Modeling and Design of Millimeter-Wave Networks for Highway Vehicular Communication
aa r X i v : . [ c s . I T ] A ug Modeling and Design of Millimeter-Wave Networksfor Highway Vehicular Communication
Andrea Tassi, Malcolm Egan, Robert J. Piechocki and Andrew Nix
Abstract —Connected and autonomous vehicles will play apivotal role in future Intelligent Transportation Systems (ITSs)and smart cities, in general. High-speed and low-latency wirelesscommunication links will allow municipalities to warn vehiclesagainst safety hazards, as well as support cloud-driving solu-tions to drastically reduce traffic jams and air pollution. Toachieve these goals, vehicles need to be equipped with a widerange of sensors generating and exchanging high rate datastreams. Recently, millimeter wave (mmWave) techniques havebeen introduced as a means of fulfilling such high data raterequirements. In this paper, we model a highway communicationnetwork and characterize its fundamental link budget metrics.In particular, we specifically consider a network where vehiclesare served by mmWave Base Stations (BSs) deployed alongsidethe road. To evaluate our highway network, we develop a newtheoretical model that accounts for a typical scenario where heavyvehicles (such as buses and lorries) in slow lanes obstruct Line-of-Sight (LOS) paths of vehicles in fast lanes and, hence, actas blockages. Using tools from stochastic geometry, we deriveapproximations for the Signal-to-Interference-plus-Noise Ratio(SINR) outage probability, as well as the probability that a userachieves a target communication rate (rate coverage probability).Our analysis provides new design insights for mmWave highwaycommunication networks. In considered highway scenarios, weshow that reducing the horizontal beamwidth from ◦ to ◦ determines a minimal reduction in the SINR outage probability(namely, · − at maximum). Also, unlike bi-dimensionalmmWave cellular networks, for small BS densities (namely, oneBS every
500 m ) it is still possible to achieve an SINR outageprobability smaller than . . Index Terms —Vehicular communications, millimeter-wave net-works, performance modeling, stochastic geometry.
I. I
NTRODUCTION
By 2020, fifty billion devices will have connectivity ca-pabilities [1]. Among these, ten million vehicles equippedwith on-board communication systems and with a varietyof autonomous capabilities will be progressively rolled out.According to the National Highway Traffic Safety Adminis-tration (U.S. Department of Transportation) and the EuropeanCommission’s Connected-Intelligent Transportation System(C-ITS) initiative [2], [3], connectivity will allow vehiclesto engage with future ITS services, such as See-Through,Automated Overtake, High-Density Platooning, etc [4].
This work is partially supported by the VENTURER Project andFLOURISH Project, which are supported by Innovate UK under GrantNumbers 102202 and 102582, respectively.A. Tassi, R. J. Piechocki and A. Nix are with the Department of Electricaland Electronic Engineering, University of Bristol, UK (e-mail: { a.tassi,r.j.piechocki, andy.nix } @bristol.ac.uk).M. Egan is with the CITI Laboratory of the Institut National de Rechercheen Informatique et en Automatique (INRIA), Universit´e de Lyon, andInstitut National de Sciences Apliqu´ees (INSA) de Lyon, FR (e-mail:[email protected]). As identified by the European Commission’s C-ITS ini-tiative, the number of sensors mounted on each vehiclehas increased. A typical sensor setup is expected to rangefrom ultra-sound proximity sensors to more sophisticatedcamcorders and ‘Light Detection And Ranging’ (LiDAR)systems [4]. Currently, the number of on-board sensors arearound units and this number is expected to double by2020 [5]. Ideally, the higher the number of on-board sensors,the “smarter” the vehicle. However, this holds true only ifvehicles are able to exchange the locally sensed data [6]. Forinstance, multiple LiDAR-equipped vehicles may approacha road hazard and share their real-time LiDAR data withincoming vehicles by means of the road-side infrastructure.This allows the approaching vehicles to compensate for theirlack of sensor data (blind-spot removal) and, for instance,help smart cruise-control systems make decisions. As such,there are strong constraints on LiDAR data delivery, whichcan be generated at rates up to
100 Mbps . More generally,semi-autonomous and fully autonomous vehicles will requirehigh rate and low latency communication links to supportthe applications envisaged by the 5G Infrastructure PublicPrivate Partnership’s (5G-PPP). These applications include theSee-Through use case (maximum latency equal to
50 ms ),which enables vehicles to share live video feeds of theironboard cameras to following vehicles. Other applicationssuch as Automated Overtake and High-Density Platooning arealso expected to require communication latencies smaller than
10 ms [4, Table 1].Recently, communication systems operating in themillimeter-wave (mmWave) range of the wireless spectrumhave been proposed as a means of overcoming the rateand latency limitations of existing technologies [7], [8].In fact, currently commercialized mmWave systems canalready ensure up to and latencies smaller than
10 ms [9]. Table I summarizes the general performancemetrics of mmWave systems and compares them with themain technologies adopted to enable infrastructure-to-vehiclecommunications. Traditionally, ITSs rely on DedicatedShort-Range Communication (DSRC) standards, such asIEEE 802.11p/DSRC and ITS-G5/DSRC [10]–[13]. Eventhough these technologies operate in a licensed band andensure low communication latencies, their maximum realisticdata rate hardly exceeds [10]. As such, severalpapers [14], [15] suggest the adoption of 3GPP’s Long TermEvolution-Advanced (LTE-A) [16], [17], which can guaranteehigher communication rates. Nevertheless, the maximumsupported data rate is limited to
100 Mbps and end-to-endlatencies cannot go below
100 ms [6]. As a result, both
TABLE IR
ADIO ACCESS SOLUTIONS FOR VEHICULAR COMMUNICATIONS . IEEE 802.11p/DSRC,ITS-G5/DSRC [13]
LTE-A [15] mmWaveSystems [9]
FrequencyBand .
85 GHz - .
925 GHz
Spanningmultiple bands in
450 MHz - .
99 GHz 28 GHz ,
38 GHz ,
60 GHz bandsand E-band
ChannelBandwidth
10 MHz
Up to
100 MHz 100 MHz - .
16 GHz
Bit Rate -
27 Mbps
Up to
Up to
Latency ≤
10 ms 100 ms -
200 ms ≤
10 ms
MobilitySupport ≤
130 km h − ≤
350 km h − ≤
100 km h − DSRC and LTE-A cannot always meet the communicationconstraints dictated by delay and bandwidth sensitive servicesthat will be offered by future ITSs [4, Table 1].In mmWave systems, both Base Stations (BSs) and usersare equipped with large antenna arrays to achieve high arraygains via beamforming techniques [18]. As mmWave systemsoperate in the portion of the spectrum between
30 GHz and
300 GHz [9], mmWave links are highly sensitive to blockages.In particular, line-of-sight (LOS) communications are charac-terized by path loss exponents that tend to be smaller than . , while non-line-of-sight (NLOS) path loss exponents areat least equal to . [19]. Due to of the the difficulty of beamalignment, commercial mmWave solutions cannot support userspeeds greater than
100 km h − . Despite this, research to copewith mobile users is gaining momentum [20]. For instance,in the UK, the main railway stakeholders are already trialingmmWave systems with enhanced beam searching techniquesto provide broadband wireless connectivity onboard movingtrains [21]. In addition, multiple research initiatives alreadyregard mmWave systems as suitable to deploy 5G cellularnetworks [22]–[24].In this paper, we consider a typical road-side infrastruc-ture for ITSs [25]. In particular, the infrastructure-to-vehiclecommunications required by ITS services are handled by adedicated network of BSs placed on dedicated antenna mastsand or other street furniture, typically on both sides of theroad [26], [27]. We deal with a highway system where vehiclesreceive high data rate streams transmitted by mmWave BSs, al-though we do not consider scenarios where there is no roadsidedeployment of BSs where vehicle-to-vehicle communicationtechnologies may provide a more effective solution. A keyfeature of our highway system is that vehicles with differentsizes are likely to drive along the same set of lanes. In a left-hand traffic system, any slow vehicle (such as double deckerbuses or lorries) typically travels in the outermost lanes ofthe highway, while the other vehicles tend to drive along theinnermost lanes. If a larger vehicle drives between a user andits serving BS, the BS is no longer in LOS. In other words,large vehicles may act as communication blockages.We develop a new framework to analyze and designmmWave communication systems. The original contributionsof this paper are summarized as follows: • We propose the first theoretical model to characterizethe link budget requirements of mmWave networks pro- viding downlink connectivity to highway vehicles wherecommunications are impaired by large vehicles acting ascommunication blockages. Specifically, we offer designinsights that take into account how the BS and blockagedensities impact the user achievable data rate. • We show that the performance of mmWave highwaynetworks can be well approximated by our theoreticalmodel, which assumes that both BS and blockage posi-tions are governed by multiple time-independent mono-dimensional PPPs. Traditional vehicular models assumethat BSs are equally spaced [28] – thus making them inca-pable of describing irregular BS deployments. In addition,the impact of blockages is either not considered [29] orthe blockage positions are deterministically known [30],which makes the latter kind of models suitable to beincluded in large-scale network simulators but also makesthem analytically intractable. On the other hand, theproposed model is analytically tractable and allows usto predict user performance in scenarios characterizedby different BS densities, traffic intensities, antenna gainand directivity without assuming the BS and blockagepositions being known in advance. • Our numerical validation demonstrates that the proposedtheoretical model is accurate and provides the followingdesign insights: (i) a smaller antenna beamwidth does notnecessarily reduce the Signal-to-Interference-plus-NoiseRatio (SINR) outage probability, and (ii) a reduced SINRoutage probability in highway mmWave networks can beachieved even by low-density BS deployments, for a fixedprobability threshold.The remainder of the paper is organized as follows. Sec-tion II discusses the related work on mmWave and vehicularcommunication systems. Section III presents our mmWavecommunication system providing downlink coverage in high-way mmWave networks. We evaluate the network performancein terms of the SINR outage and rate coverage probabilities,which are derived in Section IV. Section V validates ourtheoretical model. In Section VI, we conclude and outlineavenues of future research.II. R
