Wireless Communication using Unmanned Aerial Vehicles (UAVs): Optimal Transport Theory for Hover Time Optimization
Mohammad Mozaffari, Walid Saad, Mehdi Bennis, Merouane Debbah
11 Wireless Communication using Unmanned AerialVehicles (UAVs): Optimal Transport Theory forHover Time Optimization
Mohammad Mozaffari , Walid Saad , Mehdi Bennis , and M´erouane Debbah Wireless@VT, Electrical and Computer Engineering Department, Virginia Tech, VA, USA,Emails: { mmozaff,walids } @vt.edu. CWC - Centre for Wireless Communications, Oulu, Finland, Email: [email protected].fi. Mathematical and Algorithmic Sciences Lab, Huawei France R & D, Paris, France, and CentraleSupelec,Universite Paris-Saclay, Gif-sur-Yvette, France, Email: [email protected].
Abstract
In this paper, the effective use of flight-time constrained unmanned aerial vehicles (UAVs) as flyingbase stations that can provide wireless service to ground users is investigated. In particular, a novelframework for optimizing the performance of such UAV-based wireless systems in terms of the averagenumber of bits (data service) transmitted to users as well as UAVs’ hover duration (i.e. flight time)is proposed. In the considered model, UAVs hover over a given geographical area to serve groundusers that are distributed within the area based on an arbitrary spatial distribution function. In thiscase, two practical scenarios are considered. In the first scenario, based on the maximum possiblehover times of UAVs, the average data service delivered to the users under a fair resource allocationscheme is maximized by finding the optimal cell partitions associated to the UAVs. Using the powerfulmathematical framework of optimal transport theory, this cell partitioning problem is proved to beequivalent to a convex optimization problem. Subsequently, a gradient-based algorithm is proposed foroptimally partitioning the geographical area based on the users’ distribution, hover times, and locationsof the UAVs. In the second scenario, given the load requirements of ground users, the minimum averagehover time that the UAVs need for completely servicing their ground users is derived. To this end, first,an optimal bandwidth allocation scheme for serving the users is proposed. Then, given this optimalbandwidth allocation, the optimal cell partitions associated with the UAVs are derived by exploiting theoptimal transport theory. Simulation results show that our proposed cell partitioning approach leads toa significantly higher fairness among the users compared to the classical weighted Voronoi diagram.Furthermore, the results demonstrate that the average hover time of the UAVs can be reduced by 64% byadopting the proposed optimal bandwidth allocation as well as the optimal cell partitioning approach.In addition, our results reveal an inherent tradeoff between the hover time of UAVs and bandwidthefficiency while serving the ground users. a r X i v : . [ c s . I T ] A p r I. I
NTRODUCTION
Recently, the use of aerial platforms such as unmanned aerial vehicles (UAVs), drones, bal-loons, and helikite has emerged as a promising solution for providing reliable and cost-effectivewireless communication services for ground wireless devices [1]–[4]. In particular, UAVs can bedeployed as flying base stations for coverage expansion and capacity enhancement of terrestrialcellular networks [2]–[10]. With their inherent attributes such as mobility, flexibility, and adaptivealtitude, UAVs have several key potential applications in wireless systems. For instance, UAVscan be deployed to complement existing cellular systems by providing additional capacity tohotspot areas during temporary events. Moreover, UAVs can also be used to provide networkcoverage in emergency and public safety situations during which the existing terrestrial networkis damaged or not fully operational. One key advantage of UAV-based wireless communication isits unique ability to provide fast, reliable and cost-effective connectivity to areas which are poorlycovered by terrestrial networks. In addition, compared to ground base stations, UAVs can moreeffectively establish line-of-sight (LoS) communication links to ground users by intelligentlyadjusting their altitude. Key examples of recent projects on employing aerial platforms forwireless connectivity include Google Loon project and Facebook’s Internet-delivery drone [11].Within the scope of these practical deployments, UAVs are being used to deliver Internet accessto emerging countries and provide airborne global Internet connectivity. Despite the severalbenefits and practical applications of using UAVs as aerial base stations, one must address manytechnical challenges such as performance analysis, deployment, air-to-ground channel modeling,user association, and flight time optimization [4] and [9].
A. Related Works
In [6], the authors performed air-to-ground channel modeling for UAV-based communicationsin various propagation environments. In [7] and [8], the authors studied the efficient deploymentof aerial base stations to maximize the coverage and rate performance of wireless networks.In [12], the authors investigated the energy-efficient path planning of a UAV-mounted cloudletwhich is used to provide offloading opportunities to ground devices. The work in [13] studied theoptimal trajectory and heading of UAVs for sum-rate maximization in uplink communications.An analytical framework for trajectory optimization of a fixed-wing UAV for energy-efficientcommunications was presented in [14]. The work in [15] jointly optimized user scheduling and
UAV trajectory for maximizing the minimum average rate among ground users. The authors in[16] derived an exact expression for downlink coverage probability for ground receivers whichare served by multiple UAVs.Another important challenge in UAV-based communications is user (or cell) association. Thework in [17] investigated the area-to-UAV assignment for capacity enhancement of heterogeneouswireless networks. However, this work is limited to the case with a uniform spatial distributionof ground users. Moreover, the work in [17] does not consider any fairness criteria that can beaffected by the network congestion in a non-uniform users’ distribution case. In addition, thiswork ignores the UAVs’ flight time constraints while determining the cell partitions associatedwith the UAVs. In [18] the optimal cell partitions associated to UAVs were determined withthe goal of minimizing the UAVs’ transmit power while satisfying the users’ rate requirements.However, in [18], the impact of flight time constraints on the performance of UAV-to-groundcommunications is not taken into account.Indeed, the flight time duration of the UAVs presents a unique design challenge for UAV-based communication systems [19] and [20]. For instance, the performance of such systemssignificantly depends on the hover time of each UAV, which is defined as the flight time duringwhich the UAV must stay in the air over a given area for providing wireless service to groundusers. In fact, with a higher hover time of the UAV, the users can receive wireless service fora longer period. Thus, by increasing the hover time, a UAV can meet higher load requirementsand serve a larger area. However, the hover time of a UAV is naturally limited due to thehighly constrained battery-provided, on-board energy, as well as flight regulations such as no-fly time/zone constraints [21]. Hence, while analyzing the UAV-based communication systems,the hover time constraints must be also taken into account. In this case, there is a need for aframework to analyze and optimize the performance of UAV-based communications based onthe hover time of UAVs. In fact, to our best knowledge, none of the previous UAV studies suchas [1]–[17], considered the hover time constraints in their analysis.
