Characterizing the Energy Trade-Offs of End-to-End Vehicular Communications using an Hyperfractal Urban Modelling
Dalia Popescu, Philippe Jacquet, Bernard Mans, Bartomiej Blaszczyszyn
11 Characterizing the Energy Trade-Offs ofEnd-to-End Vehicular Communications using anHyperfractal Urban Modelling
Dalia Popescu, Philippe Jacquet,
Fellow, IEEE,
Bernard Mans, BartΕomiej BΕaszczyszyn
Abstract
We characterize trade-offs between the end-to-end communication delay and the energy in urban vehicularcommunications with infrastructure assistance. Our study exploits the self-similarity of the location of commu-nication entities in cities by modeling them with an innovative model called βhyperfractalβ. We show that thehyperfractal model can be extended to incorporate road-side infrastructure and provide stochastic geometry tools toallow a rigorous analysis. We compute theoretical bounds for the end-to-end communication hop count consideringtwo different energy-minimizing goals: either total accumulated energy or maximum energy per node. We provethat the hop count for an end-to-end transmission is bounded by π ( π β πΌ /( π πΉ β ) ) where πΌ < and π πΉ > isthe fractal dimension of the mobile nodes process. This proves that for both constraints the energy decreases aswe allow choosing routing paths of higher length. The asymptotic limit of the energy becomes signiο¬cantly smallwhen the number of nodes becomes asymptotically large. A lower bound on the network throughput capacity withconstraints on path energy is also given. We show that our model ο¬ts real deployments where open data sets areavailable. The results are conο¬rmed through simulations using different fractal dimensions in a Matlab simulator. Index Terms
Wireless Networks; Delay; Energy; Fractal; Vehicular Networks; Urban networks.
I. I
NTRODUCTION
A. Motivation and Background
Vehicular communications, V2V (vehicle-to-vehicle), V2I (vehicle to infrastructure) or V2X (vehicle toeverything), are a key component of the 5th Generation (5G) and beyond communications. These βverticalsβrepresent one major focus of the telecommunication industry nowadays. Yet like many innovationopportunities on the horizon they arrive with signiο¬cant challenges. As the vehicular networks continueto scale up to reach tremendous network sizes with diverse hierarchical structures and node types,and as vehicular interactions become more complex with entities having hybrid functions and levelsof intelligence and control, it is paramount to provide an effective integration of vehicular networkswithin the complex urban environment. Automated and autonomous driving in such a complex andevolving environment requires sensors that generate a huge amount of data demanding high bandwidthand data rates [1]. Furthermore, an effective integration of these new types of communication in the newradio βbabbling" created by the other 5G actors such as evolved mobile broadband, ultra-reliable lowlatency communications and massive machine-type communications, requires a careful design for optimalconnectivity, low interference, and maximum security.5G NR (5th Generation New Radio) is essentially a multi-beam system, with high-frequency rangesgenerated by millimeter-wave (mmWave) technology [2]. With many GHz of spectrum to offer, millimeter-wave bands are the key for attaining the high capacity and services diversity of the NR. For a long timethese frequencies have been disregarded for cellular communications due to their large near-ο¬eld loss, andpoor penetration (blocking) through common material, yet recent research and experiments have shown
B. Mans was supported in part by the Australian Research Council under Grant DP170102794.Part of the work has been done at Lincs. Dalia Popescu was with Nokia Bell Labs, 91620 Nozay, France. Bernard Mans is with MacquarieUniversity, Sydney, Australia ([email protected]). Philippe Jacquet and BartΕomiej BΕaszczyszyn are with INRIA, France. a r X i v : . [ c s . D C ] F e b that communications are feasible in ranges of 150-200 meters dense urban scenarios with the use of suchhigh gain directional [3]. Furthermore, the tight requirements (e.g, line of sight, short-range) are easilyanswered as the embedding space of vehicular networks leads to a highly directive topology (as much asit is possible, roads are built as straight lines) [4].Given the numerous challenges of mmWave [5] and the important place the vehicular communicationshold in the new communications era, realistic modeling of the topology for accurate estimation of networkmetrics is mandatory. The research community has proposed stochastic models that usually ο¬t with highprecision cellular networks or ad hoc networks. Yet for vehicles, and more importantly, for vehiclesusing mmWave technology, this cannot be done without taking into account the crucial fact that theeffectiveness of the communications are inο¬uenced by the environmental topology. Cars are located onstreets and streets are conditioned by a world-wide common urban architecture that has interesting features.One major feature of the urban architecture that we exploit in this work is self-similarity.While it has been extensively studied in diverse research ο¬elds such as biology and chemistry, self-similarity has been only recently introduced to wireless communications, after understanding that thedevice-to-device communication topologies follow the social human topology. Self-similarity is presentin every aspect of the surrounding environment but is particularly emphasized in the urban environment.The hierarchic organization with different degrees of scaling of cities is a perfect illustration of the fractalstructure of human society [6]. Figure 1 presents a snapshot of the trafο¬c in a neighborhood of Minneapolis.Common patterns and hierarchical organizations can easily be identiο¬ed in the trafο¬c measurements andshall be further explained in this paper.Figure 1: Minneapolis trafο¬c snapshotIn this paper, we extend the "hyperfractal" model that we have introduced in [7], [8] to better capturethe impact of the network topology on the fundamental performance limits of end-to-end communicationsover vehicular networks in urban settings. The model consists of assigning decaying trafο¬c densities tocity streets, thus avoiding the extremes of regularity (e.g. Manhattan grid) and uniform randomness (e.g.Poisson point process), the ο¬tting of the model with trafο¬c data of real cities having been showcasedin [9]. The hyperfractal model exploits the self-similarity: e.g., it is characterized by a dimension that islarger than the dimension of the euclidean dimension of the embedding space, that is larger than 2 whenthe whole network lays in a 2-dimensional plane.Our previous results in [7] revealed that, for nodes, the number of hops in a routing path betweenan arbitrary source-destination pair increases as a power function of the population π of nodes when π tends to inο¬nity. However, we showed that the exponent tends to zero when the fractal dimension tendsto inο¬nity. An initial observation for this model is that the optimal path may have to go through streets oflow density where inter-vehicle distance can become large, therefore the transmission becomes expensive in terms of energy cost. Hence, in this paper, the focus will be on the study of the relationship betweenefο¬cient communications and energy costs. B. Contributions and paper organization:
Our goal is to characterize trade-offs between the end-to-end communication delay and the energy inurban vehicular communications with infrastructure assistance in modern cities.Our ο¬rst contribution is to exploit the self-similarity of the location of the trafο¬c and vehicles in citiesby modeling the communication entities and relationships with an innovative model called βhyperfractalβ(to avoid extremes of classical Poisson distribution or uniform distribution tools) and to show that thehyperfractal model can be extended to incorporate road-side infrastructure, as relays. This providesfundamental properties and tools in the framework of stochastic geometry that allow for a rigorous analysis(and are of independent interest for other studies).Our main contributions are theoretical bounds for the end-to-end communication hop count. We willconsider two different energy-minimizing goals: (i) total accumulated energy or (ii) maximum energy pernode. We will prove that the hop count for an end-to-end transmission is bounded by π ( π β πΌ /( π πΉ β ) ) where πΌ < and π πΉ > is the fractal dimension of the mobile nodes process, thus proving that for bothconstraints the energy decreases as we allow choosing routing paths of higher length. We will also showthat the asymptotic limit of the energy becomes signiο¬cantly small when the number of nodes becomesasymptotically