Moments of Interference in Vehicular Networks with Hardcore Headway Distance

Transcription

1 Moments of Interferene in Vehiular Networks with Hardore Headway Distane Konstantinos Koufos and Carl P. Dettmann arxiv:83.658v3 [s.ni] Jan 9 Abstrat Interferene statistis in vehiular networks have long been studied using the Poisson Point Proess PPP for the loations of vehiles. In roads with few number of lanes and restrited overtaking, this model beomes unrealisti beause it assumes that the vehiles an ome arbitrarily lose to eah other. In this paper, we model the headway distane the distane between the head of a vehile and the head of its follower equal to the sum of a onstant hardore distane and an exponentially distributed random variable. We study the mean, the variane and the skewness of interferene at the origin with this deployment model. Even though the pair orrelation funtion beomes ompliated, we devise simple formulae to apture the impat of hardore distane on the variane of interferene in omparison with a PPP model of equal intensity. In addition, we study the extreme senario where the interferene originates from a lattie. We show how to relate the variane of interferene due to a lattie to that of a PPP under Rayleigh fading. Index Terms Headway models, interferene modeling, stohasti geometry, vehiular networks. I. INTRODUCTION Interferene statistis in wireless networks with unknown loations of users have long been studied using stohasti geometry []. Due to its analytial tratability, the Poisson Point Proess PPP is the most ommonly employed model. A PPP with non-homogeneous intensity has been used to apture a variable intensity of users due to mobility [], [3]. The distribution of interferers in ellular uplink with a single interferer per Voronoi ell has also been approximated by non-uniform PPP [4]. Superposition of independent PPPs of different intensities is appliable to heterogeneous ellular networks [5]. By definition, a PPP assumes that two points or users an ome arbitrarily lose to eah other. This assumption may not be aurate due to physial onstraints and/or medium aess ontrol. In this regard, determinantal point proesses have been used to desribe the deployment of real-world maro base stations [6], [7], and Matèrn point proesses to model the loations of ative transmitters in arrier sensing multiple aess wireless ad ho networks [8], [9]. The point proesses suggested in [] [9] were onstruted to study either planar ellular networks or one-dimensional D ad ho networks without deployment onstraints. Therefore they are not immediately tailored to desribe the unique deployment features of vehiular networks. Vehiular networks are expeted to play a key role in improving traffi effiieny and safety in the near future [], K. Koufos and C.P. Dettmann are with the Shool of Mathematis, University of Bristol, BS8 TW, Bristol, UK. {K.Koufos, Carl.Dettmann}@bristol.a.uk This work was supported by the EPSRC grant number EP/N458/ for the projet Spatially Embedded Networks. All underlying data are provided in full within this paper. []. Using a planar two-dimensional PPP to study their performane along orthogonal streets is not aurate in the high reliability regime []. An interferene model for vehiular networks should naturally ombine two spatial models; one for the road infrastruture and another for the loations of vehiles along eah road. The Manhattan Poisson Line Proess has been a popular model for the road network, where the resulting bloks might be filled in with buildings to resemble urban distrits. In ellular systems, it has been shown that a user traveling on a street experienes disontinuous interferene at the intersetions [3]. In the absene of buildings, the interferene from other roads an be easily mapped to interferene from own road with a non-uniform density of users [4], [5]. Reently, the Poisson Line Proess has been used to model the random orientations of roads [6], [7]. In an ad ho setting, the intensities of roads and users have onfliting effets: Inreasing the intensity of roads while keeping fixed the intensity of vehiles per road inreases the interferene while, inreasing the intensity of vehiles redues the average link distane and improves overage [6]. A ommon assumption in [] [7] is that the distribution of vehiles along a roadway follows the D PPP. Similar assumption has been adopted for performane analysis over higher layers, e.g., the study in [8] jointly optimizes the transmission range and the transmission probability for maximizing transport apaity in linear networks with random aess. There are some studies, e.g. [5], [9] using Matèrn proesses to approximate the density of simultaneous transmissions under the repulsive nature of IEEE 8.p. The parent density is still PPP. Finally, onnetivity studies ombining queueing theory with random geometri graphs often make a similar assumption for exponential distribution of inter-arrivals []. A great deal of transportation researh sine the early 96 s has reognized that the distribution of headway distane the distane measured from the head of a vehile to the head of its follower [], or simply the inter-vehile distane is not exponential under all irumstanes. Different models were proposed to approximate the distribution of headway, with the auray of a partiular model being dependent on the traffi status []. Empirial studies revealed that the distribution of time headway time differene between suessive vehiles as they pass a point on the roadway [] is well-approximated by the log-normal Probability Distribution Funtion PDF under free flow [3], [4] and by the log-logisti PDF under ongested flow []. Due to the mixed traffi onditions, Cowan has proposed not only single exponential, shiftedexponential, but also mixed distribution models to desribe the distribution of headways [5].

