Nearest Neighbor and Contact Distance Distribution for Binomial Point Process on Spherical Surfaces
11 Nearest Neighbor and Contact Distance Distributionfor Binomial Point Process on Spherical Surfaces
Anna Talgat, Mustafa A. Kishk and Mohamed-Slim Alouini
Abstract —This letter characterizes the statistics of the contactdistance and the nearest neighbor (NN) distance for binomialpoint processes (BPP) spatially-distributed on spherical surfaces.We consider a setup of n concentric spheres, with each sphere S k has a radius r k and N k points that are uniformly distributed onits surface. For that setup, we obtain the cumulative distributionfunction (CDF) of the distance to the nearest point from twotypes of observation points: (i) the observation point is not apart of the point process and located on a concentric sphere witha radius r e < r k ∀ k , which corresponds to the contact distancedistribution, and (ii) the observation point belongs to the pointprocess, which corresponds to the nearest-neighbor (NN) distancedistribution. Index Terms —Stochastic geometry, binomial point process,distance distribution.
I. I
NTRODUCTION
Cellular coverage has become one of the top needs of themodern society due to its importance in various applicationssuch as healthcare, remote education, industry, and muchmore. For that reason, it is important to ensure cellularcoverage all over the globe including remote areas, ruralregions, and many other under-served locations. However, dueto the lack of infrastructure, majority of these areas receivebad coverage due to lack of incentive for network operators toinvest in these locations. Recent advances in Low Earth Orbit(LEO) satellite communications are providing a promisingsolution to the coverage problem in under-served locations [1],[2]. In particular, by deploying satellite gateways in suchregions, coverage can be significantly enhanced using satellitecommunications. This system architecture requires less ex-penses compared to typical cellular architectures. In particular,it does not require the extension of optical fibers to suchremote locations, which is typically needed to provide core-connection to the deployed base stations. This is replaced inthe new setup with the wireless link between the gateway andthe satellite. The high potential of satellite communicationshas motivated many recent works, such as [3], [4], to identifytechnological advances and highlight open problems in thisfield. It has also motivated companies such as SpaceX to getpermission to build a constellation of 4425 LEO satellitesto supply low latency communication. They have a plan forsetting 1600 satellites in 1150 km altitude orbits at the firststage [5].The spatial distribution of the LEO satellite strongly affectsthe performance of the satellite communication systems. Inthis paper, we propose to model the locations of the satellites
The authors are with King Abdullah University of Scienceand Technology (KAUST),Thuwal 23955-6900,Saudi Arabia(e-mail:[email protected];[email protected];[email protected] using tools from stochastic geometry. Stochastic geometry isone of the mathematical tools that enable tractable modeling ofvarious types of wireless networks and analyzing their proper-ties [6]. We develop a new tractable approach where we modelthe locations of the LEO satellites as a BPP on a sphere. Thedeveloped framework is essential for studying the performanceof the LEO satellite communication system. However, it isfirst needed to understand the fundamental characteristics ofthe distances emerging from this point process, which is themain contribution of this paper. .
A. Related work
Characterizing the statistics of the distances between variouscomponents of the wireless networks is essential for rigorousperformance analysis. Relevant literature has mainly focusedon spatial point processes on a 2D plane. For instance, authorsin [7]–[9] characterize the CDFs of contact and nearest-neighbor distances for Poisson hole processes and Poissoncluster process, respectively. Statistical research on point pro-cesses on the sphere could date back to the 1970s, such as astudy of random sets on the sphere by Mile’s [10]. Statisticalmethods that are developed for analyzing a distribution ofpoints on a spherical region, including modeling and esti-mating techniques for a specified model, are studied in therecent work [11]. However, statistical analysis for contact andnearest neighbor distances for point processes on sphericalsurfaces are still surprisingly underdeveloped. It is importantto point out to the difference between the analysis of pointprocesses in 3-dimensional (3D) plane, which is relativelywell-understood part of literature [12], and the analysis ofpoint