ELATED W ORK
As summarized in Table II, over the past few years,mmWave systems have been proposed as a viable alterna-tive to traditional wireless local area networks [9] or as awireless backhauling technology for BSs of the same cellularnetwork [31], [32]. Furthermore, mmWave technology hasalso been considered for deploying dense cellular networkscharacterized by high data rates [23], [34], [35]. With regardsto the vehicular communication domain, J. Choi et al. [7]pioneered the application of the mmWave technology to par-tially or completely enable ITS communications. A mmWaveapproach to ITS communications is also being supported bythe European Commission [3].As both the BS deployment and vehicle locations differover both time and in different highway regions, any highwaynetwork model must account for these variations. In thissetting, stochastic geometry provides a means of characterizing
TABLE IIR
ELATED W ORKS ON MM W AVE S YSTEMS AND V EHICULAR C OMMUNICATIONS . Ref. Radio AccessTechnology Network Topology Channel and Path LossModels Mobility CommunicationBlockages [7] mmWave Vehicle-to-vehicle,Vehicle-to-infrastructure Based on ray-tracing Vehicles moving on urban roads Not analyticallyinvestigated[23] mmWave Dense cellular network Nakagami small-scale fading;BGG path loss model Static blockages Buildings[9] mmWave Dense cellular system Based on measurements Static blockages Buildings[31] mmWave Network backhauling Constant small-scale fading(i.e., the square norm of thesmall-scale fadingcontribution is equal to ) Static blockages None[32] mmWave Cellular network withself-backhauling Constant small-scale fading;BGG path loss model Static blockages Buildings[33] mmWave Co-operative cellular network Nakagami for the signal,Rayleigh for the interferencecontribution; BGG path lossmodel Static blockages Buildings[34] mmWave Cellular network withself-backhauling Based on measurements Static blockages Indoor objects[35] mmWave Multi-tier cellular network Constant small-scale fading;Probabilistic path loss model Static blockages Buildings[28] DSRC Vehicle-to-vehicle,Vehicle-to-infrastructure Coverage-based (i.e., nopacket errors from nodeswithin the radio range) Vehicles moving on urban roads None[29] DSRC Vehicle-to-vehicle,Vehicle-to-infrastructure Rice small-scale fading Vehicles moving on urban roads None[30] DSRC Vehicle-to-vehicle Obstacle-based channel andpath loss models Vehicles moving on a highway Vehicles[36] DSRC Vehicle-to-vehicle Rayleigh small-scale fading Vehicles moving on a highway None the performance of the system by modeling BS locations via aspatial process, such as the Poisson Point Process (PPP) [37].Generally, PPP models for wireless networks are now a well-established methodology [8], [37]–[41]; however, there arechallenges in translating standard results into the contextof mmWave networks for road-side deployments due to thepresence of NLOS links resulting from blockages [23]. Inparticular, the presence of blockages has only been addressedin the context of mmWave cellular networks in urban andsuburban environments that are substantially different to ahighway deployment [23]. In particular, in mmWave cellularnetworks: (i) the positions of BSs follow a bi-dimensionalPPP, and (ii) the positions of blockages are governed bya stationary and isotropic process. Even though this is acommonly accepted assumption for bi-dimensional cellularnetworks [37], this is not satisfied by highway scenarios, whereboth blockages and BS distributions are clearly not invariantto rotations or translations. With regards to Table II, the pathloss contribution of blockages has either been modeled bymeans of the Boolean Germ Grain (BGG) principle (i.e.,only the BSs within a target distance are in LOS) or in aprobabilistic fashion (i.e., a BS is in LOS/NLOS with a givenprobability). To the best of our knowledge, no models forroad-side mmWave BS deployment accounting for vehicularblockages have been proposed to date.Given the simplicity of their topology and their high levelof automation, highway scenarios have been well investigatedin the literature [28]–[30]. In particular, [28] addresses theissue of optimizing the density of fixed transmitting nodesplaced at the side of the road, with the objective of maximizingthe stability of reactive routing strategies for Vehicular Ad-Hoc Networks (VANETs) based on the IEEE 802.11p/DSRCstack. Similar performance investigations are conducted in [29]where a performance framework jointly combining physical and Media Access Control (MAC) layer quality metrics isdevised. In contrast to [28] and [29], [30] addresses the issueof blockage-effects caused by large surrounding vehicles; oncemore, [30] strictly deals with IEEE 802.11p/DSRC communi-cation systems. The proposal in [28]–[30] is not applicable formmWave highway networks as the propagation conditions ofa mmWave communication system are not comparable withthose characterizing a system operating between .
855 GHz and .
925 GHz . Another fundamental difference with betweenmmWave and IEEE 802.11p/DSRC networks is the lack ofsupport for antenna arrays capable of beamforming as theIEEE 802.11p/DSRC stack is restricted to omnidirectional ornon-steering sectorial antennas.Highway networks have also been studied using stochasticgeometry. In particular, M. J. Farooq et al. [36] propose amodel for highway vehicular communications that relies onthe physical and MAC layers of an IEEE 802.11p/DSRCor ITS-G5/DSRC system. In particular, the key differencesbetween this paper and [36] are: (i) the focus on multi-hop LOS inter-vehicle communications and routing strategieswhile our paper deals with one-hop infrastructure-to-vehiclecoverage issues, and (ii) the adoption of devices with nobeamforming capabilities while beamforming is a key aspectof our mmWave system.III. S
YSTEM M ODEL AND P ROPOSED
BS-S
TANDARD U SER A SSOCIATION S CHEME
We consider a system model where mmWave BSs providenetwork coverage over a section of a highway, illustrated inFig. 1. The goal of our performance model is to characterizethe coverage probability of a user surrounded by severalmoving blockages (i.e., other vehicles) that may prevent atarget user to be in LOS with the serving BS. Without loss obstacle lane 2 obstacle lane 1obstacle lane 1obstacle lane 2user lane 1 high speed lane 2
NLOS BS LOS and serving BSClosest NLOS BSstandard user (theory model) ψ w ǫ i ǫ (U) τ user lane 2 * standard user (numerical validation) r xy ǫ j A L ( r ) A L ( r ) x i x ( o ) ℓ ,i O Fig. 1. Considered highway system model, composed of N o = 2 obstaclelanes in each traffic direction. of generality, we consider the scenario where vehicles driveon the left-hand side of the road . For clarity, Table IIIsummarizes the symbols commonly used in the paper. In orderto gain insight into the behavior of the model, we make thefollowing set of assumptions. Assumption 3.1 (Road Layout):
We assume that the wholeroad section is constrained within two infinitely long parallellines, the upper and bottom sides of the considered roadsection. Vehicles flow along multiple parallel lanes in only twopossible directions: West-to-East (for the upper-most lanes)and East-to-West (for the lowermost lanes). Each lane has thesame width w . For each direction, there are N o obstacle lanesand one user lane closer to the innermost part of the road. Thecloser a lane is to the horizontal symmetry axis of the roadsection, the more the average speed is likely to increase – thus,the large/tall vehicles are assumed to drive along obstacle lanesmost of the time. Vehicles move along the horizontal symmetryaxis of each lane. We use a coordinate system centered ona point on the line separating the directions of traffic. Theupper side of the road intercepts the y -axis of our system ofcoordinates at the point (0 , w ( N o + 1)) , while the bottom sideintercepts at (0 , − w ( N o + 1)) .In the following sections, we will focus on characterizingthe performance of the downlink phase of a mmWave cellularnetwork providing connectivity to the vehicles flowing inthe high speed lanes of the considered model, which ischallenging. In fact, communication links targeting users inthe high speed lanes are impacted by the largest number ofcommunication blockages (namely, large vehicles) flowing onon the outer road lanes. In addition, we adopt the standardassumption of the BSs being distributed according to a PPP. Assumption 3.2 (BS Distribution):
