Energy Optimization in Massive MIMO UAV-Aided MEC-Enabled Vehicular Networks
Emmanouel T. Michailidis, Nikolaos I. Miridakis, Angelos Michalas, Emmanouil Skondras, Dimitrios J. Vergados, Dimitrios D. Vergados
Abstract —This paper presents a novel unmanned aerial vehicle (UAV)-aided mobile edge computing (MEC) architecture for vehicular networks. It is considered that the vehicles should complete latency-critical computation-intensive tasks either locally with on-board computation units or by offloading part of their tasks to road side units (RSUs) with collocated MEC servers. In this direction, a hovering UAV can serve as an aerial RSU (ARSU) for task processing or act as an aerial relay and further offload the computation tasks to a ground RSU (GRSU). In order to significantly reduce the delay during data offloading and downloading, this architecture relies on the benefits of massive multiple-input–multiple-output (MIMO). Therefore, it is considered that the vehicles, the ARSU, and the GRSU employ large-scale antennas. A three-dimensional (3-D) geometrical representation of the MEC-enabled network is introduced and an optimization method is proposed that minimizes the weighted total energy consumption (WTEC) of the vehicles and ARSU subject to transmit power allocation, task allocation, and timeslot scheduling. The numerical results verify the theoretical derivations, emphasize on the effectiveness of the massive MIMO transmission, and provide useful engineering insights.
Index Terms —Computation offloading, energy efficiency, massive multiple-input multiple-output (MIMO), mobile edge computing (MEC), unmanned aerial vehicle (UAV), vehicular networks. I. I NTRODUCTION
ITH the emergence of the big data era at vehicular networks, the Internet of Vehicles (IoV) paradigm, and the vehicular-to-everything (V2X) information interaction, a vast number of connected automobile terminals equipped with computation and multi-communication units will pave the path
E. T. Michailidis is with the Department of Electrical and Electronics Engineering, University of West Attica, Egaleo, 12241, Greece (e-mail: [email protected]). N. I. Miridakis is with the Department of Informatics and Computer Engineering, University of West Attica, Egaleo, 12243, Greece (e-mail: [email protected]). A. Michalas is with the Department of Electrical and Computer Engineering, University of Western Macedonia, Kozani, 50131, Greece (e-mail: [email protected]). E. Skondras is with the Department of Informatics, University of Piraeus, Piraeus, 18534, Greece (e-mail: [email protected]). D. J. Vergados is with the Department of Informatics, University of Western Macedonia, Kastoria, 52100, Greece (e-mail: [email protected]). D. D. Vergados is with the Department of Informatics, University of Piraeus, Piraeus, 18534, Greece (e-mail: [email protected]). This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. for novel services [1]. The next wave of applications, including augmented reality (AR), ultra-high-quality video streaming, and autonomous driving, is expected to reach the limits of current technologies and pose strict requirements in terms of computation, latency, and throughput. For locally intra-vehicle processed applications, a large amount of energy is consumed, which in turns reduces the driving range of energy-limited electric vehicles [2]. On the other hand, it is often infeasible for resource-constrained vehicles to timely handle computation-intensive tasks. To maintain the energy consumption at a low level, while meeting critical latency demands, partly or fully task offloading to mobile edge computing (MEC) servers has been suggested [3], [4]. In this direction, road side units (RSUs) along roads and in proximity to the vehicles can facilitate the provision of MEC services. Despite such promising capabilities, attaining ubiquitous connectivity and sufficient radio coverage between vehicles and MEC servers is challenging, since ground RSUs (GRSUs) often struggle in areas with obstacles and highly mobile and disperse nodes. In this regard, hovering aerial RSUs (ARSUs) based on unmanned aerial vehicles (UAVs) can fly over connected vehicles and effectively mitigate shadowing and blockage effects thus maintaining line-of-sight (LoS) propagation [5], [6]. On the other side of the equilibrium, the massive multiple-input–multiple-output (MIMO) technology has recently received unprecedented attention as a key enabler for high data rates during data offloading/downloading and drastically reduced round-trip latency [7]. A. Background
To the best of the authors’ knowledge, only single-antenna UAV-based MEC solutions have been previously studied. Most of current work on UAV-aided MEC-enabled networks has focused on energy-aware solutions both from ground users (GUs) and UAV perspective. In [8], a UAV was deployed to assist an access point (AP) to provide MEC services to GUs and an optimization algorithm that minimizes the energy consumption was proposed. By adopting similar setups, the maximum delay [9], sum power [10], task completion time [11], average latency [12], and computation efficiency [13] were also optimized. A non-orthogonal multiple access (NOMA) scheme was studied in [14], whereas an edge-cloud system supporting virtualized network functions (VNFs) was proposed in [15]. Beyond the deterministic binary and partial task offloading, the concept of stochastic offloading was studied in [16]. In [17], the resource allocation and UAV’s
Energy Optimization in Massive MIMO UAV-Aided MEC-Enabled Vehicular Networks
Emmanouel T. Michailidis, Nikolaos I. Miridakis,
