{'fraction_non_alphanumeric': 0.03182451802951001, 'fraction_numerical': 0.009494752899713316, 'mean_word_length': 4.336421052631579, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Outage Probability of Millimeter Wave Cellular Uplink with Truncated Power Control

In this paper, using the stochastic geometry, we develop a tractable uplink modeling framework for the outage probability of both the single and multi-tier millimeter wave (mmWave) cellular networks. Each tier's mmWave base stations (BSs) are randomly located and they have particular spatial density, antenna gain, receiver sensitivity, blockage parameter and pathloss exponents. Our model takes account of the maximum power limitation and the per-user power control. More specifically, each user, which could be in line-of-sight (LOS) or non-LOS to its serving mmWave BS, controls its transmit power such that the received signal power at its serving BS is equal to a predefined threshold. Hence, a truncated channel inversion power control scheme is implemented for the uplink of mmWave cellular networks. We derive closed-form expressions for the signal-to-interference-and-noise-ratio (SINR) outage probability for the uplink of both the single and multi-tier mmWave cellular networks. Furthermore, we analyze the case with a dense network by utilizing the simplified model, where the LOS region is approximated as a fixed LOS disc. The results show that imposing a maximum power constraint on the user significantly affects the SINR outage probability in the uplink of mmWave cellular networks.