Let Φ BS = { x i } bi =1 be the one-dimensional PPP, with density λ BS of the x -components of the BS locations on the road. We assumethat BSs are located along with the upper and bottom sidesof the road section. In particular, the i -th BS lies on theupper or bottom sides with a probability equal to q = 0 . .In other words, the y -axis coordinate of the i -th BS is definedas y i = w ( − B q +1)( N o +1) , where B q is a Bernoulli randomvariable with parameter q .By Assumption 3.2 and from the independent thinningtheorem of PPP [42, Theorem 2.36], it follows that the x -axis The proposed theoretical framework also applies to road systems wheredrivers are required to drive on the right-hand side of the road. TABLE IIIC
OMMONLY USED NOTATION . N o Number of obstacle lanes per driving direction w Width of a road lane λ BS Density of the PPP Φ BS of x -components of the BS locationson the road λ O,ℓ
Density of the PPP Φ O ,ℓ of x -components of the blockages onthe ℓ -th obstacle lane τ Footprint segment of each blockage p L , p N Approximated probabilities of a BS being in LOS or NLOS withrespect to the standard user, respectively λ L , λ N Densities of the PPPs of the x -components of LOS and NLOSBSs, respectively ℓ ( r i ) Path loss component associated with the i -th BS A L Assuming the standard user connects to a NLOS BS at a distance r , it follows that there are no LOS BSs at a distance less or equalto A L ( r ) A N Assuming the standard user connects to a LOS BS at a distance r , it follows that there are no NLOS BSs at a distance less orequal to A N ( r )P L , P N Probabilities that the standard user connects to a LOS or a NLOSBS, respectively G TX , G RX Maximum transmit and receive antenna gains, respectively g T X , g RX Minimum transmit and receive antenna gains, respectively L IS , E , E ( s ) Laplace transform of the interference component determined byBSs placed on the upper (
S = U ) or bottom side (
S = B ) ofthe road that are in LOS (
E = L ) or in NLOS (
E = N ) tothe standard user, conditioned on the serving BS being in LOS( E = L ) or NLOS ( E = N ) L I , E ( s ) Laplace transform of the interference I , given that the standarduser connects to a LOS BS ( E = L ) or NLOS BS ( E = N ) P T ( θ ) SINR outage probability as a function of SINR threshold θ R C ( κ ) Rate coverage probability as a function of target rate κ coordinates of the BSs at the upper and bottom sides of theroad form two independent PPPs with density . · λ BS . Assumption 3.3 (Blockage Distribution):
We assume that the ℓ -th obstacle lane on a traffic direction and the coordinates ( x (o) ℓ,i , y (o) ℓ,i ) of blockage i , x (o) ℓ,i belongs to a one-dimensionalPPP Φ O ,ℓ with density λ o ,ℓ , for ℓ ∈ { , . . . , N o } [36]. Theterm y (o) ℓ,i is equal to wℓ , or − wℓ , depending on whether we re-fer to the West-to-East or East-to-West direction, respectively.We assume that the density of the blockages of lane ℓ in eachtraffic direction is the same. Each blockage point is associatedwith a segment of length τ , centered on the position of theblockage itself and placed onto the horizontal symmetry axisof the lane (hereafter referred to as the “footprint segment”).Obstacles can be partially overlapped and the blockage widthsand heights are not part of our modeling. The presence of largevehicles in the user lanes is regarded as sporadic hence, it isignored.From Assumption 3.3, given a driving direction, we observethat the blockage density of each obstacle lane can be different.This means that our model has the flexibility to cope withdifferent traffic levels per obstacle lane; namely, the largerthe traffic density, the larger the traffic intensity. In a real (orsimulated) scenario, the obstacle density λ o ,ℓ of a road sectionis function of the mobility model, which in turn depends on thevehicle speed, maximum acceleration/deceleration, etc. Forma logic point of view, in our theoretical model, we observethat at the beginning of a time step, process Φ O ,ℓ is sampledand a new blockage position is extracted, for ℓ = 1 , . . . , N o .In Section V, we will show that the considered PPP-basedmobility model provides a tight approximation of the investi-gated network performance, in the case of blockages movingaccording to a Krauss car-following mobility model [43].Our primary goal consists of characterizing the SINR outage and rate coverage probability of users located on the user lanes,as these are the most challenging to serve due to the factthat vehicles in the other lanes can behave as blockages. Forthe sake of tractability, our theoretical model tractable willconsider the service of a standard user placed at the origin O = (0 , of the axis. A. BS-Standard User Association and Antenna Model
Since vehicles in the slow lanes can block a direct linkbetween the standard user and each BS, it is necessary todistinguish between BSs that are in LOS with the standarduser and those that are in NLOS. BS i is said to be inLOS if the footprint of any blockage does not intersect withthe ideal segment connecting the standard user and BS i .The probability that BS i is in LOS is denoted by p i, L . Weassume that the blockages are of length τ , illustrated in Fig. 1.In the case that the ideal segment connecting BS i to thestandard user intersects with one or more footprint segments,BS i is in NLOS (this occurs with probability p i, N ) and therelation p i, N = 1 − p i, L holds. For generality, we also assumethat signals from NLOS BSs are not necessarily completelyattenuated by the blockages located in the far-field of theantenna systems. This can happen when the main lobe ofthe antenna is only partially blocked and because of signaldiffraction [44], [45]. By Assumption 3.3, we observe that theprobability p i, E for E ∈ { L , N } of BS i being in LOS ( E = L )or NLOS (
E = N ) depends on the distance from O . This isdue to the fact that the further the BS is from the user, thefurther away the center of an obstacle footprint segment needsto be to avoid a blockage.Consider Assumption 3.3 and the set of points where thesegment connecting O with BS i intersects the symmetry axisof each obstacle lane. We approximate p i, L with the probability p L that no blockages are present within a distance of τ / oneither side of the ray connecting the user to BS i . Hence, ourapproximation is independent of the distance from BS i to O and p i, L can be approximated independently of i as follows: p L ∼ = N o Y ℓ =1 e − λ o ,ℓ τ , (1)while p i, N is approximated as p N = 1 − p L . Observe also thatthe term e − λ O ,ℓ τ is the void probability of a one-dimensionalPPP of density λ o ,ℓ [42].Using the approximation in (1) and invoking the indepen-dent thinning theorem of PPP, it follows that the PPP of theLOS BSs Φ L ⊆ Φ BS and of the NLOS BS Φ N ⊆ Φ BS areindependent and with density λ L = p L λ BS and λ N = p N λ BS ,respectively. In addition, the relation Φ L ∩ Φ N = ∅ holds.Consider the i -th BS at a distance r i = p x i + y i fromthe standard user. The indicator function i, L is equal to oneif BS i is in LOS with respect to the standard user, andzero otherwise. The path loss component ℓ ( r i ) impairing thesignal transmitted by BS i and received by the standard useris defined as follows: ℓ ( r i ) = i, L C L r − α L i + (1 − i, L ) C N r − α N i (2)where α L and α N are the path loss exponents, while C L and C N are the path loss intercept factors in the LOS and NLOS cases, respectively. Terms C L and C N can either bethe result of measurements of being analytically derived by afree space path loss model; the intercept factors are essentialto capture the path loss component at a target transmitter-receiver distance, which for practical mmWave system is equalto [46]. From Assumption 3.1, we remark that relation r i ≥ w ( N o + 1) holds. Hence, for typical values of roadlane widths, path loss intercept factors and exponents, relation w ( N o + 1) ≥ max { C α L L , C α N N } holds as well. This ensuresthat both C L r − α L i and C N r − α N i are less than or equal to . Assumption 3.4 (BS-Standard User Association):
In oursystem model, the standard user has perfect channel stateinformation and always connects to the BS with index i ∗ ,which is characterized by the minimum path loss component,i.e., i ∗ = arg max i =1 ,...,b { ℓ ( r i ) } .We assume that the standard user connects to a NLOS BS ata distance r . Since w ( N o + 1) ≥ max { C α L L , C α N N } , it followsthat there are no LOS BSs at a distance less than or equal to A L ( r ) , defined as: A L ( r ) = max ( w ( N o + 1) , (cid:20) C N C L r − α N (cid:21) − α L ) . (3)We observe that A L ( r ) is the distance from O for which thepath loss component associated with a LOS BS is equal tothe path loss component associated with a NLOS BS at adistance r . In a similar way, we observe that if the standarduser connects to a LOS BS at a distance r from O , it followsthat there are no NLOS BSs at a distance smaller than or equalto A N ( r ) , defined as: A N ( r ) = max ( w ( N o + 1) , (cid:20) C L C N r − α L (cid:21) − α N ) . (4)We observe that definitions (3) and (4) prevent A L ( r ) and A N ( r ) to be smaller than the distance w ( N o + 1) between O and a side of the road.The standard user will always connect to one BS at atime, which is either the closest LOS or the closest NLOSBS. This choice is made by the standard user accordingto Assumption 3.4. In particular, is closest LOS BS is ata distance greater than A L ( r ) , the BS associated with thesmallest path loss component is the closest NLOS BS, whichis at a distance r to the standard user. Those facts allow us toprove the following lemma. Lemma 3.1:
