the underlying point process Φ is a PPP which is stationary and ergodic [34]. The key quantities to estimate given a bounded window W are λ, λ 1 , λ 2 , and ρ. A general unbiased estimator of the total density λ according to [34] isλ and similar expression for λ 1 and λ 2 follow.
We also provide two ways to estimate the overlap coefficient given in (19), which involves a cross-moment of two point processes. Second-order moments of point processes and methods of their estimation are well studied in [35]- [38] and the references therein, yet there is very little in the stochastic geometry literature on cross moments [35], [39] and their estimation. Therefore, we devise estimating the overlap coefficient in one of two ways: a) Indirectly through estimating λ 12 : This uses the fact that the overlap coefficient ρ is directly proportional to the density of the mother process Φ when the latter is Poisson, with the proportionality constant being λ 12 . Hence, we computeλ 12 as in (23), and then setρ =λ 12 λ .
b) Directly through a naive estimator: This extends the notion of sample covariance to point processes to estimate the overlap ρ between two point processes. We first apply a uniform partition W (n) of size n to the observation window W . Finally, we computeρ as: Note that the expression ofρ as given by (24) does not necessarily guarantee thatλ 1 +λ 2 −ρλ =λ, which is the case when the actual deployment patterns are not Poisson.

C. System Comparison
We consider two perspectives to study how coverage probability changes as a function of overlap between two mmWave cellular networks. For instance, increasing overlap does not imply that individual densities need to increase as well. To make the comparison simpler, we assume that the two networks have equal densities.

a) Fixed individual network densities (FID):
The operators can opt to share sites with one another while maintaining the densities of their overall deployments. In a practical sense, this might occur when decreasing operational expenditures gets a higher priority densification. For example, consider two operators with fixed densities λ 1 = λ 2 = λ 0 . To achieve an arbitrary overlap of ρ, each of the networks relocates a number of BSs accounting for ρ 1+ρ of its density into the same number of sites of the competing network. As a result, λ 1 = λ 2 = 1−ρ 1+ρ λ 0 and λ 12 = 2ρ 1+ρ λ 0 . We will refer to λ 12 /λ 0 as the sharing ratio. b) Fixed combined network density (FCD): Each operator can share sites with the competing operator by means of expanding into the sites owned by the latter; i.e., each operator retains the sites that it started out with. In a practical sense, this might occur as strategic action to extend an operator's reach in a market. For example, consider as above two operators with base densities λ 1 = λ 2 = λ 0 .
To achieve an overlap of ρ, each operator expands into ρ of the competitors network. In this case, λ 1 = λ 2 = (1 − ρ)λ 0 and λ 12 = 2ρλ 0 . Note that the highest density achieved by either operator is the sum of the individual starting densities of individual networks; i.e. 2λ 0 .
The main takeaway here is that increasing ρ under FID increases overlap but not individual densities.
In contrast, increasing ρ under FCD increases overlap and individual densities.

V. NUMERICAL RESULTS
In this section, we present numerical results that validate our coverage analysis, evaluate the accuracy our model, and compare performance metrics across a range of overlap extents. We give all numerical results for the two-operator mmWave system.
For these results, we assume that the combined bandwidth of the mmWave systems operated by the networks is 200 MHz, and that the operating frequency is 28 GHz as in [20]. The values of the rest of the parameters are set to those of their counterparts in [7], [20], [25].  We validate the analytical expressions for the probability of SINR coverage that were obtained in Section IV-A for a two-operator mmWave system with closed access and full spectrum sharing. We numerically evaluate the probability of coverage expression in (22) for a range of SINR thresholds, and we plot this against the empirical probability obtained through Monte Carlo simulation; the results are shown in Fig. 2. We consider FID with different overlap coefficient values, ρ = 0 or no infrastructure sharing, ρ = 0.4 or 57% sharing, and ρ = 1 or full sharing (by letting ρ grow large, the two networks share more sites in common but their individual densities remains constant throughout). The first thing we observe, for all considered degrees of sharing, is that plots obtained through simulation match the ones obtained through analysis; which further validates the correctness of our analysis. Moreover, we observe that increasing overlap increases coverage probability at higher SINR thresholds, yet decreases coverage at lower SINR thresholds. The reason is that, in full sharing, there are no interfering BSs closer to the user than the associated BS, which has a positive impact on coverage at low SINR thresholds. As for high SINR, the anticipated signal is received at a much higher power than that of the interfering signals combined, but this is not the case in full sharing; the associated BS is co-located with another BS, which adds yet a source of interference that is just as powerful as the intended source. To evaluate the accuracy of our two-operator PPP model, we compare the probability of SINR coverage between the PPP model and actual two-operator networks in different markets. Since mmWave systems have not been deployed yet, we "down-scale" the abscissas and ordinates of the BS locations so that the individual network densities compare to that of a typical mmWave network. We set that to 60 BSs/km 2 . This operation is known as pressing, and is essentially an affine transformation in the plane which maintains the Poisson property of the original, full-scale point process. Note that pressing a network of two operators maintains their overlap.

B. Comparison of Estimators for the overlap ρ
For each market, we first estimate the densities and overlap using (23) and (24). Then, we generate a number of realizations of our two-operator PPP model with the obtained estimates. Finally we reestimate the densities and overlap of these realizations. The reason for this procedure is two-fold.
First, mismatching estimates for ρ suggest that BS locations of actual deployments are not really PPP realizations. Second, we can verify that (24) is in fact an estimator for ρ at least for a PPP. We plot our findings in Figure 4. We filter the direct estimate of ρ-which is a quantity that evolves with the number of bins-with a simple moving average filter to highlight the general trend it follows. The first observation we make is that the two estimates are different in both datasets: the two estimates are off by 0.2 in the first dataset and by 0.1 in the second. This suggests that the BS locations are not quite PPP. The second observation is that difference of the two estimates for CMA II is less than the error for CMA I. This can be explained by the fact that CMA II is a dense, DAS-node-dominated network that extends over 1% the area covered by CMA I which is a macro-tower-dominated network; and DAS node locations are more random than those of cell towers.

