Coverage Analysis of IRS-Aided Millimeter-Wave Networks: A Practical Approach

Intelligent reflecting surfaces (IRSs) have become a popular topic in recent years for their great potential for controlling the radio link environment of wireless networks. With this controlled environment, the coverage can be increased. This paper examines the coverage analysis of IRS-aided networks, considering both two-dimensional buildings and the product-distance path loss model for the first time. Leveraging the tools from stochastic geometry, the locations of base stations (BSs), buildings, and IRSs are modeled with a homogeneous Poisson point process (PPP). A Gamma approximation for the distribution of the nearest line-of-sight (LoS)-neighbor distance is proposed, leading to a closed-form expression for the distribution of the product-distance. Feasible BSs are defined as BSs which are reachable via an IRS deployed on a specific facade of a building, and the ratio of feasible BSs is derived. Simulations are performed, which confirm the proposed analytical methods. In the numerical results, it is observed that the IRSs can introduce up to a 45% coverage boost, and the effect of the IRS length on the coverage probability is limited beyond 1.2 meters at 60 GHz.


I. INTRODUCTION
T HE demand for high data rates and ubiquitous connec- tivity grows each year.According to Ericsson's Mobility Report, the mobile data traffic growth is approximately 46% for each year, and the monthly global usage per smartphone is forecasted to reach 35 GB on average by the end of 2026 [1].This demand for higher data rates, capacity, and connectivity requires using more bandwidth which exists in higher frequency bands, like the millimeter wave (mmWave) and sub-THz range.In these high-frequency bands, the propagation characteristics of the channel become an important factor to consider.The associate editor coordinating the review of this article and approving it for publication was H. Elsawy.(Corresponding author: Abdullah Yasin Etcibaşı.) Abdullah Yasin Etcibaşı was with the Department of Electrical and Electronics Engineering, Hacettepe University, 06800 Ankara, Turkey.He is now with the Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210 USA (e-mail: etcibasi.1@osu.edu).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TWC.2023.3310664.
Digital Object Identifier 10.1109/TWC.2023.3310664 Although a large amount of bandwidth can be used in the high-frequency bands, there is a price to pay, which is the high penetration loss.The building penetration loss can be as high as 40 dB for buildings with infrared reflecting glass; a person who blocks the line-of-sight (LoS) link can cause an attenuation of around 20 dB, and the outage probability can be higher at mmWave frequencies due to the various path loss exponents of different environments [2].In [3], the authors plot the foliage penetration loss and the rain attenuation versus frequency.It can be observed that the losses become severe at higher frequencies due to the environmental effects.However, if the antenna aperture is kept constant, the antenna gain will be higher at higher frequencies [2], [3], [4].Hence we can compensate for the high free-space path loss (PL) by using a large number of antenna elements.To test this statement, some studies conducted field measurements that aim to measure the maximum distance of LoS links at high frequency by using beamforming.In [5], the authors claim that they achieved 2.148 Gbps of throughput at 1.8 km and at 39 GHz of operating frequency when the user equipment (UE) is stationary.In reference [4], a range of 1.7 km is achieved at 39 GHz in the LoS scenario.However, it is noted that in the non-line-of-sight (NLoS) case, the range decreases to approximately 200 m, and there are coverage holes in NLoS areas.Combating this problem, intelligent reflecting surfaces (IRSs) have emerged as a possible solution in recent years.
IRSs are surfaces made of small elements (specifically, each element has the size λ/5 or λ/10) whose phases are controlled in real-time, and hence, the surface as a whole can reflect the incoming wave in the desired direction [6].The very first implementations of IRSs appeared in the radar domain with the objective of reducing radar cross-section.In the literature, there are different designs of IRSs and one of the early designs uses PIN diodes to control the phase of each element [7].Manipulating electromagnetic waves through IRSs has given rise to a new wireless communication paradigm known as "smart radio environment" [8].If the radio environment is controlled intelligently, coverage holes [4] can be eliminated and the cell ranges can be improved in dense urban environments.While the primary application of IRSs is commonly thought of as creating LoS links, some major different applications can be found in [9].One of the early designs of the IRSs with multiple functions can be found in [10].Furthermore, the simple and low-cost design, ultralow power consumption, and scalability of IRSs make them a potential candidate technology to enable upcoming generations of communication networks [11], [12], [13], [14].
A full evaluation of the performance of IRS-aided networks requires the consideration of the locations of blockages, IRSs, UEs, and base stations (BSs).A sensible approach is to model the locations as randomly distributed; a network modeled as such is called a stochastic-geometry (SG) based network.While there are different approaches for modeling networks with randomly distributed nodes [15], the most common approach is to use the Poisson point process (PPP) due to its analytical tractability.In the PPP model, the number of nodes in a given area follows the Poisson distribution, with the nodes being uniformly distributed across the area.Analysis of SG-based networks is done through the typical UE (TUE) which lies at the origin.According to the Slivnyak-Mecke theorem, conditioning on a point in a homogeneous PPP (HPPP) network does not affect the statistics of the network.Therefore, each UE can be considered as a TUE which is located at the origin.An example of Poisson distributed BSs and their respective Voronoi tessellation can be seen in [16].

