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Energy Efficient Resource Allocation in Machine-to-Machine Communications with Multiple Access and Energy Harvesting for IoT

Zhaohui Yang, Wei Xu, Yijin Pan, Cunhua Pan, Ming Chen

arXiv:1711.10776v1cs.IT

TL;DR

The paper asks how to minimize total energy in EH-enabled M2M cellular networks using NOMA or TDMA while accounting for nonlinear harvesting and circuit power. It formulates joint power-control and time-allocation problems, transforms them into iteratively solvable forms, and finds that NOMA is preferable at low MTCD circuit power whereas TDMA is preferable at high circuit power.

  • Problem

    M2M networks require energy-efficient access and resource allocation because MTCD energy consumption is constrained and prior models often ignored circuit power and practical nonlinear harvesting.

  • Method

    The paper jointly optimizes transmission power and time for NOMA and TDMA under throughput, power, and energy-causality constraints, using equivalent transformations and iterative algorithms.

  • Results

    NOMA consumes less total energy at low MTCD circuit power, while TDMA is preferred at high MTCD circuit power; both strategies support minimum-throughput transmission and convex per-MTCD time-energy structure.

  • Takeaways & Limitations

    Access-strategy choice should depend on the MTCD circuit-power regime: NOMA is favored at low circuit power and TDMA at high circuit power.

Abstract

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This paper studies energy efficient resource allocation for a machine-to-machine (M2M) enabled cellular network with non-linear energy harvesting, especially focusing on two different multiple access strategies, namely non-orthogonal multiple access (NOMA) and time division multiple access (TDMA). Our goal is to minimize the total energy consumption of the network via joint power control and time allocation while taking into account circuit power consumption. For both NOMA and TDMA strategies, we show that it is optimal for each machine type communication device (MTCD) to transmit with the minimum throughput, and the energy consumption of each MTCD is a convex function with respect to the allocated transmission time. Based on the derived optimal conditions for the transmission power of MTCDs, we transform the original optimization problem for NOMA to an equivalent problem which can be solved suboptimally via an iterative power control and time allocation algorithm. Through an appropriate variable transformation, we also transform the original optimization problem for TDMA to an equivalent tractable problem, which can be iteratively solved. Numerical results verify the theoretical findings and demonstrate that NOMA consumes less total energy than TDMA at low circuit power regime of MTCDs, while at high circuit power regime of MTCDs TDMA achieves better network energy efficiency than NOMA.

I. INTRODUCTION

The paper addresses access-control and energy-consumption challenges in M2M networks by studying energy harvesting with NOMA and TDMA. It develops resource-allocation methods that incorporate nonlinear harvesting and circuit power, and finds that the preferable access strategy depends on MTCD circuit power.

  • M2M networks face massive-device access-control and MTCD energy-consumption challenges, with gateways enabling cellular connectivity at additional energy cost.
  • The paper jointly optimizes power control and time allocation for NOMA and TDMA networks with nonlinear energy harvesting and circuit power consumption.The harvesting model includes a receiver-sensitivity threshold, while circuit energy is included for MTCDs and MTCGs.
  • NOMA strategy: For NOMA, each MTCD optimally transmits at minimum throughput, and its energy consumption is convex in allocated transmission time.The resulting globally optimal time equals the maximum allowed time when it does not exceed a closed-form threshold.
  • NOMA strategy: The NOMA problem is addressed with an iterative power-control and time-allocation algorithm that handles nonsmooth harvesting and nonconvex constraints through transformations.The algorithm's convergence is strictly proved.
  • TDMA strategy: The same two structural observations hold for TDMA, whose nonconvex problem is transformed into an equivalent tractable problem solved iteratively.The paper also notes that practical NOMA effectiveness limits the number of terminals sharing one resource.
  • Numerical comparison: NOMA consumes less total energy at low MTCD circuit power, whereas TDMA performs better at high circuit power.NOMA's advantage at low circuit power is associated with lower MTCD RF transmission power, while high circuit power favors TDMA.

C. TDMA Strategy

The TDMA strategy divides the uplink into separate MTCD and MTCG transmission phases, while accounting for energy harvesting, receiver sensitivity, circuit power, and total system energy.

  • TDMA uses M+N uplink phases: MTCDs transmit individually to their serving MTCGs, followed by MTCG transmissions to the BS.
  • Each MTCD's achievable throughput and harvested energy are evaluated within the TDMA transmission phases.Harvesting may be ineffective when received power falls below the receiver sensitivity threshold P0.
  • An MTCD's transmission energy includes both RF transmission power and circuit power over its allocated time.
  • The system energy model includes MTCG transmission energy and energy harvested by MTCDs during MTCG-to-BS transmissions.
  • The total energy consumption combines the energy terms of MTCDs and MTCGs across the complete uplink period.

