Success Probability of Millimeter-Wave D2D Networks with Heterogeneous Antenna Arrays

Success Probability of Millieter-Wave D2D Networks with Heterogeneous Antenna Arrays Na Deng, Yi Sun School of Inforation & Counication Engineering Dalian University of Technology Dalian, Liaoning, 11624, China Eail:{dengna,lslwf}@dlut.edu.cn Martin Haenggi Dept. of Electrical Engineering University of Notre Dae Notre Dae, IN, 46556, USA Eail:haenggi@nd.edu Abstract This paper focuses on the success probability or, equivalently, the signal-to-interference-plus-noise ratio SINR distribution at the typical receiver in illieter wave wave device-to-device D2D networks. Unlike earlier works, we consider a ore general and realistic case where devices in the network are equipped with heterogeneous antenna arrays so that the concurrent transission beas are varying in width. Specifically, we first establish a general and tractable fraework for the target network with Nakagai fading and directional beaforing. Next, we investigate the interactions aong beas with different widths and their sensitivities to the adopted odel for the antenna pattern. In addition, to show the ipact of heterogeneous antenna arrays on the link perforance, we derive the success probability of the typical receiver as well as its bounds to get deep insights on the perforance of the network. I. INTRODUCTION The proliferation of high-speed ulti-edia applications and high-end devices exacerbates the deand for high data rate services. According to the latest visual network index VNI report fro Cisco [1], the global obile data traffic will increase nearly sevenfold between 216 and 221, reaching 49. exabytes per onth by 221, wherein ore than threefourths will be video. The need for greater capacity, and hence greater spectru utilization, has very recently led to the advent of illieter wave -wave device-to-device D2D counications to efficiently use the large bandwidth ultiple gigahertz. However, this eerging technology is still in its infancy, and it is unclear what benefits and challenges it will bring. It is clear that -wave D2D counication is ore coplicated than sub-6 GHz D2D and -wave cellular counications. Firstly, the narrow bea width of -wave and the relatively low antenna height copared with that of BSs render the -wave D2D counication even ore vulnerable to blockages. Secondly and ore iportantly, different fro the cellular BSs that are usually equipped with hoogeneous antenna arrays naely the sae nuber of antennas, the devices have their inherent diverse properties and rando locations, which eans devices in the network cannot be expected to be equipped with the sae nuber of antennas and located in a well-planned anner. Therefore, in this paper, we will present a coprehensive investigation on the heterogeneous -wave D2D networks and obtain useful insights for the further developent of -wave D2D counications. Although -wave devices offer several potential advantages for D2D networks, there has been liited application of stochastic geoetry to study the potential perforance of -wave D2D networks incorporating key features of the -wave band. The priary related works are [2] and [3]: the forer approxiated the directional beaforing by a sectored odel with the assuption of hoogeneous antenna arrays and blockage effects but considered a finite nuber of interferers in a finite network region; while the latter proposed two ore accurate antenna pattern odels with the sae assuption in [2]. In contrast, our prior work in [4] used stochastic geoetry to provide a fine-grained perforance analysis of -wave D2D networks in ters of the eta distribution, and it also considered the siplified sectored odel for the antenna pattern with unifor antenna array. To the best of our knowledge, the effect of the heterogeneity of the antenna arrays on the potential perforance of wave D2D networks has not been studied in conjunction with accurate approxiations for the actual antenna pattern. In this work, we will fill this gap with new analytical results of the success probability in a stochastic geoetry fraework. II. NETWORK MODEL It is assued that the transitters belonging to the k-th tier are distributed uniforly in the two-diensional Euclidean space R 2 according to a hoogeneous PPP Φ k of density λ k and operate at a constant transit power μ k. For all j i, Φ j and Φ i are independent. The ALOHA channel access schee is adopted, i.e., in each tie slot, D2D transitters in Φ k independently transit with probability q k. Accordingly, the distribution of the devices in -wave D2D networks is defined as Φ K k1 Φ k with density λ K k1 λ k. Each transitter is assued to have a dedicated receiver at distance r in a rando orientation, i.e., the D2D users for a K- tier Poisson bipolar network [5, Def. 5.8]. Without loss of generality, we consider a receiver at the origin that attepts to receive fro an additional transitter located at r,. Due to Slivnyak s theore [5, Th. 8.1], this receiver becoes the typical receiver under expectation over the overall PPP. To analyze the typical D2D receiver belonging to the k-th tier, we further condition on that receiver at the origin to belong to