large. This is also completed with a lower bound on the network throughput capacity withconstraints on path energy. Finally we will show that our model ο¬ts real deployments where open datasets are available. The results are conο¬rmed through simulations using different fractal dimensions andpath loss coefο¬cients, using a discrete-event simulator in Matlab.The paper is organized as follows: β’ In Section III, we ο¬rst enhance the hyperfractal model by taking into account mmWave communicationrange variations as well as the energy costs of transmission. In addition, we enrich the model byincorporating road-side infrastructure with communication relays (with radio communication rangevariations). We exploit the self-similarity of intersection locations in urban settings. β’ in Section IV, theoretical properties of the hyperfractal model are obtained to allow the character-ization of bounds within the communication model. These properties are developed within a classicstochastic geometric framework and are of interest on their own. β’ In Section V, we prove that for an end-to-end transmission in a hyperfractal setup, the energy (eitheraccumulated along the path or bounded for each node) decreases if we allow the path length toincrease. In fact, we show that the asymptotic limit of the energy tends to zero when π , the numberof nodes, tends to inο¬nity. We also prove a lower bound on the network throughput capacity withconstraints on path energy. β’ In Section VI, we further provide a ο¬tting procedure that allows computing the fractal dimension ofthe relay process using trafο¬c lights data sets. β’ Finally, Section VII validates our analytical results using a discrete-time event-based simulatordeveloped in Matlab. II. R
ELATED W ORKS
Millimeter-wave is a key technological brick of the 5G NR networks, as foreseen in the ground-breakingwork done in [10] and already proved by ongoing deployments. The research community has been alreadyinvestigating challenges that may appear and proposing innovative solutions. Vehicular communicationsare one of the areas that are to beneο¬t from the high capacity offered by the mmWave technology.In [11], the authors propose an information-centric network (ICN)-based mmWave vehicular frameworktogether with a decentralized vehicle association algorithm to realize low-latency content disseminations.The study shows that the proposed algorithm can improve the content dissemination efο¬ciency yet there are no consideration about the energy. The purpose of [12] is optimizing energy efο¬ciency in a cellularsystem with relays with D2D (device-to-device) communications using mmWave.As mmWave is highly directional and blockages raise concerns, the authors of [13] propose an onlinelearning algorithm addressing the problem of beam selection with environment-awareness in mmWavevehicular systems. The sensitivity to blockages is generally solved with the assistance of the relayinginfrastructure. The authors of [14] attempt to solve the dependency of infrastructure for relaying invehicular communications by exploiting social interactions. In [15], the problem of relay selection andpower is solved using a centralized hierarchical deep reinforcement learning based method. Yet the authorsus a simpliο¬ed highway scenario, which would not scale for a city structure.Stochastic geometry studies have shown results on the interactions between vehicles on the highwaysor in the street intersections [16], [17]. The work in [18] performs statistical studies on traces of taxisto identify a planar point process that matches the random vehicle locations. The authors ο¬nd that a LogGaussian Cox Process provides a good ο¬t for particular traces. In [19] propose a novel framework forperformance analysis and design of relay selections in mmWave multi-hop V2V communications. Moreprecisely, the distance between adjacent cars is modeled as shifted-exponential distribution.Self-similarity for urban ad hoc networks has been introduced in [7], [8], where the hyperfractal modelexploits the fractal features of urban ad hoc networks with road-side infrastructure. In [9], we presented ananalysis of the propagation of information in a vehicular network where the cars (the only communicationentities) are modeled using the hyperfractal model. As there are no relays in the intersections, as in thecurrent study, in [9] we are in a disconnected network scenario where, as the nodes are allowed to move,the network becomes connected over time with mobility. The packets are being broadcast and results ontypical metrics for delay tolerant networks are presented. There is no investigation on power or energy.The study in [7] provides results on the minimal path routing using the hyperfractal model for static nodesto model the road-side infrastructure and assumes an inο¬nite radio range. This is a concern for allowedtransmission power and network energy consumption. In contrast to this ο¬rst study, in this paper we addconstraints on these quantities to provide insights on the achievable trade-offs between the end-to-endtransmission energy and delay. III. S
YSTEM M ODEL
In this section, we recall the necessary deο¬nitions of Hyperfractal model and enhance it to be able toformalise urban settings in matter of vehicles (as mobile users) as well as communication relays (as ο¬xedinfrastructures), both supported by a deterministic structure (called the support) on which various Poissonprocesses are sampled.
A. Hyperfractals for vehicular networks
Cities are hierarchically organized [6]. The main parts of the cities have many elements in common(in functional terms) and repeat themselves across several spatial scaling. This is reminiscent of a fractal,described by Lauwerier [20] as an object that consists of an identical motif repeating itself on an ever-reduced scale. The hyperfractal model has been introduced and exploited in [7], [8] under static settingsand in [9] under mobile settings.To represent the reality while being able to analyse various features, the map model lays in the unitsquare embedded in the 2-dimensional Euclidean space where various processes are sampled. Basically,we introduce a support of the intensity measure as a deterministic set (or structure) on which respectivePoisson processes are sampled (the set where this intensity measure is not null). In this paper, the supportof the population of π nodes is a grid of streets. Let us denote this structure by X = (cid:208) β π = X π with X π : = {( π β( π + ) , π¦ ) , π = , , . . . , π + β , π¦ β [ , ] } βͺ {( π₯, π β( π + ) ) , π = , , . . . , π + β , π₯ β [ , ] } , where π denotes the level and π starts from , and π is an odd integer. Three ο¬rst levels, π = , , , aredisplayed in Figure 2a. Observe that the central "cross" X splits (cid:208) β π = X π in "quadrants" which all arehomothetic to X with the scaling factor / . (a) β¬ β p r β βο£ β βο£Ά β ο£Έ βο£· β¬ β p r β βο£ β βο£Ά β ο£Έ βο£· β¬ β p r β βο£ β βο£Ά β ο£Έ βο£· β¬ β p r β βο£ β βο£Ά β ο£Έ βο£· β¬ p r β¬ β p r β βο£ β βο£Ά β ο£Έ βο£· p r β¬ β p r β βο£ β βο£Ά β ο£Έ βο£· p r (b) Figure 2: (a) Hyperfractal support; (b) Relays process construction
B. Mobile users
Following [7], we consider the Poisson point process Ξ¦ of (mobile) users on X with total intensity(mean number of points) π ( < π < β ) having 1-dimensional intensity π π = π ( π / ) ( π / ) π (1)on X π , π = , . . . , β , with π = β π for some parameter π ( β€ π β€ ). Note that Ξ¦ can be constructed inthe following way: one samples the total number of mobiles users Ξ¦ (X) = π from Poisson distribution;each mobile is placed independently with probability π on X according to the uniform distribution andwith probability π / it is recursively located in a similar way in one the four quadrants of (cid:208) β π = X π .The process Ξ¦ is neither stationary nor isotropic. However, it has the following self-similarity property:the intensity measure of Ξ¦ on X is hypothetically reproduced in each of the four quadrants of (cid:208) β π = X π with the scaling of its support by the factor 1/2 and of its value by π / .The fractal dimension is a scalar parameter characterizing a geometric object with repetitive patterns.It indicates how the volume of the object decreases when submitted to an homothetic scaling. When theobject is a convex subset of an euclidian space of ο¬nite dimension, the fractal dimension is equal to thisdimension. When the object is a fractal subset of this euclidian space as deο¬ned in [21], it is a possiblynon integer but positive scalar strictly smaller than the euclidian dimension. When the object is a measuredeο¬ned in the euclidian space, as it is the case in this paper, then the fractal dimension can be strictlylarger than the euclidian dimension. In this case we say that the measure is hyper-fractal . Remark 1.