III. ENERGY EFFICIENT RESOURCE ALLOCATION FOR NOMA

The NOMA resource-allocation problem minimizes total network energy through joint power control and time allocation under throughput, energy-causality, and timing constraints. The analysis derives optimal transmission conditions and shows how circuit power changes the time–energy trade-off.

  • NOMA resource allocation minimizes total energy by jointly optimizing MTCD and MTCG power and transmission times.The formulation enforces payload, power, energy-causality, and total-time constraints.
  • The original NOMA problem is nonconvex, so the paper derives optimal conditions and then develops an iterative power-control and time-allocation algorithm.
  • Each MTCD optimally transmits at its minimum required throughput, whereas MTCG throughput constraints need not all be active.MTCGs may transmit additional power to ensure harvested energy covers MTCD energy consumption.
  • The optimal MTCD transmit power is non-negative, decreases with allocated transmission time, and makes each MTCD's energy a function of that time.
  • Theorem 1 establishes that MTCD energy is convex in transmission time; with positive circuit power, it first decreases and then increases.Without circuit power, energy decreases monotonically as transmission time increases.
  • When available time is not large, using the maximal transmission time is optimal; with sufficiently large available time, it is not optimal.The trade-off shifts because RF-energy reduction diminishes while circuit-energy costs increase.

B. Joint Power Control and Time Allocation Algorithm

The NOMA problem is handled by smoothing the energy-harvesting model through effective-harvesting sets and transforming the fixed-set problem into a convex one. An iterative power-control and time-allocation procedure then updates these sets and solves the resulting subproblems.

  • The non-smooth harvesting function and nonconvex objective and constraints are the two main difficulties in problem (19).
  • Effective-harvesting sets Sij identify the phases during which MTCD j can harvest energy above the receiver-sensitivity threshold.
  • With fixed Sij, the harvested-power expression becomes smooth and the original problem can be equivalently transformed.
  • For fixed transmission time, the transformed problem is convex in (q, t̄), while for fixed (q, t̄) it is linear in τ.
  • IPCTA-NOMA iteratively updates harvesting sets and alternates between convex and linear subproblems to obtain a suboptimal solution.

C. Convergence and Complexity Analysis

The NOMA algorithm is shown to converge, while the TDMA formulation is also identified as nonconvex and prepared for low-complexity iterative solution.

  • Assuming Vmax →∞, the sequence generated by IPCTA-NOMA converges.
  • The dominant NOMA subproblem has complexity O(N^3) per interior-point solution, yielding total complexity O(LNOLITN^3).
  • The TDMA energy-minimization problem is nonconvex because of its objective and constraints, motivating optimal-condition analysis and a low-complexity algorithm.

A. Optimal Conditions

The optimal-condition analysis establishes minimum-throughput transmission and convex energy behavior in transmission time for TDMA, then supports an equivalent convex formulation and iterative algorithm.

  • Each MTCD optimally transmits with its minimal throughput requirement in the TDMA formulation.
  • The MTCD energy Eij is convex with respect to transmission time tj.
  • When PC = 0, Eij decreases monotonically with tj; when PC > 0, it first decreases and then increases after T*ij.
  • For sufficiently large T, transmitting for the maximal available time T is not optimal; when T is small, it is optimal.
  • With fixed harvesting sets, the TDMA problem is equivalently transformed into a convex problem, and IPCTA-TDMA iteratively updates the sets to obtain a suboptimal solution.

C. Convergence and Complexity Analysis

The iterative TDMA procedure converges, and simulations examine convergence, parameter sensitivity, and the circuit-power-dependent trade-off between NOMA and TDMA energy consumption.

  • Assuming Vmax →∞, the sequence generated by IPCTA-TDMA converges.
  • The TDMA algorithm has total complexity O(LTD(M + N)^3), where LTD is its iteration count.
  • Both IPCTA-NOMA and IPCTA-TDMA monotonically decrease total energy and converge rapidly in the simulations.
  • PC ≤4 mW: NOMA consumes less total energy than TDMA in the test case.
  • PC ≥5 mW: TDMA achieves better energy efficiency because NOMA uses longer MTCD transmission times and therefore more circuit power.
  • Increasing the required payload raises total network energy, while increasing maximal MTCD or MTCG transmission power lowers it in the reported simulations.

