Joint Sub-Carrier and Power Allocation for Efficient Communication Of

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Joint Sub-carrier and Power Allocation for Efﬁcient Communication of Cellular UAVs Hamed Hellaoui, Miloud Bagaa, Member, IEEE, Ali Chelli, Member, IEEE and Tarik Taleb, Senior Member, IEEE

Abstract—Cellular networks are expected to be the main com- This becomes more critical as the flying UAVs can operate in munication infrastructure to support the expanding applications swarm and BVLOS. All these facts highlight the importance of Unmanned Aerial Vehicles (UAVs). As these networks are of downlink communication in cellular-based UAVs. deployed to serve ground User Equipment (UEs), several issues need to be addressed to enhance cellular UAVs’ services. In To better investigate the communication quality for cellular this paper, we propose a realistic communication model on the UAVs, some research works have performed real-field trials downlink, and we show that the Quality of Service (QoS) for the [3]–[5]. 3GPP performed relevant evaluations, considering the users is affected by the number of interfering BSs and the impact inputs from several partners. The results provided in [3] point they cause. The joint problem of sub-carrier and power allocation out that flying UAVs could have poor link quality resulting is therefore addressed. Given its complexity, which is known to be NP-hard, we introduce a solution based on game theory. First, in low throughput. This is mainly due to the close free-space we argue that separating between UAVs and UEs in terms of signal propagation, which is translated into a higher probability the assigned sub-carriers reduces the interference impact on the of interference from non-serving BSs, and thus, leads to poor users. This is materialized through a matching game. Moreover, link quality. This would result in lower Quality of Service in order to boost the partition, we propose a coalitional game (QoS) for the flying UAVs, which may not be tolerable. The that considers the outcome of the first one and enables users to change their coalitions and enhance their QoS. Furthermore, a effect of this issue on ground UEs is not significant, as the power optimization solution is introduced, which is considered in underlying communication model is different compared to that the two games. Performance evaluations are conducted, and the of the flying UAVs. As cellular networks have been so far obtained results demonstrate the effectiveness of the propositions. deployed to serve ground UEs, efficient solutions are highly Index Terms—Unmanned Aerial Vehicles (UAVs), Downlink required for cellular network-enabled UAVs. communication, Cellular Networks, Game Theory. A. Prior Works I.INTRODUCTION NMANNED Aerial Vehicles (UAVs) have become an Given their potential benefits, the usage of mobile networks U integral part of many applications in different areas (e.g., for UAVs has attracted a lot of attention. In this context, crowd surveillance, first aid, disaster management, etc.) [2]. the quality of the communication link is one of the major The next generation of UAVs will rely on cellular networks as issues as the underlying communication model of cellular a communication infrastructure. This would allow exploiting UAVs is different from that of connected ground UEs [6]. the unstopped evolution of these networks and to benefit from The authors in [7], addressed minimizing the transmit power their achievements. Cellular-based UAVs would also enable of BS-UAVs, serving ground users, while satisfying the trans- Beyond Visual Line-of-Sight applications (BVLOS), where mission rate requirements of these users. To this end, the the UAVs can fly far from their control center. However, in authors investigated the power minimization problem using order to enable these potentials, it is highly important to ensure transport theory and facility location. In [8], authors addressed efficient downlink communication of the flying UAVs. Indeed, optimizing 3D placement and mobility of BS-UAVs collecting given their critical nature, any delay in communication could data from ground IoT. In terms of sub-carriers assignment, not be tolerable and might have catastrophic consequences. the authors used a constrained K-mean clustering strategy to assign different channels to devices that are located in the proximity of each other. As for power control, while the This work was partially supported by the European Union’s Horizon 2020 transmit power of each device is computed only based on Research and Innovation Program through the 5G!Drones Project under Grant No. 857031, by the Academy of Finland 6Genesis project under Grant No. the channel gain between the device and its serving UAV 318927, and by the Academy of Finland CSN project under Grant No. 311654. for the interference-free scenario, an iterative optimization is A preliminary version of this work has been accepted for publication at the considered for the interference scenario. The problem of power IEEE WCNC 2019 [1]. H. Hellaoui and M. Bagaa are with the Department of Communica- allocation for multi-UAV enabled wireless networks has been tions and Networking, Aalto University, Espoo 02150, Finland (e-mails: addressed in [9]. The authors proposed a price-based power hamed.hellaoui@aalto.fi; miloud.bagaa@aalto.fi). allocation scheme, and modeled the interaction between the T. Taleb is with the Department of Communications and Networking, Aalto University, Espoo 02150, Finland, also with Centre for Wireless Communi- UAVs and the ground users as a Stackelberg game. The ground cations (CWC), University of Oulu, Oulu 90570, Finland, and also with the user selects optimal power strategy to maximize its own Department of Computer and Information Security, Sejong University, Seoul utility. The problem of sub-carrier selection and transmission 05006, South Korea (e-mail: tarik.taleb@aalto.fi). A. Chelli is with the Faculty of Engineering and Science, University of power for UAV-assisted communication is also addressed in Agder, 4898 Grimstad, Norway (e-mail: [email protected]). [10]. The authors proposed an iterative algorithm and used IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2

Lagrangian dual decomposition method to solve it. In order • We propose a coalitional game that builds on the results to enhance the link quality, other works focus on the optimal achieved by the matching game to further reduce the deployment of BS-UAVs. In [11], the authors addressed the outage probability of each user. This game defines the on-demand deployment of multiple aerial base stations for mechanisms allowing the players to change their coali- traffic offloading/network recovery. The 3D discrete search tions and enhance their QoS. These two games together space problem is formulated as a mixed-integer program, while will constitute two sub-games of a global game. a UAV deployment algorithm in a continuous 3D space is • A power optimization solution for the communication proposed based on an unsupervised learning technique. In model is introduced, which is considered in the two sub- [12], authors tackled the optimal deployment of multiple UAVs games. in 3D space to serve ground users. Besides, the paper also • We also provide numerical results on the achieved link considered adapting BS-UAVs’ positions as users move around quality for performing downlink communication in cel- within the network. lular UAVs. In addition to BS-UAVs, some works have addressed the quality of the communication link for cellular-connected C. Organization of the Paper drones. Authors in [13] proposed an interference-aware path planing scheme for cellular UAVs. The problem is modeled as The rest of the paper is organized as follows. Section II a dynamic game among UAVs to achieve a tradeoff between introduces the considered realistic model for cellular network- maximizing energy efficiency and minimizing latency and enabled UAVs. Section III presents the first sub-game opti- interference level caused on the ground. In [14], a communica- mization based on marching game framework. The second tion model for mobile network-enabled UAVs is proposed. The sub-game optimization, based on coalitional game, is intro- authors provided an iterative power-optimization algorithm duced in Section IV. Section V provides the global game and to enhance the link quality on the uplink. In [15], authors the proposed algorithm. Simulations and performance evalua- considered sub-carrier and power allocation for multi-UAV tions are conducted and presented in Section VI. Section VII systems. The problem is addressed by formulating a weighted concludes the paper. mean square error (MSE) problem, which can be solved via alternating optimization. In [16], the authors considered the II.SYSTEM MODELAND PROBLEM FORMULATION availability of different Mobile Network Operators (MNOs) This section describes the proposed communication model to steer each UAV’s traffic through the network, ensuring the and also presents the problem formulation. The downlink best communication link. The proposed solution is based on scenario is considered in which the connected users (UEs and the framework of coalitional games in order to associate UAVs UAVs) receive data from their serving BSs. The latter employs to their optimal MNOs. Each UAV will be assigned with sub- an orthogonal frequency division multiple access (OFDMA) carriers only from the selected MNO. In the same context, technique to serve the connected users. Consequently, intra- authors in [17] considered that a UAV could also serve as a cell interference is neglected, and the interference can be relay node to route the traffic of other UAVs and proposed caused only by non-serving BSs, as shown in Fig. 1. a solution based on linear programming to select the optimal Let U, V and B denote respectively the set of UEs, UAVs successor for each UAV. and BSs. More notations are provided in Table I. Let us also B. Contributions denote by u the serving BS and by v the receiving device. The received signal at the device v considering the sub-carrier This paper studies UAVs’ communication in the downlink c ∈ C can be expressed as scenario. We start by considering a communication model that N accounts for most of the propagation phenomena experienced p X p by wireless signals. We show that users’ QoS depends on yv = αuv Puxu + αtv Ptxt + nv, (1) the number of interfering BSs and the impact they cause. t=1 This leads us to address the joint problem of sub-carrier and where αuv refers to the channel gain between the transmitter power allocation. The major contributions of the paper are the u and the receiver v. The second term in the right-hand side of following: (1) accounts for the interference impact from non-serving BSs, • A realistic communication model for cellular-based UAVs where αtv is the fading coefficient from the interfering BS t is considered and new expressions for the outage proba- to the receiver v. The BSs u and t, respectively, transmit the bility are derived. The proposed model accounts for most symbols xu and xt by employing the power Pu and Pt on the of the propagation phenomena experienced by wireless same sub-carrier assigned to the receiving device v. As for the signals. Moreover, we show that the quality of service third term of (1), nv, it refers to a zero-mean complex additive experienced by the UAVs depends on the number of white Gaussian noise with variance N0. The received signal- interfering BSs and on the impact they cause. to-noise ratio (SNR) for the link uv, γuv, can be expressed • We argue and show that separating between the UAVs as and the UEs in terms of the assigned sub-carriers would γ = P |α |2/N . (2) reduce the interference impact on the served users. This is uv u uv 0 materialized through a matching game, where sub-carriers As the receiving device v can either be a UE or a UAV, and users are matched in a way to achieve this objective. the underlying fading characteristics are substantially different. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 3