2 To the best of our knowledge, apart from the exponential distribution, other headway models have not been inorporated into the interferene analysis of vehiular networks. In [6], the log-normal distribution along with the Fenton-Wilkinson method for approximating the distribution of multi-hop distanes has been used to study the lifetime of a link. The randomness is due to the speed and headway, while fading and interferene are negleted. Given a fixed and onstant D intensity of users or vehiles, the PPP assumes that their loations are independent. Let us now onsider a simple enhanement to the PPP, whih assumes that the headway distane is equal to the sum of a onstant hardore or traking distane and an exponentially distributed Random Variable RV. The hardore distane may model the average length of a vehile plus a safety distane. Sine the hardore distane is assumed fixed and onstant, the PDF of headways beomes shifted-exponential. The motivation for this paper is to investigate how the first three moments of interferene distribution behave under the shifted-exponential model. For instane, due to the fat that the deployment of interferers beomes more regular, the predited interferene is expeted to have lower variane as ompared to that originated from a PPP of equal intensity. The shifted-exponential distribution of headways, makes the loations of vehiles orrelated. The assoiated Pair Correlation Funtion PCF has been studied in the ontext of radial distribution funtion for hard spheres in statistial mehanis, see for instane [7], [8], and it has a omplex form. As we will disuss later, the omplexity of higher-order orrelation funtions does not allow us to alulate many more interferene moments or bound the Probability Generating Funtional PGFL [3]. Deriving the first moments of interferene an serve as an intermediate step before approximating its PDF with some simple funtion with known Laplae transform using, for instane, the method of moments. The ontributions of this paper are: For small hardore distane as ompared to the mean inter-vehile distane λ, we show that the variane of interferene at the origin an be approximated by the variane of interferene due to a PPP of equal intensity λ saled with e λ. This model allows getting a quik insight on the impat of traking distane on the variane under various traffi onditions. In addition, it shows that the distribution of interferene beomes more onentrated around the mean in omparison with that due to a PPP of intensity λ. We illustrate that large ell sizesr and traking distanes, modeling driving with high speeds at motorways, are assoiated with more onentrated distributions of interferene less oeffiient of variation and also more symmetri distributions less skewness in omparison with the distributions assoiated with urban miroells. We study the variane of interferene due to D infinite lattie to shed some light on the behavior of interferene when the traking distane beomes omparable to the mean inter-vehile distane, approximating senarios like flow of platoons of vehiles and traffi jams. We devise a simple, yet aurate, model approximating the variane exp/µ r Fig.. The vehiles are modeled as idential impenetrable disks. The vehiles outside of the ell red disks generate interferene to the base station blak square. The rest blue disks are paired with the base station. In the figure, the traking distane is illustrated equal to the diameter of the disk. under the assumptions of Rayleigh fading and small interpoint lattie distane as ompared to the ell size. The rest of the paper is organized as follows. In Setion II, we present the system set-up. In Setion III, we alulate the pair and higher-order orrelations for the new deployment model. In Setion IV, we alulate the mean, the variane and the skewness of interferene. In Setion V, we derive losed-form approximations for the variane. In Setion VI, we study the extreme ase where the interferene originates from a lattie. In Setion VII, we onlude the paper and disuss topis for future work. r II. SYSTEM MODEL Let us assume that the headway distane has two omponents: A onstant traking distane > and a free omponent following an exponential RV with mean µ. This model degenerates to the time headway model M proposed by Cowan [5], if all vehiles move with the same onstant speed towards the same diretion. We study interferene at a single snapshot. The base station is loated at the origin, and the vehiles loated in the interval [ r,r ] are assoiated to the base station, not ontributing to interferene level. The rest of the vehiles generate interferene, see Fig.. While the performane evaluation of wireless ellular networks fouses mostly on the downlink overage of a user, the uplink performane in emerging vehiular networks would be important too. Besides downlink transmissions for entertainment and infotainment servies while on-board, uplink transmissions would be ritial for traffi oordination, effiieny and safety. A valid study for the uplink should naturally inorporate power ontrol, and the onstraint that a single vehile per antenna setor transmits at a time-frequeny resoure blok, see [9]. Due to the omplex form of the PCF, we leave this modeling details for future work. In this paper, we will get a preliminary insight into the impat of orrelated user loations on the moments of interferene in the uplink. Noting that the transmission range an be far greater than the width of a road, one may argue that in roads with multiple lanes, the distribution of inter-vehile distanes mapped onto a single line may still resemble an exponential. In Fig. a, it is illustrated that for λ =.4 the distribution of inter-vehile distanes starts to onverge to that due to a PPP of equivalent intensity for more than eight lanes. On the other hand, for smaller values of λ, e.g., λ=. in Fig. b, only four lanes might be enough to ahieve quite good onvergene. Given that the produt λ is fixed, the hoie of parameters λ, does not impat the speed of onvergene. We see that for r

