Connecting flying backhauls of UAVs to enhance vehicular networks with fixed 5G NR infrastructure
11 Connecting flying backhauls of UAVs toenhance vehicular networks with fixed 5G NRinfrastructure
Dalia Popescu, Philippe Jacquet,
Fellow, IEEE,
Bernard Mans
Abstract
This paper investigates moving networks of Unmanned Aerial Vehicles (UAVs), such as drones, asone of the innovative opportunities brought by the 5G. With a main purpose to extend connectivity andguarantee data rates, the drones require an intelligent choice of hovering locations due to their specificlimitations such as flight time and coverage surface. To this end, we provide analytic bounds on therequirements in terms of connectivity extension for vehicular networks served by fixed Enhanced MobileBroadBand (eMBB) infrastructure, where both vehicular networks and infrastructures are modeled usingstochastic and fractal geometry as a macro model for urban environment, providing a unique perspectiveinto the smart city.Namely, we prove that assuming 𝑛 mobile nodes (distributed according to a hyperfractal distributionof dimension 𝑑 𝐹 ) and an average of 𝜌 Next Generation NodeB (gNBs), distributed like an hyperfractalof dimension 𝑑 𝑟 if 𝜌 = 𝑛 𝜃 with 𝜃 > 𝑑 𝑟 / and letting 𝑛 tending to infinity (to reflect megalopolis cities),then the average fraction of mobile nodes not covered by a gNB tends to zero like 𝑂 (cid:16) 𝑛 − ( 𝑑𝐹 − ) 𝑑𝑟 ( 𝜃 − 𝑑𝑟 ) (cid:17) .Interestingly, we then prove that the average number of drones, needed to connect each mobile node notcovered by gNBs is comparable to the number of isolated mobile nodes. We complete the characterisationby proving that when 𝜃 < 𝑑 𝑟 / the proportion of covered mobile nodes tends to zero.Furthermore, we provide insights on the intelligent placement of the “garage of drones”, the homelocation of these nomadic infrastructure nodes, such as to minimize what we call the “flight-to-coverage Part of this work was presented at WiSARN’2020, IEEE Conference on Computer Communications Workshops (INFOCOMWKSHPS), International Workshop on Wireless Sensor, Robot and UAV Networks, July 6th, 2020, (Online). Connecting flyingbackhauls of drones to enhance vehicular networks with fixed 5G NR infrastructure, P. Jacquet D. Popescu and B. Mans, IEEEPress, pp. 472-477, doi: 10.1109/INFOCOMWKSHPS50562.2020.9162670.B. Mans was supported in part by the Australian Research Council under Grant DP170102794.Part of the work has been done at Lincs, Paris, France. Dalia Popescu was with Nokia Bell Labs, France. Philippe Jacquetis with INRIA, France. Bernard Mans is with Macquarie University, Sydney, Australia (e-mail: [email protected]). a r X i v : . [ c s . D C ] F e b time”. We provide a fast procedure to select the relays that will be garages (and store drones) in orderto minimize the number of garages and minimize the delay. Finally we confirm our analytical resultsusing simulations carried out in Matlab. Index Terms
Drone, Enhanced Mobile BroadBand (eMBB), Flying Backhaul, mmWave, Mobile VehicularNetwork, Smart City, UAVs, V2X, 5G.
I. I
NTRODUCTION
A. Mobile networks5G New Radio (5G NR) is envisioned to offer a diverse palette of services to mobile usersand platforms, with often incompatible types of requirements. 5G will support: (i) the enhancedMobile BroadBand (eMBB) applications with throughput of order of 100 Mbit/s per user, (ii) the
Ultra Reliable Low Latency Communications (URLLC) for industrial and vehicular environmentswith the hard constraint of 1 ms latency and 99.99 % reliability, and (iii) the massive MachineType Communications (mMTC) with a colossal density in order of 100 per square km. 5G willachieve this through a new air interface and novel network architectures that will either beevolved from the current 4G systems, or be completely drawn from scratch.However, it is clear that by the time the 5G standards start to be deployed, many of theseimportant challenges will still remain opened and a truly disruptive transition from current 4Gsystems will only occur over time. One of these challenges is the ability of the network to adaptefficiently to traffic demand evolution in space and time, in particular when new frequencybands in a range much higher than ever dared before are to be exploited, e.g. Frequency Range(FR) 4 at over 52.6 GHz. A key difference from today’s 4G systems is that 5G networks will becharacterized by a massive density of nodes, both human-held and machine-type: according to theEricsson mobility report (June 2020), 5G subscriptions are forecast to reach 2.8 billion globallyby the end of 2025. A larger density of wireless nodes implies a larger standard deviation in thetraffic generation process. Network planning done based on average (or peak) traffic predictionsas done for previous systems, will only be able to bring sub-optimal results. Moreover, the“ verticals ” of major interest identified for 5G networks (automotive, health, fabric-of-the-future,media, energy), come with rather different requirements and use cases. Future networks willhandle extremely heterogeneous traffic scenarios. Figure 1: Extension of connectivity by dronesTo tackle these problems, 5G NR networks are to exhibit a network flexibility that is muchhigher than in the past: infrastructure nodes must be adaptable enough in order to be able tosmoothly and autonomously react to the fast temporal and spatial variations of traffic demand.The level of flexibility that can be achieved through such advances still faces a fundamental limit:hardware location is static and the offered network capacity on a local scale is fundamentallylimited by the density of the infrastructure equipment (radio transceivers) in the area of interest[1].This opens up the possibility of
Moving Networks , informally defined as moving nodes,with advanced network capabilities, gathered together to form a movable network that cancommunicate with its environment. Moving Networks will help 5G systems to become demand-attentive, with a level of network cooperation that will facilitate the provisioning of services tousers characterized by high mobility and throughput requirements, or in situations where a fixednetwork cannot satisfy traffic demand. A key player in a moving network is the communicationentity with the highest number of degrees of freedom of movement: the drone.
B. Contributions
In this work we initiate the study and design of moving networks by a first analysis of theprovisioning and dimensioning of a network where drones act as flying backhaul. The setup we make use is that of a smart city in which we design one of the complex 5G NR urban scenarios:drones coexist with vehicular networks and fixed telecommunication infrastructures.Making use of the innovative model called Hyperfractal [2] for representing the traffic densityof vehicular devices in an urban scenario and the distribution of static telecommunicationinfrastructure, we derive the requirements in terms of resources of unmanned aerial vehicles(UAVs) for enhancing the coverage required by the users. Informally, we compute the expectedpercentage of the users in poor coverage conditions and derive the average number of dronesnecessary for ensuring the coverage with high reliability of the vehicular nodes. Furthermore,we discuss the notion of "garage of drones", the home location of these nomadic infrastructurenodes, the drones.More specifically, our contributions are: • An innovative macro model for urban environment, providing a new perspective on smartcities, where both vehicular networks and fixed eMBB infrastructure networks are modeledusing stochastic and fractal geometry (Section II). The model allows to compute precisebounds on the requirements in term of connectivity, or lack of thereof. • A proof that the average fraction of mobile nodes not covered by the fixed infrastructure(gNBs) either tends to zero, or tends to the actual number of nodes in the network with thecharacterisation of the exact threshold (Theorem III.1 and Theorem III.2 in Section III). • A proof that the number of drones to connect the isolated mobile nodes is asymptoticallyequivalent to the number of isolated mobile nodes (Theorem III.3 in Section III). Thus,proving that, typically, one drone is sufficient to connect an isolated mobile node to thenetwork. • An introduction and analysis of the “flight-to-coverage time” for the deployed drones,helping in their optimised placement and recharge in the network (Section IV). We providea fast procedure to select the relays that will be garages (and store drones) in order tominimize the number of garages and minimize the delay. • Simulations results in Matlab that confirm our stochastic results (Section V).