the k-th tier with paraeters such as transit power, nuber of antennas, etc. chosen fro that tier. A. Blockage and Propagation Model The signal path can be either LOS/unblocked or NLOS/blocked, each with a different path loss exponent. The generalized LOS ball odel [6] is adopted to capture the blockage effect in -wave counication, which was verified to be as accurate as the epirical 3GPP blockage odel by experients in [7]. Specifically, the LOS probability of the signal path between two nodes with separation d is P LOS d p L 1d <R, 1 where 1 is the indicator function, R is the axiu length of a LOS channel, and p L [, 1] is the LOS probability if d R. Let α k,l and α k,n denote the path loss exponents of LOS and NLOS paths belonging to the k-th tier, respectively. Typical values for -wave path loss exponents can be found in [8] with approxiated ranges of α k,l [1.9, 2.5] and α k,n [2.5, 4.7]. B. Directional Beaforing Model We assue that the transitters belonging to k-th tier are equipped with a unifor linear array ULA coposed of N k antenna eleents to perfor directional beaforing and their corresponding receivers have a single antenna. It is also assued that the transitter knows the direction to the receiver so that it can point its AoD at its receiver perfectly to obtain the axiu power gain. Recently, an accurate approxiation tered cosine antenna pattern was proposed in [3], which is shown to constitute a desirable trade-off between accuracy and tractability in the perforance analysis of -wave networks. This antenna pattern approxiation is based on the cosine function with the antenna gain function { Nk cos G k ϕ 2 πn k 2 ϕ if ϕ 1/N k 2 otherwise, where ϕ dt ϱ cos φ is the cosine direction corresponding to the AoD φ of the transit signal, which is tered as the spatial AoD, with d t and ϱ representing the antenna spacing and wavelength, respectively. The antenna spacing d t is usually set to be half-wavelength to enhance the directionality of the bea and avoid grating lobes; the spatial AoD ϕ is assued to be uniforly distributed in [.5,.5], and thus the spatial AoD fro an interferer to the typical receiver is also uniforly distributed in [.5,.5], as proven in [3]. While for the ostly used sectored antenna pattern, the array gains within the half-power beawidth are assued to be the axiu power gain, and the array gains corresponding to the reaining AoDs are approxiated to be the first inor axiu gain of the actual antenna pattern. Although this siple approxiation is highly tractable, it causes significant deviations fro the actual perforance, especially when there are differences in the nuber of antennas aong different devices in the network. Array Gain db 2 15 1 5 N 64 Sectored Antenna Pattern Cosine Antenna Pattern Actual Antenna Pattern N 4 5.5.1.15.2 Spatial AoD Fig. 1. Visualization of three different antenna patterns for N 4 and N 64. In Fig. 1, we copare the cosine antenna pattern, the sectored antenna pattern, as well as the actual antenna pattern [9]. Fro the actual antenna pattern, we can observe that the first side lobe gain of N 64is within 1 db of the ain lobe gain of N 4but liited in a quite sall range of AoDs, which eans the side lobe leakage causes high interference to other devices in a very narrow range of directions. For the sectored pattern, the array gains corresponding to all the directions outside the ain lobe are assued to be equal to the first side lobe gain of the actual pattern. Obviously, this approxiation leads to deviation fro the actual antenna pattern and exaggerates the effect of side lobe leakage. The larger the nuber of antennas, the greater the deviations. It is even worse for networks where different kinds of devices are likely to be equipped with different nubers of antennas. Thus, taking both accuracy and tractability into consideration, the cosine antenna pattern is adopted in the following analysis, which akes it possible to investigate the ipact of heterogeneous antenna arrays on the perforance. C. SINR Analysis We assue that the desired link between the transitterreceiver pair is in the LOS condition with deterinistic path loss r α k,l given that the typical receiver belongs to the k- th tier. In fact, if the receiver was associated with a NLOS transitter, the link would quite likely be in outage due to the severe propagation loss and high noise power at -wave bands as well as the fact that the interferers can be arbitrarily close to the receiver. Different path loss exponents are applied to the cases of LOS and NLOS paths. We denote by l k x the rando path loss function associated with the interfering transitter location x Φ k, given by { ax{d, x } l k x α k,l w.p.p LOS x ax{d, x } α 3 k,n w.p. 1 P LOS x,