I. INTRODUCTION
A fundamental requirement for the 5G-and-beyond mobile networks is the radical increase in data rate. Recent studies have identified massive multiple-input-multiple-output (MIMO), extreme network densification, and increased bandwidth as the key technologies toward meeting this requirement [1]. The millimeter wave (mmWave) frequencies (ranging from 30−300 GHz) offers a large available bandwidth thus, making them attractive for the 5G mobile networks [1]- [3].
Meanwhile, the mmWave band has long been considered ill-suited for the cellular communication due to the excessive pathloss and the poor penetration through materials such as concrete and water. Recent capacity studies and survey measurement on mmWave technologies in [3]- [6] have shown its great promise for 5G urban small cell deployments. The recent advances in low-power CMOS RF circuit and the smaller wavelength associated with the band have further substantiated this promise. The later also makes it viable to have more miniaturized antennas within the same physical area of the transmitter and receiver [2], [6]. Further, with a large antenna array, the mmWave network can apply beamforming at the transmit and receive sides to provide array gain which compensates for the pathloss [7]. The directionality gained from beamforming will lead to a reduction in interference [5]. Hence, mmWave spectrum holds great potential for providing the high data rate (Gigabits range) expected in the upcoming 5G cellular networks [8].
Modeling and analysis of cellular networks by using stochastic geometry have recently received significant attention due to its high accuracy and tractability. In this approach, the network topology is abstracted to a point process for ease of modeling and analysis. Earlier works in this area were mainly focused on the conventional ultra-high-frequency (UHF) cellular networks [9]- [19]. In the pioneering work on using stochastic geometry for cellular networks [9], it was shown that stochastic model provides a lower bound to real cellular deployment. The work in [9] was based on the downlink of cellular networks with the single slope pathloss model. This has been extended by considering a multi-slope pathloss model in [10], [11], the multi-tier cellular networks in [12]- [14], the single tier uplink cellular networks in [15] and the multi-tier uplink cellular networks in [16]- [18].
The stochastic geometry framework developed for the UHF networks do not directly apply to the mmWave networks due to blockage effects that they suffer from and the much different pathloss model. Furthermore, directional beamforming is fundamental in the design of the mmWave cellular networks. The authors in [8], [20]- [24] have analyzed the mmWave cellular networks by using the stochastic geometry framework with the blockage effect, realistic pathloss model and the beamforming gain incorporated in their model. In particular, [8] pioneered the research work on the downlink of mmWave cellular networks by leveraging on an earlier work in [25], which characterized the blockage parameter by some random distribution. Furthermore, the proposed analytical framework in [8] also captures the significant difference between the non-line-of-sight (NLOS) and line-of-sight (LOS) pathloss characteristics. The work in [8] has been extended to the downlink multi-tier mmWave cellular networks in [20], base station (BS) cooperation in [21] and the uplink single-tier mmWave cellular networks in [22]- [24].
The uplink analysis for both the conventional UHF cellular networks and the mmWave cellular networks are deemed to be quite involved as compared to the downlink analysis due to the peruser power control and the correlation among the interferers [16]. The former is due to the fact that an interfering user could even be closer to a reference mmWave BS than the user that is tagged to the reference mmWave BS. Furthermore, regarding mmWave cellular networks, measurements showed that mmWave signals propagate with a pathloss exponent of 2 in LOS paths and a much higher pathloss exponent with additional shadowing in NLOS paths [3], [4].
This poses a further challenge since the difference in pathloss exponents could results in excessive interference from NLOS users when the per-user power control is implemented. Hence, power control must be implemented with a constraint on the maximum user transmit power in order to mitigate the interference. On the other hand, the correlation among the interferer results from the implementation of orthogonal allocation scheme that does not allow for a reuse of a channel resource within the same cell, i.e., the coupling of the mmWave BS and served user-per channel point processes [15], [16]. To ensure analytical tractability, various generative models have been proposed in [15]- [17], [22]- [24] to approximate the spatial distribution of interferers in the uplink of UHF and mmWave cellular networks.
In this paper, we present a stochastic geometry framework for modeling and analyzing the uplink in single-tier and multi-tier mmWave cellular networks. Similar to the earlier works in this area [15]- [17], [22]- [24], we rely on some approximation so as to maintain analytical tractability.
Notably, we partially ignore the correlation among the interfering users. Our model captures the correlation between the interfering users and the reference mmWave BS, which serves the typical user, but it ignores the correlation among the interfering users. As evidence from [15]- [17], this approximation holds true for the uplink of both the single and multi-tier UHF cellular networks.
The accuracy of this approach is validated for both the single and multi-tier mmWave cellular networks via Monte-Carlo simulations. We here extend the work in [24] which is based on a single-tier mmWave network and does not take into account the maximum transmit power of the user. Our proposed framework takes into account the limitation in the user transmit power, the per-user power control and the cutoff threshold for the power control. We compare the findings of our analysis with that from [16] and [24]. The comparison reveals that our analysis provides several new insights that can be leveraged for designing the mmWave networks more accurately.
The main contributions of this work are summarized as follows.
• We present a stochastic geometry framework for the signal-to-interference-and-noise-ratio (SINR) outage probability in the uplink of a single-tier mmWave cellular networks, which is generic and serving as a foundation extends to the multi-tier mmWave cellular networks.
The model takes into account the limitation in the transmit power of the user, and the network defined per-user power control and the cutoff threshold. Closed-form expressions are derived for the SINR outage probability.
• We present the asymptotic dense network analysis of the SINR outage probability in both the single and multi-tier mmWave cellular networks. The asymptotic analysis leverage on approximating an intricate LOS function as a step function.
• The analytical derivations are verified via Monte-Carlo based simulations. Results show that the maximum power constraint significantly affects the SINR outage probability. Further, contrary to the SINR outage of UHF networks, which is non-increasing in the cutoff threshold, the SINR outage probability in mmWave networks could increase over some range of cutoff thresholds for some mmWave BS density, LOS and NLOS pathloss exponents, and blockage parameter.
The rest of the paper is organized as follows. The system model of the uplink of a multi-tier mmWave cellular network with truncated channel inversion power control is presented in Section II. In Section III, the uplink modeling framework for a single-tier mmWave cellular network is presented. In Section IV, we generalize the developed framework for the multi-tier mmWave cellular networks. In Section V, we utilize a simplified system model to analyze the asymptotic behavior and performance in dense mmWave networks. Numerical and simulation results are presented in Section VI. Finally, Section VII concludes the paper.

