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Optimal Resource Allocation for Delay Minimization in NOMA-MEC Networks

Fang Fang, Yanqing Xu, Zhiguo Ding, Chao Shen, Mugen Peng, George K. Karagiannidis

arXiv:2009.06397v1eess.SPcs.IT

TL;DR

The paper addresses task-delay minimization in multi-user NOMA-MEC networks with partial offloading and joint optimization of partition ratios and transmit powers. It converts the nonconvex formulation into a quasi-convex problem solved by bisection search, and derives closed-form two-user solutions. Simulations show convergence, optimality, and improved delay for partial over full offloading.

  • Problem

    The paper studies how to minimize task delay in NOMA-MEC networks while jointly allocating partial task offloading and transmit power.

  • Method

    The nonconvex problem is transformed into an equivalent quasi-convex formulation solved by bisection search, with closed-form partition and power solutions derived for two users.

  • Results

    The proposed BSS algorithm converges within 10 iterations, and partial offloading provides better task-completion performance than full offloading.

  • Takeaways & Limitations

    The method provides a globally optimal multi-user solution and a lower-complexity closed-form solution for two-user NOMA-MEC networks.

Abstract

from arXiv · show

Multi-access edge computing (MEC) can enhance the computing capability of mobile devices, while non-orthogonal multiple access (NOMA) can provide high data rates. Combining these two strategies can effectively benefit the network with spectrum and energy efficiency. In this paper, we investigate the task delay minimization in multi-user NOMA-MEC networks, where multiple users can offload their tasks simultaneously through the same frequency band. We adopt the partial offloading policy, in which each user can partition its computation task into offloading and locally computing parts. We aim to minimize the task delay among users by optimizing their tasks partition ratios and offloading transmit power. The delay minimization problem is first formulated, and it is shown that it is a nonconvex one. By carefully investigating its structure, we transform the original problem into an equivalent quasi-convex. In this way, a bisection search iterative algorithm is proposed in order to achieve the minimum task delay. To reduce the complexity of the proposed algorithm and evaluate its optimality, we further derive closed-form expressions for the optimal task partition ratio and offloading power for the case of two-user NOMA-MEC networks. Simulations demonstrate the convergence and optimality of the proposed algorithm and the effectiveness of the closed-form analysis.

I. INTRODUCTION

The paper studies task-delay minimization in NOMA-MEC networks using partial offloading, where users share a frequency band while partitioning tasks between local and edge computation. It motivates joint optimization of task allocation and offloading power under practical energy and delay constraints.

  • MEC targets compute-intensive, ultra-low-latency applications, but practical systems still face critical energy-consumption and delay-reduction challenges.
  • NOMA-MEC enables multiple users to offload simultaneously over the same frequency band, supporting massive connectivity, low latency, and energy efficiency.
  • The paper focuses on task-delay minimization with NOMA uplink and partial offloading, unlike prior work emphasizing energy minimization or full offloading.
  • Users partition tasks into locally computed and remotely offloaded parts, with the model assuming partitionable data and parallelizable execution.
  • The network model contains multiple single-antenna users and a BS equipped with an MEC server, using SIC and centralized perfect-CSI-based resource allocation.
  • The task model includes offloading time, offloading energy, local execution, and negligible downloading time, while optimizing user-side partition ratios and transmit powers.

2) Mobile Execution Time:

The paper formulates maximum task-completion-time minimization under partial offloading, energy, and power constraints. It transforms the nonconvex problem into quasi-convex feasibility searches and derives structural results for efficient optimization.

  • Mobile Execution Time: The objective is to minimize the maximum task completion time by jointly optimizing each user’s offloading ratio and transmit power.
  • Problem Formulation: Problem (9) is nonconvex because the offloading-time term is nonconvex in the task partition ratios and transmit powers.
  • BSS Iterative Algorithm: For fixed auxiliary delay α_T, the transformed problem becomes a convex feasibility problem that can be solved iteratively by bisection search.
  • Structural Properties: The optimal allocation equalizes users’ offloading times when minimizing the maximum task completion time.
  • Structural Properties: The resulting pure-NOMA scheme lets multiple users offload simultaneously in the same frequency band.

B. BSS Iterative Algorithm

The proposed BSS algorithm converts fixed-delay feasibility checks into a bisection search over a strictly quasi-convex problem, converging to the unique minimum task delay. Its complexity depends on both bisection iterations and convex feasibility subproblems, and is lower than UDM for more than five users under the stated setting.

