Coverage and Rate Analysis for Millimeter Wave Cellular Networks

Coverage and Rate Analysis for Millimeter Wave Cellular Networks

Abstract

Millimeter wave (mmWave) holds promise as a carrier frequency for fifth generation cellular networks. Because mmWave signals are sensitive to blockage, prior models for cellular networks operated in the ultra high frequency (UHF) band do not apply to analyze mmWave cellular networks directly. Leveraging concepts from stochastic geometry, this paper proposes a general framework to evaluate the coverage and rate performance in mmWave cellular networks. Using a distance-dependent line-of-site (LOS) probability function, the locations of the LOS and non-LOS base stations are modeled as two independent non-homogeneous Poisson point processes, to which different path loss laws are applied. Based on the proposed framework, expressions for the signal-to-noise-and-interference ratio (SINR) and rate coverage probability are derived. The mmWave coverage and rate performance are examined as a function of the antenna geometry and base station density. The case of dense networks is further analyzed by applying a simplified system model, in which the LOS region of a user is approximated as a fixed LOS ball. The results show that dense mmWave networks can achieve comparable coverage and much higher data rates than conventional UHF cellular systems, despite the presence of blockages. The results suggest that the cell size to achieve the optimal SINR scales with the average size of the area that is LOS to a user.

1Introduction

The large available bandwidth at frequencies makes them attractive for fifth generation cellular networks [3]. The band ranging from 30 GHz to 300 GHz has already been considered in various commercial wireless systems including IEEE 802.15.3c for personal area networking [6], IEEE 802.11ad for local area networking [7], and IEEE 802.16.1 for fixed-point access links [8]. Recent field measurements reveal the promise of signals for the access link (between the mobile station and base station) in cellular systems [9].

One differentiating feature of cellular communication is the use of antenna arrays at the transmitter and receiver to provide array gain. As the wavelength decreases, antenna sizes also decrease, reducing the antenna aperture. For example, from the Friis free-space equation [10], a signal at 30 GHz will experience 20 dB larger path loss than a signal at 3 GHz. Thanks to the small wavelength, however, it is possible to pack multiple antenna elements into the limited space at transceivers [3]. With large antenna arrays, cellular systems can implement beamforming at the transmitter and receiver to provide array gain that compensates for the frequency-dependent path loss, overcomes additional noise power, and as a bonus also reduces out-of-cell interference [4].

Another distinguishing feature of cellular communication is the propagation environment. MmWave signals are more sensitive to blockage effects than signals in lower-frequency bands, as certain materials like concrete walls found on building exteriors cause severe penetration loss [11]. This indicates that indoor users are unlikely to be covered by outdoor base stations. Channel measurements using directional antennas [12] have revealed another interesting behavior at : blockages cause substantial differences in the paths and path loss characteristics. Such differences have also been observed in prior propagation studies at ultra high frequency bands (UHF) from 300 MHz to 3 GHz, e.g. see [13]. The differences, however, become more significant for since diffraction effects are negligible [4], and there are only a few scattering clusters [14]. Measurements in [12] showed that mmWave signals propagate as in free space with a path loss exponent of 2. The situation was different for paths where a log distance model was fit with a higher path loss exponent and additional shadowing [9]. The path loss laws tend to be more dependent on the scattering environment. For example, an exponent as large as 5.76 was found in downtown New York City [5], while only 3.86 was found on the UT Austin campus [9]. The distinguishing features of the propagation environment need to be incorporated into the any comprehensive system analysis of networks.

The performance of cellular networks was simulated in prior work [14] using insights from propagation channel measurements [5]. In [15], using the path loss law measured in the New York City, lower bounds of the distribution and the achievable rate were simulated in a 28 GHz pico-cellular system. In [14], a mmWave channel model that incorporated blockage effects and angle spread was proposed and further applied to simulate the mmWave network capacity. Both results in [14] show that the achievable rate in mmWave networks can outperform conventional cellular networks in the ultra high frequency (UHF) band by an order-of-magnitude. The simulation-based approach [15] does not lead to elegant system analysis as in [16], which can be broadly applied to different deployment scenarios.

Stochastic geometry is a useful tool to analyze system performance in conventional cellular networks [16]. In [16], by modeling base station locations in a conventional cellular network as a Poisson point process (PPP) on the plane, the aggregate coverage probability was derived in a simple form, e.g. a closed-form expression when the path loss exponent is 4. Moreover, the stochastic model was shown to provide a lower bound of the performance in a real cellular system [16]. There have been several extensions of the results in [16], such as analyzing a multi-tier network in [18] and predicting the site-specific performance in heterogeneous networks in [19]. It is not possible to directly apply results from conventional networks to networks due to the different propagation characteristics and the use of directional beamforming. There has been limited application of stochastic geometry to study cellular networks. The primary related work was in [20], where directional beamforming was incorporated for single and multiple user configurations, but a simplified path loss model was used that did not take propagation features into account.