Let d L and d N be the random variables express-ing the distance to the closest LOS and NLOS BSs from theperspective of the standard user, respectively. The ProbabilityDensity Function (PDF) of d L can be expressed as: f L ( r ) = 2 λ L rb ( r ) e − λ L b ( r ) , (5)while the PDF of d N can be expressed as f N ( r ) = 2 λ N rb ( r ) e − λ N b ( r ) , (6)where b ( r ) = p r − w ( N o + 1) . Proof:
Considering the LOS case, the proof directlyfollows from the expression of the PDF of the distance of the closest point to the origin of the axis in a one-dimensional PPPwith density λ L , which is f L ( t ) = 2 λ L e − λ L t [42, Eq. (2.12)].By applying the change of variable t ← b ( r ) we obtain (5).With similar reasoning, it is also possible to prove (6).Using Lemma 3.1, (3) and (4), the following lemma holds. Lemma 3.2:
The standard user connects to a NLOS BS withprobability P N = Z ∞ w ( N o +1) f N ( r ) e − λ L b ( A L ( r )) dr. (7)On the other hand, the standard user connects to a LOS BSwith probability P L = Z ∞ w ( N o +1) f L ( r ) e − λ N b ( A N ( r )) dr = 1 − P N . (8) Proof:
Consider the event that the standard user connectsto a NLOS BS, which is at a distance r from O . This eventis equivalent to have all the LOS BSs at a distance greaterthan or equal to A L ( r ) . From (5), it follows that P [d L ≥ A L ( r ) and d N = r ] is equal to e − λ L b ( A L ( r )) . Then if wemarginalize P [d L ≥ A L ( r ) and d N = r ] with respect to r , weobtain (7). The same reasoning applies to the proof of (8).The gain of the signal received by the standard user dependson the antenna pattern and beam steering performed by theBS and the user. Each BS and the standard user are equippedwith antenna arrays capable of performing directional beam-forming. To capture this feature, we follow [23] and usea sectored approximation to the array pattern. We detailthe sectored approximation for our highway model in thefollowing assumption. Assumption 3.5 (Antenna Pattern):
The antenna patternconsists of a main lobe with beamwidth ψ and a side lobethat covers the remainder of the antenna pattern. We assumethat the gain of the main lobe is G T X and the gain of thesidelobe is g T X . Similarly, the antenna pattern of the standarduser also consists of main lobe with beamwidth ψ and gain G RX , and a side lobe with gain g RX .The antenna of each BS and the user can be steered asfollows. Assumption 3.6 (BS Beam Steering):
Let ǫ i be the anglebetween the upper (bottom) side of the road and the antennaboresight of BS i (see Fig. 1). We assume that ǫ i takes valuesin G = [ ψ , π − ψ ] . As such, the main lobe of each BS isalways entirely directed towards the road portion constrainedby the upper and bottom side. If the standard user connectsto BS i , the BS steers its antenna beam towards the standarduser. On the other hand, if the standard user is not connectedto BS i , we assume that ǫ i takes a value that is uniformlydistributed in G . Assumption 3.7 (Standard User Beam Steering):
The angle ǫ ( U ) between the positive x -axis and the boresight of the userbeam is selected to maximize the gain of the received signalfrom the serving BS. We assume that ǫ ( U ) ∈ [ ψ , π − ψ ] or ǫ ( U ) ∈ [ π + ψ , π − ψ ] if the user is served by a BS on theupper side or the bottom side of the road respectively. Thisassumption ensures that interfering BSs on the opposite sideof the road are always received by a sidelobe, with gain g RX . We also assume that the standard user directs its antenna beamtowards the serving BS, which is then received with gain G RX .IV. SINR O UTAGE AND R ATE C OVERAGE C HARACTERIZATION
For the sake of simplifying the notation and without loss ofgenerality, we assume that the BS with index is the BS thatthe standard user is connected to, while BSs , . . . , b define theset of the interfering BSs. We define the SINR at the locationof the standard user as follows: SINR O = | h | ∆ ℓ ( r ) σ + I , where I = b X j =2 | h j | ∆ j ℓ ( r j ) . (9)Terms h i and ∆ i are the small-scale fading component andthe overall transmit/receive antenna gain associated with BS i , respectively, for i = 1 , . . . , b . The term I is the totalinterference contribution determined by all the BSs except theone connected to the standard user, i.e., the total interferencedetermined by BSs , . . . , b . From Assumption 3.6 and 3.7, itfollows that ∆ is equal to G TX G RX . Finally, σ representsthe thermal noise power normalized with respect to the trans-mission power P t .As acknowledged in [7] there is a lack of extensivemeasurements for vehicular mmWave networks, as well aswidely accepted channel models. Therefore, it is necessaryto adopt conservative assumptions on signal propagation.As summarized in Table II, several channel models havebeen proposed in the literature. Typically publicly availablesystem-level mmWave simulators [47] adopt channel modelsentirely [23] or partially [33] based on the Nakagami model,which are more refined alternatives to the widely adoptedmodels dictating constant small-scale contributions [31], [32],[35]. In particular, we adopt the same channel model in [33],which is based on the following observations: (i) becauseof the beamforming capabilities and the sectorial antennapattern, the signal is impaired only by a limited number ofscatterers, and (ii) the interfering transmissions cluster withmany scatterers and reach the standard user. Furthermore,the considered sectorial antennal model at the transmitterand receiver sides (see Assumption 3.5) significantly reducesthe angular spread of the incoming signals – thus reduc-ing the Doppler spread. Moreover, the incoming signals areconcentrated in one direction. Hence, it is likely that thereis a non-zero bias in the Doppler spectrum, which can becompensated by the automatic frequency control loop at thereceiver side [48]. For these reasons, the Doppler effect hasbeen assumed to be mitigated. Assumption 4.1 (Channel Model):
The channel between theserving BS and standard user is described by a Nakagamichannel model with parameter m , and hence, | h | followsa gamma distribution (with shape parameter m and rateequal to ). On the other hand, to capture the clustering ofinterfering transmissions the channels between the standarduser and each interfering BS are modelled as independentRayleigh channels – thus | h | , . . . , | h b | are independentlyand identically distributed as an exponential distribution withmean equal to . TABLE IVV
ALUES OF ( a, b, ∆) FOR DIFFERENT < | x | , S , E , S , E > . Configuration of Conditions on | x | Enumeration of elements < S , E , S , E > ( a, b, ∆) ∈ C | x1 | , S , E , S , E < U , L , U , L > For any | x | such that J > | x | , K, g TX G RX ) , ( K, + ∞ , g TX g RX ) , ( | x | , + ∞ , g TX g RX ) For any | x | such that J ≤ | x | , K, g TX G RX ) , ( K, + ∞ , g TX g RX ) , ( | x | , | J | , g TX G RX ) , ( | J | , + ∞ , g TX g RX ) < U , L , U , N > For any | x | such that J > x N ( r ) , J, g TX g RX ) , ( x N ( r ) , + ∞ , g TX g RX ) , ( J, K, g TX G RX ) , ( K, + ∞ , g TX g RX ) For any | x | such that J ≤ Refer to the case < U , L , U , L > ( J ≤ )and replace | x | with x N ( r ) < U , L , B , L > For any | x | ( | x | , + ∞ , g TX g RX ) , ( | x | , + ∞ , g TX g RX ) , < U , L , B , N > Refer to the case < U , L , B , L > andreplace | x | with x N ( r ) < U , N , U , L > For any | x | such that x L ( r ) > K Refer to the case < U , L , B , L > andreplace | x | with x L ( r ) For any | x | such that x L ( r ) ≤ K Refer to the case < U , L , U , L > andreplace | x | with x L ( r ) < U , N , U , N > Refer to the case < U , L , U , L >< U , N , B , L > Refer to the case < U , L , B , L > andreplace x with x L ( r ) < U , N , B , N > Refer to the case < U , L , B , L > Cases where S = B , S = B
Refer to the correspondent caseswhere S = U and S = U
Cases where S = B , S = U
Refer to the correspondent caseswhere S = U and S = B
A. Analytical Characterization of I In order to provide an analytical characterization of theinterference power at O , it is convenient to split the term I intofour different components: (i) I U , L and I U , N representing theinterference power associated with LOS and NLOS BSs placedon the upper side of the road whose positions are defined bythe PPPs Φ U , L and Φ U , N , respectively, and, (ii) I B , L and I B , N the interference power generated by LOS and NLOS BSs onthe bottom side of the road placed at the location given by thePPPs Φ B , L and Φ B , N . Overall, the total interference power isgiven by I = P S ∈{ U , B } , E ∈{ L , N } I S , E . In addition, the relations Φ L = Φ U , L S Φ B , L and Φ N = Φ U , N S Φ B , N hold.In the following result, we derive an approximation for theLaplace transform L I ( s ) of I . Theorem 4.1:
Let S = U and S = B represent the caseswhere the standard user connects to a BS on the upper or thebottom side of the road, respectively. In addition, let E = L and E = N signify the cases where the standard user connectsto a LOS or NLOS BS, respectively. The Laplace transform L I S , E , E ( s ) of I S , E , conditioned on E , for S ∈ { U , B } and E ∈ { L , N } , can be approximated as follows: L I S , E , E ( s ) ∼ = Y S ∈{ U , B } , ( a,b, ∆) ∈C | x1 | , S , E , S , E q L I S , E , E ( s ; a, b, ∆) , (10)where L I S , E , E ( s ; a, b, ∆) is defined as in (30). We definethe x -axis coordinates J = w ( N o + 1) / [tan( ǫ (U) + ψ/ and K = w ( N o + 1) / [tan( ǫ (U) − ψ/ of the points wherethe two rays defining the standard user beam intersect with a side of the road, where ǫ (U) = ∓ tan − [ w ( N o + 1) /x ] ,for S = U or B , respectively (see Fig. 8). Furthermore,let us define x L ( r ) = p (A L ( r )) − w ( N o + 1) and x N ( r ) = p (A N ( r )) − w ( N o + 1) . Different combina-tions of parameters < | x | , S , E , S , E > determine differentsequences C | x | , S , E , S , E of parameter configurations ( a, b, ∆) ,as defined in Table IV. Proof:
See Appendix A.