3 3.9.8 Simulations, hardore PPP approximation.9.8 Simulations, hardore PPP approximation.7.7 CDF =8 =4 = = CDF =8 =4 = = Inter vehile distane m a λ=.5m,=6m Inter vehile distane m b λ=.5m,=4m Fig.. Simulated CDF of inter-vehile distanes resulting from the independent superposition on the same line of point proesses of intensity λ and hardore distane. The approximating CDF using a PPP of equivalent intensity λ is Fx= e λx. simulation runs per urve. two lanes, the sharp twist of the Cummulative Distribution Funtion CDF at inter-vehile distane x = is still lear. Overall, the shifted-exponential model ould be of use for roads with few lanes, e.g., bidiretional traffi streams with restrited overtaking. In this kind of senario, the model helps avoid unrealistially small headway distanes predited by the PPP model with a high probability. Regarding hannel modeling, the propagation pathloss exponent is denoted by η. The distane-based propagation pathloss funtion is gr= r η for an interferer loated at r with r > r, and zero otherwise, to filter out vehiles inside the ell. The fast fading over eah link is Rayleigh, and its multipliative impat h on the interferene power is modeled by an exponential RV with mean unity, E{h}=. The fading samples from different vehiles are independent and identially distributed i.i.d.. The transmit power level is normalized to unity. ρ x,y / λ µ λ = /6 λ = /6, PPP λ = / λ = /, PPP λ = 5/6 λ = 5/6, PPP y x / Fig. 3. Normalized PCF ρ x,y/λµ with respet to the normalized distane y x /. The dashed lines orrespond to ρ x,y = λ, or equivalently, ρ x,y/λµ= λ. III. MOMENT MEASURES The simplest funtion inorporating the distane-dependent onstraints of a point proess is the seond-order intensity measure ρ x,y, or simply the PCF. It desribes the joint probability there are two points in the infinitesimal regions dx,dy entered at x and y respetively. In order to express ρ x,y, we have to alulate the onditional probability there is a point at y given a point at x. For a PPP, the loations of different points are independent, thusρ x,y=λ, where λ is the intensity. On the other hand, for the point proess onsidered here the distane distribution between neighbors is shifted-exponential with positive shift. Next, we show how to alulate ρ x,y for this deployment rule. We will also generalize the alulation for the n-th order orrelation funtion defined over n tuples of points; needed in the alulation of the n-th moment of interferene. Due to the stationarity of the point proess, the PCF depends on the distane separation between x and y. Let us assume y > x > and denote by ρ k y,x,k N, the branh of the PCF for y x+k,x+k+. Sine two vehiles are separated at least by the traking distane, the PCF beomes zero for distanes smaller than, and thus ρ y,x=. For distane separation between and, no other vehiles an be loated in-between. Therefore ρ y,x=λµe µy x, where λµdxdy is the probability that two vehiles are loated in the infinitesimal regions dx,dy, and e µy x is the probability that no other vehile is loated in x+, y. For distane separation between and3, at most one vehile an be loated in-between, and the PCF onsists of two terms. y,x = λµ +λ µe µz x µe µy z dz e µy x x+ λµ = e + λµ y x, µy x e µy x ρ y where µe µz x dz is the probability that a vehile is loated in the region dz entered at z x+,y. Following the same reasoning, when the distane y x is between 3 and 4, there are at most two vehiles inbetween, and the PCF has three terms. The way to alulate the probabilities for zero and one vehile between x and y has been shown. It remains to alulate the probability there are two vehiles. Let us assume that the vehiles are loated at z <z. Then z x+,y and z z +,y. The

4 4 probability that four vehiles are loated at x<z <z <y is λ y y x+ z + µ 3 e µz x e µz z e µy z dz dz. After arrying out the integration and summing up, ρ 3 y,x = λµ e µy x +λµ y x e µy x + λµ3 y x 3 e µy x 3. Similarly, we an ompute ρ k y,x for larger k. {λ k ρ k y,x= j= µ j y x j j Γje µy x j,y x+k,x+k+, otherwise, where k and Γj=j! Obviously, ρ x,y= k= ρ k y,x,y > x. For y<x, we just need to interhange x and y in. This PCF has also been studied in the ontext of statistial mehanis to desribe the density variations of partiles for D hardore fluids as ompared to the ideal PPP fluid [7], [8]. The derivation of in [7], [8] is arried out using thermodynami equations of state. It is the probability of finding a partile at a distane y x from an arbitrary fixed partile at x. We have used basi probability theory instead, to highlight the onstraints introdued by the deployment rule. The Laplae tranform of is available in [3, pp. 5]. In Fig. 3, we depit the normalized PCF. For small λ, the funtion deorrelates within few multiples of. For inreasing λ, the loations of two vehiles an remain orrelated over larger ranges. Let us onsider n points on the real line, x,x,...x n, in inreasing order. Aording to [7, Eq. 7], the higherorder intensity measure ρ n, n 3 for the shifted-exponential deployment has the following form, ρ n x,x,...x n = n λ n i= ρ x i+ x i. For instane, the third-order intensity that desribes the probability to find a triple of