The fractal dimension π πΉ of the intensity measure of Ξ¦ satisο¬es (cid:18) (cid:19) π πΉ = π thus π πΉ = log ( π ) log 2 β₯ . The fractal dimension π πΉ is greater than 2, the Euclidean dimension of the square in which it isembedded, thus the model was coined hyperfractal in [7]. Notice that when π = the model reduces tothe Poisson process on the central cross, while for π β , π πΉ β it corresponds to the uniform measurein the unit square. C. Relays
Not surprisingly, the locations of communication infrastructures in urban settings also display a self-similar behavior: their placement is dependent on the trafο¬c density. Hence we apply another hyperfractalprocess for selecting the intersections where a road-side relay is installed or the existing trafο¬c light isused as road-side unit. This process has been introduced in [7].
We denote the relay process by Ξ . To deο¬ne Ξ it is convenient to consider an auxiliary Poisson process Ξ¦ π with both processes supported by a 1-dimensional subset of X namely, the set of intersections ofsegments constituting X . We assume that Ξ¦ π has discrete intensity π ( β, π£ ) = π π π (cid:18) β π π (cid:19) β + π£ (2)at all intersections X β β© X π£ for β, π£ = , . . . , β for some parameter π π , β€ π π β€ and π > . That is,at any such intersection the mass of Ξ¦ π is Poisson random variable with parameter π ( β, π£ ) and π is thetotal expected number of points of Ξ¦ π in the model. The self-similar structure of Ξ¦ π is explained by itsconstruction: we ο¬rst sample the total number of points from a Poisson distribution of intensity π andgiven Ξ¦ π (X) = π , each point is independently placed with probability π π in the central crossings of X , with probability π π β π π on some other crossing of one of the four segments forming X and, withthe remaining probability (cid:16) β π π (cid:17) , in a similar way, recursively, on some crossing of one of the fourquadrants of (cid:208) β π = X π . This is illustrated in Figure 2b. The Poisson process Ξ¦ π is not simple: we deο¬nethe relay process Ξ as the support measure of Ξ¦ π , i.e., only one relay is installed at crossings where Ξ¦ π has at least one point. Remark 2.
Note that the relay process Ξ forms a non-homogeneous binomial point process (i.e. pointsare placed independently) on the crossings of X with a given intersection of two segments from X β and X π£ occupied by a relay point with probability β exp (β π π ( β, π£ )) . Similarly to the process of users, we can deο¬ne the fractal dimension of the relay process.
Remark 3.
The fractal dimension π π of the probability density of Ξ is equal to the fractal dimension ofthe intensity measure of the Poisson process Ξ¦ π and veriο¬es π π = ( /( β π π )) log 2 . A complete hyperfractal map with mobile nodes and relays is illustrated in Figure 3.
Figure 3: Complete hyperfractal map with mobile nodes ("+") and relays ("o")IV. H
YPERFRACTAL P ROPERTIES AND C OMMUNICATION MODEL
In this section we extract the relevant properties of the Hyperfractal model and relate them to ourcommunication model. We also provide some additional insights into these models via the framework ofthe stochastic geometry and point process. These latter results are of independent interest and allow tolay foundations for other works.
A. Number of relay nodes and asymptotic estimate
We shall now provide the proof for the average number of relays in the map, this time, exploiting the"fractal-like" properties of our model and providing useful asymptotic estimates.
Theorem IV.1.
The average total number of relays π ( π ) in the map is: π ( π ) = βοΈ π»,π π» + π ( β exp (β π π ( π», π ))) = π ( π / π π log π ) (3) Proof.
The probability that a crossing of two lines of level π» and π is selected to host a relay is exactly β exp (β π π ( π», π )) .The average number of relays on a street of level π» is πΏ π» ( π ) and satisο¬es: πΏ π» ( π ) = βοΈ π β₯ π ( β exp (β π ( π», π ) π )) . We notice that πΏ π» ( π ) = πΏ ( ( β π π ) π» π ) and that πΏ ( π ) satisο¬es the functional equation: πΏ ( π ) = β exp (β π π π ) + πΏ (cid:18) π β π π (cid:19) . It is known from [22] and [23] that this classic equation has a solution such as πΏ ( π ) = π ( π / π π ) .The average total number of relays π ( π ) in the city is obtained by summing the average number ofrelays over all streets parallel to a given direction, e.g. the West-East direction (summing on all streetswould count twice each relay). Since there are π» West-East streets at level π» : π ( π ) = βοΈ π» π» πΏ π» ( π ) = βοΈ π»,π β₯ π» + π ( β exp (β π ( π», π ) π )) = β βοΈ π = ( π + ) π ( β exp (β π π π ( ( β π π )/ ) π ) (4)and satisο¬es the functional equation π ( π ) = πΏ ( π ) + π ( π β π π ) . From the same references ([22], [23]), one gets π ( π ) = π ( π / π π log π ) Since / π π < , the number of relays is much smaller than π when π β β . (cid:3) Let us verify numerically the claim of Theorem IV.1. We generate the hyperfractal maps with severalvalues of π , π π = π and π πΉ = . Let us remind the reader that the theorem gives the expression (4) of π ( π ) as a sum with π β β . In reality, a limited number of terms of the sums sufο¬ce to approximate π ( π ) with acceptable accuracy. We denote by π πππ₯ the number of terms used to compute in the sum. Figure4 shows that the computed values of the number of relays approach the measured value for π πππ₯ = .The precision is further enhanced when π πππ₯ = . B. Communication model
As we primarily seek to understand the relationship between end-to-end communications and energycosts, we do not consider detailed aspects of the communication protocol that impact these (e.g., thedistributed aspects needed to gather position information and construct routing tables in every node). Thetransmission is done in a half-duplex way, a node is not allowed to transmit and receive during the sametime-slot. The received signal is affected by additive white Gaussian noise (AWGN) noise π and path-losswith pathloss exponent πΏ β₯ .As a consequence of the high directivity and low permeability of the waves in high frequency (6GHz,28GHz, 73 GHz as candidates for 5G NR), the next hop is always the next neighbor on a street, i.e.
200 400 600 800 1000 1200 1400 1600050010001500 Ο n nu m b e r o f r e l ays computed, k max =20computed, k max =40computed, k max =60measured Figure 4: Measured versus computed number of relays in the map for increasing values of π πππ₯ there exists no other node between the transmitter and the receiver. Indeed, while a lot of work is stilldedicated to characterising the exact overall network connectivity for mmWave communications V2Vin urban setting [24], it is known that intermediate vehicles create signiο¬cant blockage and a severeattenuation of the received power for vehicles past near neighbours [25], [26]. Thus the routing strategyconsidered is a nearest neighbor routing. In fact, we can show that, under reasonable assumptions, thisstrategy is optimal. Lemma 1.
If the noise conditions are the same around each node, then the nearest neighbor routingstrategy is optimal in terms of energy.Proof.
