1 Wireless Communication using Unmanned Aerial Vehicles (UAVs): Optimal Transport Theory for Hover Time Optimization Mohammad Mozaffari 1 , Walid Saad 1 , Mehdi Bennis 2 , and M´ erouane Debbah 3 1 Wireless@VT, Electrical and Computer Engineering Department, Virginia Tech, VA, USA, Emails:{mmozaff,walids}@vt.edu. 2 CWC - Centre for Wireless Communications, Oulu, Finland, Email: [email protected].fi. 3 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 flying base stations that can provide wireless service to ground users is investigated. In particular, a novel framework for optimizing the performance of such UAV-based wireless systems in terms of the average number 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 ground users that are distributed within the area based on an arbitrary spatial distribution function. In this case, two practical scenarios are considered. In the first scenario, based on the maximum possible hover times of UAVs, the average data service delivered to the users under a fair resource allocation scheme is maximized by finding the optimal cell partitions associated to the UAVs. Using the powerful mathematical framework of optimal transport theory, this cell partitioning problem is proved to be equivalent to a convex optimization problem. Subsequently, a gradient-based algorithm is proposed for optimally partitioning the geographical area based on the users’ distribution, hover times, and locations of the UAVs. In the second scenario, given the load requirements of ground users, the minimum average hover 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 optimal bandwidth allocation, the optimal cell partitions associated with the UAVs are derived by exploiting the optimal transport theory. Simulation results show that our proposed cell partitioning approach leads to a 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% by adopting 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 bandwidth efficiency while serving the ground users.
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Wireless Communication using Unmanned AerialVehicles (UAVs): Optimal Transport Theory for
Hover Time OptimizationMohammad Mozaffari1, Walid Saad1, Mehdi Bennis2, and Merouane Debbah3
1 Wireless@VT, Electrical and Computer Engineering Department, Virginia Tech, VA, USA,
Solving the optimization problem in (16) is challenging due to various reasons. First, the
optimization variables Ai, ∀i ∈ M, are sets of continuous partitions (as we have a continuous
area) 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 the
given two-fold integrations. In addition, due to the constraints given in (17), finding Ai becomes
more challenging. To solve the optimization problem in (16), next, we model the problem by
exploiting optimal transport theory [22].
A. Optimal Transport Theory: Preliminaries
Here, we present some primary results from optimal transport theory which will be used in the
next subsection to derive the optimal cell partitions. Optimal transport theory goes back to the
Monge’s problem in 1781 which is stated as follows [22]. Given piles of sands and holes with the
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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 minimum
total transportation cost. In general, this theory aims to find the optimal matching between two
sets of points that minimizes the costs associated with the matching between the sets. These sets
can be either discrete or continuous, with arbitrary distributions (weights). Mathematically, the
Monge optimal transport problem can be written as follows. Given two probability distributions
f1 on X ⊂ Rn, and f2 on Y ⊂ Rn, find the optimal transport map T from f1 to f2 that minimizes
the following problem:minT
∫X
c (x, T (x))f1(x)dx; 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, f1 and f2 are the source and destination
probability distributions.
Solving the Monge’s problem is challenging due its high non-linear structure [22], and the
fact that it does not necessarily admit a solution as each point of the source distribution must be
mapped to only one location at the destination. However, Kantorovich relaxed this problem by
using 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]:
minπ
∫X×Y
c (x,y)dπ(x,y), (21)
s.t.∫X
dπ(x,y) = f1(x)dx,∫Y
dπ(x,y) = f2(y)dy, (22)
where π represents the transport plan which is the probability distribution on X × Y whose
marginals are f1 and f2.