distint vehiles at x,y and w, is ρ 3 x,y,w = λ ρ x,yρ y,w. The performane assessment of wireless networks ommonly utilizes the overage probability as a metri, i.e., the probability over the ensemble of all network states that the Signal-to-Interferene-and-Noise ratio is larger than a threshold. Even if the distribution of interferene is unknown, the overage probability ould be omputed, provided that the PGFL of the point proess generating the interferene is available. For a non-poissonian point proess, this is in general diffiult to alulate. In addition, we saw that the n th order intensityρ n has inreasing omplexity for inreasingn. Beause of that, we ould not simplify the multi-dimensional integrations in [3, Eq. 4] to obtain tight bounds for the overage probability. In order to bypass the alulation of the PGFL, we may approximate the interferene by some well-known PDF with parameters, seleted for instane using the method of moments. In that ase, even two or three moments of interferene might be suffiient for a good fit. Some disussion about the PDF of aggregate interferene with a guard zone around the reeiver an be found in [3, Setion III]. In this regard, we show next how to alulate the first three moments of interferene. Mean interferene 3 r =5, =4 r =5, = µ m Fig. 4. Mean interferene with respet to the random part µ of the intensity of vehiles. 5 simulation runs per marker over a line segment of 4 km. The solid lines orrespond to validated against the simulations markers. Given the deployment senario, the higher mean interferene orresponds to pathloss exponent η=, and the lower to η=3. IV. MOMENTS OF INTERFERENCE The mean interferene at the origin an be alulated using the Campbell s Theorem for stationary proesses [33]. Given the traffi parameters µ,, the intensity λ of vehiles is onstant and equal to λ = + µ, or λ = µ +µ [5]. After averaging the distane-based propagation pathloss over the intensity of interferers, we get the mean interferene level. E{I} = λe{h} η grdr = λr r η, where the fator two is due to vehiles at negative half-axis. In Fig. 4 we onsider two senarios: i traking distane = 4 m with ell size r = 5 m modeling vehiular networks in urban street miroells, and ii = m and r = 5 m modeling driving at higher speeds hene the larger traking distane in motorway maroells. We illustrate the mean interferene for inreasing traffi onditions, i.e., inreasing the random part µ of the deployment model. For eah senario, we depit the interferene level for two hannel models, η = and η =3. The large ell size in onjuntion with the large traking distane makes the mean interferene level less sensitive to the random part µ of the traffi intensity. The mean interferene due to a PPP of intensity λ is still given by. However, this is not the ase for higher moments. We shall see that different hardore distanes result in different variane and skewness of interferene while keeping the intensity λ of vehiles fixed by varying µ= λ λ. The seond moment of interferene aepts ontributions not only from a single vehile but also from pairs. E { I } = λ g rdr + gxgyρ x,ydxdy, 3 where the fator two in front of the first term omes from the seond moment of a unit-mean exponential RV, E { h } =.

5 5 In order to alulate S = gxgyρ x,ydxdy, we substitute equation into it, remembering to interhange x and y for x>y. x+k+ S = gxgyρ k y,xdydx+ k= r x+k x k =λ k= r x k+ x+k+ k= r x+k x k λ gxgyρ k x,ydydx k gxgy j= k gxgy k= r x k+ j= µ j y x j j dydx+ Γje µy x j µ j x y j j dydx, Γje µx y j where the fator two is added to aount for x r. The alulation of S involves double integration, infinite sums and requires to filter out the vehiles within the ell. In order to simplify it, we note that for inreasing distane separation, the PCF beomes progressively equal to λ. Let us assume an integer m, and approximate ρ k y,x λ, k m similar for x>y. From the first equality in 4 we have m S x+k+ k= r x+k m x k gxgyρ k y,xdydx + 4 gxgyρ k x,ydydx + k= r x k+ x+k+ x k λ gx gy dydx+ gx gy dydx. k=m r r x+m x+k r r x k+ The last line above an also be written as x m λ gx gy dydx+ gx gy dydx. Using the exat expression of the PCF up to m+ omes at the ost of alulating the integrals x+m+ r x+m gxgyρ m y, x dydx. Therefore the higher the m is, the higher is the penalty for improving the auray. For m= we get S λµ x+ r λ r gxgy x e µy x dy+ e µx y dy x+ x x gxgy dy+ x+ gxgy dx+ gxgy dy dx. In order to see the ompliations in the alulation of higher interferene moments, we show the alulation of the third moment, whih aepts ontributions from a single user, from pairs and from triples of users. E { I 3} =6λ g 3 rdr+6 g xgyρ x,ydxdy+ 6 gxgygwρ 3 x,y,wdxdydw, where the fator six in the first term omes from the third moment of an exponential RV, E { h 3} =6, and the same fator in the seond term omes from multiplying the seond moment 5 of an exponential RV, E { h } =, with the three possible ways to selet a pair out of a triple of users. We still approximate the PCF by λ beyond. The term S = g xgyρ x,ydxdy an be expressed similar to the term S in 5. S λµ x+ r λ r g xgy x g xgy e µy x dy+ e µx y dy x+ x x g xgydy+ x+ dx+ g xgydy dx. Calulating S = gxgygwρ 3 x,y,wdxdydw is more tedious beause the third-order orrelation is equal to the produt of PCFs, ρ 3 x,y,w = λ ρ x,yρ y,w. Fortunately, the pathloss funtion g is ommon for the three users. Therefore it suffies to alulates for a partiular order and sale the result by six. S 6 gxgygwρ x,yρ y,wdxdydw+ λ 6λ gx dx gygwρ y,wdydw, where the first term orresponds to ordered users x<y<w at the same side of the ell, and the