A systematic study of network performance should incorporate the impact of blockages such as buildings in urban areas. One approach is to model the blockages explicitly in terms of their sizes, locations, and shapes using data from a geographic information system. This approach is well suited for site-specific simulations [21] using electromagnetic simulation tools like ray tracing [22]. An alternative is to employ a stochastic blockage model, e.g. [23], where the blockage parameters are drawn randomly according to some distribution. The stochastic approach lends itself better to system analysis and can be applied to study system deployments under a variety of blockage parameters such as size and density.

The main contribution of this paper is to propose a stochastic geometry framework for analyzing the coverage and rate in cellular networks. As a byproduct, the framework also applies to analyze heterogenous networks in which the base stations are distributed as certain non-homogeneous PPPs. We incorporate directional beamforming by modeling the beamforming gains as marks of the base station PPPs. For tractability of the analysis, the actual beamforming patterns are also approximated by a sectored model, which characterizes key features of an antenna pattern: directivity gain, half-power beamwidth, and front-back ratio. A similar model was also employed in work on ad hoc networks [25]. To incorporate blockage effects, we model the probability that a communication link is as a function of the link length, and provide a stochastic characterization of the region where a user does not experience any blockage, which we define as the LOS region. Applying the distance-dependent probability function, the base stations are equivalently divided into two independent non-homogenous point processes on the plane: the and the base station processes. Different path loss laws and fading are applied separably to the and case. Based on the system model, expressions for the and rate coverage probability are derived in general mmWave networks. To simplify the analysis, we also propose a systematic approach to approximate a complicated LOS function as its equivalent step function. Our analysis indicates that the coverage and rate are sensitive to the density of base stations and the distribution of blockages in mmWave networks. It also shows that dense mmWave networks can generally achieve good coverage and significantly higher achievable rate than conventional cellular networks.

A simplified system model is proposed to analyze dense mmWave networks, where the infrastructure density is comparable to the blockage density. For a general function, the LOS region observed by a user has an irregular and random shape. Coverage analysis requires integrating the SINR over this region [1]. We propose to simplify the analysis by approximating the actual region as a fixed-sized ball called the equivalent LOS ball. The radius of the equivalent LOS ball is chosen so that the ball has the same average number of base stations in the network. With the simplified network model, we find that in a dense mmWave network, the cell radius should scale with the size of region to maintain the same coverage probability. We find that continuing to increase base station density (leading to what we call ultra-dense networks) does not always improve SINR, and the optimal base station density should be finite.

Compared with our prior work in [1], this paper provides a generalized mathematical framework and includes the detailed mathematical derivations. The system model applies for a general probability function and includes the impact of general small-scale fading. We also provide a new approach to compute coverage probability, which avoids inverting the Fourier transform numerically and is more efficient than prior expressions in [1]. Compared with our prior work in [2], we also remove the constraint that the path loss exponent is 2, and extend the results in [2] to general path loss exponents, in addition to providing derivations for all results, and new simulation results.

This paper is organized as follows. We introduce the system model in Section 2. We derive expressions for the and rate coverage in a general mmWave network in Section 3. A systematic approach is also proposed to approximate general probability functions as a step function to further simplify analysis. In Section 4, we apply the simplified system model to analyze performance and examine asymptotic trends in dense mmWave networks, where outdoor users observe more than one base stations with high probability. Finally, conclusions and suggestions for future work are provided in Section 6.

2System Model

In this section, we introduce our system model for evaluating the performance of a network. We focus on downlink coverage and rate experienced by an outdoor user, as illustrated in Fig. ?. We make the following assumptions in our mathematical formulation.

We say that a base station at is to the typical user at the origin if and only if there is no blockage intersecting the link . Due to the presence of blockages, only a subset of the outdoor base stations are to the typical user.

The probability function in a network can be derived from field measurements [14] or stochastic blockage models [24], where the blockage parameters are characterized by some random distributions. For instance, when the blockages are modeled as a rectangle boolean scheme in [24], it follows that , where is a parameter determined by the density and the average size of the blockages, and is what we called the average range of the network in [24].

For the tractability of analysis, we further make the following independent assumption on the LOS probability; taking account of the correlations in blockage effects generally makes the exact analysis difficult.

Note that the probabilities for different links are not independent in reality. For instance, neighboring base stations might be blocked by a large building simultaneously. Numerical results in [24], however, indicated that ignoring such correlations cause a minor loss of accuracy in the evaluation. Assumption ? also indicates that the base station process and the process form two independent non-homogeneous PPP with the density functions and , respectively, where is the radius in polar coordinates.

By Assumption ? and Assumption ?, the directivity gain in an interference link is a discrete random variable with the probability distribution as with probability , where and are constants defined in Table 1, , and .

