Characterizing Shortest Paths in Road Systems Modeled as Manhattan Poisson Line Processes

Transcription

1 Characterizing Shrtest Paths in Rad Systems Mdeled as Manhattan Pissn Line Prcesses Vishnu Vardhan Chetlur, Harpreet S. Dhilln, Carl P. Dettmann arxiv:8.332v cs.it 28 Nv 28 Abstract In this paper, we mdel a transprtatin netwrk by a Cx prcess where the rad systems are mdeled by a Manhattan Pissn line prcess MPLP and the lcatins f vehicles and desired destinatin sites, such as gas statins r charging statins, referred t as facilities, are mdeled by independent D Pissn pint prcesses PPP n each f the lines. Fr this setup, we characterize the length f the shrtest path between a typical vehicular user and its nearest facility that can be reached by traveling alng the streets. Fr a typical vehicular user starting frm an intersectin, we derive the clsedfrm expressin fr the exact cumulative distributin functin CDF f the length f the shrtest path t its nearest facility in the sense f path distance. Building n this result, we derive an upper bund and a remarkably accurate but apprximate lwer bund n the CDF f the shrtest path distance t the nearest facility fr a typical vehicle starting frm an arbitrary psitin n a rad. These results can be interpreted as nearestneighbr distance distributins in terms f the path distance fr this Cx prcess, which is a key technical cntributin f this paper. In additin t these analytical results, we als present a simulatin prcedure t characterize any distance-dependent cst metric between a typical vehicular user and its nearest facility in the sense f path distance using graphical interpretatin f the spatial mdel. We als discuss extensin f this wrk t ther cst metrics and pssible applicatins t the areas f urban planning, persnnel deplyment and wireless cmmunicatin. Index Terms Stchastic gemetry, Manhattan Pissn line prcess, Cx prcess, path distance, shrtest path. I. INTRODUCTION Transprtatin systems have been studied by researchers frm varius fields such as gegraphy, peratins research, and recently, cmplex netwrk thery, 2. It is quite intuitive t mdel transprtatin netwrks as graphs where varius sites are mdeled as ndes f the graph and the rutes between them are represented by the edges f the graph 3. Several spatial netwrk mdels have been prpsed in the literature ranging frm simple prximity graphs where the edges f the graph are defined by a deterministic set f rules t mre sphisticated mdels where the prbability f an edge between a pair f ndes is a functin f psitin f the ndes 4, 5. While these netwrk mdels are useful in studying the tplgical prperties f the netwrk such as cnnectivity, clseness and centrality, they d nt capture sme f the gemetric aspects such as the cntinuity f street segments 2, 6. S, instead V. V. Chetlur and H. S. Dhilln are with Wireless@VT, Department f ECE, Virginia Tech, Blacksburg, VA {vishnucr, hdhilln}@vt.edu. C. P. Dettmann is with Schl f Mathematics, University f Bristl, UK carl.dettmann@bris.ac.uk. The supprt f the US Natinal Science Fundatin Grant IIS and UK Engineering and Physical Sciences Research Cuncil Grant EP/N2458/ is gratefully acknwledged. Manuscript updated: Nvember 29, 28. f taking a nde-centric apprach where the streets were cnsidered as links between varius sites in a city, we visualize these sites t be lcated n an underlying street netwrk. Althugh relatively sparse, there have been a few wrks in the stchastic gemetry literature, where street netwrks were mdeled by a set f randm lines, referred t as a line prcess in the 2D Euclidean space 7 9. A well-knwn cannical line prcess mdel in the literature is the Pissn line prcess PLP,. Inspired by the grid-like structure f rad layuts f many cities, e.g., New Yrk, as well as the nature f mathematical prblems t be studied in this paper, we limit ur attentin t a special case f PLP called Manhattan Pissn line prcess MPLP in this paper 2, 3. Further, the lcatins f vehicles, and the lcatins f varius places such as gas statins r charging statins fr electric vehicles, which will hencefrth be referred t as simply facilities, are mdeled as D Pissn pint prcess PPP n each line f the line prcess, thereby frming a dubly stchastic Pissn pint prcess r Cx prcess 4, 5. T the best f ur knwledge, this is the first paper t emply Cx prcess driven by MPLP t study urban rad transprtatin systems. S, the characterizatin f even sme f the basic prperties such as the length f the shrtest path between neighbring pints is still an pen prblem and is the main fcus f this paper. In particular, we derive the distributin f the length f the shrtest path between an arbitrarily chsen vehicle and its nearest facility in the sense f path distance. A. Prir Art We will first review sme f the well-knwn netwrk mdels in the literature. One f the first and mst imprtant netwrk mdels is the Erdős-Rényi ER graph characterized by the number f vertices and the prbability f an edge between any tw vertices in the graph 6, 7. Hwever, the small clustering cefficient f the ER graphs des nt agree with the empirical data frm many real-wrld netwrks. In 8, Barabási and Albert prpsed a mdel that exhibits preferential cnnectivity, and has imprved clustering cefficient and scale-free prperties. Spatial netwrks, in which vertices have a physical lcatin and mst links are ver shrt distances, retain a large clustering cefficient but usually withut extremely high degree ndes. Examples include the Watts-Strgatz small-wrld netwrk 9 and the Hammersley netwrk prpsed by Aldus 4, 2. In the latter, the vertices are defined by an infinite PPP and each vertex has exactly fur edges. Hwever, ne f the majr limitatins f mst f these netwrk mdels is that they d nt accurately capture the