where all l k x x Φk are independent. In addition to the distance-dependent path loss, we assue independent Nakagai fading for each path, which is a sensible odel given the LOS-dependent -wave scenarios. Different Nakagai fading paraeters and M k,n are assued for LOS and NLOS paths in the k-th tier, where and M k,n are positive integers. The power fading coefficient between node x Φ k and the origin is denoted by h x, which follows a gaa distribution GaaM, 1 M with M {, M k,n }, and all h x are utually independent and also independent of the point process. For the typical receiver, the interferers outside the LOS ball are NLOS and thus can be ignored due to the severe path loss over the large distance at least R. As a result, the analysis for the network originally coposed by the ultitier PPPs reduces to the analysis of a finite network region, naely the disk of radius R centered at the origin. Based on this odel, the interference fro tier k at the origin is I k μ k G k ϕ x h x l k xb k x, 4 x Φ k where G k ϕ x is the directional antenna gain function with spatial AoD ϕ x following 2, and B k x is a Bernoulli variable with paraeter q k to indicate whether x transits a essage to its receiver. Due to the incorporation of the blockages, the LOS transitters belonging to the k-th tier with LOS propagation to the typical receiver for a PPP Φ k,l with density p L λ k, while Φ k,n with density p N λ k is the transitter set with NLOS propagation, where p L + p N 1such that Φ k Φ k,l Φ k,n. Then, the interference fro tier k can be rewritten as I k I k,l + I k,n μ k G k ϕ x h x l k xb k x. 5 s {L,N} x Φ k,s Without loss of generality, the noise power is set to one. Conditioning on that the typical receiver belongs to the k- th tier, the corresponding receiver SINR, denoted as SINR k, is then given by SINR k S k 1+I μ k N k h x r αk,l 1+ μ i G i ϕ x h x l i xb i x, 6 i [K] x Φ i where [K] {1, 2,..., K}. III. ANALYSIS OF SUCCESS PROBABILITY A. Exact Results Our first result in this section is an exact expression for the success probability PSINR >θ conditioning on the typical receiver belonging to tier k. Theore 1. Letting ɛ k r α k,l μ k N k, the link success probability of the typical active device belonging to the k-th tier equipped with N k antennas, denoted by P k θ, is given by P k θ 1 u L u uθɛk, 7! where Lu expηu, the superscript stands for the -th derivative of Lu, and ηu u 2 p s λ i q i πr 2 N i M Mi,s i,s 4rdxdr Mi,s +uμ i N i cos 2 Mi,s. 8 x ax{r, d } αi,s L u is given recursively by L u 1 n 1 n η n ul n u, 9 where the n-th derivative of ηu follows η n u 1n 1+ 8ΓM i,s +nm Mi,s i,s p s λ i q i N i ΓM i,s μ i N i cos 2 x ax{r, d } αi,s n rdxdr Mi,s +uμ i N i cos 2 x ax{r, d } αi,s Mi,s+n. 1 Proof: See Appendix A. According to the proposed odel, devices in different tiers differ in the nuber of antennas and follow ultiple utually independent hoogeneous PPPs. Therefore, the total SINR distribution of the -wave D2D network can be coputed using the law of total probability as follows. Corollary 1. For the overall active user, the link success probability is P θ k [K] λ k q k i [K] λ iq i P k θ. 11 Proof: Let us consider the point process of all active receivers those who have active transitters and focus on the typical receiver of this point process. Based on Theore 1, which gives the link success probability conditioned on this typical receiver belonging to the k-th tier, the overall link success probability is obtained as P θ k [K] Px Φ kp k θ, 12 where Px Φ k is the probability that the typical receiver λ belongs to the k-th tier. Since Px Φ k k q k,we i [K] λiqi obtain 11. B. Bounds on Success Probability Note that though the Laplace transfor of the aggregate interference can be easily evaluated by nuerical integration, the corresponding n-th derivative needs tedious and extensive coputations, which akes the exact calculation inefficient. Thus, we obtain upper and lower bounds for the exact results by using bounds of the incoplete gaa functions.