푈퐴푉푣

′ 푑푈퐴푉 3퐷 푢푣 Non-serving Non-serving BS Serving BS BS 푈퐴푉 ℎ푣 퐵푆푢

퐵푆 ℎ푢 푈퐸푣

푈퐸 푑푢푣 푈퐴푉 푑푢푣 Interferences from non-serving BSs Fig. 1: System model (downlink scenario).

TABLE I: Summary of Notations. Notation Description B Set of BSs (|B| = B). U Set of UEs (|U| = U). V Set of UAVs (|V| = V ). A Set of users A = U ∪ V (|A| = A). C Set of sub-carriers (|C| = C). uv Link between the user v and its serving BS u. tv Link between the user v and an interfering BS t ([1, . . . , N] refer to the list of interfering BSs). Pu,c Power employed by the BS u ∈ B on the sub-carrier c ∈ C. P¯uv minimum transmission power a BS u should employ to send data to the user v. V (u) Set of users connected to the BS u ∈ B. W (c) Set of users assigned to the sub-carrier c ∈ C. S Set of coalition; S = {S1, S2,..., SC }. A coalition Sc is the set of users using the sub-carrier c ∈ C. w(S) Characteristic function of the coalition S. ΠS (v) The payoff of the player v ∈ S. 0 0 v1 {S1, S2} . {S1, S2} The transfer operation of the user v1 ∈ S1 to the coalition S2. The resulting state of the coalitions are respectively denoted by S1 and S2. 0 0 {S , S } ./v1 {S , S } v ∈ S v ∈ S 1 2 v2 1 2 The operation for exchanging the users 1 1 and 2 2. The resulting state of the coalitions are respectively denoted by S1 and S2.

For the case of a UE, the path loss expression provided by 3GPP is considered as [18] γ SINR = uv . (5) uv N P 1 + γtv UE UE t=1 PLuv = 15.3 + 37.6 log10 duv , (3) UE Theorem 1. Outage probability on the downlink for a UE where duv refers to the distance in meter between the BS u and the UE v as shown in Fig. 1. A Rayleigh distribution is A UE v fails in receiving packets from its serving BS u, assumed for the fast fading associated with the UEs. As for on the downlink, iff the SINRuv falls below a threshold γth. the mean SNR of the link between the BS u and the UE v, it UE This event, called outage, occurs with a probability Pout,uv is denoted by γ¯uv and can be expressed as that can be expressed as N PLUE UE γth X αt UE − uv P (γ ) = 1 + exp − , (6) 10 out,uv th γth 1 γ¯uv = Pu × 10 /N0. (4) γ¯uv + t=1 γ¯uv γ¯tv

The instantaneous received signal-to-interference-plus-noise where αt are unique values satisfying the following equality ratio (SINR) for the link uv can be deﬁned as (fractional decomposition) IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 4

N N −m m Auv X β1j β21 Y X αt 1 − x (1 − xB )−1 = + −1 uv m j 1 (1 − xγ¯tv) = 1 . (7) m (x − ) (x − ) x − j=1 Auv Buv t=1 t=1 γ¯tv (12) N N 0 N m Proof: See Appendix A. Y −1 xAtv −m X αt X X αt,j (1 − xBtv) (1 − ) = 1 + j . m x − m t=1 t=1 Btv t=1 j=1 x − Atv If the receiving device v is a UAV, the path loss differs (13) depending on the line-of-sight (LoS) and the non-line-of-sight The function f 0 (S) is provided as (NLoS) conditions. The LoS communication results in a better j,j n QoS compared to the NLoS one. For the proposed solution, X j0 j0−1 mγth(θpS + 1) f 0 (S) = S (θ ) λ Γ j, , (14) we consider the path loss equation provided by 3GPP as [3] j,j p p A p=1 uv  UAV 03D 28.0 + 22 log10(duv ) + 20 log10(fc)  where λp and θp denote the weight and the zero factors of the  for LoS link  n-th order Laguerre polynomials, respectively [19]. Γ(a, z) is UAV  PLuv = UAV UAV 03D the upper incomplete gamma function deﬁned as Γ(a, z) = −17.5 + (46 − 7 log (h )) log (d ) R ∞ a−1 −t  10 v 10 uv t e dt.  +20 log ( 40πfc ) z  10 3  for NLoS link, Proof: See Appendix A. (8) Theorems 1 and 2 provide the outage probability on the

UAV downlink for a ground UE and a UAV, respectively. The outage where the term hv refers to the altitude of the UAV v and UAV 03D expressions have been derived by taking into account path loss, the term duv accounts for the Euclidean distance between fast fading, and interference. This makes the considered system the BS u and the UAV v, as shown in Fig. 1. The probability LoS model realistic since it captures most of the propagation of a LoS condition Puv is determined as [3] phenomena experienced by wireless signals. These expressions are original and can not be found in the literature. This also  UAV reflects a major contribution compared to existing works on 1 if hv > 100  UAV LoS  1 if duv ≤ d1 cellular UAVs as those works do not derive the expression of Puv = −dUAV  d1 + exp uv 1 − d1 if dUAV > d . the outage probability.  dUAV p dUAV uv 1 uv 1 uv The expression of the outage probability in Theorems 1 (9) and 2 do not take the shadowing into account. The reason UAV with p1 = 4300 log10(hv ) − 3800 and d1 = behind that is the fact that the shadowing is constant during UAV UAV max(460 log10(hv ) − 700, 18). The term duv refers to the transmission time of one packet by either a UAV or a UE. the distance between the BS u and the UAV v, as illustrated in The mean power of the signal fluctuates due to shadowing NLoS Fig. 1. It is worth noting that the NLoS probability, Puv , and these fluctuations typically occur on a scale of a few NLoS LoS can be obtained as Puv = 1 − Puv . The corresponding hundred wavelengths [20, p. 69]. Thus the shadowing effect fast fading follows a Nakagami-m distribution for LoS links, should only be taken into account if the receiver moves a and a Rayleigh distribution for NLoS links. The mean SNRs distance of at least 200 wavelengths, which corresponds to a 1 of the LoS and NLoS links are denoted by Auv and Buv, distance of 30 m . In this paper, we focus on the downlink respectively, and are obtained as communication between BS and UAVs that is only used for