C. Related works
While their introduction to commercial use has been delayed and restricted to specific use-cases due to the numerous challenges (e.g. [3]), the flexibility the drones bring to the network planning by their increased degree of movement, has motivated the industry to push towardstheir introduction in wide scenarios of 5G.We refer the reader to recent relevant surveys. For instance, a thorough Tutorial on UAVCommunications for 5G and Beyond is presented in [4], where both UAV-assisted wirelesscommunications and cellular-connected UAVs are discussed (with UAVs integrated into thenetwork as new aerial communication platforms and users).Importantly, many new opportunities have been often highlighted (e.g. in [5]), including (notexhaustively): • Coverage and capacity enhancement of Beyond 5G wireless cellular networks: • UAVs as flying base stations for public safety scenarios; • UAV-assisted terrestrial networks for information dissemination; • Cache-Enabled UAVs; • UAVs as flying backhaul for terrestrial networks.Considerations for a multi-UAV enabled wireless communication system, where multiple UAV-mounted aerial base stations are employed to serve a group of users on the ground have beenpresented in [6].An overview of UAV-aided wireless communications, introducing the basic networkingarchitecture and main channel characteristics, and highlighting the key design considerationsas well as the new opportunities to be exploited is presented in [7].Grasping the interest for this new concept of communication, the research community hasstarted analyzing the details of the new communication paradigms introduced with the drones.In the beginning, drones have been considered for delivering the capacity required for sporadicpeaks of demand, such as in entertainment events, to reach areas where there is no infrastructureor the infrastructure is down due to a natural catastrophe. Yet in the new 5G NR scenariosof communication, drones are not only used in isolated cases but are considered as activecomponents in the planning of moving networks in order to provide the elasticity and flexibilityrequired by these. In this sense, using drones for the 5G Integrated Access and Backhaul (IAB)([8], [9]) is one of the most relevant use-case scenarios, ensuring the long-desired flexibility ofcoverage aimed with an adaptable cost of infrastructure.The research community has been looking at specific problems in wireless networks employingdrones. In [10], the authors prove the feasibility of multi-tier networks of drones giving some firstinsight on the dimensioning of such a network. The work done in [11] uses two approaches, a network-centric approach and a user-centric approach to find the optimal backhaul placement forthe drones. On the other hand, in [12], the authors propose a heuristic algorithm for the placementof drones. The authors of [13] analyze the scenario of a bidirectional highway and propose analgorithm for determining the number of drones for satisfying coverage and delay constraints.A delay analysis is performed in [14] on the same type of scenario. Simplified assumptionsare used in [15] to showcase the considerable improvements brought by UAVs as assistance tocellular networks. The movement of network of drones, so-called swarm is analyzed in [16] andefficient routing techniques are proposed. Drones are so much envisioned for the networks ofthe future that the authors of [17] provide solutions for a platform of drones-as-a-service thatwould answer the future operators demands.Following these observations from the state of the art, we aim to initiate a complex analysisof the use of the drones in a smart city for different purposes. To our knowledge, this is the firstanalysis which looks at the entire macro urban model of the city, incorporating both devicesand the fixed telecommunications infrastructure. In our analysis, the drones are used as a flyingbackhaul for extending the coverage to users in poor conditions.II. S
YSTEM MODEL AND SCENARIO
The communication scenario in our work has the aim to provide a flexible network architecturethat allows serving the vehicular devices with tight delay constraints. The scenario comprisesthree types of communication entities: the vehicles which we denote as the user equipment(UEs), the fixed telecommunication infrastructure called Next Generation NodeB (gNBs) andthe moving network nodes which are the drones also called UAVs.The scenario we tackle in this work is as follows. An urban network of vehicles is served by afixed telecommunication infrastructure. Due to the limited coverage capability using millimeterwave (mmwave) in urban environments, the costs of installing fixed telecommunicationinfrastructure throughout the entire city and the mobility of the vehicles in the urban area, someusers will be outside of the coverage areas. This is no longer acceptable in 5G as InternationalMobile Telecommunications Standards (IMT 2020 [18]) require for most of the users and inparticular for URLLC users, full coverage and harsh constraints for delay. UAVs (such as drones)will therefore be dynamically deployed for reaching the so-called "isolated" users, forming theflying backhaul of the network.
A. Communication Model
For the sake of simplicity, we consider the communication to be done in the FR2 (frequencyrange 2) 28GHz frequency band using beamforming, mmwave technology in half-duplex mode.This follows the current specifications, yet we foresee the modelling to be easily extended forFR3 (frequency range 3) and FR4 (frequency range 4). This leads to highly directive beamswith a narrow aperture, sensitive to blockage and interference. To this end, in our modeling weintegrate the canyon propagation model which implies that the signal emitted by a mobile nodepropagates only on the street where it stands on.We consider that drones, gNBs and UEs use the same frequency band and transmission power.The transmission range is 𝑅 𝑛 = √ 𝑛 , where 𝑛 is the number of UEs in the city. The reasoningbehind this choice of transmission range is as follows. The population of a city (in most of thecases) is proportional to the area of the city [19] and the population of cars is proportional tothe population of the city (in fact the local variations of car densities counter the local variationof population densities [20]), therefore the population of cars is proportional to the area of thecity, Area = 𝐴 · 𝑛 where 𝐴 is a constant. A natural assumption is that the absolute radio rangeis constant. But since we assume in our model that the city map is always a unit square, therelative radio range in the unit square must be 𝑅 𝑛 = 𝑅 √ 𝐴𝑛 which we simplify in 𝑅 𝑛 = √ 𝑛 . Thisleads to a limited radio range, that together with the canyon effect supports the features of themmwave communications.The mobile nodes that we treat in this modeling are vehicular devices or UEs located onstreets. The UAV will communicate with the mobile users on the RAN interface, similar to agNB to UE exchange. For drone-to-drone communication and drone-to-gNB communication, itis the Integrated Access and Backhaul (IAB) interface that is used, in the same frequency band.We consider that, for a UE to benefit from the dynamical coverage assistance provided byUAVs, it has to be already registered to the network through a previously performed randomaccess procedure (RACH) and we only treat users in connected mode, therefore the network isaware about the existence of the UE, its context and service requirements and, when the situationarrives, that it is experiencing a poor coverage.The communication scenario exploits the fact that the position of a registered UE in connectedmode is known to the network and the evolution can be tracked (speed, direction of movement).The position of the UE can be either UE based positioning or UE assisted positioning, both allowing enough degree of accuracy for a proactive preparation of resources in the identifiedtarget cell. These are assumptions perfectly inline with the objectives of Release 17 of 5G NR.As a consequence of this knowledge in the network, the gNB that is currently serving thevehicle can inform the target gNB about the upcoming arrival of the vehicle such that the targetgNB can prepare/send a drone (if necessary) for ensuring the service continuity to the UE.The drones will be dynamically deployed in order to extend the coverage of the gNBs towardsthe UEs, forming a Flying Backhaul. B. Hyperfractal Modeling for an urban scenario
As 5G is not meant to be designed for a general setup but rather have specific solutions for allthe variety of communication scenarios (in particular for verticals ), the idea of a smart city inwhich one is capable of translating different parameters and features in order to better calibratethe private network being deployed arises as a possible answer to these stringent requirements.An hyperfractal representation of urban settings [2], in particular for mobile vehicles and fixedinfrastructures (such as red lights), is an innovative representation that we chose for modellingthe UEs and eMBB infrastructures here. With this model, one is capable of taking measurementsof vehicular traffic flows, urban characteristics (streets lengths, crossings, etc), fit the data to ahyperfractal with computed parameters and compute metrics of interest. Independently, this isan interesting step towards achieving modeling of smart urban cities that can be exploited inother scenarios.