2 2 cntinuity f streets, especially in mdeling urban street netwrks. Therefre, we mdel an urban transprtatin netwrk by a Cx prcess driven by a MPLP. As will be evident frm the technical discussins, such mdels are particularly wellsuited fr the statistical analysis f the gemetric prperties, such as path lengths. The idea f using a PLP t mdel rad systems was first prpsed by Baccelli in 7 t study the handver rate in cellular netwrks. While there have been ther mdels where the rads are mdeled by the edges f a Pissn Vrni tessellatin PVT r Pissn Delaunay tessellatin PDT, PLP is ften preferred due t its analytical tractability This spatial mdel was later emplyed in several wrks in the area f wireless cmmunicatins, particularly vehicular netwrks, where the lcatins f transmitters and receivers were mdeled by a Cx prcess driven by a PLP r MPLP 9, 2, 2, Hwever, nly Euclidean distances between the pints f the Cx prcess were cnsidered in all these wrks. Nnetheless, there have been a few wrks in the literature where the distance measured alng the line segments f the path was cnsidered. In 3, the authrs have cmputed the path-lss fr a signal prpagating alng the streets f an urban netwrk mdeled by a MPLP. In 8, the authrs have mdeled a tw-tier wired netwrk by a Cx prcess driven by a PLP and characterized the mean shrtest path alng the lines between a cmpnent and its nearest neighbr in the Euclidean distance sense. Therefre, this is the first paper t derive the distributin f the length f the shrtest path between a typical pint and its nearest neighbr in the sense f path distance in a Cx prcess driven by a MPLP. Mre technical details abut ur cntributin are prvided next. B. Cntributins In this paper, we mdel the randm spatial layut f rads by a MPLP. We further mdel the randm lcatins f vehicles and facilities n each rad as independent D PPPs n each line f the MPLP. Fr this setup, we characterize the least cst path t arrive at a facility fr a typical vehicular user, which is an arbitrarily chsen vehicle in the spatial mdel pint prcess f vehicles. Owing t its analytical tractability, the cst metric that we mainly fcus n is the length f the path traveled by the typical vehicular user and its variants, such as mntnic functins f this distance. We als present techniques t cmpute ther cst metrics such as time f travel thrugh Mnte-Carl simulatins. We then discuss the insights that we gain frm the analytical results and als pssible extensins f these results t varius applicatins. Mre technical details are discussed next. Distance Distributins. We derive the cumulative distributin functin CDF f the length f the shrtest path frm a typical vehicular user t its nearest facility in the sense f path distance. It is imprtant t nte that the clsest facility t the typical vehicle in the sense f Euclidean distance is nt necessarily the ne with the least path length. Fr a typical vehicular user lcated at an intersectin, we derive the exact clsed-frm expressin fr the CDF f the shrtest path distance. Building n this result, we prpse an upper bund n the CDF f the shrtest path distance fr a typical vehicular user lcated at sme arbitrary psitin n the line and its nearest facility in the sense f path distance. We als derive a remarkably tight but apprximate lwer bund n the CDF f this distance. Simulatin Techniques. Cmplementary t the analytical characterizatins, we als present a simulatin methd t cmpute any distance-based cst metric such as travel time r fuel cnsumed by the typical vehicular user t reach the nearest facility. The graphical interpretatin f the spatial mdel enables the applicatin f sme well-knwn algrithms in the literature that wuld be useful in reducing the run-time f simulatins. Insights and Applicatins. We demnstrate the utility f ur results in addressing sme statistical prblems in the areas f transprtatin, infrastructure planning, persnnel deplyment and wireless cmmunicatin. Using ur analytical results, ne can easily determine the minimum density f facilities that is necessary t ensure that the shrtest path distance fr a typical vehicular user t its nearest facility in the sense f path distance is smaller than a desired value with sme prbability. We can als gain such insights fr travel time which is a critical metric fr estimating the respnse time f medical help r plice persnnel in emergency situatins. In wireless cmmunicatins, the analytical techniques develped in this paper culd be useful in characterizing the received signal pwer in millimeter wave mmwave frequency cmmunicatins where the dminant signal culd be the ne that travels alng the streets due t high penetratin lsses thrugh buildings. II. LINE PROCESS PRELIMINARIES Since the spatial mdel cnsidered in the paper invlves line prcesses, we intrduce this tpic briefly in this sectin. While we discuss nly thse aspects f line prcesses that are necessary fr this paper, a detailed accunt f the thery can be fund in. Line prcess. Simply put, a line prcess is just a randm cllectin f lines. In rder t define it mre frmally, first bserve that any undirected line L in R 2 can be uniquely parameterized by its signed perpendicular distance ρ frm the rigin, and the angle θ subtended by the line with respect t the psitive x-axis in cunter clckwise directin, as shwn in Fig.. The sign f ρ is negative if the rigin is t the right r abve the line. Thus, the pair f parameters ρ and θ can be represented as the crdinates f a pint n the halfcylinder C, π R, which is termed as the representatin space, as illustrated in Fig. 2. Thus, a randm cllectin f lines in R 2 can be cnstructed frm a set f pints n C. Such a set f lines generated by a Pissn pint prcess PPP n C is called a Pissn line prcess PLP. As we mentined earlier, we limit ur discussin t a special case f PLP called MPLP in which the grid-like structure f lines resembles that f rad netwrks in many cities. Als, due t its analytical tractability, this mdel has been widely used fr spatial mdeling f the rad layuts in the wireless cmmunicatins cmmunity, which served as the initial mtivatin behind this wrk 3, 27. In fact, the

3 3 L v4 L h L h2 L h3 L v3 y ρ L v θ L v2 Fig. : Illustratin f Manhattan Pissn line prcess in tw-dimensinal plane R 2. ρ L h L h2 L h3 L v2 L v π 2 L v3 L v4 Fig. 2: Illustratin f a pint prcess in representatin space C, π R. mdels used by the wireless standardizatin bdies, such as the third generatin partnership prject 3GPP, are simple variants r special instances f this mdel 28. In MPLP, the rientatins f the lines are restricted t {, π/2}, thereby btaining a set f hrizntal and vertical lines in R 2. Thus, the MPLP Φ l in R 2 can be cnstructed frm tw independent D PPPs Ψ, and Ψ π/2 alng the lines θ =, and θ = π/2, respectively, in the representatin space C. Alternatively, ne can cnstruct a MPLP by first ppulating pints alng the x and y-axes in R 2 accrding t independent D PPPs Ξ x and Ξ y and drawing vertical and hrizntal lines thrugh thse pints, respectively. This interpretatin is useful in visualizing sme f the basic prperties f MPLP which will be discussed next. In this paper, we will mainly fllw this interpretatin fr the ease f clarity and expsitin. Statinarity. Analgus t a pint prcess, a line prcess Φ l is statinary if the distributin f lines is invariant t any translatin T t,β, which crrespnds t the translatin f the rigin by a distance t in a directin that makes an angle β with respect t psitive x-axis in cunter clckwise directin. Upn applying a translatin T t,β, the representatin f a line L in C changes frm ρ, θ t ρ t sinθ β, θ. Therefre, a MPLP Φ l is statinary if the D PPPs Ψ, and Ψ π/2 are statinary r alternatively, Ξ x and Ξ y are statinary. Line density. The line density f a line prcess is defined as the mean line length per unit area. The relatinship between the line density and the density f the crrespnding pint prcess is given by the fllwing Lemma. π θ x Lemma. Fr a statinary MPLP Φ l cnstructed frm independent and hmgeneus D PPPs Ξ x and Ξ y, each with density λ l, the line density µ l is given by µ l = 2λ l. Prf: Let us cnsider a ball f radius d centered at the rigin b, d. We dente the set f hrizntal and vertical lines f Φ l by Φ lh and Φ lv, respectively. The line density µ l can nw be cmputed as µ l = πd 2 E = πd 2 E = πd 2 E ρ x Ξ x: ρ x d ν l b, d l Φ l ν l h b, d ν l v b, d l h Φ lh l v Φ lv 2 d 2 ρ 2 x πd 2 E a = πd 2 λ l2 2 d 2 ρ 2 xdρ x = 2λ l, ρ y Ξ y: ρ y d 2 d 2 ρ 2 y 2 d 2 ρ 2 ydρ y where ν dentes the ne dimensinal Lebesgue measure and a fllws frm Campbell s therem fr sums ver statinary D PPPs Ξ x and Ξ y. Lines intersecting a regin. Fr a statinary MPLP Φ l with line density µ l, the number f hrizntal and vertical lines that intersect a cnvex regin K R 2 are Pissn distributed with means µ l ν K y /2 and µ l ν K x /2, respectively, where K x and K y dente the prjectin f K nt x and y axis. A. Spatial Mdeling III. SYSTEM MODEL We mdel the spatial layut f a system f rads by a statinary MPLP Φ l {L h, L h2,..., L v, L v2,... } in R 2 in which the vertical and hrizntal lines are generated by independent hmgeneus D PPPs Ξ x and Ξ y, each having density λ l. We dente the set f hrizntal and vertical lines by Φ lh {L h, L h2,... } and Φ lv {L v, L v2,... }, respectively. We mdel the lcatins f vehicular users n each line by a D PPP with density λ v. Thus, the lcatins f vehicular users in R 2 frm a Cx prcess Φ v driven by the MPLP Φ l. As discussed already, sme mdels used in industry can be argued t be special cases r variants f this mdel 28. Further, we mdel the lcatins f the facilities f the desired type which culd be gas statins r charging statins fr electric vehicles n each line by a D PPP with density λ c thereby frming a Cx prcess Φ c, as illustrated in Fig. 3. Bth the Cx prcesses Φ c and Φ v driven by the statinary MPLP Φ l are statinary 2, 29. We will refer t an arbitrarily chsen vehicular user in the pint prcess Φ v as a typical vehicular user. We will cnsider tw types f typical vehicular users in this paper: i typical intersectin user lcated at the intersectin f a hrizntal and vertical line, and ii typical general user lcated at sme arbitrary psitin n a line. Withut lss f generality, we place the typical vehicular user at the rigin wing t the statinarity f Φ v. Thus,

