Latency Minimization for Intelligent Reflecting Surface Aided Mobile Edge Computing
Tong Bai, Cunhua Pan, Yansha Deng, Maged Elkashlan, Arumugam Nallanathan, Lajos Hanzo
11 Latency Minimization for Intelligent ReflectingSurface Aided Mobile Edge Computing
Tong Bai,
Member, IEEE , Cunhua Pan,
Member, IEEE , Yansha Deng,
Member, IEEE ,Maged Elkashlan,
Member, IEEE , Arumugam Nallanathan,
Fellow, IEEE and Lajos Hanzo,
Fellow, IEEE
Abstract —Computation off-loading in mobile edge computing(MEC) systems constitutes an efficient paradigm of supportingresource-intensive applications on mobile devices. However, thebenefit of MEC cannot be fully exploited, when the com-munications link used for off-loading computational tasks ishostile. Fortunately, the propagation-induced impairments maybe mitigated by intelligent reflecting surfaces (IRS), which arecapable of enhancing both the spectral- and energy-efficiency.Specifically, an IRS comprises an IRS controller and a largenumber of passive reflecting elements, each of which may imposea phase shift on the incident signal, thus collaboratively improv-ing the propagation environment. In this paper, the beneficialrole of IRSs is investigated in MEC systems, where single-antenna devices may opt for off-loading a fraction of theircomputational tasks to the edge computing node via a multi-antenna access point with the aid of an IRS. Pertinent latency-minimization problems are formulated for both single-device andmulti-device scenarios, subject to practical constraints imposedon both the edge computing capability and the IRS phase shiftdesign. To solve this problem, the block coordinate descent(BCD) technique is invoked to decouple the original problem intotwo subproblems, and then the computing and communicationssettings are alternatively optimized using low-complexity iterativealgorithms. It is demonstrated that our IRS-aided MEC systemis capable of significantly outperforming the conventional MECsystem operating without IRSs. Quantitatively, about
20 % computational latency reduction is achieved over the conventionalMEC system in a single cell of a
300 m radius and activedevices, relying on a -antenna access point. Index Terms —Intelligent reflecting surface, mobile edge com-puting, latency minimization.
I. I
NTRODUCTION
A. Motivation and Scope
In the Internet-of-Things (IoT) era, myriads of machines andsensors are envisioned to be connected [1]. However, sincethese devices typically have limited computing capabilities,resource-intensive applications cannot be readily supported bythese devices due to their resultant excessive computationallatency. Aimed at tackling this issue, powerful computingnodes can be deployed at the edge of the network (typically
T. Bai, C. Pan, M. Elkashlan, and A. Nallanathan are with theSchool of Electronic Engineering and Computer Science, QueenMary University of London, London, E1 4NS, U.K. (e-mail:[email protected], [email protected], [email protected],[email protected]).Y. Deng is with the Department of Engineering, King’s College London,London, WC2R 2LS, U.K. (e-mail: [email protected]).L. Hanzo is with the School of Electronics and Computer Sci-ence, University of Southampton, Southampton, SO17 1BJ, U.K. (e-mail:[email protected]).(Corresponding author: Cunhua Pan and Tong Bai) co-located with the access points (APs)) [2]. As a benefit, thecomputational latency of these resource-intensive applicationscan be reduced, by employing both local computing on thedevices and edge computing for processing these computa-tional tasks, provided that these tasks can be successfully off-loaded. This paradigm is referred to as mobile edge computing(MEC) [3]–[14]. At the time of writing, the potential of thisMEC paradigm has not been fully exploited, predominantlybecause the computation off-loading link is far from perfect.For example, the devices located at the cell edge typicallysuffer from a low off-loading success rate, and/or their compu-tation off-loading may impose higher latency than computingtheir tasks locally. Hence these devices have to rely on theirown computing resources, which is however often incapableof supporting resource-intensive applications. Therefore, it isimperative to improve the performance of MEC systems froma communications perspective.The recent advances in programmable meta-materials [15]facilitate the construction of intelligent reflecting surfaces(IRSs) [16] for enhancing both the spectral- and energy-efficiency of wireless communications. Specifically, an IRSis comprised of an IRS controller and a large number ofpassive reflecting elements. Under the instructions of the IRScontroller, each IRS reflecting element is capable of adjustingboth the amplitude and the phase of the signals reflected, thuscollaboratively modifying the signal propagation environment.The gain attained by IRSs is based on the combination of boththe virtual array gain and the reflection-aided beamforminggain. To elaborate, the virtual array gain can be achieved bycombining both the direct and IRS-reflected signals, while thereflection-aided beamforming gain is realized by proactivelycontrolling the phase shift induced by the IRS elements. Bybeneficially combining these two types of gains, the IRSbecomes capable of boosting the devices’ off-loading successrate, hence improving the potential of MEC systems. In thistreatise, our attention is focused on investigating the role ofIRSs in MEC systems.
B. Related Works1) Design of Mobile Edge Computing Systems:
At thecurrent state-of-the-art, MEC systems can be categorized into[5]: single-user [6]–[9] and multi-user systems [10]–[14].Among the design metrics of single-user MEC systems, thecomputation off-loading strategy plays a crucial role. Moreexplicitly, the binary off-loading strategy of [6] was proposedto decide whether the task is executed locally at the mobile a r X i v : . [ ee ss . SP ] M a y device or remotely at the edge-cloud node. By contrast, Wang et al. [7] conceived a partial off-loading scheme for data-partitioning oriented applications, where a fraction of the datacan be processed at the mobile device, while the rest atthe edge. However, in realistic multi-user systems, inter-userinterference is imposed both on the radio communications linkand on the computing node at the edge, which may erodethe overall performance of the MEC system. In order to copewith this hindrance, Sardellitti et al. [10] jointly optimized thetransmit precoding matrices and the computational resourcesallocated to each user in a multi-cell multi-user scenario, whileSheng et al. [17] proposed an energy-efficient algorithm tooptimize the resource allocation of terminals, radio accessnetworks, and edge servers in a multi-carrier scenario. Forthe system where the devices have to make their off-loadingdecisions locally, Chen et al. [11] provided a distributed jointcomputation off-loading and channel selection policy relyingon classic game theory. Recently, a specific user associationscheme was also developed for multi-user systems served bymultiple edge computing nodes [13], while a mobility-awaredynamic service scheduling algorithm was proposed for MECsystems [14]. Furthermore, Yang et al. [18] conceived anover-the-air computation aided federated learning algorithmfor reducing both the latency and the power consumption, andfor preserving the users’ privacy in MEC systems. At the timeof writing, the computation offloading issue of the devicesin the face of hostile communications environments has notbeen well addressed. Against this background, in this paperthe performance is improved by invoking IRSs. Let us nowcontinue by reviewing the relevant research contributions onIRSs as follows.
2) Intelligent Reflecting Surface Aided Wireless Networks:
In order to explore the benefits of IRSs in wireless com-munications, extensive research efforts have been investedinto their ergodic capacity analysis [19], channel estimation[20], and practical reflection phase shift modeling [21], aswell as into the associated phase shift design [22]–[29].Specifically, a joint design of the IRS phase shift and ofthe precoding at the AP was proposed for minimizing thetransmit power, while maintaining the target receive signal-to-interference-plus-noise ratio (SINR) [22], relying on thesophisticated techniques of the semidefinite relaxation andof alternating optimization. These investigations were thenextended to the more practical discrete phase shift setting[23]. However, the excessive computational complexity of thealgorithm developed in [22] prohibits its application in large-scale IRSs. In order to reduce the complexity, Guo et al. [24] proposed three low-complexity algorithms, while Pan etal. [26] provided a pair of majorization-minimization (MM)algorithms and complex circle manifold methods for multi-cell scenarios. Furthermore, in order to reduce the overheadduring the IRS channel estimation, Yang et al. [29] groupedthe IRS elements, where each group shares the same phaseshift coefficient, and optimized the power allocation and phaseshift in orthogonal frequency division multiplexing (OFDM)-based wireless systems. Apart from the conventional commu-nications scenarios, the role of IRSs was also investigatedboth in terms of improving physical-layer security [30]–[33], and simultaneous wireless information and power transfer(SWIPT) [34], [35], where substantial gains were achieved.These impressive research contributions inspired us to exploitthe beneficial role of IRSs in MEC systems.
C. Contributions and Organizations
Our main contributions are the employment of IRSs in MECsystems, and the joint design of computing and communica-tions for minimizing the computational latency of IRS-aidedMEC systems, detailed as follows. • New IRS-aided MEC system design and latency mini-mization problem formulation:
In order to further exploitthe potential of MEC systems, we first propose IRS-aided MEC systems, for assisting the computational taskoff-loading of mobile devices. A latency-minimizationproblem is formulated for multi-device scenarios, whichoptimizes the computation off-loading volume, the edgecomputing resource allocation, the multi-user detection(MUD) matrix, and the IRS phase shift, subject to boththe total edge computing capability and to the IRS phaseshift constraints. Owing to the coupling effect of multipleoptimization variables, the latency-minimization problemcannot be solved directly. Hence, relying on the blockcoordinate descent (BCD) technique, the original prob-lem is decoupled into two subproblems for alternativelyoptimizing computing and communications settings. • Computing design:
Given a fixed communications setting,we decouple the computation off-loading volume and theedge computing resource allocation, again using the BCDtechnique. Our analysis reveals that given a fixed edgecomputing resource allocation, the optimal computationoff-loading volume can be determined by assuming theequivalence of the latency induced by local computingand by edge computing. Given a fixed computation off-loading volume, the subproblem is proved to be a convexproblem, and the optimal edge computing resource canbe found by relying on the KKT conditions and on theclassic bisection search method. • Communications design:
Given a computing setting, theobjective function (OF) becomes available in a non-convex sum-of-ratios form, which cannot be solved usingthe algorithms developed in [22], [24], [26], [35]. Totackle this challenge, this problem is transformed to anequivalent parameterized form by introducing auxiliaryvariables. Our analysis reveals that this equivalent formcan be decomposed into a series of tractable subproblems.Then, an iterative algorithm is developed to find thesolution. In each iteration, the auxiliary variables areupdated using the modified Newton’s method, while uponreformulating this series of tractable subproblems byexploiting the equivalence between the weighted sum-rate maximization problem and the weighted mean squareerror (MSE) minimization problem, closed-form expres-sions are provided for the MUD matrix and for the IRSphase shift, using the weighted minimum MSE methodand MM algorithm, respectively. Our analysis reveals thatthe proposed algorithm exhibits a low complexity. • Study of the single-device scenario:
In order to completethe investigations, the single-device scenario is also stud-ied, where neither the edge computing resource allocationnor the multi-user interference has to be considered. Alow-complexity iterative algorithm is proposed by sim-plifying the algorithm developed in the aforementionedmulti-device scenario. • Numerical validations and evaluations:
The numericalresults verify the convergence of the proposed algorithms,and quantify the performance of our IRS-aided MECsystem in terms of its latency in diverse simulationenvironments.The rest of the paper is organized as follows. In Section II,we establish the system model and formulate the latencyminimization problem. The solution of this latency minimiza-tion problem is provided in Section III. In Section IV, weinvestigate the solution of the special case, where a singledevice is served by the MEC system. Our numerical resultsare discussed in Section V. Finally, our conclusions are offeredin Section VI.