To simplify this proof we ignore the signal attenuation due to the mobile users positioned as radioobstacles between the sender and the receiver of one hop packet transmission, although this will have animportant impact on energy. Consider the packet transmission from a node at a location π₯ to a node atlocation π¦ on the same street. If π is the noise level and πΎ is the required SNR, then the transmitter mustuse a signal of power | π¦ β π₯ | πΏ π πΎ . Assume that there is a node at position π§ between π₯ and π¦ . Transmittingfrom π₯ to π§ and then from π§ to π¦ would require a cumulated energy (| π§ β π₯ | πΏ + | π¦ β π§ | πΏ ) π πΎ which is smallerthan the required energy for the direct transmission, since | π₯ β π§ | πΏ + | π§ β π¦ | πΏ β€ (| π₯ β π§ | + | π§ β π¦ |) πΏ . (cid:3) Let us make the simplifying assumption that all nodes on a street transmit with the same nominal power π π which depends only on the number π of nodes on the street. We argue that a good approximation isto suppose that: π π = π max π πΏ (5)where π max is the transmitting power necessary for a node at one end of the street to transmit a packetdirectly to a node at the other end of the street. In other words, assume a road of inο¬nite length wherethe nodes are regularly spaced by intervals of length πΏ is the length of our street. If in this conο¬gurationevery node has a nominal power of π max , then the nominal power to achieve the same performance witha density π times larger but with the same noise values should be π max π β πΏ in order to cope with the losseffect. Thus would give expression (5) if the nodes were regularly spaced by intervals of length πΏ / π . Butsince the spacing intervals are irregular, one should cope with the largest gap πΏ π / π , this brings a smallcomplication in the evaluation of π π . But the probability that there exists a spacing larger than a givenvalue π₯ / π is smaller than π ( β π₯ππΏ ) π β€ π π₯ / πΏ . Thus we have πΏ π = π ( log π / π ) (asymptotically almostsurely, and in fact as soon as lim inf π πΏ π / log π > ), and consequently π π = π ( π max log πΏ π / π πΏ ) . Tohelp the reader, we focus on the expression (5) as we are mainly interested in the order of magnitude. Deο¬nition 1.
The end-to-end transmission delay is represented by the total number of hops the packettakes in its path towards the destination.
As the energy to transmit a packet is the transmission power per unit of time, we consider the timenecessary to send a packet as being equal to the length of a time-slot. We thus do not consider any MACprotocol for re-transmission and acknowledgment of the reception (e.g., we do not consider CSMA-likeprotocols). In any case, as it will be later observed throughout our derivations, varying the MAC protocolwould just change some constants but not the overall scaling. Therefore, from now on, we will refer to π max as the nominal power. Following this reasoning, the accumulated energy to cover a whole streetcontaining π nodes with uniform distribution via nearest neighbor routing is ππ π = π max π πΏ β . In this case,the larger the population of the street the smaller the nominal power and the smaller the energy to coverthe street.Relays stand in intersections, and thus on two streets with different values of π . We consider a relay touse two different radio interfaces, each with a transmission power according to the previously mentionedrule for each of the streets. This is a perfectly valid assumption, in line with 5G devices speciο¬cationsfor dual connectivity [27]. C. Fundamental properties of the Poisson processes Ξ¦ , Ξ¦ π , and Ξ In the following, we shall provide some fundamental tools that allow one to handle our model in atypical stochastic geometric framework. This section gives insights about the theoretical foundations ofhyperfractal point process which is of independent interest to our main results and can be used in otherworks.Let πΏ + be a geometric random variable with parameter π (i.e., P ( πΏ = π ) = π ( β π ) π , π = , , . . . )and given πΏ , let π₯ be the random location uniformly chosen on X πΏ . We call π₯ the typical mobile user of Ξ¦ . More precisely, we shall consider the point process Ξ¦ βͺ { π₯ } where π₯ is sampled as described aboveand independently of Ξ¦ .Similarly, let π + and π + be two independent geometric random variables with parameter π π andgiven ( π, π ) , let π₯ β be a crossing uniformly sampled from all the intersections of X π β© X π . We call π₯ β the typical auxiliary point of Ξ¦ π . More precisely, we shall consider point process Ξ¦ π βͺ { π₯ β } where π₯ β issampled as described above and independently of Ξ¦ π .Finding the deο¬nition of the typical relay node π is less explicit yet similar to the typical pointdeο¬nition. Informally, the conditional distribution of points βseenβ from the origin given that the processhas a point there is exactly the same as the conditional distribution of points of the process βseenβ froman arbitrary location given the process has a point at that location.We deο¬ne it as the random location on the set of the crossings of X involving the following biasingof the distribution of π₯ β by the inverse of the total number of the auxiliary points co-located with π₯ β P ( π = π₯ ) = E (cid:20) ( π₯ β = π₯ ) + Ξ¦ π ({ π₯ β }) (cid:21) / E (cid:20) + Ξ¦ π ({ π₯ β }) (cid:21) . More precisely we consider Ξ (cid:48) βͺ { π } (which distribution is given for any intersection π₯ of segments in X and a possible conο¬guration π of relays) by considering P ( π = π₯, Ξ (cid:48) = π ) = E (cid:20) ( π₯ β = π₯ ) ( supp ( Ξ¦ π ) \ { π₯ } = π ) + Ξ¦ π ({ π₯ β }) (cid:21) / E (cid:20) + Ξ¦ π ({ π₯ β }) (cid:21) where Ξ¦ π and π₯ β are independent (as deο¬ned above). Note that, in contrast to the typical points of Poissonprocesses Ξ¦ and Ξ¦ π , the typical relay π is not independent of remaining relays Ξ (cid:48) .In what follows, we shall prove that our typical points support the Campbell-Mecke formula (see [28],[29]) thus justifying our deο¬nition and also providing an important tool for future exploiting the modelin a typical stochastic geometric framework. Theorem IV.2 (Campbell-Mecke formula) . For all measurable functions π ( π₯, π ) where π₯ β X and π is arealization of a point process on X E (cid:34) βοΈ π₯ π β Ξ¦ π ( π₯ π , Ξ¦ ) (cid:35) = π E [ π ( π₯ , Ξ¦ βͺ { π₯ })] (6) E (cid:34) βοΈ π₯ π β Ξ¦ π π ( π₯ π , Ξ¦ π ) (cid:35) = π E [ π ( π₯ β , Ξ¦ π βͺ { π₯ β })] (7) and E (cid:34) βοΈ π₯ π β Ξ π ( π₯ π , Ξ ) (cid:35) = E [ Ξ (X)] E [ π ( π , Ξ (cid:48) βͺ { π })] (8) where the total expected number of relay nodes E [ Ξ (X)] admits the following representation given byTheorem IV.1 E [ Ξ (X)] = π ( π ) = β βοΈ π = ( π + ) π ( β exp (β π π π ( ( β π π )/ ) π ) . (9) Proof of Theorem IV.2.
