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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 incur high latency when communication links are imperfect. This paper jointly optimizes computing and communications in IRS-aided MEC, reporting lower device-average computational latency than conventional MEC without IRS in a specified single-cell setting.

  • Problem

    MEC’s latency benefits are limited by imperfect off-loading links, while resource-intensive tasks can exceed mobile devices’ computing capabilities.

    Source: A. Motivation and Scope · p. 1

  • Method

    The paper formulates IRS-aided MEC latency minimization and jointly designs computing and communications under edge-computing and IRS phase-shift constraints.

    Source: C. Contributions and Organizations · p. 2

  • Results

    177 ms to 139 ms: device-average computational latency versus conventional MEC without IRS in a 300 m-radius cell with a 5-antenna AP and 5 active devices.

    Source: §VI · p. 14

  • Takeaways & Limitations

    In the evaluated settings, latency reductions from increasing edge-computing capability diminish beyond about 30 × 10^9 cycle/s, so extremely powerful edge nodes are unnecessary for latency minimization.

    Source: 2) Impact of the Edge Computing… · p. 12

    Practical IRS elements provide only a limited number of discrete phase shifts, making the proposed algorithms a best-case latency bound for realistic scenarios.

    Source: A. Communications Model · p. 3

Abstract

from arXiv · show

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 … A. Communications Model

The paper proposes IRS-aided MEC and jointly optimizes computation and communications to minimize off-loading latency. It also specifies a multi-device communications model and the assumptions governing its channels and IRS phase shifts.

A. Motivation and Scope

  • MEC combines local and edge computing to support resource-intensive applications, but imperfect off-loading links can make edge execution unreliable or slower than local computing.Edge nodes are typically co-located with access points, and MEC is described in –.Source: A. Motivation and Scope · p. 1

1) Design of Mobile Edge Computing Systems:

  • Prior MEC work covers single-user and multi-user off-loading and resource allocation –,,, while hostile communications environments remain insufficiently addressed.Examples include binary off-loading, partial off-loading, and multi-user resource allocation,.Source: 1) Design of Mobile Edge Computing… · pp. 1–2

2) Intelligent Reflecting Surface Aided Wireless Networks:

  • IRS research spans channel estimation, phase-shift design, security, and SWIPT –[35], motivating its application to MEC.Prior work also studied transmit-power minimization with IRS and AP precoding, as well as discrete phase shifts.Source: 2) Intelligent Reflecting Surface… · p. 2

C. Contributions and Organizations

  • The proposed multi-device IRS-aided MEC design minimizes latency by jointly optimizing off-loading volume, edge resources, multi-user detection, and IRS phase shifts using BCD.The formulation accounts for total edge-computing capability and IRS phase-shift constraints; BCD alternates between computing and communications subproblems.Source: C. Contributions and Organizations · p. 2
  • The computing and communications subproblems use separate low-complexity iterative methods within the joint BCD design.For computing, the optimal off-loading volume equalizes local and edge latency for fixed resources; resource allocation is convex and solved using KKT conditions and bisection. For communications, an auxiliary-variable formulation, modified Newton updates, weighted minimum MSE, and an MM algorithm yield tractable updates for the MUD matrix and IRS phase shifts.Source: C. Contributions and Organizations · p. 2
  • The paper also studies a single-device case, simplifying the multi-device algorithm because edge-resource allocation and multi-user interference are absent.It proposes a low-complexity iterative algorithm for this scenario.Source: C. Contributions and Organizations · p. 3

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • The system-model section introduces communications and computing models and formulates an IRS-aided MEC latency-minimization problem.The model is presented for both communications and computing perspectives.Source: §II · p. 3

A. Communications Model

  • The communications model has K single-antenna devices off-loading through an M-antenna AP, with an N-element IRS assisting transmission.The AP and edge node are co-located, and their fiber-link latency is negligible; channels are quasi-static and perfectly estimated, while IRS elements have unit reflection amplitude and adjustable phases. Perfect channel estimates and continuous phase shifts are idealizations, so the algorithms provide best-case bounds; phase quantization is evaluated in Section V-A3.Source: A. Communications Model · p. 3, Figure 1 · p. 3

B. Computing Model · C. Problem Formulation

The computing model splits each device’s workload between local execution and edge off-loading, and the formulation minimizes weighted device latency by jointly optimizing computation and communication variables. The paper uses BCD to decouple these designs and MM to address IRS phase-shift non-convexity, yielding a locally optimal solution.