Example 4.1:
Consider the scenario where the standard userconnects to a LOS BS, i.e., E = L , and relation J > holds. We evaluate the Laplace transform of the interferenceassociated with the BSs located on the upper side of the road( S = U ) that are in LOS with respect to the standard user(
E = L ). The sequence C | x | , U , L , U , L is given by the first rowof Table IV, while C | x | , B , L , U , L consists of the same elementsof sequence C | x | , U , L , B , L (last row of Table IV). As a result, L I S , E , E ( s ) can be approximated as follows: L I S , E , E ( s ) ∼ = h L I S , E , E ( s ; | x | , K, g TX G RX ) · L I S , E , E ( s ; K, + ∞ , g TX g RX ) · L I S , E , E ( s ; | x | , + ∞ , g TX g RX ) i / · L I S , E , E ( s ; x N ( r ) , + ∞ , g TX g RX ) . (11)From Theorem 4.1 and the fact that I is defined as asum of statistically independent interference components, thefollowing corollary holds. Corollary 4.1:
The Laplace transform of I , for E = { L , N } ,can be approximated as follows: L I , E ( s ) ∼ = Y S ∈{ U , B } , E ∈{ L , N } L I S , E , E ( s ) (12) Example 4.2:
Consider the scenario where E = L , andrelation J > holds. Using Corollary 4.1, L I S , E , E ( s ) can beapproximated as follows: L I , E ( s ) ∼ = L I S , E , E ( s ; | x | , K, g TX G RX ) · L I S , E , E ( s ; x N ( r ) , J, g TX g RX ) · L I S , E , E ( s ; J, K, g TX G RX ) · (cid:0) L I S , E , E ( s ; K, + ∞ , g TX g RX ) (cid:1) · (cid:0) L I S , E , E ( s ; | x | , + ∞ , g TX g RX ) (cid:1) · (cid:0) L I S , E , E ( s ; x N ( r ) , + ∞ , g TX g RX ) (cid:1) (13) B. SINR Outage and Rate Coverage Probability Framework
The general framework for evaluating the SINR outageprobability is given in the following result.
Theorem 4.2:
Let F L ( t ) = e − λ L √ t − w ( N o +1) (14)and F N ( t ) = e − λ N √ t − w ( N o +1) (15)be the probability of a LOS or NLOS BS not being at adistance smaller than t from O , respectively. We regard P T ( θ ) to be the SINR outage probability with respect to a threshold θ , i.e., the probability that SINR O is smaller than a threshold θ . P T ( θ ) can be expressed as follows: P T ( θ ) = P L − P CL ( θ ) z }| { P [ SINR O > θ ∧ std. user served in LOS ]+ P N − P [ SINR O > θ ∧ std. user served in NLOS ] | {z } P CN ( θ ) (16)where P CL ( θ ) = (cid:12)(cid:12)(cid:12) E − m − X k =0 ( − m − k (cid:18) mk (cid:19) Z + ∞ w ( N o +1) e − vσθ ( m − k )∆1CL r α L1 · L I , E (cid:18) vθr α L ( m − k )∆ C L (cid:19) f L ( r )F N (A N ( r )) dr (17)and P CN ( θ ) = (cid:12)(cid:12)(cid:12) E − m − X k =0 ( − m − k (cid:18) mk (cid:19) Z + ∞ w ( N o +1) e − vσθ ( m − k )∆1CN r α N1 · L I , E (cid:18) vθr α N ( m − k )∆ C N (cid:19) f N ( r )F L (A L ( r )) dr , (18)represent the probability of the standard user not experienc-ing SINR outage while connected to a LOS or NLOS BS,respectively. Proof:
The result (16) follows immediately once P CL ( θ ) and P CN ( θ ) as known. In particular, the following relationholds (for E = L ): P CL ( θ ) = P " | h | ∆ ℓ ( r ) σ + I > θ ∧ std. user served in LOS ( i ) ∼ = E I Z + ∞ w ( N o +1) (cid:18) − (cid:16) − e − v ( σ +I) θ ∆1CL r α L1 (cid:17) m (cid:19) · f L ( r )F N (A N ( r )) dr (19)where v = m ( m !) − /m [23, Lemma 6] and ( i ) arise from | h | being distributed as a gamma random variable. In ad-dition, F N (A N ( r )) is defined as the probability of a NLOSBS not being at a distance smaller than A N ( r ) to O , i.e., theprobability that the standard user is not connected to a NLOSBS. The expression of F N ( t ) , as in (15), immediately followsfrom the simplification of the following relation: F N ( t ) = 1 − Z tw ( N o +1) f N ( r ) dr. (20)From the binomial theorem, we swap the integral and theexpectation with respect to I and invoke Corollary 4.1 toobtain (17). By following the same reasoning, it is alsopossible to derive expressions for P CN ( θ ) and F L ( t ) . Remark 4.1:
As the value of α N increases it is less likely thatthe standard user connects to a NLOS BS. Hence, from (4), A N is likely to be equal to w ( N o + 1) . As a result, theexponential term in (8) approaches one and, hence, P L canbe approximated as follows: P L ∼ = Z ∞ w ( N o +1) f L ( r ) dr = Z ∞ λ L e − λ L t = 1 . (21)Using this approximation, it follows that P N ∼ = 0 holds. Inaddition, since P CN is always less than or equal to P N , the relation P CN ∼ = 0 holds as well. If A N ∼ = w ( N o + 1) , therelation F N (A N ( r )) ∼ = 1 holds. For these reasons, P T ( θ ) can be approximated as follows: P T ( θ ) ∼ = 1 + m − X k =0 ( − m − k (cid:18) mk (cid:19) Z + ∞ w ( N o +1) e − vσθ ( m − k )∆1CL r α L1 · L I , L (cid:18) vθr α L ( m − k )∆ C L (cid:19) f L ( r ) dr . (22)In addition, should signals from NLOS BSs be entirely atten-uated by blockages, P N would be equal to and (22) wouldhold as well. Remark 4.2:
Should the BSs be deployed only along thehorizontal line separating the two driving directions, the valueof p L be equal to and, hence, P L = 1 . Under this circum-stances (22) would hold with minimal changes. For instance,let us focus on the East-to-West driving direction, assume thatthe standard user be located in the middle of the N o + 1 lanes,and that the origin of the coordinate system overlaps withthe standard user position. In this case, an interfering BS i isassociated with ǫ i , which takes values uniformly distributed in [0 , π ) . Let us regard with U the upper-most edge of the East-to-West driving direction, E is equal to L and hence, relation L I , L ( s ) ∼ = L I U , L , L ( s ) holds. This allows us to approximate P T as in (22), where term w ( N o + 1) has to be replaced with w ( N o + 1) / .From [41, Theorem 1] and by using Theorem 4.2, it is nowpossible to express the rate coverage probability R C ( κ ) , i.e.,the probability that the standard user experiences a rate thatis greater than or equal to κ . In particular, the rate coverageprobability is given by: R C ( κ ) = P [ rate of std. user ≥ κ ]= 1 − P T (2 κ/W − , (23)where W is the system bandwidth.V. N UMERICAL R ESULTS
A. Simulation Framework
In order to validate the proposed theoretical model, wedeveloped a novel MATLAB simulation framework capableof estimating the SINR outage and rate coverage probabilitiesby means of the Monte Carlo approach. Both our simulatorand the implementation of the proposed theoretical frameworkare available online [52].We remark that Assumption 3.1 models the highway asinfinitely long, which is not possible in a simulation. How-ever, as noted in [53], [54], the radius R of the simulatedsystem (i.e., the length of the simulated road section R )can be related to the simulation accuracy error ε , as in [53,Eq. (3.5)]. In the case of a one-dimensional PPP, the radiusis related to the simulation error by R ≥ ε − α L − . Wesuperimpose a normal approximation of the binomial propor-tion confidence interval [55] to our simulation results definedas h ˆ p − z p ˆ p (1 − ˆ p ) /n ; ˆ p + z p ˆ p (1 − ˆ p ) /n i , where ˆ p is thesimulated probability value, n is the number of Monte Carloiterations and z is the (1 − . · e ) -th quantile, for ≤ e ≤ . Inparticular, z is set equal to . , defining a confidence intervals TABLE VM
AIN SIMULATION PARAMETERS . Parameter Value
Simulated time
13 h (for Figs. 3-7),
55 h (for Fig. 2)Length of thesimulated roadsection (i.e., R )
20 km ,
100 km w . , as per [49] λ u · − Mobility model Blockages and the standard user move according to aKrauss car-following mobility model [43]; maximumacceleration and deceleration equal to . / s [50],maximum vehicle speed equal to
96 km h − (blockages) and
112 km h − (standard user).Blockagedimensions The dimensions of a double decker bus, i.e., length τ equal to . and width equal to .
52 m [51] N o , { λ o, , λ o, } { · − , · − } , i.e., one blockage every {
100 m ,
50 m } λ BS From · − to · − , with a step of · − Carrier frequency f
28 GHzC L , C N −
20 log (4 πf/c ) , which is the free space path loss indB at a distance of and c is the speed of light [46] α L . α N { , . } , as per [19] m , as per [33] φ { ◦ , ◦ } G TX {
10 dB ,
20 dB } G RX
10 dB g TX , g RX −
10 dB W
100 MHz P t
27 dBm
Thermal noisepower (i.e, σ · P t )
10 log ( k · T · W · ) dBm , where k is theBoltzmann constant and the temperature T = 290 K [18] of and n is equal to · (for Fig. 2) or · (forFigs. 3-7).We simulated scenarios where the standard user drives onthe lower-most user lane along with multiple other vehicles;in particular, we considered a number of vehicles equal to ⌊ Rλ u ⌋ , where λ u is the vehicle density driving on each userlane. In addition, a number of blockages equal to ⌊ Rλ o,i ⌋ areplaced at random on each obstacle lane i , for i = 1 , . . . , N o .During each simulated scenario, both the vehicles driving onthe user lanes and blockages move according to the Krausscar-following mobility model [43] and their maximum speedis set equal to
70 mph (i.e.,
112 km h − ) and
60 mph (i.e.,
96 km h − ), respectively as dictated by the current Britishspeed limits . In order to keep the density of the simulatedblockages constant and hence allow a fair validation of theproposed theoretical framework, we adopted the Krauss car-following mobility model with the wrap-around policy. Inparticular, when a vehicle reaches the end of the simulatedroad section, it re-enters at the beginning.BSs are positioned uniformly at random at both sides ofthe road. The simulator estimates the SINR outage probability P T and rate coverage probability R C by averaging over thetotal simulation time and across a number of BS randomlocations and steering angle configurations (of the interferingBSs); number allowing to the simulated average performancemetrics to converge to a stable value. We remark that theadoption of highly directional antennas significantly reducesthe angular spread of the incoming signals. As such, in the λ BS · P L . . . . . . . . N o = 1, Simulation N o = 1, Theory N o = 2, Simulation N o = 2, Theory p L p N Avg. Blockage Duration
Upper Side Bottom Side N o = 1 0 .