A. Network Model
We consider the uplink of a K-tier mmWave cellular network and focus on the SINR experienced by outdoor users served by outdoor mmWave BSs. Each tier's BSs are randomly located and they have particular spatial density, antenna gain, receiver sensitivity, blockage parameter and pathloss exponents. The outdoor BSs of each tier are spatially distributed in R 2 according to an independent homogeneous Poisson point process (PPP) Θ k with density λ k . The users locations (before association) are assumed to form a realization of homogeneous PPP Φ with density λ u . It is assumed that the density of the users is high enough such that each BS will have at least one user served per channel. Each BS serves a single user per channel, which is randomly selected from all the users located in its Voronoi cell by using a round-robin scheduler. As in [15]- [17], [19], [22]- [24], we assume that the active users also form PPP even after associating just one user per BS. Note that this approximation only partly ignores the correlation imposed by the system model, i.e., the coupling of the BS and served user-per channel resource point processes. The correlation between the reference mmWave BS and the typical user is captured in the derivation of the outage probability in Sections III and IV.
Each tier in the mmWave network is characterized by a non-negative blockage constant β k for k ∈ {1, . . . , K}. The parameter β k is determined by the average size and density of blockages in that tier and where the average LOS range is given by 1 β k [8], [21], [25]. The probability of a communication link in the k th tier with length r being a LOS is P(LOS k ) = e −β k r , while the probability of a link being NLOS is P(NLOS k ) = 1−P(LOS k ). The LOS and NLOS links of the k th tier have different pathloss exponents denoted by α k L and α k N , respectively, ∀k ∈ {1, . . . , K}.

B. Receiver Sensitivity and Truncated Outage
We assume that all users have an equal maximum transmit power P u . Furthermore, all mmWave BS in the k th tier have the same receiver sensitivity which is denoted by ρ k min . The received signal at the mmWave BS must be greater than the receiver sensitivity ρ k min for successful transmission in the uplink channel. Hence, each user (with either LOS or NLOS link to its serving mmWave BS) associated with the k th tier adjusts its transmit power such that the average received signal at its serving mmWave BS is equal to a predefined threshold ρ k o , where ρ k o > ρ k min . Moreover, as a result of the maximum transmit power constraint, users utilize a truncated channel inversion power control scheme, where the transmitters compensate for the pathloss in the link to the receiver to keep the average received signal power to the threshold ρ k o . Any user-mmWave BS connection that requires a transmit power that exceeds P u for the pathloss inversion will not be established, hence, such a connection experiences a truncation outage [16].

C. Beamforming Gain
For analytical tractability, we assume that all users and BSs are equipped with directional antennas with a sectorized gain pattern. The main lobe gain, side lobe gain and beamwidth of the users are G max u , G min u and ζ t , respectively, while the corresponding parameters of the k th tier BS antennas are G max bk , G min bk and ζ rk , respectively. We consider that based on channel estimation, the reference BS in the j th tier and the typical user adjust their beam steering angles to achieve the maximum array gains. As a result of this, the total directivity gain of the desired signal is Since the underlying PPP is isotropic in R 2 , we model the beam directions of the interfering link as a uniform random variable on [0, 2π]. Further, the directivity gain in the interference link G j l (interference experienced at the reference BS in the j th tier) can be approximated as discrete random variable whose probability distribution is given as a j v with probability b j v (v ∈ {1, 2, 3, 4}) [25], where a j v and b j v are defined in Table I. In general, the k th tier is characterized by a set of parameters V k whose element include the k th tier's BS density λ k , blockage parameter β k , cutoff threshold ρ k o , main lobe gain G max bk , side lobe gain G min bk , beamwidth ζ rk , LOS pathloss exponent α k L and its NLOS pathloss exponent α k N such that V k = {λ k , β k , ρ k o , G max bk , G min bk , ζ rk , α k L , α k N }, ∀k = 1, . . . , K.

III. UPLINK OF SINGLE-TIER MMWAVE CELLULAR NETWORKS
In this section, we develop our framework model for the uplink of a single-tier mmWave cellular network. In particular, we present the mmWave transmission power analysis and the SINR outage probability analysis.

