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Power Control for D2D Underlaid Cellular Networks: Modeling, Algorithms and Analysis
Namyoon Lee, Xingqin Lin, Jeffrey G. Andrews, Robert W. Heath
TL;DR
The paper addresses interference management when multiple D2D links share uplink cellular spectrum with a cellular user. It combines stochastic-geometry modeling with centralized and distributed power control, finding that the proposed methods improve network performance while the centralized approach better preserves cellular reliability. The analysis identifies cross-tier interference as the key bottleneck and derives coverage and D2D sum-rate characterizations for distributed control.
Problem
Underlaid D2D transmissions create cross-tier interference for cellular links, while D2D links also face cellular and intra-D2D interference, making successful coexistence challenging.
Method
The paper uses a stochastic-geometry random network model and develops centralized and distributed power-control algorithms, including direct-link threshold-based on-off control.
Results
The centralized approach supports multiple D2D links while guaranteeing reliable cellular communication, whereas distributed control improves throughput but is not sufficient to guarantee reliable cellular links.
Takeaways & Limitations
The analysis indicates that cross-tier interference between D2D transmissions and the cellular user, rather than D2D intra-tier interference, is the network bottleneck.
Abstract
from arXiv · showhide
This paper considers a device-to-device (D2D) underlaid cellular network where an uplink cellular user communicates with the base station while multiple direct D2D links share the uplink spectrum. This paper proposes a random network model based on stochastic geometry and develops centralized and distributed power control algorithms. The goal of the proposed power control algorithms is two-fold: ensure the cellular users have sufficient coverage probability by limiting the interference created by underlaid D2D users, while also attempting to support as many D2D links as possible. For the distributed power control method, expressions for the coverage probabilities of cellular and D2D links are derived and a lower bound on the sum rate of the D2D links is provided. The analysis reveals the impact of key system parameters on the network performance. For example, the bottleneck of D2D underlaid cellular networks is the cross-tier interference between D2D links and the cellular user, not the D2D intra-tier interference. Numerical results show the gains of the proposed power control algorithms and accuracy of the analysis.
I. INTRODUCTION
The paper models D2D links sharing uplink cellular spectrum and develops centralized and distributed power-control methods to coordinate interference. Its stochastic-geometry model and analysis target reliable cellular communication while enabling additional D2D throughput.
- Motivation: D2D communication supports direct mobile-user communication but creates cross-tier interference for cellular links and adds cellular interference to D2D links.The resulting coexistence challenge makes interference management essential.
- Results: The centralized method can support multiple successful D2D links while guaranteeing reliable communication for the existing cellular link.The paper reports that underlaid D2D links can increase network sum-throughput without unacceptable degradation to existing cellular links.
- System model: The paper introduces a hybrid stochastic-geometry model with one uplink cellular user and multiple D2D links sharing the common spectrum.D2D transmitter locations are modeled using a homogeneous PPP, with receivers at fixed distances and isotropic directions.
- Centralized power control: The centralized algorithm maximizes cellular-link SINR subject to individual target SINR constraints for D2D links.The centralized power-allocation problem is treated as convex and solved using a feasibility-set increment technique.
- Distributed power control: The distributed algorithm selects each D2D transmitter’s power from {0, Pmax,d} using only direct-link information and a common threshold.Unlike the centralized method, it avoids coordination and shared CSIT, while supporting analytical coverage and D2D sum-rate analysis.
- Scope and assumptions: The model ignores out-of-cell interference from macro users in other cells, while retaining out-of-cell D2D interference outside the circular cell region.The authors state that the model captures dominant interference, including the nearest in-cell D2D transmitter for the uplink user.
III. POWER CONTROL ALGORITHMS
The paper develops centralized and distributed power-control algorithms for D2D underlaid cellular networks, balancing cellular-link reliability with support for multiple D2D links. The centralized method uses global CSI and feasibility-aware link selection, while the distributed method uses local direct-link information and on-off transmission.
- Centralized Power Control: The centralized algorithm uses global CSI to maximize cellular-link SINR while satisfying cellular and D2D SINR constraints.
- Centralized Power Control: Centralized optimization is tractable with convex-programming tools when the feasible set is nonempty.The constraint set is feasible if the normalized channel-gain matrix satisfies ρ(F) < 1.
- Centralized Power Control: When centralized constraints are infeasible, the proposed selection method successively removes the D2D transmitter causing the largest aggregate interference.This avoids brute-force search, whose complexity grows exponentially with the number of D2D links.