1) Mobile users:
The positions of the mobile users and their flows in the urban environmentare modeled with the hyperfractal model described in [2], [21], [22]. In the following, we onlyprovide the necessary and self-sufficient introduction to the model but a complete and extendeddescription can be found in [2].The map model lays in the unit square embedded in the 2-dimensional Euclidean space. Thesupport of the population is a grid of streets. Let us denote this structure by X = (cid:208) ∞ 𝑙 = X 𝑙 . Astreet of level 𝐻 consists of the union of consecutive segments of level 𝐻 in the same line. Thelength of a street is the length of the side of the map.The mobile users are modeled by means of a Poisson point process Φ on X with total intensity(mean number of points) 𝑎 ( < 𝑎 < ∞ ) having 1-dimensional intensity 𝜆 𝑙 = 𝑎 ( 𝑝 / ) ( 𝑞 / ) 𝑙 (1) on X 𝑙 , 𝑙 = , . . . , ∞ , with 𝑞 = − 𝑝 for some parameter 𝑝 ( ≤ 𝑝 ≤ ). Note that Φ can beconstructed in the following way: one samples the total number of mobiles users Φ (X) = 𝑛 fromPoisson ( 𝑎 ) distribution; each mobile is placed independently with probability 𝑝 on X accordingto the uniform distribution and with probability 𝑞 / it is recursively located in the similar wayin one the four quadrants of (cid:208) ∞ 𝑙 = X 𝑙 .The intensity measure of Φ on X is hypothetically reproduced in each of the four quadrantsof (cid:208) ∞ 𝑙 = X 𝑙 with the scaling of its support by the factor 1/2 and of its value by 𝑞 / .The fractal dimension is a scalar parameter characterizing a geometric object with repetitivepatterns. It indicates how the volume of the object decreases when submitted to an homotheticscaling. When the object is a convex subset of an euclidian space of finite dimension, the fractaldimension is equal to this dimension. When the object is a fractal subset of this euclidian spaceas defined in [23], it is a possibly non integer but positive scalar strictly smaller than the euclidiandimension. When the object is a measure defined in the euclidian space, as it is the case in thispaper, then the fractal dimension can be strictly larger than the euclidian dimension. In this casewe say that the measure is hyper-fractal . Remark 1.
The fractal dimension 𝑑 𝐹 of the intensity measure of Φ satisfies (cid:18) (cid:19) 𝑑 𝐹 = 𝑞 thus 𝑑 𝐹 = log ( 𝑞 ) log 2 ≥ . The fractal dimension 𝑑 𝐹 is greater than 2, the Euclidean dimension of the square in whichit is embedded, thus the model was called in [24] hyperfractal . Figure 2a shows an exampleof support iteratively built up to level 𝐻 = while Figure 2b shows the nodes obtained as aPoisson shot on a support of a higher depth, 𝐻 = .
2) Fixed telecommunication infrastructure, gNBs:
We denote the process for gNBs by Ξ .To define Ξ it is convenient to consider an auxiliary Poisson process Φ 𝑟 with both processessupported by a 1-dimensional subset of X namely, the set of intersections of segments constituting X . We assume that Φ 𝑟 has discrete intensity 𝑝 ( ℎ, 𝑣 ) = 𝜌 ( 𝑝 (cid:48) ) (cid:18) − ( 𝑝 (cid:48) ) (cid:19) ℎ + 𝑣 (2)on all intersections X ℎ ∩X 𝑣 for ℎ, 𝑣 = , . . . , ∞ for some parameter 𝑝 (cid:48) , ≤ 𝑝 (cid:48) ≤ and 𝜌 > . Thatis, on any such intersection the mass of Φ 𝑟 is Poisson random variable with parameter 𝑝 ( ℎ, 𝑣 ) and 𝜌 is the total expected number of points of Φ 𝑟 in the model. The self-similar structure of (a) (b) Figure 2: (a) Hyperfractal map support; (b) Hyperfractal, 𝑑 𝐹 = , 𝑛 = , nodes; Φ 𝑟 is well explained by its construction in which we first sample the total number of pointsfrom a Poisson distribution of intensity 𝜌 and given Φ 𝑟 (X) = 𝑀 , each point is independentlyplaced with probability ( 𝑝 (cid:48) ) in the central crossing of X , with probability 𝑝 (cid:48) − 𝑝 (cid:48) on some othercrossing of one of the four segments forming X and, with the remaining probability (cid:16) −( 𝑝 (cid:48) ) (cid:17) ,in a similar way, recursively, on some crossing of one of the four quadrants of (cid:208) ∞ 𝑙 = X 𝑙 .Note that the Poisson process Φ 𝑟 is not simple and we define the process Ξ for gNBs as thesupport measure of Φ 𝑟 , i.e., only one gNB is installed in every crossing where Φ 𝑟 has at leastone point. Remark 2.
Note that fixed infrastructure Ξ forms a non-homogeneous binomial point process(i.e. points are placed independently) on the crossings of X with a given intersection of twosegments from X ℎ and X 𝑣 occupied by a gNB point with probability − exp (− 𝜌 𝑝 ( ℎ, 𝑣 )) . Similarly to the process of mobile nodes, we can define the fractal dimension of the gNBsprocess.
Remark 3.
The fractal dimension 𝑑 𝑟 of the probability density of Ξ is equal to the fractaldimension of the intensity measure of the Poisson process Φ 𝑟 and verifies 𝑑 𝑟 = ( /( − 𝑝 (cid:48) )) log 2 . III. M
AIN R ESULTS
We recall the strong requirement that the vehicles should be covered in a proportion of 𝛾 𝐹𝐶 > . . In an urban vehicular network supported by eMBB infrastructure, both modeled byhyperfractals, the authors of [25], have proven that, under no constraints on the transmissionrange, the giant component tends to include all the nodes for 𝑛 large.In this analysis, under the constraint of a transmission range limited to 𝑅 𝑛 = √ 𝑛 , there willalways be disconnected nodes and the number of gNBs to guarantee 𝛾 𝐹𝐶 would not be costeffective. Therefore, we dynamically deploy drones, to extend by hop-by-hop communication,the coverage of the gNBs, as illustrated in Figure 1.This should be done by considering the constraints in latency as well, which, in our model,translate to a maximum number of drones on which the packet is allowed to hop before reachingthe gNB. A. Connectivity with fixed telecommunication infrastructure
We assume that the infrastructure location can be fitted to a hyperfractal distribution withdimension 𝑑 𝑟 and intensity 𝜌 (to provide a realistic model [22]). In a first instance, we omit theexistence of drones. Therefore a mobile node is covered only when a gNB exists at distancelower than 𝑅 𝑛 = √ 𝑛 . In this case, the following theorem gives the average number of nodes thatare not in the coverage range of the infrastructure. Theorem III.1 (Connectivity without drones) . Assuming 𝑛 mobile nodes distributed according toa hyperfractal distribution of dimension 𝑑 𝐹 and a gNB distribution of dimension 𝑑 𝑟 and intensity 𝜌 , if 𝜌 = 𝑛 𝜃 with 𝜃 > 𝑑 𝑟 / then the average number of mobile nodes not covered by a gNB is 𝑛𝐼 𝑛 with 𝐼 𝑛 = 𝑂 (cid:16) 𝑛 − ( 𝑑𝐹 − ) 𝑑𝑟 ( 𝜃 − 𝑑 𝑟 / ) (cid:17) . Remark 4.
