Energy-Efficient UAV-Mounted RIS Assisted Mobile Edge Computing

We assume that the MEC server and the ground users are equipped with a single omni-directional antenna for each. The U-RIS is equipped with $M=M_{x}\times M_{y}$ reflective elements, forming an $M_{x}\times M_{y}$ uniform rectangular array (URA). Let $\theta_{i}[n]\in\left[0,2\pi\right),i\in\mathcal{M}\triangleq\{1,\cdots,M\}$ denote the phase of the $i$th reflecting element in time slot $n$, and $\bm{\Theta}[n]=diag\{e^{j\theta_{1}[n]},e^{j\theta_{2}[n]},\cdots,e^{j\theta_{M}[n]}\}$ be the diagonal phase array for the RIS in the $n$th time slot.

We assume the links from the ground users to the U-RIS (G-U link) and from the U-RIS to the MEC server (U-S link) follow a quasi-static block fading LoS model[9]. The link between user ${k}$ and the U-RIS in the $n$th time slot, denoted by ${\bf h}_{k}[n]\in\mathbb{C}^{M\times 1}$, can be expressed as[10]

where

$\tau[n]=\sqrt{\beta_{0}d_{k}^{-\alpha_{L}}[n]}$, where $\beta_{0}$ is the channel gain with a distance of 1 meter, $d_{k}[n]=\sqrt{\|({\bf q}[n]-{\bf w}_{k})\|^{2}+H^{2}}$ is the distance between user $k$ and the U-RIS in time slot $n$, $\alpha_{L}$ is the pass loss exponent due to the LoS transmission. Moreover, $\phi_{k}[n]$ and $\varphi_{k}[n]$ denote respectively the azimuth and elevation angles of user $k$ at time slot $n$, $\lambda$ is the wavelength of carrier and $d$ is the antenna separation. The U-S link in time slot $n$, denoted as ${\bf h}_{{\text{s}}}[n]$, can be modeled similarly.

We consider the TDMA scheme for the task offloading, which means that only one user communicates with the server in a time slot. Denote $c_{k}[n]$ as the user scheduling variable. If user $k$ is chosen to be served by the U-RIS in the $n$th time slot, we have $c_{k}[n]$=1, yielding the following constraints

Therefore, the achievable rate of user $k$ in the $n$th time slot can be expressed as

where the parameter $B$, $P_{k}$ and ${\sigma^{2}}$ denote the channel bandwidth, fixed transmission power of users and the noise variance respectively. Thus, the average achievable rate of user $k$ during the computing cycle $T$ is given by $R_{k}=\frac{1}{N}\begin{matrix}\sum_{n=1}^{N}R_{k}[n]\end{matrix}$.

To exploit the full granularity in task partitioning and computing resources, we consider the way of partial offloading[11]. Specifically, the computing tasks can be divided into arbitrary sizes, and part of the tasks can be offloaded to the server, while the remaining tasks are processed locally. Denote the total offloading and local computing tasks of user $k$ over $N$ time slots as ${l^{\text{o}}_{k}}$ and ${l^{\text{l}}_{k}}$ in bits, respectively. Let $T_{k}$ be the maximum tolerable latency of user $k$. Then we have

where $\chi_{k}$ denotes the number of CPU cycles required for processing one bit of user $k$, $f^{\text{l}}_{k}$ is user $k$’s fixed CPU frequency, and $f^{\text{o}}_{k}$ is the allocated CPU frequency to compute the task of user $k$ at the MEC server. This constraint is based on two assumptions: First, the edge computing for user $k$ does not start until $l_{k}^{o}$ bits are offloaded; second, using dynamic voltage and frequency scaling (DVFS) technique, the server can dynamically allocate its resources.