A. mmWave Transmission Power Analysis
Considering the mmWave cellular network with the truncated channel inversion scheme, each user, which could be in LOS or NLOS to its serving mmWave BS will transmit with different power in order to invert the pathloss towards its serving BS. As a result of the truncation channel inversion, not all users will be able to communicate in the uplink channel 1 . In particular, LOS and NLOS users located at a distance greater than (P u/ρ o ) 1/α L and (P u/ρ o ) 1/α N , respectively, from their associated BS are unable to communicate in the uplink direction as a result of insufficient transmit power [16]. Hence, in addition to the fact that the whole user set is divided into a subset of LOS and NLOS users based on their association with their serving mmWave BS, the LOS and NLOS user sets are further divided into a non-overlapping subset of active users and inactive users. The distribution of the transmit power of a typical user is obtained from the following theorem.
Theorem III.1: In mmWave cellular networks with truncated channel inversion power control and cutoff threshold ρ o , the probability distribution function (PDF) of the transmit power of a typical user in the uplink is given by and The δ th moment of the transmit power is thus obtained as where f P (p) is given in (1).
Further, the truncation outage probability, which is the probability that a user experience outage due to insufficient power, is expressed as where Λ(p) is given in (3). 1 Note that for all the parameters in single tier network, we have removed the subscript/superscript k used to distinguish the k th -tier parameters in a multi-tier network.

Proof: See Appendix A.
Note that 1 − e −y (1 + y) is strictly increasing in y for y > 0, hence the first term of Λ(p) in (3) is greater than the third term and Λ(p) is strictly positive for all BS density λ, blockage parameter β, cutoff threshold ρ o and pathloss exponent α N ≥ α L > 0. Consequently, increasing the cutoff threshold ρ o leads to increase in the truncation outage probability as long as α N ≥ α L . In other words, the higher the cutoff threshold the poorer the mmWave network performance in terms of the truncation outage. As we will show in the later section, a low cut-off threshold could actually deteriorate the mmWave network performance in terms of the SINR outage probability. Hence, it is essential to manage the trade-off between SINR outage and truncation outage probabilities using the cutoff threshold.
On another note, by expanding (3), it can be seen that increasing the blockage parameter value β leads to a reduction in Λ(p) for fixed mmWave BS density λ, cutoff threshold ρ, and pathloss exponent α N > α L . Thus, the truncation outage probability also increases with increasing blockage parameter β. Increasing the blockage parameter implies decreasing the average LOS range and hence we have more NLOS paths requiring a much higher transmit power to meet the receiver sensitivity requirement.
Regarding the expectation of the user transmission power, i.e., the average user transmission power, it is not straightforward to gain insights. However, from (2), we expect the plot of the average transmit power to be characterized from the LOS-based average transmit power and the NLOS-based average transmit power. We validate this observation later in the numerical results section.

B. SINR Outage Probability
For an active typical user, the SINR at its connected BS (termed as the reference mmWave BS) can be written as where the useful signal power (normalized by G max b G max u ) is equal to ρ o |g o | 2 due to the truncated channel inversion power control, Z is the set of interfering users, L(D z ) is the pathloss from the interfering users to the reference mmWave BS, σ 2 is the noise power, G z is the directivity gain on an interfering link and g z is the small-scale fading which follows a Nakagami distribution with parameter N. The SINR outage probability O s is the probability that the instantaneous SINR experienced at the reference mmWave BS is less than the target SINR θ, i.e. O s = P(SINR < θ).
Given that the average received signal at the reference mmWave BS (normalized by the directivity The SINR outage probability can be computed as where I L and I N are the interference strength from LOS and NLOS users, respectively and Noting that |g o | 2 is normalized gamma random variable with parameter N, we have the following approximation follow from the fact that |g 0 | 2 is a normalized gamma random variable with parameter N and the fact that for a constant γ > 0, the probability P(|g 0 | 2 < γ) As mentioned earlier, the location of the interfering users do not create a PPP as a result of the correlation among the users from the channel assignment process. The interfering users are thus better modeled by using soft-core processes, which can capture such correlation [27]. However, most soft-core processes lack analytical tractability [28], [29], thus making the expression for the Laplace transforms of the aggregate LOS and NLOS interference, L I L and L I N , respectively, unobtainable. Hence, we approximate the location of the interfering users with a PPP. Note that the approximation has been shown to be accurate for the UHF network when the correlation among the interfering nodes and the reference receiver is captured [15], [16]. Our model here also captures this correlation. The accuracy of our assumption will be verified later through simulations. Based on the PPP approximation, and the independent and identical distributed transmit power for the set of interfering users in the uplink channel, the SINR outage probability can be obtained from the following theorem.
Theorem III.2: The SINR outage probability in the uplink of single-tier mmWave cellular networks with truncated channel inversion power control with cutoff threshold ρ o can be expressed where a v and b v are the antenna directivity parameters defined in Section II and f P (p) is defined in (1) Proof: See Appendix B Though this approximates the SINR outage probability, we find that the expression compares very well with the simulation results in Section VI-1. Furthermore, the expression here captures the user maximum power constraint contrary to the prior result on the uplink of single tier mmWave networks in [24], which is based on an unbounded power constraint. The maximum power constraint is very important in the uplink power control of mmWave network due to the significant difference in the LOS and NLOS pathloss exponent. We show the impact of the maximum power constraint in a single-tier network later in Section VI-2.