seond term desribes the ase with the user x approximately unorrelated to the loations of y,w y<w beause it is plaed at the opposite side, thus ρ x,y λ. Sine we onsider the exat expression for the PCF up to, an be separated into four terms desribing the distane separations loser or further than between the users of eah pair {y,w} and {x,y}., the first term of S above, let us denote it by S S λ λµ y+ µgw dw e µw y + r x+ y+ y+ x+ y+ r x+ µgw dw e µw y + y+ y+ λgw dw gxgydydx + λgw dw gxgy e µy x dydx, where the fator two is due to symmetry, i.e., the three users are loated at the negative half-axis. We ontinue with the seond term of S, let us denote it by S. In the expression of S the users y and w are already ordered and plaed at the same side of the ell. After using the approximation for the PCF beyond we get S λ gxdx r µ y+ r y+ λ r y+ gygw dwdy e µw y gygwdwdy + where the fator two is again due to symmetry, i.e., the sides with respet to the ell of the user x and of the pair {y,z} are interhanged. In Fig. 5, we depit the oeffiient of variation and the skewness of interferene for two senarios; urban =4 m, r = 5 m and motorway = m, r = 5 m ells. We alulate the standard deviation as E{I } E{I}, and,

6 6 Coeffiient of variation of interferene η=3, r =5, =4 η=, r =5, =4 η=3, r =5, = η=, r =5, = Skewness of interferene η=3, r =5, =4 η=, r =5, =4 η=3, r =5, = η=, r =5, = µ m a Coeffiient of variation µ m b Skewness Fig. 5. Statistis of interferene with respet to the random part µ of the deployment. The approximations solid and dashed lines use m= and numerial integration for the terms S,S,S,S. The approximations are validated against the simulations markers. 5 simulation runs per marker. The simulations are arried out over a line segment of 4 km. Standard deviation of interferene 3 η= Simulation, r = Approximation, r = Simulation, r =5 Approximation, r =5 η= Traking distane m a Standard deviation Skewness of interferene Simulation, r = Approximation, r = Simulation, r =5 Approximation, r = Traking distane m η=3 b Skewness Fig. 6. Statistis of interferene for a fixed intensity of vehiles λ=.m. The alulations using integral-based approximation for S,S,S,S for m= are validated against the simulations markers. 5 simulation runs per marker. The simulations are arried out over a line segment of 4 km. η= the skewness as E{I3 } 3E{I}E{I }+E{I} 3, where the terms E{I } E{I} 3/ E { I },E { I 3} in 3 and 6 are evaluated numerially using the approximations for the terms S,S,S with m =. We depit the results up to µ =.m. For = m and µ=.m, we have λ= 3. For larger µ, the approximation auray with m = is poor in the motorway senario beause of long-range orrelations. The mean and the variane of interferene inrease for a lower pathloss exponent given all other parameters remain fixed. We see in Fig. 5 that the oeffiient of variation, defined as the ratio of the standard deviation over the mean, beomes smaller. Lower pathloss exponents are assoiated not only with more onentrated but also with more symmetri, less skewed, interferene distributions. Given the pathloss model, the large ell size and traking distane assoiated with the motorway senario have the same effet on the distribution of interferene. The distribution beomes more onentrated around the mean and also more symmetri between the tails in omparison with the interferene distribution assoiated with urban miroells. In Fig. 6 we have simulated the standard deviation and the skewness of interferene with respet to the traking distane, while the intensity of vehiles λ is fixed. We have used ell sizes, r = m and r =5 m, and pathloss exponents, η= and η=3. We see that the approximations for the PCFs introdue negligible errors for λ.6. The approximation for the skewness is more prone to errors beause the third moment onsists of many terms involving the PCF and also, one term with the produt of PCFs. Based on Fig. 5 and Fig. 6 we dedue that the Gaussian model for the interferene distribution would not be aurate in our system set-up. The distribution is skewed. This is in aordane with the study in [34], illustrating that a twodimensional PPP with a guard zone around the reeiver generates a positively skewed PDF for the aggregate interferene under independent log-normal shadowing. We see in Fig. 6 that larger traking distanes make the variane of interferene less for a fixed intensity of vehiles. This is intuitive beause the deployment beomes more regular. The behaviour of the skewness does not appear to be monotoni. The approximations we got so far do not provide muh insight into the behaviour of seond and third moment of interferene, due to the omplex nature of S,S,S. We would like to apture the impat of traking distane on the moments of interferene using a simple expression. Assuming an intensity λ of vehiles, how do the moments due to a hardore proess, >, sale as ompared to the respetive moments due to a PPP of equal intensity λ? Next, we assume,