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 dual
formulation for the Monge-Kantorovich problem that can lead to a tractable solution. The duality
theorem is stated as [22] and [27]:
Kantrovich Duality Theorem: Given the Monge-Kantorovich problem in (21) with two
probability measures f1 on X ⊂ Rn, and f2 on Y ⊂ Rn, and any lower semi-continuous
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cost function c(x,y), the following equality holds:
minπ
∫X×Y
c (x,y)dπ(x,y) (23)
= maxϕ,ψ
∫X
ϕ(x)f1(x)dx+
∫Y
ψ(y)f2(y)dy ; ϕ(x) + ψ(y) ≤ c(x,y), ∀(x,y) ∈ X × Y,
(24)
where ϕ(x) and ψ(y) are Kantorovich potential functions. As discussed in [22], this duality
theorem 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 coincide
with the Monge’s problem. Nevertheless, when the source distribution, f1, and the cost function
are 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 of
the 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 source
measure (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, Ai, for which the
average total data service is maximized. In our model, users have a continuous distribution, and
the locations of the UAVs can be considered as discrete points. Then, the optimal cell partitions
are obtained by optimally mapping the users to the UAVs. In fact, given (16), the cell partitions
are related to the transport map by [29]:T (v) =
∑i∈M
si1Ai(v);
∫Ai
f(x, y)dxdy = ωi
, (26)
where ωi = BiTiM∑k=1
BkTk
, as given in (17), is directly related to the hover time and the bandwidth of
the UAVs. Also, 1Ai(v) is the indicator function which is equal to 1 if v ∈ Ai, and 0 otherwise.
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Therefore, the optimization problem in (16) can be cast within the optimal transport framework
as follows. Given a continuous probability measure f of users, and a discrete probability measure
Γ =∑i∈M
ωiδsi corresponding to the UAVs, we must find the optimal transport map for which∫D J (v, T (v))f(x, y)dxdy is minimized. In this case, δsi is the Dirac function, and J is the
transportation cost function which is used in (16) and is given by:
interference which reduces the maximum data service gain that can be typically achieved by
using 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 more
UAVs 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 a
minimum 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 bandwidth
allocation schemes are considered, the optimal bandwidth allocation resulting from Proposition
2, and an equal bandwidth allocation. Clearly, by increasing the bandwidth, the total hover time
required for serving the users decreases. In fact, a higher bandwidth can provide a higher the
transmission rate and, hence, users can be serviced within a shorter time duration. From Fig. 8,
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1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
Number of UAVs
Ave
rage
hov
er ti
me
(min
)1
2
3
4
5
6
7
8
9
10
Tot
al b
andw
idth
usa
ge (
Mhz
)
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 reduction
compared to the equal bandwidth allocation. This is due the fact that, according to Proposition
2, by optimally assigning the bandwidth to each user based on its demand and location, the total
hover time of UAVs can be minimized.
Fig. 9 shows the average total hover time of the UAVs as the number of UAVs varies. This
result corresponds to the interference-free scenario in which the UAVs operate on different
frequency bands. Hence, the total bandwidth usage linearly increases by increasing the number
of UAVs. From Fig. 9, we can see that the total hover time decreases as the number of UAVs
increases. 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. In
fact, increasing the number of UAVs leads to a higher transmission rate, and lower control time
thus leading to a lower hover time. For instance, as shown in Fig. 9, when the number of UAVs
increases from 2 ot 6, the total hover time decreases by 53%. Nevertheless, deploying more
UAVs in interference-free scenario results in a higher bandwidth usage. Therefore, there is a
fundamental 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 cell
partitioning, as a result of Theorem 3 and the weighted Voronoi diagram. In both cases, we
use 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 optimal
cell partitioning approach, the average total hover time can be reduced by around 20% compared
to weighted Voronoi case. This is due to the fact that, unlike the weighted Voronoi, our approach
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0 0.1 0.2 0.3 0.4 0.50
50
100
150
200
250
300
350
400
Control time factor (α)
Ave
rage
tota
l hov
er ti
me
(min
)
Weighted VoronoiProposed approach
Fig. 10: Average hover time versus control time factor (α) for σo = 200 m.