Notation:
In this paper, scalars are denoted by italic letters.Boldface lower- and upper-case letters denote vectors andmatrices, respectively; C M × N represents the space of M × N complex matrices; III N denotes an N × N identity matrix; j denotes the imaginary unit, i.e. j = − . The maths operationsused throughout the paper are summarized in Table I. Table I: Math operations
Notation Operation xxx T the transpose of xxxXXX T the transpose of XXXxxx ∗ the complex conjugate of xxxXXX ∗ the complex conjugate of XXXxxx H the Hermitian transpose of xxxXXX H the Hermitian transpose of XXXXXX − the inverse of XXX (cid:74) the Hadamard product (cid:60){·} the real part of a complex numberarg {·} the argument of a complex number | · | the absolute value of a scalar (cid:107) · (cid:107) the 2-norm of a vector (cid:98)·(cid:99) the floor of a scalar (cid:100)·(cid:101) the ceiling of a scalardiag ( xxx ) the diagonal matrix wherethe diagonal elements are xxx diag ( XXX ) the vector whose elements arethe diagonal elements of XXX CN (0 , σ ) Circularly Symmetric Complex Gaussianassociated with zero-mean and variance σ II. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
In this section, our system model is elaborated on, from bothcommunications and computing perspectives. Following this, alatency-minimization problem is formulated for our IRS-aidedMEC system, detailed as follows.
A. Communications Model
As shown in Fig. 1, we consider an MEC system operatingin a single-cell scenario, where K single-antenna devices Figure 1: Illustration of the system model, where a N -element intelligentreflecting surface (IRS) assists the computation off-loading of K single-antenna devices to the edge computing node via the M -antenna access point. may opt for off-loading a certain fraction of or all of theircomputational tasks to an edge computing node via an M -antenna AP through the wireless transmission link. The edgecomputing node and the AP are assumed to be co-locatedand connected using high-throughput low-latency optical fiber.Then, the latency imposed by the data communication betweenthe AP and the edge computing node is deemed to be negligi-ble. An IRS comprised of N reflecting elements is placed inthe cell for assisting the devices’ computation off-loading. Weassume that both the antenna spacing at the AP and the elementspacing of the IRS are high enough so that the small-scalefading associated both with two different antennas and withtwo different reflecting elements is independent, respectively.The equivalent baseband channels spanning from the k -thdevice to the AP, and from the k -th device to the IRS, as wellas from the IRS to the AP are denoted by hhh d,k ∈ C M × , hhh r,k ∈ C N × , and GGG ∈ C M × N , respectively. These chan-nels are assumed to be perfectly estimated and quasi-static,hence remaining near-constant when devices are scheduledfor off-loading their computational tasks. As for the IRS,we simply set the amplitude reflection coefficient to forall reflection elements and denote the phase shift coefficientvector by θθθ = [ θ , θ , . . . , θ N ] T , where θ n ∈ [0 , π ) for all n ∈ { , , . . . , N } . Then, we have the reflection-coefficientmatrix of the IRS
ΘΘΘ = diag (cid:8) e jθ , e jθ , . . . , e jθ N (cid:9) , where j represents the imaginary unit. It is assumed that the IRS phaseshift setting is calculated at the AP in accordance with boththe channel and computing dynamics, which is then sent tothe IRS controller along the dedicated channel. The compositedevice-IRS-AP channel is modeled as a concatenation of thedevice-IRS link, the IRS reflection characterized by its phaseshift, and the IRS-AP link.Here, it is assumed that the computation off-loading ofthe K devices takes place over a given frequency band B within the same time resource. Upon denoting the off-loading power, and off-loading signal of the K devices, aswell as the noise vector by p t , sss = [ s , s , . . . , s K ] T , and Naturally, this assumption is idealistic. Hence the algorithm developed inthis paper can be deemed to represent the best-case bound for the latencyperformance of realistic scenarios. Due to the associated hardware limitations, only a limited number ofdiscrete phase shifts can be provided for each IRS element in practice [23].Our proposed algorithms provide the best-case bound for the latency ofrealistic scenarios. The phase-quantization effects are evaluated in Section V-A3. nnn = [ n , n , . . . , n M ] T , respectively, the signal yyy ∈ C M × received at the AP is readily formulated as yyy = √ p t HHHsss + nnn = √ p t K (cid:88) k =1 (cid:0) hhh d,k + GGG
ΘΘΘ hhh r,k (cid:1) s k + nnn, (1)where we assume n m ∼ CN (0 , σ ) for m = 1 , , . . . , M .Furthermore, we define hhh k (cid:44) hhh d,k + GGG
ΘΘΘ hhh r,k and
HHH (cid:44) (cid:2) hhh , hhh , . . . , hhh K (cid:3) . As a computational complexity compromiseat the AP, a linear MUD technique is invoked. Upon denotingthe MUD matrix by WWW ∈ C M × K , the signal recovered at theAP is obtained as ˆ sss = WWW H yyy = WWW H ( √ p t HHHsss + nnn ) . (2)As for the k -th device, its recovered signal is formulated as ˆ s k = Hk (cid:34) √ p t K (cid:88) j =1 (cid:0) hhh d,j + GGG
ΘΘΘ hhh r,j (cid:1) s j + nnn (cid:35) , (3)where k is the k -th column of the matrix WWW . Then, the SINRof the k -th device’s signal recovered is given by γ k ( k , θθθ ) = p t (cid:12)(cid:12) Hk (cid:0) hhh d,k + GGG
ΘΘΘ hhh r,k (cid:1)(cid:12)(cid:12) p t (cid:80) Kj =1 ,j (cid:54) = k (cid:12)(cid:12) Hk (cid:0) hhh d,j + GGG
ΘΘΘ hhh r,j (cid:1)(cid:12)(cid:12) + σ | Hk | . (4)Accordingly, upon assuming a perfect capacity-achievingtransmission scheme is invoked, we arrive at the maximumachievable computation off-loading rate of the k -th device,formulated as R k ( k , θθθ ) = B log (cid:2) γ k ( k , θθθ ) (cid:3) . (5) B. Computing Model
We consider the data-partitioning based application of [7],where a fraction of the data can be processed locally, while theother part can be off-loaded to the edge node. The computingmodel is detailed for the local and edge computing as follows. • Local computing:
For the k -th device, L k , (cid:96) k , and c k are used to represent its total number of bits to beprocessed, its computation off-loading volume in termsof the number of bits, and the number of CPU cyclesrequired to process a single bit, respectively. As forthe local computing, upon denoting the computationalcapability at the k -th device in terms of the numberof CPU cycles per second by f lk , the time requiredfor carrying out the local computation is formulated as D lk ( (cid:96) k ) = ( L k − (cid:96) k ) c k /f lk . • Edge computing: we denote the maximum number ofexecutable CPU cycles at the edge and the computationalcapability allocated to the k -th device by f e total and f ek ,respectively, which obey (cid:80) Kk =1 f ek ≤ f e total . Here, it isassumed that the edge computing for the k -th deviceonly begins its operation, when all its (cid:96) k bits are com-pletely off-loaded. In this case, the total latency of edgecomputing is jointly constituted by the computation off-loading, and by the edge computing, as well as by theend-to-end delay of sending the computational result back. Given that the computation result is typically ofa small size [5], the feedback latency can be negligible,upon using the technique of ultra-reliable low-latencycommunications [36]. Then, the total latency imposed bythe computation off-loading and the edge computing isgiven by D ek ( k , θθθ, (cid:96) k , f ek ) = (cid:96) k /R k ( k , θθθ ) + (cid:96) k c k /f ek .To this end, the latency of the k -th device can be readilycalculated by selecting the maximum value between thoseimposed by the local and by the edge computing, formulatedas D k ( k , θθθ, (cid:96) k , f ek ) = max (cid:8) D lk ( (cid:96) k ) , D ek ( k , θθθ, (cid:96) k , f ek ) (cid:9) (6) = max (cid:26) ( L k − (cid:96) k ) c k f lk , (cid:96) k R k ( k , θθθ ) + (cid:96) k c k f ek (cid:27) . C. Problem Formulation
In this paper, we aim for minimizing the weighted compu-tational latency of all the devices, by jointly optimizing thecomputation off-loading volume (cid:96)(cid:96)(cid:96) = [ (cid:96) , (cid:96) , . . . , (cid:96) K ] T , theedge computing resources fff e = [ f e , f e , . . . , f eK ] T allocatedto each device, the MUD matrix WWW , and the IRS phase shift θθθ . Specifically, the weighted delay minimization problem isformulated as P WWW ,θθθ,(cid:96)(cid:96)(cid:96),fff e K (cid:88) k =1 (cid:36) k D k ( k , θθθ, (cid:96) k , f ek ) s.t. ≤ θ n < π, n = 1 , , . . . , N,(cid:96) k ∈ { , , . . . , L k } , k = 1 , , . . . , K, K (cid:88) k =1 f ek ≤ f e total ,f ek ≥ , k = 1 , , . . . , K. (7a)(7b)(7c)(7d)where (cid:36) k represents the weight of the k -th device. (7a)specifies the range of the phase shift of the IRS elements; (7b)indicates that the computation off-loading volume should bean integer between and L k for the k -th device; Finally, (7c)and (7d) restrict the range of the edge computing resourcesallocated to each device. Remark 1.
In Problem P , we have a total of four optimiza-tion variables, namely, the off-loading volume, edge computingresource allocation, MUD matrix, and IRS phase shift. Theoptimization of the former two variables is related to thecomputing setting, while the optimization of the other twospecifies the communications design. The difficulties of solvingProblem P are owing to three aspects. The first one is thesegmented form of the OF. The second one is the couplingeffect between the MUD matrix WWW and the IRS phase shiftvector θθθ . The final one is that the OF is non-convex regardingthe phase shift θθθ . Hence, it is an open challenge to obtaina globally optimal solution directly. In this paper, a locallyoptimal solution is provided. Specifically, upon using thepopular BCD technique for decoupling the communicationsand computing designs, the segmented form of the OF can beeasily transformed to a more tractable form. Similarly, optimalsolutions can be provided for the MUD matrix
WWW and forthe IRS phase shift vector θθθ , after they are decoupled using the BCD technique in the communications design. To tacklethe non-convexity regarding θθθ , the Majorization-Minimization(MM) algorithm is invoked, which is capable of iterativelyapproaching a locally optimal solution at low complexity.
