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Latency Minimization for Intelligent Reflecting Surface Aided Mobile Edge Computing

Tong Bai, Cunhua Pan, Yansha Deng, Maged Elkashlan, Arumugam Nallanathan, Lajos Hanzo

arXiv:1910.07990v3eess.SP

TL;DR

MEC off-loading can be hindered by imperfect communications links, limiting support for resource-intensive mobile applications. The paper formulates IRS-aided latency-minimization problems and alternates computing and communications optimization using BCD-based low-complexity iterative algorithms. In the reported single-cell setting, IRS assistance reduces device-average computational latency from 177 ms to 139 ms relative to conventional MEC without IRSs.

  • Problem

    Imperfect computation off-loading links limit the latency benefits of MEC for resource-intensive applications on devices with limited computing capability.

  • Method

    The paper jointly designs off-loading, edge-resource allocation, MUD, and IRS phase shifts, using BCD to decouple computing and communications optimization.

  • Results

    177 ms to 139 ms device-average computational latency is reported for IRS-aided MEC versus conventional MEC without IRSs in the evaluated 300 m, 5-antenna, 5-device cell.

  • Takeaways & Limitations

    IRS assistance can reduce computational latency in the evaluated MEC setting by improving the off-loading system’s wireless propagation environment.

Abstract

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Computation off-loading in mobile edge computing (MEC) systems constitutes an efficient paradigm of supporting resource-intensive applications on mobile devices. However, the benefit of MEC cannot be fully exploited, when the communications link used for off-loading computational tasks is hostile. Fortunately, the propagation-induced impairments may be mitigated by intelligent reflecting surfaces (IRS), which are capable of enhancing both the spectral- and energy-efficiency. Specifically, an IRS comprises an IRS controller and a large number of passive reflecting elements, each of which may impose a phase shift on the incident signal, thus collaboratively improving the propagation environment. In this paper, the beneficial role of IRSs is investigated in MEC systems, where single-antenna devices may opt for off-loading a fraction of their computational 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 and multi-device scenarios, subject to practical constraints imposed on both the edge computing capability and the IRS phase shift design. To solve this problem, the block coordinate descent (BCD) technique is invoked to decouple the original problem into two subproblems, and then the computing and communications settings are alternatively optimized using low-complexity iterative algorithms. It is demonstrated that our IRS-aided MEC system is capable of significantly outperforming the conventional MEC system operating without IRSs. Quantitatively, about $20~\%$ computational latency reduction is achieved over the conventional MEC system in a single cell of a $300~\rm{m}$ radius and $5$ active devices, relying on a $5$-antenna access point.

I. INTRODUCTION

The paper introduces IRS-aided MEC to improve task off-loading over imperfect wireless links, jointly optimizing computing and communications for latency minimization in single- and multi-device settings.

  • I. INTRODUCTION: IRSs modify the wireless propagation environment through programmable reflecting elements, supporting more efficient MEC off-loading.Each IRS includes a controller and passive elements that adjust reflected-signal amplitude and phase.
  • I. INTRODUCTION: The latency-minimization formulation jointly optimizes off-loading volume, edge-resource allocation, the MUD matrix, and IRS phase shifts under practical constraints.The formulation targets multi-device scenarios and accounts for total edge computing capability and IRS phase-shift constraints.
  • I. INTRODUCTION: BCD decouples computing and communications design, enabling alternating optimization of the coupled variables.The computing design further separates off-loading volume and edge-resource allocation, while communications design uses auxiliary-variable transformations and iterative optimization.
  • I. INTRODUCTION: Low-complexity iterative algorithms are developed for multi-device and simplified single-device scenarios, with numerical evaluations verifying convergence and latency performance.The single-device case removes edge-resource allocation and multi-user interference from consideration.
  • I. INTRODUCTION: The proposed system assists single-antenna devices off-loading partial or complete tasks to an edge node through a multi-antenna access point.The system model includes an IRS with N elements, K devices, and an M-antenna access point.
  • I. INTRODUCTION: The model assumes perfect channel estimation and continuous IRS phase shifts, with both assumptions identified as best-case bounds for realistic deployments.The paper notes that practical hardware supports only limited discrete phase shifts and evaluates phase quantization separately.

B. Computing Model

The computing model partitions each device’s task between local and edge execution, then minimizes weighted latency by jointly selecting off-loading volumes, edge resources, and communications variables under system constraints.