Table 1: Probability Mass Function of and
k 1 2 3 4

Measurement results indicated that small-scale fading at is less severe than that in conventional systems when narrow beam antennas are used [5]. Thus, we can use a large Nakagami parameter to approximate the small-variance fading as found in the LOS case. Let be the thermal noise power normalized by . Based on the assumptions thus far, the received by the typical user can be expressed as

Note that the SINR in (2) is a random variable, due to the randomness in the base station locations , small-scale fading , and the directivity gain . Using the proposed system model, we will evaluate the mmWave and rate coverage in the following section.

3Coverage and Rate Analysis in General Networks

In this section, we analyze the coverage and rate in the proposed model of a general network. First, we provide some ordering results regarding different parameters of the antenna pattern. Then we derive expressions for the SINR and rate coverage probability in mmWave networks with general probability function . To simplify subsequent analysis, we then introduce a systematic approach to approximate by a moment matched equivalent step function.

3.1Stochastic Ordering of SINR With Different Antenna Geometries

One differentiating feature of cellular networks is the deployment of directional antenna arrays. Consequently, the performance of networks will depend on the adaptive array pattern through the beamwidth, the directivity gain, and the back lobe gain. In this section, we establish some results on stochastic ordering of the SINRs in the systems with different antenna geometries. While we will focus on the array geometry at the transmitter, the same results, however, also apply to the receiver array geometry. The concept of stochastic ordering has been applied in analysis of wireless systems [27]. Mathematically, the ordering of random variables can be defined as follows [27].

Next, define the at the transmitter as the ratio between the main lobe directivity gain and the back lobe gain , i.e., . We introduce the key result on stochastic ordering of the SINR with respect to the directivity gains as follows.

Proof:

From Definition ?, we need to show that for each realization of base station locations , small-scale fading , and angles and , the value of the SINR increases with and . Given , , , and , we can normalize both the numerator and denominator of (Equation 1) by , and then write where with probability , and , are constants defined in Table 1. Note that is independent of , and is a non-increasing function of . Hence, when is fixed, larger provides larger ; when is fixed, larger provides larger .

Next, we provide the stochastic ordering result regarding beamwidth as follows.

The proposition can be rigorously proved using coupling techniques. We omit the proof here and instead provide an intuitive explanation as below. Intuitively, with narrower main lobes, fewer base stations will transmit interference to the typical user via their main lobes, which gives a smaller interference power. The desired signal term in (Equation 1) is independent of the beamwidth, as we ignore the channel estimation errors and potential angle spread. Hence, based on our model assumptions, smaller beamwidths provide a better SINR performance.

We note that the ordering result in Proposition ? assumes that there is no angle spread in the channel. With angle spread, a narrow-beam antenna may capture only the signal energy arriving inside its main lobe, missing the energy spread outside, which causes a gain reduction in the signal power [29]. Consequently, the results in Proposition ? should be interpreted as applying to the case where beamwidths are larger than the angle spread, e.g. if the beamwidth is more than per the measurements in [12]. We defer more detailed treatment of angle spread to future work.

3.2SINR Coverage Analysis

The coverage probability is defined as the probability that the received is larger than some threshold , i.e., We present the following lemmas before introducing the main results on SINR coverage. By Assumption ?, the outdoor base station process can be divided into two independent non-homogeneous PPPs: the base station process and process . We will equivalently consider and as two independent tiers of base stations. As the user is assumed to connect to the base station with the smallest path loss, the serving base station can only be either the nearest base station in or the nearest one in . The following lemma provides the distribution of the distance to the nearest base station in and .

Proof:

The proof follows [24] and is omitted here.

Next, we compute the probability that the typical user is associated with either a or a base station.

Proof:

See Appendix B.

Further, conditioning on that the serving base station is (or NLOS), the distance from the user to its serving base station follows the distribution given in the following lemma.

Proof:

The proof follows a similar method as that of Lemma ?, and is omitted here.

Now, based on Lemma ? and Lemma ?, we present the main theorem on the coverage probability as follows

Proof:

See Appendix C. Though as an approximation of the SINR coverage probability, we find that the expressions in Theorem ? compare favorably with the simulations in Section 5.1. In addition, the expressions in Theorem ? compute much more efficiently than prior results in [1], which required a numerical inverse of a Fourier transform. Last, the LOS probability function may itself have a very complicated form, e.g. the empirical function for small cell simulations in [13], which will make the numerical evaluation difficult. Hence, we propose simplifying the system model by using a step function to approximate in Section . Before that, we introduce our rate analysis results in the following section.