A. Related Works
Coverage analysis of SG-based networks is examined for various system models in the literature.However, the number of works which consider IRSs in their network models is limited.The first study to examine the impact of blockage on coverage analysis in an SG-based mmWave network is [17].In this study, it is assumed that the TUE at the origin can communicate with an LoS or NLoS BS, where the decision is based on the PL of the links.Nakagami-m fading model is adopted and the signal-to-noise ratio (SNR) expressions are derived for both LoS and NLoS links.Furthermore, LoS probability expression is approximated by a LoS ball model assuming that the LoS probability is a step function whose value is zero after a certain distance.However, IRSs are not considered in this work.
In the SG-based mmWave network context, several studies consider IRSs in their network models for coverage analysis [18], [19], [20], [21], [22], [23], [24].Authors in [18] assume a downlink communication between the TUE and BSs through the direct link or the indirect link.With Rayleigh small scale fading and product-distance model for the PL law of the indirect link, they derive the signal-to-interference ratio (SIR) coverage probability, where interference from only the LoS BSs are considered.However, they do not consider blockages in their work.
In [19], the channel power distribution is derived, and derivations for the signal and interference power distributions are used to obtain coverage probability.However, this work also does not consider the blockage effect.
In [20], an SG-based based IRS-aided network is studied, where the BS serves multiple UEs through an IRS.It is assumed that the direct links between the BS and the UEs are blocked, and the UEs can only communicate through the IRS via LoS links.The outage probability and the ergodic rate of this network are analyzed by considering that the UE locations follow a HPPP.However, this work assumes that there is only one BS and one IRS in the system, which is a limited model compared with multiple randomly distributed BSs and IRSs.
In [21], a Poisson cluster process model is used for the distribution of the IRS for coverage probability and ergodic rate analysis in a non-orthogonal multiple access network scenario.The model assumes that each UE has a LoS ball, within which there is always an IRS.This model does not involve cases where there is no indirect LoS link for a UE.In addition, it is assumed in [21] that the TUE has no direct LoS BS link.In contrast, in the current paper, we adopt PPP distributions for IRSs and BSs, and do not assume predefined LoS TUE-IRS links and LoS IRS-BS links.
The SNR coverage probability in a point-to-point network with PPP distributed human-body blockages, IRSs, and buildings is investigated in [22].They consider mainly two scenarios where the size of the IRSs are small but with a high deployment ratio or large IRSs with a low deployment ratio.They show that deploying more small IRSs with high density is a better option for densely-built regions than deploying large IRSs with low density for a point-to-point network having deterministic locations of the UE and the BS.
The studies in [23] and [24] insert a line Boolean model of blockages, which assumes that the buildings have zero width, i.e., lines whose centers are distributed with HPPP.These works adopt the sum-distance model for the PL of the indirect link.In [23], LoS and IRS-aided indirect LoS connections are considered only, and the blind-spot probability of a UE is derived.In addition, coverage probability, defined as the rate of the event that the PL is below a predefined threshold, is obtained.In [24], under slightly different assumptions, the coverage probability of the system is derived.We note that the assumptions of line blockages and the sum-distance PL law for indirect links are not realistic.

B. Contributions
In this paper, we analyze the coverage probability of an SG-based IRS-aided mmWave network from a practical perspective.For the first time, we both consider the rectangular Boolean scheme of blockages and the product-distance PL model for indirect links in the coverage analysis of IRS-aided outdoor multi-cell mmWave networks.The main contributions of this study can be summarized as follows.
1) Practical Scenario Consideration: Unlike previous works [23] and [24], our study adopts a more practical scenario.This involves utilizing the PL model of indirect links with the product-distance model, which better reflects the PL behavior in outdoor environments.
In contrast, the sum-distance PL law employed in [23] and [24] does not accurately capture this behavior.
It should be noted that while the sum-distance model simplifies the calculation of the distribution of the PL of the indirect link, the use of the product-distance model presents a more challenging task.Additionally, we model blockages using a rectangular Boolean scheme, which provides a more realistic representation of the 2D environment.Unlike the line blockage model employed in [23] and [24], the rectangular model allows for the consideration of different sides and orientations of the blockages.This model yields a more accurate assessment of LoS coverage, as the line blockage model tends to be overly optimistic.2) Gamma Approximation for Nearest-LoS Neighbor Distance: The probability distribution of the nearest-LoS neighbor distance in an SG-based network is analytically challenging for utilizing it for the analysis of the indirect link distance distribution.To address this, we propose a Gamma approximation, which enables the derivation of a closed-form solution for the product-distance distribution.This approximation is novel in the context of SG-based networks HPPP distributed BSs and twodimensional buildings.Although [22] also employs the product distance model for PL, the non-random locations of the UE and the BSs in that reference do not require the distribution of the product distance.3) Feasible BS Probability Analysis: Some BSs are positioned behind buildings and cannot utilize the IRS for connectivity.We derive the probability of feasible BSs, which refers to BSs positioned within the appropriate region relative to the IRS side of the building.While feasible BS probability analyses have been conducted in [23] and [24], these studies utilized line blockage models, and the methods are not applicable to our rectangular Boolean scheme of blockages.4) Novel Insights from Numerical Results: Our research provides novel insights derived from our numerical findings at a large-scale system level.We demonstrate that there exists an IRS deployment ratio beyond which adding more IRSs on blockages yields minimal improvement in coverage probability.Moreover, we show that the impact of IRS side length on network coverage diminishes after reaching 1.2 meters.Additionally, we highlight the limitations of the line Boolean sum-distance PL model in coverage analysis, as it results in lower blockage rates and smaller PL values for IRS-aided indirect links, making it overly optimistic.