4 4 y r m x Typical general user Facilities Nearest facility in the Manhattan distance sense Shrtest path Nearest facility in the Euclidean distance sense Fig. 3: Illustratin f the system mdel. a typical intersectin user is lcated at the intersectin f a hrizntal line L x and a vertical line L y, which are aligned alng the x and y-axes, respectively. Therefre, under this cnditining mre frmally, under Palm prbability, the resulting line prcess is Φ l,int = Φ l {L x, L y }, which is a cnsequence f the Slivnyak s Therem 4, 29. Cnsequently, under Palm prbability, the translated pint prcess Φ v,int can be interpreted as the superpsitin f the pint prcess Φ v, tw D PPPs each with density λ v alng the lines L x and L y, and an atm at the rigin 9, 25. Similarly, since the pint prcess Φ c is als driven by the same MPLP Φ l, the translated pint prcess Φ c,int is the superpsitin f Φ c and tw independent D PPPs each with density λ c alng L x and L y. In case f a typical general user, withut lss f generality, we assume that it is lcated n a hrizntal line f the MPLP Φ l. Upn cnditining n the lcatin f the typical general user at the rigin and using the same argument as the typical intersectin user, the resulting line prcess is Φ l,gen = Φ l {L x }. Thus, the translated pint prcess Φ v,gen can be interpreted as the superpsitin f the pint prcess Φ v, an independent D PPP with density λ v n the line L x aligned alng the x-axis and an atm at the rigin. Similarly, the translated pint prcess Φ c,gen in this case is the superpsitin f the pint prcess Φ c and a D PPP with density λ c n L x. We dente the number f hrizntal and vertical lines that intersect a regin A R 2 by N h A and N v A, respectively. We dente the number f pints in the set A by N p A. In this paper, we will dente the randm variables by upper case letters and their crrespnding realizatins by lwer case letters. Fr example, W dentes a randm variable, whereas w dentes its realizatin. We defer the definitin f ther variables t later sectins f the paper fr better readability. B. Cst Metrics Our gal is t characterize the minimum cst invlved in traveling t a facility fr a typical vehicular user. Fr instance, cnsider a typical vehicular user that is interested in ging t the nearest gas statin. Generally speaking, ur gal in this paper is t statistically characterize the cst f traveling frm the current lcatin f this vehicle t its nearest gas statin. The main metric f interest fr us in this paper is the path distance and its variants, such as mntne functins f the path distance. In rder t be cncrete, we frmally define path distance next. Definitin. Path distance. The path distance between tw pints ax, y and bx 2, y 2 is defined as the sum f lengths f the line segments that cnstitute a path P frm a t b and is dented by la, b. Therefre, in rder t minimize the cst f travel, the typical vehicular user must travel t its nearest facility in the sense f path distance. Nte that this is true fr any cst metric which is a mntnically decreasing functin f the path distance. Thus, fr such metrics, the minimum cst f travel will crrespnd t the shrtest path distance between the typical vehicular user and its nearest facility in the sense f path distance and is dented by R m. Mathematically, it can be expressed as R m = min xi Φ c,int l, x i fr a typical intersectin user and R m = min xi Φ c,gen l, x i fr a typical general user. We will als discuss a few ther cst metrics, such as travel time, in Sectin V-C. IV. ANALYTICAL RESULTS In this sectin, we will characterize the distributin f the minimum path distance R m. First, we will determine the exact CDF f R m fr a typical intersectin user. Building n this result, we will prvide an upper bund and an apprximate lwer bund fr the CDF f R m fr a typical general user. A. Typical Intersectin User Fr a typical intersectin user lcated at the rigin, the length f the shrtest path t any pint lcated at x i, y i is simply given by z i = x i y i, which is nthing but the first rder Minkwski distance f the pint frm the rigin. If the clsest facility t the typical intersectin vehicle in the sense f path distance is at a distance r m, then there can nt be any facility at a lcatin x, y in R 2 such that x y < r m. Thus, as depicted in Fig. 4, we btain an exclusin zne B frmed by the intersectin f the halfplanes x y < r m, x y < r m, x y < r m, and x y < r m. There can be n facilities n any f the line segments inside the square regin B. In additin t L x and L y, we knw that there are a randm number f lines that intersect the regin B. Frm the cnstructin f MPLP, it fllws that the number f hrizntal and vertical lines that intersect B are Pissn distributed with mean λ l 2r m. Fr a hrizntal line lcated at a distance y l < r m frm the rigin, the length f the line segment inside B is given by 2r m 2y l. Similarly, fr a vertical line at a distance x l < r m frm the rigin, the length f the line segment inside B is 2r m 2x l. Using these prperties, we will nw derive a clsedfrm expressin fr the CDF f the shrtest path distance frm a typical intersectin user t its nearest facility in the sense f path distance in the fllwing therem. Therem. The CDF f the shrtest path distance frm a typical intersectin vehicle t its nearest facility in the sense f path distance is F Rm r m

5 5 y L y y Exclusin zne x y = r m x y = r m L x r m x y = r m x L x x w x, x 2 w 2 d x L v x y = r m Fig. 4: Illustratin f the exclusin zne fr a typical intersectin user. = exp 4λ c r m 4λ l r m 2λ l e 2λ cr m. λ c Prf: See Appendix A. B. Typical General User In this subsectin, we prpse an upper bund and an apprximate lwer bund fr the CDF f R m fr a typical general user lcated at the rigin n the hrizntal line L x aligned alng the x-axis. Remark. The key difference between the spatial setup f the typical general user and that f the typical intersectin user is that there des nt exist a line L y alng the y-axis in case f a typical general user. Mre precisely, the pint prcess is nw viewed under the reduced Palm distributin f Ξ x and Palm distributin f Ξ y. This interpretatin will allw us t derive an upper bund fr the distributin f R m fr the typical general user using the results derived fr the typical intersectin user in Therem. Upper Bund: As mentined in Remark, there is n vertical line at the rigin fr a typical general user. Cnsequently, a typical general user can mve nly alng the directin f either psitive r negative x-axis n the line L x, whereas a typical intersectin user culd mve in any f the fur directins frm the rigin. Therefre, unlike the case f the typical intersectin user where the length f the shrtest path t a facility was equal t the first rder Minkwski distance, the distance t sme facilities frm the rigin in case f the typical general user is greater than their first rder Minkwski distance, as illustrated in Fig. 5. Thus, in this case, the first rder Minkwski distance f the nearest facility frm the rigin is a lwer bund n R m. Therefre, using the result derived in Therem, we present a clsed-frm upper bund n the CDF f R m in the fllwing prpsitin. Fig. 5: Illustratin f the shrtest paths fr a typical general user. Prpsitin. Fr a typical general user, the CDF f R m is upper bunded by F Rm r m exp 2λ c r m 4λ l r m 2λ l e 2λ cr m. 2 λ c Prf: This result can be directly derived frm the expressin in Therem by deleting the line L y. This means that we remve the terms crrespnding t the vertical line L y aligned alng the y-axis. Specifically, we remve the term PN p L y B = in step a and the crrespnding expressins in the subsequent steps in the prf f Therem t btain the final expressin. As will be shwn in the later sectins f the paper, this upper bund is reasnable but nt extremely tight. We will prpse a much tighter lwer bund fr the CDF f R m next. 2 Apprximate Lwer Bund: We define a randm variable R as the shrtest path distance frm the rigin t the nearest facility lcated t the right f y-axis upn starting in the directin f psitive x-axis. Similarly, we define R 2 as the shrtest path distance t the nearest facility lcated t the left f the y-axis when the typical general user starts in the directin f negative x-axis. Thus, the verall shrtest path distance t a facility frm the rigin can be written as R m min{r i } i=,2. The inequality is due t the fact that fr sme facilities t the right and left f y-axis, the shrtest path frm the rigin wuld be the ne that starts alng the directin f negative and psitive x-axis frm the rigin, respectively, as illustrated in Fig. 6. S, in such scenaris, R r R 2 wuld be an upper bund n the shrtest path distances t thse facilities. Therefre, the CDF f R m wuld be bunded frm belw by the CDF f min{r i } i=,2. When the typical general user starts in any f the tw directins, there are tw pssible events: i the user culd reach an intersectin befre arriving at any facility, and ii the user culd arrive at a facility n the same rad befre