3.3Rate Analysis

In this section, we analyze the distribution of the achievable rate in networks. We use the following definition for the achievable rate

where is the bandwidth assigned to the typical user, and is a SINR threshold determined by the order of the constellation and the limiting distortions from the RF circuit. The use of a distortion threshold is needed because of the potential for very high SINRs in that may not be exploited due to other limiting factors like linearity in the radio frequency front-end.

The average achievable rate can be computed using the following Lemma from the coverage probability .

Proof:

See [16] and [20].

Lemma ? provides a first order characterization of the rate distribution. We can also derive the exact rate distribution using the rate coverage probability , which is the probability that the achievable rate of the typical user is larger than some threshold : The rate coverage probability can be evaluated through a change of variables as in the following lemma.

Proof:

The proof is similar to that of [30]. For , it directly follows that . Lemma ? will allow comparisons to be made between mmWave and conventional systems that use different bandwidths, as presented in Section 5.1.

3.4Simplification of Probability Function

The expressions in Theorem ? generally require numerical evaluation of multiple integrals, and may become difficult to analyze. In this section, we propose to simplify the analysis by approximating a general probability function by a step function. We denote the step function as , where when , and otherwise. Essentially, the LOS probability of the link is taken to be one within a certain fixed radius and zero outside the radius. An interpretation of the simplification is that the irregular geometry of the LOS region in Fig. ? (a) is replaced with its equivalent ball in Fig. ? (b). Such simplification not only provides efficient expressions to compute , but enables simpler analysis of the network performance when the network is dense.

We will propose two criterions to determine the given probability function . Before that, we first review some useful facts.

Proof:

The average number of base stations can be computed as

where (a) follows directly from Campbell’s formula of PPP [26]. A direct corollary of Theorem ? follows as below.

Note that Theorem ? also indicates that a typical user will observe a finite number of base stations almost surely when . Hence, if satisfies , the parameter in can be determined by matching the average number of base stations a user may observe.

In the case where is not satisfied, another criterion to determine is needed. Note that even if the first moment is infinite, the probability that the user is associated with a base station exists and is naturally finite for all . Hence, we propose the second criterion regarding the association probability as follows.

From Lemma ?, the association probability for a step function equals . Hence, by Criterion ?, can be determined as .

Last, we explain the physical meaning of the step function approximation as follows. As shown in Fig. ?(a), with a general probability function , the buildings are randomly located, and thus the actual region observed by the typical user may have an unusual shape. Although it is possible to incorporate such randomness of the size and shape by integrating over , the expressions with multiple integrals can make the analysis and numerical evaluation difficult [1]. In Fig. ?(b), by approximating the probability function as a step function , we equivalently approximate the region by a fixed ball , which we define as the equivalent ball. As will be shown in Section 4, approximating as a step function enables fast numerical computation, simplifies the analysis, and provides design insights for dense network. Besides, we will show in simulations in Section 5.1 that the error due to such approximation is generally small in dense mmWave networks, which also motivates us to use this first-order approximation of the LOS probability function to simplify the dense network analysis in the following section.

4Analysis of Dense Networks

In this section we specialize our results to dense networks. This approach is motivated by subsequent numerical results in Section 5.1 that show deployments will be dense if they are expected to achieve significant coverage. We derive simplified expressions for the and provide further insights into system performance in this important asymptotic regime.

4.1Dense Network Model

In this section, we build the dense network model by modifying the system model in Section 2 with a few additional assumptions. We say that a cellular network is dense if the average number of LOS base stations observed by the typical user is larger than , or if its LOS association probability is larger than , where and are pre-defined positive thresholds. In this paper, for illustration purpose, we will let and . Further, we say that a network is ultra-dense when . Note that also equals the relative base station density normalized by the average LOS area, in this special case, as we will explain below.

Now we make some additional assumptions that will allow us to further simplify the network model.

By Assumption ?, the probability function is approximated by its equivalent step function , and the base station process is made up of the outdoor base stations that are located inside the ball . Noting that the outdoor base station process is a homogeneous PPP with density , the average number of LOS base stations is , which is the outdoor base station density times the area of the LOS region. For ease of illustration, we call the the relative density of a mmWave network. The relative density is equivalently: (i) the average number of base stations that a user will observe, (ii) the ratio of the average LOS area to the size of a typical cell [26], and (iii) the normalized base station density by the size of the LOS ball. We will show in the next section that the SINR coverage in dense networks is largely determined by the relative density .

We show later in the simulations that ignoring NLOS base stations and the thermal noise introduces a negligible error in the performance evaluation.

Based on the dense network model, the signal-to-interference ratio (SIR) can be expressed as

Now we compute the SIR distribution in the dense network model.

4.2Coverage Analysis in Dense Networks

Now we present an approximation of the SINR distribution in a mmWave dense network. Our main result is summarized in the following theorem.

Proof:

See Appendix D. When , the expression in Theorem ? can be further simplified as follows.