In the MEC system, the total energy consumption over $N$ time slots is composed of three main parts: the energy consumed by the users for offloading and local computing, by the server for computing, and by the U-RIS flight. The energy consumption for user $k$ can be formulated as¹¹1Actually, the offloading duration is ${{l^{\text{o}}_{k}}}/{R_{k}}$, but $T_{k}$ is mainly occupied by ${{l^{\text{o}}_{k}}}/{R_{k}}$ and the energy used for transmission is relatively small[3]. So we can replace it with $T_{k}$ to simplify the model without performance loss.

where ${\varphi_{\text{u}}}$ is the switched capacitance coefficient for the users [12]. Denote ${\varphi_{\text{s}}}$ as the coefficient for the server, and the energy consumption of the server for computing user $k$’s tasks is

We adopt the novel energy consumption model for rotary-wing UAVs proposed in [13], which takes the practical thrust-to-weight ratio into consideration, i.e.,

where $\mathit{E_{\text{p}}}[n]$ is a factor of the energy consumption for the U-RIS flight during the $n$th time slot, and $\kappa[n]=({1+\frac{4m||\mathbf{a}[n]||^{2}+\rho^{2}S_{FP}^{2}||{\mathbf{v}[n]||^{4}+4m\rho S_{FP}F[n]}}{4m^{2}g^{2}}})^{\frac{1}{2}}$ with $F[n]=||\mathbf{v}[n]||\mathbf{a}[n]\cdot\mathbf{v}[n]$. Here $m$ is the total mass of the UAV and the RIS; $\mathit{P}_{0}$, $U_{tip}^{2}$, $d_{0}$, $\rho$, $A$, and $\mathit{P_{i}}$, $S_{FP}$ are all mechanical coefficients; see [13] for more details.

Generally speaking, the energy consumption of the U-RIS is much larger than that of the users and the server, but the latter two are crucial in real MEC networks. Hence, we introduce a weight factor $\alpha$ for the sum of $E_{\text{p}}[n]$, with the weighted total energy consumption formulated as

In this letter, the definition of the energy efficiency is the ratio of total computed tasks in bits to the weighted total energy consumption of the system. Our main objective is to maximize the energy efficiency of the MEC system by jointly optimizing the user scheduling $\mathbf{C}\triangleq\{c_{k}[n],k\in\mathcal{K},n\in\mathcal{N}\}$, the phase-shift matrix of the U-RIS $\bm{\Phi}\triangleq\{\bm{\Theta}[n],n\in\mathcal{N}\}$, the U-RIS’s trajectory $\mathbf{Q}\triangleq\{\mathbf{q}[n],n\in\mathcal{N}\}$, the overall computed tasks in bits $\mathbf{l}\triangleq\{{l^{\text{o}}_{k}},{l^{\text{l}}_{k}},k\in\mathcal{K}\}$ and the CPU frequency allocation for different users of server $\mathbf{f}\triangleq\{f^{\text{o}}_{k},k\in\mathcal{K}\}$. The problem can be formulated as

where (10b) denotes the phase constraint of the U-RIS, (10c) indicates that there is a threshold $I_{k}$ for the minimum offloading tasks for user $k$, and (10d) means that the total server’s CPU frequency is limited by the maximum value $C_{\text{o}}$. Problem (10) is a challenging mixed-integer non-linear fractional programming problem where the objective function and the constraints (3) and (5) are not jointly convex w.r.t. the optimization variables. In the next section, we propose an iterative algorithm to efficiently obtain a suboptimal solution.

To handle the coupled variables, we first consider the design of the phase shift $\bm{\Phi}$. If the U-RIS chooses to serve user $k$ in time slot $n$, i.e., $c_{k}[n]=1$, the achievable rate is denoted as

where $\psi_{i}[n]\!\!\!\!=\!\!\!\!\!2(m_{x}\!\!\!\!-\!\!\!\!1)\pi d/{\lambda}(\cos\phi_{s}[n]\sin\varphi_{s}[n]\!\!\!-\!\!\!\cos\phi_{k}[n]\sin\varphi_{k}[n])\!\!\!+\!\!\!2(m_{y}\!\!\!\!-\!\!\!1)\pi d/\lambda(\sin\phi_{s}[n]\sin\varphi_{s}[n]\!\!\!-\!\!\!\sin\phi_{k}[n]\sin\varphi_{k}[n])$. Clearly, if the multipath signals superimpose coherently, the rate can reach the maximum. This means that the U-RIS can play an phase alignment role by setting $\theta_{i}[n]=-\psi_{i}[n]+{\omega},\omega\in[0,2\pi],i\in\mathcal{M},n\in\mathcal{N}$, yielding the following maximum achievable rate

where ${\xi_{k}}=\frac{P_{k}\beta_{0}^{2}M^{2}}{\sigma^{2}}$. After determining the value of $\bm{\Phi}$, we relax the integer constraint (3) into a linear form, and obtain the following standard LP problem for the optimization of $\mathbf{C}$:

where $\!\check{R}_{k}[n]\!\!=\!\!B\log_{2}(1\!+\!\frac{\xi_{k}}{d_{\text{s}}^{\alpha_{L}}[n]d_{k}^{\alpha_{L}}[n]})$. This problem can be solved by the CVX solver. Finally, $c_{k}[n],\forall k,n,$ are reconstructed as binary variables via rounding.