IV. UPLINK OF MULTI-TIER MMWAVE CELLULAR NETWORKS
In this section, we extend our developed model for the uplink of a single-tier mmWave cellular network to the uplink of a multi-tier mmWave cellular network. As mentioned earlier, the k th we present the distribution of transmit power for the multi-tier mmWave cellular networks.
Afterward, we derive its SINR outage probability.

A. Distribution of the Transmit Power in the Uplink of Multi-tier mmWave Cellular Networks
Similar to the single-tier network, given the cutoff threshold for the k th tier ρ k o , LOS and NLOS users located at distances greater than pathloss exponent α k L and its NLOS pathloss exponent α k N , the PDF of the transmit power of a typical active user in the uplink of the j th tier is given by where and y is a dummy variable in (14). The δ th moment of the transmit power of a user in the j th tier is given as Further, the truncation outage probability in the uplink of mmWave cellular networks for the j th tier can be obtained as Proof: See Appendix C

B. SINR Outage Probability
For an active typical user, the SINR at its connected BS in the j th tier (termed as the reference mmWave BS) can be written as where the useful signal power (normalized by G j ) is equal to ρ j o |g o | 2 due to the truncated channel inversion power control, Z k is the set of interfering users associated with a BS in the k th tier, L(D z ) is the pathloss from the interfering users to the reference BS, σ 2 is the noise power, G j z is the directivity gain on an interfering link and g z is the small-scale fading which follows a Nakagami distribution with parameter N. We consider that each of the tiers has its own SINR threshold which is represented by θ k . Further, the average received signal at any of the BSs in the j th tier is equivalent to the cutoff threshold of the j th tier represented by ρ j o . The SINR outage probability of the j th tier can be expressed as where I k L and I k N are the aggregate interference from LOS and NLOS users of the k th tier, respectively. Note that I j L and I j N represents the LOS and NLOS co-tier interference, respectively, and I k L and I k N ∀k = j denotes the LOS and NLOS cross-tier interference, respectively. Similar to the single tier case, noting that |g o | 2 is normalized gamma random variable with parameter N, we have the following approximation In addition, the interfering users do not constitute a PPP while the transmit power of the interfering users are also correlated as in the case of the single tier. 13 However, in order to keep analytical tractability, we approximate the interfering users process as a PPP while also ignoring the correlations in the transmit power of interfering users. The SINR outage probability of a typical user in the j th tier of mmWave cellular networks can be obtained from the following theorem.
Theorem IV.2: In a K−tier mmWave network with truncated channel inversion power control where each tier is distinguished by its density λ k , cutoff threshold ρ k o , blockage parameter β k , antenna parameters, G max bk , G min bk and ζ rk , LOS pathloss exponent α k L and NLOS pathloss exponent α k N , the SINR outage probability of a typical user in the j th tier is given by a j v and b j v are the antenna directivity parameters defined in Section II and f P k (p) is defined in (12).