The results in Theorem ? generally provide a close approximation of the SINR distribution when enough terms are used, e.g. when , as will be shown in Section 5.2. More importantly, we note that the expressions in Theorem ? are very efficient to compute, as most numerical tools support fast evaluation of the gamma function in ( ?), and ( ?) only requires a simple integral over a finite interval. Besides, given the path loss exponent and the antenna geometry , , Theorem ? shows that the approximated SINR is only a function of the relative density , which indicates the SIR distribution in a dense network is mostly determined on the average number of LOS base station to a user.

4.3Asymptotic Analysis in Ultra-Dense Networks

To obtain further insights into coverage in dense networks, we provide results on the asymptotic SIR distribution when the relative density becomes large. We use this distribution to answer the following questions: (i) What is the asymptotic SIR distribution when the network becomes extremely dense? (ii) Does increasing base station density always improve SIR in a mmWave network?

First, we present the main asymptotic results as follows.

Proof:

See Appendix E. Note that Theorem ? indicates that increasing base station density above some threshold will hurt the system performance, and that the SINR optimal base station density is finite.

Now we provide an intuitive explanation of the asymptotic results as follows. When increasing the base station density, the distances between the user and base stations become smaller, and the user becomes more likely to be associated with a base station. When the density is very high, however, a user sees several base stations and thus experiences significant interference.

We note that the asymptotic trends in Theorem ? are valid when base stations are all assumed to be active in the network. A simple way to avoid “over-densification” is to simply turn off a fraction of the base stations. This is a simple kind of interference management; study of more advanced interference management concepts is an interesting topic for future work.

5Numerical Simulations

In this section, we first present some numerical results based on our analyses in Section 3 and Section 4. We conclude with some simulations using real building distributions to validate our proposed mmWave network model.

5.1General Network Simulations

In this section, we provide numerical simulations to validate our analytical results in Section 3, and further discuss their implications on system design. We assume the network is operated at 28 GHz, and the bandwidth assigned to each user is MHz. The and path loss exponents are and . The parameters of the Nakagami fading are and . We assume the LOS probability function is , where meters. For the ease of illustration, we define the notion of the average cell radius of a network as follows. Note that if the base station density is , the average cell size in the network is [26]. Therefore, the average cell radius in a network is defined as the radius of a ball that has the size of an average cell, i.e., . The average cell radius not only directly relates to the inter-site distance that is used by industry in base station planning, but also equivalently characterizes the base station density in a network; as a large average cell size indicates a low base station density in the network.

[!ht]

Figure 1:   coverage probability with different antenna geometry. The average cell radius is r_\mathrm c=100 meters. The receiver beam pattern is fixed as G_{10\mbox{dB},-10\mbox{dB},90^\circ}.
Figure 1: coverage probability with different antenna geometry. The average cell radius is meters. The receiver beam pattern is fixed as .

[!ht]

Figure 2:  LOS association probability with different average cell radii. The lines are drawn from Monte Carlos simulations, and the marks are drawn based on Lemma .
Figure 2: LOS association probability with different average cell radii. The lines are drawn from Monte Carlos simulations, and the marks are drawn based on Lemma .

First, we compare the coverage probabilities with different transmit antenna parameters in Figure 1 using Monte Carlos simulations. As shown in Figure 1, when the side lobe gain is fixed, better SINR performance is achieved by increasing main lobe gain and by decreasing the main lobe beamwidth , as indicated by the analysis in Section 3.1.

[!ht]

Figure 3:  Comparison of the  coverage between using p(x) and its equivalent step function S_{R_\mathrm{B}}(x). The transmit antenna pattern is assumed to be G_{20\mbox{dB},-10\mbox{dB},30^\circ}. It shows that the step function tends to provide a more pessimistic SINR coverage probability, but the gap becomes smaller as the network becomes more dense.
Figure 3: Comparison of the coverage between using and its equivalent step function . The transmit antenna pattern is assumed to be . It shows that the step function tends to provide a more pessimistic SINR coverage probability, but the gap becomes smaller as the network becomes more dense.

Next, we compare the LOS association probabilities with different average cell radii in Figure 2. The results show that the probability that a user is associated with a LOS base station increases as the cell radius decreases. The results in Figure 2 also indicate that the received signal power will be mostly determined by the distribution of LOS base stations in a sufficiently dense network, e.g. when the average cell size is smaller than 100 meters in the simulation.

We also compare the coverage probability with different cell radii in Fig. ?. The numerical results in Fig. ? (a) show that our analytical results in Theorem ? match the simulations well with negligible errors. Unlike in a interference-limited conventional cellular network, where SINR is almost invariant with the base station density [16], the SINR coverage probability is also shown to be sensitive to the base station density in Fig. ?. The results in Fig. ? (a) also shows that networks generally require a small cell radius (equivalently a high base station density) to achieve acceptable coverage. Moreover, the results in Fig. ? (b) show that when decreasing average cell radius (i.e., increasing base station density), mmWave networks will transit from power-limited regime into interference-limited regime; as the SIR curves will converge to the SINR curve when densifying the network.