Senior Member, IEEE , Angelos Michalas, Emmanouil Skondras, Dimitrios J. Vergados, and Dimitrios D. Vergados W rajectory were optimized for a social IoV (SIoV) scenario. The advantage of employing UAVs as MEC servers in Cyber-Physical Systems (CPSs) was outlined in [18] and the three-dimensional (3-D) UAV’s trajectory was sub-optimally optimized to extend the UAV’s endurance. In [19], the use of UAV-mounted edge nodes in Long Range Wide Area Networks (LoRaWANs) was investigated. Wireless power transfer (WPT) was also introduced to prolong network’s operation time. In this respect, optimization problems were formulated to maximize the sum completed task-input bits [20] and the UAV’s required energy [21]. An Internet of Things (IoT) scenario was examined in [22], while a multi-UAV-based MEC system was proposed in [23]. In the context of terrestrial MIMO MEC networks, a multi-antenna NOMA architecture that enables multi-user computation offloading over the same time/frequency resources was proposed in [24]. Also, the optimization of energy consumption and maximum delay, under perfect and imperfect channel state information (CSI) estimation, was studied for MIMO [25] and massive MIMO [26] systems. Moreover, single-cell [27] and multi-cell [28] MEC networks that enable the simultaneous offloading of multiple APs were previously presented. The combination of massive MIMO and millimeter wave (mmWave) frequencies in wireless local area networks (WLANs) with MEC was underlined in [29]. Besides, a cell-free system consisting of multiple single/multi-antenna APs with MEC servers and a central cloud server was described in [30]. Notwithstanding, the aforementioned works are improper for UAV-based MEC networks, since the UAVs fly in a 3-D space and above rooftops leading to peculiar link geometry and especial mobility characteristics. B. Contribution
Motivated by these observations, we investigate a massive MIMO UAV-aided MEC-enabled vehicular network. The major contributions of this paper are summarized as follows: A novel dual-MEC network architecture is proposed, where a UAV operates as an ARSU equipped with a MEC server and also as an intermediate decode-and-forward (DF) aerial relay between vehicles and an GRSU. This architecture trades on the massive MIMO transmission and the efficient use of all computing resources. Moreover, a MEC computation offloading and downloading protocol is presented with distinct operation phases. Based on this protocol, partial offloading is applied to obtain a trade-off between energy consumption and delay. In order to unlock the full potential of massive MIMO, we propose the concept of triply massive MIMO, as an extension of the doubly massive MIMO [31]. Thus, it is considered that the vehicles, ARSU, and GRSU employ two-dimensional (2-D) uniform rectangular planar arrays (URPAs) with a large number of antenna elements. Although there exist some practical barriers towards the implementation of large-scale antennas regarding mainly the power consumption and complexity, we believe that these issues will be handed in the near future. Realistic 3-D placement and mobility of the vehicles, ARSU, GRSU, and URPAs is proposed, which directly affects the communication links. Position, distance, and velocity vectors are used to accommodate the geometrical representation of the proposed network architecture and construct the massive MIMO channel matrices. A multi-variable optimization problem is formulated that intends to minimize the weighted total energy consumption (WTEC) from both the vehicles and ARSU perspective and prolong their lifetime as major network segments. The Lagrange dual method is leveraged to derive closed-form solutions for the transmit power allocation, the computation bits allocation, and the time slot scheduling. A subgradient-based algorithm is also constructed, in order to expedite the optimization process. The results depict the total computation-based and communication-based delay (TCCD) and the WTEC, point out the advantage of MIMO transmission, and validate the effectiveness of the optimization procedure. C. Structure
The remainder of this paper is organized as follows. Section II introduces the system model and outlines the geometrical, mobility, and channel characteristics. Section III presents the computation offloading and downloading model. Section IV formulates the optimization problem and derives its solution, whereas Section V provides numerical results. Finally, conclusions and future directions are given in Section VI. II. S YSTEM M ODEL
Consider a MEC-enabled vehicular network that facilitates the computing offloading of K vehicles moving along a unidirectional road segment. Each vehicle has a latency-sensitive and bit-wise-independent computation task that can be executed partly locally with the on-board computing processor and partly remotely by computation offloading to network MEC servers. In this direction, a fixed GRSU with sufficiently powerful computation capacity and grid power supply is situated along the road. It is assumed that the k -th vehicle cannot directly communicate with the GRSU owing to signal blockage or severe shadowing. Thus, an ARSU is employed to enable vehicle-to-ARSU (V2U) networking facilitating the MEC services. Contrary to GRSU, the ARSU has certain computing and energy limitations that depend on its type, weight, and battery size. To preserve its energy resources, the ARSU can determine the portion of tasks that can locally process and then act as an aerial relay forwarding the remaining part of the vehicles’ offloaded tasks to GRSU. By employing a sufficiently large data buffer, the ARSU can separately store the offloaded data and the computation results. A. Geometrical Characteristics and Mobility Model
Fig. 1 illustrates the 3-D geometrical characteristics of the proposed dual-MEC architecture. To aid our analysis, the subscripts k , U , and R, where k K are associated with the k -th vehicle, the ARSU, and the GRSU, respectively. he ( x , y , z ) axes designate the global coordinate system (GCS), which controls the position of each network segment, with the projection U O of the ARSU’s array center U O positioned at the origin ( x = 0, y = 0, z = 0), where O denotes the projection of a point O onto the xy plane. Since the height of k -th vehicle (GRSU) is relatively low compared to that of the ARSU, k O R O is almost identical to . k R O O
It is considered that the k -th vehicle, the ARSU, and the GRSU employ uniform rectangular planar arrays (URPAs). The URPAs are defined by the local coordinate system (LCS), the origins of which are at URPAs’ centers. More specifically, the k -th vehicle is equipped with an URPA with k kx ky L L L antennas spanning kx L rows along the x -axis and ky L columns along the y -axis of the LCS with equal inter-element spacing , k where k L The position of URPAs in the GCS is specified by the transformation from GCS to LCS, whereas the angles / 2, / 2 x (slant angle), / 2, / 2 y (downtilt angle), and
0, 2 z (bearing angle) designate a 3-D counterclockwise rotation of LCS with reference to GCS and can also describe ARSU’s roll, pitch, and yaw, respectively [32]. Then, a sequence of rotations assigns the URPAs’ orientation. For the position vector of antenna element kmm A with kx m L ky m L we obtain , k kmm mm A RA where
X x Y y Z z x xx x
R R R R cos 0 sin cos sin 00 1 0 sin cos 0 ,sin 0 cos 0 0 1 y y z zz zy y (1) , ,0