  • B. Computing Model: Each device divides a data-partitioning task between local processing and edge off-loading.L_k denotes total bits, ℓ_k the off-loaded bits, and c_k the CPU cycles required per bit.
  • B. Computing Model: Local-computing latency depends on the device’s locally processed bits, required cycles per bit, and local CPU capability.The local computing model uses L_k, ℓ_k, c_k, and f_k^l.
  • B. Computing Model: Edge-computing latency combines wireless off-loading, edge execution, and result feedback, with feedback treated as negligible.Edge execution begins after all ℓ_k bits have been off-loaded, and edge resources are allocated across devices under a total capability constraint.
  • B. Computing Model: A device’s total latency is modeled as the maximum of its local-computing and edge-computing latencies.This captures parallel local and edge processing under partial off-loading.
  • B. Computing Model: The weighted latency problem jointly optimizes off-loading volumes, edge-resource allocations, the MUD matrix, and IRS phase shifts.The constraints enforce continuous phase ranges, integer off-loading volumes, and feasible edge-resource allocations.
  • B. Computing Model: The original optimization is difficult because of segmented off-loading, coupling between the MUD matrix and IRS phases, and phase-shift non-convexity.BCD decouples computing and communications, while MM iteratively approaches a locally optimal IRS phase-shift solution.

III. JOINT OPTIMIZATION OF COMPUTING AND COMMUNICATIONS SETTING

The joint design alternates between computing and communications variables using BCD, solving each block with tractable subproblems and iterative updates. The computing block optimizes off-loading and edge-resource allocation while communications variables remain fixed.

  • Joint BCD framework: BCD alternates optimization of computing and communications settings until the objective function converges.The computing block updates off-loading volume and edge-resource allocation; the communications block updates the MUD matrix and IRS phase shift.
  • Computing design: Given fixed W and θ, the computing block decouples optimization of off-loading volume ℓ and edge resource allocation f^e.The two variables are optimized alternately through Algorithm 1.
  • Computing design: Given W, θ, and f^e, Proposition 1 provides the optimal number of off-loaded bits.The off-loading volume is optimized using the proposition, while the communications setting and edge allocation are held fixed.
  • Computing design: The fixed-off-loading edge-resource subproblem is convex and can therefore be solved using KKT conditions.Slater’s condition is stated to hold, and the Lagrange multiplier is obtained using bisection search.
  • Computing design: Algorithm 1 iteratively updates ℓ and f^e, then outputs their optimal values for fixed W and θ.Its complexity is dominated by computing the edge allocation and performing the bisection search.

B. Joint Optimization of the MUD Matrix and the IRS Phase Shift Coefficient While Fixing the Computing Settings

With computing variables fixed, the communications design optimizes the MUD matrix and IRS phase shifts under phase constraints. The resulting objective is non-convex because of segmented latency terms and a sum of fractional functions.

  • Problem formulation: Fixing ℓ and f^e reformulates Problem P0 as a communications-design problem over W and θ.The IRS phase shifts satisfy 0 ≤ θ_n < 2π.
  • Problem challenges: The communications problem is difficult because max creates segmented D_k(w_k,θ), while the objective becomes a non-convex sum of ratios.These two properties motivate the subsequent equivalent reformulation.
  • Problem transformation: The reformulated problem replaces D_k with an equivalent quantity and removes constant terms without changing the phase-shift constraints.The phase variables remain constrained by 0 ≤ θ_n < 2π.
  • Problem transformation: An equivalent formulation imposes ϖ_kℓ_k R_k(w_k,θ) ≤ β_k for each device alongside the IRS phase constraints.This introduces auxiliary β-related constraints for the communications optimization.
  • Problem transformation: Proposition 3 links a solution of the transformed problem to KKT conditions through an auxiliary multiplier vector λ.The relationship holds when β and λ are set to their corresponding optimal values.

1. Initialization

The communications subproblem is transformed into a tractable iterative procedure. It alternates optimization of W and θ with updates of auxiliary variables β and λ until convergence.