For any given $\mathbf{C}$ and $\bm{\Phi}$, problem (10) can be recast as

Note that problem (14) is difficult to solve due to the non-convex objective function and constraint (5). To obtain an approximate solution, we first consider convexifying the constraint. By introducing the slack variables ${\mathbf{y}}=\left\{y_{k}[n],\forall k,n\right\}$, ${\bf p}=\left\{p[n],\forall n\right\}$, where $y_{k}[n]\geq\parallel{\bf q}[n]-{\bf{w}}_{k}\parallel^{2}+H^{2}$, $p[n]\geq\parallel{\bf q}[n]-{\bf{w}}_{\text{s}}\parallel^{2}+H^{2}$, and the auxiliary variable ${\bf D}=\{d_{k},\forall k\}$, constraint (5) can be transformed into

where $\gamma_{0}[n]=\log_{2}(1+\frac{\xi_{k}}{(p[n])^{\alpha_{L}/2}(y_{k}[n])^{\alpha_{L}/2}})$. Since $y_{k}[n]$ and $p[n]$ can be increased to reduce the objective value, the constraints of $y_{k}[n]$ and $p[n]$ must hold with equality at the optimal solution to problem (14); hence, constraint (5) can be equivalently relaxed into (15) without loss of optimality. Note that $\gamma_{0}[n]$ is convex with respect to $y_{k}[n]$ and $p[n]$, so we apply the first-order Taylor expansion of $\gamma_{0}[n]$ at the given point $(p^{(t)}[n]$, $y_{k}^{(t)}[n])$ in the $t$th iteration to convert the non-convex constraint (15a) to a convex form as follows

where

and $\mathit{Y}_{k}^{(t)}[n]=(y_{k}[n]-y_{k}^{(t)}[n])$.

We next deal with the non-convex objective function. Note that $E_{\text{p}}[n]$ and $E_{k}^{\text{s}}$ are the two non-convex terms in the denominator of the objective function. To tackle the non-convexity of $E_{\text{p}}[n]$, we successively apply the inequalities $\bm{a}\!\cdot\!\mathbf{b}\!\leq\!||\bm{a}||||\mathbf{b}||,\bm{a},\mathbf{b}\!\in\!{{\mathbb{R}^{2}}}$ and $(a^{2}+b^{2})^{\frac{1}{2}}\!\leq\!(a+b),a,b\in{{\mathbb{R}_{+}}}$ for the non-convex term $F[n]$ and $\sqrt{((\kappa[n])^{2}+\frac{||\mathbf{v}[n]||^{4}}{4v_{0}^{4}})^{\frac{1}{2}}-\frac{||\mathbf{v}[n]||^{2}}{2v_{0}^{2}}}$ in $E_{\text{p}}[n]$, respectively. Finally we get a convex upper bound $E_{\text{up}}[n]$ of $E_{\text{p}}[n]$, expressed as

To handle the non-convexity of $E_{k}^{\text{s}}$, similarly we introduce the slack variable $\mathbf{u}=\{u_{k},\forall k\}$ satisfying $u_{k}\geq{l^{\text{o}}_{k}}(f^{\text{o}}_{k})^{2}$. By inequality transformation and applying first-order Taylor expansion at given point ${l^{\text{o}}_{k}}^{(t)}$ in the $t$th iteration, we have

Therefore, problem (14) can be approximated as

This problem is quasi-convex because the objective function consists of a linear numerator as well as a convex denominator and all constraints are convex. Therefore, it can be efficiently solved by existing fractional programming methods, such as the Dinkelbach algorithm.

Based on the solutions obtained in the previous two subsections, the designed overall algorithm for problem (10) is summarized in Algorithm 1. The complexity of the proposed algorithm is $\mathcal{O}(N^{3.5}\log({1}/{\epsilon}))$. In general, Algorithm 1 yields a lower bound of the original problem, and its performance in improving system energy efficiency is verified in Section IV.