Tk k kmm mm mm x y A is the antenna vector in the LCS, kx k kxkmm kx k kx L m m Lx L m m L (2) , ky k kykmm ky k ky L m m L y L m m L (3) and T denotes the transpose operation. By replacing the indices, the position vectors Upp A and Rqq A of antenna elements App A and , Rqq A respectively, are similarly defined. Based on Fig. 1, , ,0 kpp mm pp mm D D A is the distance vector between A kmm and A ,
Upp ,0 Upp kU pp
D D A is the distance vector between the k -vehicle array center and A ,
Upp / tan ,0, TkU U k U h h D is the distance vector between the k -vehicle and ARSU array centers, U h is the altitude of the ARSU antenna array, and k is the elevation angle of ARSU relative to . k O The other distance vectors can be similarly defined. Moreover, cos , sin , 0
Tk k k k v v and U v , , , , , cos cos , sin cos , sin TU U xy U z U xy U z U z v denote the velocity vectors of the k -vehicle and the ARSU, respectively, k U v v is the instantaneous velocity of the k -vehicle (ARSU), , k U xy is the azimuth angle that controls the moving direction of the k -vehicle (ARSU) in the azimuth domain, and , U z is the elevation angle that characterizes possible rising, diving, and hovering operations of the ARSU. The total energy of ARSU is limited. Thus, the flying period is restrained by . U T For convenience, adequately small constant is used to divide U T into N timeslots. In each timeslot n N the k -th vehicle and the ARSU can be considered to be static, whereas their antenna position vectors are updated, respectively, as k kmm mm k n n n A A v and U Upp pp U n n n A A v with kmm t A k kmm mm n n A A and . U U Upp pp pp t n n A A A
Note that the distance vectors are also updated accordingly. Based on [18], we model the energy consumption during flight in the n -th timeslot for a fixed-wing ARSU as , fl fw U xy U zU xy cE n c n c nn v vv (4) z y R O k v x R O U O Upp A R , qq pp D UR D L Ux
1 2 δ U δ U Rqq A L Rx
1 2 L Ry δ R δ R R O ARSU URPA GRSU URPA A Upp A Rqq A Rqq ,0 qq D k U O k kmm A L kx
1 2 L ky δ k δ k -th vehicle URPA k A kmm ,0 pp D , pp mm D A kmm k O kU D k O k O L Uy , U xy v U v , U xy , U z , U z v Fig. 1. The geometrical characteristics of the proposed massive MIMO UAV-aided MEC-enabled vehicular network, where an ARSU assists the k -th vehicle execute its offloaded computing task and also acts as an aerial relay to further transmit part of this task to an GRSU for computing. here c and c are constants depending on the ARSU's weight, wing area, and air density, c is a constant associated with ARSU's descending/ascending, , U xy n v and , U z n v are the horizontal and vertical ARSU velocity vector, respectively, with , , , U U xy U z n n n v v v and is the Euclidean norm. For a rotary-wing ARSU, the energy consumption during flight can be modelled as [18] U xyfl rw r U xy nE n P d s G nv v v
U xy U xy U z n nP P nv v v v v (5) where P and P describe the blade profile power and induced power, respectively, P controls the descending/ascending power, tip v is the tip speed of rotor blade, v is the mean rotor induced velocity, r d is the fuselage drag ratio, s is the rotor solidity, ρ is the air density, and G is the rotor disc area. B. Wireless Transmission Model
In each timeslot the k -th vehicle and the ARSU move over a small distance. Thus, the channel coefficients are keeping unchanged and can be estimated using uplink and downlink orthogonal pilot sequences (prior to data transmission) at the start of each timeslot [33]. Overall, the channel is described by a series of channel snapshots for different placement of the k -th vehicle and the ARSU in each timeslot. It is assumed that the V2U, ARSU to GRSU (U2R), GRSU to ARSU (R2U), and ARSU to k -vehicle (U2V) channels are dominated by LoS links within the short frame , U T as also indicated by recent measurements in several propagation environments [34]-[36]. Then, the massive MIMO channel between the k -vehicle and the ARSU in the n -th time slot can be described by the matrix U k
L LkU n G containing the channel coefficients , pp mm g n between antenna elements A kmm and A .
Upp Based on [31], [33], we obtain , , pp mm kU pp mm g n n h n and , kU kU kU n n n G H F where kU kU n n D is the free space path-loss component, is the channel gain at a reference distance d is the path-loss exponent, which is assumed identical to all links and closely follows the exponent observed in free-space path-loss i.e., 2 [37], , , , exp 2 , pp mm pp mm pp mm h n j f n n (6) , , , U kkU pp mm k Upp mm U kkU pp mm n n n n nf n n n n D A A v vD A A (7) , U kkU pp mmpp mm n n nn
D A A (8) , is the inner product operator, λ is the carrier wavelength, , U k kU pp mm L L n h n H is a U k
L L matrix, and k k kU kU L L n n F is a k k L L diagonal matrix. Note that kU n is assumed to be constant over many coherence time intervals and is invariable over k L and , U L since , . kU k U n D By using (6)-(8) and properly replacing the indices, the channel matrices for the other links can be similarly defined. However, in the U2R and R2U cases, the channel coefficients are only affected by the movement of ARSU, since GRSU is static. III. C OMPUTATION O FFLOADING AND D OWNLOADING M ODEL
We define , , k k k k l c b the computation task of the k -th vehicle, where k c is the number of required central processing unit (CPU) cycles per bit, k b is the task-input data size (in bits), k k c b is the total required number of CPU cycles, and k is the proportionality ratio between offloaded data and computed results. The maximum CPU frequency at the k -th vehicle and the ARSU is denoted as ,max k f and ,max , U f respectively, with ,max ,max , k U f f whereas U c denotes the number of required CPU cycles per bit at the ARSU. Since the computational resources of the k -th vehicle are limited, more computing power is required to accomplish its task within the maximum allowable latency (task deadline) . k U T In this regard, partial task offloading is exploited. Hence, the k -th vehicle offloads to ARSU and GRSU (via relaying) part of its task in each timeslot. The computation task at the k -vehicle in a given timeslot is partitioned as , , , ,min , k k l k U k R k b n b n b n b n b n (9) where , , , , , 0, k l k U k R b n b n b n (10) , , k l b n , , k U b n and , k R b n are the computation bits allocated for local computing, offloading to ARSU for computing, and offloading to GRSU for computing via ARSU, respectively. Besides, ,min k b are the minimum task bits that should be periodically completed in each timeslot. Note that the case that k U T is only considered in this paper . k A. Transmission Delay and Computation Delay