B. Computing Model

  • The data-partitioning model lets each device process part of its application locally and off-load the remainder to the edge node.For device k, the model denotes total bits by Lk, off-loaded bits by ℓk, and required CPU cycles per bit by ck.Source: B. Computing Model · p. 4
  • Local latency is formulated using the device’s CPU-cycle capability, while edge computation is allocated per device under a maximum edge-cycle capacity.Edge processing for a device starts only after all its off-loaded bits arrive; feedback latency is treated as negligible, and edge latency includes off-loading and computation.Source: B. Computing Model · p. 4
  • Each device’s latency is the maximum of its local-computing latency and its edge-computing latency.This captures the parallel local and edge processing paths in the model.Source: B. Computing Model · p. 4

C. Problem Formulation

  • The objective minimizes weighted latency across devices by jointly optimizing off-loading volumes, edge resource allocations, the MUD matrix, and IRS phase shifts.The device-specific weights are denoted by ϖk.Source: C. Problem Formulation · p. 4
  • The constraints bound each IRS phase shift to [0, 2π), require integer off-loading volumes from 0 to Lk, and restrict per-device edge computing resources.Source: C. Problem Formulation · p. 4
  • BCD decouples computing and communications designs, while MM iteratively addresses the non-convex IRS phase shifts at low complexity to obtain a locally optimal solution.The formulation is difficult because of the segmented objective, coupling between the MUD matrix and IRS phase shifts, and phase-shift non-convexity.Source: C. Problem Formulation · pp. 4–5

III. JOINT OPTIMIZATION OF COMPUTING AND COMMUNICATIONS SETTING … 1) Optimization of

The section uses block coordinate descent (BCD) to optimize computing and communications settings in alternating steps. With communications fixed, it optimizes off-loading volume and edge-resource allocation through decoupled updates, including a convex resource-allocation subproblem.

III. JOINT OPTIMIZATION OF COMPUTING AND COMMUNICATIONS SETTING

  • BCD alternates between optimizing computing variables with communications fixed and optimizing the MUD matrix and IRS phase shift with computing fixed.The process continues until the objective function converges.Source: §III · p. 5

Computing Resource Allocation While Fixing the Communications Settings

1) Optimization of

  • Given fixed communications and edge-resource allocation, Proposition 1 gives the optimal number of off-loaded bits.The selected value makes local and edge computation latency equivalent.Source: 1) Optimization of · p. 5
  • Algorithm 1 iteratively updates off-loading volume and edge-resource allocation, using bisection to obtain the multiplier, until termination.It outputs the optimized off-loading volume and edge-resource allocation for the fixed MUD matrix and IRS phase shift.Source: 1) Optimization of · p. 5
  • Problem P1-E is convex, enabling its edge-resource allocation solution to be characterized by KKT conditions.The optimal allocation and nonnegative Lagrange multiplier must satisfy those conditions.Source: 1) Optimization of · p. 5
  • For a given multiplier, edge-resource allocation is derived directly, and bisection finds the optimal multiplier using the monotonicity of total allocated resources.The multiplier search uses a termination coefficient.Source: 1) Optimization of · pp. 5–6

B. Joint Optimization of the MUD Matrix and the IRS Phase Shift Coefficient While Fixing the Computing Settings · 1) Problem Transformation:

With the off-loading and computing settings fixed, the paper transforms the joint MUD-matrix and IRS-phase optimization to address its segmented latency terms and non-convex sum-of-ratios objective.

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

  • Given off-loading vector ℓ and edge-resource allocation f^e, P0 is reformulated with each IRS phase constrained to 0 ≤ θ_n < 2π.Source: B. Joint Optimization of the MUD… · p. 6
  • The transformation addresses segmented D_k(w_k, θ) terms and an objective that sums fractional functions of W and θ.These features make Problem P2 a non-convex sum-of-ratios optimization problem.Source: B. Joint Optimization of the MUD… · p. 6

1) Problem Transformation:

1. Initialization · 3. Output optimal W ∗and θ∗

The method transforms the sum-of-ratios problem into a parameterized subtractable form, alternates optimization of W and θ with updates to β and λ, and initializes and updates these variables iteratively.