II. SYSTEM MODEL
In this section, we introduce the system model that we use in our SNR coverage probability analysis.To properly evaluate the coverage probability in a mmWave IRS-aided network, we use the SG model.Some preliminaries on SG models are in [23].
We consider that the locations of BSs form an HPPP Φ BS with density λ BS on the R 2 plane, and all BSs have the same normalized transmit power P t = 0 dBW.Blockages, typically buildings, form a Boolean scheme of rectangles [25] whose centers follow an HPPP Φ b with density λ b .We assume that the blockage distribution is stationary and isotropic, which means that the distribution is invariant under the motions of translation and rotation.We define a blockage with three parameters: its length, width, and orientation.The expected value of the length and width of blockages are defined as E[L] and E[W ], respectively.The orientations of the buildings, denoted by θ b , are uniformly distributed in [0, π].
IRSs are deployed optimally 1 to the buildings with a ratio of 0 < µ < 1.Hence, by the independent p-thinning of an HPPP, the buildings containing IRSs follow an HPPP, Φ IRS , with density λ b µ.UEs are distributed as a stationary point process (PP), independent of the BSs and buildings on the plane, and the TUE is assumed to be located at the origin.Since the UEs are distributed independently and follow a stationary PP, the downlink SNR experienced by the TUE has the same distribution as the aggregate one [17].
We assume that communication is done through LoS links; in other words, radio waves cannot penetrate buildings and the LoS component is dominant in the channel.The LoS probability in an SG-based network is derived in [25], which is p(r) = exp(−βr), where ) for outdoor UEs.In this paper, we assume that both the TUE and the BSs are located outdoors based on the results in [17].
The TUE can only be associated with one BS.This connection can be made either with a direct LoS-link or an indirect LoS-link.The direct LoS-link is the link between the LoS BS and the TUE.The indirect LoS-link is created with two independent LoS links: one between the TUE and the LoS IRS, and the other between the IRS and the BS.These possible connections can be seen in Fig. 1.If there are multiple LoS BSs, we assume that the TUE is associated with the nearest LoS BS.Similarly, if there are multiple IRSs that have a LoS link with the TUE, the nearest LoS IRS is chosen.Afterward, the nearest LoS BS with respect to this chosen IRS is selected, which forms an indirect UE-IRS-BS link.Throughout the paper, we will refer to the link between the TUE and the nearest LoS BS as the UE-BS link, the link between the TUE and the nearest LoS IRS as the UE-IRS link, and the link between the IRS and the nearest LoS BS as the IRS-BS link.For example, in Fig. 1, the TUE has LoS connections to BS3, BS4, IRS2, and IRS3.The UE-BS link (direct link) is the link between the TUE and BS4 because the nearest LoS BS to the TUE is BS4.On the other hand, the indirect LoS-link is the link UE-IRS2-BS2, since the nearest LoS IRS to the origin is IRS2 and the nearest LoS BS to IRS2 is BS2.The TUE can only be associated with either BS4 or BS2, and this selection is made by comparing the PL of these two links.In other words, if the TUE sees both the direct LoS BS and the indirect LoS BS, then it decides which one to connect to by comparing their PL values.However, if it sees only a direct LoS BS or an indirect LoS BS, then it connects to the available BS without any further comparison.We also assume that the indirect link can only be made through one IRS; we do not consider links using multiple IRS hops.
In the literature, two commonly used path loss (PL) models are found for IRS-assisted communication: the product-distance PL model and the sum-distance PL model [26].The sum-distance PL model assumes that the IRS acts as a perfect electric conductor.However, this model is only applicable to near-field scenarios, making it impractical Fig. 1.UE Association: In our model, we assume that the TUE connects to BS4 using the direct link or to BS2 using IRS2.
for outdoor cellular systems [27]. 2 Experimental evidence presented in [27] demonstrates that the product-distance PL model is more suitable for far-field applications.In our study, we adopt the PL expression derived in [28], and after slight rearrangement, we obtain the following expression: where z represents the product of d U I and d IB , which are the distances of the UE-IRS and IRS-BS links, respectively.The IRS is modeled as a uniform planar array with its elements arranged on a rectangular The constant C involves the transmit and receive antenna gains and the free space path loss, given by C = ( λ √ GtGr 4π ) 2 .The size of each IRS element is considered as √ 0.5λ× √ 0.5λ and is accounted by the (2/λ) 2 term in Eq. ( 1). 3 The angle between the normal vector of the IRS and the incident wave is denoted as The PL for the direct link follows the free-space PL model: Taking small-scale fading into account, the complex baseband representation of the received signal for the direct link can be expressed as: In the case of the indirect link, the received signal is given by: where x is the transmitted signal, n represents the additive white Gaussian noise with zero mean and variance σ 2 n , Φ = diag{[e jϕ1 , e jϕ2 , • • • , e jϕ N ]} denotes the reflection coefficient matrix where ϕ i ∈ [0, 2π) is the phase shift of i th element of the IRS.Furthermore, h D denotes the UE-BS channel, the vectors h 2 and h 1 represent the UE-IRS and IRS-BS channels, respectively.The scaling parameter 1/N is introduced for the normalization of the signal power.Note that, for an increasing number of IRS elements, N , there will be an array gain of N .However, adopting [28], we have already accounted for this array gain in the PL expression in Eq. ( 1), so we need the 1/N normalization in Eq. ( 4).Since we consider an LoS-dependent network, where all channels are modeled by Rician fading, and share the same shape K and scale Ω parameters.We also assume perfect channel state information, optimal array response vectors, and optimal reflection coefficient matrix design, ensuring that the phase shifts are aligned perfectly with the phase of the cascaded channel.Hence the SNR 4 expression for the direct link is given by ( For the indirect link, we have where h 1,n and h 2,n denote the channel coefficients between the UE and the n th element of the LoS IRS, and the n th element of the IRS and the LoS BS, respectively.The channel coefficients are modeled independently across different IRS elements, as in [31], [32], and [33].