C. Validation of Model with Actual BS Deployments
To assess how accurately our model reflects the performance of an actual two-operator system, we compare the probability of coverage of the two systems while constraining the two networks to have the same individual densities and overlap. We do this experiment for two sets of actual deployments.
We first estimate λ 1 , λ 2 and λ 12 as described in Section IV-B. Figure 5 shows the views of base station deployments in CMAs I and II. The figure on the left shows base stations of networks A and B, and the one on the right shows those of operators B and C. Figure 6 shows the SINR probability of coverage attained by a typical user connecting to Operator A (left) and Operator C (right). Note that the overlap coefficent estimates used in evaluating the SINR coverage probability formula were obtained indirectly. We can see that the SINR coverage plots for our two-operator PPP model match reasonably well with those for the actual deployments. Even though our model produces an almost identical coverage curve for CMA II, it produces a coverage probability curve that slightly deviates from the actual deployment at a wide range of SINR.

D. Comparison of Sharing Schemes
We compare the probability of rate coverage between three infrastructure sharing scenarios as well as the single operator system under FID and FCD. A single operator system comprises of the network of a single operator that shares neither infrastructure nor spectrum. In the sharing scenarios, we consider ρ = 0, 0.4, 1. Figure 7 compares the probability of rate coverage between the different systems under FCD. A single system is made of the network of a single operator that shares neither infrastructure nor spectrum.
In the sharing scenarios, we consider ρ = 0, 0.4, 1. We observe that by fixing the total density to a base value λ 0 and increasing the overlap between the base stations of the two operators, the probability of coverage increases for all rate thresholds. At low rate thresholds, the performance of a single-operator network with half the bandwidth and half as many BSs excels over shared networks with ρ = 0.4 at the least. This is explained by having no interfering BSs closer to the user than the associated BS.  increase in rate coverage. If each network further expand into 40% of the competitors' network, i.e.
an overlap coefficient of 0.4, their networks will provide up to 800% increase in rate coverage.
Additionally, we see that these trends remain the same regardless of the types of small-scale and large-scale fading considered. This strengthens our earlier assumption that signals undergo Rayleigh fading. rate thresholds for deployments with more prominent sharing and the opposite trend at high thresholds follows the same reasoning we made before in relation to Figure 2. Second, the single user system outperforms systems with sharing for low rate thresholds, which is also be observed in Figure 7 for the FCD sharing strategy. This directly affects planning for deployments that communicate with devices through very low rates, such us IoT devices that wake up infrequently to send status updates or small measurements.  nor infrastructure with other competitors. We observe that FCD gives a higher rate than the two strategies. In particular, the median rate an operator achieves under FCD is almost double the median rate achievable by a single operator at any individual density. Even though the interference in FCD is higher due to the presence of transmitting base stations in equal density for the second operator, the used spectrum is double. Moreover, sharing gives an almost steady median rate for FID. The reason is that the number of interfering BSs of the opposite network remains the same with varying the overlap coefficient. The only difference that changing overlap makes is whether the interferers of the opposite network are co-located with those of the home network.

VI. CONCLUSION
In this paper, we proposed a novel mathematical framework for modeling BS locations of a multioperator cellular mmWave system, and provided analytical expressions for the SINR coverage probability as a performance metric for an arbitrary network. For a tractable evaluation of system performance, we narrowed the scope to a two-operator scenario. For the two-oprator scenario, we provided a method   to fit actual deployments with our model and estimate its necessary parameters, and validated the model with realistic deployments of cellular systems. Additionally, we suggested different schemes that describe increased infrastructure sharing: FID and FCD. We showed that the median rate in FID does not change appreciably as the overlap between the networks varies, unlike in FCD that witnesses a steadily increasing median rate in parallel to increasing overlap (or equivalently, increasing the density of the individual networks). We saw that the FCD strategy outperforms the single operator system for the same BS density, which is possible due to the availability of double the spectrum resources. We also observed that, for both FID and FCD, the single operator system with half the total bandwidth excels in the low rate threshold regime. That is, it is able to provide low data rates to more users than some of the two-operator systems under FID or FCD.
APPENDIX A

PROOF OF PROPOSITION 1
The user connects to Network 1 through L T . The Laplace transform of I L and I N , the interference from the LOS and NLOS brackets, is given as follows: We derive the expression for L I L (L I N follows similarly). Since the point processes representing Finally, since the LOS and NLOS blocks are independent, we have that L I L +I N (s) = L I L (s) · L I L (s).

APPENDIX B PROOF OF PROPOSITION 2
According to the thinning theorem for a PPP [29], Φ 1 and Φ 2 are PPPs (Note that they are not independent) with respective densities aλ and (1 − b)λ. Moreover, we claim that ρ = a − b. To see this, consider the independently-marked point processΦ = k≥0 δ (X k , U k ) , where δ is the Dirac measure, and { U k } are the iid marks, and U k ∼ U(0, 1), ∀k. Given sets A and B in the Euclidean plane, and I, J ⊂ [0, 1], the intensity measure and the second moment measure ofΦ are given in [29] as where vol ( · ) is the Lebesgue measure of a set taken with respect to the appropriate number of dimensions. Note that (28) holds true: the ground point process Φ is a PPP on R 2 × [0, 1], then the independently marked point processΦ is also a PPP [29]. Now let I = [0, a] and J = [b, 1]. The correlation between the number of sites of Network 1 in a set A and Network 2 in a set B is