B. Contributions
The main contribution of this paper is to develop a novel framework for optimized UAV-to-ground communications under explicit UAVs’ hover time constraints. In particular, we considera network in which multiple UAVs are deployed as aerial base stations to provide wireless service to ground users that are distributed over a geographical area based on an arbitrary spatialdistribution. We investigate two key practical scenarios:
UAV communication under hover timeconstraints , and
UAV communication under load constraints . In the first scenario, given themaximum possible hover time of UAVs that is imposed by the limited on-board energy of UAVsand flight regulations, we maximize the average number of bits (data service) that is transmittedto the users under a fair resource allocation scheme. To this end, given the hover times and thespatial distribution of users, we find the optimal cell partitions associated to the UAVs. In thiscase, using the powerful mathematical framework of optimal transport theory [22], we proposea gradient-based algorithm that optimally partitions the geographical area based on the users’distribution as well as the UAVs’ hover times and locations. In the second scenario, given theload requirements of ground users, we minimize the average hover time needed for completelyserving the users. To this end, we introduce an optimal bandwidth allocation scheme as well asoptimal cell partitions for which the average hover time of UAVs is minimized. Our results forthe first scenario show that our proposed cell partitioning approach leads to a significantly higherfairness among the users compared to the classical weighted Voronoi approach. For the secondscenario, the results show that the average hover time can be reduced by 64% by adopting theproposed optimal bandwidth allocation and cell partitioning approach. Furthermore, our resultsreveal an inherent tradeoff between the hover time of UAVs and bandwidth efficiency whileservicing the ground users.The rest of this paper is organized as follows. In Section II, we present the system model.In Section III we investigate Scenario 1 in order to maximize data service. Section IV presentsScenario 2 for minimizing the hover time of UAVs. Simulation results are presented in SectionV and conclusions are drawn in Section VI.II. S
YSTEM M ODEL
Consider a geographical area
D ⊂ R within which a number of wireless users are locatedaccording to a given distribution f ( x, y ) in the two-dimensional plane. In this area, a set M of M UAVs are used as aerial base stations to provide wireless service for the ground users .Let s i = ( x i , y i , h i ) be the three-dimensional (3D) coordinate of each UAV i ∈ M with h i For wireless backhauling of aerial networks, satellite and WiFi are considered as the two feasible candidates [21]. being the altitude of UAV i . We consider a downlink scenario in which each UAV adopts afrequency division multiple access (FDMA) technique to provide service for the ground usersas done in [2] and [23]. Let P i and B i be, respectively, the maximum transmit power and thetotal available bandwidth for UAV i . Moreover, we use A i , as shown in Fig. 1, to denote thepartition of the geographical area which is served by UAV i . In this case, all users located incell partition A i will be connected to UAV i . Hence, the geographical area is divided into M disjoint partitions each of which is associated with one of the UAVs. Let τ i be the hover time ofUAV i , defined as the time duration that a UAV uses to hover (stop) over the corresponding cellpartition to service the ground users. During the hover time, the UAV must initiate connectionsto the ground users, perform required computations, and transmit data to the users. Let T i bethe effective data transmission period during which a UAV services the users. In general, theeffective data transmission time is less than the total hover time. Consequently, we considera control time as g i ( . ) , a function of the number of users in A i , to represent the portion ofthe hover time that is not used for the effective data transmission. This control time naturallycaptures the total time that a UAV i needs to spend for computations, setting up connections,and control signaling. Intuitively, the control time will increase when the number of users in thecorresponding cell partition increases.In our model, we use the term data service to represent the amount of data (in bits) that eachUAV transmits to a given user. Clearly, the data service depends on several factors such as theeffective data transmission time (which is directly related to flight time), and the transmissionbandwidth. Therefore, here, the effective data transmission times and bandwidth of the UAVsare considered as resources which are used for servicing the users. Given this model, to pro-vide service for the ground users using UAVs, we consider two scenarios. The first scenario, Scenario 1 , can be referred to as
UAV communications under hover time constraints . In thiscase, given the maximum possible hover times (imposed by the energy and flight limitations ofeach UAV), we maximize the average data service to the users under a fair resource allocationpolicy by optimal cell partitioning of the area. Here, we optimally partition the geographical areabased on the hover times and the spatial distribution of users. In Scenario 1, given the maximumpossible hover time of each UAV, the total data service under user fairness considerations ismaximized. In fact, Scenario 1 corresponds to resource-limited communication scenarios inwhich the amount of resources (e.g. hover times and bandwidth) is not sufficient to completely
UAV i ( , , ) i i i i s x y h h d R ( , ) v x y Cell partition associated to UAV i ( , ) f x y users distribution : hover time i Fig. 1: System model.meet the demands. An example of such scenario is when battery-limited UAVs are deployed inhotspots with high number of users and demands. The second scenario,
Scenario 2 , is referredto as
UAV communication under load constraints . In this case, our goal is to completely meetthe demands of ground users by properly adjusting the hover time of the UAVs. In particular,given the load requirement of each user at a given location, we minimize the average hovertime needed for completely serving the ground users. As a result, the load requirement of theground users will be satisfied with a minimum average hover time of the UAVs. In this case, byminimizing the hover time, one can minimize the energy consumption of the UAVs as well asthe time needed to completely serve the ground users. Such analyses in Scenario 2 are primarilyuseful is emergency situations in which all users need to be quickly served by the UAVs.
A. Air-to-ground path loss model
The air-to-ground signal propagation is affected by the obstacles and buildings in the envi-ronment. In this case, depending on the propagation environment, air-to-ground communicationlinks can be either LoS or non-line-of-sight (NLoS). In general, while designing a UAV-basedcommunication system, a complete information about the exact locations, heights, and the numberof obstacles may not be available [6]. In such case, one must consider the randomness associatedwith the LoS and NLoS links [8], [6]. Clearly, the probability of having LoS communicationlinks depends on the locations, heights, and the number of obstacles, as well as the elevationangle between a given UAV and it’s served ground user. In our model, we consider a widelyused probabilistic path loss model provided by International Telecommunication Union (ITU-R)[24], and the work in [6]. In this case, the path loss between UAV i and a given user at location ( x, y ) can be given by [8], and [6]: Λ i ( x, y ) = (cid:0) πf c d o c (cid:1) (cid:0) d i ( x, y ) /d o (cid:1) µ LoS , LoS link, (cid:0) πf c d o c (cid:1) (cid:0) d i ( x, y ) /d o (cid:1) µ NLoS , NLoS link, (1)where µ LoS and µ NLoS are different attenuation factors considered for LoS and NLoS links. Here, f