III. SNR COVERAGE PROBABILITY
In this paper, our aim is to characterize the coverage probability where T denotes the SNR threshold, the required SNR to establish communication.
We examine the coverage probability considering four mutually exclusive events: A D1 , A D2 , A I1 , and A I2 .We obtain the conditional coverage probability given these events and combine them to obtain the overall coverage probability as follows.
The events A D1 , A D2 , A I1 , and A I2 are determined by the existence of direct or indirect LoS links, and the corresponding PLs.The events and distributions in (8) are defined below.
• A D1 : event that TUE has both a direct LoS link and an indirect LoS link, and the PL of the direct link is less than that of the indirect link, • A D2 : event that TUE has a direct LoS BS link, and no indirect LoS BS link, • A I1 : event that TUE has both a direct LoS link and an indirect LoS link, and the PL of the indirect link is less than that of the direct link, • A I2 : event that TUE has an indirect LoS BS, and no direct LoS BS, ) and f I (z|A I2 ): conditional distributions of the product distances for indirect links.Note that the TUE establishes connection via the direct LoS BS in the cases of A D1 and A D2 , and via the indirect LoS BS in the cases of A I1 and A I2 .
In the sequel, we will examine each component of ( 8), starting with the distance distributions f D (x) and f I (z).These findings will then be integrated according to Eq. ( 8) to obtain the coverage probability.
Theorem 1 (Distribution of the Nearest LoS Node Distance [23]): Consider nodes (BSs or IRSs) distributed according to an HPPP with intensity G/(2π).Then, the distribution of the distance of the LoS node closest to the origin given that the TUE sees at least one LoS node is given by where dx. Proof: With the independent thinning theorem of a PPP, the intensity of the LoS nodes will be p(r)G/(2π).Similar to [23], the nearest neighbor distance distribution is an inhomogeneous PPP (IPPP) network with the intensity p(r)G/(2π), which results in (9).
Since the intensities of BSs and IRSs are known, the direct corollary of Theorem 1 gives the UE-BS and UE-IRS distance distributions.
Corollary 1.1: Distribution of the distance of the UE-BS link, f d U B (x), and the distribution of the distance of the UE-IRS link, f d U I (x), can be obtained by substituting 2πλ BS and 2πλ b µ as the variable G in Eq. ( 9), respectively.
Our goal is to obtain the distribution of the product distance, which will be based on the distribution of the distance of the nearest LoS node in Eq. ( 9).However, Eq. ( 9) is too complex for analytical tractability as it involves an exponential term within an exponential function, thus we resort to approximate forms.Among the possible approximate distributions, we observe that the shape of the Gamma distribution is similar to the actual distribution in Eq. (9).
The verification of the Gamma approximation will be given in the sequel by first comparing the exact distribution with the Gamma approximation, and also later in the numerical results section, where the analytical coverage probability based on this approximation is compared with the simulation results.Therefore, this is a suitable approximation both in terms of tractability and in terms of its accuracy verified by numerical analysis.