0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
Interference factor (β)
Ave
rage
tota
l hov
er ti
me
(min
)
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 time
difference 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 = 0.5.
In Fig. 11, we show the impact of interference on the hover time of UAVs. Clearly, the total
hover time increases as the interference between the UAVs increases. This is due to the fact that
a lower SINR leads to a lower transmission rate and, hence, a given UAV needs to hover for
a longer time in order to completely service its users. For instance, the average hover time in
the 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 interference
mitigation 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 wireless
networks while taking into account the flight time constraints of UAVs. In particular, we have
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investigated two UAV-based communication scenarios. First, given the maximum possible hover
times of UAVs, we have maximized the average data service to the ground users under a fair
resource allocation policy. To this end, using tools from optimal transport theory, we have
determined the optimal cell partitions associated with the UAVs. In the second scenario, given
the load requirements of users, we have minimized the average hover time of UAVs needed to
completely serve the users. In this case, we have derived the optimal cell partitions as well as
the optimal bandwidth allocation to the users that lead to the minimum hover time. The results
have shown that, using our proposed cell partitioning approach, the users receive higher fair data
service compared to the classical Voronoi case. Moreover, our results for the second scenario
have revealed that the average hover time of UAVs can be significantly reduced by using our
proposed approach.
APPENDIX
A. Proof of Theorem 3
As shown in Proposition 2, there exist optimal cell partitions Ai, i ∈M which are solutions
to the optimization problem in (50). Now, we consider two optimal partitions Al and Am, and
a point vo = (xo, yo) ∈ Am. Also, let Bε(vo) be the intersection of Am with a disk that has a
center vo and radius ε > 0. To characterize the optimal solution of (50), we first generate new
cell partitions Ai (a variation of optimal partitions) as follows:Am = Am\Bε(vo),
Al = Al ∪Bε(vo),
Ai = Ai, i 6= l,m.
(60)
Also, let aε =∫Bε(vo)
f(x, y)dxdy, and ai =∫Aif(x, y)dxdy. Considering the optimality of
Ai, i ∈M, we have:∑i∈M
∫Ai
[Nu(x, y)
CBii (x, y)
+ qi(x, y)
]f(x, y)dxdy + gi(ai)
(a)
≤∑i∈M
∫Ai
[Nu(x, y)
CBii (x, y)
+ qi(x, y)
]f(x, y)dxdy + gi(ai),
∫Al
[Nu(x, y)
CBll (x, y)
+ ql(x, y)
]f(x, y)dxdy + gl(al) +
∫Am
[Nu(x, y)
CBmm (x, y)
+ qm(x, y)
]f(x, y)dxdy + gm(am)
≤∫Al∪Bε(vo)
[Nu(x, y)
CBll (x, y)
+ ql(x, y)
]f(x, y)dxdy + gl(al + aε)
29
+
∫Am\Bε(vo)
[Nu(x, y)
CBmm (x, y)
+ qm(x, y)
]f(x, y)dxdy + gm(am − aε)∫
Bε(vo)
[Nu(x, y)
CBmm (x, y)
+ qm(x, y)
]f(x, y)dxdy + gm(am)− gm(am − aε)
≤∫Bε(vo)
[Nu(x, y)
CBll (x, y)
+ ql(x, y)
]f(x, y)dxdy + gl(al + aε)− gl(al), (61)
where (a) comes from the fact that Ai is optimal and, hence, any variation of that (Ai) cannot
lead to a better solution.