c is the carrier frequency, c is the speed of light, and d o is the free-space reference distance. d i ( x, y ) = (cid:112) ( x − x i ) + ( y − y i ) + h i is the distance between UAV i and an arbitrary ground userlocated at ( x, y ) . For the UAV-user link, the LoS probability is given by [6]: P LoS ,i = b (cid:18) π θ i − (cid:19) b , (2)where θ i = sin − ( h i d i ( x,y ) ) is the elevation angle (in radians) between the UAV and the user. Also, b and b are constant values reflecting the environment impact. Note that, the NLoS probabilityis P NLoS ,i = 1 − P LoS ,i . Clearly, considering d o = 1 m, and K o = (cid:16) πf c c (cid:17) the average path loss is K o d i ( x, y )[ P LoS ,i µ LoS + P NLoS ,i µ NLoS ] . Hence, the received signal power from UAV i will be: ¯ P r,i ( x, y ) = P i K o d i ( x, y ) [ P LoS ,i µ LoS + P NLoS ,i µ NLoS ] , (3)where P i is the UAV’s transmit power. Then, the received SINR for a user located at ( x, y ) whileconnecting to UAV i will be: γ i ( x, y ) = ¯ P r,i ( x, y ) I i ( x, y ) + σ , (4)where I i ( x, y ) = β (cid:80) j (cid:54) = i ¯ P r,j ( x, y ) is the received interference at location ( x, y ) stemming fromall UAVs except UAV i . We also consider a weight factor ≤ β ≤ to adjust the amount ofinterference and capture the impact of any interference mitigation technique. Naturally, β = 1 and β = 0 , respectively, correspond to the full interference and interference-free scenarios.Clearly, the throughput of a user located at ( x, y ) if it connects to UAV i is: C i ( x, y ) = B ( x, y ) log (1 + γ i ( x, y )) , (5)where B ( x, y ) is the bandwidth allocated to the user at ( x, y ) .Subsequently, the total data service for the user provided by the UAV will be: L i ( x, y ) = T i C i ( x, y ) , (6)where T i is the effective transmission time of UAV i . Also, L i ( x, y ) represents the total numberof bits transmitted to the user located at ( x, y ) . Note that, the data service offered to each ground user depends on a number of key parameters such as the location of the user and the servingUAV, the bandwidth allocated to the user, and the effective data transmission time of the UAV, T i .Here, we consider the available bandwidth and effective data transmission times as the resources used by the UAVs to serviced the ground users. Clearly, the amount of resources that each usercan receive depends on several parameters such as the total number of users, cell partitions aswell as bandwidth and hover times of the UAVs. Given this model, we next analyze Scenario 1.III. S CENARIO
1: O
PTIMAL C ELL P ARTITIONING FOR D ATA S ERVICE M AXIMIZATION WITHFAIR RESOURCE ALLOCATION
In this section, our goal is to find the optimal cell partitions that maximize the average dataservice to the ground users based on the UAVs’ hover times and the spatial distribution of theground users. In this case, each cell partition is assigned to one UAV, and the users within the cellpartition must be serviced by the corresponding UAV. We note that, in classical cell partitioningapproaches such as Voronoi and weighted Voronoi diagrams [25], the spatial distribution ofusers is not taken into account. As a result, some partitions can be highly congested with usersand, hence, each user will receive significantly lower amount of resources than those in lesscongested partitions. Thus, such classical cell partitioning approaches can lead to a highly unfairdata service for the users. In our cell partitioning approach, however, while maximizing the totaldata service, we ensure that the resources are equally shared among all the users. Hence, ourapproach avoids creating unbalanced cell partitions and, thus, it leads to a higher level of fairnesscompared to the classical Voronoi approach. In the following, we present the details of our cellpartitioning approach based on the UAVs’ hover times and the spatial distribution of users.Let τ i be the hover time of UAV i during which it provides service for the users located in thecorresponding cell partition, A i . The hover time is composed of the effective data transmissiontime and the control time. To ensure a fair resource allocation policy, we consider the followingfairness criterion: T i = τ i − g i (cid:18)(cid:90) A i f ( x, y ) d x d y (cid:19) , ∀ i ∈ M , (7)where g i is the control time which depends on the number of the users located in A i . Note that, theaverage number of users within each cell partition A i is linearly proportional to (cid:82) A i f ( x, y ) d x d y .In other words, given the spatial distribution of users, f ( x, y ) , and the total number of users, N ,the average number of users in partition A i is equal to N (cid:82) A i f ( x, y ) d x d y [26]. From (5) and (6) which are used to compute the amount of data service, we can see that the value T i B i canbe considered as the resources that UAV i uses to service users in A i . In this case, under a fairresource allocation policy, we should have: T i B i (cid:82) A i f ( x, y ) d x d y = T j B j (cid:82) A j f ( x, y ) d x d y , ∀ i (cid:54) = j ∈ M , (8)where (8) ensures that a UAV with more resources (bandwidth and hover time) will serve ahigher number of users.Now, using (8) and considering the fact that (cid:82) D f ( x, y ) d x d y = M (cid:80) k =1 (cid:82) A k f ( x, y ) d x d y = 1 , wehave the following constraint on the number of users in each partition: (cid:90) A i f ( x, y ) d x d y = B i T iM (cid:80) k =1 B k T k , ∀ i ∈ M . (9)As we can see from (9), the number of users in each generated optimal partition will dependon the UAVs’ resources. Clearly, when the UAVs have the same hover times and bandwidths,(7)-(9) lead to (cid:82) A i f ( x, y ) d x d y = M , ∀ i ∈ M . This case implies that the identical UAVs willservice equally-loaded cell partitions.Given (5), (6), and (9), we can write the average data service at location ( x, y ) ∈ A i as: L i ( x, y ) = T i B i N (cid:82) A i f ( x, y ) d x d y log (1 + γ i ( x, y )) = (cid:32) N M (cid:88) k =1 B k T k (cid:33) log (1 + γ i ( x, y )) . (10)Now, we formulate an optimization problem for maximizing the average data service byoptimal partitioning of the target area. The data service maximization problem is given by: max A i , i ∈M M (cid:88) i =1 (cid:90) A i (cid:32) N M (cid:88) k =1 B k T k (cid:33) log (1 + γ i ( x, y )) f ( x, y ) d x d y, (11)s.t. (cid:90) A i f ( x, y ) d x d y = B i T iM (cid:80) k =1 B k T k , ∀ i ∈ M , (12) γ i ( x, y ) ≥ γ th , if ( x, y ) ∈ A i , ∀ i ∈ M , (13) A l ∩ A m = ∅ , ∀ l (cid:54) = m ∈ M , (14) Note that, given hover times of the UAVs, τ i , ∀ i ∈ M , we can compute T i , ∀ i ∈ M by solving the system of equations in(7) and (8). (cid:91) i ∈M A i = D , (15)where (12) captures the constraint on the load of each cell partition. Also, (13) is the necessarycondition for connecting each user to a UAV i . (14) and (15) ensure that the cell partitions aredisjoint and their union covers the entire target area D .Considering the constraint in (13), we define the function q i ( x, y ) = (cid:16) γ i ( x,y ) γ th (cid:17) n with n beinga large number (i.e. tends to + ∞ ), and, then, we substract q i ( x, y ) from the objective functionin (11). Clearly, when (13) is violated, q i ( x, y ) tends to + ∞ and, hence, point ( x, y ) will not beassigned to UAV i or equivalently ( x, y ) / ∈ A i . Therefore, whenever the problem is feasible, wecan remove (13) while penalizing the objective function in (11) by q i ( x, y ) . Now, by defining λ = N M (cid:80) k =1 B k T k , and ω i = B i T iM (cid:80) k =1 B k T k , the maximization problem in (11) can be rewritten as thefollowing minimization problem: min A i , i ∈M M (cid:88) i =1 (cid:90) A i − ( λ log (1 + γ i ( x, y )) − q i ( x, y )) f ( x, y ) d x d y, (16)s.t. (cid:90) A i f ( x, y ) d x d y = ω i , ∀ i ∈ M , (17) A l ∩ A m = ∅ , ∀ l (cid:54) = m ∈ M , (18) (cid:91) i ∈M A i = D . (19)Solving the optimization problem in (16) is challenging due to various reasons. First, theoptimization variables A i , ∀ i ∈ M , are sets of continuous partitions (as we have a continuousarea) which are mutually dependent. Second, to perfectly capture the spatial distribution of users, f ( x, y ) is considered to be a generic function of x and y and, this leads to the complexity of thegiven two-fold integrations. In addition, due to the constraints given in (17), finding A i becomesmore challenging. To solve the optimization problem in (16), next, we model the problem byexploiting optimal transport theory [22]. A. Optimal Transport Theory: Preliminaries