6 6 y y L v r 2 L x Shrter path x L x x w w x B x Fig. 6: Illustratin f the scenari in which the shrtest path t a facility t the left f the y-axis is the ne that starts in the directin f psitive x-axis. reaching an intersectin. Therefre, R i can be mathematically written as { Xi W i, E,i ccurs, R i = 3 D i, E,i ccurs. We nw define all the randm variables appearing in 3. Fr ntatinal cnsistency, we dente the scenari in which the typical general user starts in the psitive x-axis directin by the subscript i = and the scenari in which the user starts in the negative x-axis directin by the subscript i = 2. The event in which the typical general user wuld reach an intersectin befre arriving at a facility is dented by E,i and its cmplementary event is dented by E,i. D i dentes the distance t the first facility frm the rigin and X i dentes the distance t the first intersectin frm the rigin in the directin f travel, as shwn in Fig. 5. W and W 2 dente the shrtest first rder Minkwski distance frm the intersectin t facilities lcated t the right and left f the y-axis, respectively. As the lcatins f facilities n each line fllws a D PPP with mean λ c, the distance D i is expnentially distributed with mean λ c. Thus, the CDF and PDF f D i are CDF: F Di d i = exp λ c d i, 4 PDF: f Di d i = λ c exp λ c d i. 5 Since the pints at which the vertical lines crss the x-axis frm a D PPP with density λ l, the distance X i als fllws an expnential distributin with mean λ l. Thus, the CDF and PDF f X i are CDF: F Xi x i = exp λ l x i, 6 PDF: f Xi x i = λ l exp λ l x i. 7 We will nw fcus n cmputing the distributin f R i. Nte that R and R 2 are identically distributed due t symmetry abut the y-axis. Therefre, it is sufficient t derive the marginal distributin f ne f them. Withut lss f generality, let us cnsider R. Since R is defined separately Fig. 7: Illustratin f the exclusin zne fr the case w < x. fr the events E, and E,, we will derive the CDF f R cnditined n these events and then btain the verall CDF using law f ttal prbability. We will begin by deriving the prbability f ccurrence f the events E, and E, in the fllwing Lemma. Lemma 2. The prbability f ccurrence f the events E, and E, are given by as PE, = λ l and PE, = λ c. 8 λ l λ c λ l λ c Prf: The prbability f the event E, can be calculated PE, = P D > X = E X PD > x X = = = λ l λ l λ c. F D x f X x dx e λcx λ l e λ lx dx Since the events E, and E, are cmplementary, the prbability f ccurrence f E, can be calculated as PE, = PE,. This cmpletes the prf. We will fcus n the cmputatin f the CDF f W cnditined n the event E, next. The cnditining n E, implies that there des nt exist any facility between the rigin and the intersectin. This additinal infrmatin abut the distributin f facilities in the interval, X n L x must be included in the cmputatin f the cnditinal CDF f W. Since the distance X is randm, we will derive the intermediate results by additinally cnditining n X and we will take expectatin ver X in the final step in the cmputatin f the marginal CDF f R. Similar t the prcedure fllwed in the derivatin f Therem, we will cnsider an exclusin zne B frmed by the intersectin f the half-planes x x y < w, x x y < w, x x y < w, x x y < w, and x >. Nte that the shape f the exclusin regin B depends n the values

7 7 y L v λ l 3 2e 2λ cx e 2λcw 4e λcxw. 2λ c x w x x w w B h2 L x x Prf: See Appendix B. Having determined all the cmpnents required t cmpute the CDF f R cnditined n E,, we will nw prceed t the derivatin f CDF f R cnditined n E, in the fllwing Lemma. Lemma 4. Cnditined n the event E,, the CDF f R is given by B h B h2 F R r E, = exp λ l λ c r. 2 Prf: The cnditinal CDF f R can be cmputed as Fig. 8: Illustratin f the exclusin zne fr the case w x. f w with respect t x. While B is a square fr w < x, it is a pentagn fr w x, as shwn in Figs. 7 and 8, respectively. S, we will derive the cnditinal CDF f W fr the tw cases w < x and w x separately. We knw that there can nt be any facility n any f the line segments inside B. In additin t L x, there exists a randm number f hrizntal lines abve and belw the line L x that intersect the regin B. Likewise, in additin t the vertical line f the intersectin L v, there exists a randm number f vertical lines that intersect the regin B. Hwever, cnditined n the event E,, the distributin f vertical lines t the left f L v is nt the same as the distributin f lines t the right f L v. Since the first intersectin t the right f the rigin is at a distance x, there can nt be any vertical line that intersects L x in the interval, x, as shwn in Fig. 7. S, we just need t fcus n the set f vertical lines that intersect the regin B x = B {x > x }. We will nw derive a clsed frm expressin fr the cnditinal CDF f W in the fllwing Lemma. Lemma 3. The CDF f W cnditined n E, and X is given by { FW,w E,, x, w < x, F W w E,, x = F W,2w E,, x, w x, 9 where and F W,w E,, x = exp F W,2w E,, x = exp 3λ c w 3λ l w 3λ l e 2λ cw, 2λ c 3λ c λ l w λ c x F R r E, = P R < r E, = P R > r, E, P E, a = P D > r, D < X P E, = P E, E X P r < D < x X = F D x F D r f X x dx P E, r = λ l λ c e λ cr e λcx λ l e λ lx dx λ c r = exp λ l λ c r, where a fllws frm the cnditin fr the ccurrence f the event E,. With this, we have derived all the intermediate results required t cmpute the CDF f R. Cmbining these results given in Lemmas 2, 3, and 4, we will nw derive the marginal CDF f R in the fllwing Lemma. Lemma 5. The CDF f R is F R r = r r 2 F W,r x E,, x PE, X f X x dx r 2 F W,2r x E,, x PE, X f X x dx PE, F R r E,. 3 Prf: The CDF f R can be cmputed as F R r = P R < r a = PR < r, E, PR < r, E, = E X PR < r, E, X PR < r, E, b = E X PR < r E,, X PE, X = = PR < r E, PE, Px W < r E,, X PE, X f X x dx PE, PR < r E, F W r x E,, x PE, X f X x dx PE, F R r E,