APPENDIX A PROOF OF LEMMA 2

Appendix A derives the inverse matrix needed to solve the MTCD transmission-power equations, then establishes that the resulting power is non-negative and decreases with transmission time.

  • The appendix rewrites the MTCD power equations in matrix form and introduces a lemma for powers of matrix W_i.The lemma covers powers through the number of MTCDs in the group and shows the corresponding final power is a zero matrix.
  • Mathematical induction verifies the stated expression for every relevant power of W_i.The proof establishes the basis, induction hypothesis, and induction step before confirming the lemma.
  • Substituting the matrix-power result into the inverse-matrix expression yields the MTCD transmission power.The resulting expression is obtained by combining the appendix equations for the inverse matrix and the system model.
  • The derived MTCD transmission power is non-negative and decreases with transmission time.This follows because e^x − 1 is non-negative for x ≥ 0 and decreases with t_i in the derived expression.

APPENDIX B PROOF OF THEOREM 1

Appendix B proves that MTCD energy is convex in transmission time and characterizes its minimizing time using derivative behavior and perspective-function convexity.

  • The proof analyzes the first- and second-order derivatives of the MTCD energy with respect to transmission time.The derivative limits and monotonicity determine whether energy decreases throughout the feasible interval or has an interior minimizer.
  • Perspective-function arguments establish convexity of the transformed energy components with respect to transmission time.The proof applies convexity of f_ijl and g_ij and their perspective functions after the variable transformation.
  • Because both transformed components are convex, E_ij is convex with respect to t_i.The conclusion follows directly from the decomposition in (B.6).
  • When the derivative changes sign, the minimizing transmission time can be found using the bisection method.Energy decreases up to T*_{ij} and increases afterward; otherwise, it decreases for all feasible transmission times.

APPENDIX C PROOF OF THEOREM 2

Appendix C proves the transmission-time structure of the optimal solution by exploiting convex MTCD energy, constructing feasible alternatives, and using contradiction arguments around the maximum-time constraint.

  • When the relevant transmission time is below T*_n, reducing the associated powers lowers the objective while preserving feasibility.This supports the theorem’s conclusion that the total transmission time equals T in the first regime.
  • The theorem’s second part states that transmitting for the maximal time is not optimal when T exceeds TUpp.The appendix explains this through a contradiction based on a special solution with total transmission time less than T.
  • The total energy of all MTCDs served by MTCG i is convex in transmission time.This follows by summing the convex individual energies E_ij.
  • Group energy decreases up to T*_i and increases after T*_i.Lemma 5 derives this unimodal behavior from the first-order derivative of aggregate energy.
  • For fixed powers and transmission times, the remaining energy-minimization problem is linear and can be solved by the simplex method.The appendix obtains this subproblem by substituting the system constraints into the objective and energy-causality constraints.
  • A constructed feasible solution and contradiction argument show that the maximum transmission-time constraint is inactive when T is at least TUpp.The proof compares the original solution with a solution using the groupwise minimizing times and an optimized remaining-time allocation.

APPENDIX D PROOF OF THEOREM 3

Appendix D establishes convexity of the transformed optimization problem by proving convexity of its feasible constraints and objective, then shows that fixing part of the variables yields a linear subproblem.

  • With τ fixed, the feasible set of problem (27) is convex in (q, t̄).The linear constraints are immediate, while the remaining constraints are converted using convexity and composition properties.
  • The transformed constraint functions are convex because the relevant perspective terms are convex and the transformed utility is concave.The proof uses that −ū(x) is convex and applies composition rules to the nonlinear constraints.
  • The objective function is convex in (q, t̄).Substitution into the objective expresses it through convex transformed energy terms and −ū(x).
  • With (q, t̄) fixed, the remaining constraints are linear in t_{N+i}, making problem (27) a linear problem.The appendix replaces the corresponding constraints with their equivalent linear form.

APPENDIX E PROOF OF THEOREM 4

The appendix establishes convergence of IPCTA-NOMA by showing that each iteration does not increase the nonnegative total energy value. It also proves the convex reformulation used in the iterative procedure.

  • Convergence proof: IPCTA-NOMA updates (q,t) so the total energy value is non-increasing at every iteration.
  • Convergence proof: Because total energy is nonnegative and non-increasing, IPCTA-NOMA converges to a finite lower bound.
  • Convex reformulation: Introducing new non-negative variables and substituting them transforms problem (29) equivalently into problem (34).
  • Convex reformulation: Problem (34) is convex because its objective and nonlinear constraints use convex perspective-function constructions, while the remaining constraints are linear.
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