Contrary to previous work on conventional single-antenna MEC networks (e.g., [8], [22]), the use of massive MIMO can meaningfully shorten the offloading transmission time thus rendering the duration of data offloading, computing, and downloading comparable. It is assumed that the downloading transmission time for the R2U channel can be omitted, since here are no transmit power constraints at the GRSU side. Also, it is assumed that the computing time at the GRSU is negligible, whereas the decision time for task partitioning is very short compared to the entire latency and can be neglected as well. To implement the data offloading and downloading processes, while avoiding interference among the vehicles, each timeslot is fairly divided into K equal durations Kk k n with . K kk n Then, in the time duration off off dow dow, , , , , k k k U k cU k U k R n n n n n n five operation phases are considered, as shown in Fig. 2. More specifically, off , / k k UR kU n b n R is the transmission time for offloading the bits , , , k UR k U k R b n b n b n from the k -th vehicle to the ARSU (Phase 1); off, , / k U k R UR n b n r is the transmission time for offloading , k R b n from ARSU to GRSU via relaying (Phase 2); , , ,max / k cU U k U U n c b n f is the computation delay at the ARSU (Phase 3); dow, , / k U k U Uk n b n R is the transmission time for downloading , , k U k k U b n b n from the ARSU to the k -th vehicle (Phase 4); and dow, , / k R k R Uk n b n R is the transmission time for downloading , , k R k k R b n b n from the ARSU to the k -th vehicle (Phase 5). Note that off 2min , ,0 2 0 01 log 1 k U L L k kU lkU kl p n nr n B B N L (11) is the achievable rate of the V2U massive MIMO channel, off k p n is the transmit power in Phase 1, B is the allocated bandwidth, N is the variance of the additive white Gaussian noise (AWGN) at ARSU, and min ,, 1 k U L LkU l l n are the singular values of . kU n G It is noted that the achievable rates , UR r , RU r and Uk r of the U2R, R2U, and U2V massive MIMO channels, respectively, can be obtained by using (11) and appropriately replacing the indices. Each vehicle can simultaneously carry out local computing and bits offloading, whereas the local computation delay , , ,max / k cl k k l k n c b n f can span a timeslot. Overall, the following time allocation constraints should be satisfied: ,off off dow dow, , , , k clk k U k cU k U k R nn n n n n K K (12) off off dow dow, , , , . k k U k cU k U k R n n n n n K (13) It is assumed that the bits offloading (or bits downloading) cannot outreach the rate capabilities of the massive MIMO channels. Thus, we obtain the following constraints: Without loss of generality, we assume that N is the variance of the AWGN at any network node. Timeslot 1
Timeslot n Timeslot N / T N off k n off, k U n , k cU n dow, k U n dow, k R n Bits offloading to ARSU from k -th vehicle from ARSUfrom ARSUBits offloadingto GRSU to k -th vehicle(no relaying) Bits downloading from ARSUto k -th vehicle(relaying)Bits downloading Phase 1
Phase 2
Phase 3
Phase 4
Phase 5
Computing at ARSU n / K Task Deadline Flying period U T k n K n , k cl n Fig. 2. The proposed computation bits offloading/downloading protocol.
It is assumed that the bits offloading (or bits downloading) cannot outreach the rate capabilities of the massive MIMO channels. Thus, we obtain the following constraints: off, , k UR k kU b n n r n (14) off, , , k R k U UR b n n r n (15) dow, , , k U k U Uk b n n r n (16) dow, , . k R k R Uk b n n r n (17) B. Energy Consumption
The energy consumed for data offloading (or computed data downloading) in the Phases 1, 2, 4, and 5, respectively, can be expressed as off off off off off,max k k k k k
E n p n n P n (18) off off off off off, , , , ,max , k U k U k U k A k U
E n p n n P n (19) dow dow dow dow dow, , , , ,max , k U k U k U k U k U
E n p n n P n (20) dow dow dow dow dow, , , , ,max , k R k R k R k R k R
E n p n n P n (21) where off , , k U p n dow , , k U p n and dow , k R p n are the transmit powers in Phases 2, 4, and 5, respectively, and off,max , k P off, ,max , k U P dow, ,max , k U P and dow, ,max k R P are the maximum transmit powers in Phases 1, 2, 4, and, 5, respectively. During task processing, energy for computation is also consumed. The power consumption of the CPU at the k -th vehicle and at the ARSU is modeled as
3, ,max k cl k k
P f and
3, ,max , k cU U U P f respectively [38], where k and U are the effective capacitance coefficient that rely on the chip architecture at the k -th vehicle and at the ARSU, respectively. In each timeslot, the energy consumption for local intra-vehicle and ARSU computing can be, respectively, expressed as [38]
33 2, , , , , k cl k cl k cl k k k l E n P n c b n (22)
33 2 2, , , , . k cU k cU k cU U U k U E n P n c K b n (23) V. P ROBLEM F ORMULATION AND O PTIMIZATION
Considering a dual-MEC UAV-aided massive MIMO vehicular network, a novel multi-variable optimization problem is formulated to minimize the WTEC. This problem is explicitly subjected to physical layer parameters, such as transmit power allocation from each vehicle and the ARSU, as well as the massive MIMO uplink and downlink data rates. Also, this problem accounts for timeslot scheduling and task allocation. Towards this end, the optimization problem is formulated as , , 1 1
P1 : min
N Ktotal k k U Un k
E w E n w E n
B P τ (24a) s.t. (9), (10), (12-21) (24b) where total E is the WTEC, , , , , , , k l k U k R b n b n b n B off off dow dow, , , , , , , k k U k U k R p n p n p n p n P off off, , , k k U n n τ dow dow, , , k U k R n n are the optimizing variables, k w and U w are the weight factors of energy consumption of ARSU and k -th vehicle, respectively, and off, k k cl k E n E n E n
33 2 off off, , k k k l k k c b n p n n (25) off dow dow, , , ,1 3off off 3 2 2, , ,1 KU k U k cU k U k RkK k U k U U U k Uk
E n E n E n E n E np n n c K b n dow dow dow dow, , , , k U k U k R k R p n n p n n (26) are the energy consumption of the k -vehicle and the ARSU, respectively, in each timeslot, including the energy consumed for offloading/downloading and bits computation. The weights can be modified according to energy demands and trade-offs and also provide priority/fairness among the vehicles. One observes that total E in problem (P1) is an increasing function of the offloaded data. Hence, offloading should be realized, only if local computation violates latency constraints. Lemma 1:
Problem (P1) is a convex problem.