8 8 λ c λ l TABLE I: KL-divergence f the apprximate jint PDF f R and R 2 cmputed as the prduct f their marginal PDFs frm their empirical jint PDF. x 2 x x 2 x r c = F r W,r x E,, x PE, X f X x dx 2 r 2 F W,2r x E,, x PE, X f X x dx PE, F R r E,, where a fllws frm the law f ttal prbability, b fllws frm the applicatin f Bayes therem, and c fllws frm substituting 9 in the previus step. In rder t cmpute the exact CDF f min{r, R 2 }, we need t determine the jint distributin f R and R 2, which is nt quite tractable due t the peculiar cupling induced by the underlying line prcess. Therefre, in the interest f analytical tractability, we assume R and R 2 t be independent and derive the apprximate CDF f min{r, R 2 }. This assumptin is strngly supprted by the empirical evidence btained frm Mnte-Carl simulatins. Fr different sets f λ l and λ c, we determined the Kullback-Leibler KL divergence f the apprximate jint PDF f R and R 2 cmputed as the prduct f their marginal PDFs frm their actual jint PDF, as shwn in Table I. It can be bserved that the KL divergence fr different cmbinatins f λ l and λ c is quite small. Thus, we can infer that the dependence between R and R 2 is minimal. S, using the assumptin f independence between R and R 2, we derive an apprximate lwer bund fr the CDF f R m in the fllwing therem. Therem 2. Fr a typical general user, the CDF f R m is apprximately lwer bunded by F Rm r m F R r m 2, 4 where F R is the marginal distributin f R given in Lemma 5. Prf: The CDF f R m can be cmputed as F Rm r m = PR m < r m P min{r, R 2 } < r m = P R > r m, R 2 > r m a PR > r m PR 2 > r m b = F R r m 2, where a fllws frm assuming R and R 2 t be independent, and b fllws frm the identical distributin f R and R 2. This cmpletes the prf. V. RESULTS AND DISCUSSION In the previus sectin, we have derived the analytical expressins fr the CDF f the shrtest path distance t the Fig. 9: Illustratin f the equivalent weighted graph cnstructed frm the spatial setup. The ndes crrespnding t facilities are indicated by circles and the ndes crrespnding t intersectins are indicated by squares. nearest facility fr the typical intersectin user and typical general user. We will fcus n the cmputatin f the empirical CDF f this distance and verificatin f ur analytical results using Mnte-Carl simulatins in this sectin. We will first present a simulatin methd t cmpute any distancedependent metric using the spatial mdel described in Sectin III. We then verify the accuracy f analytical results by numerically cmparing them with the results frm Mnte- Carl simulatins. We will als discuss the extensin f this wrk t study ther cst metrics in the cntext f varius applicatins. A. Simulatin Techniques In this subsectin, we will present the prcedure t characterize the minimum cst f travel fr a typical vehicular user using Mnte Carl simulatins. This methd is applicable t any cst metric that depends n the spatial separatin between the typical vehicular user and the facilities, such as path distance, travel time r fuel cnsumed. The prblem f finding the minimum cst can be psed as a classical shrtest path prblem by interpreting the setup as a weighted graph where the weights f the edges crrespnd t the cst assciated with traveling between thse pints. In rder t run Mnte-Carl simulatins, we first generate a realizatin f the spatial mdel and calculate the cst f travel frm the typical user at rigin t all the facilities. The next step is t cnstruct an equivalent undirected weighted graph f the spatial setup, as illustrated in Fig. 9, by fllwing these steps: The lcatins f the facilities, typical user, and the intersectins f rads are cnsidered as the ndes f a graph G. Any tw ndes f the graph are cnnected if and nly if they are lcated n the same line and are als adjacent. The weight f an edge f the graph G is equal t the cst incurred t travel between the pints crrespnding t the end pints f the edge in R 2. Nw, we can easily find the minimum cst frm the typical user surce nde t any facility destinatin nde using Dijkstra s algrithm n the weighted graph, thereby eliminating the burden f determining all pssible rutes t a facility frm the typical user 3. We can then find the verall minimum cst t a facility by cmparing the minimum cst f travel t

9 9 a b c d Fig. : The fur different regimes f the spatial mdel: a Dense Rads-Dense Facilities DR-DF, b Dense Rads-Sparse Facilities DR-SF, c Sparse Rads-Dense Facilities SR-DF, and d Sparse Rads-Sparse Facilities SR-SF..8 DR-DF SR-DF DR-SF.8 DR-DF SR-DF.6 SR-SF.6 DR-SF.4.4 SR-SF Fig. : CDF f the minimum shrtest path distance fr a typical intersectin user fr the fur regimes: DR-DF λ l = km, λ c = 3 facilities/km, SR-DF λ l = km, λ c = 3 facilities/km, DR-SF λ l = km, λ c =.5 facilities/km, and SR-SF λ l = km, λ c =.5 facilities/km. Fig. 2: CDF f the minimum shrtest path distance fr a typical general user fr the fur regimes: DR-DF λ l = km, λ c = 5 facilities/km, SR-DF λ l = km, λ c = 5 facilities/km, DR-SF λ l = km, λ c =.5 facilities/km, and SR-SF λ l = km, λ c =.5 facilities/km. all the facilities in the spatial netwrk. This prcess is repeated ver several realizatins f the spatial mdel t btain an empirical CDF f the minimum travel cst between a typical user and a facility. Remark 2. Althugh the Dijkstra s algrithm directly yields the shrtest path t a destinatin nde, it is still time cnsuming t apply the algrithm t every destinatin nde. Hwever, if the cst metric is the path distance, we can emply a simple strategy that wuld significantly reduce the run-time f simulatins. Assuming that the shrtest path distance t the ndes are determined sequentially, we first srt the destinatin ndes in the increasing rder f their Euclidean distances frm the rigin. Then, we cmpute the shrtest path distance t each destinatin nde in that rder by applying Dijkstra s algrithm until we encunter a destinatin nde whse Euclidean distance E i is greater than the shrtest path distance M j f any f the previus ndes, i.e., E i > min{m, M 2,..., M i }. Thus, this technique will help in further reducing the cmputatinal lad by aviding the cmputatin f the shrtest path t every destinatin nde. B. Numerical Results We first simulate the spatial mdel described in Sectin III in MATLAB. We then cmpute the empirical CDF f R m and cmpare it with the CDF btained frm the analytical expressins given in Therems and 2. Fr exhaustive numerical analyses and cmparisns, we define these fur brad regimes based n the densities f rads and facilities: i Dense Rads-Dense Facilities DR-DF crrespnding t large values f λ l and λ c, ii Dense Rads-Sparse Facilities DR- SF crrespnding t large values f λ l and small values f λ c, iii Sparse Rads-Dense Facilities SR-DF crrespnding t small values f λ l and large values f λ c, and iv Sparse Rads-Sparse Facilities SR-SF crrespnding t small values f λ l and λ c, as illustrated in Fig.. Nte that Fig. is nly fr illustratin purpse and the actual simulatin values crrespnding t these cnfiguratins are prvided alng with the results in Figs. and 2. Fr a typical intersectin

