Joint Offloading and Computing Optimization in Wireless Powered Mobile-Edge Computing Systems

Motivation

• Low-power IoT devices require enhanced computation capability for latency-sensitive applications, while sustainable energy supply remains challenging.MEC enables task offloading to AP-integrated edge servers, and WPT provides wireless energy delivery.Source: §I · p. 1

System and framework

• The proposed system uses a multi-antenna AP to wirelessly charge multiple users, which execute tasks locally or through partial offloading.Each task can be partitioned into locally computed and offloaded bits.Source: §I · p. 2, §II · p. 3
• The design jointly optimizes AP energy beamforming, user CPU frequencies, offloaded bits, and offloading time allocation.The offloaded-bit variable equals the offloading rate multiplied by the allocated offloading time.Source: §I · p. 2

Optimization objective

• The latency-constrained formulation minimizes the AP’s total energy consumption and yields an optimal solution in semi-closed form.The optimal offloading rate depends on channel power gain and circuit power, while each user retains strictly positive local computation.Source: §I · p. 2

Evaluation

• Numerical results show that the proposed joint design significantly reduces energy consumption compared with benchmark schemes.The paper reports merits over schemes without joint WPT, local computing, and offloading optimization.Source: §I · p. 2

Energy transfer

• The AP uses energy transmit beamforming to charge single-antenna users, whose harvested energy supports both local computing and offloading.The harvested signal is rectified into DC energy and stored in user batteries.Source: A. Energy Transmit Beamforming from AP… · p. 3

Task partitioning

• Each user partitions R_i input bits into ℓ_i offloaded bits and q_i locally computed bits, with R_i = ℓ_i + q_i.The partition is assumed not to add computation input bits.Source: B. Energy Consumption at Users for… · p. 3

TDMA protocol

• TDMA divides the block into 2K slots: users offload sequentially, then receive computation results after MEC processing.The AP’s MEC computation time is treated as negligible because of its sufficient computing capability.Source: 1) Computation Offloading from Users… · pp. 3–4

Offloading model

• With MRC reception, each user’s achievable offloading rate depends on bandwidth, effective channel power gain, noise power, and the coding gap.The paper sets Γ = 1 for simplicity.Source: 1) Computation Offloading from Users… · p. 4

Energy model

• User offloading energy includes transmit power and constant circuit power, while AP energy is modeled as proportional to the total offloaded bits.The proportionality coefficient α depends on AP transceiver, MEC-server chip, and CPU-frequency characteristics.Source: 1) Computation Offloading from Users… · p. 4

Model considerations

• The model assumes CSI is available for energy beamforming, although CSI acquisition consumes user energy and creates an energy-versus-accuracy tradeoff.The paper notes that feedback overhead can be made very small with existing acquisition methods.Source: 1) Computation Offloading from Users… · p. 4

Local-computing model

• Each user's local computation requires C_i CPU cycles per input bit, and DVFS adjusts the per-cycle CPU frequencies.The total cycles for q_i bits are C_iq_i.Source: 2) Local Computing at Users · p. 4

Latency constraint

• Local execution must finish within the block length T, so its execution time cannot exceed T.Source: 2) Local Computing at Users · pp. 4–5

Modeling assumption

• Continuous CPU-frequency variables provide a performance upper bound because practical frequencies are integer-valued from finite sets.Discrete-frequency modeling would generally produce an NP-hard mixed-integer design problem.Source: 2) Local Computing at Users · p. 4

Energy model

• Under a low-CPU-voltage assumption, local-computing energy is modeled using the effective capacitance coefficient κ_i and CPU frequencies.κ_i depends on the chip architecture at user i.Source: 2) Local Computing at Users · p. 5

Energy constraint

• Users' total consumed energy cannot exceed harvested energy, assuming sufficiently large storage and replenishment by the end of each block.This is a total-energy constraint rather than conventional within-block energy causality.Source: C. Energy Harvesting Constraints at… · p. 5, A. Problem Formulation · p. 5

Problem formulation

• Each task is partitioned into locally computed bits q_i and offloaded bits ℓ_i satisfying q_i = R_i − ℓ_i.The task must be completed before the end of the block.Source: A. Problem Formulation · p. 5
• The AP minimizes total energy by jointly optimizing transmit covariance Q, local CPU frequencies, offloaded bits, and users' time allocations.The objective includes AP energy for computation and wireless power transfer.Source: A. Problem Formulation · p. 5