Here, we present some primary results from optimal transport theory which will be used in thenext subsection to derive the optimal cell partitions. Optimal transport theory goes back to theMonge’s problem in 1781 which is stated as follows [22]. Given piles of sands and holes with the Fig. 2: Transport map between two probability distributions.same volume, what is the best move (transport map) to entirely fill up the holes with the minimumtotal transportation cost. In general, this theory aims to find the optimal matching between twosets of points that minimizes the costs associated with the matching between the sets. These setscan be either discrete or continuous, with arbitrary distributions (weights). Mathematically, theMonge optimal transport problem can be written as follows. Given two probability distributions f on X ⊂ R n , and f on Y ⊂ R n , find the optimal transport map T from f to f that minimizesthe following problem: min T (cid:90) X c ( x , T ( x )) f ( x ) d x ; T : X → Y , (20)where c ( x , T ( x )) denotes the cost of transporting a unit mass from a location coordinate x ∈ X to a location y = T ( x ) ∈ Y . Also, as shown in Fig. 2, f and f are the source and destinationprobability distributions.Solving the Monge’s problem is challenging due its high non-linear structure [22], and thefact that it does not necessarily admit a solution as each point of the source distribution must bemapped to only one location at the destination. However, Kantorovich relaxed this problem byusing transport plans instead of maps, in which one point can go to multiple destination points.The relaxed Monge’s problem is called Monge-Kantorovich problem which is written as [22]: min π (cid:90) X × Y c ( x , y ) d π ( x , y ) , (21)s.t. (cid:90) X d π ( x , y ) = f ( x ) d x , (cid:90) Y d π ( x , y ) = f ( y ) d y , (22)where π represents the transport plan which is the probability distribution on X × Y whosemarginals are f and f .The Monge-Kantorovich problem has two main advantages compared to the Monge’s prob-lem. First, it admits a solution for any semi-continuous cost function. Second, there is a dualformulation for the Monge-Kantorovich problem that can lead to a tractable solution. The dualitytheorem is stated as [22] and [27]: Kantrovich Duality Theorem : Given the Monge-Kantorovich problem in (21) with twoprobability measures f on X ⊂ R n , and f on Y ⊂ R n , and any lower semi-continuous cost function c ( x , y ) , the following equality holds: min π (cid:90) X × Y c ( x , y ) d π ( x , y ) (23) = max ϕ,ψ (cid:26)(cid:90) X ϕ ( x ) f ( x ) d x + (cid:90) Y ψ ( y ) f ( y ) d y ; ϕ ( x ) + ψ ( y ) ≤ c ( x , y ) , ∀ ( x , y ) ∈ X × Y (cid:27) , (24)where ϕ ( x ) and ψ ( y ) are Kantorovich potential functions. As discussed in [22], this dualitytheorem provides a tractable framework for solving the optimal transport problems. In particular,we will use this theorem to tackle our optimization problem in (16).We note that, in general, the solutions for the Monge-Kantorovich problem do not coincidewith the Monge’s problem. Nevertheless, when the source distribution, f , and the cost functionare continuous, these two problems are equivalent [28]. In addition, the optimal transport map, T : x → y , is linked with the optimal Kantorovich potential functions by: T ( x ) = { y | ϕ ∗ ( x ) + ψ ∗ ( y ) = c ( x , y ) } , (25)where ϕ ∗ ( x ) and ψ ∗ ( y ) are the optimal potential functions corresponding dual formulation ofthe Monge-Kantorovich problem.Given this optimal transport framework, we can solve our optimization problem in (16). In par-ticular, we model this problem as a semi-discrete optimal transport problem in which the sourcemeasure (users’ distribution) is continuous while the destination (UAVs’ distribution) is discrete. B. Optimal Cell Partitioning
Using optimal transport theory, we can find the optimal cell partitions, A i , for which theaverage total data service is maximized. In our model, users have a continuous distribution, andthe locations of the UAVs can be considered as discrete points. Then, the optimal cell partitionsare obtained by optimally mapping the users to the UAVs. In fact, given (16), the cell partitionsare related to the transport map by [29]: (cid:40) T ( v ) = (cid:88) i ∈M s i A i ( v ); (cid:90) A i f ( x, y ) d x d y = ω i (cid:41) , (26)where ω i = B i T iM (cid:80) k =1 B k T k , as given in (17), is directly related to the hover time and the bandwidth ofthe UAVs. Also, A i ( v ) is the indicator function which is equal to 1 if v ∈ A i , and 0 otherwise. Therefore, the optimization problem in (16) can be cast within the optimal transport frameworkas follows. Given a continuous probability measure f of users, and a discrete probability measure Γ = (cid:80) i ∈M ω i δ s i corresponding to the UAVs, we must find the optimal transport map for which (cid:82) D J ( v , T ( v )) f ( x, y ) d x d y is minimized. In this case, δ s i is the Dirac function, and J is thetransportation cost function which is used in (16) and is given by: J ( v , s i ) = J ( x, y, s i ) = q i ( x, y ) − λ log (1 + γ i ( x, y )) . (27)Clearly, the cost function, J , and the source distribution, f , are continuous. As a result, theMonge’s problem coincides with the Monge-Kantorovich problem. Next, we propose a solutionto (16) by exploiting the dual formulation of the Monge-Kantorovich problem. Theorem 1.
The optimization problem in (16) is equivalent to the following unconstrainedmaximization problem: max ψ i ,i ∈ M (cid:40) F ( ψ T ) = M (cid:88) i =1 ψ i ω i + (cid:90) D ψ c ( x, y ) f ( x, y ) d x d y (cid:41) , (28)where ψ T is a vector of variables ψ i , ∀ i ∈ M , and ψ c ( x, y ) = inf i J ( x, y, s i ) − ψ i . Proof:
We use the Kantorovich duality theorem (23) in which f ( x, y ) and Γ = (cid:80) i ∈M ω i δ s i are two probability measures, and J ( v , s ) is the cost function. Clearly, due to the continuity of f ( x, y ) and J ( v , s i ) , the Monge’s problem is equivalent to the Monge-Kantorovich problem. min T (cid:90) D J ( v , T ( v )) f ( x, y ) d x d y (29) = max ϕ,ψ (cid:40)(cid:90) D ϕ ( v ) f ( x, y ) d x d y + (cid:90) S ψ ( s ) (cid:88) i ∈M ω i δ s − s i d s ; ϕ ( v ) + ψ ( s ) ≤ J ( v , s ) (cid:41) (30) = max ϕ,ψ (cid:40)(cid:90) D ϕ ( x, y ) f ( x, y ) d x d y + M (cid:88) i =1 ψ ( s i ) ω i ; ϕ ( x, y ) + ψ ( s i ) ≤ J ( x, y, s i ) , ∀ i ∈ M (cid:41) . (31)Note that, to maximize (31) given any ψ , we need to choose a maximum value for ϕ . Consideringthe fact that ϕ ( x, y ) + ψ ( s i ) ≤ J ( x, y, s i ) must be satisfied for all ( x, y ) ∈ D and s i ∈ S , themaximum allowable value of ϕ is given by: ϕ ( x, y ) = ψ c ( x, y ) = inf s i J ( x, y, s i ) − ψ ( s i ) , (32)where ψ c is called the c-transform of ψ . Now, considering ψ i = ψ ( s i ) , (31) and (32) lead to: min T (cid:90) D J ( v , T ( v )) f ( x, y ) d x d y = max ψ i ,i ∈ M (cid:40) F ( ψ T ) = M (cid:88) i =1 ψ i ω i + (cid:90) D ψ c ( x, y ) f ( x, y ) d x d y (cid:41) , (33) ψ c ( x, y ) = inf i J ( x, y, s i ) − ψ i . (34)As a result, the optimization problem in (16) is reduced to (28) with a set of M optimizationvariables, ψ i , ∀ i ∈ M . This proves the theorem.Theorem 1 shows that the complex optimal cell partitioning problem in (16) can be transformedto a tractable optimization problem with M variables. In other words, by solving (28), one canuse the optimal values of ψ i , ∀ i ∈ M to find the optimal cell partitions. Using Theorem 1, wecan further proceed to solve (16) by presenting the following theorem: Theorem 2.
