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Resource Allocation Under Channel Uncertainties for Relay-Aided Device-to-Device Communication Underlaying LTE-A Cellular Networks
Monowar Hasan, Ekram Hossain, Dong In Kim
TL;DR
The paper addresses resource allocation for relay-assisted D2D communication when channel gains and interference are uncertain. It develops a robust distributed allocation method that preserves convexity and uses a chance-constraint trade-off, finding that relaying significantly improves network performance beyond a distance threshold.
Problem
The paper addresses resource allocation for multi-user, multi-relay D2D communication when link gains and interference are uncertain while cellular and D2D QoS must be satisfied.
Method
It formulates a worst-case robust allocation problem, solves it with a gradient-aided distributed algorithm, and uses chance constraints to trade robustness against optimality.
Results
Beyond a distance threshold, relay-aided D2D communication significantly improves network performance relative to direct D2D communication.
Takeaways & Limitations
The framework maintains convexity under bounded channel uncertainty while allocating relay resources for end-to-end rate and QoS.
Abstract
from arXiv · showhide
Device-to-device (D2D) communication in cellular networks allows direct transmission between two cellular devices with local communication needs. Due to the increasing number of autonomous heterogeneous devices in future mobile networks, an efficient resource allocation scheme is required to maximize network throughput and achieve higher spectral efficiency. In this paper, performance of network-integrated D2D communication under channel uncertainties is investigated where D2D traffic is carried through relay nodes. Considering a multi-user and multi-relay network, we propose a robust distributed solution for resource allocation with a view to maximizing network sum-rate when the interference from other relay nodes and the link gains are uncertain. An optimization problem is formulated for allocating radio resources at the relays to maximize end-to-end rate as well as satisfy the quality-of-service (QoS) requirements for cellular and D2D user equipments under total power constraint. Each of the uncertain parameters is modeled by a bounded distance between its estimated and bounded values. We show that the robust problem is convex and a gradient-aided dual decomposition algorithm is applied to allocate radio resources in a distributed manner. Finally, to reduce the cost of robustness defined as the reduction of achievable sum-rate, we utilize the \textit{chance constraint approach} to achieve a trade-off between robustness and optimality. The numerical results show that there is a distance threshold beyond which relay-aided D2D communication significantly improves network performance when compared to direct communication between D2D peers.
I. INTRODUCTION
The paper studies relay-assisted D2D communication when direct links are unfavorable and channel information is uncertain. It formulates robust resource allocation for multiple cellular and D2D links, preserving convexity and enabling distributed optimization.
- Motivation: The study addresses longer distances, poor propagation, and interference that can limit direct D2D communication while maintaining cellular and D2D QoS.
- Motivation: Relay-assisted D2D communication supports distant or poorly connected D2D pairs through LTE-A Layer-3 relays.Relay resources are shared by D2D links and two-hop cellular links.
- Contributions: Under uncertain channel information, the robust formulation remains convex and is solved with a distributed polynomial-time algorithm.
- Contributions: A robustness–optimality trade-off mechanism addresses the achievable sum-rate reduction introduced by robust resource allocation.
- Contributions: The resource allocation problem covers multiple D2D links and multiple cellular links supported by multiple relays, rather than a single D2D link.
IV. RESOURCE BLOCK (RB) AND POWER ALLOCATION IN RELAY NODES
The nominal relay-node problem allocates resource blocks and transmit power to maximize each UE's two-hop capacity while enforcing power, interference, and QoS constraints.
- Objective: Each relay allocates RBs and power to maximize system capacity, defined as the minimum achievable data rate over two hops.
- Resource allocation: The RB allocation indicator identifies whether RB n is assigned to UE u_l.
- Resource allocation: The achievable sum-rate is accumulated over the RBs allocated to each UE.
- Rate model: The formulation uses first- and second-hop unit-power SINRs and corresponding interference levels to determine each UE's rate.
- Constraints: Constraints enforce one-UE-per-RB assignment, power budgets, interference thresholds, and minimum QoS requirements.
B. Continuous Relaxation and Reformulation
The nominal mixed-integer problem is relaxed through time sharing, producing a convex formulation whose allocation solution is globally optimal for the original problem and asymptotically optimal with many RBs.
- Relaxation: The original resource allocation problem is a computationally intractable mixed-integer nonlinear program.