1. Initialization

  • Initialization sets t2 = 0, chooses ζ and ϵ in (0, 1) and θ(0) satisfying (7a), then computes W(0), R(0), λ(0), and β(0).Source: §1 · p. 6
  • Each iteration updates W and θ using Algorithm 3, then updates λ and β.Source: §1 · p. 6
  • The iteration selects i(t2+1) as the smallest positive integer satisfying the stated condition.Source: §1 · p. 6

3. Output optimal W ∗and θ∗

  • If (W ∗, θ∗) solves P2-E3 and satisfies (26) for β = β∗ and λ = λ∗, then (W ∗, θ∗, β∗) solves P2-E2 for λ = λ∗.Source: §3 · p. 6
  • The sum-of-ratios Problem P2-E1 is transformed into parameterized subtractable Problem P2-E3, then solved by optimizing W and θ and updating β and λ with modified Newton’s method until convergence.Source: §3 · p. 6
  • Given β, λ, and off-loading vector ℓ, optimizing W and θ reduces to weighted sum-rate maximization, which is handled through weighted MSE minimization with auxiliary weights Υk.Source: §3 · p. 7
  • Problem P2-E4 is more tractable because its objective is convex in each optimization variable when the other is fixed, enabling BCD optimization.Source: §3 · p. 7

2) MUD Matrix Design: · 3) Auxiliary Variable Design: · 4) IRS Phase Shift Coefficient Design:

The design alternates between optimizing the MUD and auxiliary variables and optimizing the IRS phases. The phase update is cast as a non-convex unit-modulus problem and solved with a majorization-minimization procedure.

2) MUD Matrix Design:

  • With the phase vector and auxiliary variable fixed, the MUD vector is obtained by setting the objective’s first-order derivative to zero, yielding an equivalent weighted-MSE minimization.This gives the MUD update for Problem P2-E4.Source: 2) MUD Matrix Design · p. 7

3) Auxiliary Variable Design:

  • With the phase vector and MUD fixed, the auxiliary variable is chosen to minimize the objective; under an MMSE MUD, the system’s SINR and MSE are related.The passage states the SINR–MSE relationship but does not provide its complete expression.Source: 3) Auxiliary Variable Design · p. 7

4) IRS Phase Shift Coefficient Design:

  • With the auxiliary variable and MUD fixed, the IRS phases are optimized after removing objective terms independent of the phase vector, subject to 0 ≤ θ_n ≤ 2π.The phase bounds apply for every IRS element n.Source: 4) IRS Phase Shift Coefficient Design · p. 7
  • Representing each phase by φ_n = e^(jθ_n) converts the phase design to a unit-modulus constraint, making the resulting problem non-convex.The transformed constraint is |φ_n| = 1 for every element n.Source: 4) IRS Phase Shift Coefficient Design · pp. 7–8
  • The MM algorithm builds an upper-bound surrogate and alternates its construction with minimization to generate feasible unit-modulus phase vectors.It starts from a feasible initial phase vector.Source: 4) IRS Phase Shift Coefficient Design · p. 8
  • The MM iteration terminates when the relative objective change is at most ε or the iteration count reaches t_max.The stopping condition is given for the phase-design MM algorithm.Source: 4) IRS Phase Shift Coefficient Design · p. 8

C. Overall Algorithm to Solve Problem P0

Algorithm 4 applies block coordinate descent (BCD) to solve Problem P0. Its objective value decreases across Steps 2 and 3 and is bounded below, ensuring convergence; simulations show rapid convergence.