Given (28), F is a concave function of variables ψ i , i ∈ M . Also, we have: ∂F∂ψ i = ω i − (cid:90) D i f ( x, y ) d x d y, (35)where D i = { ( x, y ) | J ( x, y, s i ) − ψ i ≤ J ( x, y, s j ) − ψ j , ∀ j (cid:54) = i } . Proof:
Clearly, M (cid:80) i =1 ψ i ω i is a linear function of ψ i . Also, given any i ∈ M , J ( x, y, s i ) − ψ i isa linear function of ψ i . Let z ( ψ T ) = inf i J ( x, y, s i ) − ψ i with ψ T being a vector of all variables ψ i , i ∈ M . Then, we can observe that the hypograph of z ( ψ T ) , a set of points below z ( ψ T ) ,is a convex set. Subsequently, considering the fact that a function is concave if and only if itshypograph is convex, we prove the concavity of z ( ψ T ) . Finally, since multiplying z ( ψ T ) by apositive probability density function f ( x, y ) , and taking integration over ( x, y ) does not violatethe concavity, F is also a concave function of ψ T .To find the derivative of F with respect to ψ i , we first compute ∂ψ c ∂ψ i . Clearly, based on (34),we have: ∂ψ c ∂ψ i = − , if J ( x, y, s i ) − ψ i ≤ J ( x, y, s j ) − ψ j , ∀ j (cid:54) = i, , otherwise . (36)Then, by defining D i = { ( x, y ) | J ( x, y, s i ) − ψ i ≤ J ( x, y, s j ) − ψ j , ∀ j (cid:54) = i } , the derivative of F ,given in (28), will be: ∂F∂ψ i = ω i − (cid:90) D i f ( x, y ) d x d y . (37)This proves the theorem.Theorem 2 shows the concavity of F as a function of ψ T . Thus, the optimal values forvariables ψ i , ∀ i ∈ M , can be obtained by maximizing F . Then, given the optimal ψ i , ∀ i ∈ M ,equations (25), and (26) are used to determine the optimal cell partitions corresponding to the Algorithm 1
Gradient method for optimal cell partitioning Inputs: f ( x, y ) , ρ , τ i , s i , ∀ i ∈ M . Outputs: ψ ∗ i , A i , ∀ i ∈ M . Set initial values for ψ Tt , ( t = 1 ) . while (cid:13)(cid:13)(cid:13) ∇ F ( ψ Tt ) (cid:13)(cid:13)(cid:13) > ρ do Set k = 1 , (cid:15) = 1 . Update ψ Tt +1 = ψ Tt + ε k ∇ F ( ψ Tt ) , k, t ∈ N . if F ( ψ Tt ) < F ( ψ Tt +1 ) then Go to Step 12 . else Go to Step 17 . end if while F ( ψ Tt ) < F ( ψ Tt +1 ) do k → k + 1 . (cid:15) k = 2 k − (cid:15) . Update ψ Tt +1 = ψ Tt + ε k ∇ F ( ψ Tt ) , k, t ∈ N . end while while F ( ψ Tt ) > F ( ψ Tt +1 ) do k → k + 1 . (cid:15) k = 2 − k +1 (cid:15) . Update ψ Tt +1 = ψ Tt + ε k ∇ F ( ψ Tt ) , k, t ∈ N . end while t → t + 1 . end while ψ ∗ i = ψ Tt ( i ) , ∀ i ∈ M . A i = (cid:8) ( x, y ) | J ( x, y, s i ) − ψ ∗ i ≤ J ( x, y, s j ) − ψ ∗ j , ∀ j (cid:54) = i (cid:9) , ∀ i ∈ M . optimization problem in (16). In this case, using the first derivative of F provided in (35), wepropose a gradient-based method to determine the optimal vector ψ T that leads to the optimalcell partitions. Here, using the gradient descent method is simple in terms of implementation anddoes not require computing the Hessian matrix of F which is needed in the Newton methods.In fact, given the intractable expression of F in (28), finding its second derivative is challengingand, thus, we adopt the gradient-based approach.The proposed algorithm for finding the optimal cell partitions is shown as Algorithm 1 andproceeds as follows. The inputs are the distribution of users, hover times, locations of the UAVs, and ρ > which is the threshold based on which the algorithm stops. In Algorithm 1, we firstinitialize vector ψ Tt with t being the iteration number. Next, using (35), we compute ∇ F ( ψ Tt ) . InStep 6, we update ψ Tt using step size (cid:15) k . The appropriate step size at each iteration is determinedthrough Steps 7 to 21. In this case, the algorithm stops when the condition in is not satisfied.Clearly, due to the concavity of F , the optimal solution to (28) is attained. Finally, based on theoptimal vector ψ T , the optimal cell partitions are determined using Steps 24 and 25.In summary, we proposed a framework for maximizing the average data service to ground userswhile considering some level fairness between the users. To this end, we used tools from optimaltransport theory to determine the optimal cell partition associated to each UAV that services theusers in the cell partition. In the next section, we investigate Scenario 2 in which the minimumaverage hover times of UAVs needed for completely servicing the users are determined.IV. S CENARIO
2: M
INIMUM H OVER T IME F OR MEETING L OAD R EQUIREMENTS
Our goal is to meet the load requirements of the ground users while minimizing the averagehover times of the UAVs. In particular, given the demand of each user, we find the minimumrequired average hover time of the UAVs to completely serve the users. The hover time of eachUAV depends on the distribution and load of the users, bandwidth allocation between users, andthe cell partition that is assigned to the UAV. Next, we first derive an expression for the averagehover time needed to serve any arbitrary cell partition under an optimal bandwidth allocationbetween the users. Then, we determine the optimal cell partitions for which the average hovertimes required for completely servicing the entire target area is minimized. We note that, given aspecified partition A i , the hover time of the UAV i needed for serving the users depends on thebandwidth allocation strategy. Hence, we first derive the minimum average hover time of eachUAV that can be attained by optimal bandwidth allocation to the users in the given cell partition. Proposition 1.
Let u ( x, y ) be the load (in bits) of a user located at ( x, y ) . The minimum averagehover time of UAV i for serving partition A i that can be achieved by optimally allocating thebandwidth to the users, is given by: τ i = (cid:90) A i N u ( x, y ) C B i i ( x, y ) f ( x, y ) d x d y + g i (cid:18)(cid:90) A i f ( x, y ) d x d y (cid:19) , (38)where N is the total number of users, C B i i = B i log (1 + γ i ( x, y )) , and g i ( . ) is the additionalcontrol time which is a function of the number of users in the cell partition. Proof:
Let Y be an arbitrary set of Y users in a cell partition A i . We denote the load andbandwidth allocated to user r by u r and W r . Then, the time needed to serve user r is given by: t r = u r W r E r , (39)where E r is the spectral efficiency (bit/s/Hz) at the user’s location. Clearly, to completely serveall the users in A i , the hover time of UAV i must be max t r + g Y , with g Y being the additionalcontrol time. In this case, the hover time can be minimized by an optimal bandwidth allocationas follows: min max W r , r =1 ,...,Y t r + g Y , (40)s.t. Y (cid:88) r =1 W r = B i . (41)The minmax problem in (40) can be transformed to: min Z + g Y , (42)s.t. Z ≥ u r W r E r , ∀ r ∈ Y , (43) Y (cid:88) r =1 W r = B i . (44)Now, using (43) and (44), we have Z ≥ B i Y (cid:80) r =1 u r E r . Hence, the minimum hover time under anoptimal bandwidth allocation will be Y (cid:80) r =1 u r B i E r + g Y . Furthermore, it can be shown that W r = B i u r E r / Y (cid:80) k =1 u k E k is an optimal bandwidth allocation to user r . Clearly, Y (cid:80) r =1 u r B i E r is also equal to thetotal time needed for sequentially serving the users using the entire bandwidth B i . Therefore,given a cell partition A i and the users’ distribution, f ( x, y ) , the minimum average hover timeof UAV i can be given by: τ i = (cid:90) A i N u ( x, y ) B i log (1 + γ i ( x, y )) f ( x, y ) d x d y + g i (cid:18)(cid:90) A i f ( x, y ) d x d y (cid:19) . (45)Finally, considering C B i i = B i log (1 + γ i ( x, y )) , this proposition is proved.From (38), we can see that the hover time increases as the number of users increases. In fact,for a higher number users, both the total data transmission time and the additional control timeincrease. Moreover, the hover time increases as the load of the users increases. In particular,for an equal load of users, the hover time increases sublinearly by increasing the load as thecontrol time does not depend on the load here. From (38), we can see that the hover time can be reduced by increasing the transmission rate. In addition, (38) implies that the UAV must hoverfor a longer time over a partition with higher users’ density (i.e. f ( x, y ) ).Next, we minimize the total average hover time by solving the following optimization problem: min A i , i ∈M M (cid:88) i =1 (cid:90) A i N u ( x, y ) C B i i ( x, y ) f ( x, y ) d x d y + g i (cid:18)(cid:90) A i f ( x, y ) d x d y (cid:19) , (46)s.t. γ i ( x, y ) ≥ γ th , if ( x, y ) ∈ A i , ∀ i ∈ M , (47) A l ∩ A m = ∅ , ∀ l (cid:54) = m ∈ M , (48) (cid:91) i ∈M A i = D , (49)where the objective function in (46) represents the total average hover time needed for providingservice for the target area. Following an approach that is similar to the one we used in (11), ouroptimization problem in (46) can be rewritten as: min A i , i ∈M M (cid:88) i =1 (cid:90) A i (cid:20) N u ( x, y ) C B i i ( x, y ) + q i ( x, y ) (cid:21) f ( x, y ) d x d y + g i (cid:18)(cid:90) A i f ( x, y ) d x d y (cid:19) , (50)s.t. A l ∩ A m = ∅ , ∀ l (cid:54) = m ∈ M , (51) (cid:91) i ∈M A i = D , (52)where q i ( x, y ) = (cid:16) γ i ( x,y ) γ th (cid:17) n with n → + ∞ .Solving (50) is challenging as the optimization variables A i , ∀ i ∈ M , are sets of continuouspartitions which are mutually dependent. Moreover, since g i is a generic function of A i and f ( x, y ) , this problem is intractable. Next, we use optimal transport theory to completely charac-terize the optimal solution. To this end, we first prove the existence of the solution to (50) forany semi-continuous function g i , ∀ i ∈ M . Note that, in general, (50) does not necessary admitan optimal solution when the semi-continuity of g i does not hold. Proposition 2.