Success Probability 1.9.8.7.6.5.4.3 Upper Bound per 13.2 Lower Bound per 13 w. β 1 k Exact w. N [4,16,64] per 7.1 Exact w. N [4,64,256] per 7 Exact w. N [16,64,256] per 7 2 15 1 5 5 1 15 2 25 θ db Fig. 2. The success probability for different configurations of antenna arrays. Success Probability 1.9.8.7.6.5.4.3.2.1 Upper Bound per 13.5 per 7.3 per 7.9 per 7 2 15 1 5 5 1 15 2 25 θ db Fig. 3. The success probability for different LOS probabilities. Theore 2. Let β k [Γ1+ ] 1/ and ˆP k θ 1 1 +1 Lu uθβk ɛ k. 13 For K-tier Poisson -wave D2D counication networks, the link success probability of the active device belonging to the k-th tier P k θ is upper bounded by ˆP k θ, while a lower bound on P k θ, denoted by ˇP k θ, is achieved by setting β k 1in 13. Proof: It is known fro [1] that 1 [1 exp x] M ΓM,x 1 [1 exp βx] M, 14 where β [Γ1+M] 1/M, ΓM,x ΓM,x/ΓM, and the equality holds only if M 1. Based on this inequality, the lower and upper bounds on the link success probability are obtained as follows. Letting β k [Γ1+ ] 1/ and ˆP k θ be the upper bound on P k θ, wehave [ ˆP k θ 1 E 1 exp θβ k ɛ k 1 + I ] 1 M k,l 1 [ 1 +1 E exp θβ k ɛ k 1 + I ] 1 +1 Lu uθβk ɛ k. 15 By substituting 8 into 15, we obtain the upper bound for the link success probability. Fro 14, the lower bound for the link success probability ˇP k θ is then obtained by setting β k 1in 15. Reark 1. Copared with the exact results for the SINR distribution, both bounds give uch sipler expressions without requiring the derivatives for Lu at u, where Lu is the product of ultiple exponential functions with integral expressions in the exponents. Thus the effort for the coputation of the SINR distribution is significantly reduced. Siilar to the exact results, we can obtain bounds for the overall link success probability by Corollary 1. IV. NUMERICAL RESULTS In this section, we give soe nuerical results of the success probability for the heterogeneous -wave D2D networks, where K 3, λ i.1, μ i 2, q i 1, α i,l 2.5, α i,n 4, M i,l 4, M i,n 2, i [K], r 2, d 1, R 2 are default values. Fig. 2 illustrates the success probability as a function of θ for different configurations of antenna arrays in a 3-tier -wave D2D network. It can be seen that the upper bound 13 derived for the success probability is quite tight, and the horizontal gap between the bounds and the exact curve is nearly constant, with the upper bound less than.5 db and the lower bound about 2.2 db away. Moreover, it is also observed that the configuration with larger antenna arrays perfors better in ters of the success probability, since larger antenna arrays produce narrower transission beas, which liit the interference signal to a certain direction, causing less interference to the receivers. Coparing the curves corresponding to the cobinations of antenna arrays [4, 64, 256] and [16, 64, 256], there is a critical point at θ 1 db, where the success probability with [4, 64, 256] is quite close to but saller than that with [16, 64, 256]. This is because when the SINR threshold is large, the successful transissions ostly occur at the transitters with larger antenna arrays e.g., N 3 256. In this case, the desired signal between two cases is alost at the sae level while the interference suffered in the case of N [4 64 256] is ore severe than that in the case of N [16 64 256]. Fig. 3 shows the ipact of LOS probability p L on the success probability in a 3-tier -wave D2D network, where N 1 4, N 2 16, and N 3 64. It is observed that the link