- Distributed On-Off Power Control Algorithm: The distributed algorithm requires no coordination and selects D2D power from {0, Pmax,d} using only direct-link quality and threshold Gmin.A D2D transmitter uses Pmax,d when its direct-link gain exceeds Gmin and otherwise remains silent.
- Distributed On-Off Power Control Algorithm: Increasing Gmin reduces inter-D2D interference but also lowers the transmission probability and can reduce the D2D sum rate.Thus, the threshold or transmission probability is central to distributed sum-rate performance.
IV. COVERAGE PROBABILITY ANALYSIS FOR DISTRIBUTED POWER CONTROL
This section derives analytical cellular-coverage expressions under distributed power control using stochastic-geometry assumptions. The analysis identifies how D2D density and power affect cellular coverage and validates the expression against simulation.
- Coverage Probability Analysis: The analysis models D2D transmit powers as i.i.d. and independent of the uplink user’s transmit power.
- Coverage Probability Analysis: Theorem 1 provides an analytical formula for uplink cellular coverage probability.
- Coverage Probability Analysis: Cellular coverage depends on D2D transmitter density and the distribution of D2D transmit power.The analysis specifically highlights λ and a power-related moment as D2D-network parameters affecting coverage.
- Coverage Probability Analysis: Higher D2D-transmitter density decreases cellular coverage because it causes more interference to the cellular link.
- Coverage Probability Analysis: The analytical coverage expression closely matches Monte Carlo simulation across the tested target-SINR range and two D2D densities.The reported densities are λ ∈ {0.00002, 0.00005}.
- Coverage Probability Analysis: A lower bound on cellular coverage is derived for arbitrary path-loss exponent and noise-limited cases using moments rather than full distributions.
B. Optimal Cellular Link Power Control Strategy
The optimal cellular-link strategy depends on user distance and power constraints, with on-off power control maximizing coverage. Closed-form characterization is available in the interference-limited regime.
- Optimal Cellular Link Power Control Strategy: The optimal power allocation is obtained from an infinite-dimensional distribution optimization problem with a simple structure.The paper reports that this structure permits finding the optimal distribution despite the general difficulty of such problems.
- Optimal Cellular Link Power Control Strategy: Theorem 2 states that conditional on uplink-user distance, on-off power control maximizes cellular-link coverage probability.The optimal non-degenerate power level is denoted p⋆(d).
- Optimal Cellular Link Power Control Strategy: In the interference-limited regime, the optimal cellular transmit power has a closed-form expression subject to the peak-power constraint.The implemented power is the minimum of the unconstrained solution and Pmax,c, with binary transmission probability governed by the average-power constraint.
- Optimal Cellular Link Power Control Strategy: Near the cell center, the cellular user uses constant average transmit power and coverage decreases exponentially with target threshold β0.
- Optimal Cellular Link Power Control Strategy: At mid-cell distances, on-off control increases transmit power proportionally to d^α, compensating for path loss while coverage decreases linearly with β0.
- Optimal Cellular Link Power Control Strategy: At the cell edge, the peak-power constraint forces transmission at Pmax,c with probability Pavg,c, and coverage again decreases linearly with β0.
- Optimal Cellular Link Power Control Strategy: For the example parameters, coverage is approximately 0.743 at d = R/2 and 0.3298 at d = 0.7R.At d = R, the optimal cellular transmit power reaches 0.2 W, the maximum power.
C. D2D Link Coverage Probability
The paper derives exact and approximate coverage-probability expressions for a typical D2D link using SINR distributions and validates them against Monte Carlo simulations.
- The derivation computes Laplace transforms of interference while modeling transmitter locations, fading, and link distances probabilistically.
- The typical D2D coverage probability is obtained by deriving the ccdf of SINR_k and applying a theorem-based expression.
- A closed-form coverage expression is derived for fixed D2D distance, fixed D2D transmit power, constant cellular transmit power, and σ2 = 0.
- Fig. 3 evaluates coverage with dk,k = 50 m, p0 = 100 mW, pk = 0.1 mW, R = 500 m, and λ ∈{0.00002, 0.00005}.
- The exact and approximate analytical expressions agree accurately with Monte Carlo simulations, especially for α = 4.
V. SUM RATE ANALYSIS OF D2D LINKS
This section analyzes D2D-link sum rate under the proposed on-off power control and identifies the threshold that maximizes it.
- The analysis characterizes the optimal on-off power-control threshold for maximizing the D2D-link sum rate.
