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Optimal Power Allocation for Outage Minimization in Fading Channels with Energy Harvesting Constraints

Chuan Huang, Rui Zhang, Shuguang Cui

arXiv:1212.0075v2cs.NIcs.PF

TL;DR

The paper studies outage-minimizing power allocation in fading channels subject to energy-harvesting constraints over time. It derives globally optimal offline solutions despite non-convexity, identifies a save-then-transmit structure, and develops optimal and suboptimal online schemes using dynamic programming.

  • Problem

    The paper addresses outage minimization for transmissions constrained by the energy harvested over time in fading channels.

  • Method

    The paper derives globally optimal offline allocations and develops optimal and suboptimal online schemes using dynamic programming.

  • Results

    The outage-minimization power-allocation problems are generally non-convex, while optimal offline allocation follows a save-then-transmit protocol.

  • Takeaways & Limitations

    For N = 1, the results revisit the classic outage-capacity problem for fading channels.

Abstract

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This paper studies the optimal power allocation for outage minimization in point-to-point fading channels with the energy-harvesting constraints and channel distribution information (CDI) at the transmitter. Both the cases with non-causal and causal energy state information (ESI) are considered, which correspond to the energy harvesting rates being known and unknown prior to the transmissions, respectively. For the non-causal ESI case, the average outage probability minimization problem over a finite horizon is shown to be non-convex for a large class of practical fading channels. However, the globally optimal "offline" power allocation is obtained by a forward search algorithm with at most $N$ one-dimensional searches, and the optimal power profile is shown to be non-decreasing over time and have an interesting "save-then-transmit" structure. In particular, for the special case of N=1, our result revisits the classic outage capacity for fading channels with uniform power allocation. Moreover, for the case with causal ESI, we propose both the optimal and suboptimal "online" power allocation algorithms, by applying the technique of dynamic programming and exploring the structure of optimal offline solutions, respectively.

I. INTRODUCTION

The paper formulates finite-horizon outage minimization for delay-constrained transmissions over fading channels with energy-harvesting constraints, considering non-causal and causal energy-state information. It derives offline and online allocation methods, including threshold, forward-search, dynamic-programming, and lower-complexity schemes.

  • I. INTRODUCTION: Energy harvesting imposes a cumulative constraint: energy consumed by any time must not exceed energy harvested by then.
  • I. INTRODUCTION: The paper studies constant-rate, delay-constrained transmission over N EH periods of M blocks, with receiver CSI and transmitter CDI, under non-causal or causal ESI.Non-causal ESI reveals all Qi before transmission; causal ESI reveals Q1 through Qi at period i.
  • I. INTRODUCTION: For N = 1, outage probability is generally concave-convex, making power-controlled minimization non-convex, yet a one-dimensional search obtains the global optimum.The optimum is uniform power above a rate- and CDI-determined threshold and on-off transmission below it; the on-off scheme improves low-power outage and is asymptotically optimal as M grows.
  • I. INTRODUCTION: For N > 1 with non-causal ESI, a forward-search algorithm finds the globally optimal offline allocation using at most N one-dimensional searches.The optimal profile is non-decreasing over time and has a save-then-transmit structure; a lower-complexity suboptimal algorithm avoids exhaustive searches.
  • I. INTRODUCTION: For causal ESI with N > 1, the optimal online policy is formulated as an MDP and solved by dynamic programming, while a suboptimal scheme offers a more flexible performance-complexity tradeoff.

II. SYSTEM MODEL AND PROBLEM FORMULATION … 1) Non-causal ESI:

The paper models block-fading transmission across N energy-harvesting periods, each containing M communication blocks, and formulates finite-horizon outage minimization under non-causal energy-state information. The non-causal formulation assumes all harvesting rates are known beforehand and yields structural properties for optimal power allocation.