7 7 in addition, a small traking distane as ompared to the ell size r. Under small λ and r, we will relate the standard deviation of interferene to that due to a PPP, and draw useful remarks. The more ompliated study about the behaviour of skewness, and the seletion of appropriate models to desribe the PDF of interferene are left for future work. V. CLOSED-FORM APPROXIMATION FOR THE VARIANCE The ontribution to the seond moment of interferene due to pairs of vehiles at distanes larger than is given by the seond term in equation 5. Let us denote it by S >. After substituting the propagation pathloss funtion we get x S > =λ x η y η dydx+ x η gydydx a r x+ = E{I} +λ r x+ r x dydx x η y η + r + r dydx x η y η b = E{I} +4λ x η y η dydx r x+ = E{I} + λ r η b+ η + η η b η F η,η,η; b, where a follows from λ r r x η y η dydx = E{I}, b from symmetry, in we substitute b = r, and F is the Gaussian hypergeometri funtion [35, pp. 556]. Let us denote by S < the first term of S in 5, i.e., the ontribution to the seond moment from pairs of vehiles at distane separation less than. Due to the ommon pathloss funtion g over the users, the ontributions to S < for y>x and x<y are equal for <r, and thus S < = 4λµ x+ r x+ After integrating in terms of y we have S < = 4λ µ η r 7 x η y η e µy x dydx. 8 Γ η,+xµ Γ η,+xµ x η e µ+x dx, 9 whereγa,x= t a x e dt is the inomplete Gamma funtion. t We annot express the above integral in terms of wellknown funtions. In order to approximate it, we expand the integrand around µx+. For a fixed λ and >, we have µ= λ λ >λ. In addition, x+>r. Therefore the expansion should be valid forλr, i.e., the average number of vehiles within the ell must be high. After expanding up to the first-order term and arrying out the integration we get S < = r 4λ e µ xx+ η µ + ηe µ +µ µ x+ 4λ e µ F η,η,η, r η r η λe µ +µ F η,η+,η+, µr η dx + r. For positive λ, the relation λ = orresponds to a lattie with inter-point distane = λ, and the relation λ = orresponds to a PPP of intensity λ. We would like to approximate the variane of interferene for small, while λ remains fixed. We start from the approximation of the term S < in, we substitute µ = λ, and expand around λ, up to seond-order. λ r η S < λ F η,η,η, r r η F η η,η+,η+, r. After substituting 5 into 3, noting that S=S > +S <, arrying out the integration desribing the ontribution to the seond moment from a single vehile, we get Var{I} 4λr η η +S >+S < E{I}. Next, we substitute 7 and in. Var{I} 4λr η η + λ r η η b+ + η η b F η,η,η; b E{I} +λ r η η b F η,η,η, b b F η,η+,η+, b. η After expanding up to seond order in b we have Var{I} 4λr η η λ+λ r η. 3 If we approximate the PCF one step further m=3 instead of m=, and repeat the same proedure, we end up with Var{I} 4λr η λ+ λ +λ r η. 4 η The leading order term, r η, in 3 and 4, will dominate the variane for r. In addition, for small λ, we an use the expansion of the exponential funtion around zero, e λ λ+ λ, to get Var{I} 4λr η η e λ. 5 The above approximation relates in a simple manner the variane of interferene due to a PPP of intensity λ, with the variane of interferene due to a hardore proess of equal intensity, for small λ. Introduing a traking distane, while keeping the intensity λ fixed, makes the deployment more regular, and this results in exponential redution e λ for the variane of interferene, or equivalently,e λ, for the standard deviation. The linear redution in logarithmi sale of the standard deviation with respet to is evident in Fig. 6. Using the approximation for the variane in 5, the oeffiient of η variation an be read as η λr e λ. This approximation for the orrelation oeffiient agrees with the illustrations by Fig. 5a, and allows us to draw the following onlusion: The distribution of interferene beomes more onentrated around

8 8 Standard deviation of interferene x Simulations PPP approximation after PPP approximation after Standard deviation of interferene.5 x Simulations PPP approximation after PPP approximation after Traking distane m a r = m Traking distane m b r = 5 m Fig. 7. Standard deviation of interferene with respet to the traking distane. The intensity is λ=.m. 5 simulation runs per marker. Pathloss exponent η=3. For the PPP approximation after we alulate S > from 7, and S < numerially from 8. In the numerial alulation of PPP approximation after 4, similar but more integrals to 7 and 8 are involved. The dashed line at the top orresponds to a PPP of intensity λ. The dashed line at the bottom orresponds to a lattie with inter-point distane λ. The details for the alulation of the variane due to a lattie are given in Setion VI..45 x 3 x 4 5. Standard deviation of interferene Simulations PPP approximation after PPP approximation after Standard deviation of interferene Simulations PPP approximation after PPP approximation after Traking distane m a r = m Traking distane m b r = 5 m Fig. 8. Standard deviation of interferene with respet to the traking distane. The intensity of vehiles is λ=.5m. See the aption of Fig. 7 for parameter settings and explanation of the legends. the mean for smaller pathloss exponent η, larger ell size r, and inreasing hardore distane while λ remains fixed. In Fig. 7, we have simulated the standard deviation of interferene for high traffi onditions λ =.m, i.e., on average one vehile per m. We depit the results for λ,.8. We see that the losed-form models 3 5 are indeed valid for small. The model in 4 provides a good fit also for realisti traking distanes. This is beause it uses the exat PCF up to 3 instead of. The onsidered distanes,,8 m are muh smaller than the ell size r, thereby the expansions around b are aurate too. For traking distanes>6 m, the model using the exat PCF only up to starts to fail, beause of larger range orrelations. In Fig. 8, we repliate the results of Fig. 7 for lower traffi intensity, on average, one vehile per 4 m. The average inter-vehile distane beomes omparable to the ell size r, and the feasible traking distanes span a muh larger range. We depit the results up to =5 m, or equivalently λ,.65. We dedue that the models 3 5 do not fail due to the approximation of the PCF. We also note that the soure of error is the approximation in b rather than the expansion around λ. The