A. Sum Rate of D2D Links
The paper derives the D2D sum rate from active-link interference and the typical-link SIR distribution, with the approximation governed by two interference-related factors.
- Active D2D links are selected when |h_k,k|2d_k,k^-α ≥ G_min, giving an active-link count of |S| = ˜λπR2.
- Assuming Gaussian signal transmission from active links, the achievable D2D sum rate is formulated using their aggregate interference.
- The ergodic rate of a typical D2D link is rewritten using the SIR distribution and then approximated through the distribution of link interference.
- The approximated typical-link ergodic rate is determined by the Laplace transform of total interference and the effect of uplink interference.
B. Optimizing D2D ON-Off Threshold
The paper optimizes the distributed on-off threshold through transmission probability and analyzes how D2D transmission capacity varies with SINR targets and network parameters.
- B. Optimizing D2D ON-Off Threshold: The D2D on-off threshold G_min is optimized by first solving for the transmission probability P_s that maximizes the approximated transmission capacity.
- B. Optimizing D2D ON-Off Threshold: Although the objective is not concave, its unique optimum allows P_s to be obtained from the first-order optimality condition.
- B. Optimizing D2D ON-Off Threshold: The transmission capacity depends on the target SINR β, path-loss exponent α, D2D density λ, and D2D distance d_k,k.
- B. Optimizing D2D ON-Off Threshold: When β is small, all D2D transmitters are scheduled, matching the performance without power control.
- B. Optimizing D2D ON-Off Threshold: For sufficiently large β, D2D links transmit with probability P_s^*, mitigating inter-D2D interference.
- B. Optimizing D2D ON-Off Threshold: When β exceeds the indicated threshold, D2D transmission capacity becomes independent of λ and increases linearly with R2.
- B. Optimizing D2D ON-Off Threshold: Analytical D2D sum-rate results match Monte Carlo simulations under the listed α = 4, distance, radius, and power settings.
VI. SIMULATION RESULTS
The simulations compare the proposed power-control methods with no power control using cellular and D2D coverage probability.
- The numerical evaluation measures performance gains from the proposed power-control methods relative to no power control.
1) Simulation Setup:
The simulations examine sparse and dense D2D deployments and compare coverage under different power-control methods. Centralized control protects cellular coverage, while distributed control improves coverage under specified SINR conditions.
- Simulation Setup:: D2D transmitters follow a PPP with λ ∈ {0.00002, 0.00005}, corresponding to average link counts E[K] ∈ {15, 39}.The cellular user is uniformly placed within R = 500 m, and D2D receivers are fixed 50 m from their transmitters.
- Coverage Probability Comparison in Sparse D2D Link Deployment:: In sparse deployment, centralized control achieves nearly perfect cellular coverage at low target SINR while supporting 48% active D2D links at β = 3 dB.
- Coverage Probability Comparison in Sparse D2D Link Deployment:: At β = 15 dB, on-off control provides 13% cellular-link and 5% D2D-link coverage gains over no power control.
- Coverage Probability Comparison in Dense D2D Link Deployment:: In dense deployment, centralized admission control prevents significant cellular degradation as D2D density increases, while D2D coverage deteriorates from increased intra-D2D interference.
- Conclusions: The conclusion reports that centralized control improves cellular throughput while preserving reliable uplink communication, whereas distributed control cannot guarantee reliable cellular links.
APPENDIX
The appendix derives uplink coverage by conditioning on transmit power and distance, evaluating interference transforms, and then removing the conditioning.
- The derivation uses the exponential fading distribution to express the desired-link success probability.
- Conditioned on uplink transmit power and transmitter-to-base-station distance, the interference term is evaluated through its Laplace transform.
- De-conditioning over transmit power and distance yields the uplink coverage probability.
- The analysis derives the distribution and moments of X = p^-1 d^α, including an explicit computation of E[d^α].
- Convexity of φ(x) for x ≥ 0 permits applying Jensen’s inequality in the coverage analysis.
C. Proof of Corollary 2
The corollary proof establishes an interior optimizer for the relevant function and concludes that binary power control is optimal for the relaxed problem.
- The function φ(p) is positive and continuous for p > 0, with limiting behavior implying a non-degenerate maximizer p⋆(d) ∈ (0, ∞).
- The proof temporarily ignores a constraint and reformulates the conditional cellular-link coverage optimization as a relaxed problem.
- The relaxed optimum concentrates the solution at p⋆(d), yielding the conclusion that binary power control is optimal.
- The appendix invokes Slivnyak’s theorem to derive the ccdf of SINR_k before completing the proof.