78 0 .
22 0 .
19s 2 . N o = 2 0 .
69 0 .
31 0 .
23s 2 . Fig. 2. Probability P L that the standard users connects to a LOS BS as afunction of λ BS , for N o = { , } and α N = 4 . simulated scenarios, we assume the standard user is equippedwith an automatic frequency control loop compensating for theDoppler effect [48]. In addition, the simulated channel followsAssumption 4.1.With regards to Table V, we consider N o = { , } obstaclelanes per driving direction. For N o = 2 , we assume differenttraffic intensities by setting densities { λ o , , λ o , } as per rowsix of Table V. Furthermore, we consider a typical highwaylane width w [49].In Section III-A, we approximated the probabilities p L and p N for a BS of being in LOS or NLOS with respect to thestandard user, respectively. It is worth noting that approxima-tion (1) has been invoked only in the derivation of the proposedtheoretical model. In contrast, in the simulated scenarios a BSis in NLOS only if the ideal segment connecting the standarduser and the BS intersects with one or more vehicles in theobstacle lanes and not just with their footprint segments.Communications between the standard user and the servingBS are impaired only by large vehicles (namely, trucks,double-decker buses, etc.) driving on the obstacle lanes.Specifically, we consider blockages with length ( τ ) and widthof a double-decker bus [51]. Without loss of generality, boththe proposed theoretical framework and our simulations con-sider bi-dimensional scenarios. Although it is always possibleto deploy BSs having an antenna height sufficient to preventthe vehicles in the obstacle lanes to behave as blockages, itis not always feasible in practice. For instance, in a -laneroad section ( N o = 2 ) with w = 3 . where the standarduser drives in the middle of the lower-most user lane and theuser antenna height is . , the BS antenna height should begreater than . to allow a blockage-free scenario, whichis at least twice as much the antenna height in a typicalLTE-A urban deployment [56]. Therefore, we assume that theBS antenna height is , which means that vehicles in theobstacle lanes always behave as blockages. For this reason, wedo not further consider the height of the vehicles in our study.All the remaining simulation parameters are summarized inTable V. B. Theoretical Model Assessment
In order to numerically study our mmWave highway net-work and assess the accuracy of our theoretical model, wefirst consider α N = 4 and a road section with a length θ (dB) P T ( θ ) − . . . . . . . . ψ = 30 ◦ , G TX = 10dB ψ = 90 ◦ , G TX = 10dB ψ = 30 ◦ , G TX = 20dB ψ = 90 ◦ , G TX = 20dBSimulationTheory . . . . . . . . (a) λ BS = 10 − , x -ISD =
100 m θ (dB) P T ( θ ) − . . . . . . . . ψ = 30 ◦ , G TX = 10dB ψ = 90 ◦ , G TX = 10dB ψ = 30 ◦ , G TX = 20dB ψ = 90 ◦ , G TX = 20dBSimulationTheory . . . . . . . . (b) λ BS = 4 · − , x -ISD =
250 m
Fig. 3. SINR outage probability P T as a function of the threshold θ , for N o = 1 , α N = 4 , ψ = { ◦ , ◦ } and G TX = {
10 dB ,
20 dB } . R = 100 km , which ensures a simulation accuracy errorof at least − . . In addition, the adoption of a relativelysmall but realistic value of α N makes more likely for thestandard user to a connect to an NLOS BS [23] and hence,allows us to effectively validate the proposed LOS/NLOS userassociation model (see Lemma 3.2). Considering the density λ BS of process Φ BS , we ideally project the BSs onto the x -axis and we define their projected mean Inter-Site-Distance( x -ISD) as /λ BS .Let us consider a Fig. 2 shows the probability of the standarduser connecting to a LOS BS as a function of λ BS for one andtwo obstacle lanes on each side of the road. The equivalent x -ISD spans between ( λ BS = 2 · − ) and
50 m ( λ BS =2 · − ). In particular, as typically happens, we observe that P L is significantly greater than P N . Specifically, if N o = 1 then, for λ BS = 4 · − , the simulated value of P L is equalto . . When N o increases to , the simulated value of P L reduces to . , for λ BS = 4 · − .Fig. 2 also compares our approximated theoretical expres-sion of P L , as in (8), with the simulated one. We notethat (8) overestimates P L , and, hence, (7) underestimates P N .However, we observe that: (i) for λ BS ∈ [2 · − , − ] , theoverestimation error is smaller than . ), and (ii) for densescenarios ( λ BS > − ), it never exceeds . . Generally, weobserve that the proposed theoretical model follows the trendof the simulated values. From Fig. 2, we also conclude that P L may have a non-trivial minimum. In our scenarios, this isparticularly evident when N o = 2 . θ (dB) P T ( θ ) − . . . . . . . . ψ = 30 ◦ , G TX = 10dB ψ = 90 ◦ , G TX = 10dB ψ = 30 ◦ , G TX = 20dB ψ = 90 ◦ , G TX = 20dBSimulationTheory . . . . . . . . (a) λ BS = 10 − , x -ISD =
100 m θ (dB) P T ( θ ) − . . . . . . . . ψ = 30 ◦ , G TX = 10dB ψ = 90 ◦ , G TX = 10dB ψ = 30 ◦ , G TX = 20dB ψ = 90 ◦ , G TX = 20dBSimulationTheory . . . . . . . . (b) λ BS = 4 · − , x -ISD =
250 m
Fig. 4. SINR outage probability P T as a function of the threshold θ , for N o = 2 , α N = 4 , ψ = { ◦ , ◦ } and G TX = {
10 dB ,
20 dB } . Remark 5.1:
As we move from sparse to dense scenarios,it becomes more likely for a NLOS BS to be closer to thestandard user; thus P L decreases. However, this reasoningholds up to a certain value of density. In fact, at some point,the BS density becomes so high that it becomes increasinglyunlikely not to have a LOS BS that is close enough to serve thestandard user. This phenomenon may determine a non-trivialminimum in P L .The table superimposed to Fig. 2 lists the (simulated) valuesof p L , p N and the average duration of a blockage eventimpairing transmissions from BSs on the upper and bottomside of the road. In particular, we observe that a blockageevent can occur with a probability greater than . and canlast up to .