Proof:
From (24a) (25), and (26), we can conclude that the objective function of problem (P1) is convex with respect to , P , , k l b n and , , k U b n since its Hessian matrix is positive semidefinite. Also, the expressions in constraints (9), (10), (12), (13), and (18)-(21) are linear. Moreover, the right-hand-side of (14)-(17) are concave, since , log 1 / f x t t x t with t is concave [39]. Therefore, Problem (P1) is a convex problem. To solve Problem (P1), the Lagrangian dual method is leveraged. First, the non-negative dual variables
6, , 1 k n are introduced, each associated with one of the constraints in (9), (13)-(17). Then, the Lagrange function of problem (P1) can be expressed as in (27), shown at the bottom of this page, where , x , x , x , x , x and x denote the sets of
1, , , k n
2, , , k n
3, , , k n
4, , , k n
5, , , k n
6, , , k n respectively. Thus, the dual function of problem (P1) can be written as , , , , , min , , , , , , , , B P τ x x x x x x B P τ x x x x x x (28a) s.t. (10), (12), (18)-(21) (28b) Based on the results on [22], , , , , , x x x x x x is bounded, provided that
3, , 4, , 6, , 1, , k n k n k n k k n Then, the dual problem of problem (P1) can be expressed as
33 2 off off1 2 3 4 5 6 ,1 1 3off off 3 2 2 dow dow dow dow, , , , , , ,1 11, , ,min 1, , ,1 1 1 , , , , , , , , = + N K k k k k l k k kn kN KU k U k U U U k U k U k U k R k Rn kN K Kk n k k n k ln k k w c b n w p n n w p n n c K b n p n n p n nb n b n
B P τ x x x x x x
2, , 3, , 5, , 1, , ,,max1 1 1 2, , off off3, , 4, , 6, , 1, , , 2, , 2, , ,1 1 1 1 1 1 1 1dow2, , ,1
N N K k n U k n k n k k n k UUn n kN K N K N K N Kk nk n k n k n k k n k R k n k k n k Un k n k n k n kK k n k Un k c b nfb n n nKn off 2min , ,dow off2, , , 3, , 0 2 0 01 1 1 1 1 1off 2min , , ,off dow4, , , 0 2 5, , , 00 01 1 1 1 1 log 1log 1 l k UU R
L LN N K N K k kU lk n k R k n k kn k n k lL LN K N Kk U UR lk n k U k n k UUn k l n k p n nn n B B N Lp n nn B n BB N L dow 2min , , ,2 0 01dow 2min , , ,dow6, , , 0 2 0 01 1 1 og 1log 1 .
U kU k
L L k U Uk lUlL LN K k R Uk lk n k R Un k l p n nB N Lp n nn B B N L (27)
P1-dual: max , , , , , x x x x x x x x x x x x (29a) s.t. , , , , , 0 x x x x x x (29b)
3, , 4, , 6, , 1, , k n k n k n k k n (29c) Since problem (P1) is convex, the Slater’s condition is satisfied [39]. Owing to the strong duality between (P1) and (P1-dual), we obtain the optimal solution of problem (P1) by solving its dual problem, i.e., problem (P1-dual). For arbitrary values of , x we obtain the dual function by solving the problem defined in (29a)-(29c). This problem can be rewritten into a set of KN independent subproblems. We further decompose these subproblems into several subproblems as off off off off2, ,, L1 : min k k k k k n kn p n w p n n off 2min , ,off3, , 0 2 0 01 log 1 k U L L k kU lk n k kl p n nn B B N L s.t. (12), (18) off off, , off off, 2, , ,, L2 : min k U k U
U k U k n k Un p n w p n n off 2min , , ,off4, , , 0 2 0 01 log 1 U R
L L k U UR lk n k U Ul p n nn B B N L s.t. (12), (19) dow dow, , dow dow, 2, , ,, L3 : min k U k U
U k U k n k Un p n w p n n dow 2min , , ,dow5, , , 0 2 0 01 log 1 U k
L L k U Uk lk n k U Ul p n nn B B N L s.t. (12), (20) dow dow, , dow dow, 2, , ,, L4 : min k R k R
U k R k n k Rn p n w p n n dow 2min , , ,dow6, , , 0 2 0 01 log 1 U k
L L k R Uk lk n k R Ul p n nn B B N L s.t. (12), (21) ,
33 2, 1, , ,
L5 : min k l k k k k l k n k lb n w c b n b n s.t. (10), (12) ,
33 2 2,
L6 : min k U
U U U k Ub n w c K b n
2, , 3, , 5, , 1, , ,,max k n U k n k n k k n k UU c b nf s.t. (10), (12) ,
3, , 4, , 6, , 1, , ,
L7 : min k R k n k n k n k k n k Rb n b n s.t. (10) Since these subproblems are convex, their solutions satisfy the Karush–Kuhn–Tucker (KKT) conditions. The Lagrangian of subproblem (L1) can be written as off off1 1, , 2, , 3, , 4, , 2, ,off 2min , ,off3, , 0 2 0 01off off off off1, , 2, , 3, , , , , log 1 k U k n k n k n k n k k k n kL L k kU lk n k klk n k k n k k n k k w p n np n nn B B N Ln n p n n off off off off4, , ,max , k n k k k k P n p n n (30) where
1, , , k n
2, , , k n
3, , , k n and
4, , k n are non-negative Lagrange multipliers associated with the constraints off k n off / , k n K off off k k p n n and off off off off,max , k k k k p n n P n respectively, which are specified in (12) and (18). Based on KKT, the complementary slackness conditions can be expressed as off1, , k n k n off2, , k n k n off off3, , k n k k p n n and off off off off4, , ,max k n k k k k P n p n n
Thus, the optimal transmit power at Phase 1 can be obtained by applying KKT conditions, setting the derivative of the Lagrangian of subproblem (L1) with respect to off k p n to zero, and using numerical solving techniques. Then, off* k n can be obtained by substituting off* k p n into subproblem (L1) and is expressed as in (31a), shown at the bottom of the next page. Using (31a), we also obtain: off* off* off* . k k k E n p n n (31b) By using similar KKT-based methods, the solutions to the remaining subproblems (L2)-(L4) can be directly computed. To derive a closed-form solution and provide insights into problem (L1), the special, but common, case of highly correlated rank-1 LoS massive MIMO channels is studied, which leads to the lower bound of the achievable rate [29], [33]. Then, kU n H has one non-zero singular value. Next, the upper bound of the achievable rate is investigated, where this channel is full-rank and all of its singular values are nonzero and equal. Unlike in Rayleigh fading channels, in this special case, orthogonality among the spatially multiplexed signals can be attained, under strict geometrical constraints and specific orientation of the arrays [32]. Proposition 1: The optimal transmit power and offloading time at Phase 1 for the lower bound of the achievable rate can be, respectively, obtained as off,max
3, ,off* 0, 0 0 ln 2 Φ k Pk n kk lb k kU
N Lp n B w n (32a) Although it is infeasible to adjust the placement of URPAs in highly mobile vehicular scenarios, the special case of full-rank channels is studied, since it corresponds to the theoretical upper bound of the achievable rate. nd as in (32b), shown at the bottom of this page, where min , 2 ,1 k U
L LHkU kU kU kU ll n tr n n n G G and H and tr represent the conjugate transpose and trace of matrix, respectively. Moreover, the optimal transmit power and offloading time at Phase 1 for the upper bound of the achievable rate can be, respectively, obtained as off,max off* off*, , 0 min , k Pk ub k U k lb p n L L p n (33a) and as in (33b), shown at the top of the next page.