- Relaxation: Time-sharing factors relax the binary RB-assignment constraint by representing the portion of time an RB is assigned to a UE.
- Convex reformulation: The relaxed problem is convex because its objective is concave, the QoS constraint is convex, and the remaining constraints are affine.
- Optimality: The power and RB allocation obtained from the relaxed formulation is a globally optimal solution to the original problem.
- Scope: The formulation uses perfect channel and interference information for the nominal problem before uncertainty is introduced.
V. ROBUST RESOURCE ALLOCATION
The robust formulation models uncertain link gains and interference with bounded uncertainty sets while retaining convexity through protection functions and norm-based reformulation. The resulting problem can be expressed as a conic quadratic program under ellipsoidal uncertainty.
- Uncertainty model: Uncertainty affects link gains and aggregated interference in both communication hops.
- Robust formulation: The robust rate expression approximates achievable rate under receiver channel uncertainty for mathematical tractability.Information-theoretic lower and upper rate bounds exist, but the paper uses the stated rate expression.
- Convexity: The robust counterpart remains convex because the relevant uncertainty sets are compact and convex and the maximum over a convex set is convex.
- Protection function: Protection functions replace explicit uncertainty sets to express robust constraints in closed form.
B. Uncertainty Set and Protection Function
The paper models channel gains and interference as bounded deviations from nominal estimates, then converts robust constraints into tractable convex forms using protection functions and dual norms.
- Uncertainty sets: Uncertain channel gains and interference are represented by norm-based sets measuring differences between actual and estimated values.The sets cover first- and second-hop channel gains and experienced interference.
- Protection functions: Protection functions incorporate uncertainty bounds into the robust optimization constraints.Their values depend on the uncertain parameters and allow the problem to be rewritten deterministically.
- Protection functions: Dual norms replace maximization over uncertainty regions, preserving convexity of the robust formulation.The protection functions for the uncertainty sets are derived through dual norms.
- Convex reformulation: For linear norms with order α ≥ 2, the protection functions become deterministic linear norms of order β and remove nonlinear max operators.This yields the standard convex optimization problem P5.
- Convex reformulation: With ellipsoidal uncertainty, α = 2 and β = 2, so P5 becomes a conic quadratic programming problem.The paper then develops a distributed gradient-aided algorithm to solve it efficiently.
A. Algorithm Development
The proposed algorithm jointly allocates resource blocks and transmit power through relay-local updates, using gradient-based multiplier adjustment until the network rate converges.
- Allocation rules: The optimal relay power over each resource block is obtained from a water-filling equation, while resource-block assignment follows the allocation rule in (12).The power expression uses the protection-related quantity defined by (24).
- Distributed algorithm: Each relay independently allocates resource blocks and power to its associated user equipments using gradient-based updates.The algorithm initializes multipliers, updates allocation variables, and iterates across relays.
- Distributed algorithm: The relays aggregate achievable network rate and stop when successive iterations differ by less than the convergence tolerance.Resources are then assigned and the average achievable data rate is calculated.
- Distributed algorithm: L3 relays can schedule independently and obtain other-relay power and channel information through the X2 interface without increasing signaling overhead at the eNB.This information supports distributed resource allocation.
- Convergence and complexity: The algorithm approaches a sum-rate whose successive-iteration difference is below arbitrary ε > 0 with polynomial complexity in |U_l| and N.The stated overall complexity is O(|U_l|N + T|U_l|N) when T iterations are required.
C. Cost of Robust Resource Allocation
The paper treats the loss from worst-case robustness as a controllable cost and uses chance constraints to balance uncertainty protection against achievable network sum-rate.
- Robustness cost: The cost of robustness is defined as the reduction in achievable sum-rate between nominal and robust resource allocation.The reduction is denoted R_Δ.
- Robustness cost: The achievable-rate reduction depends on the uncertainty-set size and can be controlled by adjusting the uncertainty bounds.Proposition 5 relates the reduction to optimal dual variables and uncertainty parameters.
- Robustness–optimality trade-off: Worst-case robust allocation may be overly conservative because actual uncertainty does not always reach its worst case.The paper therefore seeks a robustness–sum-rate trade-off.
- Chance constraints: Chance constraints replace selected uncertain constraints with convex safe approximations under independent identically distributed uncertainty assumptions.Bernstein approximations are used for the first- and second-hop constraints.