III. J
OINT O PTIMIZATION OF C OMPUTING AND C OMMUNICATIONS S ETTING
The joint optimization of computing and computationssettings is realized relying on the BCD technique. The pivotalidea of the BCD technique is to optimize one of the variableswhile fixing the other variables in an alternating manner,until the convergence of the OF is achieved. In the rest ofthis section, the joint optimization of the off-loading volumeand of the edge computing resource allocation is presentedwhile fixing the communications setting, followed by the jointoptimization of the MUD matrix and of the IRS phase shiftwhile fixing the computing setting. Our goal is the jointoptimization both of the communications and of the computingdesign.
A. Joint Optimization of the Off-loading Volume and the EdgeComputing Resource Allocation While Fixing the Communica-tions Settings
Given an MUD matrix
WWW and an IRS phase shift vector θθθ ,Problem P can be simplified to P (cid:96)(cid:96)(cid:96),fff e K (cid:88) k =1 (cid:36) k D k ( (cid:96) k , f ek ) s.t. (7b) , (7c) , (7d) . (8a)The optimization of (cid:96)(cid:96)(cid:96) and fff e can be decoupled, relying on theaforementioned BCD technique, detailed as follows.
1) Optimization of (cid:96)(cid:96)(cid:96) : The value of (cid:96)(cid:96)(cid:96) can be optimized, withthe aid of the proposition below.
Proposition 1.
Given an MUD matrix
WWW , and an IRS phaseshift coefficient vector θθθ . as well as an edge computingresource allocation vector fff e , the optimal number of off-loaded bits is given by (cid:96) ∗ k = arg min ˆ (cid:96) k ∈ (cid:8) (cid:98) ˆ (cid:96) ∗ k (cid:99) , (cid:100) ˆ (cid:96) ∗ k (cid:101) (cid:9) D k (ˆ (cid:96) k ) , (9) where (cid:98)·(cid:99) and (cid:100)·(cid:101) represent the floor and ceiling operations,respectively, and ˆ (cid:96) ∗ k is selected for ensuring that the value of D lk (ˆ (cid:96) k ) becomes equivalent to that of D ek (ˆ (cid:96) k ) , i.e. ˆ (cid:96) ∗ k = L k c k R k f ek f ek f lk + c k R k (cid:0) f ek + f lk (cid:1) . (10) Proof:
See Appendix A. (cid:4)
2) Optimization of fff e : Here, the edge computing resourceallocation fff e is optimized, while fixing the MUD matrix WWW ,the IRS phase shift coefficient vector θθθ , and the off-loadingvolume (cid:96)(cid:96)(cid:96) . Specifically, upon substituting (10) into the OF ofProblem P , the problem can be reformulated as: P - E : min fff e K (cid:88) k =1 (cid:36) k ( L k c k R k + L k c k f ek ) f ek f lk + c k R k ( f ek + f lk ) s.t. (7c) , (7d) . (11a) Algorithm 1
Joint optimization of (cid:96)(cid:96)(cid:96) and fff e , given WWW and θθθ
Input: hhh r,k , hhh d,k , GGG , B , p t , σ , K , (cid:36) k , L k , c k , f lk , f e total , t max , (cid:15) , WWW , θθθ , and ˆ fff e satisfying (7c) and (7d) Output:
Optimal (cid:96)(cid:96)(cid:96) ∗ and fff e ∗ , given WWW and θθθ
1. Initialization initialize t = 0 , (cid:15) (0)1 = 1 , fff e (0) ← ˆ fff e calculate RRR using (5)
2. Joint optimization of (cid:96)(cid:96)(cid:96) and fff e while (cid:15) ( t )1 > (cid:15) && t < t max do • calculate (cid:96)(cid:96)(cid:96) ( t +1) using (10) • calculate fff e ( t +1) and µ by using (16) and the bisection searchmethod, respectively • (cid:15) ( t +1)1 = (cid:12)(cid:12) obj (cid:0) (cid:96)(cid:96)(cid:96) ( t ,fff e ( t (cid:1) − obj (cid:0) (cid:96)(cid:96)(cid:96) ( t ,fff e ( t (cid:1)(cid:12)(cid:12) obj (cid:0) (cid:96)(cid:96)(cid:96) ( t ,fff e ( t (cid:1) • t ← t + 1 end while3. Output optimal (cid:96)(cid:96)(cid:96) ∗ and fff e ∗ (cid:96)(cid:96)(cid:96) ∗ ← (cid:96)(cid:96)(cid:96) ( t ) and fff e ∗ ← fff e ( t ) Problem P - E can be proved to be a convex optimizationproblem following the proposition below. Proposition 2.
Problem P - E is a convex optimization prob-lem. Proof:
See Appendix B. (cid:4)
Since Problem P - E is convex and the Slater’s condition[37] is satisfied , the Karush–Kuhn–Tucker (KKT) may beimposed on the problem for finding its optimal solution.Specifically, the Lagrangian function associated with Problem P - E is given by L ( fff e , µ, ννν ) = K (cid:88) k =1 (cid:36) k ( L k c k R k + L k c k f ek ) c k R k f lk + ( f lk + c k R k ) f ek + µ (cid:18) K (cid:88) k =1 f ek − f e total (cid:19) , (12)where the variable µ is the non-negative Lagrange multiplier,while the optimal edge computing resource allocation vector fff e ∗ and the optimal Lagrange multiplier µ ∗ should satisfy thefollowing KKT conditions, for k = 1 , , . . . , K : ∂ L ∂f ek = − (cid:36) k L k c k R k (cid:2) c k R k f lk + ( f lk + c k R k ) f ek ∗ (cid:3) + µ ∗ = 0 ,µ ∗ (cid:18) K (cid:88) k =1 f ek ∗ − f e total (cid:19) = 0 ,f ek ∗ ≥ . (13)(14)(15)The value of f ek can be directly derived from (13) for a given µ , which is written as f ek = (cid:113) (cid:36) k L k c k R k µ − c k R k f lk f lk + c k R k , k = 1 , . . . , K. (16)In order to ensure f ek ≥ in (16), we have (cid:113) (cid:36) k L k c k R k µ − c k R k f lk ≥ , which is reformulated as µ ≤ (cid:36) k L k c k f lk . Given In words, the Slater’s condition for convex programming states that strongduality holds if all constraints are satisfied and the nonlinear constraints aresatisfied with strict inequalities. that µ (cid:54) = 0 in (16), the optimal µ ∗ can be found in therange of ( µ l , µ u ] = (cid:16) , min k (cid:0) (cid:36) k L k c k f lk (cid:1)(cid:105) to ensure (14), us-ing the well-known bisection search method associated withthe termination coefficient of (cid:15) , because (cid:80) Kk =1 f ek can beproved to be monotonically decreasing with respect to µ .The procedure of solving Problem P is summarized inAlgorithm 1. The complexity of Algorithm 1 is dominatedby calculating fff e ( t +1) using (16) and by calculating µ usingthe bisection search method. Its complexity is on the order of O (cid:0) log ( µ u − µ l (cid:15) ) K (cid:1) . Thus the total complexity of Algorithm 1is O (cid:0) t max log ( µ u − µ l (cid:15) ) K (cid:1) . B. Joint Optimization of the MUD Matrix and the IRS PhaseShift Coefficient While Fixing the Computing Settings
Given an off-loading volume vector (cid:96)(cid:96)(cid:96) and an edge comput-ing resource allocation vector fff e , Problem P is reformulatedas P WWW ,θθθ K (cid:88) k =1 (cid:36) k D k ( k , θθθ ) s.t. ≤ θ n < π, n = 1 , , . . . , N. (17a) Remark 2.