C. Overall Algorithm to Solve Problem P0

  • Algorithm 4 uses BCD to solve Problem P0, with its computational complexity mainly determined by Steps 2 and 3.The complexity of those steps is analyzed in the preceding subsections.Source: C. Overall Algorithm to Solve Problem P0 · pp. 8–9
  • Algorithm 4 is guaranteed to converge because Steps 2 and 3 decrease the objective value, which is bounded below by the total edge computing resource constraint.The objective decrease is guaranteed in both steps, and the resource constraint supplies its lower bound.Source: C. Overall Algorithm to Solve Problem P0 · p. 9
  • Simulations show that Algorithm 4 converges rapidly, which the paper presents as evidence of the algorithms’ low complexity.The computational complexity is mainly dependent on Steps 2 and 3.Source: C. Overall Algorithm to Solve Problem P0 · p. 9

IV. SPECIFIC CASE STUDY: THE SINGLE-DEVICE SCENARIO

The single-device case simplifies MEC optimization by removing edge-resource allocation and multi-user interference, while enabling a single-ratio formulation. The paper alternates between communication-variable optimization and an offloading decision update.

IV. SPECIFIC CASE STUDY: THE SINGLE-DEVICE SCENARIO

  • With one device, all edge-computing resources can be assigned to it, the sum-of-ratios objective becomes a single ratio, and multi-user interference is absent.These simplifications make the joint optimization more tractable.Source: §IV · p. 9
  • After optimizing w and θ, the algorithm updates the offloading decision by relaxing ℓ and then integerizing it.The relaxed ℓ is selected to minimize latency for fixed w and θ.Source: §IV · p. 9
  • BCD optimizes the receiver w and IRS phase vector θ in turn: w follows maximum-ratio combining, while θ aligns the direct and reflected signal phases.For a given w, the phase choice satisfies arg{w^Hh_d} = arg{w^HGΘh_r}.Source: §IV · p. 9

V. NUMERICAL RESULTS

The numerical study evaluates IRS-aided MEC algorithms in a single-cell setting across single-, two-, and multi-device scenarios, examining algorithm properties and latency performance.

V. NUMERICAL RESULTS

  • The single-cell setup uses a 300 m AP coverage radius with the IRS deployed at the cell edge.For the multi-device scenario, devices are uniformly distributed within a circle whose size and location are prescribed by simulation parameters.Source: §V · p. 10
  • The communications model includes i.i.d. zero-mean, unit-variance complex Gaussian fading, distance-dependent path loss, and additive white Gaussian noise.Task and computing variables follow uniform distributions, with default settings referenced in Table II.Source: §V · p. 10
  • The study compares optimized IRS phases, random IRS phases, and operation without an IRS across the device scenarios.With IRS optimizes off-loading, edge computing allocation, the MUD matrix, and phase shifts; RandPhase randomizes phase shifts, while Without IRS sets the IRS composite channel to zero.Source: §V · p. 10
  • Figure 2 presents convergence for Algorithm 5 in the single-device scenario and Algorithm 4 in the two-device scenario.The caption specifies a total edge computing capability of 50 × 10^9 cycle/s; both devices are set at 280 m in the two-device case.Source: Figure 2 · p. 10, §V · p. 10

A. Properties of the Proposed Algorithms … 2) Impact of the Edge Computing Capability:

The proposed algorithms converge quickly, show limited sensitivity to initialization, and retain much of the continuous-phase performance under quantization. In the single-device scenario, IRS gains grow with the number of reflecting elements, while latency benefits from greater edge computing capability diminish beyond a threshold.

1) Convergence:

2) Impact of the Initialization Settings:

  • Across 100 random initializations, maximum and minimum latency are nearly identical for one device, while their multi-device gap is 2%–17%, indicating near-optimal performance.The maximum and minimum locally optimal results are treated as worst-case and potentially globally optimal, respectively.Source: 2) Impact of the Initialization Settings · pp. 10–11, Figure 4 · p. 11

3) Impact of the Phase Quantization:

  • Because practical IRS hardware offers limited discrete phase shifts, the continuous phase design can be rounded to the nearest discrete phase, causing quantization loss.The simulations compare continuous, 1-bit, and 2-bit phase-shift assumptions.Source: 3) Impact of the Phase Quantization · p. 11, Figure 5 · p. 11
  • The latency decreases with more discrete IRS phase shifts, and the continuous-versus-2-bit performance gap is 1%–5%, with quantization loss negligible for as few as four phase shifts.The figure compares continuous, 1-bit, and 2-bit phase-shift assumptions.Source: 3) Impact of the Phase Quantization · pp. 11–12, Figure 5 · p. 11