  • A. System Model: The system spans N energy-harvesting periods with M unit-length communication blocks each, while harvesting rates remain constant within each period.The harvesting process varies more slowly than channel fading, motivating constant rates over the M blocks in each period.
  • A. System Model: The channel is block-fading: channel gains are i.i.d. across communication blocks, unknown to the transmitter, and perfectly known at the receiver.The transmitted signal uses power P_i,j, and the receiver experiences independent CSCG noise with unit variance.
  • A. System Model: All NM blocks transmit at rate R, with outage probability F(P_i,j) determined by transmit power, fading distribution, and rate, and strictly decreasing in power.Thus, increasing transmit power lowers the outage probability under the model assumptions.
  • B. Problem Formulation: The paper formulates finite-horizon average outage minimization for both non-causal and causal energy-state information.The non-causal case assumes the energy-harvesting rate levels for all N periods are known before transmission.
  • 1) Non-causal ESI:: Under non-causal energy-state information, each block’s transmit power is constrained by cumulative harvested energy across periods, with unit block length normalizing energy and power.The resulting problem minimizes average outage over the NM communication blocks subject to the energy-harvesting constraints.
  • 1) Non-causal ESI:: An optimal non-causal allocation can be chosen non-decreasing over time, although it may not be unique, and it exhausts all available energy by the final block.Non-uniqueness arises because swapping decreasing power values preserves feasibility and objective value; strict decrease of F makes final energy use optimal.

2) Causal ESI: · III. OPTIMAL POWER ALLOCATION FOR THE CASE OF N = 1 · A. Properties of Outage Probability Function

With causal ESI, the battery state is Markov and only the current harvesting rate is known, yielding an MDP formulation. For N = 1, outage-function properties support allocation algorithms for both causal and non-causal ESI, especially for Type B fading channels.

  • 2) Causal ESI:: The battery state {B_i,j} is first-order Markov, with zero initial storage, while only the current harvesting rate Q_n is known and future rates remain random.The resulting group of problems is an MDP whose optimal solution is studied through dynamic programming.
  • III. OPTIMAL POWER ALLOCATION FOR THE CASE OF N = 1: For N = 1, the paper derives optimal and lower-complexity suboptimal power allocations that apply to both causal and non-causal ESI.The derivation first establishes outage-probability properties and then applies them to Problems (P1) and (P2).
  • A. Properties of Outage Probability Function: For Weibull fading, the outage probability is non-convex for every fading parameter β, which controls channel diversity and includes Rayleigh fading at β = 2.The Weibull model therefore represents practical channels with different diversity orders.
  • A. Properties of Outage Probability Function: Proposition 3.1 shows that the Weibull outage function is concave on [0, P_b] and convex for P > P_b.This follows from the sign change of the second derivative at P_b.
  • A. Properties of Outage Probability Function: A unique P_a > P_b exists such that the outage curve remains above the line joining (0, 1) and (P_a, F(P_a)); bisection computes it.The paper notes that P_a generally lacks a closed-form expression but can be approximated within a prescribed tolerance.
  • A. Properties of Outage Probability Function: Type B outage functions have a concave-convex shape with unique 0 < P_b ≤ P_a, whereas Type A functions are the special case P_a = P_b = 0.Weibull, Rician, Nakagami, and double Rayleigh fading generally yield Type B functions; arbitrary outage functions may fit neither type.

B. Optimal Power Allocation with N = 1

For N = 1, the relaxed outage-minimization problem yields the globally optimal allocation for the original problem: at most one block uses power below P_b, while all larger positive powers are equal. The solution requires at most a one-dimensional search and shows that uniform allocation can be sub-optimal in low-power or high-outage regimes.

  • Optimality for N = 1: The relaxed Problem (P3) solution is also optimal for Problem (P1) when N = 1.A non-decreasing optimal solution satisfies the omitted energy constraints, making the relaxation tight.
  • Optimality for N = 1: The optimal profile has at most one strictly positive power below P_b, while all powers above P_b are identical.Thus, solving P3 reduces to finding the exceptional block count and its lower power value.
  • Solution method: Only a one-dimensional search is needed to compute the optimal allocation when Q_1 < P_a, although exhaustive search is required because monotonicity cannot be guaranteed.The problem is otherwise non-convex when F(P_j) is non-convex, such as for Type B fading.
  • Low-power regime: Uniform allocation can be sub-optimal for Type B fading when Q_1 < P_a, corresponding to a low-power or high-outage regime.In this regime, on-off allocation achieves the minimum outage probability.

C. Suboptimal Power Allocation with N = 1 · IV. OFFLINE POWER ALLOCATION FOR THE CASE OF N > 1

For N = 1, the threshold P_a motivates an on-off allocation that is asymptotically optimal as M grows, while the next section extends offline allocation to N > 1 with non-causal ESI.