A. Gamma Approximation
Consider the distribution given in Eq. ( 9).To approximate the distribution f X (x) with the Gamma probability density function Γ(k, θ), we choose the shape parameter k and scale parameter θ such that we fit the first and second-order moments of the Gamma distribution to those of f X (x).Let Y ∼ Γ Y (y; k y , θ y ) and X ∼ f X (x).
Theorem 2: For the random variable X ∼ f X (x), the expected value of X is given by dt is the upper incomplete Gamma function.
Proof: See Appendix A. Theorem 3: For the random variable X ∼ f X (x), the second non-central moment of X is given by Proof: The proof follows the same steps used in the proof of Theorem 2.
Hence, the shape and scale parameters of the Gamma approximation for f X (x) are determined as follows: θ y = var(X) where , and E[X] and E[X 2 ] are given in Eq. ( 10) and (11), respectively.The Gamma approximation of the distribution of UE-BS and UE-IRS distances can be found by using Corollary 1.1 and Eq. ( 12)-( 13) as follows: where the pairs (k d , θ d ) and (k 1 , θ 1 ) are calculated using ( 12)-( 13) with G = 2πλ BS and G = 2πλ b µ, respectively.The validity of this approximation is demonstrated in Section IV.
Note that not all LoS IRS-BS links form an indirect TUE-IRS-BS link.An indirect TUE-IRS-BS can be created if and only if the TUE and the BS (both are LoS to the IRS) see the same facade of the building in which the IRS is deployed.We call this IRS-BS link as a feasible link, and the BSs satisfy this condition feasible BSs.In the following section, we propose a method to derive the ratio of feasible BSs within a given area.

B. Probability of Feasible BSs
We consider a square map in which all the BSs and buildings are located, with the length of each side equal to L m .We assume that the TUE is associated with the nearest IRS, and we take the center of the building on which this IRS is deployed as a reference point.Using the rotational and placement invariance property, we assume that the building is at the center of the map. 5 Furthermore, we assume that the IRS building is located at the center of a circle with a radius of r 1 , and the TUE is located anywhere on this circle, as depicted in Fig. 2a.
We define the events B k , C k , and D, where, B k for k = 1, . . ., 8 represents the event that the TUE lies in the k th region shown in Fig. 2a, or equivalently, k th arc shown in Fig. 2b, C k denotes the presence of a BS in region k, and D indicates the event that there is a feasible BS.We will now examine the probabilities of these events. 5This approximation holds better for larger map lengths.
Lemma 1: The probability that the TUE lies on the k th arc given the radius r 1 is given by where φ 1 = cos −1 (a/r 1 ), and φ 2 = cos −1 (b/r 1 ).
Proof: The proof follows directly from the arc lengths shown in Fig. 2a.
Lemma 2: The probability that a BS lies in the k th region is given by where a = E[W ]/2, and b = E[L]/2.Proof: Since the BSs are distributed as PPP, and the regions are well defined with a, b, and L m , these probabilities can be found by just using the areas of the rectangular regions.
By using Lemma 1 and Lemma 2, the probability of feasible BSs can be found.
Proposition 1: For HPPP-distributed BSs with intensity λ BS , HPPP-distributed rectangular blockages with intensity λ b , and IRS deployment onto the blockages with rate µ in a square region, the probability that the BSs are feasible according to the nearest IRS-deployed blockage is given by where f R1 (r 1 ) ∼ fd U I (x) is the distribution of the distance of UE-IRS link, and Proof: The proof is based on combining the results of Lemma 2 and Lemma 1, using the regions and corresponding arcs in Fig. 2a.
From the perspective of the LoS IRS, only the feasible BSs are considered suitable.Therefore, the density of LoS BSs is reduced by a factor of ξ.Consequently, by applying pthinning to the HPPP Φ BS , the feasible BSs form an HPPP with a density of λ BS ξ.We will use this result to obtain the approximate probability density function of the distance between the LoS IRS and the feasible BS.
Corollary 3.1: The probability density function of the distance between the feasible IRS-BS link can be approximated by the Gamma distribution as where the pair (k 2 , θ 2 ) is calculated using Eq. ( 12)-( 13) with G = 2πλ BS ξ.