Proof: See Appendix D
It has been shown in [8] that the LOS probability function can be approximated by a step function in a dense mmWave network. Hence, in the next section, we propose to simplify our uplink system model and the subsequent analysis by using a step function approximation of the LOS probability function as well.

V. ANALYSIS OF THE UPLINK OF DENSE MMWAVE NETWORKS
In this section, we present the analysis for the uplink of a dense mmWave cellular network.
The motivation for the dense network analysis is based on the fact that mmWave cellular networks must be dense in order to achieve its forecasted gain [8]. Here we approximate the LOS probability P(LOS k ) by using a step function such that the LOS probability P(LOS k ) is taken to be 1 when the link is within a circular disc B(0, R B ) centered at the reference mmWave BS and 0 when outside the disc. Next, we present uplink signal to interference ratio (SIR) distribution for the dense multi-tier mmWave networks, which we later degrade to the single tier scenario.

A. Outage Analysis in the Uplink of Dense Multi-tier mmWave Networks
The dense mmWave network will be interference limited with mainly LOS interferers limiting its performance. Hence, we ignore both the noise power and the NLOS interfering users in the analysis. Further, according to [3], the signal power from LOS interferers are nearly deterministic, hence we also ignore the small-scale fading. Consequently, the SIR at the BS that the typical user connects to in the j th tier can be expressed from (17) as As mentioned earlier, for the dense deployment, the LOS interferers are dominant and the SINR outage probability in the j tier can thus be approximated as where I k L = z∈Z k ∩B(0,R B ) P z G z L(D z ) is the interference power received at the reference BS from users connected to BSs in the k th tier. Note that the average receive signal at the reference BS normalized by the directivity gain G j is equivalent to the cutoff threshold ρ j o . The SINR outage probability can be approximated as where the dummy variable h in (a) is used to denote normalized gamma variable with parameter L. Note that the distribution of the normalized gamma variable converges to an identity when its parameters tend to infinity, (b) follows from [26] such that the probability P(|h| 2 < γ) is In the following theorem, we summarize the main result for the SIR outage distribution in a multi-tier network.
Theorem V.1: The SINR outage probability in the j th tier of a K-tier mmWave cellular network with truncated channel inversion power control can be approximated as is the upper incomplete gamma function, a j v and b j v are the antenna directivity parameters defined in Section II, η = L(L!) − 1 L , L is used in the approximation and f P k (p) is defined as Proof: See Appendix E.
For a single-tier dense mmWave network, the approximation of SINR outage probability can be obtained from the following corollary.

Corollary V.2:
The SINR outage probability in the j th tier of a K-tier mmWave cellular network with truncated channel inversion power control can be approximated as Proof: The proof follows directly from proof of the multi-tier mmWave cellular networks and is omitted here.

VI. NUMERICAL RESULTS
In this section, we present numerical results to illustrate our analytical findings for both the single-tier and two-tier mmWave cellular networks. Unless otherwise stated, we set the BS densities λ 1 = 10 BS/km 2 and λ 2 = 2λ 1 , the maximum transmit power P u = 1 W, receiver sensitivity, and the BS antenna gain. Note that this validation is essential since the cumulative distribution of the SINR is based on the assumption that the active user constitute a PPP and that the transmit powers of the users are independent. Further, independent LOS probability was assumed. The approach in this paper, however, captures the correlation between the location of the reference mmWave BS and that of the interfering users. It also captures the correlation between the typical user's (served by the reference mmWave BS) transmit power and the interfering users' transmit powers.
2) Comparison with the case without maximum power constraint: In Fig. 3, we compare our analysis with the one presented in [24], which does not incorporate the maximum user power constraint. Note that the analysis in [24] is for the single-tier, and hence the comparison presented in Fig. 3 is also based on a single tier. As it can be seen, the maximum power constraint significantly affects the SINR outage probability. The figure shows that the SINR outage probability derived in [24] does not vary with P u , since the maximum power constraint is ignored in [24]. In Fig. 4, we plot the SINR outage probability for the uplink of a single-tier mmWave networks with truncated channel inversion power control for SINR threshold θ = 20, 25 and 30dB, and BS densities λ 1 = 10, 100 BS/km 2 . It can be seen that the SINR  outage probability of mmWave deviates from that of the UHF network presented in [16]. More specifically, four sections can be identified from the plot for the BS density λ 1 = 10 BS/km 2 :