 PLUAV command and control trafﬁc, which is characterized by small LoS − u  Auv = Puv × Pu/N0 × 10 10 packet sizes (maximum of 1250 bytes). Such packets can PLUAV (10) LoS − u be transmitted in 0.1 s with a throughput of 100Kbps2. In  Buv = (1 − Puv ) × Pu/N0 × 10 10 . the 3GPP speciﬁcation [3], it is reported that the maximum UAV speed would not exceed 160 Km/h (44.44 m/s). Thus, Theorem 2. Outage probability on the downlink for a UAV during the time required for a packet transmission (0.1 s), the The outage probability for a UAV v served by the BS u is UAV moves a distance of 4.44 m (4.4 m 30 m). Therefore, expressed as the shadowing can be considered constant during a packet

m j −j N transmission from a BS to a UAV. Similarly, for the downlink UAV X (−1) m X 0 Pout,uv(γth) = β1j Γ(j) + αtfj,1(Btv) communication between a BS and UE, it is possible to show (j − 1)! Auv j=1 t=1 that the UE cannot move more than 200 wavelengths (30 m) N m j0 !! X X αt,j0 (−1) γth during a packet transmission, and thus, the shadowing can be − f 0 (A /m) − β B 1 + exp(− ) 0 j,j tv 21 uv (j − 1)! Buv t=1 j0=1 considered constant when evaluating the outage for the BS- N 0 N m j !! UE communication. A measurement campaign on multimedia X α X X αt,j (−1) t − Γ(j) , (11) γth 1 j + γth m (j − 1)! t=1 Buv Btv t=1 j=1 + 1 Buv Atv The carrier frequency is set to 2GHz; thus, the wavelength equals 0.15 0 m. where β1j, β21, αt and αt,j are unique values satisfying the 2The throughput for command and control traffic has been specified in two following equations (fractional decomposition): [3] as equal to 100Kbps. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 5 traffic in LTE [21] shows that the maximum packet size, for model, which considers most of the propagation phenomena video call traffic, reaches 1387 Bytes, which is transmitted characterizing wireless communication. with an average throughput of 1.21 Mbps. This implies that As discussed previously, the QoS on the downlink is such a packet has a transmission time of 9.1 ms. Thus, this affected by the number of interfering BSs and the impact packet witnesses a constant value of the shadowing because they cause. For this purpose, the joint problem of sub-carrier the UE would move less than 0.3 m during the transmission assignment with power optimization is considered. We propose of such a packet if we assume a speed of motion of the UE a solution based on the framework of game theory. The game of 110 km/h (UE is outdoor in a car). is divided into two independent sub-games: a matching sub- By carefully analyzing the outage expressions, we can game and a coalitional sub-game. The matching sub-game, conclude that the outage probabilities depend on the number G1, is executed at first. It aims at matching the users to of interfering BSs and on the impact they are causing as the sub-carriers in a way to meet the preferences of each reflected by the terms γ¯tv for Theorem 1 and Atv and Btv other. The preferences are defined in a way to optimize the for Theorem 2. Minimizing the outage probability would be outage probability of the users and to enhance the downlink translated into optimizing the sub-carrier assignment and the communication of cellular-based UAVs. The second sub-game, transmission power on each sub-carrier. These two parameters G2, builds on the outcome assignment of the matching game influence both the number of interfering BSs and their impact. to boost this optimization. It is based on the framework of Let Pc = [P1,c,P2,c,...,PB,c] be the transmission power coalitional games and defines operations to enable users to to be used by the B BS for the sub-carrier c. We define change their coalitions in order to enhance their QoS. The Pout,uv(Pc) as power optimization is taken into account in the two sub-games. UAV The following sections explain our proposed solution in more Pout,uv(Pc) if v ∈ V Pout,uv(Pc) = UE (15) detail. Pout,uv(Pc) if v ∈ U, UAV UE III.MATCHING SUB-GAME OPTIMIZATION where Pout,uv(Pc) and Pout,uv(Pc) are the outage probabilities for the link uv provided by Theorems 1 and 2 respectively. The framework of many-to-one matching game [22] is adopted to model the problem. A related projection of this Pc is the transmission power to be used by the set of BSs on the sub-carrier c assigned to the user v. To characterize the framework is the college admissions problem where colleges selected sub-carrier for the communication between a BS u are receiving the applications of students and each college has a fixed quota for accommodation. Both colleges and serving a user v, the decision Boolean variable xuv,c is defined as students have an order of preference. The goal is to model the interactions between these entities and to achieve a matching 1 If the sub-carrier c is used for the link uv x = stability satisfying as much as possible their preferences. uv,c 0 Otherwise. The game G is defined as G = (A, C, v, ), where (16) 1 1 c A = U∪V is the set of UEs and UAVs (considered as students) Consequently, the joint problem of sub-carrier assignment while C is the set sub-carriers (considered as colleges). v and and power optimization can be expressed as c are, respectively, the preference relation for a user v ∈ A and a sub-carrier c ∈ C. The goal of the game is to associate each user v ∈ A to a sub-carrier c ∈ C, while meeting certain min max Pout,uv(Pc) (17) v∈A constraints and preferences. Indeed, each sub-carrier c has a quota of only B (number of BSs). Moreover, a sub-carrier can s.t. accommodate at most one user connected to a given BS. This X ∀u ∈ B, ∀v ∈ V (u); xuv,c = 1 (18) is due to the fact that each of the connected users to a given c∈C BS will be served by different sub-carriers. Let W (c) be the X set of users assigned to the sub-carrier c. Consequently, the ∀u ∈ B, ∀c ∈ C; xuv,c ≤ 1 (19) matching game should maintain the following condition: v∈V (u) T ∀u ∈ B, ∀c ∈ C; 0 ≤ Pu,c ≤ Pmax. (20) ∀c ∈ C, ∀u ∈ U; | W (c) V (u)| ≤ 1. (21) The objective function in (17) is to minimize the outage To define the order of preference, one would rely on probability for the the set of users connected to the cellular equations (6) and (11) as they express the objective function network. Both sub-carrier assignment and power optimization to minimize (minimizing the outage probability for each user). are considered for minimizing the outage probability. (18) However, as it can be seen from those equations, computing ensures that each user v is assigned with one sub-carrier the outage probability for a user depends on the terms γ¯tv (for from its serving BS u. Here, V (u) refers to the set of users a UE) or Atv and Btv (for a UAV). These terms reflect the connected to the BS u, as mentioned in Table I. (19) captures interference part from the non-serving BSs t using the same the fact that a sub-carrier is used at most by one user. (20) sub-carrier as v. In other words, an interdependence is present is the feasibility constraint for the transmission power. The between users to specify a sub-carrier, based on (6) and (11), formulated optimization problem is a non-linear program, as it depends on the choice of the other users in using the which is complex to solve, especially for large networks. same sub-carrier or not. This makes the direct usage of those This complexity is inherent from the proposed communication equations for defining the order of preference very complex. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 6