C. Product-Distance Distribution
The SNR expression of the indirect link ( 6) has the product of two distances: the UE-IRS distance and the IRS-BS distance.Hence, we need the distribution of this product-distance to analyze the coverage probability in (8).To that end, we introduce multiplicative convolution.
Lemma 3 (Multiplicative Convolution [34]): The multiplicative convolution of two functions Z 1 and Z 2 is defined as , where U, V > 0 are independent 6 random variables represent the length of the UE-IRS link and the IRS-BS link, respectively.Then, the distribution of the product-distance f I (z) is equal to the multiplicative convolution of f d U I (u) and f d IB (v). Proof: dF where (a) utilizes the fact that U > 0, and (b) applies the Leibniz rule.Obtaining a closed-form solution to (23) using the exact distributions for the UE-IRS and IRS-BS link lengths based on ( 9) is too complex.Hence we use the approximate forms given in (15) and (20).We employ the multiplicative convolution property of the Mellin Transform (MT) to obtain the closedform solution.
Theorem 5 ( [34]): The MT of the multiplicative convolution of functions Z 1 and Z 2 is equal to the product of their individual MTs: where M[f ](s) = ∞ 0 x s−1 f (x) dx, denotes the MT.By employing Theorem 5 and the approximate forms of f d U I (u) and f d IB (v), an approximate closed-form expression can be obtained for the distribution of the product-distance. 6According to the Slivnyak's theorem, conditioning on a point does not change the distribution of the remaining process.
Proposition 2: For the distribution of the UE-IRS distance fd U I (u) ∼ Γ U (u; k 1 , θ 1 ), U ≥ 0, and the distribution of the IRS-BS distance fd IB (u) ∼ Γ V (v; k 2 , θ 2 ), V ≥ 0, the product-distance distribution UV ∼ fI (z) is given by where , is the modified Bessel function of the second kind, and is the modified Bessel function of the first kind.
Proof: See Appendix B. For the derivations in the sequel, we also require the cumulative distribution function (CDF) of fI (z).
Lemma 4: The CDF of the product-distance given in Proposition 2 is given by where 2 , and θ m = θ 1 θ 2 .Proof: See Appendix C.

D. Conditional Distribution of the Distance of the Links
Having derived the distributions of the distance for the direct link and the product-distance for the indirect link, we now focus on (8), where we need the conditional distributions of these distances given the events mentioned in (8).First, we examine the probabilities of these four events.Let where we used the void probability of a PPP [23] in steps (a), (b), and (c).Then, we have: Note that the events A ′ I and A ′′ I are not mutually exclusive.The probabilities P(A ′ D ) and P( ÂI ) will be used to obtain the probabilities of the four events in (8).For the events A D1 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
and A I1 , we also need to find the probabilities of A D,P L and A I,P L .This is because when the TUE has access to both a direct BS and an indirect BS, the decision regarding which one to connect to is based on a comparison of their PL values.We first obtain the conditional probabilities of A D,P L and A I,P L , given the distances.
Lemma 5: The probability that the PL value of the direct link is lower than or equal to the PL value of the indirect link, given that the UE-BS distance equals x, is given by: Proof: See Appendix D. Lemma 6: The probability that the PL value of the indirect link is lower than the PL value of the direct link, given that the product-distance equals z, is given by: where χ(z, θ i ) = 2z/(N λ cos θ i ), and γ(•, •) is the lower incomplete Gamma function.Proof: where step (a) follows from the fact that 14), substituting the Gamma CDF in (35), we get (34).The probabilities P(A D,P L ) and P(A I,P L ) are obtained by applying the total probability theorem to (33) and (34), and averaging over x and z, respectively.
Proposition 3: The probability that the PL value of the direct link is lower than or equal to the PL value of the indirect link is Proof: See Appendix E. Using the definitions in ( 27)-( 32), the probabilities of the four events in (8) can be derived as follows.
where (•), indicates the complement of an event.
Next, the conditional distributions of the distances given the events are obtained.
Proposition 4: For the events given in ( 8), the conditional distributions of the direct and indirect links' distances are given by Proof: The proof follows directly from the Bayesian expansion.

E. Conditional SNR Coverage Probabilities
We have derived all the components in the coverage probability in (8), except for the probabilities that the SNR is larger than the threshold given the distances, which will be investigated next.
Theorem 6: The probability that the SNR of the direct link is greater than a threshold given that the distance of the direct link is given by where Proof: The proof directly follows from the fact that the small-scale fading of the direct link has Rician fading with the parameters K and Ω.
For the indirect link, we assume that the individual channels of IRS elements are independent and identically distributed (i.i.d.) with Rician distribution, as we mentioned in Section II.Hence, we can apply the central limit theorem to find the sum of the products of these channels.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Then, U is distributed as a Gaussian with mean m u , and variance where and Λ 2 = K+1 Ω .Proof: See Appendix F. The conditional SNR coverage probability of the indirect link can be computed using Lemma 7.
Proposition 5: The probability that the SNR of the indirect link is greater than a threshold given the product-distance of the indirect link is where T I (T, z, θ i ) = N 2 σ 2 n T P L I (z, θ i ), and Q(x) is the Q-function. Proof: where we used (47) to obtain (51).By applying the total probability theorem with respect to θ i to (51), the proof can be completed.
To conclude, the SNR coverage probability can be obtained by substituting (37)-( 40), (41)-( 44), (45), and (50) into (8).As the expressions for the integrands in (8) are derived, the integrals can be evaluated efficiently in order to obtain a final result for the SNR coverage probability.