models 3 5 are still valid for small traking distanes. For realisti values of, r z Fig. 9. One-dimensional lattie. The base station, blak square, is loated at the origin. Interferene is due to points outside of the ell, red disks. The RVs z,z represent distanes between the ell border and the lattie point nearest to it generating interferene. they give muh more aurate preditions than the PPP. VI. INTERFERENCE DUE TO A LATTICE In the previous setion, we onstruted simple losed-form models for the variane of interferene due to a hardore proess. These models fail to desribe the variane with longrange orrelations, i.e., λ. Due to its high omplexity, we leave this study for future, and study the extreme senario, λ =, to get a preliminary insight. For λ =, the loations of vehiles form a lattie. Studying the moments of interferene due to infinite latties is also a preliminary step before inorporating more ompliated deployments in our r z r

9 9 analysis, e.g., Cowan M3 [5]. Aording to this model, finite latties of geometrially distributed sizes are separated by exponentially distributed gaps, modeling bunhes of vehiles with gaps in-between the bunhes. The performane of lattie networks not only D has been studied in [36]. Over there, the loation of the reeiver assoiated to the transmitter at the origin is optimized to maximize the ahievable rate. Given the reeiver s loation, the interferene from all points Z\{o} beomes deterministi. In our system set-up, the soures of randomness are the Rayleigh fading and the distane z between the ell border r and the nearest point to it generating interferene, see Fig. 9. Sine we know the loations of all interferers given z,, the PCF beomes an infinite series of Dira delta funtions, and this will greatly simplify the derivation of higher-order moments. For instane, we will show that the double integration gxgyρ x,ydxdy in the alulation of seond moment degenerates to single integration in terms of z. The instantaneous interferene at the base station due to the lattie points loated at the positive half-axis is I = h k gx k = h k gr +z+k, k= k= where h k,x k are the fading oeffiient and loation for the k-th point respetively, and z is a uniform RV, z=u,. The Moment Generating Funtion MGF of interferene is Φ I s= e si f h f x dhdx, where h, x are the vetors of fading oeffiients and user loations respetively, and f h,f x are the assoiated PDFs. The mean interferene an be alulated by evaluating the first derivative of the MGF at s=. E{I} = Φ I = h k gx k f h f x dhdx s s= k= a = h k gr +z+kf h f z dhdz b = k= = = +η k= gr +z+kf z dz r +z+k η dz k= ζ η, r +z dz, 6 where the fator two has been added to aount for lattie points in the negative half-axis, a is due to the fat that given z, the loations of all points beome nonrandom, b follows from independent fading oeffiients and E{h k }=, and ζn,x= k= k +x n is the Hurwitz Zeta funtion. After arrying out the integration in 6, E{I} = ζη,q ζη,+q η η a = r η η, 7 where q = r, and in a we have used the identity for onseutive neighbors ζn,x = ζn,+x+x n. Due to the Campbell s Theoreom [33], the mean interferene an also be alulated by averaging the distanebased pathloss over the intensity of lattie points E{I} = λ r+r η dr = λr η η, where the intensity λ=. In order to alulate the seond moment of interferene, we need to onsider expliitly the interferene originated from the negative half-axis. For that we have to identify the onditional Probability Mass Funtion PMF of the distane z between the ell border r and the nearest lattie point to it generating interferene, given the distane z, see Fig. 9. Let us denote ǫ= r r. The onditional PMF beomes equal to z = ǫ z with probability ǫ, and equal to z = ǫ z with probability ǫ. For presentation larity, we will assume that the diameter of the ell, r, is an integer multiple of the inter-point distane, i.e., ǫ =. In that ase, z = z with probability one. Extensions and numerial results for a positive ǫ will be given. The seond moment of interferene an be alulated by evaluating the seond derivative of the MGF at s=. E { I } = Φ I s= s = h k gx k + h m gx m f h f x dhdx = k= m= h k gx k + h k h m gx k gx m f h f x dhdx, k= k= k,m where the sum over m desribes the interferene from the negative half-axis, and the fator two in front of the square term is due to symmetry. After expanding the square term we have E { I } = h k gx k + h k h k gx k gx k + a = b = k=m= k=k k h k h m gx k gx m f h f x dhdx gx k + k= k=m= k= k=k k gx k gx m gx k gx k + f x dx gx k f x dx+ gx k gx k f x dx+ k=k = }{{}}{{} J J gx k gx m f x dx, k= m= } {{ } J 3 where a is due to E { h k} =, E{hk }=, and independent fading among the users, and in b we have added k =k in the seond sum so that the sum over k goes over all positive integers similar to k and subtrat it from the first sum. The distanes z,z to the ell borders are in general unequal z z. Therefore J J 3 k goes over the positive half- while m spans the negative half-axis. The term J an be alulated