10 user, as expected, the CDF btained frm the expressin in matches exactly with the CDF btained frm Mnte- Carl simulatins fr all the cnfiguratins as depicted in Fig.. In case f a typical general user, while the upper bund given in Prpsitin is nt extremely tight in all the regimes, the apprximate lwer bund given in Therem 2 clsely fllws the empirical CDF, as shwn in Fig. 2. The remarkable accuracy f the lwer bund fllws frm the careful cnstructin f tw apprximately independent randm variables R and R 2, as discussed in detail in Sectin IV. C. Cst Metrics and Applicatins In this sectin, we will discuss sme f the ther useful cst metrics related t transprtatin systems. We will als discuss the applicability and direct extensins f the results presented in this paper t address sme f the prblems related t infrastructure planning, persnnel deplyment and wireless cmmunicatins. Transprtatin Systems: Mdeling spatial layut f rads as MPLP, we have cmprehensively analyzed the shrtest path distance between a typical vehicular user and its nearest facility in the sense f path distance. Sme f the ther metrics that culd easily be studied are minimum travel time t a facility fr a typical vehicular user and the fuel cnsumed t reach a facility. The minimum travel time t a facility can be cmputed as the sum f the ratis f length f each rad segment alng the path t the maximum speed permitted n the crrespnding segment. Assuming that the maximum speed allwed u max is the same n all the rads, the CDF f the minimum travel time t a facility T m can be directly btained frm ur analytical results as F Tm t m = F Rm u max t m. Fr instance, assuming a speed limit f 4 km/hr, which is usually the case in dwntwn areas in majr cities, we btain the CDF fr the minimum time f travel as shwn in Fig. 3. Hwever, fr an arbitrary distributin f velcities n different rads, the exact analysis is nt straightfrward. This is because the path that yields the shrtest path distance t a facility is nt necessarily the quickest path t reach a facility. Hwever, we can btain the lwer bund fr the minimum travel time by cnsidering the same maximum velcity acrss all the rads. Similarly, we can btain an upper bund n the minimum travel time by cnsidering the same minimum velcity n all the rads. Our preliminary investigatin indicates that these bunds may nt be very tight and hence mtivates future wrk in this directin. Hwever, the simulatin prcedure presented in Sectin V-A is still valid and can be applied t cmpute the exact minimum travel time using Mnte-Carl simulatins. In this case, the weight f an edge f the graph wuld be the minimum time taken t travel between the pints crrespnding t the end-pints f the edge. Similarly, we culd als characterize the fuel cnsumptin f a typical vehicular user, as it is primarily a functin f the distance traveled and the fuel efficiency f the vehicle. 2 Infrastructure Planning/ Persnnel Deplyment: As we have stated earlier in the paper, the facilities that we cnsider in ur system mdel culd be gas statins r ther infrastructure Fig. 3: CDF f the minimum travel time t a facility fr a typical general user λ l = km, λ c =.5 facilities/km, and u max = 4 km/hr. CDF FTm t m Increasing λ c =.2,.25,.33,.5,, 2 facilities/km Min. Time f Travel T m mins. Fig. 4: CDF f the minimum travel time t a facility fr a typical general user λ l = km, and u max = 4 km/hr. Using the analytical results presented in the paper, we can gain insights int the minimum distance t these facilities as a functin f the density f rads and the density f facilities n each rad. This wuld help in infrastructure planning t ensure that the minimum distance r minimum travel time t these facilities fr a typical vehicular user des nt exceed a desired threshld. Fr example, frm Fig. 4 where we plt the CDF f minimum travel time fr different values f the density f facilities, we can infer that the minimum travel time wuld be less than minute with at least a prbability f 7% if the value f λ c is greater than.5 facilities/km. This analysis wuld be very useful in the study f the typical respnse time fr first respnders r plice persnnel t arrive at the site f an emergency r an accident. 3 Wireless Cmmunicatin: The 5G New Radi NR standard fr mbile cmmunicatins includes the use f mmwave frequency spectrum 3 GHz - 3 GHz. Therefre, it is imprtant t understand the perfrmance f the netwrk

11 Euclidean distance Path distance Fig. 5: CDF f the distance t the clsest transmitting nde in the sense f Euclidean distance and path distance λ l = km, and λ c =.25 ndes/km. at these higher frequencies. Unlike the cellular netwrks perating in sub-6 GHz frequencies, where the receiver usually cnnects t its clsest transmitting nde in the Euclidean distance sense, the prpagatin at higher frequencies is quite sensitive t blckages, thereby resulting in high attenuatin f the signal. Therefre, in sme urban scenaris with dense buildings and tall structures, it is quite pssible that the dminant cmpnent f the received signal is the ne that travels alng the rads with diffractin arund the crners at intersectin 3. Further, the strngest signal received frm the desired transmitter is the ne that travels alng the shrtest path. As is evident frm Fig. 5, the CDF f the distance t the clsest nde in the Euclidean distance sense is significantly different frm that f the clsest nde in the path distance sense. Therefre, the analytical techniques develped in this paper culd be useful in characterizing the received signal pwer in such netwrks. VI. CONCLUSION Transprtatin systems have been mstly studied using spatial netwrk mdels where varius sites were mdeled as ndes f a graph and the rutes between thse sites were mdeled as edges f the graph. While these mdels are useful in studying the tplgical prperties f the netwrk, they d nt accurately capture sme f the gemetric aspects such as cntinuity f streets. S, in this paper, we have mdeled the spatial layut f rads by a MPLP and the lcatins f vehicles and facilities by independent and hmgeneus D PPPs. Fr this setup, we first derived the clsed-frm expressin fr the CDF f the shrtest path distance fr a typical intersectin user t its nearest facility in the sense f path distance. Building n this result, we derived an upper bund and a much tighter but apprximate lwer bund fr the CDF f the shrtest path distance fr a typical general user. In additin t these analytical results, we als discussed a simulatin methd t characterize any distance-dependent metric thrugh graphical interpretatin f the spatial mdel. We verified the accuracy f ur analytical results by cmparing them with the results btained frm Mnte-Carl simulatins. We then discussed sme key insights ffered by ur analytical results fr infrastructure planning and persnnel deplyment. This wrk has several extensins. First f all, the spatial mdel cnsidered in the paper can be used t study ther useful metrics such as rute-length efficiency statistic which is defined as a functin f the rati f the shrtest path distance between a pair f pints t the crrespnding Euclidean distance between thse pints 4. While we have derived the results fr an urban netwrk by mdeling the streets as a MPLP, the analytical prcedure and the cnstructin f bunds can be extended t a mre general setup where the rad netwrk is mdeled by a PLP. Anther useful directin f wrk wuld be t explre the tempral behavir f the netwrk by cnsidering flw f vehicles alng the streets mdeled by a MPLP r a PLP. Als, the discussin n applicatins f ur results in transprtatin, infrastructure planning, and wireless cmmunicatin in Sectin V-C culd mtivate future wrk in all these areas. A. Prf f Therem APPENDIX The CDF f R m can be cmputed as F Rm r m = PR m > r m = PN p Φ l,int B = a = P N p {L x Φ lh } B = N p {L y Φ lv } B = b = PN p L x B = PN h B \ L x = n hl n hl = N p Φ lh B = N h B \ L x = n hl N p L y B = n vl = PN v B \ L y = n vl N p Φ lv B = N v B \ L y = n vl c = PN p L x B = PN h B \ L x = n hl n hl = n hl P N p L hj B = PN p L y B = n vl = j= nvl N v B \ L y = n vl P N p L vk B = d = e 2λcrm rm e 2λcrm n hl = k= e 2λ lr m 2λ l r m n hl n hl! exp λ c 2r m 2y dy r m n vl = e 2λ lr m 2λ l r m n vl n vl! nhl