The quantity 𝐼 𝑛 is the probability that a mobile node is isolated and thus the theoremproves that the proportion of non covered mobile nodes tends to zero.Proof. Let a mobile node be on a road of level 𝐻 at the abscissa 𝑥 . For the mobile node to becovered, it is necessary for a gNB to exist on the same street in the interval (cid:2) 𝑥 − √ 𝑛 , 𝑥 + √ 𝑛 (cid:3) .We shall first estimate the probability 𝐼 𝑛 ( 𝑥, 𝐻 ) that a mobile node is not covered. Let 𝑁 𝑉 ( 𝑥 ) be the number of intersections of level 𝑉 ( i.e. with a street of level 𝑉 ) which are within distance 𝑅 𝑛 of abscissa 𝑥 on the road of level 𝐻 . Since the placement of gNBs follows a Poisson processof mean 𝜌 𝑝 ( 𝐻, 𝑉 ) : 𝐼 𝑛 ( 𝑥, 𝐻 ) = exp (cid:32) − 𝜌 ∑︁ 𝑉 𝑁 𝑉 ( 𝑥 ) 𝑝 ( 𝐻, 𝑉 ) (cid:33) . (3)In order to get an upper bound 𝐼 𝑛 ( 𝐻 ) on 𝐼 𝑛 ( 𝑥, 𝐻 ) , a lower bound of the quantities 𝑁 𝑉 ( 𝑥 ) is necessary. For any given integer 𝑉 ≥ , the intersections of level equal or larger than 𝑉 areregularly spaced at frequency 𝑉 + on the axis. Only half of them are exactly of level 𝑉 , regularlyspaced at frequency 𝑉 (as a consequence of the construction process). Therefore a lower boundof 𝑁 𝑉 ( 𝑥 ) is (cid:98) 𝑉 √ 𝑛 (cid:99) .Let 𝑉 𝑛 be the smallest integer which satisfies 𝑉 𝑛 ≥ √ 𝑛 . Therefore the lower bound of 𝑁 𝐻 ( 𝑥 ) is 𝑉 − 𝑉 𝑛 when 𝑉 ≥ 𝑉 𝑛 and 0 when 𝑉 < 𝑉 𝑛 . Thus the upper bound 𝐼 𝑛 ( 𝐻 ) of 𝐼 𝑛 ( 𝑥, 𝐻 ) is 𝐼 𝑛 ( 𝐻 ) = exp (cid:32) − 𝜌 ∑︁ 𝑉 ≥ 𝑉 𝑛 𝑉 − 𝑉 𝑛 𝑝 ( 𝐻, 𝑉 ) (cid:33) with the expression 𝑝 ( 𝐻, 𝑉 ) = ( 𝑝 (cid:48) ) ( 𝑞 (cid:48) / ) 𝐻 + 𝑉 we get 𝐼 𝑛 ( 𝐻 ) = exp (cid:16) − 𝜌 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝑉 𝑛 + 𝐻 (cid:17) . (4)We now have 𝑉 𝑛 ≥ log 𝑛 thus ( 𝑞 (cid:48) / ) 𝑉 𝑛 ≤ 𝑛 − 𝑑𝑟 / and with 𝜌 = 𝑛 𝜃 we obtain 𝐼 𝑛 ( 𝐻 ) ≤ exp (− 𝑛 𝜃 − 𝑑 𝑟 / 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 ) . From here we can finish the proof since the proportion 𝐼 𝑛 of isolatedmobile nodes is: 𝐼 𝑛 = ∑︁ 𝐻 𝜆 𝐻 𝐼 𝑛 ( 𝐻 ) = ∑︁ 𝐻 𝑝𝑞 𝐻 𝐼 𝑛 ( 𝐻 )≤ ∑︁ 𝐻 𝑝𝑞 𝐻 exp (cid:16) − 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 𝑛 𝜃 − 𝑑 𝑟 / (cid:17) (5)and we prove in the appendix (via a Mellin transform) that ∑︁ 𝐻 𝑝𝑞 𝐻 exp (cid:16) − 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 𝑦 (cid:17) = 𝑝 ( 𝑝 (cid:48) ) − 𝛿 log ( / 𝑞 (cid:48) ) Γ ( 𝛿 ) 𝑦 − 𝛿 ( + 𝑜 ( )) (6)For 𝑦 → ∞ with 𝛿 = 𝑑 𝐹 − 𝑑 𝑟 / . We complete the proof by using 𝑦 = 𝑛 𝜃 − 𝑑 𝑟 / . (cid:3) Let us now study the second regime of 𝜃 , 𝜃 < 𝑑 𝑟 / . Theorem III.2.
For 𝜃 < 𝑑 𝑟 / , the proportion of covered mobile nodes tends to zero. For 𝑢 𝑖 positive for all 𝑖 : let 𝑃 ( 𝑢 , 𝑢 , . . . , 𝑢 𝑉 , . . . ) = 𝐸 (cid:104)(cid:206) 𝑉 𝑢 𝑁 𝑉 ( 𝑥 ) 𝑉 (cid:105) , andlet 𝑃 𝑉 ( 𝑢 , . . . , 𝑢 𝑉 − ) = 𝑃 ( 𝑢 , . . . , 𝑢 𝑉 − , , , . . . ) . Lemma 1.
For all integer 𝑉 , 𝑃 𝑉 ( 𝑢 , . . . , 𝑢 𝑉 − ) ≥ − 𝑉 √ 𝑛 .Proof. Naturally 𝑃 𝑉 ( 𝑢 , 𝑢 , . . . , 𝑢 𝑉 − ) ≥ 𝑃 𝑉 ( , . . . , ) . The quantity 𝑃 𝑉 ( , . . . , ) is the probab-ility that for all 𝑖 < 𝑉 , 𝑁 𝑖 ( 𝑥 ) = . This is equivalent to the fact that 𝑥 is always at distance largerthan 𝑅 𝑛 to any intersection of level higher than 𝑉 . Since there are 𝑉 of such intersections, themeasure of the union of all these intervals is smaller than 𝑉 /√ 𝑛 . (cid:3) Lemma 2.
Assuming for all integer 𝑖 , 𝑢 𝑖 ≤ , Let 𝑉 𝑛 = (cid:100) log √ 𝑛 (cid:101) we have 𝑃 ( 𝑢 , . . . ) ≥ (cid:206) 𝑉 ≥ 𝑉 𝑛 𝑢 𝑉 − 𝑉𝑛 + 𝑖 𝑃 𝑉 𝑛 ( 𝑢 , . . . , 𝑢 𝑉 𝑛 − ) .Proof. Indeed for 𝑉 ≥ 𝑉 𝑛 we have 𝑁 𝑉 ( 𝑥 ) ≤ 𝑉 − 𝑉 𝑛 + , thus 𝑃 ( 𝑢 , . . . ) ≥ (cid:214) 𝑉 ≥ 𝑉 𝑛 𝑢 𝑉 𝑉 − 𝑉 𝑛 + 𝐸 (cid:104) 𝑢 𝑁 ( 𝑥 ) · · · 𝑢 𝑁 𝑉𝑛 − ( 𝑥 ) 𝑉 𝑛 − (cid:105) . (cid:3) Lemma 3.
Let 𝑉 ≤ 𝑉 𝑛 , the following holds: 𝑃 𝑉 ( 𝑢 , . . . 𝑢 𝑉 − ) ≥ 𝑢 𝑉 − 𝑃 𝑉 − ( 𝑢 , . . . , 𝑢 𝑉 − ) .Proof. For
𝑉 < 𝑉 𝑛 we have 𝑁 𝑉 ( 𝑥 ) ≤ . (cid:3) Proof of theorem III.2.