3) Effect of the BS Density:
1) a decrease in SINR outage probability can be seen for the cutoff threshold ρ 1 o ranging from −100 to −50 dBm with a slow descent region observed for ρ 1 0 ranging from −85 to −75 dBm; 2) a fairly stable outage probability can be observed for ρ 1 o ranging from −50 to −31 dBm; 3) an increase in SINR outage probability can be seen for ρ 1 o ranging from −31 to −18 dBm, and 4) a decrease in SINR outage probability can be seen for ρ 1 o ranging from −18 to 0 dBm. This observation is as a result of the large difference in the pathloss exponent of the LOS and NLOS propagation path, with each having its dominance region which also depends on the BS density and blockage parameter. The latter specifies the LOS range. Further, the receiver sensitivity also specifies the density of active LOS and NLOS users and consequently, the interference received at the reference BS. It can also be observed from Fig. 4 that for the same SINR threshold, increasing the BS density leads to an increase in the SINR outage probability. Fig. 5, we show the effect of blockages on the SINR outage probability. Based on the LOS probability function e −β 1 r , a lower β 1 yields a larger number of LOS interfering user. Hence, the interference power increases when β 1 is lowered leading to a higher SINR outage probability for a lower β 1 , as it can be seen in Fig. 5. Fig. 6 compares the truncation outage probability for the uplink of mmWave and UHF cellular networks for BS density λ = 1, 10 and 100 BS/km 2 .

5) Truncation Outage Probability:
The truncation outage probability of UHF networks has been defined in [16]. It can be seen that similar to the UHF case, increasing the cutoff threshold increases the outage probability since more users are unable to communicate due to insufficient transmit power. These results are in line with the insights previously drawn on equations (3) and (5). Furthermore, for BS densities λ = 1, 10, the truncation outage of mmWave networks experience a slow growth region as the cutoff threshold increases before its saturation contrary to UHF networks, which does not experience a slow growth region. The slow growth region is due to the difference in the truncation outage probability for LOS and NLOS links at a given cutoff threshold. Meanwhile, for a high BS density of λ = 100, the truncation outage probability of mmWave converges to that of UHF with α = 2 since more paths becomes LOS as the BS density increases. As expected, Fig. 6 shows that the truncation outage of mmWave networks reduces with as the BS density increases. This observation is due to the shortening of the average link lengths as the BS density is increased. Fig. 7 shows the effect of blockages on the truncated outage probability. As the average line of sight, which is proportional to the inverse of the blockage parameter value 1 β , increases the truncation outage probability reduces as much lower transmit power is required to meet the receiver sensitivity requirement when the density of blockages and the average size of blockages are much lower. These results are also in line with the insights previously drawn on equations The dense network approximation becomes more accurate as L increases. Further, Fig. 10, compares the multi-tier dense network approximation in Theorem V.2 with the exact expression in Theorem IV.2 while focusing on a two-tier network. The first tier's BS density is obtained from λ 0 = λ 1 πR 2 B while the second tier's BS density λ 2 = 2λ 1 and L equals 10. It can be seen that similar to the single-tier network, the dense network approximation of the SINR outage probability is also fairly accurate for the multi-tier network. Fig. 11, we plot the average transmit power of the users against the cutoff threshold ρ 1 0 for the single-tier mmWave network. It can be observed that for the case with BS density λ = 1, the average transmit power increases with the cutoff threshold for ρ 1 o ranging from −100 to −75 dBm and it then falls for ρ 1 o ranging from −75 to −40 dBm. Afterwards average transmit power then rises with the cutoff threshold until its saturation. Note that increasing ρ 1 o the user need to transmit a higher power to invert the pathloss and maintain a high threshold at the serving BS [16]. However, each user is constrained to a maximum power Given that the pathloss exponent α is a random variable which takes the values α L and α N with probability e −βr and 1 − e −βr , respectively, where r is the length of the link. Because the transmit power of a typical user P s = ρ o r αs s for s = {L, N}, r s ∈ Φ is a function mapping r s in R 2 to P s in R, according to the Mapping theorem [28, Thm 2.34], P s forms an inhomogeneous PPP with the intensity measure Λ(P ) = Λ L (P ) + Λ N (P ), where