1) Impact of the Number of Reflecting Elements:

2) Impact of the Edge Computing Capability:

  • Latency falls sharply as edge computing capability increases at low capacity, but gains diminish near 30 × 10^9 cycle/s as off-loading latency becomes dominant.The study concludes that an extremely powerful edge node is unnecessary for latency minimization.Source: 2) Impact of the Edge Computing… · p. 12, Figure 7 · p. 12

3) Impact of the Device Location: … VI. CONCLUSIONS

The paper finds that IRS phase design, device location, edge resources, and interference shape latency in IRS-aided MEC. With IRS reduces device-average latency from 177 ms to 139 ms versus Without IRS at five devices.

3) Impact of the Device Location:

  • With IRS extends the latency benefit to 100 m, versus less than 20 m with random phases; in Fig. 11, d1=220 m and 280 m yield equal latency.For the single-device case, latency peaks at d=260 m and then falls as the composite device-IRS-AP link becomes dominant; the equivalent-latency locations depend on channel path-loss exponents.Source: 3) Impact of the Device Location · pp. 12–13

1) Impact of the Number of IRS Elements:

  • With IRS, Device 2 outperforms Device 1, while Device 1 has lower latency under Without IRS and RandPhase; more IRS elements give Device 2 a higher gain.Device 2 is closer to the IRS, where the composite channel dominates off-loading, and IRSs can change the devices’ latency ranking.Source: 1) Impact of the Number of IRS Elements · p. 13

C. Multi-Device Scenario; 2) Impact of the Edge Computing Capability:

4) Impact of the Path Loss Exponent:

  • Increasing the IRS path-loss exponent raises device latency and removes the With IRS latency-intercept point.The paper attributes this to reduced IRS array and beamforming gain and recommends IRS placement that avoids obstacles to achieve lower αui and αia.Source: 4) Impact of the Path Loss Exponent · p. 13

5) Impact of the Number of Devices:

  • Device-average latency increases with the number of devices because each receives fewer edge resources and less beamforming gain.More powerful edge computing can address the resource reduction, while additional IRSs can form stronger beams.Source: 5) Impact of the Number of Devices · pp. 13–14

VI. CONCLUSIONS

6) Impact of Inter-Cell Interference:

APPENDIX A THE PROOF OF PROPOSITION 1 · APPENDIX B THE PROOF OF PROPOSITION 2 · APPENDIX C THE PROOF OF PROPOSITION 3

The appendices prove Proposition 1 using a relaxation of the integer task variable, establish strict convexity for P1-E, and use KKT conditions to prove Proposition 3’s conclusions.

APPENDIX A THE PROOF OF PROPOSITION 1

  • The integer task variable ℓk is relaxed to ˆℓk ∈ [0, Lk], and its integer optimum is obtained by applying an operation to the relaxed solution.The relaxed delay is minimized at the value specified in the proof; the supplied passage does not identify the integer operation.Source: Appendix A · pp. 14–15
  • The relaxed delay ˆDk(ˆℓk) achieves its minimum at the value specified in the proof.This result is used in the proof before recovering an integer value of ℓk.Source: Appendix A · p. 15

APPENDIX B THE PROOF OF PROPOSITION 2

  • P1-E is strictly convex because its objective is convex in f^e_k and constraints (7c) and (7d) are linear.The proof establishes nonnegative second derivative with respect to f^e_k.Source: Appendix B · p. 15

APPENDIX C THE PROOF OF PROPOSITION 3

  • For P2-E2, the proof introduces a non-negative Lagrange multiplier λk and states KKT conditions for a solution (W*, θ*, β*).The phase variables satisfy 0 ≤ θ*k ≤ 2π for k = 1, 2, . . . , K.Source: Appendix C · p. 15
  • Equations (60), (61), and (66) are the KKT conditions of P2-E3 when λ = λ* and β = β*, proving Proposition 3’s first conclusion.The proof identifies these equations with P2-E3’s KKT conditions under the stated multiplier and variable assignments.Source: Appendix C · p. 15
  • The proof states that Proposition 3’s second conclusion follows by the same procedure.The supplied passage does not include further details of that procedure.Source: Appendix C · p. 15

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