The following identity holds: 𝐼 𝑛 ( 𝐻 ) = 𝑃 (cid:16) 𝑒 − 𝜌𝑝 ( 𝐻, ) , . . . , 𝑒 − 𝜌𝑝 ( 𝐻,𝑉 ) , . . . (cid:17) . (7)Clearly, with Lemma 2 we get 𝐼 𝑛 ( 𝐻 ) ≥ exp (cid:32) − 𝜌 ∑︁ 𝑉 ≥ 𝑉 𝑛 𝑝 ( 𝐻, 𝑉 ) 𝑉 − 𝑉 𝑛 + (cid:33) × 𝑃 𝑉 𝑛 (cid:16) 𝑒 − 𝜌𝑝 ( 𝐻, ) , . . . , 𝑒 − 𝜌𝑝 ( 𝐻,𝑉 𝑛 − ) (cid:17) . Since 𝜃 < 𝑑 𝑟 / , the first right hand factor tends to one as the quantity (cid:205) 𝑉 ≥ 𝑉 𝑛 𝜌 𝑝 ( 𝐻, 𝑉 ) = 𝑂 (cid:16) 𝑛 𝜃 − 𝑑 𝑟 / 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 (cid:17) tends to zero.Using lemma 3 the second term is lower bounded for any integer 𝑘 < 𝑉 𝑛 by: 𝑘 (cid:214) 𝑖 = exp (− 𝜌 𝑝 ( 𝐻, 𝑉 𝑛 − 𝑖 )) (cid:16) − 𝑉 𝑛 − 𝑘 /√ 𝑛 (cid:17) (8) Each of the terms 𝜌 𝑝 ( 𝐻, 𝑉 𝑛 − 𝑖 ) = 𝑂 ( 𝑛 𝜃 − 𝑑 𝑟 / ( 𝑞 (cid:48) / ) − 𝑖 ) tends to zero, thus exp (− 𝜌 𝑝 ( 𝐻, 𝑉 − 𝑖 )) tendsto 1. Since 𝑉 𝑛 ≤ √ 𝑛 thus − 𝑉 𝑛 − 𝑘 /√ 𝑛 ≥ − − 𝑘 . For any fixed 𝑘 we have lim inf 𝑛 𝐼 𝑛 ( 𝐻 ) ≥ − − 𝑘 uniformly in 𝐻 . Since 𝑘 can be made as large as possible thus − 𝑘 as small as wewant, the 𝐼 𝑛 ( 𝐻 ) to tend to 1 uniformly in 𝐻 . (cid:3) B. Extension of connectivity with drones
Now that we have analyzed the connectivity properties of the network and observed thesituation when nodes are not covered. We have discovered two regimes, in function of theexisting gNB infrastructure features: 𝜃 > 𝑑 𝑟 / and 𝜃 < 𝑑 𝑟 / . Let us provide the necessarydimensioning of the network in terms of UAVs for ensuring the required services for the firstcase. As in the second case the number of isolated nodes is overwhelming, we consider dronesare not a desirable, cost-effective solution for connectivity. We have, therefore, also identifiedthe regime for which drones are to be deployed. Theorem III.3 (Connectivity with drones) . Assume 𝑛 mobile nodes distributed according to ahyperfractal distribution of dimension 𝑑 𝐹 and a gNB distribution of dimension 𝑑 𝑟 and intensity 𝜌 . If 𝜌 = 𝑛 𝜃 with 𝜃 > 𝑑 𝑟 / then the average number of drones needed to cover the mobile nodesnot covered by gNBs is 𝐷 𝑛 = 𝑛𝐼 𝑛 (cid:16) + (cid:205) 𝑘 ≥ ( 𝑘 + ) − 𝑑 𝐹 exp (− 𝑝 (cid:48) 𝑞 (cid:48) 𝑛 𝜃 − 𝑑𝑟 / 𝑘 𝑑 𝑟 ) (cid:17) . Remark 5. when 𝑑 𝐹 > and 𝜃 < 𝑑 𝑟 / the distribution of the number of drones tends to bea power law of power − 𝑑 𝐹 . When 𝜃 > 𝑑 𝑟 / the average number of drones is asymptoticallyequivalent to the average number of isolated nodes. The probability that an isolated node needsmore than one single drone decays exponentially. Lemma 4.
Let 𝐼 ( 𝐻, 𝑅 ) be an upper bound of the probability that an interval of length 𝑅 on aroad of level 𝐻 does not contain any gNB. We have 𝐼 ( 𝐻, 𝑅 ) ≤ exp (cid:16) − 𝜌 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 𝑅 𝑑 𝑟 / (cid:17) . (9) Proof.
This is an adaptation of the estimate of 𝐼 𝑛 ( 𝐻 ) in the proof of theorem III.1, where wereplace /√ 𝑛 by 𝑅 . Notice that 𝐼 𝑛 ( 𝐻 ) = 𝐼 ( 𝐻, 𝑅 𝑛 ) . (cid:3) Let us now look at the possibility of having gNBs at a Manhattan distance 𝑅 . By Manhattandistance we consider the path from one mobile node to a gNB is • either, the segment from the mobile node to the gNB if they are on the same road; in thiscase the distance is the length of this segment, and is called a one leg distance; • or, composed of the segment from the mobile node to the intersection to the road of thegNB and the segment from this intersection to the gNB; in this case the distance is the sumof the lengths of the two segments, and is called a two-leg distance. Lemma 5.
Let 𝐽 ( 𝑅 ) be the probability that a mobile node has no gNB at two-leg distance lessthan 𝑅 , 𝐽 ( 𝑅 ) ≤ exp (cid:16) − 𝜌 𝑝 (cid:48) 𝑞 (cid:48) 𝑅 𝑑𝑟 + 𝑑 𝑟 / (cid:17) .Proof. Let 𝐽 ( 𝐻, 𝑅 ) be the probability that there is no relay on a road of level 𝐻 at a two-legdistance smaller than 𝑅 . From the mobile nodes the maximal gap to the next intersection oflevel 𝐻 is − 𝐻 + , we have 𝐽 ( 𝐻, 𝑅 ) ≤ (cid:214) 𝐻 𝑖 ≤ 𝑅 𝐼 ( 𝐻, 𝑅 − 𝑖 − 𝐻 ) Each factor 𝐼 ( 𝐻, 𝑅 − 𝑖 − 𝐻 ) comes from the fact that from the intersection of abscissa 𝑖 − 𝐻 thetwo segments apart of the perpendicular road of length 𝑅 − 𝑖 − 𝐻 should not contain any relay.The power comes from the fact that we have to consider two intersections apart at distance 𝑖 − 𝐻 from the mobile node. We consider 𝐻 ≥ 𝐻 𝑅 = (cid:100) log / 𝑅 (cid:101) . For 𝐻 < 𝐻 𝑅 we will simplyassume 𝐽 ( 𝐻, 𝑅 ) ≤ . For 𝐻 ≥ 𝐻 𝑅 we have 𝐽 ( 𝐻, 𝑅 ) ≤ exp (cid:32) − 𝜌 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 ∑︁ − 𝐻 𝑖 ≤ 𝑅 ( 𝑅 − 𝑖 − 𝐻 ) 𝑑 𝑟 / (cid:33) with the fact that 𝑉 𝑅 ∑︁ 𝑖 = ( 𝑅 − 𝑖 − 𝑉 ) 𝑑 𝑟 / ≥ 𝐻 ∫ 𝑅 − / 𝐻 𝑥 𝑑𝑟 / 𝑑𝑥 = 𝐻 ( 𝑅 − − 𝐻 ) 𝑑 𝑟 / + + 𝑑 𝑟 / ≥ 𝐻 𝑅 + 𝑑 𝑟 / + 𝑑 𝑟 / we obtain that 𝐽 ( 𝐻, 𝑅 ) ≤ exp (cid:18) − 𝜌 𝑝 (cid:48) ( 𝑞 (cid:48) ) 𝐻 𝑅 + 𝑑 𝑟 / + 𝑑 𝑟 / (cid:19) The overall evaluation 𝐽 ( 𝑅 ) is made of the product of all the 𝐽 ( 𝐻, 𝑅 ) since the intersection andgNBs positions are independent: 𝐽 ( 𝑅 ) = (cid:214) 𝐻 𝐽 ( 𝐻, 𝑅 )≤ exp (cid:32) − 𝜌 𝑝 (cid:48) ∑︁ 𝐻 ≥ 𝐻 𝑅 ( 𝑞 (cid:48) ) 𝐻 𝑅 + 𝑑 𝑟 / + 𝑑 𝑟 / (cid:33) = exp (cid:18) − 𝜌 𝑝 (cid:48) ( 𝑞 (cid:48) ) 𝐻 𝑅 𝑅 + 𝑑 𝑟 / + 𝑑 𝑟 / (cid:19) With the estimate that ( 𝑞 (cid:48) ) 𝐻 𝑅 ≥ 𝑞 (cid:48) 𝑅 𝑑 𝑟 / − we get 𝐽 ( 𝑅 ) ≤ exp (cid:16) − 𝜌 𝑝 (cid:48) 𝑞 (cid:48) 𝑅 𝑑𝑟 + 𝑑 𝑟 / (cid:17) . (cid:3) Lemma 6.
