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Joint Power and Time Allocation for NOMA-MEC Offloading

Zhiguo Ding, Jie Xu, Octavia A. Dobre, H. Vincent Poor

arXiv:1807.06306v1cs.IT

TL;DR

The paper addresses how NOMA should be used for energy-efficient MEC offloading when users have different deadlines. It derives closed-form joint power and time allocations for OMA, pure NOMA, and hybrid NOMA, then establishes deadline-dependent strategy choices. The analysis shows NOMA is at least as good as OMA when D_n < 2D_m, while OMA is best when D_n ≥ 2D_m.

  • Problem

    The paper examines which of OMA, pure NOMA, or hybrid NOMA should serve two-user MEC offloading under differing latency requirements.

  • Method

    The paper applies geometric programming to derive closed-form joint optimal power and time allocations for the three offloading strategies.

  • Results

    NOMA outperforms or matches OMA when D_n < 2D_m, whereas OMA outperforms hybrid and pure NOMA when D_n ≥ 2D_m.

  • Takeaways & Limitations

    Deadline conditions determine whether OMA, pure NOMA, or hybrid NOMA should be selected for MEC offloading.

Abstract

from arXiv · show

This paper considers non-orthogonal multiple access (NOMA) assisted mobile edge computing (MEC), where the power and time allocation is jointly optimized to reduce the energy consumption of offloading. Closed-form expressions for the optimal power and time allocation solutions are obtained and used to establish the conditions for determining whether conventional orthogonal multiple access (OMA), pure NOMA or hybrid NOMA should be used for MEC offloading.

I. INTRODUCTION

The paper studies energy-efficient MEC offloading with NOMA in a two-user setting, comparing OMA, pure NOMA, and hybrid NOMA. Closed-form power and time allocations reveal when each strategy is preferable.

  • Existing NOMA-MEC comparisons with OMA rely on simulation, while analytical work assumes fixed bandwidth allocation.
  • Hybrid NOMA lets one user offload part of its task during another user’s slot and the remainder during its own slot.
  • The paper derives closed-form optimal time and power allocations for OMA, pure NOMA, and hybrid NOMA using geometric programming.
  • Hybrid-NOMA-MEC is superior to OMA-MEC for demanding latency requirements, whereas OMA-MEC is preferred when a user’s task is delay tolerant.

II. SYSTEM MODEL

The system models users offloading latency-critical, inseparable tasks to an MEC server, with two users sharing a resource block. NOMA allows simultaneous offloading, while hybrid NOMA combines shared and exclusive transmission intervals.

  • Users with limited computational capabilities offload computationally intensive, latency-critical, and inseparable tasks to the MEC server.
  • Each task is characterized by the number of contained nats N_k and its computation deadline D_k.
  • The server schedules two users, m and n, on the same resource block, ordered by deadlines with D_m ≤ D_n.
  • OMA assigns each user a dedicated offloading slot, serving user m first because its deadline is more demanding.
  • NOMA allows simultaneous offloading during D_m, while user m matches OMA performance when decoded at the second SIC stage.
  • Hybrid NOMA has user n share D_m with user m and then transmit alone during an additional interval T_n.

III. NOMA-ASSISTED MEC OFFLOADING

The NOMA-MEC problem minimizes user n’s offloading energy subject to rate and deadline constraints. The analysis distinguishes latency regimes, focusing first on D_n < 2D_m and then treating D_n ≥ 2D_m.

  • The optimization minimizes D_mP_n,1 + T_nP_n,2, the energy consumed by user n across shared and exclusive transmission intervals.
  • Server-side computation energy and the energy and time needed to return task outcomes are omitted from the model.
  • The rate constraint ensures that user n offloads N nats within D_m + T_n, while the deadline constraint requires T_n + D_m ≤ D_n.
  • When D_n = D_m, OMA requires infinite power whereas NOMA requires finite power.
  • The analysis first considers D_n < 2D_m to avoid the trivial OMA-solution case, then discusses D_n ≥ 2D_m.

A. Finding the Optimal Solutions for Pn,1 and Pn,2

For D_n < 2D_m, geometric programming and KKT conditions yield closed-form optimal powers as functions of T_n, after which the optimal time allocation is determined.

  • The formulation is transformed so geometric programming can be applied to the NOMA-MEC objective and constraints.
  • For fixed T_n, logarithmic variables y_i = ln x_i convert the problem into an equivalent form with an exponential objective.
  • KKT conditions are applied to the transformed problem to obtain the optimal solution.
  • The optimal power expressions are obtained after fixing T_n, with the resulting solutions used to determine the optimal T_n.
  • Lemma 1 expresses the optimal P_n,1 and P_n,2 as closed-form functions of T_n under D_n < 2D_m.