  • Problem transformation: The transformed objective is strictly quasi-convex, enabling fixed-α_T convex feasibility subproblems.The original constraint set remains nonconvex, but fixing α_T makes the relevant constraints convex.
  • BSS procedure: BSS decreases α_T when the feasibility problem is feasible and increases it when infeasible.This search progressively brackets the optimal delay α*_T.
  • Convergence: The algorithm converges to the unique optimal solution because the objective is strictly quasi-convex.The bisection interval is repeatedly reduced during the search.
  • Complexity analysis: The total complexity combines bisection iterations with the cost of solving each convex feasibility subproblem.The bisection iteration count depends logarithmically on the initial delay range and target accuracy, while ellipsoid iterations depend on ϵ2.
  • Complexity comparison: For ϵ2 = 0.001, the proposed algorithm is lower-complexity than UDM when the user number exceeds five.The comparison assumes similar bisection iteration counts.

IV. CLOSED-FORM OPTIMAL SOLUTION DERIVATION FOR THE TWO-USER CASE

For two-user NOMA-MEC networks, the paper transforms the optimization into a convex problem and derives closed-form task partition and power solutions using KKT conditions. The resulting solution reduces complexity relative to the general multi-user BSS algorithm and can support larger networks through two-user clustering.

  • Problem formulation: The two-user formulation targets minimizing the maximum task delay by optimizing offloading ratios and transmit powers.The analysis uses equalities linking offloading and local-computing times, with feasibility depending on available user energy.
  • Convex reformulation: The nonconvex two-user problem is equivalently transformed into a convex problem.Equality constraints are used to rewrite the formulation before deriving the solution.
  • Proof note: The convexity proof for the transformed problem is omitted because of limited space.This is stated as a proof-level limitation in the derivation.
  • Closed-form derivation: KKT conditions yield closed-form optimal offloading powers and task partition ratios through four cases.The derivation relies on convexity and satisfaction of Slater’s condition; the Lambert W function appears in the expressions.
  • Complexity reduction: The closed-form solution significantly reduces complexity compared with the proposed general BSS algorithm.It is obtained from users’ channel gains and computing capabilities.
  • Extension to larger networks: For many users, clustering them into two-user groups allows the closed-form solution to be applied within each cluster.The paper suggests matching theory, game theory, or machine learning for grouping.

V. TASK DELAY MINIMIZATION FOR LIMITED COMPUTING RESOURCES MEC SERVERS

With limited MEC-server computing resources, server-side computation time becomes part of task delay and affects the offloading scheme. The paper retains the BSS approach and applies the two-user closed-form powers to this setting.

  • Resource-limited setting: Limited MEC-server resources make computation time longer for the same task than with considerable server capacity.The BS’s MEC server computation can no longer be ignored in the delay model.
  • Server computation model: Server computation time increases with the offloaded task ratio and decreases with server CPU frequency.C_S denotes CPU cycles per bit and f_S denotes CPU frequency.
  • Optimization impact: Including server computation time affects the offloading scheme and adds server computation energy to the delay-minimization model.The model introduces the server effective capacitance coefficient κ_S for CPU cycles.
  • Solution method: The BSS algorithm remains applicable because the limited-resource problem is likewise quasi-convex.Its quasi-convexity can be established by steps similar to those used previously.
  • Two-user solution: The two-user closed-form powers p*_1 and p*_2 remain applicable in the limited-resource scenario.The resulting minimum-delay performance is demonstrated in Fig. 2.

VI. SIMULATION RESULTS AND DISCUSSION

Simulations evaluate convergence, optimality, delay, rate, and efficiency for the proposed NOMA-MEC resource-allocation schemes against analytical and benchmark solutions. The results show rapid convergence, delay advantages in selected settings, and trade-offs between NOMA and OFDMA resource configurations.

  • Convergence and optimality: The BSS algorithm converges within 10 iterations and its convergence point matches the optimal analytical solution.The simulations therefore support both the practicality and optimality of the proposed algorithm.
  • Delay comparison: The proposed scheme achieves better delay-minimization performance than the UDM method because it optimizes task assignment and offloading power.The comparison uses equal task lengths and common energy and power limits.
  • NOMA and OFDMA comparison: With one resource block, NOMA consistently outperforms OFDMA in task delay and sum rate, while OFDMA with two resource blocks performs best overall.The comparison is made as maximum power varies.
  • Efficiency comparison: With the same number of resource blocks, NOMA has higher energy efficiency than OFDMA, and NOMA achieves the highest power efficiency among the compared schemes.OFDMA with two resource blocks has higher energy efficiency than NOMA with one resource block, showing the effect of resource allocation.
  • User number and power: Task completion time increases with user number, while increasing the transmit-power limit from 0.01 W to 0.02 W reduces completion time.This comparison concerns NOMA partial offloading and NOMA full offloading under the stated simulation parameters.

VII. CONCLUSION

The paper formulates NOMA-MEC task-delay minimization as a nonconvex problem, transforms it into an equivalent quasi-convex problem, and solves the multi-user case with bisection search. For two users, it derives closed-form optimal power allocation and offloading-task ratios, with simulations demonstrating convergence and optimality.