First, consider the process of users Ξ¦ . The Campbell-Mecke formula and theSlivnyak theorem [30] for the non-stationary Poisson point processes Ξ¦ give E (cid:34) βοΈ π₯ π β Ξ¦ π ( π₯ π , Ξ¦ ) (cid:35) = β« X E [ π ( π₯, Ξ¦ βͺ { π₯ })] π ( ππ₯ ) , (10)where π ( ππ₯ ) is the intensity measure of the process Ξ¦ . Specifying this intensity measures the right-handside term of (10), thus this becomes β βοΈ π = β« X π E [ π ( π₯, Ξ¦ βͺ { π₯ })] π ( β π ) π πππ₯. In the above expression, one can recognize E [ π ( π₯ , Ξ¦ βͺ { π₯ })] which concludes the proof of (6). Theproof of (7) follows the same lines. Consider now the relay process Ξ . By the deο¬nition of Ξ , one canexpress the left-hand side of (8) in the following way: E (cid:34) βοΈ π₯ π β Ξ π ( π₯ π , Ξ ) (cid:35) = E (cid:34) βοΈ π₯ π β Ξ¦ π π ( π₯ π , supp ( Ξ¦ π )) Ξ¦ π ({ π₯ π }) (cid:35) , where supp ( Ξ¦ π ) denotes the support of Ξ¦ π . Using (7), we thus obtain: E (cid:34) βοΈ π₯ π β Ξ π ( π₯ π , Ξ ) (cid:35) = π E (cid:20) π ( π₯ β , supp ( Ξ¦ π βͺ { π₯ β })) + Ξ¦ π ({ π₯ β }) (cid:21) . (11)By the deο¬nition of the joint distribution of π₯ β and supp ( Ξ¦ π βͺ { π₯ β }) the right-hand side of (11) is equalto π E (cid:20) + Ξ¦ π ({ π₯ β }) (cid:21) E [ π ( π , Ξ (cid:48) βͺ { π })] . This completes the proof of (7) with E [ Ξ (X)] = π E (cid:20) + Ξ¦ π ({ π₯ β }) (cid:21) . (cid:3) V. M
AIN R ESULTS
We now provide our theoretical bounds for the end-to-end communication hop count. The number ofmobile nodes is exactly π , where π is an integer which runs to inο¬nity. A. Energy vs Delay
Given that the transmitting power is dependent on the average density of the nodes on the streets andthat the transmission power per node is limited by the protocols to a value of π max , the connectivity isrestricted. We introduce the following notions and notations. Let π‘ be a node and let π ( π‘ ) be the nominaltransmission energy of this node. Deο¬nition 2.
Let T be a sequence of nodes that constitutes a routing path. The path length is π· (T ) = |T | .The relevant energy quantities related to the paths are: β’ The path accumulated energy is the quantity πΆ (T ) = (cid:205) π‘ βT π ( π‘ ) . β’ The path maximum energy is the quantity π (T ) = max π‘ βT π ( π‘ ) . The path accumulated energy is of interest as we want to optimize the quantity of energy expended inthe-end-to-end communication, and respectively, the path maximum power as we want to ο¬nd the pathwhich maximum power does not exceed a given threshold depending on the energy sustainability of thenodes or the protocol. For example, it is unlikely that a node can sustain a nominal power of π max equalto the power needed to transmit in a range corresponding to the entire length of a street. In this case it isnecessary to ο¬nd a path that uses streets with enough population to reduce the node nominal power andcommunication range (due to the mmWave technology limitations). Deο¬nition 3. β’ Let πΊ ( π, E ) be the set of all nodes connected to the central cross with a path accumulated energynot exceeding E . β’ Let πΊ π ( π, E ) be the subset of πΊ ( π, E ) , where the path to the central cross should not go throughmore than π ο¬xed relays. Deο¬nition 4.
Let πΊ (cid:48) ( π, E ) and πΊ (cid:48) π ( π, E ) be the respective equivalents of πΊ ( π, E ) and πΊ π ( π, E ) but withthe consideration of the path maximum power instead of accumulated energy.B. Path accumulated energy The following theorem gives the asymptotic connectivity properties of the hyperfractal in function ofthe accumulated energy and in function of the path maximum power. This shows that for π large, evenfor some sequences of energy thresholds E π tending to zero, the sets πΊ ( π, E π ) asymptotically dominatethe network. The same holds for the sets sequence πΊ (cid:48) ( π, E π ) . Theorem V.1.
In an urban network with π mobile nodes following a hyperfractal distribution, the followingholds: lim π ββ E (cid:26) | πΊ ( π, π β πΎ π max ) | π (cid:27) = (12) for πΎ < πΏ β and lim π ββ E (cid:26) | πΊ (cid:48) ( π, π β πΎ π max ) | π (cid:27) = (13) for πΎ < πΏ where πΏ is the pathloss coefο¬cient. The following lemma ensures the existence of nodes in a street (with proof in the Appendix). Lemma 2.
There exists π > such that, for all integers π» and π , the probability that a street of level π» contains less than ππ π» / nodes or more than ππ π» nodes is smaller than exp (β πππ π» ) . The following corollary gives a result on the scaling of the number of nodes in a segment of street andthe accumulated energy, getting us one step closer to the results we are looking for.
Corollary 1.
Let < π β€ , assume an interval corresponding to a fraction π of the street length. If theinterval is on a street of level π» , the probability that it contains less than ππ π» π / nodes and it is coveredwith accumulated energy greater than π ( ππ π» ) β πΏ π max is smaller than π β πππ π» π .Proof. This is a slight variation of the previous proof. If we denote by π π» ( π, π ) the number of nodes onthe segment, we have E [ π π‘π π» ( π,π ) ] = ( + π π» π ( π π§ β )) π . The previous proof applies by replacing π π» by ππ π» . The accumulated energy has the expression π max π π» ( π,π ) π πΏπ» ( π ) . Further applying the previous reasoningto each of the random variables π π» ( π ) and π π» ( π, π ) gets the result. (cid:3) Throughout the rest of the paper, we only consider the cases where π πΉ > and π π < π πΉ β , i.e. ( / π ) < /( β π π ) .The following theorem is the main result of our paper and shows that increasing the path lengthdecreases the accumulated energy. In fact, for π β β , the limiting energy goes to zero. Theorem V.2.
In a hyperfractal with π nodes, with nodes of fractal dimension π πΉ and relays of fractaldimension π π , the shortest path of accumulated energy πΈ π = π πΈ π ( β πΏ )( β πΌ ) π max , where π πΈ > and πΌ < ,between two nodes belonging to the giant component πΊ ( π, πΈ π ) , passes through a number of hops : π· π = π ( π β πΌ /( π πΉ β ) ) (14)Although the source and the destination belong to πΊ ( π, πΈ π ) , it is not necessary that all the nodesconstituting the path also belong to πΊ ( π, πΈ π ) , i.e., the path may include nodes that are more than onehop from the central cross. Remark 4.
We have the identity (cid:18) πΈ π π max (cid:19) /( πΏ β ) π· π πΉ β π = π ( π π πΉ β ) . (15)Let us now prove the theorem. Proof.