The challenges of solving Problem P are due totwo aspects. The first one is the segmented form of D k ( k , θθθ ) that is caused by the operation max as detailed in (6) , whilethe second issue is that the OF is the summation of fractionalfunctions, with respect to WWW and θθθ as shown in the OF ofProblem P - E below, which makes the problem a non-convexsum-of-ratios optimization. In order to tackle these two issues,we transform the problem as follows.1) Problem Transformation: As detailed in Proposition 1,the optimal solution of Problem P results in D k = D lk = D ek .Hence upon replacing D k by D ek and removing the constantterms, Problem P is reformulated as: P - E WWW ,θθθ K (cid:88) k =1 (cid:36) k (cid:96) k R k ( k , θθθ ) s.t. ≤ θ n < π, n = 1 , , . . . , N. (18a)It is then rewritten as the following equivalent form: P - E WWW ,θθθ,βββ K (cid:88) k =1 β k s.t. (cid:36) k (cid:96) k R k ( k , θθθ ) ≤ β k , k = 1 , , . . . , K, ≤ θ n < π, n = 1 , , . . . , N. (19a)(19b)The following proposition may assist us in solving Prob-lem P - E . Proposition 3. If ( WWW ∗ , θθθ ∗ , βββ ∗ ) is the solution of Problem P - E , a λλλ ∗ = [ λ , λ , . . . , λ K ] exists that ( WWW ∗ , θθθ ∗ ) satisfiesthe KKT conditions of the following problem, when we set βββ = βββ ∗ and λλλ = λλλ ∗ P - E WWW ,θθθ K (cid:88) k =1 λ k (cid:2) (cid:36) k (cid:96) k − β k R k ( k , θθθ ) (cid:3) s.t. ≤ θ n < π, n = 1 , , . . . , N. (25a) Algorithm 2
Joint optimization of
WWW and θθθ , given (cid:96)(cid:96)(cid:96) and fff e Input: hhh r,k , hhh d,k , GGG , B , p t , σ , (cid:36) k , (cid:96)(cid:96)(cid:96) , and fff e Output:
Optimal
WWW ∗ and θθθ ∗ , given (cid:96)(cid:96)(cid:96) and fff e
1. Initialization initialize t = 0 , ζ ∈ (0 , , (cid:15) ∈ (0 , , and θθθ (0) satisfying (7a)calculate WWW (0) and
RRR (0) using (31) and (34), respectivelycalculate λλλ (0) and βββ (0) using (26)
2. Joint optimization of
WWW , θθθ , λλλ and βββ repeat • update WWW ( t +1) and θθθ ( t +1) using Algorithm 3 • update λλλ ( t +1) and βββ ( t +1) as follows λ ( t +1) k = λ ( t ) k − ζ i ( t χ k (cid:0) λ ( t ) k (cid:1) R k (cid:0) ( t +1) k , θθθ ( t +1) (cid:1) , (20)and β ( t +1) = β ( t ) − ζ i ( t κ k (cid:0) β ( t ) k (cid:1) R k (cid:0) ( t +1) k , θθθ ( t +1) (cid:1) , (21)where i ( t +1) is the smallest integer among i ∈ { , , , . . . } satisfying K (cid:88) k =1 (cid:12)(cid:12)(cid:12)(cid:12) χ k (cid:18) λ ( t ) k − ζ i χ k ( λ ( t ) k ) R k ( ( t +1) k , θθθ ( t +1) ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) + K (cid:88) k =1 (cid:12)(cid:12)(cid:12)(cid:12) κ k (cid:18) β ( t ) − ζ i κ k ( β ( t ) ) R k ( ( t +1) k , θθθ ( t +1) ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 − (cid:15) ζ i ) K (cid:88) k =1 (cid:104)(cid:12)(cid:12) χ k (cid:0) λ ( t ) k (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) κ k (cid:0) β ( t ) (cid:1)(cid:12)(cid:12) (cid:105) . (22) • t ← t + 1 until the following conditions are achieved λ ( t ) k R k ( ( t ) k , θθθ ( t ) ) − , (23) β k R k ( ( t ) k , θθθ ( t ) ) − (cid:36) k (cid:96) k = 0 (24)
3. Output optimal
WWW ∗ and θθθ ∗ WWW ∗ ← WWW ( t ) and θθθ ∗ ← θθθ ( t ) Furthermore, ( WWW ∗ , θθθ ∗ ) also satisfies the following equations,when we set βββ = βββ ∗ and λλλ = λλλ ∗ λ k = R k ( ∗ k ,θθθ ∗ ) , k = 1 , , . . . , K,β k = (cid:36) k (cid:96) k R k ( ∗ k ,θθθ ∗ ) , k = 1 , , . . . , K. (26) Correspondingly, if ( WWW ∗ , θθθ ∗ ) is a solution to Problem P - E and satisfies (26) when we set βββ = βββ ∗ and λλλ = λλλ ∗ , ( WWW ∗ , θθθ ∗ , βββ ∗ ) is the solution of Problem P - E associatedwith the Lagrange multiplier λλλ = λλλ ∗ . Proof:
See Appendix C. (cid:4)
To this end, the sum-of-ratios form in Problem P - E has been transformed to a parameterized subtractable formin Problem P - E , which can be solved in two steps [38]–[40]: the first step is to obtain WWW ∗ and θθθ ∗ by solving Problem P - E , given βββ and λλλ ; the second step is to update βββ and λλλ using the modified Newton’s method until the convergence isachieved. The procedure is summarized in Algorithm 2, wherewe have χ k ( λ k ) = λ k R k ( ∗ k , θθθ ∗ ) − , k = 1 , , . . . , K, (27) κ k ( β k ) = β k R k ( ∗ k , θθθ ∗ ) − (cid:36) k (cid:96) k , k = 1 , , . . . , K. (28)The complexity of Algorithm 2 is analyzed at the end ofSection III-B.Let us now focus our attention on the first step of solv-ing Problem P - E , i.e. optimizing WWW ∗ and θθθ ∗ , given aset of βββ and λλλ as well as an off-loading volume vec-tor (cid:96)(cid:96)(cid:96) . In this case, Problem P - E can be simplified to max WWW ,θθθ (cid:80) Kk =1 λ k β k R k ( k , θθθ ) subject to (25a), which constitutesa weighted sum-rate maximization problem. As revealed in[41], maximizing the weighted sum-rate can be accomplishedvia weighted MSE minimization. The latter problem is easierto handle, because it is convex regarding each optimizationvariable, while fixing others. As such, we focus our attentionon constructing the corresponding weighted MSE minimiza-tion problem. Specifically, following Theorem I in [41], weintroduce an auxiliary weight variable Υ k for the k -th deviceand formulate the corresponding weighted MSE minimizationproblem as: P - E WWW ,θθθ K (cid:88) k =1 (cid:2) Υ k e k ( WWW , θθθ ) − λ k β k log ( λ − k β − k Υ k ) − λ k β k (cid:3) s.t. ≤ θ n < π, ∀ n ∈ { , , . . . , N } , (29a)where the mathematical expression of { Υ k } is given in Sec-tion III-B3 and e k represents the MSE of the k -th user, whichis given by e k ( WWW , θθθ ) (cid:44) E (cid:2) (ˆ s k − s k )(ˆ s k − s k ) H (cid:3) = (cid:2) √ p t Hk ( hhh d,k + GGG
ΘΘΘ hhh r,k ) − (cid:3) × (cid:2) √ p t Hk ( hhh d,k + GGG
ΘΘΘ hhh r,k ) − (cid:3) H + p t K (cid:88) j (cid:54) = k Hk ( hhh d,j + GGG
ΘΘΘ hhh r,j )( hhh d,j + GGG
ΘΘΘ hhh r,j ) H k + σ Hk k . (30)As such, compared to Problem P - E , Problem P - E becomes more tractable, because given an IRS phase shift co-efficient vector, the OF of Problem P - E is convex regardingan optimization variable, while fixing the other one. Again, theBCD technique is invoked for solving this problem as follows.
2) MUD Matrix Design:
In Problem P - E , fixing thephase shift coefficient vector θθθ and the auxiliary variable Υ k ,the MUD vector can be obtained by forcing the first-orderderivative of the OF with respect to k as . After several stepsof mathematical manipulations, it is readily observed that theabove minimization is equivalent to minimizing the weightedMSE. Then, the MUD vector is given by [42, Sec. 6.2.3] k = √ p t JJJ − ( hhh d,k + GGG
ΘΘΘ hhh r,k ) , (31)where JJJ = p t (cid:80) Kj =1 ( hhh d,j + GGG
ΘΘΘ hhh r,j )( hhh d,j + GGG
ΘΘΘ hhh r,j ) H + σ III M .
3) Auxiliary Variable Design:
Fixing θθθ and k , the optimalauxiliary variable can be obtained by minimizing the OF ofProblem P - E with respect to Υ k , given by Υ k = λ k β k ( e k ) − . (32) Furthermore, substituting (31) into (30), the MSE becomes e MMSE k = 1 − p t ( hhh d,k + GGG
ΘΘΘ hhh r,k ) H JJJ − ( hhh d,k + GGG
ΘΘΘ hhh r,k ) . (33)Bearing in mind that the relationship between the SINR andthe MSE of the system equipped with the minimum meansquare error (MMSE) MUD is given by γ k = ( e MMSE k ) − − [42, Sec. 6.2.3], (5) may be reformulated as R k = − B log (cid:0) e MMSE k (cid:1) . (34)
4) IRS Phase Shift Coefficient Design:
In this subsection,we focus our attention on optimizing the reflection phaseshift coefficients θθθ , while fixing the auxiliary variable Υ k andthe MUD matrix WWW . Specifically, by substituting (30) intothe OF of Problem P - E and removing the terms that areindependent of the phase shift coefficient vector θθθ , Problem P - E is reformulated as: P - E θθθ K (cid:88) k =1 K (cid:88) j =1 Υ k p t Hk hhh j hhh Hj k − K (cid:88) k =1 Υ k √ p t hhh Hk k − K (cid:88) k =1 Υ k √ p t Hk hhh k s.t. ≤ θ n ≤ π, ∀ n ∈ { , , . . . , N } , (35)where the first and the second terms in the OF can berespectively formulated in expansion forms as K (cid:88) k =1 K (cid:88) j =1 Υ k p t Hk hhh j hhh Hj k = K (cid:88) k =1 K (cid:88) j =1 (cid:0) Υ k p t Hk GGG
ΘΘΘ hhh r,j hhh
Hr,j
ΘΘΘ H GGG H k +Υ k p t Hk hhh d,j hhh Hr,j
ΘΘΘ H GGG H k +Υ k p t Hk GGG
ΘΘΘ hhh r,j hhh
Hd,j k + Υ k p t Hk hhh d,j hhh Hd,j k (cid:1) , (36)and K (cid:88) k =1 Υ k √ p t hhh Hk k = K (cid:88) k =1 (cid:0) Υ k √ p t hhh Hd,k k + Υ k √ p t hhh Hr,k
ΘΘΘ H GGG H k (cid:1) . (37)Upon defining AAA (cid:44) (cid:80) Kk =1 Υ k p t GGG H k Hk GGG , BBB (cid:44) (cid:80) Kj =1 hhh r,j hhh Hr,j , CCC (cid:44) (cid:80) Kk =1 (cid:80) Kj =1 Υ k p t hhh r,j hhh Hd,j k Hk GGG , and
DDD (cid:44) (cid:80) Kk =1 Υ k √ p t hhh r,k Hk GGG , Problem P - E may be rewrit-ten as: P - E θθθ tr (ΘΘΘ H AAA
ΘΘΘ
BBB ) + tr (cid:2) ΘΘΘ H ( CCC − DDD ) H (cid:3) + tr (cid:2) ΘΘΘ(
CCC − DDD ) (cid:3) s.t. ≤ θ n ≤ π, ∀ n ∈ { , , . . . , N } . (38)Defining φφφ (cid:44) [ φ , . . . , φ N ] T where φ n = e jθ n , and vvv = (cid:2) [ CCC − DDD ] , , . . . , [ CCC − DDD ] N,N (cid:3) T , we havetr (ΘΘΘ H AAA
ΘΘΘ
BBB ) = φφφ H ( AAA (cid:12)
BBB ) φφφ, (39) Algorithm 3
Joint optimization of
WWW and θθθ , given λλλ and βββ
Input: hhh r,k , hhh d,k , GGG , B , p t , σ , (cid:36) k , t max , (cid:15) , λλλ and βββ Output:
Optimal
WWW ∗ and θθθ ∗ , given λλλ and βββ
1. Initialization initialize t = 0 , (cid:15) (0)3 = 1 , θθθ (0) satisfying (7a)
2. Joint optimization of
WWW and θθθ while (cid:15) ( t )3 > (cid:15) && t < t max do • calculate WWW ( t +1) using (31) • calculate ΥΥΥ ( t +1) using (32) • calculate θθθ ( t +1) by solving Problem P - E with the aid ofthe MM algorithm • (cid:15) ( t +1)3 = (cid:12)(cid:12) obj (cid:0) WWW ( t ,θθθ ( t (cid:1) − obj (cid:0) WWW ( t ,θθθ ( t (cid:1)(cid:12)(cid:12) obj (cid:0) WWW ( t ,θθθ ( t (cid:1) • t ← t + 1 end while3. Output optimal WWW ∗ and θθθ ∗ , given λλλ and βββWWW ∗ ← WWW ( t ) and θθθ ∗ ← θθθ ( t ) where (cid:12) represents the Hadamard product, andtr (cid:2) ΘΘΘ H ( CCC − DDD ) H (cid:3) = vvv H φφφ ∗ , tr (cid:2) ΘΘΘ(
CCC − DDD ) (cid:3) = φφφ T vvv. (40)Further defining ΨΨΨ (cid:44)
AAA (cid:12)
BBB , we may equivalently rewriteProblem P - E as: P - E φφφ f ( φφφ ) = φφφ H ΨΨΨ φφφ + 2 (cid:60) (cid:8) φφφ H vvv ∗ (cid:9) s.t. | φ n | = 1 , ∀ n ∈ { , , . . . , N } . (41)Problem P - E is a non-convex one because of the unitmodulus constraint on φ n . In the following, the MM algorithm[43] is invoked for solving this problem, which has twosteps. In the majorization step, we construct a continuoussurrogate function g ( φφφ | φφφ t ) , which represents the upperboundof f ( φφφ ) . Then in the minimization step, φφφ is updated by φφφ t +1 ∈ arg min φφφ g ( φφφ | φφφ t ) . As such, we may initialize φφφ thatsatisfies the constraint (41), and then use the MM algorithm togenerate a sequence of feasible vectors { φφφ t } , where t refers tothe iteration index. Now the surrogate function is constructedwith the aid of the proposition below. Proposition 4.