processes solely distributed on a spherical surface. Onlyvery recently, during writing this paper, a new work has tackledthis problem while modeling the location of the satellites as aBPP on a sphere [13]. The main differences between this paperand [13] is: (i) we study a more general model where pointsare randomly located on multiple concentric spheres, whichresembles the scenario of having the satellites at multiplealtitudes, and (ii) in addition to deriving the contact distancedistribution, we also derive the nearest neighbor distributionfor a satellite on the k th sphere, which is an important metricthat has its own value for studying routing between LEOsatellites. A deterministic version of the setup considered inthis paper was studied in [14], [15] with the objective ofoptimizing the LEO satellite constellations.. B. Contributions
The main contributions of this work are as follows. First,we model n concentric spheres with N k points uniformlydistributed on each sphere ∀ k . Then we use tools from a r X i v : . [ c s . I T ] J un Fig. 1. System model for n level of spheres concentric with the Earth. stochastic geometry to provide a new tractable model forstudying distance distributions in satellite networks locatedon spherical surfaces. In particular, we model the location ofpoints as a spherical BPP to study the distribution of nearest-neighbor distance for two different locations of observationpoint which are (i) observation point is not a part of the pointprocess and located on the Earth, and (ii) observation point isa part of the point process and located on k th sphere. Closed-form expressions for the distance distributions are derivedand verified using Monte-Carlo simulations. Finally, with theassistance of numerical results, various system-level insightsare drawn and discussed.II. S YSTEM M ODEL
As stated above, the analysis in this paper is motivated bythe recent advances in the area of LEO satellite communicationsystems. Hence, our objective is to provide a model thatcaptures two kinds of communication links: (i) links betweengateways on the earth and LEO satellites, and (ii) inter-satellitelinks between LEO satellites. For the former, it is important toderive the distribution of the distance between a point on theearth and its nearest LEO satellite. For the latter, in order tostudy backhaul communication between LEO satellites, it isimportant to derive the distribution of the distance betweena given LEO satellite and its nearest neighbor. Given thatcurrent LEO satellite deployment plans are considering variousaltitudes with various number of satellites at each altitude, weconsider a system composed of n concentric spheres, denotedby S k ⊂ R , ∀ ≤ k ≤ n . On each sphere, a point process Φ k composed of N k points are uniformly distributed. Eachsphere is defined by the altitude a k (altitude of k th spherefrom the surface of Earth) and the radius r k = r e + a k ,where r e is the radius of the earth. Hence, the consideredpoint process is defined as Φ = (cid:83) nk =1 Φ k on (cid:83) nk =1 S k . Wedenote its corresponding counting measure by N , such that N ( A ) denotes the number of points in Φ falling in the region A ⊆ (cid:83) nk =1 S k . For each BPP Φ k , fixed number N k of pointsare independently and uniformly distributed on a sphere S k defined as S k ∆ = { ( r k , ϕ , θ ): r k = r e + a k , ≤ ϕ ≤ π , ≤ θ < π } ,where the ( r k , ϕ , θ ) represent the spherical coordinates in R .The nearest neighbor or contact distance (depending onthe definition of the observation point) is the distance from Fig. 2. Observation point located on the i th sphere. the observation point to the nearest point in Φ and is given by D . The corresponding distribution F D ( d ) ∆ = P ( D < d ) is thenearest neighbor or contact distance distribution function. A. Scenario-1 Description
The observation point is located on the Earth. The corre-sponding distribution is F D ( d ) ∆ = P ( D < d ) = 1 − n (cid:89) k =1 P ( D k ≥ d ) , where P ( D k ≥ d ) = ¯ F D k ( d ) = 1 − F D k ( d ) is the complemen-tary cumulative distribution function (CCDF) of the contactdistance D k from the observation point to the nearest pointon k th sphere.By definition, we know that if d < a k then F D k ( d ) = 0 . For d > a k , F D k ( d ) is the probability that the number of pointson a given spherical cap A k at height h ( d, k, is greater thanzero: F D k ( d ) ∆ = P ( D k < d ) = P ( N ( A k ) > . Hence, the CCDF of D k can be computed as follows. P ( D k ≥ d ) = P ( N ( A k ) = 0) = [ P ( z k < z ( d, k, N k , where z k = r k cos ϕ , and z ( d, k,