The main part of our proof is to consider the case when the source, denoted by π π» , and thedestination, π π , both stand on two different segments of the central cross. In this case, we consider theenergy constraint πΈ π . We can easily extend the result to the case when the source and the destinationstand anywhere in the giant component πΊ ( π, πΈ π ) by taking πΈ π as energy constraint and the theoremfollows.When π π» and π π are on the central cross, there exists a direct path that takes the direct route by stayingon the central cross, more speciο¬cally, in Figure 5 a), the segments [ π π΄ ] , [ π΄π ] , [ ππΆ ] , [ πΆ π· ] . Then, thepath length is of order Ξ ( π ) while the accumulated energy of order Ξ ( π β πΏ ) π max .In order to signiο¬cantly reduce the order of magnitude of the path hop length, one must consider adiverted path with three ο¬xed relays, as indicated in Figure 5 a). The diverted path proceeds into twostreets of level π₯ . Let T be the path. It is considered that π₯ = πΌ log π log ( /( β π π )) for πΌ < . The path is madeof two times two segments: the segment of street [ π π΄ ] on the central cross which corresponds to thedistance from the source to the ο¬rst ο¬xed relay to a street of level π₯ , and then the segment [ π΄π΅ ] betweenthis relay and the ο¬xed relay to the crossing street of level π₯ . The second part of the path is symmetric andcorresponds to the connection between this relay and the destination through segment [ π΅πΆ ] and [ πΆ π· ] .Denote by πΏ ( π₯, π¦ ) the distance from an arbitrary position on a street of level π¦ to the ο¬rst ο¬xed relayto a street of level π₯ . The probability that a ο¬xed relay exists at a crossing of two streets of respective mH mVx a) x H V x mV y b) x Aβ Bβ Cβ Dβ E mH SA S B CD FG O O Figure 5: a) Diverted path with three ο¬xed relays (left), b) ο¬ve ο¬xed relays (right).level π₯ and π¦ is β exp (β π π π ( π₯, π¦ )) . Since the spacing between the streets of level π₯ is β π₯ , it is knownfrom [7] that πΏ ( π₯, π¦ ) β€ β π₯ β exp (β π π π ( π₯, π¦ )) where π π is the effective number of relays in the map (reminding that π π = π to simplify). The probabilitythat the two streets of level π₯ have a ο¬xed relay at their crossing is β exp (β π π π ( π₯, π₯ )) . With the condition π = π , one gets π π π ( π₯, π₯ ) = π β πΌ log ( / π )/ log ( /( β π π )) > π β πΌ since ( / π ) < log ( /( β π π )) . Thereforethe probability that the relay does not exist decays exponentially fast. Since the accumulated energy ofthe path, πΈ (T ) , satisο¬es with high probability πΈ (T ) = π ( πΏ ( π₯, ) π β πΏ π max ) + π ( ( ππ π₯ ) β πΏ π max ) and the average number of nodes of the path, π· (T ) , satisο¬es with probability tending to 1, exponentiallyfast: π· (T ) = π ( πΏ ( π₯, ) π ) + π ( ππ π₯ ) . Then, with the value π₯ = πΌ log π log ( /( β π π )) , we detect that the main contributor of the accumulated energy arethe segments [ π΄π΅ ] and [ π΅πΆ ] , namely πΈ (T ) = π (cid:16) π ( β πΏ )( β πΌ ) (cid:17) . and let π πΈ such that πΈ (T ) β€ π πΈ π ( β πΏ )( β πΌ ) . The main contributor in hop count in the path is in fact in the parts which stands on the central cross,namely [ π π» π΄ ] and [ π π πΆ ] : π· (T ) = π (cid:16) π β πΌ /( π πΉ β ) (cid:17) . (cid:3) In Theorem V.2, it is always assumed that πΈ π β , since πΌ < . In this case, π· π spans from π ( π β /( π πΉ β ) ) to π ( π ) (corresponding to a path staying on the central cross). When the fractal dimension π πΉ is large it does not make a large span. In fact, if πΈ π is assumed to be constant, i.e. πΌ = , then wecan have a substantial reduction in the number of hops, as described in the following theorem. Theorem V.3.
In a hyperfractal with π nodes, with nodes of fractal dimension π πΉ and relays of fractaldimension π π , the shortest path of accumulated energy πΈ π = π£ πΈ π max with π£ πΈ > , between two nodesbelonging to the giant component πΊ ( π, πΈ π ) , passes through a number of hops : π· π = π (cid:16) π β ππ ( + / ππΉ ) (cid:17) The theorem shows the achievable limits of number of hops when the constraint on the path energy islet loose. In fact, this allows taking the path with ο¬ve ο¬xed relays (Figure 5). The condition on π£ πΈ > comes from the 5 relays plus the step required to escape the giant component. Remark 5.
When π π β then π· π = π ( π /( π πΉ + ) ) , and the hyperfractal model is behaving like a hypercubeof dimension π πΉ + . Notice that in this case π· π tends to be π ( ) when π πΉ β β . Proof.
In the proof of Theorem V.2, it is assumed that π₯ < log π log ( /( β π π )) in order to ensure that the numberof hops on the route of level π₯ tends to inο¬nity. However, we can rise the parameter π₯ in the range log π log ( /( β π π )) β€ π₯ < log π ( / π ) .We have ππ π₯ β . In this case, πΈ (T ) β π max since the streets of level π₯ are empty of nodes withprobability tending to 1. Let us denote π₯ = π½ log π ( / π ) with π½ < . We have π· (T ) = π ( πΏ ( π₯, ) π ) = π ( π β π½ / π π ) . Clearly, π½ cannot be greater than 1 as, in this case, the two streets of level π₯ will not hold aο¬xed relay with high probability and the packet will not turn at the intersection. Therefore the smallestorder that one can obtain on the diverted path with three relays is limited to π β / π π , which is not theclaimed one.To obtain the claimed order, one must use the diverted path with ο¬ve ο¬xed relays, as shown in ο¬gure 5 b).The diverted path is composed by the segments: [ π π΄ (cid:48) ] , [ π΄ (cid:48) πΈ ] , [ πΈ πΉ ] , [ πΉπΊ ] , [ πΊπΆ (cid:48) ] and [ πΆ (cid:48) π· (cid:48) ] . It is shownin [7] that the order can be decreased to π β /(( + / π πΉ ) π π ) . (cid:3) C. Path maximum power
The next results revisit the previous theorems on the path accumulated energy in the correspondingcase of the imposed constraint on the path maximum power.
Theorem V.4.
The shortest path of maximum power less than π π = π β πΏ ( β πΌ ) π max with πΌ < , betweentwo nodes belonging to the giant component πΊ (cid:48) ( π, π π ) , passes through a number of hops: π· π = π (cid:16) π β πΌ /( π πΉ β ) (cid:17) It is important to note that although the orders of magnitude of path length π· π are the same inboth Theorem V.2 and Theorem V.4, the results consider two different giant components: (accumulated) πΊ ( π, πΈ π ) and (maximum) πΊ (cid:48) ( π, π π ) . Remark 6.
We have the identity (cid:18) π π π max (cid:19) / πΏ π· π πΉ β π = π ( π π πΉ β β πΏ ) . (16) Theorem V.5.
Let the maximum path transmitting power between two points belonging to the giantcomponent, πΊ (cid:48) ( π, π π ) be π π = π max . The number of hops π· π on the shortest path is π (cid:16) π β /( π π ( + / π πΉ ) (cid:17) . This theorem gives the path length when no constraint on transmitting power exists (the maximumtransmitting power allowed is the highest power for a transmission between two neighbors in thehyperfractal map). We obtain here the same results of [7], where an inο¬nite radio range is considered,which is not a feasible result for mmWave technology deployments.
D. Remarks on the network throughput capacity
Let us consider the scaling of the network throughput capacity with constraints on the energy. In [31],the authors express the throughput capacity of random wireless networks as: π ( π ) = Ξ (cid:18) π (cid:205) π β πΊ π π ( π ) (cid:205) π, π β πΊ π π π (cid:19) . (17)where π ( π ) is the throughput capacity, deο¬ned as the expected number of packets delivered to theirdestinations per slot, π π ( π ) is the expected transmission rate of each node π among all the nodes π and πΊ is the giant component. In the following, denote by πΆ the transmission rate of each node.Using our results of Theorems V.2 and V.4 and substituting them in (17), we obtain the followingcorollary on a lower bound of the network throughput capacity with constraints on path energy. Corollary 2.