Denoting the maximum eigenvalue of
ΨΨΨ by ˆ λ max and given a solution φφφ t at the t -th iteration, we havethe inequality below f ( φφφ ) ≤ φφφ H ˆ λ max III N φφφ − (cid:60) (cid:8) φφφ H (ˆ λ max III N − ΨΨΨ) φφφ t (cid:9) +( φφφ t ) H (ˆ λ max III N − ΨΨΨ) φφφ t + 2 (cid:60) (cid:8) φφφ H vvv ∗ (cid:9) . (42) Proof:
See [26], [44]. (cid:4)
Here, the terms on the right side of (42) is defined by oursurrogate function g ( φφφ | φφφ t ) . Then, Problem P - E at the t -thiteration is reformulated as P - E φφφ g ( φφφ | φφφ t ) s.t. | φ n | = 1 , ∀ n ∈ { , , . . . , N } . (43)Since ( φφφ t ) H (ˆ λ max III N − ΨΨΨ) φφφ t is a constant for a given φφφ t andwe have φφφ H ˆ λ max III N φφφ = M ˆ λ max , Problem P - E can be Algorithm 4
Joint Optimization of (cid:96)(cid:96)(cid:96) , fff e , WWW and θθθ
Input: hhh r,k , hhh d,k , GGG , B , p t , σ , (cid:36) k , L k , c k , f e total , and (cid:15) Output:
Optimal (cid:96)(cid:96)(cid:96) , fff e , WWW and θθθ
1. Initialization initialize t = 0 , (cid:15) (0)4 = 1 initialize θθθ (0) satisfying (7a) and fff e (0) satisfying (7c) and (7d)calculate WWW (0) using (31)
2. Joint optimization of (cid:96)(cid:96)(cid:96) and fff e , given WWW ( t ) and θθθ ( t ) calculate (cid:96)(cid:96)(cid:96) ( t +1) and fff e ( t +1) using Algorithm 1
3. Joint optimization of
WWW and θθθ , given (cid:96)(cid:96)(cid:96) ( t +1) and fff e ( t +1) calculate WWW ( t +1) and θθθ ( t +1) using Algorithm 2
4. Convergence checking (cid:15) ( t )4 = (cid:12)(cid:12) obj (cid:0) (cid:96)(cid:96)(cid:96) ( t ,fff e ( t ,WWW ( t ,θθθ ( t (cid:1) − obj (cid:0) (cid:96)(cid:96)(cid:96) ( t ,fff e ( t ,WWW ( t ,θθθ ( t (cid:1)(cid:12)(cid:12) obj (cid:0) (cid:96)(cid:96)(cid:96) ( t ,fff e ( t ,WWW ( t ,θθθ ( t (cid:1) if (cid:15) ( t )4 > (cid:15) && t < t max holds then t = t + 1 Go to Step 2 else integerize (cid:96) ( t +1) by (9)Output the optimal (cid:96)(cid:96)(cid:96) ∗ , fff e ∗ , WWW ∗ and θθθ ∗ end if equivalently written as P - E φφφ (cid:60) (cid:110) φφφ H (cid:2) (ˆ λ max III N − ΨΨΨ) φφφ t − vvv ∗ (cid:3)(cid:111) s.t. | φ n | = 1 , ∀ n ∈ { , , . . . , N } . (44)Then, the optimal solution of Problem P - E is readily givenby φφφ t +1 = e j arg { (ˆ λ max III N − ΨΨΨ) φφφ t − vvv ∗ } . (45)Accordingly, the optimal solution to Problem P - E can beobtained as θθθ t +1 = arg { (ˆ λ max III N − ΨΨΨ) φφφ t − vvv ∗ } . (46)The termination condition of the MM algorithm is given by (cid:12)(cid:12) f ( φφφ t +1 ) − f ( φφφ t ) (cid:12)(cid:12) /f ( φφφ t +1 ) ≤ (cid:15) or t ≥ t maxMM . The procedureof solving Problem P - E is summarized in Algorithm 3.The complexity of Algorithm 3 is dominated by its Step 2.Specifically, the complexity of calculating WWW ( t +1) by (31) ison the order of O (cid:0) max { KM , KM N } (cid:1) ; the complexity ofcalculating ΥΥΥ ( t +1) by (32) is on the order of O ( K ) . Withregard to the calculation of θθθ ( t +1) using the MM algorithm,the complexity of calculating the eigenvalue λ max of ΨΨΨ ison the order of O ( N ) , while for each iteration of the MMalgorithm, the main complexity lies in the calculation of φφφ t +1 in (45), whose complexity is on the order of O ( N ) . Hence thecomplexity of the MM algorithm is O ( N + t maxMM N ) . Sum-ming these three terms together, we obtain the total complexityof Algorithm 3 as O (cid:0) max { N + t maxMM N , KM , KM N } (cid:1) .Finally, the complexity of Algorithm 2 is mainly dependenton updating WWW ( t +1) and θθθ ( t +1) using Algorithm 3, becauseall other steps are given by explicit mathematical expressions. C. Overall Algorithm to Solve Problem P Based on the above discussions, we provide the detaileddescription of the BCD algorithm used for solving Problem P in Algorithm 4. Note that a decreasing OF value of Problem P is guaranteed in Step 2 and Step 3. Furthermore, the OFvalue has a lower bound due to the constraint on the totaledge computing resources. Hence, Algorithm 4 is guaranteedto converge.The computational complexity of Algorithm 4 is mainlydependent on its Step 2 and Step 3, whose complexitieshave been analyzed in the above subsections. Furthermore,the simulation results in Section V show that Algorithm 4converges rapidly, which demonstrates the low complexity ofour algorithms.IV. S PECIFIC C ASE S TUDY : T HE S INGLE -D EVICE S CENARIO
In order to fully characterize the IRS-aided MEC system,a special case is investigated in this section, where a singledevice is served by the MEC system. The optimization prob-lem of the single-device scenario becomes much simpler forthe following reasons. Firstly, the edge computing resourceallocation no longer has to be considered, because all the edgecomputing resources can be assigned to this single device.Secondly, the sum-of-ratios form in Problem P - E becomesa single-ratio form, which implies that the optimization prob-lem is more tractable. Thirdly, the multi-user interference doesnot have to be considered, when the detection vector and theIRS phase shift coefficient vector are optimized. The jointoptimization is detailed as follows.Problem P can be simplified for the single-device scenarioas P D ( ) s.t. ≤ θ n < π, n = 1 , , . . . , N,(cid:96) ∈ { , , . . . , L } , (47a)(47b)where the OF D ( ) becomes D ( ) = max (cid:26) ( L − (cid:96) ) cf l , (cid:96)R ( ) + (cid:96)cf e total (cid:27) . (48)As illustrated in Proposition 1, for a given set of and θθθ , D ( ) achieves its minimum value when (cid:96) is selected toensure ( L − (cid:96) ) cf l = (cid:96)R ( ) + (cid:96)cf e total . Therefore, the optimal valueof the relaxation of (cid:96) is given by ˆ (cid:96) ∗ = LcRf e total f e total f l + cR (cid:0) f e total + f l (cid:1) . (49)Then, Problem P is reformulated as P - E (cid:96)R ( ) + (cid:96)cf e total s.t. ≤ θ n < π, n = 1 , , . . . , N, (50a)which is equivalent to P - E R ( ) s.t. ≤ θ n < π, n = 1 , , . . . , N. (51a)Substituting (4) and (5) into the OF of Problem P - E , andtaking several steps of mathematical manipulation, Problem Algorithm 5
Joint Optimization of (cid:96) , and θθθ proposed forthe single-user scenario Input: hhh r , hhh d , GGG , B , p t , σ , L , c , f e total , and (cid:15) Output:
Optimal (cid:96) , and θθθ
1. Initialization initialize t = 0 , (cid:15) (0)5 = 1 initialize θθθ (0) satisfying (7a)calculate (0) using (53)
2. Joint optimization of and θθθ repeat • calculate θθθ ( t +1) and ( t +1) using (55) and (53), respectively • (cid:15) ( t )5 = (cid:12)(cid:12) obj (cid:0) ( t ,θθθ ( t (cid:1) − obj (cid:0) ( t ,θθθ ( t (cid:1)(cid:12)(cid:12) obj (cid:0) ( t ,θθθ ( t (cid:1) , where the objrefers to the OF of Problem P - E • t = t + 1 until (cid:15) ( t )5 ≤ (cid:15) || t > t max
3. Optimization of (cid:96)(cid:96)(cid:96) calculate ˆ (cid:96) ( t +1) using (49)integerize (cid:96) ( t +1) by (9) P - E may be equivalently transformed into P - E p t (cid:12)(cid:12) H (cid:0) hhh d + GGG
ΘΘΘ hhh r (cid:1)(cid:12)(cid:12) σ | H | s.t. ≤ θ n < π, n = 1 , , . . . , N. (52a)Again, the BCD technique is invoked for optimizing and θθθ in Problem P - E . Specifically, given a θθθ , can be optimizedfollowing the well-known maximum ratio combining (MRC)criterion [45], which is given by = √ p t ( hhh d + GGG
ΘΘΘ hhh r ) /σ, (53)while for a given , we have the following inequality for theOF of Problem P - E , p t (cid:12)(cid:12) H (cid:0) hhh d + GGG
ΘΘΘ hhh r (cid:1)(cid:12)(cid:12) σ | H | ≤ p t (cid:12)(cid:12) H hhh d (cid:12)(cid:12) σ | H | + p t (cid:12)(cid:12) H GGG
ΘΘΘ hhh r (cid:12)(cid:12) σ | H | . (54)The equality in (54) holds only when the IRS phase shift co-efficient obeys arg { H hhh d } = arg { H GGG
ΘΘΘ hhh r } . Accordingly,the reflection phase shift vector θθθ may be readily obtained as θθθ = arg { H hhh d } − arg { diag { H GGG } hhh r } . (55)In Algorithm 5, we provide the overall algorithm that is usedfor solving our optimization problem for the single-devicescenario.The complexity of Algorithm 5 is dominated by calculating θθθ ( t +1) and ( t +1) using (55) and (53), whose complexitiesare on the order of O (cid:0) max { M N, N } (cid:1) and of O ( M N ) ,respectively. Hence the complexity of Algorithm 5 is on theorder of O ( M N ) .V. N UMERICAL R ESULTS
In this section, the benefits of deploying the IRS in a MECsystem are evaluated, relying on our algorithms developed inSection III and IV. We consider a single-cell MEC system forboth the single-device and two-device as well as multi-device scenarios. As shown in Fig. 3, the AP’s coverage radius is R = 300 m and the IRS is deployed at the cell edge. Thelocation of the device is specified both by d and by d inthe single-device scenario, while in the two-device scenario,the devices’ locations are specified by ( d , d ) and ( d , d ) ,respectively. Furthermore, in the multiple-device scenario, itis assumed that the devices are uniformly distributed withina circle, whose size and location are prescribed by its radius r , as well as d and d , respectively. The default value of theseparameters are set in the “Location model” block of Table II.As for the communications channel, we consider both thesmall scale fading and the large scale path loss. Specifically,the small scale fading is i.i.d. and obeys the complex Gaussiandistribution associated with zero mean and unit variance, whilethe path loss in dB is given byPL = PL − α log (cid:16) dd (cid:17) , (56)where PL is the path loss at the reference distance d ; d and α represent the distance of the communications link and itspath loss exponent, respectively. Here we use α ua , α ui and α ia to denote the path loss exponent of the link between thedevice and the AP, that of the link between the device andthe IRS, as well as that of the link between the IRS and theAP, respectively. The zero-mean additive white Gaussian noiseassociated with the variable of σ is imposed on the off-loadedsignal. The default settings of these parameters are specified inthe “Communications model” block of Table II. The variables L k , c k and f lk obey the uniform distribution, whose ranges aregiven in the “Computing model” block of Table II. Table II: Default simulation parameter setting