Solution method

• The non-convex formulation is transformed into a convex problem and solved through Lagrange duality, strong duality, and the ellipsoid method.The dual function decomposes into one AP covariance subproblem and K user time-and-offloading subproblems.Source: A. Problem Formulation · p. 5, B. Optimal Solution to Problem (P1) · p. 6, 1) Evaluating the Dual Function · p. 6

Optimal structure

• At optimality, all CPU-cycle frequencies for a user are identical and satisfy f_i,1 = ... = f_i,C_i(R_i−ℓ_i) = C_i(R_i−ℓ_i)/T.Source: B. Optimal Solution to Problem (P1) · p. 5

Algorithm

• Algorithm 1 obtains optimal time allocations, offloaded bits, local CPU frequencies, and the AP covariance matrix.The covariance matrix requires an additional computation after solving the dual-based subproblems.Source: 3) Finding the Optimal Primary… · p. 7

Optimal properties

• If a user's energy-harvesting constraint is slack, optimal offloading is zero and the user computes all tasks locally.This follows because offloading would incur additional MEC-server computation energy at the AP.Source: 3) Finding the Optimal Primary… · p. 7
• Every user optimally retains some local computation, so ℓ_i < R_i and fully offloading all bits is suboptimal.Source: 3) Finding the Optimal Primary… · p. 7

Evaluation setup

• The evaluation compares joint optimization with local-only, full-offloading-only, isotropic-WPT, and separate MEC-WPT benchmark schemes.Source: §IV · p. 8
• The simulations use ζ = 0.3, N = 4 AP antennas, C_i = 10^3 cycles/bit, p_c,i = 10^-4 W, and α = 10^-4 J/bit.Other baseline parameters include σ^2 = 10^-9 W and B = 2 MHz.Source: §IV · p. 8
• Results average over 500 randomized channel realizations under a Rayleigh fading model with path-loss exponent 3.Source: §IV · p. 8

Homogeneous-user setup

• The proposed joint design achieves the lowest average AP energy consumption among the evaluated schemes for homogeneous users.The comparison uses identical user distances of d_i = 5 meters and identical computation loads R = R_i.Source: A. Case with Homogeneous Users · p. 8

Time block length

• As T increases, local-computing-only energy consumption decreases and approaches the proposed joint design, while full-offloading-only consumption remains almost unchanged for T ≥0.1 sec.The full-offloading behavior is attributed to an offloading time fixed around 0.1 sec to save circuit energy.Source: A. Case with Homogeneous Users · p. 9

User number

• The proposed joint-design gain becomes more significant as K increases, while equal offloading-time allocation approaches its performance for K ≥15.With many users, full offloading suffers because users share the finite time block and offloading energy rises drastically.Source: A. Case with Homogeneous Users · p. 9, Figure 4 · p. 9

Computation bits

• All six schemes consume more energy as R increases, while full offloading increasingly outperforms local computing and approaches the proposed joint design at large R.The paper attributes this to lower offloading energy per bit than local-computing energy in the large-R case.Source: A. Case with Homogeneous Users · p. 9

Offloading bandwidth

• For offloading schemes, AP energy consumption decreases as B increases, and offloading becomes preferable to local computing at larger bandwidth.At B ≤3 MHz, local computing outperforms full offloading; this ordering does not hold for large B.Source: A. Case with Homogeneous Users · pp. 9–10, Figure 6 · p. 9

Distance heterogeneity

• The proposed joint design achieves the lowest energy consumption among six schemes as the far-user distance d2 increases.The comparison uses R1 = R2 = 20 kbits for both users.Source: B. Case with Heterogeneous Users · p. 10
• The proposed joint design achieves a significant performance gain over the separate-design scheme when d2 > 4 meters.Source: B. Case with Heterogeneous Users · p. 10

Offloading adaptation

• The far user offloads significantly more bits than the near user, especially at larger d2.As d2 increases, near-user offloading decreases while far-user offloading increases.Source: B. Case with Heterogeneous Users · p. 10