0) = r k − h ( d, k, . WithPythagoras’ theorem, we can easily derive the expression of h ( d, k, . Assuming that the communication between anypoint on the earth and an LEO satellite requires a Line-of-Sight (LoS), the maximum distance, d max ( k, , that can betaken from the observation point also forms a spherical cap A max , k with height h max ( k, . When the number of pointsin A max , k is zero, it means that there are no points in S k that have an LoS with the observation point. Hence, for thatscenario, we assume that D k = ∞ . As a result, the CCDF of D k for d > d max ( k, is P ( D k ≥ d ) = P ( N ( A max , k ) = 0) = [ P ( z k < z max ( k, N k , where z max ( k,
0) = r e . Combining all the conditions whichare d < a k , a k < d < d max ( k, and d > d max ( k, , we canderive F D k ( d ) ∀ k . B. Scenario-2 Description
The observation point is located on the S i , and the point ispart of the point process. So, the corresponding distribution is F D ( d ) = 1 − (cid:2) i − (cid:89) k =1 P ( D k,i ≥ d ) (cid:3)(cid:2) P ( D i ≥ d ) (cid:3)(cid:2) n (cid:89) k = i +1 P ( D k,i ≥ d ) (cid:3) , TABLE ISUMMARY OF NOTATION
Notation Description Φ k ; N k BPP modeling the locations of point; number of point on k th sphere S k ; r k ; a k k th sphere with radius r k and altitude a k to the surface of the Earth. h ( d, k, i ) height of spherical cap, ( k, i ) represent the spheres where nearest distance and observation point are located respectively. h max ( k, i ) height of spherical cap formed by the maximum distance d max ( k, i ) . d(m) F D ( d ) TheoreticalSimulation
Fig. 3. Scenario-1: CDF of nearest-neighbor distance for different numberof multi-level spheres with values: a k circle = [1110 1150 1275 1325] and N k circle =[50 40 25 15]; a k square = [1110 1150 1275 1325 1500 1700] and N k square =[75 65 55 45 25 15]; a k diamond = [1110 1150 1275 1325] and N k diamond =[105 85 60 35]. where D k,i is the distance between the observation point andthe nearest point on S k , and D i is the distance between theobservation point and the nearest point on the same sphere S i .Here, the CCDFs correspond to NN distance distribution for(a) below the i th sphere, (b) on the i th sphere and (c) above the i th sphere respectively. Fig.2. shows the system model for thecase (c). As can be seen from the figure, we have two sphericalcaps formed on S k with corresponding heights h ( d, k, i ) and h max ( k, i ) . We follow the same procedure as Scenario-1 toderive the complete CDF for each case where conditions are | a k − a i | < d, | a k − a i | < d < d max ( k, i ) and d > d max ( k, i ) for (a) and (c) and d < d max ( k, i ) and d > d max ( k, i ) for case(b). Also, we get h ( d, k, i ) , h max ( i, i ) and d max ( k, i ) for eachcase separately by using Pythagoras’ theorem.III. D ISTANCE D ISTRIBUTION
In this section, we determine the distribution of the nearestdistance from a specified observation point for a general BPP.
Theorem 1 (Scenario-1: Contact distance distribution) . F D ( d ) ∆ = P ( D < d ) = 1 − n (cid:89) k =1 P ( D k ≥ d ) , (1) where the CCDF of D k is P ( D k ≥ d ) = , d < a k (cid:2) − π arccos (1 − d − a k r e r k )) (cid:3) N k , a k ≤ d ≤ d max ( k, (cid:2) − π arccos( r e r k ) (cid:3) N k , d > d max ( k, , where d max ( k,
0) = (cid:112) r e a k + a k . d(m) F D ( d ) TheoreticalSimulation
Moving the observation point on S , S , S and S respectively. Fig. 4. Scenario-2: CDF of the distance to the nearest neighbor for a setupcomposed of 4 spheres as follows: S = [1000 500], S = [1325 400], S =[1625 325] and S = [2000 280], where S k = [ a k N k ] . Proof: See Appendix A.
Theorem 2 (Scenario-2: Nearest neighbor distance distribu-tion) . F D ( d ) ∆ = P ( D < d ) = 1 − n (cid:89) k =1 P ( D k ≥ d ) , (2) where the CCDFs of D k are described below.For k = i , P ( D i ≥ d ) = (cid:2) − π arccos (1 − d r i ) (cid:3) N i − , d < d max ( i, i ) (cid:2) − π arccos (1 − r e r i ) (cid:3) N i − , d > d max ( i, i ) , where d max ( i, i ) = 2 (cid:112) r i − r e .For k (cid:54) = i , P ( D k ≥ d ) = , d < | a k − a i | (cid:2) − π arccos (1 − d − ( a i − a k ) r i r k ) (cid:3) N k , | a k − a i | ≤ d ≤ d max ( k, i ) (cid:2) − π arccos (1 − ( r i + r k ) − d ( k,i )2 r i r k ) (cid:3) N k , d > d max ( k, i ) , where d max ( k, i ) = (cid:112) r k − r e + (cid:112) r i − r e .Proof: See Appendix B. IV. N
UMERICAL R ESULTS
In this section, we provide numerical results for the deriveddistance distributions. As shown in Fig. 3 and Fig. 4, Theorems1 and 2 are perfectly matching with simulation, which affirmsthe accuracy of our analysis.In Fig.3, we plot the CDF of the contact distance forthree different system setups, as described in the caption. Weobserve that the number of satellites on each sphere, when thealtitudes are fixed, have a high influence on the distribution ofthe contact distance.