Description Parameter and ValueLocation model R = 300 m d = d = d = 10 m Communication model Bandwidth = 1 MHz PL = 30 dB , d = 1 m α ua = 3 . , α ui = 2 . , α ia = 2 . M = 5 p t = 1 mW σ = 3 . × − mW Computing model L k = [250 , c k = [700 , / bit f lk = [4 , × cycle / s Weight (cid:36) k = 1 /K Convergence criterion (cid:15) = 0 . The following subsections detail our simulation results, interms of the properties of our proposed algorithm and ofthe latency performance in both the single-device, and two-device, as well as multi-device scenarios in various simulationenvironments. The following three schemes are considered: • With IRS:
The off-loading volume, edge computing re-source allocation, MUD matrix, and IRS phase shift areoptimized relying on Algorithm 4 and Algorithm 5 in themulti-device and single-device scenarios, respectively. • RandPhase:
The off-loading volume, edge computingresource allocation, as well as MUD matrix are optimizedusing Algorithm 4 and Algorithm 5 in multi-device and L a t e n c y ( m s ) N = 10N = 20N = 40 (a) L a t e n c y ( m s ) N = 10N = 20N = 40 (b)Figure 2: Convergence of the algorithms for (a) the single-device scenariousing Algorithm 5 and (b) the two-device scenario using Algorithm 4. Theparameters are set as follows: f e total = 50 × cycle / s ; c k = 750 cycle, L k = 300 Kb , and f lk = 0 . × cycle / s for all devices. (a): d = 280 m ;(b): d = d = 280 m . single-device scenarios, respectively, while skipping thestep of designing the IRS phase shift, which is randomlyset obeying the uniform distribution in the range of [0 , π ) . • Without IRS:
The composite channel
GGG
ΘΘΘ hhh r,k taking intoaccount the IRS is set to . The off-loading volume, edgecomputing resource allocation, and the MUD matrix aredesigned following Algorithm 4 and Algorithm 5 in themulti-device and single-device scenarios, respectively. A. Properties of the Proposed Algorithms
In this subsection, the properties of Algorithm 4 and 5 areinvestigated, with the aid of numerical results.
1) Convergence:
Fig. 2 shows the device-average latencyversus the number of iterations under various settings of theIRS phase shift number, i.e. N = 10 , , and , for boththe single-device and multi-device scenarios. We have thefollowing two observations. Firstly, a larger number of phaseshifts leads to a slightly slower convergence, especially forthe multi-device scenario. This is because more optimizingvariables are involved. Secondly, the proposed algorithms arecapable of achieving a convergence within iterations, whichvalidates its practical implementation.
2) Impact of the Initialization Settings:
As elaborated on inRemark 1, locally optimal results are provided by our proposedalgorithms. Hence, the results obtained are directly dependenton the initialization settings of Algorithm 4 and 5. In orderto clarify its impact, Fig. 4 presents the latency performanceunder different initialization settings both for single- andmulti-device scenarios. Specifically, for each realization of (a) (b) (c)Figure 3: Top-view setting of (a) the single-device scenario, (b) the two-device scenario, and (c) the multiple-device scenario. L a t e n c y ( m s ) MaxMin (a) L a t e n c y ( m s ) MaxMin (b)Figure 4: Simulation results of the the maximum and minimum latency versusthe realization index obtained under random initialization settings for (a)the single-device scenario using Algorithm 5 and (b) the two-device scenariousing Algorithm 4. “Max” and “Min” refer to the maximum and minimumvalue, respectively. The parameters are set as follows: N = 40 . (a): d =280 m ; (b): d = d = 280 m . the wireless channels and computing tasks to be processed, locally optimal results are obtained using our proposedalgorithms, where each of the initializations is randomly set.Among these locally optimal results, the maximum latencyvalue that can be deemed to be the worst-case result usingour proposed algorithms is labeled as “Max” in Fig. 4, whilethe minimum latency value is labeled by “Min” in Fig. 4which may resemble the globally optimal result. It is shownthat these two values are almost identical for the single-devicescenario, while their gap ranges from to in the multi-device scenario, which implies that our proposed algorithmsare capable of approaching the optimal performance.
3) Impact of the Phase Quantization:
Due to the associatedhardware limitation, only a limited number of discrete IRSphase shifts can be provided in practice [23], which pro-hibits the direct implementation of our proposed algorithms.An intuitive practical solution to this issue is to round thecontinuous phase shift obtained to its nearest discrete phase L a t e n c y ( m s ) Cont.2 bit1 bit (a) L a t e n c y ( m s ) Cont.2 bit1 bit (b)Figure 5: Simulation results of the latency versus the realization indexunder different assumptions of IRS phase shifts for (a) the single-devicescenario and (b) the two-device scenario. “Cont.”, “1-bit” and “2-bit” referto the assumptions of continuous, 1-bit, and 2-bit phase shifts, respectively.The parameters are set as follows: N = 40 . (a): d = 280 m ; (b): d = d = 280 m . shift. Naturally, a performance loss is imposed, owing to theassociated quantization effect. Fig. 5 evaluates the impactof phase quantization on the latency, where three practicalassumptions are considered. Specifically, under the assumptionof continuous phase shifts, the phase shift of each IRS elementcan be set as an arbitrary value in the interval of [0 , π ] ;Determined by a 1-bit control signal, the phase shift of eachIRS element has to be either or π under the assumption of1-bit phase shift; for a 2-bit control signal, the phase shift ofeach IRS element has to be one of the values in the set of (cid:8) , π , π, π (cid:9) . Particular to the schemes under discrete phaseshift assumptions, the values of WWW , (cid:96)(cid:96)(cid:96) , and fff e are updatedbased on the quantized phase shifts and then the latency iscalculated accordingly. We have the following observations.Firstly, as expected, the latency decreases upon increasing thenumber of discrete phase shifts. Secondly, the performancegap between the schemes under the assumptions of continuousphase shifts and -bit phase shifts ranges from to , L a t e n c y ( m s )
10 20 30 40 50 60 70 80 90 100N
With IRSRandPhaseWithout IRS
Figure 6: Simulation results of the latency versus the number of the IRSelements in the single-device scenario, where we set d = 280 m and f e total =50 × cycle / s . which implies that the quantization loss becomes negligiblefor as few as four phase shifts in practice. B. Single-Device Scenario
Fig. 6-8 present the latency versus various parameter set-tings in the single-device scenario, discussed as follows.
1) Impact of the Number of Reflecting Elements:
Fig. 6presents the latency versus the number of the reflecting ele-ments, for the various phase shift design schemes. Our obser-vations are as follows. Firstly, the performance gap betweenthe schemes “Without IRS” and “RandPhase” becomes higherupon increasing the number of reflecting elements, whichimplies that the IRS is capable of assisting the computationoff-loading even without carefully designing the phase shift.This is because the received SINR can be improved bydeploying an IRS for computation off-loading. The gain wastermed as the virtual array gain in Section I. Secondly, theperformance gain of the scheme “With IRS” over the scheme“RandPhase” is around
11 ms when we set N = 10 , while itbecomes
46 ms when we have N = 100 . This implies that asophisticated design of the IRS phase shift response providesa beamforming gain, and that increasing the number of IRSelements leads to a higher reflection-based beamforming gain.Combining these two types of gains together, IRSs are capableof efficiently reducing the latency in MEC systems.
2) Impact of the Edge Computing Capability:
Fig. 7 showsthe latency versus the edge computing capability, for variousIRS phase shift schemes. Our observations are as follows.For all these three schemes, the increase of f e total drasticallyreduces the latency when f e total is of a small value, while thereduction of the latency becomes smaller when f e total reaches acertain threshold value, say × cycle / s . This is becausethe latency imposed by the edge computing dominates when f e total is of a small value, whereas the latency imposed bycomputation off-loading plays a dominant role when f e total reaches a high value. Therefore, it is not necessary to equip theedge computing node with an extremely powerful computingcapability for latency minimization. L a t e n c y ( m s ) f etotal (10 cycle/s) With IRSRandPhaseWithout IRS
Figure 7: Simulation results of the latency versus the edge computingcapability in the single-device scenario, where we set d = 280 m and N = 40 . L a t e n c y ( m s ) d (m) With IRSRandPhaseWithout IRS
Figure 8: Simulation results of the latency versus the device location in thesingle-device scenario, where we set N = 40 and f e total = 50 × cycle / s .