The optimization problem in (50) generally admits an optimal solution.
Proof:
Let a i = (cid:82) A i f ( x, y ) d x d y , then we also define a unit simplex as follows: E = (cid:40) a = ( a , a , ..., a M ) ∈ R M ; M (cid:88) k =1 a i = 1 , a i ≥ , ∀ i ∈ M (cid:41) . (53)Clearly, given any vector a , problem (50) can be considered as a classical semi-discrete optimaltransport problem. In particular, considering f ( v ) = f ( x, y ) , and c ( v , s i ) = u ( x,y ) C Bii ( x,y ) + q i ( x, y ) , (50) can be transformed to: min T (cid:90) D c ( v , s ) f ( v ) d v , s = T ( v ) , (54)where T is the transport map which is associated to cell partitions A i by : (cid:40) T ( v ) = M (cid:88) i =1 s i A i ( v ); (cid:90) A i f ( v ) d v = a i (cid:41) . (55)Clearly, as discussed in Section III, the optimal transport problem in (54) admits a solution.Hence, for any a ∈ E , the problem in (50) has an optimal solution. Since E is a unit simplexin R M which is a non-empty and compact set, the problem admits an optimal solution over theentire E . Thus, the proposition is proved.Next, we completely characterize the solution space of problem (50) which allows finding theoptimal cell partitioning and the average hover time of each UAV. Theorem 3.
The optimal hover time of UAV i required to completely service the target area isgiven by: τ ∗ i = (cid:90) A ∗ i N u ( x, y ) C B i i ( x, y ) f ( x, y ) d x d y + g i (cid:32)(cid:90) A ∗ i f ( x, y ) d x d y (cid:33) , (56)where A ∗ i is the optimal cell partition given by: A ∗ i = (cid:40) ( x, y ) | N u ( x, y ) C B i i ( x, y ) + q i ( x, y ) + g (cid:48) i ( a i ) ≤ N u ( x, y ) C B j j ( x, y ) + q j ( x, y ) + g (cid:48) j ( a j ) , ∀ j (cid:54) = i ∈ M (cid:41) , (57)where a i = (cid:82) A i f ( x, y ) d x d y , and N is the total number of users. Proof:
See Appendix A.Using Theorem 3, we can find the optimal cell partitions as well as the minimum hover timeneeded to completely service the users. In fact, the target area is optimally partitioned in a waythat the average hover time that the UAVs use to serve their users is minimized. Note that, forthe special case where g (cid:48) i = 0 , the result in (64) corresponds to the classical weighted Voronoidiagram. In this case, the users are assigned to the UAVs based on the maximum received signalstrength criterion. Consequently, the users can be served with a maximum rate and, hence,the total required hover time is minimized. However, in general, the classical weighted Voronoidiagram is not optimal [26] as the effect of control time is ignored while generating cell partitions.From (57), we can see that there is a mutual dependency between a i and A i , ∀ i ∈ M .Therefore, solving (57) does not have an explicit form and, hence, an iterative-based approachis needed to find a solution to (57). Next, given the results of Theorem 3, we present an iterative Algorithm 2
Iterative algorithm for optimal cell partitions and hover times Inputs: f ( x, y ) , u ( x, y ) , Z , g i , s i , ∀ i ∈ M . Outputs: A ∗ i , τ ∗ i , ∀ i ∈ M . Set t = 1 , generate an initial cell partitions A ( t ) i , and set φ ( t ) i ( x, y ) = 0 , ∀ i ∈ M . while t < Z do Compute φ ( t +1) i ( x, y ) = (1 − /t ) φ ( t ) i ( x, y ) , if ( x, y ) ∈ A i ( t ) , − (1 − /t ) (cid:16) − φ ( t ) i ( x, y ) (cid:17) , otherwise . Compute a i = (cid:82) D (cid:16) − φ ( t +1) i ( x, y ) (cid:17) f ( x, y ) d x d y , ∀ i ∈ M . t → t + 1 . Update cell partitions using (64). end while A ∗ i = A ( t ) i , Compute τ ∗ i using (38) based on A ∗ i , ∀ i ∈ M . algorithm based on [29] and shown in Algorithm 2, that solves (57) and finds the optimal cellpartitions and the average hover time of each UAV. This algorithm guarantees the convergenceto the optimal solution within a reasonable number of iterations [29]. In addition, this algorithmis practical to implement as its complexity grows linearly with the size of the area D .Algorithm 2 for finding the optimal cell partitions as well as the average hover times proceedsas follows. The inputs are load and distribution of the users, locations of the UAVs, control timefunction, and the number of iterations, L . Here, we use t to represent the iteration number. First,we generate initial cell partitions A ( t ) i , and set φ ( t ) i ( x, y ) = 0 , ∀ i ∈ M , with φ ( t ) i ( x, y ) being apre-defined parameter that will be used to update the cell partitions. Next, we update φ ( t +1) i ( x, y ) ,and compute a i in step 6. Then, in step 8, we update cell partitions by using (64). Finally, atthe end of the iteration, the optimal cell partitions and the minimum average hover time of theUAVs are determined. V. S IMULATION R ESULTS AND A NALYSIS
For our simulations, we consider a rectangular area of size m × m in which theground users are distributed according to a two-dimensional truncated Gaussian distributionwhich is suitable to model a hotspot area and is given by [30]: f ( x, y ) = 1 η exp (cid:16) L x − µ x √ σ x (cid:17) exp (cid:16) L y − µ y (cid:112) σ y (cid:17) , (58) Table I:
Simulation parameters.
Parameter Description Value f c Carrier frequency 2 GHz P i UAV transmit power 0.5 W N o Noise power spectral -170 dBm/Hz N Number of ground users 300 µ LoS
Additional path loss to free space for LoS 3 dB µ NLoS
Additional path loss to free space for NLoS 23 dB B Bandwidth 1 MHz α Control time factor 0.01 h UAV’s altitude 200 m u Load per user 100 Mb µ x , µ y Mean of the truncated Gaussian distribution 250 m, 330 m b , b Environmental parameters (dense urban) 0.36, 0.21 [6] where η = 2 πσ x σ y erf (cid:16) L x − µ x √ σ x (cid:17) erf (cid:16) L y − µ y √ σ y (cid:17) , and the size of the area is L x × L y . Also, µ x , σ x , µ y , and σ y are the mean and standard deviation values of the x and y coordinates, and erf( z ) = √ π z (cid:82) e − t d t . In this case, ( µ x , µ y ) represents the center of the hotspot, and the densityof the users around the center is inversely proportional to the values σ x and σ y . In our simulations,we consider σ x = σ y = σ o . Note that, although we consider the truncated Gaussian distributionof users, our analysis can also accommodate any any other arbitrary distribution. Moreover, wedeploy the UAVs based on a grid-based deployment with an altitude of 200 m. Unless statedotherwise, we consider a full interference scenario with an interference factor β = 1 . For thecontrol time function, we consider g i ( N a i ) = α ( N a i ) , with α being an arbitrary constant factor.This function is a reasonable choice in our model as it is a superlinear function of the numberof users and its value can be adjusted by factor α . However, any arbitrary continuous controlfunction can also be considered in our model. The simulation parameters are listed in Table I.We compare our results, obtained based on the proposed optimal cell partitioning approach, withthe classical weighted Voronoi diagram baseline. Note that, all statistical results are averagedover a large number of independent runs. Next, we present the results corresponding to Scenario1 and Scenario 2, separately. A. Results for Scenario 1
Fig. 3 shows the proposed optimal cell partitions and the classical weighted Voronoi diagram.In this case, we consider 5 UAVs that provide service for the non-uniformly distributed groundusers (truncated Gaussian distribution with σ o = 1000 m). Moreover, in Scenario 1, we assume X−coordinate (m) Y − c oo r d i na t e ( m )
200 400 600 800 10002004006008001000 (a) Proposed optimal cell partitions.