  • C. Suboptimal Power Allocation with N = 1: The threshold P_a determines the allocation structure when Q_1 < P_a, making it central to the optimal power allocation.The allocation uses mostly identical nonzero powers near P_a, with at most one exception below P_b.
  • C. Suboptimal Power Allocation with N = 1: When Q_1 < P_a, optimal nonzero powers are identical except possibly one below P_b and are as close to P_a as possible.This structure motivates an on-off two-level allocation.
  • C. Suboptimal Power Allocation with N = 1: For N = 1, the scheme captures the optimal allocation structure through an on-off two-level strategy that can perform close to optimal allocation.Uniform power is used in active blocks, while the number of active blocks is determined.
  • C. Suboptimal Power Allocation with N = 1: The proposed N = 1 scheme uniformly allocates power across “on” blocks and selects how many blocks are active, avoiding exhaustive search.For Q_1 ≥ P_a, it transmits with power Q_1 in all M blocks.
  • C. Suboptimal Power Allocation with N = 1: The on-off power allocation is asymptotically optimal for Problem (P1) with N = 1 as M goes to infinity.Its active-state power converges to P_a in this limit.
  • IV. OFFLINE POWER ALLOCATION FOR THE CASE OF N > 1: The offline allocation analysis for N > 1 derives optimal and suboptimal solutions for Problem (P1) under non-causal energy-state information.The supplied passage identifies this as the scope of the general-case section.

A. Optimal Offline Power Allocation with N > 1

For N > 1, the optimal non-decreasing allocation has a save-then-transmit, on-off structure: silence is followed by transmission that eventually uses powers above Pb and increases after energy exhaustion. Algorithm II computes this globally optimal profile with at most N one-dimensional searches and reduced complexity.

  • Optimal power-profile structure: The optimal profile initially stays silent, may use below-Pb power once, then transmits above Pb with increases after harvested energy is exhausted.This structure follows from the restriction to at most one positive below-Pb power and the conditions governing consecutive powers above Pb.
  • Optimal algorithm: Algorithm II computes the globally optimal non-decreasing solution for Problem (P1) with N > 1 by determining when transmission starts and the initial power parameters.The algorithm uses a forward search over EH periods to identify the period containing a possible below-Pb transmission block, after which the remaining allocation is computed efficiently.
  • Transmission policy: Because the fading-channel outage objective is non-convex, the optimal strategy is on-off transmission that transmits only when available power is sufficiently large.This contrasts with the concave throughput setting and applies to the considered Type B outage probability function.
  • Complexity: The algorithm repeats one-dimensional searches at most N times, has O(N^2) computation apart from those searches, and reduces computation relative to optimizing over NM communication blocks.The one-dimensional searches are the main computational burden.

B. Suboptimal Offline Power Allocation with N > 1 · V. ONLINE POWER ALLOCATION · A. Optimal Online Power Allocation

For N > 1, Algorithm III provides a lower-complexity offline allocation that is asymptotically optimal as M grows, while causal-ESI online allocation is solved optimally through dynamic programming. The online formulation accounts for time-coupled battery states and recursively computes the minimum average outage probability.

  • B. Suboptimal Offline Power Allocation with N > 1: Algorithm III searches for the next possible power-exhausting EH period, then uses best-effort transmission when bPi ≥ Pa and on-off transmission otherwise.The on-off case ensures that allocated power is equal to or larger than Pa.
  • B. Suboptimal Offline Power Allocation with N > 1: Algorithm III offers a lower-complexity solution for Problem (P1) with N > 1 and is asymptotically optimal as M goes to infinity.It combines the N = 1 suboptimal allocation structure with the optimal solution’s EH-period search indices.
  • V. ONLINE POWER ALLOCATION: With only causal ESI and N > 1, the paper derives the optimal online solution using dynamic programming and also proposes a lower-complexity suboptimal algorithm.The approach builds on the N = 1 results while addressing the multi-period causal-ESI setting.
  • A. Optimal Online Power Allocation: Online power variables cannot generally be optimized independently across EH periods because the battery states are coupled over time.This coupling motivates the dynamic-programming formulation for Problem (P2) with N > 1.
  • A. Optimal Online Power Allocation: For Problem (P2.n), dynamic programming recursively computes the minimum average outage probability from JN(QN, BN) through Jn(Qn, Bn).The recursion applies for 1 ≤ n ≤ N given initial states Qn and Bn = Bn,1.
  • A. Optimal Online Power Allocation: The online dynamic programs eliminate the first M − 1 EH constraints by selecting a non-decreasing optimal power allocation.Each subproblem retains M EH constraints before this reduction.
  • A. Optimal Online Power Allocation: Problems (P4.i) are solved by applying Theorem 3.1 with fixed Bi+1 and searching over Bi+1 from 0 to Bi + MQi.This procedure solves the MDPs by dynamic programming while reusing the N = 1 results.