3) Impact of the Device Location:
Fig. 8 depicts the latencyversus the device location, equipped with various IRS phaseshift schemes. Our observations are as follows. In the casewhere no IRS is employed, the latency increases upon increas-ing the distance between the AP and the device. In the casewhere the IRS’s phase shift is randomly set, the advantage ofusing the IRS becomes visible when the distance between thedevice and the IRS is less than
20 m . By contrast, the benefitof the IRS becomes notable for a much larger coverage of
100 m for the “With IRS” scheme. This observation impliesthat a sophisticated design of the IRS phase shift response iscapable of extending the coverage of the IRS. Furthermore,the latency reaches its maximum value at d = 260 m andthereafter becomes smaller for the “With IRS” scheme. This isbecause the direct device-AP link dominates the computationoff-loading when the device’s location obeys d ≤
260 m , whilethe composite device-IRS-AP link plays a dominant role, whenwe have d ≥
260 m . This observation further consolidates thata higher gain can be achieved in the near-IRS area, where the L a t e n c y ( m s )
10 20 30 40 50 60 70 80 90 100N
With IRSRandPhaseWithout IRSAverageDevice 1Device 2
Figure 9: Simulation results of the latency versus the number of the IRSelements in the two-device scenario, where we set d = 260 m , d = 280 m ,and f e total = 50 × cycle / s . composite device-IRS-AP link dominates the computation off-loading. C. Multi-Device Scenario
Fig. 9-12 present the latency in the two-device scenario, andFig. 13-14 show the latency in the multiple-device scenario,which are discussed as follows.
1) Impact of the Number of IRS Elements:
Fig. 9 depictsthe latency versus the number of the IRS elements in the two-device scenario, equipped with various phase shift schemes.Apart from the insights obtained in the single-device scenario,we also have the following observations. Firstly, Device 2outperforms Device 1 for the “With IRS” scheme, whilstDevice 1 has a lower latency both for the “Without IRS”and “RandPhase” schemes compared to Device 2. This isin accordance with the comparative relationship between thedevices located at d = 260 m and at d = 280 m in terms ofthe latency using those three phase shift schemes, as shown inFig. 8. This also implies that IRSs may change the latencyranking of the devices in MEC systems. Secondly, uponincreasing the number of IRS elements, Device 2 obtains ahigher gain than Device 1. This is because Device 2 is locatedcloser to the IRS, where the composite device-IRS-AP channeldominates the computation off-loading. Again, this impliesthat given a specific path loss exponent, a higher array andpassive beamforming gain may be achieved if the device islocated closer to the IRS.
2) Impact of the Edge Computing Capability:
Fig. 10presents the latency versus the edge computing capability inthe multi-device scenario, equipped with various phase shiftschemes. This scenario follows similar trends to the single-device case illustrated in Fig. 7.
3) Impact of the Device Location:
Fig. 11 plots the latencyversus the location of Device 1, while fixing the location ofthe AP, the IRS, and Device 2. As for the “With IRS” scheme,the curve of Device 1’s latency intercepts that of Device 2 at d = 220 m and d = 280 m , which implies that the devicesat these two locations have the same channel gain. In other L a t e n c y ( m s ) f etotal (10 cycle/s) With IRSRandPhaseWithout IRSAverageDevice 1Device 2
Figure 10: Simulation results of the latency versus the edge computingresource in the two-device scenario, where we set d = 260 m , d = 280 m ,and N = 40 . L a t e n c y ( m s ) d (m) With IRSRandPhaseWithout IRSAverageDevice 1Device 2
Figure 11: Simulation results of the latency versus the user location in thetwo-device scenario, where we set d = 280 m , N = 40 , and f e total =50 × cycle / s . words, the IRS is capable of assisting the device at d = 280 m to achieve the same latency as the device at d = 220 m .Note that the specific values of these two equivalent-latencylocations are dependent on the specific values of the path lossexponents of the device-IRS, IRS-AP, and device-AP channels,as presented below.
4) Impact of the Path Loss Exponent:
Fig. 12 illustratesthe latency versus the path loss exponent value associatedwith the IRS. It can be observed that the intercept pointdisappears, when α IRS is changed for the “With IRS” scheme.Furthermore, the latency of devices increases upon increasing α IRS . This is because higher α IRS leads to a lower array andbeamforming gain by the IRS. This provides important insightsfor engineering design: the location of the IRS should becarefully selected to avoid obstacles, for achieving a lower α ui and α ia .
5) Impact of the Number of Devices:
Fig. 13 shows thelatency versus the number of devices in the cycle in multi-device scenario. It can be readily observed that the device- L a t e n c y ( m s ) α IRS
With IRSRandPhaseWithout IRSAverageDevice 1Device 2
Figure 12: Simulation results of the latency versus the path loss exponent inthe two-device scenario, where we set α ui = α ia = α IRS . The parametersare set as follows: d = 220 m , d = 280 m , N = 40 , and f e total =50 × cycle / s . L a t e n c y ( m s ) K With IRSRandPhaseWithout IRS
Figure 13: Simulation results of the device-average latency versus the numberof devices K . The parameters are set as follows: d = 280 m , d = 10 m , r = 10 m , N = 40 , and f e total = 50 × cycle / s . average latency increases upon increasing the number of de-vices in the IRS-aided MEC system. This is partially becauseof the reduced edge computational resources allocated to eachdevice and partially due to the reduced beamforming gainachieved at each device. The former issue may be overcomeby equipping the edge node with more powerful computingcapability, while the latter problem can be solved by deployingmore IRSs in the MEC system for forming stronger beams.Nonetheless, compared to the “Without IRS” scheme, our“With IRS” scheme is capable of reducing the device-averagelatency from 177 ms to 139 ms, when we have devices in theMEC system. This again validates the benefits of our proposedsystem.
6) Impact of Inter-Cell Interference:
In realistic scenarios,inter-cell interference (ICI) also degrades computation off-loading. To quantify the impact of ICI, Fig. 14 presents thelatency versus the ICI-to-noise power ratio, where the BS isassumed to know the power of the received interference but L a t e n c y ( m s ) With IRSRandPhaseWithout IRS
Figure 14: Simulation results of the device-average latency versus the ICI-to-noise ratio. The parameters are set as follows: K = 3 , d = 280 m , d = 10 m , r = 10 m , N = 40 , and f e total = 50 × cycle / s . not the specific signal transmitted from other cells. Observethat the benefit of employing IRSs in MEC systems decreasesupon increasing the ICI-to-noise power ratio. To elaborate,when computation off-loading is used in the face of strongICI, the fraction of tasks that can be off-loaded becomesmarginal. In this case, the wireless devices have to rely on theirown computing capabilities. In other words, the potential ofIRSs may not be fully exploited. This observation suggestsan important insight for engineering design: the spectrumallocation of adjacent cells has to be carefully managed forminimizing the ICI in IRS-aided MEC systems.VI. C ONCLUSIONS
In order to reduce the computational latency, an IRSwas proposed for employment in MEC systems. Based onthis model, a latency-minimization problem was formulated,subject to practical constraints on the total edge computingcapability and IRS phase shifts. Sophisticated algorithms weredeveloped for optimizing both the computing and commu-nications settings. The benefits of using IRSs in the MECsystem were evaluated under various simulation environments.Quantitatively, the device-average computational latency wasreduced from
177 ms to
139 ms , compared to the conventionalMEC system operating without IRSs in a single cell associatedwith the cell radius of
300 m , a -antenna access point and active devices. Furthermore, the rapid convergence of our pro-posed algorithm was confirmed numerically, which validatestheir benefits. As our future work, an energy-minimizationbased design will be conceived for IRS-aided MEC systems.A PPENDIX AT HE PROOF OF P ROPOSITION ˆ (cid:96) k ∈ [0 , L k ] is used to represent the relaxation [46] ofthe integer value (cid:96) k ∈ { , , . . . , L k } . Furthermore, given thevalues of WWW , θθθ and fff e , we define the delay associated with ˆ (cid:96) k to be ˆ D k (ˆ (cid:96) k ) (cid:44) max (cid:8) D lk (ˆ (cid:96) k ) , D ek (ˆ (cid:96) k ) (cid:9) , which can bereformulated from (6) as a segmented form below ˆ D k (ˆ (cid:96) k ) = ( L k − ˆ (cid:96) k ) c k f lk , ≤ ˆ (cid:96) k ≤ L k c k R k f ek f ek f lk + c k R k ( f lk + f ek ) , ˆ (cid:96) k R k + ˆ (cid:96) k c k f ek , L k c k R k f ek f ek f lk + c k R k ( f lk + f ek ) < ˆ (cid:96) k ≤ L k . (57)A glance at (57) reveals that ˆ D k (ˆ (cid:96) k ) decreases upon in-creasing ˆ (cid:96) k in the range of ˆ (cid:96) k ∈ (cid:20) , L k c k R k f ek f ek f lk + c k R k (cid:0) f ek + f lk (cid:1) (cid:21) ,while ˆ D k (ˆ (cid:96) k ) increases upon increasing ˆ (cid:96) k in the rangeof ˆ (cid:96) k ∈ (cid:20) L k c k R k f ek f ek f lk + c k R k (cid:0) f ek + f lk (cid:1) , L k (cid:21) . Therefore, it is readilyinferred that ˆ D k (ˆ (cid:96) k ) achieves its minimum value when weset ˆ (cid:96) k = L k c k R k f ek f ek f lk + c k R k (cid:0) f ek + f lk (cid:1) , which is denoted by ˆ (cid:96) ∗ k . Bearingin mind that the optimal value of (cid:96) k has to be an integer,it may be obtained by carrying out the operation (cid:96) ∗ k =arg min ˆ (cid:96) ∈ (cid:8) (cid:98) ˆ (cid:96) ∗ k (cid:99) , (cid:100) ˆ (cid:96) ∗ k (cid:101) (cid:9) D k (ˆ (cid:96) k ) . This completes the proof.A PPENDIX BT HE PROOF OF P ROPOSITION P - E with respect to f ek by Φ - E , which is calculated as Φ - E = 2 (cid:36) k L k c k R k ( f lk + c k R k ) (cid:2) f ek f lk + c k R k (cid:0) f ek + f lk (cid:1)(cid:3) . (58)Since the values of (cid:36) k , c k , R k , f lk are all positive and wehave L k ≥ , f ek ≥ , it may be readily demonstrated that Φ - E ≥ . Hence the OF is a convex function with respectto f ek . Furthermore, the constraint functions (7c) and (7d) areall of linear forms. Hence, Problem P - E is shown to be astrictly convex problem.A PPENDIX CT HE PROOF OF P ROPOSITION P - E is given by L ( WWW , θθθ, βββ, λλλ ) = K (cid:88) k =1 β k + K (cid:88) k =1 λ k (cid:2) (cid:36) k (cid:96) k − β k R k ( k , θθθ ) (cid:3) (59)where { λ k } is the non-negative Lagrange multiplier. If ( WWW ∗ , θθθ ∗ , βββ ∗ ) is the solution of Problem P - E , there exists λλλ ∗ satisfying the following KKT conditions ∂ L ∂θ k = − λ ∗ k β ∗ k (cid:79) R k ( ∗ k , θθθ ∗ ) = 0 , k = 1 , , . . . , K,∂ L k = − λ ∗ k β ∗ k (cid:79) R k ( ∗ k , θθθ ∗ ) = 0 , k = 1 , , . . . , K,∂ L ∂β k = 1 − λ ∗ k R k ( ∗ k , θθθ ∗ ) = 0 , k = 1 , , . . . , K,λ ∗ k (cid:2) (cid:36) k (cid:96) k − β ∗ k R k ( ∗ k , θθθ ∗ ) (cid:3) = 0 , k = 1 , , . . . , K,λ ∗ k ≥ , k = 1 , , . . . , K,(cid:36) k (cid:96) k − β ∗ k R k ( ∗ k , θθθ ∗ ) ≤ , k = 1 , , . . . , K, ≤ θ ∗ k ≤ π, k = 1 , , . . . , K. (60)(61)(62)(63)(64)(65)(66) Since we have R k ( k , θθθ ) > , (62) is equivalent to λ ∗ k = 1 R k ( ∗ k , θθθ ∗ ) , ∀ k ∈ { , , . . . , K } , (67)and then (63) is equivalently written as β ∗ k = (cid:36) k (cid:96) k R k ( ∗ k , θθθ ∗ ) , ∀ k ∈ { , , . . . , K } . (68)Furthermore, Eq. (60), (61) and (66) are exactly the KKTconditions of Problem P - E , when we set λλλ = λλλ ∗ and βββ = βββ ∗ . This proves the first conclusion of Proposition3. Following the same procedure, the second conclusion ofProposition 3 may also be readily shown.R EFERENCES[1] M. R. Palattella, M. Dohler, A. Grieco, G. Rizzo, J. Torsner, T. Engel,and L. Ladid, “Internet of things in the 5G era: Enablers, architecture,and business models,”
IEEE J. Sel. Areas Commun.