In Fig.4, we plot the CDF of the distance to the nearestneighbor distance as the observation point moves from S to S n where n = 4 . We notice that the distance to the nearestneighbor gets higher as the observation point moves from innerspheres to outer spheres.V. C ONCLUSION
In this letter, we presented a stochastic geometry frameworkto model the spatial distribution of LEO satellite communica-tion systems. For that setup, we derived the exact analyticalexpressions for the CDFs of the nearest neighbor and the con-tact distance where fixed numbers of points are independentlyand uniformly distributed on a number of concentric spheres.The provided setup can be used in various applications suchas (i) studying the coverage probability of the LEO-aidedcommunication networks and (ii) studying the routing amongLEO satellites. A
PPENDIX
A. Proof of Theorem 1
The proof of Theorem 1 and 2 follow the same steps. • If d < a k , we have P ( D k ≥ d ) = 1 ∀ k . • If a k < d < d max ( k, , then we have the contact distancedistribution, P ( D k ≥ d ) = P ( N ( A k ) = 0)= [ P ( z k < z ( d, k, N k = [ P ( r k cos ϕ < z ( d, k, N k = [ P ( ϕ > arccos z ( d, k, r k ) + P ( ϕ < − arccos z ( d, k, r k )] N k = [1 − π arccos( z ( d, k, r k )] N k , where h ( d, k,
0) = d − a k r e and d max ( k,
0) = (cid:112) r e a k + a k are easily derived by Pythagoras’ theorem and z ( d, k,
0) = r k − h ( d, k, . • If d > d max ( k, , skipping all the intermediate steps we get P D k ≥ d ) = P ( N ( A max , k ) = 0) = [ P ( z k < z max ( k, N k = [1 − π arccos( z max ( k, r k )] N k , where z max ( k,
0) = r e ∀ k . This concludes the proof.
B. Proof of Theorem 2
For S k and k (cid:54) = i : • If d < | a k − a i | , we have P ( D k ≥ d ) = 1 . • If | a k − a i | < d < d max ( k, i ) , P ( D k ≥ d ) = P ( N ( A k ) = 0) = [ P ( z k < z ( d, k, i ))] N k = [1 − π arccos( z ( d, k, i ) r k )] N k , where h ( d, k, i ) = d − ( a i − a k ) r i , z ( d, k, i ) = r k − h ( d, k, i ) ,and d max ( k, i ) = (cid:112) r k − r e + (cid:112) r i − r e . • If d > d max ( k, i ) , P ( D k ≥ d ) = P ( N ( A max , k ) = 0) = [ P ( z k < z max ( k, i ))] N k = [1 − π arccos( z max ( k, i ) r k )] N k , where h max ( k, i ) = ( r i + r k ) − d ( k,i )2 r i and z max ( k, i ) = r k − h max ( k, i ) . For S k and k = i , the observation point is part of the pointprocess Φ i , the remaining point process becomes a sphericalBPP with N i − points. • If d < d max ( i, i ) , then P ( D i ≥ d ) = P ( N ( A i ) = 0) = [ P ( z i < z ( d, i, i ))] N i − = [1 − π arccos( z ( d, i, i ) r i )] N i − , where h ( d, i, i ) = d r i , d max ( i, i ) = 2 (cid:112) r i − r e and with z ( d, i, i ) = r i − h ( d, i, i ) . • If d > d max ( i, i ) , then P ( D i ≥ d ) = P ( N ( A max , i ) = 0) = [ P ( z i < z max ( i, i ))] N i − = [1 − π arccos( z max ( i, i ) r i )] N i − , where h max ( i, i ) = r e r i and z max ( i, i ) = r i − h max ( i, i ) .This concludes the proof..R EFERENCES[1] S. Dang, O. Amin, B. Shihada, and M.-S. Alouini, “What should 6Gbe?”
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