  • Initialization and iteration: Algorithm 2 initializes θ, W, R, λ, and β, then repeatedly updates W and θ through Algorithm 3 and updates λ and β.The procedure returns W* and θ* after convergence.
  • Initialization and iteration: The transformed communications problem is solved by first optimizing W and θ for fixed β and λ, then updating β and λ.The updates use a modified Newton’s method until convergence.
  • Initialization and iteration: For fixed β and λ, the W-and-θ subproblem becomes weighted sum-rate maximization.Weighted sum-rate maximization is handled through an equivalent weighted MSE minimization formulation.
  • Initialization and iteration: Weighted MSE minimization is tractable because the objective is convex in each optimization variable when the others are fixed.The formulation introduces an auxiliary weight variable Υ_k for each device and uses the MSE e_k.

2) MUD Matrix Design:

The MUD and IRS phase-shift design is handled through weighted-MSE updates and majorization-minimization under unit-modulus constraints. The resulting algorithms produce iterative feasible updates and have complexity driven largely by matrix and IRS-phase operations.

  • MUD matrix design: For fixed θ and w_k, the auxiliary variable Υ_k is optimized by minimizing the weighted-MSE objective.The MSE is related to SINR under MMSE MUD through γ_k = (e_k^MMSE)^−1.
  • IRS phase-shift design: The phase-shift subproblem is formed by fixing Υ_k and W and removing objective terms independent of θ.The phase coefficients remain bounded by 0 ≤ θ_n ≤ 2π.
  • IRS phase-shift design: The transformed phase problem uses φ_n = e^{jθ_n} and imposes the unit-modulus constraint |φ_n| = 1.This constraint makes Problem P2-E7 non-convex.
  • Algorithmic procedure: Algorithm 3 alternates updates of W, Υ, and θ, using MM for θ, and outputs W* and θ* for fixed λ and β.The iteration terminates when the objective change satisfies the stated condition.
  • IRS phase-shift design: MM constructs an upper-bounding surrogate g(φ|φ^t), then minimizes it to generate a feasible sequence of phase vectors.The majorization and minimization steps are repeated over iteration index t.
  • Complexity: The MM phase update has complexity O(N^3 + t_max^MM N^2), making it a principal contributor to Algorithm 3’s complexity.The N^3 term computes the maximum eigenvalue, while each MM iteration contributes O(N^2).

C. Overall Algorithm to Solve Problem P0

Algorithm 4 applies BCD to Problem P0, with alternating updates whose objective value decreases and is bounded below, guaranteeing convergence with low practical complexity.

  • Algorithm 4 uses BCD to solve Problem P0 through alternating optimization steps.The objective value decreases in Steps 2 and 3.
  • A lower bound imposed by the total edge computing resources guarantees convergence of Algorithm 4.
  • Simulation results show that Algorithm 4 converges rapidly, supporting the low complexity of the proposed algorithms.

IV. SPECIFIC CASE STUDY: THE SINGLE-DEVICE SCENARIO

The single-device case simplifies latency optimization by eliminating edge-resource allocation and reducing the objective to a single ratio. The proposed solution alternates optimization of offloading, receive combining, and IRS phases under phase-shift constraints.

  • With one served device, all edge-computing resources can be assigned to it, eliminating edge-resource allocation from the optimization.
  • The single-device formulation becomes a single-ratio problem instead of the sum-of-ratios form in Problem P2-E1.
  • For fixed receive combiner and IRS phases, the relaxed offloading volume is minimized by selecting it according to Proposition 1, then integerizing it.
  • Algorithm 5 jointly optimizes offloading, receive combining, and IRS phases, subject to 0 ≤ θ_n < 2π for every reflecting element.
  • Block coordinate descent alternates receive-combiner and IRS-phase optimization, with the combiner obtained using maximum ratio combining for fixed IRS phases.
  • The complexity of Algorithm 5 is on the order of O(MN^2), dominated by updating the IRS phases and receive combiner.

V. NUMERICAL RESULTS

The numerical study evaluates IRS-aided MEC in single-cell single-device, two-device, and multi-device settings. It compares optimized IRS phases with random-phase and no-IRS baselines under prescribed channel, computing, and deployment assumptions.

  • The simulations consider single-device, two-device, and multi-device scenarios in a single-cell MEC system.
  • The AP coverage radius is R = 300 m, with the IRS deployed at the cell edge.
  • The With IRS scheme jointly optimizes offloading, edge-resource allocation, multiuser detection, and IRS phase shifts.
  • The RandPhase scheme optimizes the computing and communication variables while using randomly selected IRS phases.
  • Default settings include total edge computing capacity of 50 × 10^9 cycle/s, task size L_k = 300 Kb, and local computing frequency f_l_k = 0.5 × 10^9 cycle/s.
  • The Without IRS baseline sets the IRS-assisted composite channel to zero and optimizes the remaining system variables.