Now, we multiply both sides of the inequality in (61) by 1aε
. Then, we take the limit when
ε→ 0, and we use the following equality:
limε→0
1
aε
∫Bε(vo)
[Nu(x, y)
CBll (x, y)
+ ql(x, y)
]f(x, y)dxdy = lim
ε→0
∫Bε(vo)
[Nu(x,y)
CBll (x,y)
+ ql(x, y)
]f(x, y)dxdy∫
Bε(vo)f(x, y)dxdy
=Nu(x, y)
CBll (xo, yo)
+ ql(xo, yo). (62)
Subsequently, following from (61), we have:
Nu(x, y)
CBmm (xo, yo)
+ qm(xo, yo) + g′l(am) ≤ Nu(x, y)
CBll (xo, yo)
+ ql(xo, yo) + g′l(al). (63)
Note that, (63) provides the condition under which a point (xo, yo) is assigned to partition m
rather than l. Therefore, the optimal cell partitions can be characterized as:
A∗i =
(x, y)| Nu(x, y)
CBii (x, y)
+ qi(x, y) + g′i(ai) ≤Nu(x, y)
CBj
j (x, y)+ qj(x, y) + g′j(aj),∀j 6= i ∈M
.(64)
Finally, using (38), the optimal average hover time of UAV i is:
τ ∗i =
∫A∗
i
Nu(x, y)
CBii (x, y)
f(x, y)dxdy + gi
(∫A∗
i
f(x, y)dxdy
), (65)
which proves the theorem.
REFERENCES
[1] D. Orfanus, E. P. de Freitas, and F. Eliassen, “Self-organization as a supporting paradigm for military UAV relay networks,”IEEE Communications Letters, vol. 20, no. 4, pp. 804–807, 2016.
[2] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Unmanned aerial vehicle with underlaid device-to-device communi-cations: Performance and tradeoffs,” IEEE Transactions on Wireless Communications, vol. 15, no. 6, pp. 3949–3963, June2016.
[3] A. Merwaday and I. Guvenc, “UAV assisted heterogeneous networks for public safety communications,” in Proc. of IEEEWireless Communications and Networking Conference Workshops (WCNCW), March 2015.
[4] Y. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with unmanned aerial vehicles: opportunities and challenges,”IEEE Communications Magazine, vol. 54, no. 5, pp. 36–42, May 2016.
30
[5] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Mobile unmanned aerial vehicles (UAVs) for energy-efficient Internetof Things communications,” available online: arxiv.org/abs/1703.05401, 2017.
[6] A. Hourani, S. Kandeepan, and A. Jamalipour, “Modeling air-to-ground path loss for low altitude platforms in urbanenvironments,” in Proc. of IEEE Global Communications Conference (GLOBECOM), Austin, TX, USA, Dec. 2014.
[7] M. M. Azari, F. Rosas, K. C. Chen, and S. Pollin, “Joint sum-rate and power gain analysis of an aerial base station,” inProc. of IEEE Global Communications Conference (GLOBECOM) Workshops, Dec. 2016.
[8] E. Kalantari, H. Yanikomeroglu, and A. Yongacoglu, “On the number and 3D placement of drone base stations in wirelesscellular networks,” in Proc. of IEEE Vehicular Technology Conference, Sep. 2016.
[9] I. Bor-Yaliniz and H. Yanikomeroglu, “The new frontier in ran heterogeneity: Multi-tier drone-cells,” IEEE CommunicationsMagazine, vol. 54, no. 11, pp. 48–55, 2016.
[10] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Efficient deployment of multiple unmanned aerial vehicles for optimalwireless coverage,” IEEE Communications Letters, vol. 20, no. 8, pp. 1647–1650, Aug. 2016.