12 2 rm exp λ c 2r m 2x dx nvl r m rm = e 2λcrm e 2λ lr m exp 2λ l e 2λcrm y dy rm e 2λcrm e 2λ lr m exp 2λ l e 2λcrm x dx = exp 4λ c r m 4λ l r m 2λ l e 2λ cr m, λ c where a fllws frm the fact that the distributin f hrizntal and vertical lines are independent, b fllws frm cnditining n the number f hrizntal and vertical lines intersecting the regin B, c fllws frm the independent distributin f facilities n the lines, and d fllws frm the Pissn distributin f the number f lines intersecting B and the vid prbability f D PPP n each line. B. Prf f Lemma 3 The cnditinal CDF f W can be cmputed as F W w E,, x = PW > w E,, X = PN p B = E,, X. 5 As we had discussed earlier, the shape f the exclusin zne B is different fr the tw cases w < x and w x and hence we will handle these tw cases separately. We will first cnsider the case w < x. In this case, B is a square regin and we nw need t determine the prbability that there are n pints inside this square regin centered at an intersectin, as shwn in Fig. 7. By expressing the cnditinal vid prbability in 5 as the prduct f vid prbabilities f independent individual cmpnents, as in the prf f Therem, we btain PN p B = E,, X = PN p L x B = E,, X N h B \ L x = n hl E,, X n hl = N p Φ lh B = N h B \ L x = n hl, E,, X N p L v B = E,, X N v B x = n vl E,, X n vl = N p Φ lv B x a = PN p L x B = E,, X N h B \ L x = n hl n hl = = N v B x = n vl, E,, X N p Φ lh B = N h B \ L x = n hl N p L v B = n vl = PN v B x = n vl b N p Φ lv B x = PN p L x B = E,, X N h B \ L x = n hl n hl = n hl P N p L hj B = j= N p L v B = nvl k= P c = e λcw N p L vk B x n hl = w = N v B x = n vl n vl = = e 2λ lw 2λ l w n hl n hl! PN v B x = n vl exp λ c 2w 2y dy w nhl e λ lw λ e 2λcw l w n vl n n vl = vl! w exp λ c 2w 2x dx nvl w w = e λcw e 2λ lw exp 2λ l e 2λcw y dy w e 2λcw e λ lw exp λ l e 2λcw x dx = exp 3λ c w 3λ l w 3λ l e 2λ cw, 6 2λ c where a fllws frm the fact that the distributin f pints n the randm hrizntal lines, randm vertical lines intersecting B x, and the line L v is independent f E, and X, b fllws frm the independent distributin f pints ver different lines, and c fllws frm the Pissn distributin f lines and the vid prbability f D PPPs n thse lines. Substituting 6 in 5, we btain the expressin fr the cnditinal CDF f W fr the case w < x. We will nw cnsider the case where w x, where the exclusin regin B is a pentagn as depicted in Fig. 8. The length f the hrizntal line segment inside B depends n the distance f the line frm the rigin. Fr a hrizntal line L h which intercepts the y-axis at y h such that y h < w x, the length f the line segment inside B is given by x w y h. On the ther hand, if y h w x, then the length f line segment inside B is 2w y h. S, we partitin the set f hrizntal lines that intersect B int tw sets: i the set f hrizntal lines that intersect the regin B h = B { y < w x }, and ii the set f hrizntal lines that intersect the regin B h2 = B { y w x }. As B h2 is cmpsed f tw nn-cntiguus regins B h 2 = B {y w x } and B h 2 = B {y w x }, we will handle them separately in ur analysis. Thus, the cnditinal vid prbability fr the

13 3 case w x can be cmputed as PN p B = E,, X = PN p L x B = E,, X N h Bh \ L x = nh E,, X n h = N p Φ lh B h = N h B h \ L x = n h, E,, X n h2 = P N h B h 2 = nh2 E,, X N p Φlh B h2 = Nh B h2 = n h2, E,, X N h B h 2 = nh3 E,, X n h3 = N p Φlh B h2 = Nh B h2 = n h3, E,, X N p L v B = E,, X N v B x = n vl E,, X n vl = N p Φ lv B x a = N v B x = n vl, E,, X = PN p L x B = E,, X PN p L v B = N h Bh \ L x = nh X b n h = N p Φ lh B h = N h B h \ L x = n h, X n h2 = P N h B h 2 = nh2 X N p Φlh B h2 = Nh B h2 = n h2, X N h B h 2 = nh3 X n h3 = N p Φlh B h2 = Nh B h2 = n h3, X n vl = PN v B x = n vl X N p Φ lv B x = N v B x = n vl, X, = PN p L x B = E,, X PN p L v B = N h B h \ L x = n h X n h = n h2 = nh P N p L hi B h = X i= PN h B h 2 = n h2 X c n h3 = n vl = n h2 j= P N p L hj B h 2 = X PN h B h 2 = n h3 X n h3 P N p L hj B h 2 = X j= PN v B x = n vl nvl k= P N p L vk B x = = e λcwx e 2λcw e 2λ lw x 2λ l w x n h n n h = h! w x nh dy exp λ c x w y w x e λ lx λ l x n h w nh2 2 λc2w 2y dy e n n h2 = h2! w x x e λ lx λ l x n h w nh3 3 λc2w 2y dy e n n h3 = h3! w x x e λ lw λ l w n vl w nvl λc2w 2x dx e n n vl = vl! w = exp 3λ c λ l w λ c x λ l 2λ c 3 2e 2λ cx e 2λcw 4e λcxw, 7 where a fllws frm the fact that the distributin f pints n the randm hrizntal lines, vertical lines intersecting B x and the line L v are independent f E,, b fllws frm the independent distributin f pints ver lines, and c fllws frm the Pissn distributin f lines and the vid prbability f D PPPs n each f thse lines. Substituting 7 in 5, we btain the expressin fr the cnditinal CDF f W fr the case w x. This cmpletes the prf. REFERENCES J. Lin, Spatial analysis and mdelling f urban transprtatin netwrks, Ph.D. dissertatin, KTH Ryal Institute f Technlgy, S. Marshall, J. Gil, K. Krpf, M. Tmk, and L. Figueired, Street netwrk studies: frm netwrks t mdels and their representatins, Netwrks and Spatial Ecnmics, pp. 5, M. Barthélemy, Spatial netwrks, Physics Reprts, vl. 499, n. -3, pp., 2. 4 D. J. Aldus and J. Shun, Cnnected spatial netwrks ver randm pints and a rute-length statistic, Statist. Sci., vl. 25, n. 3, pp , Aug C. P. Dettmann, O. Gergiu, and P. Pratt, Spatial netwrks with wireless applicatins, Cmptes Rendus Physique, vl. 9, n. 4, pp , S. Marshall, Line structure representatin fr rad netwrk analysis, Jurnal f Transprt and Land Use, vl. 9, n., 25.