Let 𝐷 ( 𝐻, 𝑅 ) be the probability that for a mobile node on a road of level 𝐻 thereis no gNB at Manhattan distance less than 𝑅 . We have 𝐷 ( 𝐻, 𝑅 ) ≤ 𝐼 ( 𝐻, 𝑅 ) 𝐽 ( 𝑅 ) . Let 𝑘 be aninteger and 𝑃 𝑛 ( 𝐻, 𝑘 ) be the probability that a mobile node on road of level 𝐻 needs 𝑘 or moredrones to be connected to the closest gNB. We have 𝑃 𝑛 ( 𝐻, 𝑘 ) ≤ 𝐼 𝑛 ( 𝐻, 𝑘 𝑅 𝑛 ) 𝐽 ( ( 𝑘 − ) 𝑅 𝑛 ) (10) Proof.
The fact that 𝐷 ( 𝐻, 𝑅 ) ≤ 𝐼 ( 𝐻, 𝑅 ) 𝐽 ( 𝑅 ) comes from the fact that probability that there isno relay at distance 𝑅 is equal to the product of the probabilities of the event: (i) there is norelay at one-leg distance smaller than 𝑅 ( 𝐼 ( 𝐻, 𝑅 ) ), (ii) there is no relay at a two-leg distancesmaller than 𝑅 ( 𝐽 ( 𝑅 ) ).The expression for 𝑃 𝑛 ( 𝐻, 𝑘 ) comes from the fact that to have 𝑘 or more drones we need nogNB within one leg Manhattan distance 𝑘 𝑅 𝑛 and no gNB within two leg distance ( 𝑘 − ) 𝑛 , for 𝑘 ≥ since we have to lay an extra drone at road intersection. (cid:3) Proof of Theorem III.3.
The average number of drones needed to connect a mobile nodes ona road of level 𝐻 to the closest gNB is 𝐿 𝑛 ( 𝐻 ) = (cid:205) 𝑘 ≥ 𝑃 ( 𝐻, 𝑘 ) . Thus 𝐿 𝑛 ( 𝐻 ) ≤ 𝐼 𝑛 ( 𝐻 ) + (cid:205) 𝑘 ≥ 𝐼 𝑛 ( 𝐻, 𝑘 𝑅 𝑛 ) 𝐽 ( 𝑘 𝑅 𝑛 ) Therefore the average total number 𝐷 𝑛 of drones is given by: 𝐷 𝑛 = 𝑛 ∑︁ 𝐻 𝜆 𝐻 ∑︁ 𝑘 ≥ 𝑃 ( 𝐻, 𝑘 )≤ 𝑛 ∑︁ 𝐻 𝜆 𝐻 ∑︁ 𝑘 ≥ 𝐼 𝑛 ( 𝐻, 𝑘 𝑅 𝑛 ) 𝐽 ( ( 𝑘 − ) 𝑅 𝑛 )≤ 𝑛𝐼 𝑛 + ∑︁ 𝑘 ≥ ∑︁ 𝐻 𝑛𝐼 𝑛 ( 𝐻, ( 𝑘 + ) 𝑅 𝑛 ) exp (− 𝜌 𝑝 (cid:48) 𝑞 (cid:48) 𝑘 𝑅 𝑑 𝑟 𝑛 𝑘 𝑑 𝑟 ) Figure 3: Garage elimination algorithmBy an adaptation of (6) we have (cid:205) 𝐻 𝜆 𝐻 𝐼 𝑛 ( 𝐻, ( 𝑘 + ) 𝑅 𝑛 ) ∼ ( 𝑘 + ) − 𝛿𝑑 𝑟 / 𝐼 𝑛 and by the fact that 𝜌𝑅 𝑑 𝑟 𝑛 = 𝑛 𝜃 − 𝑑 𝑟 / we get the claimed result. (cid:3) Remark 6.
The fact that the number of drones to connect the isolated mobile nodes isasymptotically equivalent to the number of isolated mobile nodes when 𝜃 > 𝑑 𝑟 / , is optimalwhen drone sharing is not allowed. We conjecture that the isolated mobile nodes are so dispersedthat sharing the connectivity of a drone is unlikely. In the other case when 𝑑 𝑟 / < 𝜃 < 𝑑 𝑟 / thenumber of drones per isolated node tends to be finite as soon as 𝑑 𝐹 > . IV. G
ARAGES OF DRONES
While the fixed infrastructure is robust and can serve the vehicle requirements with a properplanning, the cost of installing and operating fixed infrastructure is substantial. In addition, as inmany situations in telecommunications, the planning is often over-dimensionned in order to copewith "worst case scenarios” (e.g., day versus night traffic). In this sense, our scenario furtheroffloads the traffic towards a flexible, “ephemeral” infrastructure, advancing towards the so-called“moving networks of drones”. We will now propose a procedure for trimming the number ofrequired relays, by converting some into garages of drones which will be dynamically deployedand launched to meet the requirements of the mobile users. The drones are to be seen as “mobilerelays”, with the possibility to build chains of drones that will serve the user with hop-by-hopcommunication through a highly reconfigurable Integrated Access and Backhaul (IAB).Although it has been envisioned for the drones to be shared between operators, it is unlikelythis will be done in an early phase: each operator will own and maintain its own fleet of drones. Furthermore, as the UAVs are a cost effective and flexible solution for extending the coverage,the places for storing and charging them, what we call "garages", are to be located in the samesites as the gNBs, owned by the operators.We now provide a first straight-over insight on the home locations of the nomadicinfrastructure.We define as the " flight-to-coverage " time, the time necessary for a drone to leave the garageand be in a distance lower than 𝑅 𝑛 to the UE and to the gNB, such as to be able to form thebackhaul. In order to fulfill the service requirements, the flight-to-coverage time should be lowerthan the allowed delay. Again, the drone needs not to be hoovering over the UE or the gNB butjust have them in the range of its CSI-RS (Channel State Information Reference Signal) beams.We thus transform the flight-to-coverage time into constraints on connectivity (at all time), fora chain of drones of maximum length 𝑘 . We want to select the relays that will be garages (andstore drones) in order to minimize the number of garages and delay, i.e. under the constraint thatat most 𝑘 hops are needed to forward the packets (and that the chains of drones are no longerthan 𝑘 ).We now describe informally an algorithm we use to select relays to become garages. Similarto a dominating set problem, we first make every relay a garage. We then check the garages inan arbitrary order 𝐴 , 𝐴 , . . . . We then eliminate sequentially a garage if it has four relays atManhattan distance less than 𝑅 , one in each of the four quadrants, e.g., as in Figure 3. We callthese relays the "covering" relays. They have the property that every mobile node at Manhattandistance smaller than 𝑅 to the eliminated garage is necessary at Manhattan distance smaller than 𝑅 to at least one of the covering relay. We call this property the covering transfer property.Note that with the garage elimination heuristic, we eliminate the garage 𝐴 ℓ iff it has a coveringset made of four relays of index smaller than ℓ . Lemma 7.
If a mobile node is at Manhattan distance smaller than 𝑅 to at least one relay, thenit is at Manhattan distance smaller than 𝑅 to at least one non-eliminated garage.Proof. Since there are no road with null density, the mobile node can be in any point of themap which is at Manhattan distance smaller than 𝑅 from a relay. Let us suppose that there issuch point 𝑋 such that all relays at distance smaller than 𝑅 are eliminated garage. Let denote 𝑖 the smallest index among the indices of the relays at distance smaller than 𝑅 . Since we supposethat its garage has been eliminated, the relay 𝐴 𝑖 is covered by four relays of smaller index. Let (H,V) Figure 4: Garage serving model 𝑗 < 𝑖 be the index of the covering relay which is in the same quadrant of 𝐴 𝑖 as the point 𝑋 .Since 𝐴 𝑗 and 𝑋 are at Manhattan distance smaller than 𝑅 of each other, this contradicts the factthat 𝑖 is the smallest index of the relays within distance 𝑅 from 𝑋 . Therefore 𝑋 has necessarilyat least one non-eliminated garage within distance 𝑅 . (cid:3) Consequently, we will eliminate quickly all relays which have themselves a neighboringrelay in distance of less than 𝑅 , therefore, there is no uncovered segment of road betweenthe two relays. The remaining garages should hold drones to ensure connectivity within thedelay tolerated.Figure 4 illustrates an operating model scenario with a garage of drones. A garage selectedon a relay at an intersection of levels ( 𝐻, 𝑉 ) should be able to send its drones to all the carsarriving from the closest relays, forming together chains of drones of maximum length 𝑘 . Achain of drones will be constructed with drones coming from both of the neighboring garages,as the passing of a car through a relay is announced to the neighboring relay through the wiredF1 interface such that the following garage in the direction of movement can send its drones tomeet the car to ensure the coverage along the path.A garage should be able to hold enough drones to ensure the continuity of service for allupcoming vehicles on the lines of the hyperfractal support crossing the area where the dronescan move in an acceptable time. Note that as the arriving traffic flow in the "cell" served by thegarage is a mixture of Poisson point processes on lines, the average incoming traffic the garage is serving can be computed therefore, given the distance towards the neighboring relays.We assume that the average speed of a drone is considered to be comparable to the maximumspeed of a car. The drones of a garage can thus be in three possible states: in route towards thecar they will be serving, serving a car, and coming back from the service.Consequently, this procedure leads to the property that, within the "cell" around a garage, thegraph is connected over time using the drones (as per Figure 4). We will refer to this cell as a"moving network of drones". V. N UMERICAL E VALUATIONS
In this section, we first provide some simulations on the connectivity of gNBs, UEs anddrones. We then provide some simulations on the garage locations and their properties.The numerical evaluations are run in a locally built simulator in MatLab, following thedescription provided in Section II for both the stochastic modelling of the location of the entitiesand the communication model. The map length is of one unit and a scaling is performed in orderto respect the scaling of real cities as well as communication parameters.