Specifically, comparing the curves for meters and meters in Fig. ? (a), we find that increasing base station density generally improve the SINR in a sparse network; as increasing base station density will increase the LOS association probability and avoid the presence of coverage holes, i.e. the cases that a user observes no LOS base stations. A comparison of the curves for meters and meters, however, also indicates that increasing base station density need not improve SINR, especially when the network is already sufficiently dense. Intuitively, increasing base station density also increases the likelihood to be interfered by strong LOS interferers. In a sufficiently dense network, increasing base station will harm the SINR by adding more strong interferers.

[!ht]

Figure 4:  Rate coverage comparison between mmWave and conventional cellular networks. The mmWave transmit antenna pattern is assumed to be G_{10\mbox{dB},-10\mbox{dB},30^\circ}. We assume the conventional system is operated at 2 GHz with a cell radius of 500 m, and the transmit power of the conventional base station is 46 dBm.
Figure 4: Rate coverage comparison between mmWave and conventional cellular networks. The mmWave transmit antenna pattern is assumed to be . We assume the conventional system is operated at 2 GHz with a cell radius of 500 m, and the transmit power of the conventional base station is dBm.

Now we apply Theorem 3 to compare the SINR coverage with different LOS probability functions . We approximate the negative exponential function by its equivalent step function . Applying either of the criteria in Section 3.4, the radius of the equivalent LOS ball equals 200 meters. As shown in Figure 3, the step function approximation generally provides a lower bound of the actual SINR distribution, and the errors due to the approximation become smaller when the base station density increases. The approximation of step function also enables faster evaluations of the coverage probability, as it simplifies expressions for the numerical integrals.

We provide rate results in Figure 4, where the lines are drawn from Monte Carlos simulations, and the marks are drawn based on Lemma ?. In the rate simulation, we assume that 64 QAM is the highest constellation supported in the networks, and thus the maximum spectrum efficiency per data stream is 6 bps/Hz. In Figure 4, we compare the rate coverage probability between the mmWave network and a conventional network operated at 2 GHz. The mmWave bandwidth is 100 MHz (which conceivably could be much larger, e.g. 500 MHz [4]), while we assume the conventional system has a basic bandwidth of 20 MHz, which can be potentially extended to 100 MHz by enabling carrier aggregation [33]. Rayleigh fading is assumed in the UHF network simulations. We further assume that conventional base stations have perfect channel state information, and apply spatial multiplexing (44 single user MIMO with zero-forcing precoder) to transmit multiple data streams. More comparison results with other techniques can be found in [34]. Results in Figure 4 shows that, due to the favorable SINR distribution and larger available bandwidth at mmWave frequencies, the mmWave system with a sufficiently small average cell size outperforms the conventional system in terms of providing high data rate coverage.

5.2Dense Network Simulations

Now we show the simulation results based on the dense network analysis in Section 4. First, we illustrate the results in Theorem ? with the simulations in Figure 5. In the simulations, we include the base stations and thermal noise, which were ignored in the theoretical derivation. The expression derived in Theorem ? generally provides a lower bound of the coverage probability. The approximation becomes more accurate when more terms are used in the approximation, especially when . We find that the error due to ignoring NLOS base stations and thermal noise is minor in terms of the SINR coverage probability, primarily impacting low SINRs.

Figure 5:  Coverage probability in a dense mmWave network. The mmWave transmit antenna pattern is assumed to be G_{10\mbox{dB},-10\mbox{dB},30^\circ}. We assume R_\mathrm{B}=200 m, and the relative base station density \rho=4. N is the number of terms we used to approximate the coverage probability in Theorem .
Figure 5: Coverage probability in a dense mmWave network. The mmWave transmit antenna pattern is assumed to be . We assume m, and the relative base station density . is the number of terms we used to approximate the coverage probability in Theorem .

Next, we compare the coverage probability with different relative base station density when dB. Recall that is the base station density normalized by the size of the LOS region. In ?(a), the path loss exponent is assumed to be . We compute the coverage probability from dB meters to dB with a step of 1 dB. The analytical expressions in Theorem ? are much more efficient than simulations: the plot takes seconds to finish using the analytical expression, while it approximately takes an hour to simulate 10,000 realizations at each step. As shown in Fig. ? (a), although there is some gap between the simulation and the analytical results in the ultra-dense network regime, both curves achieve their maxima at approximately , i.e., when the average cell radius is approximately 1/2 of the LOS range . Moreover, when the base station density grows very large, the coverage probability begins to decrease, which matches the asymptotic results in Theorem ?. The results also indicate that networks in the environments with dense blockages, e.g. the downtown areas of large cities where the LOS range is small, will benefit from network densification; as they are mostly operated in the region where the relative density is (much) smaller than the optimal value , and thus increasing by densifying networks will improve SINR coverage.