In order to set a preference order for the users, we propose before, the interference terms reflect the mean SNR and to rely on minimizing the interference impact over the link depend on the power employed by the non-serving BSs. tv for each user v. We first evaluate the interference amount To reduce the interference impact, each sub-carrier would over the links tv. The associated terms reflect the mean SNR prefer having users requiring less transmission power from from the interfering BS to the user v (γ¯tv in the case of a UE, their serving BSs. In addition, a sub-carrier accommodating and Atv and Btv in the case of a UAV). The formulas for the users requiring large transmission powers would admit mean SNR are expressed in (4) and (10). Therefore, the terms fewer users. The power value P¯uv is used to compare related to the links tv depend on the power employed by the between users’ transmission power. This represents the interfering BS for transmitting data to the served users. The basis for sub-carriers to order their preferences of the users. power value would be larger than the one employed to serve Considering the first rule, the users are not able to know users in an interference-free situation. Indeed, if a user fails which sub-carriers they are gathered in, so they can make in receiving packets from its serving BS in an interference- their choices and set the preferences. In order to overcome free environment, it will definitely fail in the presence of this issue, we consider that the sub-carriers are ordered (C = interference. We consider this fact to estimate the minimum [c1 . . . cC ]). Consequently, users from the same type can easily transmission power employed by a BS u to serve its user v. be gathered in the same sub-carriers by choosing to be near to Lemma 1. A UE v will fail in receiving data from its serving one of the two ends. We consider that the UEs would prefer BS u iff the transmission power employed by the latter is less being in the first sub-carriers while the UAVs would rather be than a value given as interested in the last ones. As for the second rule, a sub-carrier uses the transmission power P¯uv to order its preferences from PLUE − uv the candidate users of each BS. It prefers at most one user per P¯uv = SNRthN0/10 10 , (22) BS, thus maintaining condition (21). where, SNRth reflects the receiver sensitivity. As for a UAV Based on this discussion, we define the preference relation v, the value is given as for users/sub-carriers and the admission criteria as follows. UAV UAV PLuv PLuv ¯ LoS − 10 NLoS − 10 Puv = SNRthN0/(Puv 10 + Puv 10 ). Definition 1. Ordering of preferences and admission criteria (23) For a UE v ∈ U, the preference order of the sub-carriers is defined as follows: Equations (22) and (23) of the above lemma can be derived UE v v v from equations (4) and (10), respectively. They provide the P ref (v) ⇐⇒ c1 c2 · · · cC . (24) minimum transmission power a BS u should employ to send For a UAV v ∈ V, the preference order of the sub-carriers is data to the user v. Below this value, the user will definitely defined as follows fail in receiving the data. A numerical evaluation of P¯uv for UAV v v v both UEs and UAVs, considering the same SNRth, shows that P ref (v) ⇐⇒ cC · · · c2 c1. (25) UAVs require less transmission power compared to ground UEs. This result is due to the better channel condition charac- Each sub-carrier c ∈ C receives applications of the candidate terizing aerial communication. Consequently, the interference users and prefers at most one user per BS. The preference impact, over the link tv, is bigger when the interfering BS t relation c is defined between two users v1 and v2 connected is serving a UE compared to the case where it is serving a to the same BS u (v1, v2 ∈ V (u)) as follows UAV. We exploit this fact to order the preferences in a way ¯ ¯ v1 c v2 ⇐⇒ Puv1 < Puv2 . (26) to reduce the interference impact. The following two rules are The above definition allows ordering the preferences for introduced. both users and sub-carriers. As discussed before, UEs prefer 1) As per the previous discussion, users of the same type being in the first sub-carriers while the UAVs would rather (either UEs or UAVs) would prefer to be gathered in the be interested in the last ones. This is, respectively, reflected v same sub-carriers. Indeed, the heterogeneity in terms of by equations (24) and (25) of Definition 1. Here, ci cj channel conditions between flying UAVs and ground UEs means that the user v prefers the sub-carrier ci on cj. On the is behind the issue of cellular UAVs. Due to the close other hand, the sub-carriers use the transmission power P¯uv to free-space signal propagation, a flying UAV might require establish the preference order of the candidate users connected less power to receive data on the downlink than what is to a BS u. As previously discussed, the preferred user is the required for a UE on the ground having the same distance. one having the smallest value of P¯uv. This is materialized Consequently, having UEs and UAVs on the same sub- through equation (26) of Definition 1. carrier might result in considerable interference on UAVs The matching will result in assigning the UEs to the first from the BSs serving UEs. Separating between UAVs and sub-carriers and the UAVs to the last ones. When there are UEs in the assigned sub-carriers would reduce this impact. some BSs having less users than the number of sub-carriers, This constitutes the preference of the users. the matching will end up with some free sub-carriers in 2) The sub-carriers will receive the application of the users. between the two types of users. However, this means that a The admitted users within a given sub-carrier will be sub-carrier in the middle might have no users while others are subject to interference from non-serving BSs whose served filled. In order to overcome this situation, we introduce virtual users are also admitted in this sub-carrier. As discussed users in the network in a way to complete the network. These IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 7 virtual users are considered to complete the number of users (11). On the other hand, when the transmission powers of for each BS. They will be removed once the matching is done. the interfering BSs are unchanged (over the links tv), the Moreover, in order to enable sub-carriers to accommodate outage probability of the user v decreases as the transmission users requiring large transmission powers to admit fewer users power from the serving BS increases (over the link uv). (rule 2), the virtual users are distributed to have more large This can be derived from the outage probability definition, ¯ PN values of the Puv parameter. A half-normal distribution is used which is Pout(γth) = P (γuv/(1 + t=1 γtv) ≤ γth). This to distribute P¯uv between the smallest and the biggest values. consideration is exploited to propose the power optimization One important notion in matching game is the stability. It described through Algorithm 1. defines the situation where players are matched and have no Algorithm 1 presents the power optimization procedure for incentive to change their association. The matching algorithm a sub-carrier c ∈ C of all the BSs. Initially, the maximum should lead to a stable situation, which is defined as follows. transmission power is used (line 1). For each BS u ∈ B, if it is serving a user v through the sub-carrier c (lines 5 and 6), Definition 2. Matching stability a power optimization is considered; The transmission power The matching is said to be unstable if there are two users v1 of the BS u is reduced until having the maximum outage and v2 connected to the same BS u (v1, v2 ∈ V (u)) such as v1 probability smaller than a threshold Pout,th, while maintaining is matched to c1 and v2 is matched to c2, although v2 prefers the power larger than the minimum value, P¯uv (lines [8-10]). c1 to c2 and c1 prefers v2 to v1. Formally, this is expressed As explained in the previous paragraph, when the transmission as powers from the interfering BSs are unchanged, the outage probability for the user v increases as the transmission power ∃u ∈ B, ∃v1, v2 ∈ V (u), ∃c1, c2 ∈ C; v1 from the serving BS decreases. Since changing the transmis- v1 ∈ W (c1) and v2 ∈ W (c2) and c2 c1 and v1 c2 v2. (27) sion power of a BS also influences the outage probability of users subject to interference, this power optimization is repeated. Lines [11-14] evaluate if the transmission power has Once the sub-carriers are matched with the users, a power changed after executing lines [8-10]. The variable ’Stable’ is optimization procedure is considered. This would allow reduc- used to characterize the stable situation, where no BS needs ing the interference impact from non-serving BSs, after the to reduce its transmission power. This will allow reducing matching process, while maintaining the outage probability the transmission power of the BSs while maintaining desired for the concerned users under a certain threshold. The power outage probabilities. optimization procedure will be considered for each sub-carrier The parameters P max, P¯ and P stp denote, respectively, of the set of BSs. As it can be noticed form (6) and (11), the uv the maximum value for the transmission power, the minimum outage probability for a user is impacted by the transmission value for the transmission power, and the unit value for power employed by the serving and interfering BSs. This is reducing the power. The complexity of Algorithm 1 in the referred respectively by the terms γ¯ and γ¯ in Theorem 1 max ¯ uv tv worst case scenario is O(S˙ ∗ B ∗ ( P −Puv + 1)), where and by the terms A ,B ,A and B in Theorem 2. This P stp uv uv tv tv S˙ shows the interdependence between the transmission powers reflects the number of unstable solutions found before the effective one. Indeed, in the worst case scenario, lines [8-10] that should be employed by the different BSs, and also the max ¯ will be executed P −Puv + 1 times for each BS u ∈ B. complexity of deriving these powers from equations (6) and P stp It is difficult to determine the value of S˙, as it depends on the outage probability expressions, reflected in Theorems 1 Algorithm 1 Power optimization for users using c ∈ C. and 2, which are very complex. We, therefore, provide in stp max the performance evaluation section some obtained results for Input: Pout,th,SNRth,P ,P max ˙ 1: Pˆc = P S along with the considered parameters. In the other hand, max ¯ 2: while True do P −Puv B O(( P stp +1) ) can be considered as the upper bound of 3: Stable = True the complexity (at most, reducing the power will be evaluated 4: for each BS u ∈ B do max ¯ T P −Puv 5: if W (c) V (u) 6= null then P stp + 1 times at each BS). 6: v = W (c) T V (u) 7: P˙c = Pˆc stp 8: while Pout,uv [P˙1,c, P˙2,c,..., P˙u,c − P ,..., P˙B,c] ≤ IV. COALITIONAL SUB-GAME OPTIMIZATION ˙ stp ¯ Pout,th and Pu,c − P > Puv do The matching game provided in the previous section will stp 9: P˙c = [P˙1,c, P˙2,c,..., P˙u,c − P ,..., P˙B,c] 10: end while result in sub-carrier assignment with power optimization. This 11: if Pˆc 6= P˙c then assignment is a heuristic and might not be optimal. The ˆ ˙ 12: Pc = Pc objective of the second sub-game is to boost this optimization 13: Stable = False 14: end if by progressively reducing the outage probability for each 15: end if user. To this end, we consider the framework of coalitional 16: end for game. The cooperative nature of this game would allow the 17: if Stable then 18: Break enforcement of the optimization achieved in the first sub-game. 19: end if Formally, the game G2 is defined as G2 = (A, S, w); with 20: end while A being the set of users (UEs and UAVs) and S the set of 21: return Pˆc coalitions. A coalition Sc represents the set of users using the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 8 sub-carrier c ∈ C. The initial coalition assignment is indeed the result of the first matching game. Having said that, the  Π 0 (v ) ≥ Π (v ) and  S 1 S1 1 number of coalitions is exactly the number of sub-carriers.  2 Π 0 (v ) ≥ Π (v )  S 2 S2 2 We can therefore write S = {S1, S2,..., SC } with Sc ⊆ A.  1  (31.1)  We note that each two coalitions involves entirely two different  And T  set of players; i.e. ∀S1, S2 ∈ S : S1 6= S2 =⇒ S1 S2 = ∅  0  ∀v ∈ S T S :Π 0 (v) ≥ Π (v) and  1 1 S S1  1 (a user connected to a BS will be served using only one sub-  T 0 ∀v ∈ S2 S2 :Π 0 (v) ≥ ΠS2 (v) carrier). w represents the characteristic function and is defined 0 0  S2 {S , S } ./v1 {S , S }⇐⇒ by the payoff of each player, w(S ) = (Π (v), v ∈ S ), as 1 2 v2 1 2 (31.2) c Sc c  And   ˆ  Π 0 (v1) > ΠS (v1) or ΠSc (v) = 1 − Pout,uv(Pc). (28)  S 1  2 Π 0 (v2) > ΠS2 (v2) or  S1  0 ∃v ∈ S T S :Π 0 (v) > Π (v) or  1 1 S S1 As it can be seen from (28), the outage probability is  1  0 ∃v ∈ S T S :Π 0 (v) > Π (v) . considered in the definition of the payoff of the corresponding  2 2 S S2  2 player belonging to a coalition. This is coupled with power (31.3) (31) optimization, Pˆc, leading therefore to an optimized outage probability for all the players in the coalition. As for the benefit The definition in (30) means that a player v1 will be of a coalition, it is defined as transferred from the coalition S1 to the coalition S2, resulting 0 0 X respectively into two coalitions S1 and S2, if the gain of this w(S ) = (Π (v)) . (29) c Sc operation on the first coalition is larger than the loss on the v∈Sc second coalition. Indeed, transferring a player from a coalition (without exchange) would be translated into withdrawing an In the context of coalitional game, the users change from interferer from the coalition. This would enhance the outage one coalition to another in order to have better payoff. Within probability of the coalition’s users and increase the underlying our framework, we define two types of operations to enable utility function. On the other hand, the receiving coalition will users to change their coalitions and increase their benefits. have a new user causing interference and leading therefore These operations are user transfer and user exchange. In the to decreased utility function. The transfer operation would first case, a user belonging to a coalition will be transferred to be approved only when the gain is greater than the loss. As another one. This could happen when the receiving coalition for the exchange rule, the definition in (31) means that two does not have another user connected to the same BS as players v1 ∈ S1 and v2 ∈ S2 will be exchanged, and leading 0 0 the candidate user. In the second case, two users connected to two coalitions S1 and S2, if at least the payoff of one to the same BS will be exchanged from their respective player will be increased after the operation (through conditions coalitions. The execution of these operations would result in in (31.3)), while that of all other players remain unaffected two new coalitions instead of the originals3. The transfer or (including both the exchanged players (31.1) and the other the exchange would be useful if it leads to enhanced payoff for players (31.2)). We note that while the transfer rule focus on the players belonging to the new resulting coalitions. Formally, the global benefit and payoff of the coalitions, the exchange we define the transfer and the exchange rules as follows: rule is more centered on the individual profit of the users. The users change from one coalition to another as defined in Definition 3. Transfer and exchange rules the above definition. This process is repeated iteratively. The situation (partition of coalitions) where there is no incentive