A. Connectivity of gNBs, UEs and Drones.
We first give some visual insight on the connectivity variation with the fractal dimension ofthe gNBs and 𝜃 . Figure 5a shows in red ∗ the locations of vehicular UEs for 𝑛 = and 𝑑 𝐹 = and in black circles the locations of the gNBs for 𝑑 𝑟 = and 𝜌 = 𝑛 . We are, therefore, in thefirst regime of 𝜃 , 𝜃 > 𝑑 𝑟 / . On the other hand, Figure 5b displays, a snapshot of a network forthe same 𝑑 𝐹 , 𝑑 𝑟 and 𝑛 , the second regime of 𝜃 , with 𝜃 < 𝑑 𝑟 / , as more precisely 𝜃 = / .Notice how the number of gNB falls drastically for the second regime of 𝜃 . This generates,as expected, and graphically visible in Figure 6, numerous disconnected UEs. For this regime,the number of isolated UEs is overwhelming, as clearly shown in Figure 6b.We now look at what happens for a higher fractal dimension of the fixed telecommunicationinfrastructures. Figure 7a shows a snapshot of a network with 𝑛 = vehicular UEs (in red ∗ )and 𝑑 𝐹 = and gNBs with 𝑑 𝑟 = . and 𝜌 = 𝑛 (in black circles). This is here in the first regimeof 𝜃 , 𝜃 > 𝑑 𝑟 / while Figure 7b displays, for the same 𝑑 𝐹 , 𝑑 𝑟 and 𝑛 , the second regime of 𝜃 , 𝜃 < 𝑑 𝑟 / , more precisely, in this case, 𝜃 = / . (a) 𝜃 > 𝑑 𝑟 / (b) 𝜃 < 𝑑 𝑟 / Figure 5: UEs and gNBs, 𝑑 𝐹 = , 𝑑 𝑟 = , 𝑛 = (a) 𝜃 > 𝑑 𝑟 / (b) 𝜃 < 𝑑 𝑟 / Figure 6: Snapshot of UEs not covered, 𝑑 𝐹 = , 𝑑 𝑟 = Similarly to Figure 6, Figure 8 shows a snapshot of the isolated UEs for the two regimes of 𝜃 for 𝑑 𝑟 = . . Notice that the number of isolated nodes is significantly higher for a large fractaldimension of the eMBB infrastructure, even for the first regime of 𝜃 .Let us now look at the validation of Theorem III.1 on the number of isolated nodes, thisis an important parameter estimation as it gives the operator insight on the requirements fordimensioning the network. Figure 9 shows the number of UEs that are not covered by a gNBwhen we vary the total number of devices and for two values of the fractal dimension of thegNBs: 𝑑 𝑟 = and 𝑑 𝑟 = . In both cases, the fractal dimension of the nodes is 𝑑 𝐹 = and 𝑛 = 𝜌 ,therefore 𝜃 = .Notice that the bound we provided concurs with the simulation results. For the extension (a) 𝜃 > 𝑑 𝑟 / (b) 𝜃 < 𝑑 𝑟 / Figure 7: UEs and gNBs, 𝑑 𝐹 = , 𝑑 𝑟 = . , 𝑛 = (a) 𝜃 > 𝑑 𝑟 / (b) 𝜃 < 𝑑 𝑟 / Figure 8: Snapshot of UEs not covered, 𝑑 𝐹 = , 𝑑 𝑟 = . of this work and to provide an insight for the potential users of the tools in this work, wesuggest using the expression in (5) for a tighter bound or the expression in (6) of III.1 if a closeexpression is desired. For instance, in this plot, we have used the latter.Next, for three values of fractal dimension of gNBS, 𝑑 𝑟 = , 𝑑 𝑟 = and 𝑑 𝑟 = respectively,yet for a case of 𝜃 = / , we show in Figure 10, that the number of isolated nodes tends to theactual number of nodes in the network, as stated in Theorem III.2.This confirms again that, in the case when 𝜃 < 𝑑 𝑟 / , the eMBB infrastructure alonecannot provide the required connectivity and consequently the number of disconnected nodes isoverwhelming.Figure 11 illustrates the result proved in Theorem III.3: the number of drones required to ensure
200 400 600 800 1000 nr nodes n r o f un c o v e r ed node s theoreticalsimulated (a)
200 400 600 800 1000 nr nodes n r o f un c o v e r ed node s theoreticalsimulated (b) Figure 9: (a) 𝑑 𝑟 = , 𝑑 𝐹 = ; (b) 𝑑 𝑟 = , 𝑑 𝐹 =
200 400 600 800 1000 nr nodes n r o f un c o v e r ed node s nr nodesd r =3d r =4d r =5 Figure 10: Proportion of isolated nodes for 𝜃 < 𝑑 𝑟 / connectivity for the isolated UEs (when 𝜃 > 𝑑 𝑟 / ) behaves asymptotically like the number ofisolated nodes. B. Location and Size of Drone Garages.