IV. NUMERICAL RESULTS
In this section, we will provide some key numerical results to understand the impact of IRSs on the mmWave networks from the coverage perspective; but first, we will investigate the accuracy of the Gamma approximation given in Section III-A.In Fig. 3, it can be seen that the Gamma approximation of the Fig. 3. Probability density function of the nearest BS distance using the exact distribution in (9), Gamma approximation in (14), and Monte Carlo simulation.The unit of λ BS is BS/m 2 .

TABLE I NUMERICAL RESULTS PARAMETERS
nearest-LoS neighbor distances fits quite close to the exact distribution. 7Also, observe that the simulation results follow the exact plots very well.
There are 4 main parameters in our simulations, which are λ BS , µ, ρ, and L I .The definitions of the first two parameters are given in Section II.The parameter ρ is the percentage of the map area covered by the buildings, and L I is the length of a side of the square IRSs in meters, L I = 0.7071 λ √ N .The values of the system parameters we used in our simulations are given in Table I.The reference value of ρ = 0.26 is utilized based on the findings presented in [17].We compare our results for IRS-aided networks with the network with no IRS, where the coverage probability performance result reduces to In Fig. 4, we present the coverage probability performance results for three different cases.First, we show the IRS-aided network performance, next we show the no-IRS network performance.Thirdly, we show the coverage performance in an IRS-aided network given the UE is connected via a direct UE-BS link.We compare our analytical results with the simulation results.We observe that the analytical results are close to the simulations, validating the approximations made in the derivations.Comparing IRS-aided case to no-IRS case, we observe the gains provided by the IRS.For low target SNR values, the performance is dominated by the geographical network density parameters, since the SNR exceeding the threshold condition is almost always satisfied given that there is a direct or indirect link.For larger values of target SNR, we observe the combined effects of small-scale Rician fading and large-scale blockage on the performance.The coverage probability for an IRS-aided network given just the direct connection is lower than that of no-IRS case.The reason for that is that even when there is a UE-BS link, there are cases where there is an indirect link with a lower PL value, meaning the 1 × 1m IRS can compete with the direct link.
The impact of the IRS deployment on the network coverage depends on the BS density.To observe this, we present the coverage increase percentage plots in Fig. 5.In Fig. 5a, the BS density is λ BS = 3/km 2 , and in Fig. 5b, λ BS = 11/km 2 .The SNR threshold is 0 dB, and the IRS side length is L I = 1 m.In these plots, the y-axes show the percentage increase of the coverage probabilities provided by the IRS, with respect to the no-IRS network.This percentage increase is plotted as a function of the IRS deployment ratio, µ.Comparing Fig. 5a with Fig. 5b, we observe that the performance increase due to IRS is larger for smaller λ BS .This result is expected because in scenarios with a high BS density, there is little room for improvement by the IRS.The plots are given for different values of building area percentage, ρ.Increasing ρ has two effects on the IRS performance gain.First, IRS density λ b µ increases with the blockage density, providing more opportunities for the UE-IRS links, improving the probability of the IRS-aided indirect link, and the percentage increase.Secondly, increasing blockage density reduces the occurrence of IRS-BS links, diminishing the probability of the indirect link, and the percentage increase.Note that, even for a deployment ratio of µ = 0.1, the coverage boost of the IRS varies between 25 to 35 percent.The gain increases with increasing µ.We observe that the gains start the diminish beyond µ ≃ 0.30, so deployment ratio of 30 percent is sufficient.The slight decrease in the IRS gain for very large values of µ is due to the fact that the nearest IRS to the UE is chosen, which can be sub-optimal for a small fraction of cases.
In Fig. 6, we present performance results investigating the impact of the IRS size.We provide coverage probability curves as a function of the required SNR for IRS side lengths L I ranging from 25 cm to 2 meters.The no-IRS network performance is also shown for comparison.These results are for base station density λ BS = 7/km 2 , blockage area ratio ρ = 0.35, and IRS deployment ratio µ = 0.3.For low target SNR values, the coverage probability performances for all IRS lengths are close, because small IRSs are sufficient for these SNRs.For larger SNRs, we start to observe the coverage probability differences.However, for IRS sizes larger than L I = 1.2 m, we observe smaller differences.Hence, it can be stated that an IRS size of L I = 1.2 m is sufficient at 60 GHz.
Next, we investigate the impact of θ i , the angle between the normal of an IRS and the incident wave of the UE-IRS link.Remember that θ i determines the path loss of the indirect link in (1), and throughout the derivation we average results over uniform distributed θ i ∼ U (0, π/2).In Fig. 7, we illustrate the impact of the incident angle by assuming a constant θ i .For comparison, we also show the random θ i result and the no-IRS result.Examining the coverage probability curves, as θ i approaches π/2, the coverage performance curve deviates markedly from our random model and approaches the no-IRS curve.We observe that, if in further analyses, constant   incident IRS angle assumption is useful, the value of the θ i can be taken as 0.35π for the ease of derivations.
As we have mentioned in Section I-A, some previous work on the coverage analysis of IRS-aided SG-based networks considers the line Boolean model for blockages, and the sum-distance PL model for the indirect link, both of which are unrealistic assumptions.Next, we compare this approach with the current paper where realistic rectangular blockage and product-distance PL models are used.Here, we take the blockage density λ b = 90.9/km 2 for both models, which corresponds to the building area ratio of ρ = 0.26 for our model.In order to approximate the line blockage model with sum-distance PL, we use a modified version of our model with E[W ] = 1 m (so that it does not contradict with the L I = 1 m IRS deployment) and P L I ∼ (d U I + d IB ).For the line blockage sum-distance PL model, we observe the following.For low SNRs, the IRS cannot provide much gain.That is because of the low probability of the buildings blocking direct links for the line blockage model.This result is not realistic because it underestimates the ratio of blocked UE-BS links.For large SNRs, the IRSs contribute in the form of large SNR boosts, due to the small PL of the sum-distance PL model.However, this observed gain is misleading because, in reality, the PL will be much larger.Hence, it can be stated that our proposed analysis is much more practical than the line Boolean blockage sum-distance PL model.