B. Suboptimal Online Power Allocation … VII. CONCLUSION

The paper develops offline and online power-allocation methods for outage minimization under energy-harvesting constraints, including a q-period look-ahead scheme for causal energy information. Numerical results show that online methods can closely approach offline performance, while optimal offline allocation exhibits a save-then-transmit structure.

  • B. Suboptimal Online Power Allocation: The q-period look-ahead algorithm uses the current battery state and predicted harvested-energy means for the next q − 1 periods under a discrete-time first-order Markov EH process.The prediction window satisfies q ≥2, and the exact future-energy distribution need not be known when only its mean values are available.
  • B. Suboptimal Online Power Allocation: The suboptimal online procedure repeatedly computes the current period’s allocation using either the optimal or suboptimal algorithm until the N-th energy-harvesting period.When q = 1, it reduces to greedy allocation that exhausts all stored harvested energy at the end of each harvesting period.
  • A. The Case of N = 1: For N = 1, optimal allocation has outage probability no larger than uniform allocation for Q1 < Pa, and both optimal and suboptimal schemes converge as M approaches infinity.As M increases, the minimum outage probability converges to its M →∞ value, while the limiting optimal curve connects [0, 1] and [Pa, F(Pa)].
  • VII. CONCLUSION: The paper finds that outage probability is generally non-convex in transmit power for most practical fading channels, making the power-allocation problem non-convex.Globally optimal solutions are derived by exploiting outage-probability properties and the causality structure of the energy-harvesting constraints.
  • VII. CONCLUSION: The optimal offline allocation follows a save-then-transmit protocol, while causal energy-information settings admit optimal and suboptimal online schemes based on dynamic programming and offline-solution structure.For N = 1, the results revisit the classic outage-capacity problem with new observations.

APPENDIX A · PROOF OF PROPOSITION 3.2 · APPENDIX B

The proof establishes Proposition 3.2 by analyzing the slope of lines joining (0, 1) to the outage-probability curve. It shows the slope is lower-bounded, ensuring the desired point exists.

  • PROOF OF PROPOSITION 3.2: The proof defines S(P) as the slope of the line through (0, 1) and (P, F(P)) on the outage-probability curve.
  • PROOF OF PROPOSITION 3.2: A lower bound on S(P) implies that the desired point can be found at the corresponding bound.
  • PROOF OF PROPOSITION 3.2: The argument examines the slope function for all P > 0.
  • PROOF OF PROPOSITION 3.2: As P approaches infinity, S(P) approaches 0 because its numerator remains bounded.
  • PROOF OF PROPOSITION 3.2: For sufficiently large A > 0, defining S(0) = 0 makes S(P) bounded above and below on [0, A].
  • PROOF OF PROPOSITION 3.2: Beyond A, S(P) remains lower-bounded because it increases while staying non-positive, completing the proof.

PROOF OF THEOREM 3.1

The proof establishes geometric properties of the concave-convex Type B function and uses them to reduce the optimization to finitely many endpoint candidates. It then orders these candidates to identify the optimum for Q1 < Pa, with Q1 ≥ Pa following as the special case k0 = 0.