We also simulate with other LOS path loss exponents in Fig. ? (b). The results show that the optimal base station density is generally insensitive to the change of the path loss exponent. When the LOS path loss exponent increases from 1.5 to 2.5, the optimal cell size is almost the same. The results also illustrate that the networks with larger path loss exponent have better SINR coverage in the ultra-dense regime when . Intuitively, signals attenuate faster with a larger path loss exponent, and thus the inter-cell interference becomes weaker, which motivates a denser deployment of base stations in the network with higher path loss.

Table 2: Achievable Rate with Different BS Densities
Carrier frequency 28 GHz 28 GHz 28 GHz 28 GHz 2 GHz 2 GHz
Base station density Ultra dense Dense Intermediate Sparse - -
Relative density 16 4 1 0.45 - -
Spectrum efficiency (bps/Hz) 5.5 5.8 4.3 2.7 4.6 4.6
Signal bandwith (MHz) 100 100 100 100 20 100
Achievable rate (Mpbs) 550 580 430 270 92 459

Finally, we compare the spectral efficiency and average achievable rates as a function of the relative density in Table 2. We find with a reasonable amount of density, e.g. when the relative density is approximately 1, the system can provide comparable spectrum efficiency as the conventional system at UHF frequencies. With high density, rates that can be achieved are an order of magnitude better than that in the conventional networks, due to the favourable SINR distribution and larger available bandwidth at mmWave frequencies.

5.3Comparison with Real-scenario Simulations

Now we compare our proposed network models with the simulations using real data. In the real-scenario simulations, we use the building distribution on the campus at The University of Texas at Austin. We also apply a modified version of the base station antenna pattern in [35] with a smaller beam width of . The directivity gain at the base station is dB. The mobile station is assumed to use uniform linear array with 4 antennas. When applying our analytical models, we fit the parameters of the LOS probability functions to match the building statistics [24], and use the sectored model for beamforming pattern. We also assume the mmWave base stations are distributed as a PPP with m. As shown in Fig. ?, though some deviations in the high SINR regime, our analytical models generally show a close characterization of the reality. The deviation is explained as follows: the proposed analytical model computes the aggregated SINR coverage probability, averaging over all realizations of building distributions over the infinite plane, while the real-scenario curve only considers a specific realization of buildings in a finite snapshot window. In this case, our model overestimates the coverage probability in the low SINR regime, and underestimates in the high SINR regime, as both signals and interference become more likely to be blocked in the real scenario simulation. We have found in other simulation examples that the reverse can also be true. Our model should be viewed as a characterization of the average distribution and does not necessarily lower or upper bound the distribution for a given realization.

6Conclusions

In this paper, we proposed a stochastic geometry framework to analyze coverage and rate in networks for outdoor users and outdoor infrastructure. Our model took blockage effects into account by applying a distance-dependent probability function, and modeling the base stations as independent inhomogeneous and point processes. Based on the proposed framework, we derived expressions for the downlink SINR and rate coverage probability in cellular networks, which were shown to be efficient in computation and also a good fit with the simulations. We further simplified the blockage model by approximating the random LOS region as a fixed-size equivalent LOS ball. Applying the simplified framework, we analyzed the performance and asymptotic trends in dense networks.

We used numerical results to draw several important conclusions about coverage and rate in mmWave networks.

  • SINR coverage can be comparable to conventional networks at UHF frequency when the base station density is sufficiently high.

  • Achievable rates can be significantly higher than in conventional networks, thanks to the larger available bandwidth.

  • The SINR and rate performance is largely determined by the relative base station density, which is the ratio of the base station density to the blockage density.

  • A transition from a power-limited regime to an interference-limited regime is also observed in mmWave networks, when increasing base station density.

  • The optimal SINR and rate coverage can be achieved with a finite base station density; as increasing base station density need not improve SINR in a (ultra) dense mmWave network.

In future work, it would be interesting to analyze the networks with overlaid microwave macrocells, as mmWave systems will co-exist with base stations operated in UHF bands. It would be another interesting topic to incorporate mmWave hardware constraints in the system analysis, and investigate the performance of mmWave networks applying analog/ hybrid beamforming [36] or using low-resolution A/D converters at the receivers [37].

Acknowledgement

The authors would like to thank Dr. Xinchen Zhang for his valuable feedback on early drafts of this paper.

Appendix A

We provide two useful inequalities in the following lemmas. The first lemma approximates the tail probability of a gamma random variable.

The following inequality will be used in the dense network analysis.