Proof:
The lower bound of the achievable rate can be expressed as [29], [33] off, 0 2 0 0 log 1 . k kUkU lb k p n nr n B B N L (34) Using (34) instead of (11), and solving the equation off1 1, , 2, , 3, , 4, , , , , / 0, k n k n k n k n k p n we obtain the optimal solution in (32a). Then, the optimal solution in (32b) can be obtained by substituting off* , k p n into subproblem (L1). Also, the upper bound of the achievable rate is expressed as [29], [33] off, 0 2 0 0 min , log 1 .min , k kUkU ub k U k k U p n nr n B L L B N L L L (35) Similarly, using (35), one can obtain (33a) and then (33b). Remark 1: From (32a) and (33a), one concludes that there exists a linear relationship between the optimized transmitted power of a rank-1 and that of a full-rank channel, whereas off* off*, , . k lb k ub p n p n As min , k U L L increases off*, k ub p n linearly increases and leads to increased WTEC. Clearly, the massive MIMO channels and the system geometry directly affect the energy optimization. In addition, larger weight values correspond to a decreased optimized transmit power.
Next, we provide recommendations for energy-efficient task allocation. By solving subproblem (L5) with the aid of KKT conditions, we obtain the following optimal solution ,max
1, ,*, 3 0 .3 k k f ck nk l k k k b n w c (36) Moreover, by solving subproblem (L6) via KKT conditions, the optimal solution can be written as ,max ,* ,3, ,max 0 , , 0,30, <0 U U f ck n k nk U U U U U k n b n K w c f (37) where , ,max 1, , 3, , 5, , 2, , . k n U k n k n k n k k n U f c Using (36) and (37), we obtain the optimal computation delays * *, , ,max / k cl k k l k n c b n f and * *, , ,max / . k cU U k U U n c b n f In addition, by solving subproblem (L7), the optimal solution reads as
3, , 4, , 6, , 1, ,*, 3, , 4, , 6, , 1, ,
0, 0 ,, 0 k n k n k n k k nk R k n k n k n k k n b n (38) where is an arbitrary non-negative constant. Remark 2: The expressions in (36) and (37) indicate that the weight values directly control the division of the task-input bits and the corresponding computation delays. As k w and U w increase, less task-input data is processed locally and at ARSU, respectively. Also, *, k U b n decreases, as k increases. Moreover, the vehicles and the ARSU would choose to offload data to ARSU and GRSU, respectively, for computing, as far as *, k k l b n b n and *, , , k UR k U b n b n respectively. off* 2min , ,off* 2, , 3, , 0 2 0 01 off* 2min , ,off* off* 2, , 3, , 0 2 0 01off* 2, , log 1 00, , log 1 00, k Uk U L L k kU l k k k n k n kl L L k kU lk k k k n k n klk k k p n nw p n BK B N Lp n nt n w p n BK B N Lw p n off* 2min , ,, 3, , 0 2 0 01 ,log 1 0 k U L L k kU ln k n kl p n nB B N L (31a) off*,off*, 2, , 3, , 0 2 0 0off*,off* off*, , 2, , 3, , 0 2 off*, 2, , 3, , 0 , log 1 00, , log 1 00, lo k lb kUk k lb k n k n k k lb kUk lb k k lb k n k n kk k lb k n k n p n nw p n BK B N Lp n nn w p n BK B N Lw p n B off*,2 ,g 1 0 k lb kU k p n nB N L (32b) off*,off*, 2, , 3, , 0 2 0 0off*,off* off*, , 2, , 3, , 0 2 0 0 , min , log 1 0min ,0, , min , log 1 0min ,0, k ub kUk k ub k n k n k U k k Uk ub kUk ub k k ub k n k n k U k k U p n nw p n B L LK B N L L Lp n nn w p n B L LK B N L L L off*,off*, 2, , 3, , 0 2 0 0 . min , log 1 0min , k ub kUk k ub k n k n k U k k U p n nw p n B L L B N L L L (33b) Henceforth, predetermined dual variables are considered. The next step of the optimization procedure is to obtain the optimal dual variables and solve problem P1-dual. Since problem P1-dual is generally non-differentiable, the ellipsoid method [39] is adopted to obtain a solution. The subgradient of the objective function can be represented by , , , , , ,
TT T T T T T x x x x x x where off 2min , ,off1 , 0 2 0 01 log 1 , k U
L L k kU lk UR k kl p n nb n n B B N L x (39a) off 2min , , ,off2 , , 0 2 0 01 log 1 , U R
L L k U UR lk R k U Ul p n nb n n B B N L x (39b) dow 2min , , ,dow3 , , 0 2 0 01 log 1 , U k
L L k U Uk lU R k U Ul p n nb n n B B N L x (39c) dow 2min , , ,dow4 , , 0 2 0 01 log 1 , U k
L L k R Uk lk R k R Ul p n nb n n B B N L x (39d) , k k l k U k R b n b n b n b n x (39e) off off dow dow6 , , , , . k k U k cU k U k R n n n n n x (39f) Since * τ and *, k g b n are not unique, we formulate the following linear programming problem: , off, 1 1 P2 : min k R
N K k k U Ub n n k w E n w E n τ (40a) s.t. (10), (12), (13), (18)-(21) (40b) * *, , , ,min k l k U k R k b n b n b n b n (40c) * off off*, , k U k R k kU k b n b n n r p n (40d) off off*, , , k R k U UR k U b n n r p n (40e) * dow dow*, , , k U k k U Uk k U b n n r p n (40f) dow dow*, , , k R k k R Uk k R b n n r p n (40g) where off dow dow, , ,1 . KU k U k U k Rk
E n E n E n E n Thus, problem (P2) should be solved, in order to obtain the optimal solution to primal problem (P1). Based on the previous results and observations, the subgradient-based Algorithm 1 is proposed to optimally solve this problem. The complexity and running time of Algorithm 1 depends on the number of time slots and the number of vehicles. More importantly, the main complexity of Algorithm 1 lies in steps 4 to 6, where the complexity is , O KN , O KN and
O K N [39]. respectively. Hence, Algorithm 1 has a total complexity of . O K N
Finally, in Step 9, the complexity mainly depends on solving problem (P2) by CVX [40].