  • Geometric observations: For the Type B function, designated secant lines lie above the relevant function values, providing the geometric inequalities used throughout the proof.For 0 ≤ X1 ≤ X2 ≤ X3 ≤ Pa, L1(X0) ≥ L2(X0); for Pa ≤ X0 with 0 ≤ X1 ≤ X0 ≤ X2, L3(X0) ≥ F(X0).
  • Lemma B.1: Lemma B.1 shows that the relevant block-allocation objective attains its minimum at an endpoint, with the applicable endpoint determined by whether Pa is below MQ1/k.The proof considers profiles with one variable power and identical remaining powers, then compares the resulting outage values using the Type B geometry.
  • Case Q1 < Pa: For Q1 < Pa, the optimal value of Problem (P3) is restricted to one of the candidate values {pk}, k = 1, · · · , M.The candidates correspond to profiles with one initial zero-power block count and the remaining power allocated equally across the other blocks.
  • Case Q1 < Pa: The candidate outage values are ordered on either side of k0, yielding the optimal power allocation and its range through (19) and Proposition 3.3.Specifically, pM > pM−1 > · · · > pk0+1 and pk0 < pk0−1 < · · · < p1.
  • Case Q1 ≥ Pa: The Q1 ≥ Pa case is a special case of Q1 < Pa obtained by setting k0 = 0, so its proof is analogous and completes Theorem 3.1.The paper explicitly states that the remaining case follows by the same argument and omits the repeated details.

APPENDIX C ALGORITHM FOR THE OPTIMAL POWER ALLOCATION OF PROBLEM (P1) WITH N > 1

For N > 1, Algorithm II recursively searches feasible power profiles across energy-harvesting periods, expanding the search when needed until it reaches an optimal allocation. The resulting profile is non-decreasing over time and follows a save-then-transmit structure.

  • Search procedure: Algorithm II recursively searches across energy-harvesting periods, using Case I for best-effort transmission and Case II to locate and optimize a lower-power communication block.Case II enlarges the search region when the candidate allocation may improve outage performance, while checking energy feasibility and outage reduction.
  • Optimality checks: The procedure repeats its checks until the N-th energy-harvesting period or until enlargement cannot improve performance, then accepts the resulting profile as optimal.Search updates continue only when the newly derived profile satisfies the relevant energy-harvesting constraint and yields lower outage probability.
  • Resulting power profile: The optimal solution of Problem (P1) is non-decreasing over time and has a save-then-transmit structure.When initial power is much smaller than P_a, the transmitter remains silent while saving energy, then transmits continuously with non-decreasing power once accumulated energy is sufficiently close to P_a.

APPENDIX D · PROOF OF THEOREM 4.1

The appendix proves Theorem 4.1 by comparing the optimal solution with Algorithm II through power-changing blocks and exhaustive first-difference cases. The argument uses energy-harvesting constraint equalities, waterfilling, convexity, and contradictions to exclude every nonoptimal profile.

  • PROOF OF THEOREM 4.1: The proof repeatedly establishes that relevant energy-harvesting constraints are achieved with equality at block boundaries.This equality is stated for later power-changing blocks and is also used in contradiction arguments to rule out alternative profiles.
  • PROOF OF THEOREM 4.1: The proof aligns the optimal profile with Algorithm II until their first differing communication block, then analyzes three possible locations for that difference.The cases are ordered before, at, and after the first power-changing block, with additional subcases at the boundary.
  • PROOF OF THEOREM 4.1: When power is saved at the first differing block, the proof reallocates it over subsequent blocks in a waterfilling manner while preserving identical allocations across appropriate block ranges.The construction progressively merges ranges when the newly obtained power exceeds later block values.
  • PROOF OF THEOREM 4.1: The constructed alternative corresponds to an iteration of Step (2.2.2.4) in Algorithm II and is therefore shown nonoptimal by the algorithm’s search procedure.This excludes the case where the alternative differs through a range of later communication blocks.
  • PROOF OF THEOREM 4.1: A newly defined problem beginning at the second power-changing block has an optimal value no larger than the outage probability of the compared profile, while the full-horizon profile outperforms its solution.Because the newly defined problem is a special case of the final Algorithm II iteration, the corresponding case cannot be optimal.
  • PROOF OF THEOREM 4.1: If the first difference occurs after the initial power-changing block, the allocation remains in the convex regime, so the case is excluded using an argument analogous to Theorem 1 in.The proof states that power values satisfy P ∗ i,j ≥ Pb in this case.
  • PROOF OF THEOREM 4.1: Combining the case analyses proves Theorem 4.1.The conclusion follows after excluding all possible locations of the first difference between the profiles.
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