A user v1 ∈ S1 served by a BS u would be transferred to to do a change is said to be stable. another coalition S2 (does not have another user served by Deﬁnition 4. Coalitions stability T the same BS; i.e., V (u) S2 = ∅), resulting respectively into 0 0 A partition of coalitions is said to be stable when the two coalitions S1 and S2, iff: transfer and exchange rules can not be applied (deﬁned in

0 0 0 0 v1 equations (30) and (31)), i.e., {S1, S2} . {S1, S2} ⇐⇒ w(S1) − w(S1) > w(S2) − w(S2). 0 0 0 0 (30)  v1 ∀S1, S2 ∈ S, ∀v1 ∈ S1; @S1, S2 ⊆ A : {S1, S2} . {S1, S2} and 0 0 0 0 Two users v1 ∈ S1 and v2 ∈ S2 served by the same BS  v1 ∀S1, S2 ∈ S, ∀v1 ∈ S1, v2 ∈ S2; @S1, S2 ⊆ A: {S1, S2}./v2 {S1, S2} u (v1, v2 ∈ V (u)) would be exchanged, resulting into two (32) 0 0 coalitions S1 and S2 respectively, iff: V. GLOBAL GAMEAND GENERAL ALGORITHM Based on the framework of game theory, we propose two sub-games to enhance communication in cellular UAVs. While the matching sub-game provides an initial assignment of users to sub-carriers with power optimization, the coalitional game 3The number of coalitions is always the same. We mean by ’new coali- boosts this optimization. Algorithm 2 shows the execution tions’ the original ones after executing the operations (transfer or exchange). steps of the global game. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 9

Algorithm 2 Global game. lines 6 and 11 respectively. The target sub-carriers accept the Input: A = U S V, B users as per their preference order defined in equation (26). // 1st sub-game The admission of UEs and UAVs is shown in lines [7-8] and 1: while A\ S W (c) 6= ∅ do c∈C lines [12-13], respectively. Each rejected user will apply for S 2: T = A\ W (c) its next preferred sub-carrier. c∈C 3: QUE = T T U The coalitional sub-game considers the outcome of the UAV T 4: Q = T V matching game as initial partition of coalitions (line 17). // UEs apply for their preferred sub-carriers using P ref UE (v) 5: if QUE 6= ∅ then Thereafter, the transfer and exchange operations will be at- 6: c = lower sub-carrier not yet targeted by UEs tempted. For each two coalitions S1 and S2 and each BS u ∈ B UE UE T 7: D = {v˙ ∈ Q , v˙ c v ∀v ∈ Q V (u ˙ )} (lines [20-21]), the operations will be attempted depending 8: W (c) = W (c) S D 9: end if on whether the BS has players in these coalitions. If the BS // UAVs apply for their preferred sub-carriers using P ref UAV (v) has a player in each coalition, an exchange operation will be 10: if QUAV 6= ∅ then attempted (lines [22-27]). However, if the BS has a player in 11: c = upper sub-carrier not yet targeted by UAVs UAV UAV T only one coalition, the transfer operation will be attempted to 12: D = {v˙ ∈ Q , v˙ c v ∀v ∈ Q V (u ˙ )} 13: W (c) = W (c) S D the other one (lines [28-33] and lines [34-39]). As defined 14: end if in Definition 3, the transfer and exchange operations will // Rejected users will apply for their next preferences 15: end while be approved only when they lead to enhanced payoff. The 16: P = Pˆ // for each c ∈ C using Algorithm 1 variable ’Stable’ is used to characterize the stability situation // 2nd sub-game as defined in Definition 4; i.e., the situation where no exchange 17: Initialize S from C 18: while True do or transfer rules need to be applied. 19: Stable = True As shown in Algorithm 2, the execution of the global 20: for each two coalitions S1, S2 do game starts with the matching sub-game, then the coalitional 21: for each BS u ∈ B do T T 22: if V (u) S1 6= ∅ and V (u) S2 6= ∅ then one. The convergence of the global game depends on the T T 23: let v1 = V (u) S1 and v2 = V (u) S2 convergence of the two sub-games. In what follows, we prove // try exchanging v1 and v2 0 0 the convergence of the proposed matching and coalitional sub- v1 24: if {S1, S2} ./v2 {S1, S2} then 0 0 games. 25: S1 = S1; S2 = S2 26: Stable = False Theorem 3. Matching game convergence 27: end if T 28: else if V (u) S1 6= ∅ then The matching game of Algorithm 2 (1st sub-game) is T 29: let v1 = V (u) S1 guaranteed to converge to a stable matching. // try the transfer of v1 0 0 v1 30: if {S1, S2} . {S1, S2} then Proof: From Definition 2, a matching would be unstable 0 0 31: S1 = S1; S2 = S2 if a user and a sub-carrier consider the change an improve- 32: Stable = False 33: end if ment. This is materialized by equation (27). Let us recall the T 34: else if V (u) S2 6= ∅ then instability situation; T 35: let v2 = V (u) S2 // try the transfer of v2 0 0 v2 ∃u ∈ B, ∃v1, v2 ∈ V (u), ∃c1, c2 ∈ C; 36: if {S2, S1} . {S2, S1} then 0 0 v ∈ (c ) and v ∈ (c ) and c v1 c and v v , 37: S1 = S1; S2 = S2 1 W 1 2 W 2 2 1 1 c2 2 | {z } | {z } | {z } 38: Stable = False (33.a) (33.b) (33.c) 39: end if (33) 40: end if 41: end for 42: end for where the part (33.a) corresponds to a matching achieved after 43: if Stable then executing the algorithm, while the parts (33.b) and (33.c) 44: Break reflect the instability. Let us suppose that equation (33) is 45: end if 46: end while correct after the execution of the first sub-game of Algo- rithm 2. According to this Algorithm, the users connected to each BS apply for the sub-carriers according to their order of preferences. This is defined by equation (24) for the UEs and The matching sub-game will assign a sub-carrier c ∈ C (25) for the UAVs. to each user u ∈ A (line 1). This assignment respects At most, two sub-carriers will receive the candidates as the the ordering of preferences provided in Definition 1. In the UEs prefer the first sub-carriers while the UAVs prefer the last logic of the assignment, users (UEs or UAVs) apply for ones. Each sub-carrier prefers the users having the smallest their preferred sub-carriers, while the latter accept or reject value of P¯uv from each BS, as defined in (26). The users that the candidates according to their preferences. Lines 3 and 4 got rejected will apply for their next preferred sub-carriers. identify, respectively, the UEs and the UAVs not yet assigned Having said that, the UEs of each BS will be in the first sub- to a sub-carrier. As per the preference order of the users, the carriers while the UAVs will be in the last ones. Moreover, most preferred sub-carrier for the UEs is the lower one while the UEs are ordered in the sub-carriers according to their the upper one is the most preferred sub-carrier for the UAVs P¯uv values while the UAVs according to the inverse of this (equations (24) and (25) respectively). This is materialized in parameter. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 10