We run some simulations in order to validate the physical distance requirements betweenmobile nodes, relays and garages. In order to have realistic figures we have assumed that thepractical range of emission with mmWave is of order 100m, and that the density of mobile nodesis around 1,000 per square kilometer. This would lead to 𝑅 𝑛 = √︁ / 𝑛 , equivalent to a squareof 1 square kilometer has a side length of 10 times the radio range and contains 1,000 mobilenodes. We fix 𝑑 𝐹 = and 𝑑 𝑟 ≈ . (with 𝑝 𝑟 = . ).We first compute the distribution of the distance of the mobile nodes to their closest basestations expressed in hop count in Figure 12. A hop count of one means that the mobile nodes is
200 400 600 800 1000 nr nodes n r o f d r one s drones uppper bounddrones measuredisolated nodes (a) 𝑑 𝑟 = , 𝑑 𝐹 = ;
200 400 600 800 1000 nr nodes n r o f d r one s drones uppper bounddrones measuredisolated nodes (b) 𝑑 𝑟 = , 𝑑 𝐹 = Figure 11: Number of drones for connectivity (a) 𝜃 = 𝑑 𝑟 / ; (b) 𝜃 = . 𝑑 𝑟 / ; (c) 𝜃 = . 𝑑 𝑟 / ; Figure 12: distribution of distance to base stations in hop countin direct range to a base station. A hop count of 𝑘 ( 𝑘 integer) means that the mobile nodes wouldneed 𝑘 − drones to let it connected to its closest base station. We display the distribution forvarious values of 𝑛 (green: 𝑛 = , ; blue: 𝑛 = , ; red: 𝑛 = , ; brown: 𝑛 = , ;black 𝑛 = , ).Secondly, in Figure 13, we compute the average number of isolated nodes (in blue) ( i.e. themobile nodes not at a direct range to a base station), and at the same time the average numberof drones (in brown) to connect them to the closest base station. The two numbers are given asa fraction of the total number of mobile nodes present in the map.In Figure 14 we display the map of the garage locations after reduction on base stations.The reduction is made according to various coverage radii (the parameter 𝑅 respectively) whichvaries from radio ranges 5 (500m) to 80 (8km). We can see as expected that the garage densitydecreases when the coverage radius increases. The parameters are 𝑛 = , and 𝜃 = . 𝑑 𝑟 . (a) 𝜃 = 𝑑 𝑟 / ; (b) 𝜃 = . 𝑑 𝑟 / ; (c) 𝜃 = . 𝑑 𝑟 / (noticethe change of scale); Figure 13: proportion of isolated nodes (blue), proportion of drones (brown), versus the totalnumber 𝑛 of mobile nodesThe coverage radius impacts the delay at which the drones can move to new mobile nodes,although many of these moves could be easily predicted from the aim and trajectory of themobile nodes. Figure 15 shows the variation of size of the garage set as function of the coverageradius. The parameters are 𝑛 = , ; in brown 𝜃 = 𝑑 𝑟 / ; in blue 𝜃 = . 𝑑 𝑟 / ; and in green 𝜃 = . 𝑑 𝑟 / . When the coverage radius is zero, every relay is a garage and we get the initialnumber of relays. We notice that when the coverage radius tends to infinity the limit density ofgarages is not bounded and increases with the number of relays. We conjecture that it increasesas the logarithm of this number. On the right subfigure, we display the size of the garage setwhen the city map is considered on a torus without border. In this case the garage size decreasesto 1 when the coverage radius increases. Figure 16 gives examples of garage maps in a torus.Figure 17 shows the distribution of distance of the mobile nodes to the closest garage. Ingreen is for coverage radius 𝑅 𝑛 , in blue for coverage radius 𝑅 𝑛 , in red 𝑅 𝑛 , in brown 𝑅 𝑛 ,in black 𝑅 𝑛 . We notice that despite the coverage radius increases the typical distance to theclosest garage does not grow too much in comparison because the residual density of garageprevents. Remember that the distance to the closest garage is larger than the number of dronesneeded to connect the mobile node to the closest relay, which is given by figure 12. The distanceto the closest garage gives an indication on how fast drones can be moved towards new mobilenodes. (a) coverage distance 𝑅 𝑛 (b) coverage distance 𝑅 𝑛 (c) coverage distance 𝑅 𝑛 Figure 14: Map of drone garages (black circles), among the base stations (green crosses), fordifferent coverage distances. (a) Garage number in a map with border (b) in a Torus map with no border
Figure 15: Size of the garage set as function of coverage radius, in brown 𝜃 = 𝑑 𝑟 / , in blue 𝜃 = . 𝑑 𝑟 / , and in green 𝜃 = . 𝑑 𝑟 / . (a) coverage distance 𝑅 𝑛 (b) coverage distance 𝑅 𝑛 (c) coverage distance 𝑅 𝑛 Figure 16: Map of drone garages (black circles), among the base stations (green crosses), fordifferent coverage distances in a torus map. (a) 𝑛 = , , 𝜃 = . 𝑑 𝑟 / (b) 𝑛 = , , 𝜃 = . 𝑑 𝑟 / Figure 17: distribution of distances to closest garage (in hop count) for various coverage radii.VI. C
ONCLUDING R EMARKS
This work has provided a study of the connectivity properties and dimensioning of the movingnetworks of drones used as a flying backhaul in an urban environment with vehicular users.By making use of the hyperfractal model for both the vehicular networks and for thefixed eMBB infrastructures, we have derived analytic bounds on the requirements in terms of connectivity extension: we have proved that for 𝑛 mobile nodes (distributed according to ahyperfractal distribution of dimension 𝑑 𝐹 ) and an average of 𝜌 gNBs (of dimension 𝑑 𝑟 ) if 𝜌 = 𝑛 𝜃 with 𝜃 > 𝑑 𝑟 / , the average fraction of mobile nodes not covered by a gNB tends to zero like 𝑂 (cid:16) 𝑛 − ( 𝑑𝐹 − ) 𝑑𝑟 ( 𝜃 − 𝑑𝑟 ) (cid:17) . Furthermore, for the same regime of 𝜃 , we have obtained that the numberof drones to connect the isolated mobile nodes is asymptotically equivalent to the number ofisolated mobile nodes. This gives insights on the dimensioning of the flying backhauls andalso limitations of the usage of UAVs (second regime of 𝜃 ). This work has also initiated thediscussions on the placement of the home locations of the drones, what we called the “garage ofdrones". We have provided a fast procedure to select the relays that will be garages (and storedrones) in order to minimize the number of garages and minimize the delay.Our simulations results concur with our bounds, and illustrate the step-change of regime basedon 𝜃 . Our simulations also show how this can be exploited to have as few garages as possible,while having drones servicing efficiently the mobile vehicles with limited delay. Hence makingthe scenario attractive for further study and possible implementation.Overall our results have provided a realistic stochastic communication model for studyingthe development of 5G in smart cities. The interest of such an innovative framework wasdemonstrated by the computation of exact bounds and the identification of particular behaviours(such as the characterisation of a threshold). It is also a step towards constructing a “smart citymodeling” framework that can be exploited in other urban scenarios.A PPENDIX
Proof that 𝑓 ( 𝑦 ) = (cid:205) 𝐻 𝑝𝑞 𝐻 exp (cid:0) − 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 𝑦 (cid:1) = 𝑝 ( 𝑝 (cid:48) ) − 𝛿 log ( / 𝑞 (cid:48) ) Γ ( 𝛿 ) 𝑦 − 𝛿 ( + 𝑜 ( )) . We use thetechnique in [26] by the Mellin transform 𝑓 ∗ ( 𝑠 ) = ∫ ∞ 𝑓 ( 𝑦 ) 𝑦 𝑠 − 𝑑𝑦 , which is defined for somecomplex number 𝑠 such that (cid:60)( 𝑠 ) > . Indeed since the Mellin transform of exp (− 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 𝑦 ) is ( 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 ) 𝑠 Γ ( 𝑠 ) where Γ ( 𝑠 ) is the Euler “Gamma" function defined for (cid:60)( 𝑠 ) > , thus 𝑓 ∗ ( 𝑠 ) = (cid:205) 𝐻 𝑝𝑞 𝐻 ( 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 ) − 𝑠 Γ ( 𝑠 ) = 𝑝 ( 𝑝 (cid:48) ) − 𝑠 − 𝑞 ( 𝑞 (cid:48) / ) − 𝑠 Γ ( 𝑠 ) as long as (cid:60)( 𝑠 ) < 𝛿 (thus the sum (cid:205) 𝐻 𝑝𝑞 𝐻 ( 𝑝 (cid:48) ( 𝑞 (cid:48) / ) 𝐻 ) − 𝑠 absolutely converges).The asymptotic of function 𝑓 ( 𝑦 ) is obtained by the inverse Mellin transform as explainedin [26] as the residues of function of 𝑓 ∗ ( 𝑠 ) 𝑦 − 𝑠 on the main pole 𝑠 = 𝛿 which lead to the claimedasymptotic expression. To the risk to be pedantic the reference [26] also mentions that thereare additional poles on the complex numbers 𝛿 + 𝑖𝑘 𝜋 / log ( 𝑞 (cid:48) / ) for 𝑘 integer which lead tonegligible fluctuations of the main asymptotic term. R EFERENCES [1] Q. L. Gall, B. Blaszczyszyn, E. Cali, and T. En-Najjary, “Relay-assisted device-to-device networks: Connectivity anduberization opportunities,” in , 2020, pp. 1–7.[2] D. Popescu, P. Jacquet, B. Mans, R. Dumitru, A. Pastrav, and E. Puschita, “Information dissemination speed in delaytolerant urban vehicular networks in a hyperfractal setting,”
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