[11] Facebook, “Connecting the world from the sky,” Facebook Technical Report, 2014.[12] S. Jeong, O. Simeone, and J. Kang, “Mobile edge computing via a UAV-mounted cloudlet: Optimal bit allocation and
path planning,” available online: https://arxiv.org/abs/1609.05362., 2016.[13] F. Jiang and A. L. Swindlehurst, “Optimization of UAV heading for the ground-to-air uplink,” IEEE Journal on Selected
Areas in Communications, vol. 30, no. 5, pp. 993–1005, June 2012.[14] Y. Zeng and R. Zhang, “Energy-efficient UAV communication with trajectory optimization,” IEEE Transactions on Wireless
Communications, to appear 2017.[15] Q. Wu, Y. Zeng, and R. Zhang, “Joint trajectory and communication design for UAV-enabled multiple access,” available
online: https://arxiv.org/abs/1704.01765, 2017.[16] V. V. Chetlur and H. S. Dhillon, “Downlink coverage analysis for a finite 3D wireless network of unmanned aerial vehicles,”
available online: arxiv.org/abs/1701.01212, 2017.[17] V. Sharma, M. Bennis, and R. Kumar, “UAV-assisted heterogeneous networks for capacity enhancement,” IEEE Commu-
nications Letters, vol. 20, no. 6, pp. 1207–1210, June 2016.[18] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Optimal transport theory for power-efficient deployment of unmanned
aerial vehicles,” in Proc. of IEEE International Conference on Communications (ICC), May 2016.[19] S. Niu, J. Zhang, F. Zhang, and H. Li, “A method of UAVs route optimization based on the structure of the highway
network,” International Journal of Distributed Sensor Networks, 2015.[20] K. Dorling, J. Heinrichs, G. G. Messier, and S. Magierowski, “Vehicle routing problems for drone delivery,” IEEE
Transactions on Systems, Man, and Cybernetics: Systems, vol. 47, no. 1, pp. 70–85, Jan 2017.[21] S. Chandrasekharan, K. Gomez, A. Al-Hourani, S. Kandeepan, T. Rasheed, L. Goratti, L. Reynaud, D. Grace, I. Bucaille,
T. Wirth, and S. Allsopp, “Designing and implementing future aerial communication networks,” IEEE CommunicationsMagazine, vol. 54, no. 5, pp. 26–34, May 2016.
[22] C. Villani, Topics in optimal transportation. American Mathematical Soc., 2003, no. 58.[23] O. Lysenko, S. Valuiskyi, P. Kirchu, and A. Romaniuk, “Optimal control of telecommunication aeroplatform in the area
of emergency,” Information and Telecommunication Sciences, no. 1, 2013.[24] ITU-R, “Rec. p.1410-2 propagation data and prediction methods for the design of terrestrial broadband millimetric radio
access systems,” Series, Radiowave propagation, 2003.[25] F. Aurenhammer, “Voronoi diagramsa survey of a fundamental geometric data structure,” ACM Computing Surveys (CSUR),
vol. 23, no. 3, pp. 345–405, 1991.[26] A. Silva, H. Tembine, E. Altman, and M. Debbah, “Optimum and equilibrium in assignment problems with congestion:
Mobile terminals association to base stations,” IEEE Transactions on Automatic Control, vol. 58, no. 8, pp. 2018–2031,Aug. 2013.
[27] F. Santambrogio, “Optimal transport for applied mathematicians,” Birkauser, NY, 2015.[28] L. Ambrosio and N. Gigli, “A users guide to optimal transport,” in Modelling and optimisation of flows on networks.
Springer, 2013, pp. 1–155.[29] G. Crippa, C. Jimenez, and A. Pratelli, “Optimum and equilibrium in a transport problem with queue penalization effect,”
Advances in Calculus of Variations, vol. 2, no. 3, pp. 207–246, 2009.[30] H. Ghazzai, “Environment aware cellular networks,” available online: http://repository.kaust.edu.sa/kaust/handle/10754/344436,
Feb. 2015.[31] M. C. Achtelik, J. Stumpf, D. Gurdan, and K. M. Doth, “Design of a flexible high performance quadcopter platform
breaking the MAV endurance record with laser power beaming,” in Proc. of IEEE International Conference on IntelligentRobots and Systems, Sep. 2011.
[32] R. Jain, D.-M. Chiu, and W. R. Hawe, A quantitative measure of fairness and discrimination for resource allocation inshared computer system. tech. rep., Digital Equipment Corporation, DEC-TR-301, 1984, vol. 38.