X−coordinate (m) Y − c oo r d i na t e ( m )
200 400 600 800 10002004006008001000 1234567x 10 −5 UAV 1UAV 2UAV 3UAV 4UAV 5 (b) Weighted Voronoi diagram.
Fig. 3: Cell partitions associated to UAVs given the non-uniform spatial distribution of users.
200 400 600 800 100000.20.40.60.81 σ o (m) J a i n ’ s i nde x Proposed cell partitioningWeighted Voronoi
Fig. 4: Jain’s fairness index for average data service to users.that the maximum hover time of each UAV is 30 minutes which corresponds to the typicalhover time for quadcopter UAVs [31]. In Fig. 3, areas shown by a darker color have a higherpopulation density. As we can see from Fig. 3b, the cell partitions associated with UAVs 4 and 5have significantly more users than cell partition 1. Therefore, given the limited hover times, userslocated at cell partitions 4 and 5 cannot be fairly served by UAVs. However, in the proposedoptimal cell partitioning case (obtained by Algorithm 1), the cell partitions change such that theaverage data service under a fair resource allocation constraint is maximized. For instance, asshown in Fig. 3a, the size of cell partitions 4 and 5 decreases compared to the weighted Voronoidiagram. As a result, the proposed cell partitions lead to a higher level of fairness among theusers than the weighted Voronoi case.To show how fairly the users can be served in different cases of cell partitioning, we use the Jain’s fairness index. This metric can be applied to any performance metric such as rate or serviceload and it is maximized when all users receive an equal service. Here, we compute the Jain’sindex based on the data service that is offered to each user. The Jain’s index is given by [32]: F Jain ( l , l , ..., l N ) = (cid:32) N (cid:88) i =1 l i (cid:33) × (cid:16) N N (cid:88) i =1 l i (cid:17) − , (59)where N is the number of users, and l i is the data service to user i . Clearly, /N ≤ F Jain ≤ ,with F Jain = 1 /N and F Jain = 1 indicating the lowest and highest level of fairness.Fig. 4 shows the Jain’s fairness index for different values of σ o which is given in (58). In thisfigure, as σ o increases the spatial distribution of users becomes closer to a uniform distribution.As we can see from this figure, the minimum Jain’s index corresponding to the proposed cellpartitioning method is above 0.5. However, in the weighted Voronoi case, it can decrease to 0.18for a highly non-uniform distribution of users with σ o = 200 m. This is due to the fact that,in the Voronoi case, users located in highly congested partitions receive lower service than thepartitions with low number of users. In the proposed approach, however, the resources (hovertime and bandwidth) are fairly shared between the users thus leading to a higher fairness index.From Fig. 4 we can also observe that, for higher values of σ o (more uniform distribution), thefairness index for the proposed approach becomes closer to the weighted Voronoi case.Fig. 5 shows the average number of users in each cell partition. Clearly, in the Voronoi case,the average number of users per cell significantly varies for different cell partitions. For instance,the average number of users in cell 5 is three times higher than cell 3. Consequently, comparedto cell 5, users in cell 3 will receive lower data service from their associated UAV. However,in the proposed approach, the cell partitions associated with the UAVs are formed such thatthe number of users per cell be proportional to the bandwidth and hover time of the UAVs.In this case, given equal bandwidth and hover times of UAVs, the cell partitions contains anequal number of users. Therefore, our approach avoids generating unbalanced cell partitions and,hence, it leads to a higher level of fairness compared to the classical Voronoi approach.Fig. 6 shows the average total data service as a function of the interference factor, β used in(4). From this figure we can see that, as the interference between UAVs decreases, the total dataservice that they can provide to the ground users increases. For instance, by decreasing β from1 (full interference case) to 0.1, the total data service increases by a factor of 3 when 5 UAVsare deployed. Moreover, Fig. 6 shows that the service gain achieved by using a higher number Cell number A v e r age nu m be r o f u s e r s pe r c e ll Weighted VoronoiProposed cell partitioning
Fig. 5: Average number of users per cell partition.
Interference factor ( β ) T o t a l da t a s e r v i c e ( G b ) Fig. 6: Average data service versus interference factor.of UAVs is significant only when the interference between the UAVs is highly mitigated (lowvalues of β ). For example, increasing the number of UAVs from 5 to 10 can lead to 56% dataservice gain for β = 0 . , while this gain is only 5% in the full interference case. Therefore,deploying more UAVs is beneficial in terms of data service if the interference between the UAVsis properly mitigated.In Fig. 7, we show how the total data service changes as the maximum hover time of the UAVsincreases. As expected, by increasing the hover time of each UAV, the users can be served fora longer time and, hence, the total data service increases. Fig. 7 also compares the performanceof deploying 5 UAVs versus 10 UAVs. Interestingly, we can see that the 5-UAVs case with40 minuets hover time (for each UAV) outperforms the 10-UAVs case with 30 minuets hovertime. As a result, in this case, deploying UAVs that has a 33% higher hover time is preferredthan doubling the number of UAVs. In fact, increasing the number of UAVs results in a higher
30 35 40 45 50 55 60020406080100120140
Maximum hover time of each UAV (min) T o t a l da t a s e r v i c e ( G b )
10 UAVs5 UAVs
Fig. 7: Total data service versus the maximum hover time of each UAV.
Bandwidth (Mhz) A v e r age t o t a l ho v e r t i m e ( m i n ) Optimal bandwidth allocationEqual bandwidth allocation
Fig. 8: Average hover time versus bandwidth.interference which reduces the maximum data service gain that can be typically achieved byusing more UAVs. Therefore, depending on system parameters, using more capable UAVs (i.e.with longer flight time) to service ground users can be more beneficial than deploying moreUAVs with shorter flight times.