  • The original task-delay minimization problem is nonconvex, but is equivalently transformed into a quasi-convex problem.
  • The proposed bisection-search algorithm efficiently obtains the globally optimal solution for the multi-user case.
  • For two-user NOMA-MEC networks, closed-form optimal power-allocation and offloading-task-ratio expressions reduce complexity and provide analytical insight.
  • Simulations demonstrate convergence and optimality and provide an effective task-delay minimization solution for a single-cell NOMA-MEC network.
  • Applying the solution to more complicated multi-cell networks is identified as future work involving user grouping or pairing to decouple optimization.

APPENDIX A PROOF OF LEMMA 1

The appendix analyzes simultaneous offloading by multiple users over one subchannel under SIC decoding ordered by channel gains, then derives the offloading sum rate and time expressions used in the proof.

  • M users simultaneously offload through one subchannel, with the base station decoding signals in decreasing channel-gain order.
  • The proof rewrites equal products through exponential substitutions involving the task-related variables.
  • The appendix derives expressions for the users’ cumulative rates, offloading sum rate, and offloading time.
  • The derivation proceeds by checking the first-user case and extending the result inductively to establish Lemma 1.

APPENDIX B PROOF OF PROPOSITION 1

The proof establishes that an optimal solution to the offloading-time problem occurs when the relevant users’ offloading times are equal, using contradiction and power adjustments.

  • The appendix represents the offloading-time minimization problem using the relationship established by Lemma 1.
  • The proof assumes an optimal solution with unequal offloading times and identifies the larger-time user for contradiction.
  • The resulting adjustment increases the other user’s offloading time until a power value equalizes the relevant offloading times.
  • Therefore, the optimal solution can only be obtained when the offloading times satisfy the equality condition for the users.
  • Increasing one user’s power decreases its offloading time while leaving the other user’s power fixed.

APPENDIX C THE PROOF OF PROPOSITION 2

The appendix proves quasi-convexity by showing that the sublevel sets are convex, including convexity of the key two-user constraint through its Hessian.

  • A function is quasi-convex when all of its sublevel sets are convex, so the proof analyzes the set SαT.
  • For αT > 0, the relevant sublevel set is rewritten as an inequality whose constraints are examined individually.
  • The constraints in (46b) are linear equalities in β1 and β2 and therefore define convex sets.
  • For the two-user case, the key function combines a linear task term with a logarithmic rate term and is analyzed using its Hessian.
  • The Hessian is positive definite, making the function convex; intersecting the convex sublevel sets proves that problem (13) is quasi-convex.

APPENDIX D PROOF OF PROPOSITION 3

The appendix proves that the optimal solution to problem (49) occurs only under the stated offloading-time condition. The proof uses contradiction by showing that adjusting β_m can reduce offloading time while satisfying the constraints.

  • Proof strategy: The proof assumes an optimum at a specified offloading time and uses contradiction to test alternative β_m values.The alternatives are chosen above or below β_m while satisfying the energy and power constraints.
  • Monotonicity: Increasing β_m increases offloading time and decreases local energy consumption in the considered case.
  • Contradiction cases: A feasible β̂ greater than β_m yields lower offloading time, contradicting optimality of the assumed solution.
  • Conclusion: Therefore, the optimal solution to problem (49) can only be obtained under the appendix’s specified offloading-time condition.
  • Contradiction cases: A parallel argument considers β̂ less than β_m and again derives a lower offloading time than the assumed minimum latency.

APPENDIX E DERIVATION OF THE OPTIMAL SOLUTION TO PROBLEM

The appendix derives the optimal solution to problem (21) by rewriting it as a convex problem and applying KKT conditions. The resulting solution is organized into four cases determined by active constraint multipliers.

  • KKT formulation: Problem (21) is rewritten into an equivalent formulation whose Lagrangian and constraints support KKT-based optimization.The derivation lists primal feasibility, dual feasibility, stationarity, and complementary-slackness conditions.
  • KKT formulation: Because the reformulated problem is convex and satisfies Slater’s condition, its KKT conditions are necessary and sufficient for optimality.
  • Case analysis: The multiplier conditions are divided into cases such as λ2 > 0, λ5 = 0 and λ2 = 0, λ5 > 0, with further splits based on λ4 and λ6.
  • Closed-form solutions: The derivation obtains optimal solutions including (26) and (32) under the corresponding feasibility and multiplier conditions.
  • Case analysis: Several multiplier configurations are shown to reduce to previously analyzed cases, so they do not require separate optimal-solution expressions.
  • Conclusion: Overall, the optimal solution of problem (21) is concluded to consist of four cases.
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