68 s .Fig. 3 shows the effect of the SINR threshold θ on the out-age probability P T ( θ ) , for N o = 1 , several antenna beamwidth ψ and a range of BS transmit antenna gains G TX . Here, thevehicular receive antenna gain is set to G RX = 10 dB . InFig. 3a, the x -ISD is fixed at
100 m . It should be noted thatthe proposed theoretical model, as in Theorem 4.2, not onlyfollows the trend of the simulated values of P T ( θ ) but alsoit is a tight upper-bound for our simulations for the majorityof the values of θ . In addition, the deviation between theory The standard user drives in the East-to-West direction. Hence, the East-to-West blockages have an (average) relative speed equal to
16 km h − (namely,
112 km h − −
96 km h − ). For blockages with a length equal to . ablockage event is expected to last about . , which is close to the result ofour simulations. The same reasoning applies to West-to-East blockages. and simulation is negligible when θ ∈ [ − ,
15 dB] or θ ∈ [ − ,
10 dB] , for G TX = 10 dB or G TX = 20 dB ,respectively. On the other hand, that deviation gradually in-creases as θ becomes larger. Nevertheless, the maximum MeanSquared Error (MSE) between simulation and theory is smallerthan . · − . Overall, we observe the following facts: • Changing the beamwidth from ◦ to ◦ alters the SINRoutage probability only by a maximum of · − . Thiscan be intuitively explained by noting that the servingBS is likely to be close to the vertical symmetry axis ofour system model. From Assumption 3.7, the standarduser aligns its beam towards the serving BS. As such,the values of J and K (see Theorem 4.1) do not largelychange on passing from ψ = 30 ◦ to ψ = 90 ◦ . Thus,for the interference component to become substantial, thevalue of ψ should be quite large. • Overall, we observe that when the beamwidth increases,so does P T . Intuitively, that is because the standard useris likely to receive a large interference contribution viathe main antenna lobe. • Increasing the value of the maximum transmit antennagain (from
10 dB to
20 dB ) results in a reduction of theSINR outage probability that, for large values of θ , can begreater than . . This clearly suggests that the incrementon the interfering power is always smaller than or equalto the correspondent increment on the signal power. Thisis mainly because of the directivity of the consideredantenna model and the disposition of the BSs.Fig. 3b refers to the same scenarios as in Fig. 3a except forthe x -ISD that is equal to
250 m . In general, we observe thatthe comments to Fig. 3a still hold. Furthermore, the impact ofthe value of ψ on P T becomes negligible. Intuitively, this canbe explained by noting that the number of interfering BSs thatare going to be received by the standard user at the maximumantenna gain decreases as λ BS decreases. However, as the BSdensity decreases (the BSs are more sparsely deployed), itbecomes more likely (up to a certain extent) that the numberof interfering BSs remains the same, even for a beamwidthequal to ◦ .Fig. 4 refers to the same scenarios as Fig. 3 with twoobstacle lanes on each side of the road ( N o = 2 ). In additionto the discussion for Fig. 3, we note the following: • For the smallest value of the antenna transmit gain( G TX = 10 dB ), both the simulated and the proposedtheoretical model produce values of P T that are negligi-bly greater that those when N o = 1 . • For x -ISD =
100 m and G TX = 20 dB , the SINRoutage is slightly greater that the correspondent case asin Fig. 3a. In particular, for θ ≥
25 dB , we observe anincrement in the simulated P T bigger than · − . • As soon as we refer to a sparser network scenario, x -ISD =
250 m , the conclusions drawn for Fig. 3b alsoapply for Fig. 4b. Hence, the impact of ψ on P T vanishes.From Fig. 3 and Fig. 4, we already observed that theproposed theoretical model, as in Theorem 4.2, follows wellthe trend of the corresponding simulated values, and it ischaracterized by an error that is negligible for the most λ BS · P T ( θ )
10 30 50 70 90 110 130 15000 . . . . . N o = 1, θ = 5dB, Simulation N o = 1, θ = 5dB, Theory N o = 2, θ = 5dB, Simulation N o = 2, θ = 5dB, Theory N o = 1, θ = 15dB, Simulation N o = 1, θ = 15dB, Theory N o = 2, θ = 15dB, Simulation N o = 2, θ = 15dB, Theory
50 60 70 80 9000 . . . . . . Fig. 5. SINR outage probability P T as a function of the BS density λ BS ,for θ = { ,
15 dB } dB, N o = { , } , α N = 4 , ψ = 30 ◦ and G TX = 20 dB . κ (Mbps) R C ( κ ) . . . . . . G TX = 10dB, Simulation G TX = 10dB, Theory G TX = 20dB, Simulation G TX = 20dB, Theory Fig. 6. Rate coverage probability R C as a function of the threshold κ , for α N = 4 , ψ = 30 ◦ , G TX = {
10 dB ,
20 dB } , λ BS = 4 · − , N o = 2 . important values of θ (e.g., θ ≤
20 dB ). These facts arefurther confirmed by Fig. 5, which shows the value of P T as a function of λ BS , for θ = 5 dB or
15 dB , and ψ = 30 ◦ . Inparticular, as also shown in Fig. 3 and Fig. 4, as θ increases thedeviation between the simulations and the theoretical modelincreases. However, the MSE between theory and simulationnever exceeds · − in Figs. 4a and 4b. Furthermore, Fig. 5allows us to expand what was already observed for Fig. 3 andFig. 4: • As expected, P T increases as N o increases. However,when N o passes from to , P T increases no morethan · − . Hence, we conclude that the network isparticularly resilient to the number of obstacle lanes. • The impact of λ BS on the value of P T more evident forsparse scenarios – λ BS ≤ · − and λ BS ≤ · − ,for θ = 5 dB and θ = 15 dB , respectively. Otherwise, theimpact of λ BS is reasonably small, if compared to whathappens in a typical bi-dimensional mmWave cellularnetwork [23]. This can be justified by the same reasoningprovided for Fig. 3a. • As the value of λ BS increases, the interference componentprogressively becomes dominant again and hence, P T isexpected to increase. In Fig. 5, this can be appreciatedfor N o = 2 and θ = 15 dB .Let us consider again Fig. 5. In the considered scenarios, it ispossible to achieve a value of P T smaller than . for valuesof λ BS ∼ = 2 . · − .Fig. 6 shows the rate coverage probability as a functionof the rate threshold κ , for ψ = 30 ◦ , λ BS = 4 · − and N o = 2 . From (23), we remark that the expression of R C λ BS · P T ( θ )
10 30 50 70 90 110 130 15000 . . . . . N o = 1, θ = 5dB, Simulation N o = 1, θ = 5dB, Theory N o = 2, θ = 5dB, Simulation N o = 2, θ = 5dB, Theory N o = 1, θ = 15dB, Simulation N o = 1, θ = 15dB, Theory N o = 2, θ = 15dB, Simulation N o = 2, θ = 15dB, Theory
50 60 70 80 9000 . . . . . . Fig. 7. SINR outage probability P T as a function of the BS density λ BS ,for θ = { ,
15 dB } dB, N o = { , } , α N = 5 . , ψ = 30 ◦ and G TX = 20 dB . Simulation results obtained for R = 20 km . directly follows from P T . For this reason, we observe that thegreater the gain G TX , the higher the value of R C . Finally, weobserve that the MSE between simulations and the proposedtheoretical approximation is smaller than . · − .For completeness, our model was validated by considering α N = 5 . and a significantly shorter highway section, namely R = 20 km . We observe that the considered value of α N isamong the highest NLOS path loss exponent that has everbeen measured in an outdoor performance investigation [23].In particular, Fig. 7 compares simulation and theoretical resultsfor the same transmission parameters and road layout as inFig. 5. The bigger NLOS path loss exponent determines bigger P T values than the correspondent cases reported in Fig. 5 (theabsolute difference is bigger than . · − ). Nevertheless,what observed for Fig. 5 applies also for Fig. 7. In particular,we conclude that the proposed theoretical model remains validfor shorter road sections.VI. C ONCLUSIONS AND F UTURE D EVELOPMENTS
This paper has addressed the issue of characterizing thedownlink performance of a mmWave network deployed alonga highway section. In particular, we proposed a novel theoreti-cal framework for characterizing the SINR outage probabilityand rate coverage probability of a user surrounded by largevehicles sharing the other highway lanes. Our model treatedlarge vehicles as blockages, and hence, they impact on thedeveloped LOS/NLOS model. One of the prominent featuresof our system model is that BSs are systematically placed atthe side of the road, and large vehicles are assumed to drivealong parallel lanes. Hence, unlike a typical stochastic geom-etry system, we assumed that both BS and blockage positionsare governed by multiple independent mono-dimensional PPPsthat are not independent of translations and rotations. Thismodeling choice allowed the proposed theoretical frameworkto model different road layouts.We compared the proposed theoretical framework withsimulation results, for a number of scenarios. In particular,we observed that the proposed theoretical framework canefficiently describe the network performance, in terms ofSINR outage and rate coverage probability. Furthermore, weobserved the following fundamental properties: • Reducing the antenna beamwidth from ◦ to ◦ doesnot necessarily have a disruptive impact on the SINRoutage probability, and hence, on the rate coverage prob-ability. • In contrast with bi-dimensional mmWave cellular net-works, the network performance is not largely impactedby values of BS density ranging from moderately sparseto dense deployments. • Overall, for a fixed SINR threshold, a reduced SINRoutage probability can be achieved for moderately sparsenetwork deployments.A