In a hyperfractal with π nodes, fractal dimension of nodes π πΉ , and πΌ < and πΆ thetransmission rate of each node when either β’ πΈ π = π (cid:16) π ( β πΏ )( β πΌ ) π max (cid:17) is the maximum accumulated energy of the minimal path between any pairof nodes in the giant component πΊ ( π, πΈ π ) or β’ π π = π ( π β πΏ ( β πΌ ) π max ) is the maximum path power of the minimal path between any pair of nodesin the giant component πΊ (cid:48) ( π, π π ) ,a lower bound on the network throughput is: π ( π ) = Ξ© (cid:16) πΆπ πΌππΉ β (cid:17) (18) Remark 7.
We notice that with πΌ < and π πΉ > we have π ( π ) of order which can be smaller than π / which is less than the capacity in a random uniform network with omni-directional propagation asdescribed in [32]. Remark 8.
When πΌ = , i.e. with no energy constraint πΈ π = π πΈ π max the path length can drop down to π· π = π (cid:16) π β /(( + / π πΉ ) π π ) (cid:17) and, in this case, we have π ( π ) = Ξ© ( π /(( + / π πΉ ) π π ) ) which tends to be in π ( π ) when π πΉ β β and π π β . In this situation the capacity is of optimal order since π· π tends to be π ( ) . VI. F
ITTING THE HYPERFRACTAL MODEL FOR MOBILE NODES AND RELAYS TO DATA
The hyperfractal models for mobile nodes and for relays have been derived by making observations onthe scaling of trafο¬c densities and the scaling of the infrastructures, with road lengths, distances betweenintersections which allow rerouting of packets, etc . Let us emphasize again that in the deο¬nition of thehyperfractal model, there is neither an assumption nor a condition on geometric properties (in the senseof geometric shape, strait lines, intersection angles, etc ). Our description of a hyperfractal starting fromthe support X splitting the space into four quadrants is an example, (e.g., split by three to ο¬t a Kochsnowο¬ake).In our previous works [9], we have introduced a procedure which allows transforming trafο¬c ο¬owmaps into hyperfractal by computing the fractal dimension π πΉ of each trafο¬c ο¬ow map then quantify themetrics of interest. We shall revisit the theoretical foundation of this procedure in order to compute thefractal dimension π π of the relay placement since as observed from Section III, the placement of relaysis dependent on the density of mobile nodes. A. Theoretical Foundation: computation of the fractal dimension of the relays
In this work, we state that the relaying infrastructure placement also follows a distribution with parameterof a fractal dimension, π π . We thus now present a procedure for the computation of the fractal dimensionof the road-side infrastructure in a city.The criteria for computing the fractal dimension of the road-side infrastructure are similar to the criteriaused for computing the fractal dimension of the trafο¬c ο¬ow map. The ο¬tting procedure exploits the scalingbetween the length of different levels of the support X π and the scaling of the 1-dimensional intensity perlevel, π π . The difο¬culty is that the roads rarely have an explicit level hierarchy since the data we haveabout cities are in general about road segment lengths and average mobile nodes densities. To circumventthis problem, we do a ranking of the road segments in the decreasing order of their mobile density. If S is a segment we denote π (S) its density and πΆ π (S) the accumulated length of the segment ranked before S ( i.e. of larger density than π (S) ). For π > we denote π ( π ) = π ( πΆ β π ( π )) . Formally πΆ β π ( π ) is the roadsegment π with the smallest density such that πΆ π (S) β€ π . The hyperfractal dimension will appear in theasymptotic estimate of π ( π ) when π β β via the following property: π ( π ) = Ξ (cid:16) π β π πΉ (cid:17) . (19) (a) Trafο¬c lights data in Adelaide. -2 -1 (b) Computation of π π Figure 6: Data ο¬tting for AdelaideThe procedure for the computation of the fractal dimension of the relays is similar to the ο¬ttingprocedure for mobile nodes, [9] and has the following steps. First, we consider the set of road intersection I deο¬ned by the pair of segments (S , S ) such that S and S intersect. Let π , π be two real numberswe deο¬ne π ( π , π ) as the probability that two intersecting segments S and S such that πΆ π (S ) β€ π and πΆ π (S ) β€ π contains a relay. The hyperfractality of the distribution of the relay distribution implieswhen π , π β β : π ( π , π ) = Ξ (cid:16) ( π π ) β π π / (cid:17) . (20)Since the probability is not directly measurable we have to estimate it via measurable samples. Indeedlet π ( π , π ) be the number of intersections (S , S ) β I such that πΆ π (S ) β€ π and πΆ π (S ) β€ π and let π ( π , π ) be the number of relays between segments (S , S ) such that πΆ π (S ) β€ π and πΆ π (S ) β€ π .One should have: π ( π , π ) π ( π , π ) = Ξ (cid:16) ( π π ) β π π / (cid:17) (21)and from here get the fractal dimension of the relay process. B. Data Fitting Examples
Using public measurements [33], we show that the data validates the hyperfractal scaling of relaysrepartition with density and length of streets. While trafο¬c data is becoming accessible, the exact lengthof each street is difο¬cult to ο¬nd, therefore the ο¬tting has been done manually.Figure 6a shows a snapshot of the trafο¬c lights locations in a neighborhood of Adelaide, together withtrafο¬c densities on the streets, when available. As the roadside infrastructure for V2X communicationshas not been deployed yet or not at a city scale, we will use trafο¬c light data as an example for RSUlocation. It is acceptable to assume the RSUs will have similar placement, as, themselves, trafο¬c lightsare considered for the location of the RSUs on having radio devices mounted on top of them [34]. Byapplying the described ο¬tting procedure and using equation (21) the estimated fractal dimension of thetrafο¬c lights distribution in Adelaide is π π = . . In Figures 6b we show the ο¬tting of the data for thedensity repartition function.Note that it is the asymptotic behavior of the plots that are of interest (i.e., the increasing accumulateddistance with decreasing density therefore decreasing the probability of having a relay installed) since thescaling property comes from the roads with low density, thus the convergence towards the rightmost partof the plot is of interest.