Algorithm 1
Optimal Solution to Problem (P1) 1:
Set the values of the system parameters and the value of the tolerant threshold ε . 2: Initialize the iteration index, the dual variables x and the ellipsoid. Then, obtain the channel matrices using (6)-(8) and decompose these matrices via singular value decomposition (SVD) to obtain the singular values. Repeat Solve subproblems (L1)-(L4) with KKT conditions and obtain * P and * . τ Then, use (36) and (37) and obtain *, k l b n and *, , k U b n respectively. Calculate the WTEC. Solve problem P1-dual and calculate the subgradients of the objective function and the constraints. 6:
Update x according to the ellipsoid method. End Repeat until convergence. Let x x Update * P by solving subproblems (L1)-(L4) with KKT conditions. Use (36) and (37) and obtain *, k l b n and *, , k U b n respectively . Then, obtain *, k R b n and * τ by solving problem (P2) by CVX. Finally, obtain the minimum WTEC. D EFINITION , N OTATION , AND V ALUES OF S YSTEM P ARAMETERS
Parameter Value
Number of vehicles: K λ k -th vehicle, ARSU, and GRSU array, respectively: , , k U R L L L
36, 36, 36 Inter-element spacing at k -th vehicle, ARSU, and GRSU antenna array, respectively: , , k U R λ /2 Slant, downtilt, and bearing angle, respectively: , , x y z π /3, π /4, π /3 Initial elevation angle of ARSU relative to , O , O , O , R O respectively: , , , R π /3, π/4, π/6, π /3 Velocity and moving direction of k -th vehicle in the azimuth domain, respectively: , k v k
60 km/h, π/3 Velocity and moving direction of ARSU in the azimuth (elevation) domain, respectively: , , , U U xy U z v
10 m/s, π /3 ( π /9) Initial height of ARSU: U h
10 m [8] Fixed-wing parameters: , c , c c [18] Rotary-wing parameters: , tip v , v , r d , s , , G , P , P P
12 30 0.4 / 8, sG G α d β -50 dB [22] Task deadline (flight duration of ARSU): U T off,max , k P off, ,max , k U P dow, ,max , k U P dow, ,max k R P
35 dBm [22] Variance of AWGN at k -th vehicle, ARSU, and GRSU: N -130 dBm/Hz [22] Bandwidth for uplink (or downlink): B k -th vehicle (ARSU): ,max ,max k U f f k -th vehicle (ARSU): k U c c cycles/bit [22] CPU capacitance coefficient at k -th vehicle (ARSU): k U -27 [22] Weight for energy consumption for k -th vehicle (ARSU): k U w w k V. N UMERICAL R ESULTS AND D ISCUSSION
In this section, numerical results are presented to illustrate the TCCD N KTCCD kn k n and the WTEC for different values of the key system parameters and under latency constraints. The effectiveness of the massive MIMO and the optimization method is also studied, whereas the convergence performance of the proposed algorithm is also evaluated. The results take into account the number of antennas, the number of vehicles, the computation task size, the relative location of the ARSU with respect to the vehicles (GRSU), the time horizon , U T the velocity and weight of ARSU, and the proportionality ratio between offloaded data and computed results. The vector kV B is utilized to represent the set of required computation data (in Mbits), in which the k -th entry stands for the required computation task for the k -th vehicle per timeslot. Without loss of generality, it is assumed that , k U R L L L whereas the vehicles have the identical task requirement. Unless otherwise stated, the values of system parameters are listed in Table I. Two main computation schemes are studied as: (i)
Non-optimized scheme:
In each timeslot, it is considered that the computation delay spans half of the maximum allowable time duration. Thus, using (12), , ,max / 2 k l k k b f c bits are locally computed, , , k UR k k l b n b n b bits are transmitted to ARSU, , ,max / 2 k U U U b f c K bits are computed at ARSU, and , , , k R k UR k U b n b n b bits are transmitted to GRSU. Also, the maximum transmit power is used. (ii) Optimized scheme:
To demonstrate the benefits of jointly optimizing the computation and communication procedures, in each timeslot, the optimized number of computation bits is used, as discussed in Section IV. Also, the optimized transmit power is used and the optimized time duration is associated with each operation phase. Fig. 3 shows the non-optimized and optimized TCCD of Phases 1-5 as a function of the number of vehicles for a rotary-wing ARSU, URPAs with different number of antenna elements, and task requirement k b n per time slot. Clearly, the delay substantially decreases with the number of antennas and grows with the number of vehicles. Besides, the number of supported vehicles changes with the number of antenna elements. To provide computing services to three vehicles, while satisfying the stringent latency constraints, URPAs with at least 16-elements are required. Meanwhile, the optimized scheme supports a larger number of vehicles, when compared with the non-optimized one, thus revealing the effectiveness of our optimization method. Fig. 4 shows the non-optimized and optimized TCCD as a function of the number of antenna elements for a rotary-wing ARSU and varying task requirement . V B One observes that the TCCD significantly decreases, as the number of antennas increases, owing to the higher data rates and the lower transmission delay. As the number of antennas increases from 16 to 64, up to 1 Mbits and 1.55 Mbits can be supported for 11the non-optimized and optimized scheme, respectively. Also, using URPAS with relatively small dimensions, e.g., 36-element URPAs of 0.45 m x 0.45 m size for the commonly used 2 GHz carrier frequency, up to 0.82 Mbits can be computed per time slot. Besides, using a similar setup and single-antennas less than 0.2 Mbits can be timely executed. Thus, the benefits and feasibility from integrating large-scale antennas on size-constrained conventional vehicles and currently available commercial off-the-shelf UAVs is affirmed. Overall, a reasonable number of antennas should be employed, according to the amount of offloaded data, in order to attain acceptable TCCD, while satisfying practical antenna size constraints. By using mmWave frequency bands, which are potentially available for air-to-ground-communications [41], the antenna arrays can be even more compact and more demanding tasks can be handled. Fig. 5 investigates the impact of the altitude of a hovering rotary-wing ARSU on the non-optimized WTEC for 