For two sub-carriers c1 and c2, the order can be either c1 addition, as each derived partition is optimized compared before c2 or the inverse. Let us suppose that c1 is ordered to the previous (transfer and exchange are approved only before c2 (the same logic holds if c2 has an order before when there is enhancements), the final partition is optimal. c1) and consider the part (33.a) of equation (33), which we Consequently, proving Theorem 4 comes down to proving that are supposing its correctness. As per the previous paragraph the derived partitions do not repeat. To this end, we consider (the order of the users in the sub-carriers), three situations are two types of operations: transfer and exchange. In the case of possible for v1 and v2, which are the following a transfer operation, equation (30) is considered. This equation  v1 ∈ U and v2 ∈ V (34.1) can also be written as   Or 0 0 0 0  {S , S } .v1 {S , S }⇐⇒w(S ) + w(S ) > w(S ) + w(S ) (33.a) =⇒ v , v ∈ U and P¯ ≤ P¯ (34.2) (34) 1 2 1 2 1 2 2 1 1 2 uv1 uv2 (36)  Or  which implies that the resulting coalitions, together, have bet-  v1, v2 ∈ V and P¯uv ≥ P¯uv (34.3) 1 2 ter payoff than the originals. In addition, user transfer from one If we consider the two situations (34.1) and (34.2), v1 ∈ coalition to another does not affect the other coalitions (i.e., U, they are in contradiction with the part (33.b), c v1 c . 2 1 ∀Si ∈ S \ {S1, S2}, w(Si) remains the same). Consequently, Indeed, if c1 is ordered before c2, UEs would prefer c1 on we can write the following c2 (equation (24)). Now, the only situation that might hold is (i) (i+1) X X (34.3). As c1 is ordered before c2, and v1 has bigger value of S → S ⇐⇒ w(S) < w(S). (37) the P¯uv parameter than v2 (UAVs are ordered according the S∈S(i) S∈S(i+1) ¯ inverse of the parameter Puv, as per the previous discussion). On the other hand, the exchange operation is ruled by Nevertheless, this also contradicts the part (33.c), v v , as 1 c2 2 equations (31). As it is clearly stated by these equations, the a sub-carrier prefers users with smaller value of the parameter ¯ exchange is approved only when at least the payoff of one Puv. Consequently, regarding equation (33), if (33.a) holds, player is increased while those of the other players, in the (33.b) or (33.c) do not. This proves the incorrectness of the two coalitions, remain unchanged. Besides, as in the case of instability assumption and concludes the proof of Theorem 3. transfer, the other coalitions are not affected by the exchange operation. These facts guarantee the equivalence in equation As per the above proof, the matching sub-game is stable and (37) even for the exchange operation. This means that each converges to a final partition. We provide in the following the derived partition is different from the previous and therefore convergence proof of the coalitional sub-game, which implies, not repeated. Moreover, the sum of benefits of the coalitions together with the matching stability, the convergence of the increases after each transformation. This concludes the proof global game. of Theorem 4. Theorem 4. Coalitional game convergence Algorithm 2 reflects the global game which is composed Starting from an initial partition of the players on the of two sub-games with different complexities. The first sub- coalitions S = {S1, S2,..., SC }, the coalitional game of game (lines [1 - 15]) represents the matching game, and its Algorithm 2 (2nd sub-game) is guaranteed to converge to a complexity is O(A) (matching each user to a sub-carrier). This final and optimal partition. A complexity can be further reduced to O( 2∗B ) if the UEs and Proof: The initial partition of the players on the coalitions the UAVs at each BS u ∈ B are being matched in parallel S is the result of the first sub-game. This coalition will be the to the sub-carriers, as presented in Algorithm 2. As for the subject of transformations applied sequentially. Let us express second sub-game (lines [18 - 46]), it reflects the coalitional this sequence by the following equation sub-game and its complexity in the worst-case scenario is O(S¨ ∗ C(C − 1) ∗ B), where S¨ is the number of unstable S(0) = S(G1) → S(1) → S(2) → S(3) ... (35) solutions found before reaching the effective one. The term where S(0) is the initial state of the coalitions which is C(C − 1) refers to the number of possible combinations the result of the matching game (S(G1)). → reflects the of two coalitions, while B is the number of BSs. Indeed, applied transformation which can be a transfer or an exchange the transfer/exchange operation will be evaluated between operation. S(i) is the state of the coalitions after the ith every two coalitions for users belonging to the same BS transformation. As the number of coalitions, the number of u ∈ B. It is very difficult to determine the value of S¨ as it players and the number of states a player can have in a depends on the outage probabilities, which are very complex. coalition (used transmission power) are limited, the number Therefore, we provide in the performance evaluation section a of partitions is also limited. We therefore define the following discussion on the results associated with S¨ and the considered lemma. parameters. On the other hand, O((C(C − 1))B) can be considered as the upper bound of the complexity (at most, Lemma 2. The convergence of the sequence of equation C(C − 1) combinations will be evaluated at each BS). We (35) is guaranteed when the transformations do not lead to can see that the first sub-game is associated with a lower repeated partitions. complexity compared to the second one. While the first game The above lemma is based on the fact that the number is meant to provide a quick solution reflected in a heuristic, of partitions is limited. If the produced partitions are not the second game starts with the achieved partition and boosts repeated, this would lead to a final partition S(final). In the optimization to further reduce the outage probabilities. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 11

Therefore, we will also compare the complexity of the second TABLE II: Summary of simulation parameters. game with the brute-force search solution. Parameter Value Execution of the Algorithm Simulation area 1000m x 1000m Algorithm 2 intends to be executed in a centralized way by Altitude of the UAVs 22.5 - 300 m [3] the cellular network providing communication to the users. BSs 10 BSs (each with 11 sub- It aims to optimize the allocation of power and sub-carriers carriers) to the connected users in a way to achieve enhanced QoS. P max, P stp 20w, 0.1w (resp.) The latter is represented by the outage probabilities for UEs fc 2 GHz and UAVs. Executing this Algorithm requires information on N0 -130 dBm [20] the base stations and also on the connected users. On the Nakagami parameter m 2 users’ side, the geographical position of each UE and UAV P 0.1 is required. On the base stations’ side, their geographical out,th positions are required in addition to the used sub-carriers and transmission powers that will be fixed by the Algorithm. All results were obtained by using 50 UEs and 50 UAVs in the this information is mainly provided by the cellular network. simulation. Fig. 2 shows that the outage increases as the value Indeed, the latter owns the base stations and has access to of γ increases. The parameter γ indicates the threshold that their geographical positions, their transmission powers and th th the SINR should exceed in order to have successful reception their used sub-carriers. Furthermore, the geographical position of the packets. It represents the receiver sensitivity threshold. of the users can be known in advance (e.g., prior to its As γ increases, the receiver’s ability to detect weak signals flight, the UAV operator submits the flight plan to the UTM th decreases. This highlights the importance of choosing a good -Unmanned aerial system Traffic Management- which can be value for γ . In the rest of the evaluations, the value 10−1 communicated to the mobile operator) or be provided by the th is considered for the parameter γ . Moreover, we can also mobile network using MPS (Mobile Positioning System). All th see from Fig. 2 that on average, the outage probability for the this information will feed the Algorithm to allow the cellular UAVs is larger than that of the UEs. This is mainly due to the network to provide enhanced QoS to the served users. On better channel condition characterizing aerial communication, the other hand, the proposed solution allows optimizing the which is translated into more interference on the UAVs. This outage probability for the connected users without considering complies with the results of real-field evaluations and shows their mobility (or using prior knowledge on their mobility). the issue of downlink communication in cellular UAVs. Extending the current solution to cover the mobility of the users will be addressed as future work. To further investigate this problem, we illustrate in Fig. 3 the impact of the number of devices (UEs and UAV) on the average outage probability. We notice that for both UEs VI.PERFORMANCE EVALUATION and UAVs, the outage probability increases as the number of This section provides the performance evaluation of the pro- deployed devices increases. However, the outage probability of posed solution for efficient communication in cellular-based UAVs increases faster compared to the outage probability of UAVs. The solution aims to enhance the QoS for the UAVs, UEs. This shows that as the number of devices in the network reflected by the outage probability, by jointly optimizing the increases, it becomes challenging to maintain a good link sub-carrier and power allocation. The simulation parameters quality for UAVs as they are more sensitive to interference. are summarized in Table II. Therefore, efficient solutions for downlink communication of Fig. 2 illustrates the average outage probability of UAVs UAVs are highly required. and UEs for different values of the parameter γth. These In Fig. 4, we evaluate the performance of our proposed