IEEE Signal Process. Mag. , vol. 31, pp. 45–55, Nov. 2014.[4] W. Shi, J. Cao, Q. Zhang, Y. Li, and L. Xu, “Edge computing: Visionand challenges,”
IEEE Internet Things J. , vol. 3, pp. 637–646, March2016.[5] Y. Mao, C. You, J. Zhang, K. Huang, and K. B. Letaief, “A surveyon mobile edge computing: The communication perspective,”
IEEECommun. Surv. Tutor. , vol. 19, pp. 2322–2358, April 2017.[6] W. Zhang, Y. Wen, K. Guan, D. Kilper, H. Luo, and D. O. Wu, “Energy-optimal mobile cloud computing under stochastic wireless channel,”
IEEE Trans. Wireless Commun. , vol. 12, pp. 4569–4581, Sep. 2013.[7] Y. Wang, M. Sheng, X. Wang, L. Wang, and J. Li, “Mobile-edge com-puting: Partial computation offloading using dynamic voltage scaling,”
IEEE Trans. Commun. , vol. 64, pp. 4268–4282, Oct. 2016.[8] J. Ren, G. Yu, Y. Cai, and Y. He, “Latency optimization for resourceallocation in mobile-edge computation offloading,”
IEEE Trans. WirelessCommun. , vol. 17, pp. 5506–5519, Aug. 2018.[9] T. Bai, J. Wang, Y. Ren, and L. Hanzo, “Energy-efficient computationoffloading for secure UAV-edge-computing systems,”
IEEE Trans. Veh.Technol. , vol. 68, pp. 6074–6087, June 2019.[10] S. Sardellitti, G. Scutari, and S. Barbarossa, “Joint optimization of radioand computational resources for multicell mobile-edge computing,”
IEEE Trans. Signal Inf. Process. Netw. , vol. 1, pp. 89–103, June 2015.[11] X. Chen, L. Jiao, W. Li, and X. Fu, “Efficient multi-user computationoffloading for mobile-edge cloud computing,”
IEEE/ACM Trans. Netw. ,vol. 24, pp. 2795–2808, Oct. 2015.[12] X. Lyu, W. Ni, H. Tian, R. P. Liu, X. Wang, G. B. Giannakis, andA. Paulraj, “Optimal schedule of mobile edge computing for Internet ofThings using partial information,”
IEEE J. Sel. Areas Commun. , vol. 35,pp. 2606–2615, Nov. 2017.[13] Y. Dai, D. Xu, S. Maharjan, and Y. Zhang, “Joint computation offloadingand user association in multi-task mobile edge computing,”
IEEE Trans.Veh. Technol. , vol. 67, pp. 12313–12325, Dec. 2018.[14] T. Ouyang, Z. Zhou, and X. Chen, “Follow me at the edge: Mobility-aware dynamic service placement for mobile edge computing,”
IEEE J.Sel. Areas Commun. , vol. 36, pp. 2333–2345, Oct 2018.[15] T. J. Cui, M. Q. Qi, X. Wan, J. Zhao, and Q. Cheng, “Codingmetamaterials, digital metamaterials and programmable metamaterials,”
Light: Science & Applications , vol. 3, no. 10, p. e218, 2014.[16] M. Di Renzo, M. Debbah, D.-T. Phan-Huy, et al. , “Smart radio environ-ments empowered by reconfigurable AI meta-surfaces: an idea whosetime has come,”
EURASIP J. Wireless Commun. Netw. , vol. 2019, pp. 1–20, May 2019.[17] M. Sheng, Y. Wang, X. Wang, and J. Li, “Energy-efficient multiuserpartial computation offloading with collaboration of terminals, radioaccess network, and edge server,”
IEEE Trans. Commun. , vol. 68,pp. 1524–1537, March 2020. [18] K. Yang, T. Jiang, Y. Shi, and Z. Ding, “Federated learning via over-the-air computation,” IEEE Trans. Wireless Commun. , vol. 19, pp. 2022–2035, March 2020.[19] Y. Han, W. Tang, S. Jin, C.-K. Wen, and X. Ma, “Large intelli-gent surface-assisted wireless communication exploiting statistical CSI,”
IEEE Trans. Veh. Technol. , vol. 68, pp. 8238–8242, Aug. 2019.[20] B. Zheng and R. Zhang, “Intelligent reflecting surface-enhanced OFDM:Channel estimation and reflection optimization.” [Online]. Available:https://arxiv.org/abs/1909.03272.[21] S. Abeywickrama, R. Zhang, and C. Yuen, “Intelligent reflecting surface:Practical phase shift model and beamforming optimization.” [Online].Available: https://arxiv.org/abs/1907.06002.[22] Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wirelessnetwork via joint active and passive beamforming,”
IEEE Trans. WirelessCommun. , pp. 1–1, 2019.[23] Q. Wu and R. Zhang, “Beamforming optimization for intelligent re-flecting surface with discrete phase shifts.” [Online]. Available: https://arxiv.org/abs/1810.03961.[24] H. Guo, Y.-C. Liang, J. Chen, and E. G. Larsson, “Weighted sum-rate optimization for intelligent reflecting surface enhanced wirelessnetworks.” [Online]. Available: https://arxiv.org/abs/1905.07920.[25] J. Ye, S. Guo, and M.-S. Alouini, “Joint reflecting and precoding designsfor SER minimization in reconfigurable intelligent surfaces assistedMIMO systems.” [Online]. Available: https://arxiv.org/abs/1906.11466.[26] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, andL. Hanzo, “Intelligent reflecting surface for multicell MIMO communi-cations.” [Online]. Available: https://arxiv.org/abs/1907.10864.[27] G. Zhou, C. Pan, H. Ren, K. Wang, W. Xu, and A. Nallanathan, “Intelli-gent reflecting surface aided multigroup multicast MISO communicationsystems.” [Online]. Available: https://arxiv.org/abs/1909.04606.[28] C. Huang, G. C. Alexandropoulos, C. Yuen, and M. Debbah, “Indoorsignal focusing with deep learning designed reconfigurable intelligentsurfaces.” [Online]. Available: https://arxiv.org/abs/1905.07726.[29] Y. Yang, B. Zheng, S. Zhang, and R. Zhang, “Intelligent reflectingsurface meets OFDM: Protocol design and rate maximization.” [Online].Available: https://arxiv.org/abs/1906.09956.[30] M. Cui, G. Zhang, and R. Zhang, “Secure wireless communication viaintelligent reflecting surface,”
IEEE Wireless Commun. Lett. , pp. 1–1,2019.[31] H. Shen, W. Xu, S. Gong, Z. He, and C. Zhao, “Secrecy ratemaximization for intelligent reflecting surface assisted multi-antennacommunications,”
IEEE Commun. Lett. , vol. 23, pp. 1488–1492, Sep.2019.[32] D. Xu, X. Yu, Y. Sun, D. W. K. Ng, and R. Schober, “Resource allocationfor secure IRS-assisted multiuser MISO systems.” [Online]. Available:https://arxiv.org/abs/1907.03085.[33] J. Chen, Y.-C. Liang, Y. Pei, and H. Guo, “Intelligent reflecting surface:A programmable wireless environment for physical layer security.”[Online]. Available: https://arxiv.org/abs/1905.03689.[34] Q. Wu and R. Zhang, “Weighted sum power maximization for intelligentreflecting surface aided SWIPT.” [Online]. Available: https://arxiv.org/abs/1907.05558.[35] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, andL. Hanzo, “Intelligent reflecting surface enhanced MIMO broadcastingfor simultaneous wireless information and power transfer.” [Online].Available: https://arxiv.org/abs/1908.04863.[36] M. Bennis, M. Debbah, and H. V. Poor, “Ultrareliable and low-latencywireless communication: Tail, risk, and scale,”
Proc. IEEE , vol. 106,pp. 1834–1853, Oct. 2018.[37] S. Boyd and L. Vandenberghe,
Convex optimization
IEEE Trans. Signal Process. , vol. 62, pp. 741–751, March 2013.[40] Y. Pan, C. Pan, Z. Yang, M. Chen, and J. Wang, “A caching strategytowards maximal D2D assisted offloading gain,”
IEEE Trans. MobileComput. , pp. 1–1, 2019.[41] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, “An iteratively weightedMMSE approach to distributed sum-utility maximization for a MIMOinterfering broadcast channel,”
IEEE Trans. Signal Process. , vol. 59,pp. 4331–4340, Sep. 2011.[42] L.-L. Yang,
Multicarrier communications . John Wiley & Sons, 2009. [43] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algo-rithms in signal processing, communications, and machine learning,”
IEEE Trans. Signal Process. , vol. 65, pp. 794–816, Feb. 2016.[44] J. Song, P. Babu, and D. P. Palomar, “Sequence design to minimizethe weighted integrated and peak sidelobe levels,”
IEEE Trans. SignalProcess. , vol. 64, pp. 2051–2064, Apri 2015.[45] A. Goldsmith,
Wireless Communications . Cambridge University Press,2005.[46] M. L. Fisher, “The Lagrangian relaxation method for solving integerprogramming problems,”