Energy imbalance

• As d2 increases, near-user residual energy rises dramatically while the far user uses all available energy when d2 > d1.The increased AP energy consumption mainly satisfies the far user’s energy requirement.Source: B. Case with Heterogeneous Users · p. 10

Task-size heterogeneity

• When the far user’s task size R2 increases, all schemes consume more AP energy, while local-only and separate-design schemes become much worse for R2 > 20 kbits.The figure considers R1 = 20 kbits and d2 = 6 meters.Source: B. Case with Heterogeneous Users · p. 10, Figure 8 · p. 11

Energy imbalance

• Heterogeneous locations or task sizes still produce unbalanced energy demand and supply, motivating user cooperation between near and far users.Far or computation-heavy users require more transmit power, whereas nearby users with lighter tasks may have surplus energy.Source: B. Case with Heterogeneous Users · pp. 10–11

• The framework jointly optimizes energy beamforming, offloading, and computing in wireless powered multiuser MEC systems.Source: §V · p. 11
• The latency-constrained design minimizes the AP’s total energy consumption subject to individual user computation-latency constraints.Source: §V · p. 11
• Lagrange duality yields the optimal resource allocation solution in semi-closed form.Source: §V · p. 11
• Numerical results demonstrate the merits of the proposed joint design over alternative benchmark schemes.Source: §V · p. 11

• If a user offloads all task bits, ℓi = Ri, its local CPU frequency is zero.Source: A. Proof of Lemma 3.1 · p. 11
• For the nontrivial case, the proof considers partial offloading with 0 ≤ ℓi < Ri for every user.Source: A. Proof of Lemma 3.1 · p. 11
• Jensen’s inequality establishes convexity-based equalities that characterize the optimal computation allocation.Source: A. Proof of Lemma 3.1 · p. 11
• For fixed time and offloaded-bit variables, optimal CPU frequencies are chosen as small as possible while satisfying the latency lower bound.Source: A. Proof of Lemma 3.1 · p. 11
• At the optimum, the CPU frequencies across the relevant segments are equal.Source: A. Proof of Lemma 3.1 · p. 11

• Assuming F(λ) is not positive semidefinite provides an eigenvector ξ associated with a negative eigenvalue.Source: B. Proof of F (λ) ⪰0 · p. 11
• Setting Q = τξξ^H ⪰ 0 with τ tending to infinity creates a feasible direction for the matrix optimization problem.Source: B. Proof of F (λ) ⪰0 · p. 11
• This construction makes the objective unbounded below, so boundedness of the dual function requires F(λ) ⪰ 0.Source: B. Proof of F (λ) ⪰0 · p. 11

• When λ_i = 0, t_i* = 0 is optimal for problem (16).Source: C. Proof of Lemma 3.2 · pp. 11–12
• The multipliers γ_i, ν_i, and η_i correspond to the constraints ℓ_i ≤ R_i, ℓ_i ≥ 0, and t_i ≥ 0, respectively.Source: C. Proof of Lemma 3.2 · p. 12
• The KKT conditions provide necessary and sufficient conditions for the optimal primal-dual point.Source: C. Proof of Lemma 3.2 · p. 12
• β′(x) is the first-order derivative of β(x), and the derivatives in the KKT conditions are taken with respect to t_i*.Source: C. Proof of Lemma 3.2 · p. 12
• The inverse of y = β(x) − xβ′(x) is used to derive the optimal t_i* for cases with positive offloading time and nonzero p_c,i.Source: C. Proof of Lemma 3.2 · p. 12

• The constraint F(λ) ⪰ 0 can be equivalently expressed as a scalar inequality constraint.Source: D. Proof of Lemma 3.3 · p. 12
• At a query point λ1, the scalar constraint value is determined by the smallest eigenvalue π(λ1) of F(λ1).Source: D. Proof of Lemma 3.3 · p. 12
• An eigenvector v1 corresponding to the smallest eigenvalue of F(λ1) is used to evaluate π(λ1).Source: D. Proof of Lemma 3.3 · p. 12
• The subgradient of F(λ) is obtained using the eigenvector associated with the smallest eigenvalue.Source: D. Proof of Lemma 3.3 · p. 12
• The subgradient has a zero entry because π(λ) is independent of μ.Source: D. Proof of Lemma 3.3 · p. 12