Fig. 2: Outage probabilities Pout,uv(γth) for different threshold Fig. 3: Effect of increasing the number of UEs and UAVs on values γth. the outage probability. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 12

UEs 8 UAVs

4 Number of users 3

0 1 2 3 4 5 6 7 8 9 10 11 Subcarriers Fig. 5: User distribution on the sub-carriers after running the Fig. 4: Evaluation of the outage probability for the proposed game. solutions.

Fig. 6: Comparison between the complexity of brute-force Fig. 7: Comparison of the proposed solution with brute-force search with the coalitional game search (10 BSs). solution in reducing the average outage probability of UAVs. other baseline solutions. Moreover, the gain of using our Additionally, we compare our proposed algorithm to two solution becomes more significant as the number of devices baseline solutions. The first baseline solution uses random in the network increases. In this way, if a predefined QoS is assignment of the sub-carriers, while the second baseline required for the set of UAVs, our proposed solution allows solution is inspired from graph theory and utilizes maxi- us to accommodate more devices in the network compared mum independent set for sub-carrier assignment. The second to the baseline solutions. Compared to the random partition, baseline solution was proposed in [1]. The average outage the first sub-game allowed to reduce the outage probability by probability is evaluated for different values of the number of 13.72% and 35.11% when considering 70 and 100 users in the devices in the network, and the obtained results are presented network, respectively. As for the second sub-game, it allowed in Fig. 4. The line labeled ’power optimization’ refers to the reducing the outage probability by 57.62% and 48.71% when case when directly performing the power optimization on the considering 70 and 100 users in the network, respectively. random partition without considering the rules for assigning We have also evaluated the number of unstable solutions for the sub-carriers in the matching sub-game. This allows us power optimization before the effective one. This reflects the to verify the validity of the proposed rules. By comparing parameter S˙ considered in the definition of the complexity max ¯ ˙ P −Puv this result with that from considering the first sub-game, of Algorithm 1 as O(S ∗ B ∗ ( P stp + 1)). Through the we can see that the proposed rules show their effectiveness, different evaluations, S˙ was on average equal to 6,42. It has especially when the network is saturated. From this figure, also reached 14 as a maximum value. This result complements we also notice that our proposed solution allows us to reduce the evaluation of the complexity of Algorithm 1 provided in the average outage probability significantly compared to the Section III. IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 13

Fig. 5 provides the distribution of users on the sub-carriers sub-games to tackle this complex problem. This complexity after executing Algorithm 2 (Global game). As detailed in the is inherent from the underlying realistic model. The first previous section, the last sub-game is based on the framework sub-game is based on matching game theory. It implements of coalitional game. It allows users to change their coalitions the proposition that separating between the ground UEs and in a way to enhance their payoffs. The obtained results show the flying UAVs in terms of the assigned sub-carriers would that, after reaching stability, more UEs are assigned to the first reduce the interference impact and enhance the QoS. The sub-carriers while more UAVs are assigned to the last ones. second sub-game is based on coalitional game. It organizes This comes to support the idea defended in this paper that the players based on the outcome of the first game and defines separating between UAVs and UEs in terms of the assigned the mechanisms allowing them to change their coalitions and sub-carriers would enhance the quality of communication. The to enhance the users’ QoS. Furthermore, a power optimization heterogeneity of the channel conditions between the two types algorithm is proposed for this communication model, which is of users is translated into different power requirements to be considered in the two sub-games. The performed evaluations employed by the serving BS. Consequently, having different demonstrated the effectiveness of the proposed solution in types of users in the same sub-carriers could increase the enhancing communication in cellular UAVs. interference impact. APPENDIX A As mentioned in Section V the matching sub-game is very APPENDIX:PROOFOF THEOREMS 1 AND 2 quick and its complexity is O(A). This complexity can even This appendix provides the proofs of Theorems 1 and 2. The A be reduced to O( 2∗B ) when users belonging to different BSs outage probability expressions on the downlink are derived are assigned in parallel. The second sub-game is meant to for both a UE and a UAV. This probability is defined as further reduce the outage probabilities by performing a series Pout(γth) = P (SINR ≤ γth). Let us recall the expression of of transfer/exchange operations starting from the partition the SINR which is given as achieved by the first one. The complexity of the second sub- ¨ ¨ γuv γuv γuv game is O(S ∗ C(C − 1) ∗ B), where S is the number of SINRuv = = = (A.1) N 0 P 1 + I I unstable solutions found before reaching the effective one. 1 + γtv Through the different evaluations, the average value of S¨ t=1 was 4.36 and has reached 6 as a maximum value. We have PN where, the term I = t=1 γtv includes all the interfering compared the complexity of the second sub-game (upper BSs. Consequently, the outage probability for the ink uv can bound, O((C(C − 1))B)) with that of the brute-force search be written as B γuv solution. The complexity of the latter is O((C!) ). Indeed, C! Pout(γ ) = P (SINR ≤ γ ) = P ≤ γ th th I0 th corresponds to the number of combinations of the users on the Z ∞ 0 sub-carriers of a given BS. The evaluation is reflected in Fig. 6 = EI0 P (γuv ≤ γthy|I = y) = Fγuv (γthy)PI0 (y)dy (A.2) 1 (each BS has 11 users each assigned to a sub-carrier). As it can be seen from this figure, the colaitional game is associated where the term Fγ (x) refers to the Cumulative Distri- with a very low complexity compared to the brute-force uv bution Function (CDF) of γuv (computed as Fγ (x) = search. Furthermore, we have also implemented the algorithm R x uv 0 0 Pγuv (y)dy), while PI (y) is the Probability Density Func- that looks for the optimal solution (brute-force search) and tion (PDF) of I0. The expression of these functions differs on compared the achieved average outages probabilities with that whether the receiver equipment v is a UE or a UAV. from the proposed solution. The obtained results are shown in The receiver equipment v is a UE Fig. 7. This evaluation is considered for a small network of 10 Let us compute the Moment Generating Function (MGF) and BSs, where each one has 3 sub-carriers. As we can see, the the PDF of γuv. Their expressions can be given as proposed solution achieves a near-optimal solution that has the same trend as the optimal solution generated by the brute-force UE −1 Mγuv (s) = (1 − sγ¯uv) (A.3) search, especially when the network is not complete (some UE 1 x sub-carriers are free). Pγuv (x) = exp − . (A.4) γ¯uv γ¯uv

VII.CONCLUSION The MGF of I includes all the interfering BSs and can be deduced as In this paper, we addressed the issue of downlink com- N N N Y Y −1 X αt MI (s) = Mγ (s) = (1 − sγ¯tv) = (A.5) munication in cellular network-enabled UAVs. Given the lack tv s − 1 of realistic communication models that reﬂect the underlying t=1 t=1 t=1 γ¯tv phenomena experienced by wireless signals when communi- where αt is the same as in Theorem 1, satisfying (7), and is cating with ﬂying UAVs, we introduced a model that accounts obtained using fractional decomposition (multinomial theorem for path loss, fast fading, and interference. The analysis of [23]). The PDF PI (x) of I can be obtained, by computing the this model showed that the quality of service experienced by inverse Laplace transform of MI (s) in (A.5), as the UAVs is affected by the number of interfering BSs and " N # N the impact they cause. We, therefore, considered the joint −1 −1 X αt X x PI (x) =L [MI (s)] = L 1 = αt(−1) exp − . s − γ¯tv problem of sub-carrier assignment and power optimization. t=1 γ¯tv t=1 Leveraging the framework of game theory, we proposed two (A.6) IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 14