Appendix B

Proof of Lemma ?: For , let be the distance from the typical user to its nearest base station in . Note that it is possible that the user observes no base stations in . The user is associated with a base station in if and only if it has a LOS base station, and its nearest base station in has smaller path loss than that of the nearest base station in . Hence, it follows that

where is the probability that the user has at least one LOS base stations, (a) follows that by Lemma ?, and is the probability density function of . Next, note that

where denotes the ball centered at the origin of radius . Substituting (Equation 4) for (Equation 3) gives Lemma ?.

Appendix C

Proof of Theorem ?: Given that the user is associated with a base station in , by Slivnyak’s Theorem [26], the conditional coverage probability can be computed as

where and are the interference strength from the tiers of LOS and NLOS base stations, respectively. Next, noting that is a normalized gamma random variable with parameter , we have the following approximation

where , is from Lemma ? [38] in Appendix A, follows from Binomial theorem and the assumption that is an integer, and follows from the fact that and are independent. Now we apply concepts from stochastic geometry to compute the term for LOS interfering links in (Equation 5) as

where in is a normalized gamma random variable with parameter , , and for , and are defined previously in Table 1; (c) is from computing the Laplace functional of the PPP [26]; is by computing the moment generating function of a gamma random variable .

Similarly, for the NLOS interfering links, the small-scale fading term is a normalized gamma variable with parameter . Thus, we can compute as

Then, we obtain ( ?) from (Equation 5) by the linearity of integrals.

Given the user is associated with a base station, we can also derive the conditional coverage probability following same approach as that of Thus, we omit the detailed proof of ( ?) here.

Finally, by the law of total probability, it follows that

Appendix D

Proof Sketch of Theorem ?: For a general , the coverage probability can be computed as

where is the interference power given that the distance to the user’s serving base station is . Next, the probability can be approximated as

In , the dummy variable is a normalized gamma variable with parameter , and the approximation in follows from the fact that a normalized Gamma distribution converges to identity when its parameter goes to infinity, i.e., [39], where is the Dirac delta function. In , it directly follows from Lemma ? by taking .

Next, we can compute as

where (c) follows from computing the Laplace functional of the PPP [26], and (d) follows from changing variable as . Hence, ( ?) directly follows from substituting ( ?) for (Equation 6) and letting .

When , the steps above hold true till (Equation 7), which can be further simplified as

where is from computing integration by part, (f) follows from Lemma ? by letting . Lastly, ( ?) follows from substituting (Equation 8) for (Equation 6) and letting .

Appendix E

Proof Sketch of Theorem ?: First, we show that the exact SIR distribution in our dense network model is unchanged when the relative density is fixed. Let be the path loss gain in the -th link. Then the SIR expression in (Equation 2) can be rewritten as , where is the directivity gain in the -th link. Further, using displacement theorem [26] and the method in [1], we can show that forms a PPP on the interval with density measure function . Next, for , we define an class of equivalent networks as the networks with base station density and LOS range . Note that all networks in the equivalent class have the same relative density as the original network. Then we can show that a scaled version of path loss gain process in an equivalent network has the exact same distribution, more specifically, the measure density function, as in the original network. Thus, all equivalent networks have the same SIR distribution as the original network; as the scaling constant cancels in the SIR expression. So far, we have shown that the SIR distribution in dense networks is unchanged when the relative density is fixed.

Given the SIR invariant property with respect to , one way to investigate the asymptotic SIR when is to fix as a constant and examine the SIR performance when . In other words, when , the asymptotic SIR in the original network has the same distribution as the SIR in its asymptotic equivalent network, which has a base station density of , and an infinitely large . Let be the SIR in the asymptotic equivalent network. Note that in the asymptotic equivalent network, base stations form a homogeneous PPP with density on the entire plane.

Next, for any realization of the path loss process , the SIR in the asymptotic equivalent network can be lower and upper bounded by assuming all interfering links achieve the maximum directivity gain and the minimum directivity gain , respectively, as

For , an lower bound of the SIR coverage probability in the asymptotic equivalent network can be computed as for ,

where follows from Proposition 10 in [31].

Finally, for , we show that the upper bound of in (Equation 9) converges to zero in probability. By [40], the distribution of the upper bound expression in (Equation 9) is invariant with the small-scale fading distribution in the asymptotic equivalent network, and thus has the same distribution as in the Rayleigh-fading network investigated in [18], where independent Rayleigh fading is assumed in each links. Consequently, we compute the coverage probability for the upper bound in (Equation 9) as follows: for all ,

where , is obtained from (Equation 9), (c) is from the union bound, (d) follows from the fact that the upper bound in (Equation 9) has the same distribution as in the Rayleigh-fading network in [18], and the similar algebra in [18], and (e) is from the fact that is infinity when . By now, we have shown that an upper bound of SIR in the asymptotic equivalent network, which also upper bounds the SIR in the original network when , converges to zero in probability.

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