B. Finding the Optimal Solution for Tn

After substituting Lemma 1’s optimal power allocation, the reduced energy is expressed as a function of T_n. Its monotonicity determines the optimal T_n, with T_n^*<D_m under D_n<2D_m.

  • The original problem becomes an equivalent optimization after substituting the optimal solution from Lemma 1.
  • The normalized energy g_Tn omits the constant |h_n|^-2 and depends on T_n through y_1^* and y_2^*.
  • The derivative analysis shows g_x(x) is monotonically non-increasing, so g_Tn is also monotonically non-increasing.
  • Therefore, the optimal T_n is obtained from the endpoint solution specified after the monotonicity result.
  • Under the considered condition D_n<2D_m, the optimal allocation satisfies T_n^*<D_m.

1) For the superiority of NOMA over OMA:

The comparison depends on user n’s deadline: NOMA is at least as energy-efficient as OMA when D_n<2D_m, whereas OMA is better when D_n≥2D_m.

  • 1) For the superiority of NOMA over OMA:: When D_n<2D_m, the derived energy gap satisfies f_Tn(T_n)≤0, so NOMA matches or outperforms OMA.
  • 1) For the superiority of NOMA over OMA:: For D_n≥2D_m, user n has less demanding latency requirements and T_n can exceed D_m because T_n=D_n−D_m.
  • 1) For the superiority of NOMA over OMA:: When D_n≥2D_m, the hybrid-NOMA solution is feasible only for T_n<D_m, while its energy decreases monotonically with T_n.
  • 1) For the superiority of NOMA over OMA:: OMA achieves the lower energy bound when D_n≥2D_m and therefore consumes less energy than hybrid NOMA.
  • 1) For the superiority of NOMA over OMA:: OMA also outperforms pure NOMA in this regime, so OMA is preferable to both NOMA variants when D_n≥2D_m.

IV. NUMERICAL RESULTS

Simulations compare NOMA-MEC and OMA-MEC, verify Lemma 1’s power allocation, and examine how performance changes with D_n. NOMA gains are strongest for small D_n, while increasing D_n makes hybrid NOMA approach OMA.

  • NOMA-MEC yields a significant performance gain over OMA-MEC, particularly when D_n is small.As D_n approaches D_m, OMA’s available offloading interval D_n−D_m becomes nearly zero and its energy becomes prohibitively large.
  • The power allocation from Lemma 1 produces the lowest energy among the possible choices of (P_n,1,P_n,2).
  • NOMA and OMA performance becomes quite similar when D_n becomes large.
  • As D_n increases, the power allocated during D_m approaches zero, degrading hybrid NOMA relative to OMA.

V. CONCLUSIONS

The paper applies NOMA to MEC and obtains optimal power and time allocation solutions using geometric programming, with analytical and simulation results demonstrating superior performance over OMA-MEC.

  • NOMA is applied to MEC to optimize power and time allocation for offloading energy consumption.The paper reports optimal allocation solutions obtained using geometric programming.

APPENDIX A PROOF OF LEMMA 1

The appendix compares hybrid NOMA, pure NOMA, and OMA through KKT-based cases and energy expressions, showing hybrid NOMA has the smallest energy consumption under the considered conditions.

  • 1) Hybrid NOMA: The λi = 0, ∀i ∈{1, 2} case is hybrid NOMA because both users have non-zero NOMA powers.The proof analyzes this case through KKT conditions and derives feasible optimal solutions for y1 and y2.
  • 2) Pure NOMA: The λ1 = 0 and λ2 ≠ 0 case is pure NOMA because only user m receives non-zero NOMA power.The condition y2 = 0 corresponds to allocating all NOMA power to Dm.
  • 3) OMA: The λ1 ≠ 0 and λ2 = 0 case is OMA because only user n receives non-zero NOMA power.All power is allocated to Tn in this case.
  • 4) Comparisons among the three cases: Hybrid NOMA requires less energy consumption than pure NOMA.This comparison follows from the nonnegative Tn condition and the derived energy bound.
  • 4) Comparisons among the three cases: Hybrid NOMA requires the smallest energy consumption among the compared cases.The proof combines the energy comparisons and the monotonicity of the hybrid-NOMA energy function.
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