success probability deteriorates with the increase of p L. The reason is that a high LOS probability eans the propagation environent suffers fro less blockage and, accordingly, the interfering signal experiences less propagation loss than that in the blocked case. As a result, the aggregate interference at receivers will becoe ore severe, thereby decreasing the success probability. V. CONCLUSION In this paper, we analyzed the perforance of -wave D2D networks where devices are diversified in their directional antenna arrays. Interestingly, we found that the first side lobe gain of a larger antenna array can be close to the ain lobe gain of a saller one erely in a liited spatial direction, and deonstrated that the ostly used sectored odel cannot reflect this very iportant feature and thus is not suitable for -wave networks coposed of increasingly diverse devices. In contrast, the cosine antenna pattern has superior accuracy and siilar analytical tractability. By adopting this consine antenna pattern, we derived the success probability of the typical receiver and provided tight bounds to siplify the exact results. It was observed that the introduction of large antenna arrays in -wave networks bring iense benefits in ters of the success probability reliability, which can not only iprove the desired signal but also significantly reduce the interference. Overall, the results provide valuable engineering insights to help network operators deploy -wave D2D networks that satisfy stringent reliability requireents. ACKNOWLEDGMENT The work of N. Deng and Y. Sun has been supported by National Natural Science Foundation of China 617171, Fundaental Research Funds for the Central Universities DUT16RC3119, and the work of M. Haenggi has been supported by the U.S. NSF grant CCF 152594. APPENDIX A. Proof of Theore 1 Proof: The link success probability of a device belonging to the k-th tier, denoted by P k θ, is expressed as [ ] P k θ E Γ,θɛ k 1 + I 1 1 [ ] E e θɛ k1+i θɛ k1 + I! u L u uθɛk! where Γx, y Γx, y/γx is the noralized incoplete gaa function, Lu E[e ui+1 ] is the Laplace transfor of the interference and noise, and the superscript stands for the -th derivative of Lu. Due to the independence of the K tiers, we have Lu e u L Ii,s u, 16 where L Ii,s u follows as L Ii,s u E[exp ui i,s ] [ ] q i E Mi,s +1 q i 1+uμi G i ϕ x l i x/m i,s x Φ i,s 2 exp p s λ i q i πr 2 N i 4M Mi,s i,s rdxdr Mi,s +uμ i N i cos 2 Mi,s.17 x ax{r, d } αi,s Letting Lu expηu and thus L 1 u η 1 ulu, L u can be calculated recursively according to the forula of Leibniz for the higher-order derivative of the product of two functions, given by L u d 1 1 1 du L1 u η n ul n u, n n 18 where the n-th derivative of ηu is easily given by 1. REFERENCES [1] Cisco, Cisco visual networking index: Global obile data traffic forecast update, 216-221, http://www.cisco.co/c/en/us/ solutions/collateral/service-provider/visual-networking-index-vni/ obile-white-paper-c11-52862.pdf, Feb. 217. [2] K. Venugopal, M. C. Valenti, and R. W. Heath, Device-to-device illieter wave counications: Interference, coverage, rate, and finite topologies, IEEE Transactions on Wireless Counications, vol. 15, no. 9, pp. 6175 6188, Sept 216. [3] X. Yu, J. Zhang, M. Haenggi, and K. B. Letaief, Coverage analysis for illieter wave networks: The ipact of directional antenna arrays, IEEE Journal on Selected Areas in Counications, vol. 35, no. 7, pp. 1498 1512, July 217. [4] N. Deng and M. Haenggi, A fine-grained analysis of illieter-wave device-to-device networks, 217, accepted at IEEE Transactions on Counications. Available on IEEE Xplore Early Access. [5] M. Haenggi, Stochastic geoetry for wireless networks. Cabridge University Press, 212. [6] S. Singh, M. N. Kulkarni, A. Ghosh, and J. G. Andrews, Tractable odel for rate in self-backhauled illieter wave cellular networks, IEEE Journal on Selected Areas in Counications, vol. 33, no. 1, pp. 2196 2211, Oct 215. [7] J. G. Andrews, T. Bai, M. N. Kulkarni, A. Alkhateeb, A. K. Gupta, and R. W. Heath, Modeling and analyzing illieter wave cellular systes, IEEE Transactions on Counications, vol. 65, no. 1, pp. 43 43, Jan 217. [8] T. S. Rappaport, G. R. MacCartney, M. K. Saii, and S. Sun, Wideband illieter-wave propagation easureents and channel odels for future wireless counication syste design, IEEE Transactions on Counications, vol. 63, no. 9, pp. 329 356, Sept 215. [9] C. A. Balanis, Antenna Theory: Analysis and Design. Hoboken, NJ, USA: John Wiley & Sons, 25. [1] H. Alzer, On soe inequalities for the incoplete gaa function, Matheatics of Coputation, vol. 66, no. 66, pp. 771 778, 1997.