14 4 7 F. Baccelli, M. Klein, M. Leburges, and S. Zuyev, Stchastic gemetry and architecture f cmmunicatin netwrks, Telecmmunicatin Systems, vl. 7, n., pp , Jun C. Glaguen, F. Fleischer, H. Schmidt, and V. Schmidt, Analysis f shrtest paths and subscriber line lengths in telecmmunicatin access netwrks, Netwrks and Spatial Ecnmics, vl., n., pp. 5 47, Mar V. V. Chetlur and H. S. Dhilln, Cverage analysis f a vehicular netwrk mdeled as Cx prcess driven by pissn line prcess, IEEE Trans. n Wireless Cmmun., t appear. S. N. Chiu, D. Styan, W. S. Kendall, and J. Mecke, Stchastic gemetry and its applicatins. Jhn Wiley & Sns, 23. R. E. Miles, Randm plygns determined by randm lines in a plane, Prceedings f the Natinal Academy f Sciences, vl. 52, n. 4, pp. 9 97, C. Chi and F. Baccelli, Pissn Cx pint prcesses fr vehicular netwrks, IEEE Trans. n Veh. Technlgy, vl. 67, n., pp. 6 65, Oct Y. Wang, K. Venugpal, R. W. Heath, and A. F. Mlisch, Mmwave vehicle-t-infrastructure cmmunicatin: Analysis f urban micrcellular netwrks, IEEE Trans. n Veh. Technlgy, M. Haenggi, Stchastic Gemetry fr Wireless Netwrks. Cambridge University Press, V. V. Chetlur and H. S. Dhilln, Dwnlink cverage analysis fr a finite 3D wireless netwrk f unmanned aerial vehicles, IEEE Trans. n Cmmun., t appear. 6 P. Erdös and A. Rényi, On randm graphs, I, Publicatines Mathematicae Debrecen, vl. 6, pp , , On the evlutin f randm graphs, in Publicatin f the Mathematical Inst. f the Hungarian Acad. f Sci.,, 96, pp R. Albert and A.-L. Barabási, Statistical mechanics f cmplex netwrks, Rev. Md. Phys., vl. 74, pp , Jan D. J. Watts and S. H. Strgatz, Cllective dynamics f small-wrld netwrks, nature, vl. 393, n. 6684, p. 44, D. Aldus and P. Diacnis, Hammersley s interacting particle prcess and lngest increasing subsequences, Prbability Thery and Related Fields, vl. 3, n. 2, pp , Jun F. Vss, C. Glaguen, F. Fleischer, and V. Schmidt, Distributinal prperties f Euclidean distances in wireless netwrks invlving rad systems, IEEE Jurnal n Sel. Areas in Cmmun., vl. 27, n. 7, pp , Sep C. Glaguen, F. Fleischer, H. Schmidt, and V. Schmidt, Simulatin f typical Cx Vrni cells with a special regard t implementatin tests, Mathematical Methds f Operatins Research, vl. 62, n. 3, pp , , Fitting f stchastic telecmmunicatin netwrk mdels via distance measures and Mnte Carl tests, Telecmmunicatin Systems, vl. 3, n. 4, pp , Apr V. V. Chetlur and H. S. Dhilln, Success prbability and area spectral efficiency f a VANET mdeled as a Cx prcess, IEEE Wireless Cmmun. Letters, vl. 7, n. 5, pp , Oct C. Chi and F. Baccelli, An analytical framewrk fr cverage in cellular netwrks leveraging vehicles, IEEE Transactins n Cmmunicatins, vl. 66, n., pp , Oct F. Baccelli and X. Zhang, A crrelated shadwing mdel fr urban wireless netwrks, in Prc., IEEE INFOCOM. IEEE, 25, pp X. Zhang, F. Baccelli, and R. W. Heath, An indr crrelated shadwing mdel, in Prc., IEEE Glbecm, 25, pp GPP TR , Study n LTE-based V2X services, Jul F. Mrlt, A ppulatin mdel based n a Pissn line tessellatin, in Prc., Mdeling and Optimizatin in Mbile, Ad Hc and Wireless Netwrks, May 22, pp E. W. Dijkstra, A nte n tw prblems in cnnexin with graphs, Numerische mathematik, vl., n., pp , 959. Vishnu Vardhan Chetlur received the B. E. Hns. degree in Electrnics and Cmmunicatins Engineering frm the Birla Institute f Technlgy and Science BITS Pilani, India, in 23. After his graduatin, he wrked as a design engineer at Redpine Signals Inc. fr tw years. He is currently a Ph.D. student at Virginia Tech, where his research interests include wireless cmmunicatin, smart cities, and stchastic gemetry. He has held shrt-term visiting psitins at Bell Labs, Qualcmm Inc., and Philips Research India. He graduated tp f his class in the department f Electrical Engineering at BITS and was awarded the institute Silver medal fr being ranked secnd in the whle institute. He was als a recipient f the BITS merit schlarship fr his excellence in academics. Harpreet S. Dhilln S M 3 received the B.Tech. degree in Electrnics and Cmmunicatin Engineering frm IIT Guwahati, India, in 28; the M.S. degree in Electrical Engineering frm Virginia Tech, Blacksburg, VA, USA, in 2; and the Ph.D. degree in Electrical Engineering frm the University f Texas at Austin, TX, USA, in 23. In academic year 23-4, he was a Viterbi Pstdctral Fellw at the University f Suthern Califrnia, Ls Angeles, CA, USA. He jined Virginia Tech in August 24, where he is currently an Assistant Prfessr and Steven O. Lane Junir Faculty Fellw f Electrical and Cmputer Engineering. He has held shrt-term visiting psitins at Bell Labs, Samsung Research America, Qualcmm Inc., and Plitecnic di Trin. His research interests include cmmunicatin thery, stchastic gemetry, gelcatin, and wireless ad hc and hetergeneus cellular netwrks. Dr. Dhilln is a Clarivate Analytics Highly Cited Researcher and has cauthred five best paper award recipients including the 26 IEEE Cmmunicatins Sciety CmSc Heinrich Hertz Award, the 25 IEEE CmSc Yung Authr Best Paper Award, the 24 IEEE CmSc Lenard G. Abraham Prize, and tw cnference best paper awards at IEEE ICC 23 and Eurpean Wireless 24. He was named the 28 Cllege f Engineering Faculty Fellw and the 27 Outstanding New Assistant Prfessr by Virginia Tech. His ther academic hnrs include the 23 UT Austin Wireless Netwrking and Cmmunicatins Grup WNCG leadership award, the UT Austin Micrelectrnics and Cmputer Develpment MCD Fellwship, and the 28 Agilent Engineering and Technlgy Award. He currently serves as an Editr fr the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, and the IEEE WIRELESS COMMUNICATIONS LETTERS. Carl P. Dettmann received the BSc Hns. and the PhD degrees in physics frm the University f Melburne, Australia, in 99 and 995, respectively. Fllwing research psitins at New Suth Wales, Nrthwestern, Cpenhagen, and Rckefeller Universities, he mved t the University f Bristl, Bristl, United Kingdm, where he is nw prfessr and deputy directr f the Institute f Prbability, Analysis, and Dynamics. He has published mre than 25 internatinal jurnal and cnference papers in cmplex and cmmunicatins netwrks, dynamical systems, and statistical physics. He is a Fellw f the Institute f Physics and serves n its fellwship panel. He has delivered many presentatins at internatinal cnferences, including a plenary lecture at Dynamics Days Eurpe and a tutrial at the Internatinal Sympsium n Wireless Cmmunicatin Systems.