15 20 25 30 35 40024681012 x 10 nr hops ene r g y Figure 7: Minimum accumulated end-to-end energy versus hops for a transmitter-receiver pair (ο¬xed andallowed number of hops in red circles, and maximum number of hops in black stars).VII. N
UMERICAL E VALUATION
We evaluate the accuracy of the theoretical ο¬ndings in different scenarios by comparing them to resultsobtained by simulating the events in a two-dimensional network. We developed a MatLab discrete timeevent-based simulator following the model presented in Section III. The length of the map is 1000 and,therefore, π max is just πΏ , where πΏ is the pathloss coefο¬cient that will be chosen to be , or , inline with millimeter-wave propagation characteristics. Figure 7 shows the trade-offs between accumulatedend-to-end energy and hop count for a transmitter-receiver pair by selecting randomly pairs of vehiclesin a hyperfractal map with π = , pathloss coefο¬cient πΏ = , fractal dimension of nodes π πΉ = . andfractal dimension of relays π π = . The plot shows the minimum accumulated energy for the end-to-endtransmission for a ο¬xed and allowed number of hops, π , in red circle markers. Note that the energy doesnot decrease monotonically as forcing to take a longer path may not allow to take the best path. Howeverwhen considering the minimum accumulated energy of all paths up to a number of hops , the black starmarkers in Figure 7, the energy decreases and exhibits the behavior claimed in Theorem V.2. That is, theminimum accumulated energy is indeed decreasing when the number of hops is allowed to grow (and theend-to-end communication is allowed to choose longer, yet cheaper, paths).Let us further validate Theorem V.2 through simulations performed for 100 randomly chosen transmitter-receiver pairs in hyperfractal maps with various conο¬gurations. We run simulations for different valuesof the number of nodes, π = nodes and nodes respectively, different values of pathloss, πΏ = and πΏ = and different conο¬gurations of the hyperfractal map. The setups of the hyperfractal maps are:node fractal dimension π πΉ = . and relay fractal dimension π π = . for the ο¬rst setup and π πΉ = . and π π = . for the second setup.The results exhibited in Figure 8 are obtained by computing, for each of the transmitter-receiver pair, theminimum accumulated end-to-end energy for a path smaller than π , then averaging over the 100 results.The left-hand sides of the Figures 8 (a) and 8 (b) show the variation of the minimum path accumulatedenergy for the path with the increase of the number of hops in a hyperfractal setup of π πΉ = . and π π = for π = in 8 (a) and π = in 8 (b). The ο¬gures illustrate that, indeed, allowing the hopcount to grow decreases the energy considerably. The decay of the maximum accumulated energy withthe allowed number of hops is even more visible in logarithmic scale in the right side of the same ο¬gures.The decays remain substantial when changing the hyperfractal setup to π πΉ = . , π π = . . Figures 8 (c)and 8 (d) show the results for π = and π = in the new setup. The decay is dramatic when lookingin logarithmic scale. Even though there can be oscillations around the linearly decreasing characteristic,as seen in Figure 8 (d), left-hand side, the global behavior stays the same, decreasing, as better noticedin logarithmic scale in Figure 8 (d), right-hand side.When changing the pathloss coefο¬cient to πΏ = , the effect of Theorem V.2 remains, as illustrated inFigure 9 for a hyperfractal setup of π πΉ = . , π π = , π = nodes.To validate the results of Theorem V.4 on the variation of path length with the imposed constraint onmaximum energy per node, we choose randomly 100 transmitter-receiver pairs belonging to the central
15 20 25 30 350.60.811.21.4x 10 k hops m i n i m u m ene r g y k hops m i n i m u m ene r g y (a) ( π πΉ , π π , π ) = . , . ,
22 24 26 28 30 32200250300350400 k hops m i n i m u m ene r g y k hops m i n i m u m ene r g y (b) ( π πΉ , π π , π ) = . , . , k hops m i n i m u m ene r g y k hops m i n i m u m ene r g y (c) ( π πΉ , π π , π ) = . , . ,
20 30 40 50 60 7012345x 10 k hops m i n i m u m ene r g y k hops m i n i m u m ene r g y (d) ( π πΉ , π π , π ) = . , . , Figure 8: Minimum accumulated end-to-end energy versus hops, averaging over 100 transmitter-receiverpairs, πΏ = , linear scale left side of sub-ο¬gures, logarithmic scale right side of sub-ο¬gures
10 15 20 25 30 35 40123456 x 10 k hops m i n i m u m ene r g y (a) Linear k hops m i n i m u m ene r g y (b) Logarithmic scale Figure 9: Minimum accumulated end-to-end energy versus hops, averaging over 100 transmitter-receiverpairs, πΏ = cross and compute the shortest path by applying a constraint on the maximum transmission energy ofnodes belonging to the path. The hyperfractal setups are: nodes fractal dimension π πΉ = . , relays fractaldimension π π = . , pathloss coefο¬cient πΏ = and we vary the number of nodes, π to be either π = or π = . For both values of π , Figure 10 (a) conο¬rms that decreasing the constraint of path maximumenergy increases the path length.Changing the fractal dimensions does not change the behavior, as illustrated in Figure 10 (b). Thehyperfractal conο¬gurations are: nodes fractal dimension π πΉ = . , relays fractal dimension π π = ,pathloss coefο¬cient πΏ = and we vary the number of nodes, π to be either π = or π = . Again,making a tougher constraint on the path maximum energy leads to the increase of the path length, showingthat achievable trade-offs in hyperfractal maps of nodes with RSU.VIII. C ONCLUSION
This paper presented results on the trade-offs between the end-to-end communication delay and energyspent on completing a transmission in millimeter-wave vehicular communications in urban settings by path length pa t h m a x i m u m ene r g y n=500n=800 (a) π πΉ = . , π π = . path length pa t h m a x i m u m ene r g y n=500n=800 (b) π πΉ = . , π π = Figure 10: Path Maximum Energy trade-off with delay (i.e. path length)exploiting the βhyperfractalβ model. This model captures self-similarity as an environment characteristic.The self-similar characteristic of the road-side infrastructure has also been incorporated.Analytical bounds have been derived for the end-to-end communication hop count under the constraintsof total accumulated energy, and maximum energy per node, exhibiting the achievable trade-offs ina hyperfractal network. The work presented a lower bound on the network throughput capacity withconstraints on path energy. Further examples of model ο¬tting with data have been given. The analyticalresults have been validated using a discrete-time event-based simulator developed in Matlab.A
PPENDIX AP ROOFS
A. Proof of Lemma 2Proof.
Let π π» ( π ) be the number of nodes contained in the street of level π» .Let π§ be a real number. By Chebyshevβs inequality, we have: E [ π π§π π» ( π ) ] = ( + ( π π§ β ) π π» ) π If π§ > : π (cid:18) π π» ( π ) < ππ π» (cid:19) = π (cid:16) π β π§π π» ( π ) > π π§ππ π» / (cid:17) β€ E [ π β π§π π» ( π ) ] π β π§ππ π» / Therefore E [ π β π§π π» ( π ) ] π β π§ππ π» / = exp ( π ( log ( + ( π β π§ β ) π π» ) + π§π π» / )) . For | π§ | bounded there exists π > such that | π π§ β | β€ π | π§ | and there exists π such that π π§ β β€ π§ + ππ§ .For | π₯ | bounded there exists π such that log ( + π₯ ) β€ π₯ β ππ₯ . From these steps we obtain that, forsufο¬ciently small | π§ | , one has: log ( + ( π β π§ β ) π π» ) + π§ π π» β€ β π§ π π» + ππ π» π§ β ππ π» π§ β€ β ππ π» . which settles that E [ π β π§π π» ( π ) ] π β π§ππ π» / β€ π β πππ π» . (22)The proof of the second part of the lemma proceeds via similar reasoning, by using the inequality: π ( π π» ( π ) > ππ π» ) β€ E [ π π§π π» ( π ) ] π π§ππ π» . (23) (cid:3) R EFERENCES [1] Z. Zhang, L. Wang, Z. Bai, K. S. Kwak, Y. Zhong, X. Ge, and T. Han, βThe analysis of coverage and capacity in mmwave vanet,β in , 2018, pp. 221β226.[2] 3GPP TS 38.211 V15.3.0 (2018-09), βPhysical channels and modulation (release 15),β in
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