B. Results for Scenario 2
Here, we present the results for Scenario 2 in which the users are completely serviced using aminimum hover time. In this case, we consider a 10 Mb data service requirement for each user.In Fig. 8, we show the total hover time versus the transmission bandwidth. Two bandwidthallocation schemes are considered, the optimal bandwidth allocation resulting from Proposition2, and an equal bandwidth allocation. Clearly, by increasing the bandwidth, the total hover timerequired for serving the users decreases. In fact, a higher bandwidth can provide a higher thetransmission rate and, hence, users can be serviced within a shorter time duration. From Fig. 8, Number of UAVs A v e r age ho v e r t i m e ( m i n ) T o t a l band w i d t h u s age ( M h z ) Fig. 9: Average hover time versus number of UAVs and bandwidth usage.we can see that, the optimal bandwidth allocation scheme can yield a 51% hover time reductioncompared to the equal bandwidth allocation. This is due the fact that, according to Proposition2, by optimally assigning the bandwidth to each user based on its demand and location, the totalhover time of UAVs can be minimized.Fig. 9 shows the average total hover time of the UAVs as the number of UAVs varies. Thisresult corresponds to the interference-free scenario in which the UAVs operate on differentfrequency bands. Hence, the total bandwidth usage linearly increases by increasing the numberof UAVs. From Fig. 9, we can see that the total hover time decreases as the number of UAVsincreases. A higher number of UAVs corresponds to a higher number of cell partitions. Hence,the size of each cell partition decreases and the users will have a shorter distance to the UAVs.In addition, lower control time is required during serving a smaller and less congested cell. Infact, increasing the number of UAVs leads to a higher transmission rate, and lower control timethus leading to a lower hover time. For instance, as shown in Fig. 9, when the number of UAVsincreases from 2 ot 6, the total hover time decreases by 53%. Nevertheless, deploying moreUAVs in interference-free scenario results in a higher bandwidth usage. Therefore, there is afundamental tradeoff between the hover time of UAVs and the bandwidth efficiency.Fig. 10 shows the impact of control time on the total hover time for the proposed cellpartitioning, as a result of Theorem 3 and the weighted Voronoi diagram. In both cases, weuse the optimal bandwidth allocation scheme. Clearly, as the control time factor, α , increases,the total hover time also increases. From Fig. 10, we can see that, using our proposed optimalcell partitioning approach, the average total hover time can be reduced by around 20% comparedto weighted Voronoi case. This is due to the fact that, unlike the weighted Voronoi, our approach Control time factor ( α ) A v e r age t o t a l ho v e r t i m e ( m i n ) Weighted VoronoiProposed approach
Fig. 10: Average hover time versus control time factor ( α ) for σ o = 200 m. Interference factor ( β ) A v e r age t o t a l ho v e r t i m e ( m i n ) Load per user= 100 MbLoad per user= 200 Mb
Fig. 11: Average hover time versus interference factor.also minimizes the control time while generating the cell partitions. We note that, the hover timedifference between these two cases increases as α increases. In particular, as shown in Fig. 10,our approach yields around 32% hover time reduction when α = 0 . .In Fig. 11, we show the impact of interference on the hover time of UAVs. Clearly, the totalhover time increases as the interference between the UAVs increases. This is due to the fact thata lower SINR leads to a lower transmission rate and, hence, a given UAV needs to hover fora longer time in order to completely service its users. For instance, the average hover time inthe full interference case ( β = 1 ) is 4.5 times larger than the interference-free case in which β = 0 . Therefore, one can significantly reduce the hove time of UAVs by adopting interferencemitigation techniques such as using orthogonal frequencies and scheduling of UAVs.VI. CONCLUSIONS
In this paper, we have proposed a novel framework for optimizing UAV-enabled wirelessnetworks while taking into account the flight time constraints of UAVs. In particular, we have investigated two UAV-based communication scenarios. First, given the maximum possible hovertimes of UAVs, we have maximized the average data service to the ground users under a fairresource allocation policy. To this end, using tools from optimal transport theory, we havedetermined the optimal cell partitions associated with the UAVs. In the second scenario, giventhe load requirements of users, we have minimized the average hover time of UAVs needed tocompletely serve the users. In this case, we have derived the optimal cell partitions as well asthe optimal bandwidth allocation to the users that lead to the minimum hover time. The resultshave shown that, using our proposed cell partitioning approach, the users receive higher fair dataservice compared to the classical Voronoi case. Moreover, our results for the second scenariohave revealed that the average hover time of UAVs can be significantly reduced by using ourproposed approach. A PPENDIX
A. Proof of Theorem 3
As shown in Proposition 2, there exist optimal cell partitions A i , i ∈ M which are solutionsto the optimization problem in (50). Now, we consider two optimal partitions A l and A m , anda point v o = ( x o , y o ) ∈ A m . Also, let B (cid:15) ( v o ) be the intersection of A m with a disk that has acenter v o and radius (cid:15) > . To characterize the optimal solution of (50), we first generate newcell partitions (cid:98) A i (a variation of optimal partitions) as follows: (cid:98) A m = A m \ B ε ( v o ) , (cid:98) A l = A l ∪ B ε ( v o ) , (cid:98) A i = A i , i (cid:54) = l, m. (60)Also, let a ε = (cid:82) B ε ( v o ) f ( x, y ) d x d y , and (cid:98) a i = (cid:82) (cid:98) A i f ( x, y ) d x d y . Considering the optimality of A i , i ∈ M , we have: (cid:88) i ∈M (cid:90) A i (cid:20) N u ( x, y ) C B i i ( x, y ) + q i ( x, y ) (cid:21) f ( x, y ) d x d y + g i ( a i ) ( a ) ≤ (cid:88) i ∈M (cid:90) (cid:98) A i (cid:20) N u ( x, y ) C B i i ( x, y ) + q i ( x, y ) (cid:21) f ( x, y ) d x d y + g i ( (cid:98) a i ) , (cid:90) A l (cid:34) N u ( x, y ) C B l l ( x, y ) + q l ( x, y ) (cid:35) f ( x, y ) d x d y + g l ( a l ) + (cid:90) A m (cid:20) N u ( x, y ) C B m m ( x, y ) + q m ( x, y ) (cid:21) f ( x, y ) d x d y + g m ( a m ) ≤ (cid:90) A l ∪ B ε ( v o ) (cid:34) N u ( x, y ) C B l l ( x, y ) + q l ( x, y ) (cid:35) f ( x, y ) d x d y + g l ( a l + a (cid:15) ) + (cid:90) A m \ B ε ( v o ) (cid:20) N u ( x, y ) C B m m ( x, y ) + q m ( x, y ) (cid:21) f ( x, y ) d x d y + g m ( a m − a (cid:15) ) (cid:90) B ε ( v o ) (cid:20) N u ( x, y ) C B m m ( x, y ) + q m ( x, y ) (cid:21) f ( x, y ) d x d y + g m ( a m ) − g m ( a m − a (cid:15) ) ≤ (cid:90) B ε ( v o ) (cid:34) N u ( x, y ) C B l l ( x, y ) + q l ( x, y ) (cid:35) f ( x, y ) d x d y + g l ( a l + a (cid:15) ) − g l ( a l ) , (61)where ( a ) comes from the fact that A i is optimal and, hence, any variation of that ( (cid:98) A i ) cannotlead to a better solution.Now, we multiply both sides of the inequality in (61) by a ε . Then, we take the limit when ε → , and we use the following equality: lim ε → a ε (cid:90) B ε ( v o ) (cid:34) N u ( x, y ) C B l l ( x, y ) + q l ( x, y ) (cid:35) f ( x, y ) d x d y = lim ε → (cid:82) B ε ( v o ) (cid:20) Nu ( x,y ) C Bll ( x,y ) + q l ( x, y ) (cid:21) f ( x, y ) d x d y (cid:82) B ε ( v o ) f ( x, y ) d x d y = N u ( x, y ) C B l l ( x o , y o ) + q l ( x o , y o ) . (62)Subsequently, following from (61), we have: N u ( x, y ) C B m m ( x o , y o ) + q m ( x o , y o ) + g (cid:48) l ( a m ) ≤ N u ( x, y ) C B l l ( x o , y o ) + q l ( x o , y o ) + g (cid:48) l ( a l ) . (63)Note that, (63) provides the condition under which a point ( x o , y o ) is assigned to partition m rather than l . Therefore, the optimal cell partitions can be characterized as: A ∗ i = (cid:40) ( x, y ) | N u ( x, y ) C B i i ( x, y ) + q i ( x, y ) + g (cid:48) i ( a i ) ≤ N u ( x, y ) C B j j ( x, y ) + q j ( x, y ) + g (cid:48) j ( a j ) , ∀ j (cid:54) = i ∈ M (cid:41) . (64)Finally, using (38), the optimal average hover time of UAV i is: τ ∗ i = (cid:90) A ∗ i N u ( x, y ) C B i i ( x, y ) f ( x, y ) d x d y + g i (cid:32)(cid:90) A ∗ i f ( x, y ) d x d y (cid:33) , (65)which proves the theorem. R EFERENCES [1] D. Orfanus, E. P. de Freitas, and F. Eliassen, “Self-organization as a supporting paradigm for military UAV relay networks,”
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