10 as in equation 7, i.e., onditioning in terms of z, integrating the Zeta funtion and using its onseutive neighbors identity J = gr +z+k f z dz = k= r +z+k η dz k= = ζη,q ζη,+q η η = λr η η. 8 In a similar manner, the terms J and J 3 an be expressed as J = r +z+k η r +z+k η dz k= k = = η ζ η, r +z dz. J 3 = 9 r +z+k η r + z+m η dz k=m= = η ζ η, r +z ζ η, r + z dz. For positive ǫ, the alulation of J,J and J 3 requires to average over the PMF of z. The terms J and J would also inlude integrals of sums over the negative half-axis; instead of saling by two the orresponding integrals over the positive half-axis. Due to the fat that the RV z is also uniform, z = U,, the terms J and J for ǫ > ends up equal to equations 8 and 9. The term J 3 ontains the rossterms, over the two axes, thus it requires to average over the PMF of the RV z given z. The term J 3 for ǫ > will be larger than that in 9, refleting the extra randomness introdued by the onditional PMF. Reall that for an arbitrary ǫ,z = ǫ z for z ǫ and z = ǫ z for z> ǫ. Therefore the term J 3 beomes J 3 = η+ ǫ ζ η+ ǫ η, r +z ζ ζ η, r +z ζ η, r + ǫ z dz+ η, r + ǫ z dz. For ǫ=, equation degenerates to the expression of J 3 in 9. Finally, the variane of interferene an be read as Var{I} = λr η η +J λr η +J 3. η In Fig., the integral-based alulation of the variane, see with the termj 3 alulated in, is verified with the simulations. We inlude also the alulations with the term J 3 alulated in 9, i.e., ǫ= {,r }. The impat of positive ǫ beomes more prominent for higher inter-point distane while keeping the ell size r fixed. The terms J, J 3 are diffiult to express in losed-form, see [37] for some reent work involving integrals of produts of Zeta funtions. A high preision evaluation of the Hurwitz Zeta funtion is also an issue beause the funtion is an infinite sum [38]. In order to derive a losed-form approximation for, we note that for large q = r,ǫ = and Rayleigh fading, the variane of interferene due to a lattie an be well-approximated by λr η η. This is beause the variane 4λr due to a PPP under Rayleigh fading, η η, aepts equal ontributions, λr η η, due to fading and due to random user loations. The variane of interferene due to a lattie with inter-point distane muh less than the ell radius should be random mostly due to the fading, i.e., λr η η. In Fig., we see that the orresponding urve due to a PPP of intensity λ essentially overlaps with the urve depiting the integrationbased results for a lattie with = λ and ǫ =. Their differene not possible to notie it in the figure is the standard deviation of interferene due to a lattie without fading. It might be useful to derive a losed-form approximation for the differene of the varianes for ǫ > and ǫ =. We reall it is only the term J 3 that depends on ǫ. Therefore we will expand J 3 for large q in 9 and, and take their differene. With large q, the argument of the Zeta funtion beomes also large, thus it an be well-approximated by an integral instead of a sum. Starting from 9 we get J 3 = η ζη, q+x ζη, q+ xdx η k+q+x η dk k+q+ x η dk dx = η η q+x η q+ x η dx. For q x, we may do first-order expansion. J 3 η η q η η x q η r η η 6η r +6r = 3 η. q η η x q η dx After approximating in a similar manner the term J 3 in and subtrat it from the above, we end up with ǫǫ r η. Therefore the variane of interferene due to a lattie of interpoint distane an be approximated as Var{I} r η η +ǫ ǫr η, where for ǫ=, the variane has been approximated by the variane due to a PPP of intensity. The auray of is illustrated in Fig., where it essentially overlaps with the integration-based results. Using ǫ= in indiates that under Rayleigh fading and large ell size, a lattie of intensity λ= an at most inrease by r η 4 the variane of interferene due to a PPP of intensity λ. This approximation is also available in Fig., dashed yan urve. In Fig. 7 and Fig. 8, the seleted values of ell size, r, and intensity λ result in ǫ=. We an also observe over there the approximately -relation of the standard deviations of interferene due to a PPP and due to a lattie of equal intensity under Rayleigh fading. The third moment of interferene originated from a lattie an be alulated in a similar manner. The alulation is more

11 Standard deviation of interferene 3.4 x Simulations Integration Integration, ε= PPPλ/ PPPλ/+r η /4 Standard deviation of interferene 3 x Simulations Integration Integration, ε= PPPλ/ PPPλ/+r η / Distane m between lattie points a r = m Distane m between lattie points b r = 5 m Fig.. Standard deviation of interferene originated from a lattie. 5 6 simulation runs per marker. Pathloss exponent η=3. The integration orresponds to equation, where the terms J in 9 and J 3 in are evaluated numerially. The integration with ǫ= alulates J 3 numerially from 9. The standard deviation of interferene due to a PPP of intensity λ is λ η r η, where λ=. umbersome beause triples of sums are involved but it does not ome with any new insights. The third-order orrelation degenerates to one-dimensional integral with respet to z. A low-omplexity approximation for the skewness, similar to the one in equation for the variane, is also possible. VII. CONCLUSIONS In this paper we have shown that introduing small traking distane as ompared to the mean inter-vehile distane λ in D vehiular networks redues the variane of interferene exponentially, e λ, with respet to the variane due to a PPP of equal intensity λ. Sine the mean interferene levels under the two deployment models are equal, the oeffiient of variation is redued by e λ, and the interferene distribution beomes more onentrated around its mean. Assuming that the hardore distane is fixed, the exponential orretion makes sense to use espeially under dense traffi, large λ. In addition, the distribution of interferene for small λ remains positively-skewed, indiating that the gamma distribution would probably provide better fit than the normal distribution. We have also studied the extreme senario where the interferene originates from infinite D lattie to get some first insight into the properties of interferene due to the flow of platoons of vehiles. 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