- Robustness–optimality trade-off: Reducing Θ increases robustness, whereas increasing Θ increases network sum-rate.Thus, the trade-off parameters regulate robustness against uncertainty and optimality.
E. Sensitivity Analysis
The sensitivity analysis evaluates the proposed allocation scheme under a 3GPP propagation model, varying D2D geometry and uncertainty assumptions across simulated network realizations.
- Simulation setup: Performance is evaluated with an event-driven MATLAB simulator using distance-dependent path loss, shadow fading, and multipath Rayleigh fading.UE-to-relay and relay-to-D2D links use a specified 3GPP path-loss model.
- Simulation setup: The simulated network contains one three-sectored 700 m × 700 m cell, one relay per sector, and uniformly distributed cellular and D2D users.The parameters D_r,d and D_d,d vary across simulations.
- Uncertainty assumptions: The first-hop channel-gain uncertainty bound is set so its error does not exceed 50% of the nominal value.The simulations also specify estimated interference and uncertainty matrices for the relevant links.
- D2D geometry: D2D transmitters and receivers are placed on the perimeter of a circle with radius D_r,d while maintaining peer distance D_d,d.This geometry is the basis for varying D2D placement and separation.
- Simulation setup: Results are averaged over 250 realizations of user locations and link gains.This averaging is applied to the simulated scenarios.
B. Results
The proposed distributed algorithm converges quickly and maintains a robustness–optimality trade-off. Relay-aided D2D communication improves data rates especially for distant D2D links, while approaching an asymptotic upper bound with lower complexity.
- Convergence of the proposed algorithm: Less than 20 iterations are sufficient for convergence when the step size is sufficiently small.The convergence result is reported for the proposed algorithm under the tested parameter settings.
- Robustness–optimality trade-off: A trade-off parameter below 0.2 makes the robustness cost highly sensitive, whereas larger values increase network sum-rate but reduce robustness.Adjusting the parameter within the reported range provides a trade-off between optimality and robustness.
- Effect of relaying: The proposed relay-based scheme greatly improves D2D data rates as distance increases and performs close to the asymptotic upper bound with significantly less complexity.The comparison uses average achievable D2D data rates for the proposed and reference schemes.
- Effect of relaying: A distance margin exists beyond which relaying improves network performance, including relay–D2D distances of at least 80 m for distant D2D UEs.The reported gain concerns aggregated data rate under varying relay–D2D UE distances.
- Effect of relaying: Under channel uncertainty, rate gain is reduced compared with perfect channel information, but relay-aided communication still benefits increasingly distant D2D links.Direct communication performs better at shorter distances, while its performance deteriorates as D2D distance increases.
VIII. CONCLUSION
The paper develops a resource-allocation framework for relay-assisted D2D communication under channel-gain uncertainty. Simulations indicate that relaying significantly improves network performance for distant D2D UEs beyond a distance threshold.
- The framework models uncertainty in relay-to-relay link gains and interference at receiving network nodes as bounded differences between actual and nominal values.
- Maintaining the robust formulation's convexity, the paper proposes a polynomial-time distributed algorithm for radio-resource allocation.
- The chance-constraint trade-off mechanism balances robustness against the reduction in achievable network sum-rate.
- Beyond a distance threshold, relaying D2D traffic significantly improves network performance compared with direct D2D communication.
- The proposed framework could be extended by including delay as a QoS parameter and D2D discovery alongside resource allocation.
APPENDIX A POWER AND RB ALLOCATION FOR NOMINAL PROBLEM
The appendix derives resource-allocation conditions for the nominal problem using KKT conditions, Lagrangian duality, and sensitivity analysis. It then connects perturbations in robust constraints to changes in the optimal value.
- KKT optimality conditions define the Lagrangian and associate multipliers with QoS, resource-allocation, and other nominal-problem constraints.
- Differentiating the Lagrangian with respect to transmit power and RB-allocation variables yields conditions for power and RB allocation.
- The derivation rewrites constraints using the inequality ∥y∥2 ≤ ∥y∥1 to obtain a more tractable formulation.
- Convexity permits construction of a Lagrange dual function whose derivatives produce the power- and RB-allocation expressions.
- Local sensitivity analysis and a Taylor expansion relate perturbations in robust constraints to the change in the optimal objective value.
- Non-negative Lagrange multipliers imply that achievable sum-rate is lower with imperfect channel information than with perfect information.