Considering the fundamental theorem of transformation of Using (A.6) and the fundamental theorem of transformation random variables, PI0 is obtained as 0 of random variables [24], the PDF P 0 (y) of I is computed I N as X 0 y − 1 N P 0 (y) = α (−1) exp − X y − 1 I t B PI0 (y) = αt(−1) exp − . (A.7) t=1 tv γ¯tv t=1 N m j UE X X (−1) m(y − 1) j−1 As for the CDF Fγ (x) of γuv, it is determined from (A.4) + αt,j exp − (y − 1) . uv (j − 1)! A as t=1 j=1 tv Z x (A.14) UE UE x Fγuv (x) = Pγuv (y)dy = 1 − exp − . (A.8) 0 γ¯uv As for the CDF of γuv, it is computed as Z x m j Consequently, the outage probability can be computed as UAV UAV X (−1) Fγuv (x) = Pγuv (y)dy = β1j Z ∞ 0 (j − 1)! UE j=1 P (γ ) = F (γ y)P 0 (y)dy out th γuv th I ! 1 Z x my Z x y Z ∞ j−1 γthy y exp − dy − β21 exp − dy (A.15) = 1 − exp − PI0 (y)dy 0 Auv 0 Buv 1 γ¯uv | {z } | {z } K1 K2 N  −j γth X αt m mx = 1 + exp − .  K1 = Γ(j) − Γ j, γ 1 (A.9) with Auv Auv γ¯uv th + (A.16) t=1 γ¯uv γ¯tv x  K2 = Buv 1 − exp − . Buv Note that (A.9) is the outage probability provided in Theo- Finally, the outage probability can be computed as rem 1. m −j Z ∞ X (−1)j m The receiver equipment v is a UAV Pout(γ )= Fγ (γ y)P 0 (y)dy β1j Γ(j)− th uv th I (j − 1)! A Aerial communication is characterized by both LoS and NLoS 1 j=1 uv Z ∞ ! Z ∞ conditions. As defined in (10), the terms Auv and Buv reflect mγthy γthy Γ j, PI0 (y)dy −β21Buv 1− exp(− )PI0 (y)dy the mean SNR related to the two conditions. Let us compute 1 Auv 1 Buv UAV the MGF M (s) of γuv. (A.17) γuv m " j −j " N n X (−1) m X 0 X = β1j Γ(j) + α BtvλpΓ (j − 1)! A t −m uv t=1 p=1 UAV sAuv −1 j=1 Mγ (s) = Mγ Mγ = 1 − (1 − sBuv) uv LoS,uv NLoS,uv m ! N m j0 n mγth(θpBtv + 1) X X (−1) X j0 m j, − α 0 (A /m) t,j 0 tv X β1j β21 Auv (j − 1)! = + (A.10) t=1 j0=1 p=1 (s − m )j (s − 1 ) j=1 Auv Buv !## N j0−1 mγth(θp(Atv/m) + 1) X 0 λpθ Γ j, − β21Buv 1 + α where β and β are the same as in Theorem 2, satisfying p A t 1j 21 uv t=1 (12). They are obtained using fractional decomposition. The exp(− γth ) N m j −j ! UAV UAV Buv X X (−1) γth m γth PDF of Pγ (x) can be obtained from the MGF Mγ (s) γ 1 − αt,j + exp − Γ(j) uv uv th + (j − 1)! Buv Atv Buv of γ using the inverse Laplace transform as Buv Btv t=1 j=1 uv (A.18) m j −j N X (−1) m X 0 m " #! " # = β1j Γ(j) + αtfj,1(Btv) UAV X −1 1 −1 1 (j − 1)! Auv P (x) = β1j L + β21L j=1 t=1 γuv (s − m )j (s − 1 ) j=1 Auv B 0 !! uv N m (−1)j γ m X X th j − αt,j0 fj,j0 (Atv/m) − β21Buv 1 + exp − X j−1 mx (−1) x (j0 − 1)! B = β1j x exp(− ) − β21 exp(− ). (A.11) 0 uv A (j − 1)! B t=1 j =1 j=1 uv uv N 0 N m j !! X α X X αt,j (−1) t − Γ(j) . γth 1 j As for the MGF of I it is computed as + γth m (j − 1)! t=1 Buv Btv t=1 j=1 + Buv Atv N N −m (A.19) Y Y −1 sAtv MI (s) = Mγ (s) = (1 − sBtv) 1 − tv m t=1 t=1 The integrals in (A.17) involve the Gamma function with N 0 N m X α X X αt,j = t + (A.12) the exponential function. The Laguerre polynomial, defined s − 1 (s − m )j R ∞ −x Pn t=1 Btv t=1 j=1 Atv as 0 e f(x)dx = p=1 λpf(θp), is used to perform a numerical evaluation, where λ and θ are the weight and where α0 and α are the same as in Theorem 2, satisfying p p t t,j the zero factors of the n-th order Laguerre polynomials, (13). They are obtained using fractional decomposition. Now, respectively. The result in (A.18) is obtained using a change we can compute the PDF of I as of variable. Note that the expression of fj,j0 (S) is provided in   equation (14). The result in (A.19) is the same as the outage N 0 N m −1 −1 X αt X X αt,j probability expression presented in Theorem 2. PI (x) = L [MI (s)] = L  +  s − 1 (s − m )j t=1 Btv t=1 j=1 Atv Note that Theorem 1 is a special case of Theorem 2. Using N N m j j−1 the expression of the outage in (A.19), we can derive the X 0 x X X (−1) x mx = −αt exp − + αt,j exp − . Btv (j − 1)! Atv following 4 special cases. t=1 t=1 j=1 LoS NLoS LoS (A.13) Special Case 1: Puv = 0, Puv = 1, Ptv = 0, NLoS and Ptv = 1. This implies that in (A.10) the parameters IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 15

β1j = 0 and β21 = −1/Buv, whereas in (A.13) the parameters [5] X. Lin, V. Yajnanarayana, S. D. Muruganathan, S. Gao, H. Asplund, 0 H. Maattanen, M. Bergstrom, S. Euler, and Y. . E. Wang, “The sky is αt,j = 0 and α = −1/Btv. Thus, the outage probability for t not the limit: Lte for unmanned aerial vehicles,” IEEE Communications special case 1 can be computed using (A.19) as Magazine, vol. 56, no. 4, pp. 204–210, April 2018. [6] H. Hellaoui, O. Bekkouche, M. Bagaa, and T. Taleb, “Aerial control N 0 ! system for spectrum efficiency in uav-to-cellular communications,” IEEE γth X αt Communications Magazine, vol. 56, no. 10, pp. 108–113, OCTOBER Pout(γth) = 1 + exp − γ 1 , (A.20) Buv th + 2018. t=1 Buv Btv [7] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Optimal transport which matches with the outage probability provided in theory for power-efficient deployment of unmanned aerial vehicles,” in Theorem 1. 2016 IEEE International Conference on Communications (ICC), May LoS NLoS LoS 2016, pp. 1–6. Special Case 2: Puv = 1, Puv = 0, Ptv = 1, and [8] ——, “Mobile unmanned aerial vehicles (UAVs) for energy-efficient NLoS Ptv = 0. This implies that in (A.10) the parameters β1j 6= Internet of things communications,” IEEE Transactions on Wireless Communications, vol. 16, no. 11, pp. 7574–7589, Nov. 2017. 0 and β21 = 0, whereas in (A.13) the parameters αt,j 6= 0 and 0 [9] X. Liu, L. Li, F. Yang, X. Li, W. Chen, and W. Xu, “Price-based αt = 0. Thus, the outage probability for special case 2 can be power allocation for multi-uav enabled wireless networks,” in 2019 28th computed using (A.19) as Wireless and Optical Communications Conference (WOCC), 2019, pp. 1–5. [10] Z. Li, Y. Wang, M. Liu, R. Sun, Y. Chen, J. Yuan, and J. 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