V. CONCLUSION
In this paper, we investigated the coverage probability of IRS-aided SG-based networks with buildings that have a rectangular Boolean model and product-distance PL model for the indirect link.We propose a Gamma approximation for the nearest-LoS neighbor distance distribution and derive the product-distance distribution.Based on the SG-based model and the Rician small-scale fading, we obtain the coverage probability of the IRS-aided network.We investigate the gains provided by the IRS with varying model parameters.Then, we compare our model with the one in the literature which is the line Boolean sum-distance model, and show that that model is not suitable for incorporating realistic effects of the IRS-aided network, justifying our model and derivations.Future works might consider 3D buildings and heights of IRS and BS, and the extension of results to the multiple-input and multiple-output (MIMO) case.Another future work might be to consider utilizing a hard-core point process model to differentiate between the far-field and near-filed users relative to IRSs.
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where we employed a change of variables u = e −βx , ln(u) − 1 = τ and −τ (k + 1) = m, in steps (a), (c), and (d), respectively.In step (e), we used the fact that Γ(s + 1, x) = s Γ(s, x) + x s e −x , and in step (b) we used the Maclaurin expansion of exp(−γu ln(u) + γu) given by, Since the integral in (b) is defined between 0 and 1, this approximation gives a proper result.

APPENDIX B
By Theorem 4, the distribution of the product-distance can be approximated as, where we applied the MT and the inverse MT (IMT, M −1 ), in step (a) and (b), respectively, and in step (c), we used Theorem 5.
To derive the closed-form equation of (55), we need to find the MT of f d U I (u) and f d IB (v), then the IMT of multiplication of these transforms gives us the product-distance distribution.

M[ fd
where in step (a), we employed change of variable x/θ D = τ .By applying some basic mathematical manipulations to (60), the expression (36) can be obtained.

APPENDIX F
Let f H (x) be as defined in (46) and W be Hence, where Λ 2 = K+1 Ω , and in step (a) we used the series expansion of the 0 th order modified Bessel function of 1 st kind given by, where ℜ{β} > 0, ℜ{γ} > 0. Now, we can find the mean value and the variance of U using (61).For the variance, the non-central second moment of W is required.With the same approach used in the proof of m u , the E[W 2 ] is given by, Hence, by using (62), the variance of u can be found as,

Manuscript received 19
February 2023; revised 26 May 2023 and 14 August 2023; accepted 26 August 2023.Date of publication 7 September 2023; date of current version 11 April 2024.This work was supported by Türk Telekom within the framework of the 5G and Beyond Joint Graduate Support Program coordinated by the Information and Communication Technologies Authority.

Fig. 2 .
Fig. 2. Auxiliary figures for the analysis of feasible BSs.(a) The map, the circle on which the TUE is located, φ 1 and φ 2 , and eight different regions in which the TUE and BSs can be located.(b) Angles of the regions and corresponding arc lengths.

Lemma 7 :
Define U = N n=1 |h 1,n | |h 2,n | where the |h 1,n | and |h 2,n | are i.i.d.random variables with the Rician distribution given by

Fig. 5 .
Fig. 5. Coverage probability increase of the IRS-aided network with respect to the no-IRS network, as a function of the IRS deployment ratio for (a) λ BS = 3/km 2 , (b) λ BS = 11/km 2 .

Fig. 6 .
Fig. 6.SNR coverage probability as a function of target SNR, for various IRS sizes.