8) Average User Transmit Power: In
[21] and the intensity function λ(P ) = ∂Λ(P ) ∂P = λ L (P ) + λ N (P ), where and λ N = 2πλ Therefore, the PDF of P is given by The PDF of P has been normalized as a result of the truncated channel inversion power control.
Furthermore, the η th moment of P is thus given by Pu 0 p η f P (p) and the theorem is obtained.

B. Proof of Theorem III.2
Noting that the average interference received from any interfering user (normalized by G) is strictly less than ρ o . Consequently, the sum interference received at the reference BS from LOS interferers can be expressed from (6) as where Φ L is a PPP of LOS interferers, and 1 1(.) is an indicator function which takes the values of one when (.) is true and zero otherwise. Consequently, the Laplace transform of the aggregate interference from LOS interferer L I L in (8) can be computed as where (b1) follows from the independence of Φ L , g z , G z and P z , (b2) follows from the probability generation functional (PGFL) of the PPP [9] and the independence of the interference link directivity gain G z with probability distribution G z = a v with probability b v , (b3) follows from from computing the moment generating function of a gamma random variable g, (b4) is obtained by changing the variables y = r/(sna v P ) 1 α L while f P is given in (1).
Similarly, the for the NLOS interfering links, L I N can be computed as where B = (sna v ρ) Given that y k = min m k ∈Φ k (||u − m k || α k s ) is used to select the serving BS in the k th tier. Then and Noting that the user connects to the BS that provides the maximum average received signal, we can follow the same approach for the multi-tier UHF network in [16]. The typical user u connects to the reference BS from the j th tier, then y j = min k (y k ). The transmit power of the typical user connected to the reference user in the j th tier is given by P j = ρ j o min k (y k ) where P j ≤ P u . Consequently, the cumulative distribution function (CDF) of the transmit power can be expressed as and the PDF of the transmit power is given as f P j (p) = dF P j (p) dp (38) Noting that the average interference received from any interfering user from the k th tier is less than ρ k o . The sum interference received at the reference BS in the j th tier from LOS users in the k th can be expressed as where Φ k L is a PPP of LOS interfering users from the k th tier. The indicator function is used to capture the correlation among the location of the interfering users and the location of the reference BS. Hence, the Laplace transform of the aggregate interference from LOS users in the k th tier received by the reference mmWave BS in the j th tier L I k L can be computed as Similarly, the Laplace transform of the aggregate interference from NLOS users in the k th tier received by the reference BS in the j th tier L I k N can be expressed as The proof is based on the key assumptions in Appendix B such that where (e1) follows from computing the PGFL of the PPP Φ k L , (e2) follow from a change of variable w = sla j v P k r −α j L and (e3) follows from the simplified PDF of the transmit power over the LOS region. The PDF of the uplink transmit power in the k th tier of dense mmWave networks can be obtained by noting that P k = ρ k o r α L k , where r k is the k th tier uplink distance in the dense deployment which follows a Rayleigh distribution f r k (r) = 2πλre −πλr 2 , 0 ≤ r ≤ ∞.
Consequently, following the same approach in Appendix C we obtain the PDF of the k th tier transmit power as Finally, the approximation of the SINR outage probability in the uplink of a multi-tier mmWave network given in (23) can be obtained by substituting (43) for (22) and with λ 0 = λπR 2 B .