A. Properties of the Proposed Algorithms

The proposed algorithms are evaluated for convergence, initialization sensitivity, phase quantization, IRS size, and edge computing capability. Results show convergence and latency benefits from IRS phase-shift design and larger IRS configurations, while excessive edge computing capacity yields diminishing latency reductions.

  • Convergence: The proposed algorithms achieve convergence, although larger numbers of IRS phase shifts slightly slow convergence, especially in the multi-device scenario.The slowdown is attributed to the larger number of optimization variables.
  • Impact of Initialization Settings: 2% to 17% is the latency gap between the worst and best locally optimal multi-device results across random initializations.In the single-device scenario, maximum and minimum latency values are almost identical.
  • Impact of Phase Quantization: 1% to 5% is the performance gap between continuous and 2-bit IRS phase shifts.Latency decreases as the number of discrete phase shifts increases, and the quantization loss becomes negligible with four phase shifts.
  • Impact of the Number of Reflecting Elements: 46 ms is the latency gain of “With IRS” over “RandPhase” at N = 100, compared with around 11 ms at N = 10.The results associate sophisticated phase-shift design with beamforming gain and larger IRSs with higher reflection-based gain.
  • Impact of Edge Computing Capability: 30 × 10^9 cycle/s is the approximate edge-computing threshold beyond which increasing capability produces smaller latency reductions.At low capability, edge-computing latency dominates; at high capability, computation off-loading latency dominates.

3) Impact of the Device Location:

Device location affects IRS-aided MEC latency through the balance between direct and composite device–IRS–AP links. IRS phase design, element count, path loss, device density, and interference further shape these latency outcomes.

  • Location and phase shifts: A designed IRS phase response extends the IRS benefit across a 100 m device-location coverage, compared with less than 20 m for random phases.Without IRS, latency increases with AP–device distance; with the designed IRS scheme, the composite link dominates when d ≥ 260 m.
  • IRS elements and device comparison: IRS assistance can reverse device latency rankings: Device 2 outperforms Device 1 with designed phases, while Device 1 is better without IRS or with random phases.Device 2 also gains more as the number of IRS elements increases because it is closer to the IRS.
  • Equivalent-latency locations: An IRS can make a device at d = 280 m achieve the same latency as one at d = 220 m, although these equivalent locations depend on channel path loss exponents.The equivalence is associated with equal channel gain under the examined phase-shift scheme.
  • Path loss: Increasing αIRS raises device latency because it reduces the IRS array and beamforming gain, and the equivalent-latency intercept can disappear.The result motivates careful IRS placement to avoid obstacles.
  • Multiple devices: Device-average latency increases with the number of active devices because each device receives fewer edge resources and less beamforming gain.More powerful edge computing can address resource allocation, while additional IRSs can strengthen beams.
  • Inter-cell interference: Increasing the ICI-to-noise ratio decreases the IRS benefit because strong interference leaves only a marginal fraction of tasks suitable for off-loading.The passage recommends careful adjacent-cell spectrum management.
  • Overall evaluation: The IRS-aided scheme reduces device-average latency from 177 ms to 139 ms versus conventional MEC without IRS in the stated five-device setting.The evaluation uses a 300 m cell radius and a 5-antenna access point.

APPENDIX A THE PROOF OF PROPOSITION 1

The appendix establishes optimality properties for the computational subproblem and characterizes the communication subproblem through convexity and KKT conditions.

  • Off-loading optimization: The relaxed off-loading variable achieves its minimum delay at ˆℓk = LkckRkfe.The integer off-loading value is then obtained by applying the stated integer operation.
  • Computing-resource optimization: The objective is convex in fe_k, and the relevant constraints are linear, making Problem P1-E strictly convex.This supports obtaining the unique optimum through the convex formulation.
  • Communication optimization: The communication subproblem is characterized using a Lagrangian with non-negative multipliers and KKT conditions, including 0 ≤ θ*_k ≤ 2π.The appendix states that the resulting equations are exactly the KKT conditions of Problem P2-E3.
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