PPENDIX AP ROOF OF T HEOREM E = { L , N } , the Laplace transform of I S , E can beexpressed as: L I S , E , E ( s ) = E Φ S , E Y j ∈ Φ S , E E h E ∆ (cid:16) e − s | h j | ∆ j ℓ ( r j ) (cid:17) (24) ( i ) = exp − E ∆ E h Z + ∞ w ( N o +1) (1 − e − sh ∆C E r − α E ) · rqλ E p r − w ( N o + 1) dr ! (25)where E Φ S , E represents the expectation with respect to thedistance of each BS in Φ S , E from O . Similarly, operators E ∆ and E h signify the expectation with respect to the overallantenna gain and the small-scale fading gain associated withthe transmissions of each BS, respectively. For the sake ofcompactness, from ( i ) onward we refer to | h | simply as h . We observe that equality ( i ) arises from the definitionof a probability generating functional (pgfl) of a PPP [42,Definition 4.3] and the mapping theorem applied to Φ S , E [42,Theorem 2.34]. In addition, the pgfl allows us to drop therelation to a specific BS j in the terms expressing a distanceof a BS to O , its channel and antenna gains. For this reason,in the integrand function, we simply refer to terms r , h and ∆ .Let a and b be two real numbers greater than or equal to w ( N o + 1) and such that a ≤ b . With regards to (25), wecondition with respect to a specific value of h and ∆ , and weapproximate the following term : Z ba (1 − e − sh ∆C E r − α E ) 2 rqλ E p r − w ( N o + 1) dr ( i ) = Z √ b − w ( N o +1) √ a − w ( N o +1) (cid:16) − e − sh ∆C E ( t + w ( N o +1) ) − α E / (cid:17) qλ E dt ( ii ) ∼ = Z ba (cid:16) − e − sh ∆C E t − α E (cid:17) qλ E dt (26) ( iii ) = − qλ E Z b − α E a − α E (1 − e − sh ∆C E x ) α − x − α − − dx ( iv ) = Θ( h, ∆) z }| { qλ E h (1 − e − sh ∆C E x ) x − α − i b − α E x = a − α E − qλ E Z b − α E a − α E sh ∆C E x − α E e − sh ∆C E x dx | {z } Λ( h, ∆) , (27) For clarity, we define [ f ( x )] bx = a = f ( b ) − f ( a ) . where ( i ) arises from the change of variable t ← p r − w ( N o + 1) , while ( ii ) assumes that w ( N o + 1) is equal to (see Section V-B for the validation of theproposed theoretical framework). Equality ( iii ) arises byapplying the changes of variable y ← t α E and then x ← y − .In addition, in ( iv ) , we resort to an integration by parts.With regards to (27), we keep the conditioning to ∆ andcalculate the expectation of Θ( h, ∆) and Λ( h, ∆) , with respectto h . From Assumption 4.1, it should be noted that we refer toa Rayleigh channel model, and, hence, the following relationholds: E h [Θ( h, ∆)] = 2 qλ E (cid:20) x − α − (cid:18) − s ∆C E x + 1 (cid:19)(cid:21) b − α E x = a − α E . (28)Term E h [Λ( h, ∆)] can be found as follows: E h [Λ( h, ∆)] = − qλ E Z b − α E a − α E x − α E · Z ∞ sh ∆C E e − ( s ∆C E x +1) h dh dx (29) = − qλ E Z b − α E a − α E x − α E ∂∂x (cid:18) − s ∆C E x + 1 (cid:19) dx ( i ) = − qλ E ( s ∆C E ) α E Z − ( s ∆C E b − α E +1) − − ( s ∆C E a − α E +1) − (cid:18) − t − (cid:19) − α E dt ( ii ) = − qλ E ( s ∆C E ) α E " t ( − t − ) − α E Γ (cid:18) α E + 1 (cid:19) · ˜ F (cid:18) α E , α E + 1; 1 α E + 2; − t (cid:19) − ( s ∆C E b − α E +1) − t = − ( s ∆C E a − α E +1) − , where ( i ) arises from the change of variable t ← − s ∆C E x +1 .Let us signify with ˜ F ( a, b ; c ; z ) = P ∞ k =0 { a } k { b } k { c } k z k k ! theGauss hypergeometric function . We observe that the inte-gral as in equality ( i ) is closely related to that as in [58,Eq. (3.228.3)], and after some manipulations we have equal-ity ( ii ) . From the approximation in (26), it follows that a ∼ = p a − w ( N o + 1) and b ∼ = p b − w ( N o + 1) .Hence, we observe that L I S , E , E ( s ) , conditioned on the gain ∆ (see (9)), can be expressed as follows: L I S , E , E ( s ) ∼ = L I S , E , E ( s ; a, b, ∆) (cid:12)(cid:12)(cid:12) a =0 ,b =+ ∞ (30) = exp − (cid:16) E h [Θ( h, ∆)] + E h [Λ( h, ∆)] (cid:12)(cid:12)(cid:12) a =0 ,b =+ ∞ (cid:17)! . Let us focus on the transmit antenna gain of the j -thinterfering BS, which has a PDF that depends on the distance r j and the orientation of the beam ǫ i . To take into account theexact formulation of the BS transmit antenna gain would makethe performance model intractable. As such, we instead makethe approximation that the transmit antenna gain is alwaysequal to g TX . We observe that z is always a real number, which allows us to significantlyreduce the complexity of the whole numerical integration process [57]. ǫ (U) J K r x w ( N o + ) xy O Fig. 8. Case where the standard user is served by a BS from the upper sideof the road.
With regards to (24), we observe that after conditioning onthe standard user being connected to a BS at a distance r from O , then the receive antenna gain g of the interfering BSs j isdetermined by the parameter list < | x | , p, r, S , E , S , E > ,where: (i) | x | is the absolute value of the x -axis coordinateof BS , (ii) p captures the fact that the interfering BS is ata location on the positive (right-hand side of y -axis, RX ) ornegative side (left-hand side of y -axis, LX ) of the x -axis, and(iii) r is the distance of the interfering BS to O .Let us consider the x -axis coordinates J and K of the pointswhere the two rays defining the antenna beam of the standarduser intersect the side of the road, as shown in Fig. 8. Thereceive antenna gain of the interfering BS also depends on: (i)the fact the standard user connects to a BS on the upper/bottomside of the road ( S ∈ { U , B } ), that can be in LOS/NLOS( E ∈ { L , N } ) with respect to the standard user, (ii) thevalues of S and E , and (iii) the specific configuration of thevalues of | x | , J and K . By invoking the same approximationas in (26), we say that r ∼ = p r − w ( N o + 1) and thefollowing parameters determine the receiver gain: • S = U , E = L , S = U and
E = L - we divide this caseinto the following subcases: – If the value of | x | is such that J > - we observethat there are no LOS BSs at a distance smaller than | x | . Hence, it follows that g = G RX if (cid:26) | x | ≤ r ≤ Kp = RX (31) g = g RX if (cid:26) K ≤ r ≤ + ∞ p = RX or (cid:26) | x | ≤ r ≤ + ∞ p = LX (32) – If J ≤ - by following the same reasoning as before,in addition to the case as in (31), it follows that g = g RX if (cid:26) K ≤ r ≤ + ∞ p = RX or (cid:26) | J | ≤ r ≤ + ∞ p = LX (33) g = G RX if (cid:26) | x | ≤ r ≤ | J | p = LX (34) • S = U , E = L , S = U and
E = N - we apply thesame reasoning as before by bearing in mind that it isimpossible for a NLOS BS to be at a distance that issmaller than A N ( r ) to O . Equivalently, it is impossiblefor a NLOS BS to be associated with a x -axis coordinatesmaller than x N ( r ) = p (A N ( r )) − w ( N o + 1) . Inparticular, for J ≤ , the value of g can be derived asin (31) and (33)-(34), where term | x | is replaced by x N ( r ) . On the other hand, for J > , the value of g canbe expressed as follows: g = g RX if (cid:26) x N ( r ) ≤ r ≤ J or K ≤ r ≤ + ∞ p = RX or (cid:26) x N ( r ) ≤ r ≤ + ∞ p = LX (35) g = G RX if (cid:26) J ≤ r ≤ Kp = RX (36) • S = U , E = L , S = B and
E = L - fromAssumption 3.7, we observe that g is always equal to g RX . In addition, we note that it is not possible to havea LOS BS at a distance smaller than r . Hence, we haveonly two possible configurations: g = g RX if (cid:26) | x | ≤ r ≤ + ∞ p = RX or (cid:26) | x | ≤ r ≤ + ∞ p = LX (37) • S = U , E = L , S = B and
E = N - similarly to theprevious case, we observe that g is equal to g RX and itis not possible to have a NLOS BS at a distance smallerthan x N ( r ) . Hence, we have the following cases: g = g RX if (cid:26) x N ( r ) ≤ r ≤ + ∞ p = RX or (cid:26) x N ( r ) ≤ r ≤ + ∞ p = LX (38) • With regards the remaining parameter combinationswhere S = U , E = N , we observe the following cases: – S = U , E = L - we define x L ( r ) = p (A L ( r )) − w ( N o + 1) . If x L ( r ) > K , referto (37) and replace | x | with x L ( r ) . Otherwise, referto (31)-(34) and replace | x | with x L ( r ) . – S = U , E = N - refer to (31)-(34). – S = B , E = L - refer to (37) and replace | x | with x L ( r ) . – S = B , E = N - refer to (37). • By following the above approach, it is possible to de-rive all the remaining configurations. In particular, thecharacterization of g , for a parameter configuration where S = B and S = B ( S = B and S = U ) follows exactlythe same rule of the corespondent parameter list, where S = U and S = U ( S = U and S = B ).The aforementioned parameter configurations are also summa-rized in Table IV. With regards to parameter p , we observethat the probability P [ p ] of p being equal to DX or RX is . .Consider (30), all the elements are in place to explicitlycalculate the expectation with respect to ∆ . In particular, itfollows that L I S , E ( s ) can be expressed as: L I S , E , E ( s ) ( i ) ∼ = exp (cid:18) − E ∆ (cid:18) E h [Θ( h, ∆)+Λ( h, ∆)] (cid:12)(cid:12)(cid:12) a =0 ,b =+ ∞ (cid:19)(cid:19) ( ii ) ∼ = exp − X S ∈{ U , B } ( a,b, ∆) ∈C | x1 | , S , E , S , E P [ p ] (cid:18) E h [Θ( h, ∆) + Λ( h, ∆)] (cid:12)(cid:12)(cid:12) a,b, ∆ (cid:19) = Y S ∈{ U , B } , ( a,b, ∆) ∈C | x1 | , S , E , S , E exp (cid:18) − (cid:18) E h [Θ( h, ∆)] (cid:12)(cid:12)(cid:12) a,b, ∆ + E h [Λ( h, ∆)] (cid:12)(cid:12)(cid:12) a,b, ∆ (cid:19)(cid:19) = Y S ∈{ U , B } , ( a,b, ∆) ∈C | x1 | , S , E , S , E q L I S , E , E ( s ; a, b, ∆) , (39)where ( i ) is (30). From the previous discussion, for a given | x | , ∆ can either be equal to g TX g RX or g TX G RX . In par-ticular, the value of ∆ is determined by the list of parameters < | x | , S , E , S , E > , where terms a and b are the minimumand maximum distance r to an interfering BS, respectively.We define sequence C | x | , S , E , S , E . This sequence consistsof all the possible parameter configurations ( a, b, ∆) . Forinstance, if S = U , E = L , S = U , E = L and
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