1, K b n and varying initial elevation angle of the ARSU with respect to the vehicle, i.e., , and GRSU, i.e., . R As U h increases, the ARSU draws away from both the vehicle and GRSU and more energy is consumed. It is also evident that the WTEC fairly increases as decreases, since the quality of the V2U and U2V channels in terms of the path-loss and correlation is somehow degraded. However, changing R is even less influential and negligibly affects the WTEC. Previous results on single-antenna configurations [8], stated that the UAV should be closed to ground nodes to ensure low offloading/downloading energy consumption and support large task sizes. Nevertheless, in this paper, the use of massive MIMO promises meaningfully lower transmission delays and enhanced rates that obliterate such indications and compensate the increased path-loss observed in larger distances. Therefore, this paper suggests that the ARSU should not necessarily approach the vehicles and/or GRSU to attain satisfactory WTEC and/or meliorate possible side effects of unstable U h due to obstacles and wind/pressure variableness. By avoiding aimless movements, a significant amount of propulsion energy can be saved thus extending the endurance of ARSU. Fig. 3. The non-optimized and optimized TCCD as a function of the number of vehicles for varying number of antennas.
Fig. 4. The non-optimized and optimized TCCD as a function of the task requirement per time slot for varying number of antennas.
Fig. 5. The non-optimized WTEC as a function of the altitude of the ARSU for varying elevation angle of the ARSU.
Figs. 6 illustrates the non-optimized and optimized WTEC as a function of the task requirement per time slot for a fixed-wing ARSU and six computing scenarios, including local computing, full offloading and partial offloading. Apparently, the GRSU is not necessary to assist on computation for small values of task bits, e.g., 0.1 Mbits per time slot. However, local computing is subject to a maximum computing capability ,max / , t k k f c according to (12). Besides, exploiting only the ARSU for computing leads to even greater WTEC, since wireless transmission consumes additional energy, whereas (23) indicates that the energy consumption for ARSU computing exponentially increases with K. In order to extend the supported number of task bits, the vehicles and the ARSU may cooperatively handle the computation process with an acceptable growth of the WTEC. On the other hand, demanding computation tasks presuppose the participation of the GRSU for efficient edge computing and slight energy cost. As the task bits increase the vehicles should tend to transmit ideally the entire amount of task bits to the GRSU via the ARSU. Clearly, the curves of the optimized schemes outperform the non-optimized ones and depict the advantages 12of partial offloading by capitalizing on the local computation resources, as well as the MEC resources at ARSU and GRSU. Also, the difference between the non-optimized and optimized schemes enlarges with the task requirement, while the common scenario of rank-1 channels constitutes the most energy-efficient solution. Fig. 7 shows the curves of the optimized WSEC as a function of the task completion time (flying period) for a fixed-wing ARSU for varying , U v , U w and , U when the task requirement is , , 0.6,0.6,0.6 V b b b B Mbits and . One observes that the consumed energy drastically and linearly increases, as the stringent deadline increases. It is also obvious that WTEC increases as U v and U w step up, since the propulsion energy contributes more to the WTEC. Highly-intensive computation tasks, e.g., video-rendering applications and delivery of 360 o videos, may lead to k [42]. However, owing to the enhanced spectral efficiency offered by the massive MIMO channels, changing k negligibly affects the WTEC. This is not the case for conventional single-antenna scenarios [8]. Fig. 6. The non-optimized and optimized WTEC as a function of the task requirement per time slot for different computing scenarios.
Fig. 7. The optimized WTEC as a function of the task completion time for varying velocity of the ARSU, weight factor of the ARSU, and task size ratio of output data to input data.
Fig. 8. The optimized WTEC as a function of the number of iterations for varying task requirement and number of antenna elements.
Finally, Fig. 8 investigates the convergence efficiency of the proposed Algorithm 1 and demonstrates the optimized WTEC for a fixed-wing ARSU and tolerant threshold as a function of the iteration index. It can be seen that the proposed optimized WTEC scheme nearly converges after about 7 iterations, regardless of the task sizes and the number of antennas, thus achieving computational effectiveness. VI. C ONCLUSIONS AND F UTURE R ESEARCH D IRECTIONS
In this paper, a novel WTEC optimization problem for a delay-constrained massive MIMO UAV-aided MEC-enabled vehicular network has been formulated. This problem has been decomposed into multiple convex subproblems that have been solved by the Lagrangian dual method and an efficient subgradient-based algorithm. Capitalizing on the convenient form of the closed-form solutions, numerical calculations have been carried out to illustrate the mathematical derivations. We showed that the number of antennas determines the number of supported vehicles and the size of offloaded data, under latency constraints. It has been also demonstrated that the vehicles may perform local computation for low task requirements. As task bits increase, partial task offloading is necessary. Since the velocity and weight factor of ARSU control the propulsion energy, the proposed approach has underlined that massive MIMO can counterbalance the distance-dependent path-loss and reduce purposeless mobility of ARSU. This work can be expanded into various fertile research areas. As a pre-determined ARSU’s trajectory is considered, the 3-D trajectory optimization is envisioned as a future work. Multiple ARSUs and GRSUs can be utilized along with learning-based methods for intelligent control, in order to extend the network range. Also, massive connectivity can be ensured by adopting NOMA, while mm-wave frequencies can further increase the array gain. R
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