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Cooperative Energy Harvesting Networks with Spatially Random Users
Zhiguo Ding, H. Vincent Poor
TL;DR
The paper asks how to characterize outage in a cooperative network when user locations are spatially random. It analyzes outage and models cooperation as a canonical coalitional game; in the high-SNR regime, the grand coalition is stable.
Problem
The paper characterizes outage for cooperative users while accounting for spatial randomness in their locations.
Source: §I · p. 1Method
The paper analyzes outage in a multiple-pair network with an energy-harvesting relay and models user cooperation as a canonical coalitional game.
Source: §I · p. 1Results
In the high-SNR regime, the grand coalition is stable, and simulations show larger coalitions have fewer user errors and higher payoff, with the grand coalition performing best.
Source: B. Characterizing the addressed game · p. 2, §IV · p. 3Takeaways & Limitations
In the addressed high-SNR scenario, users in singleton sets seek to join existing coalitions, and the grand coalition is a stable system point.
Source: B. Characterizing the addressed game · p. 2
Abstract
from arXiv · showhide
This paper considers a cooperative network with multiple source-destination pairs and one energy harvesting relay. The outage probability experienced by users in this network is characterized by taking the spatial randomness of user locations into consideration. In addition, the cooperation among users is modeled as a canonical coalitional game and the grand coalition is shown to be stable in the addressed scenario. Simulation results are provided to demonstrate the accuracy of the developed analytical results.
I. INTRODUCTION
The paper studies energy harvesting for cooperative communication, especially in energy-constrained networks without natural light or wind. It accounts for random user locations when characterizing outage and models cooperation as a coalitional game.
I. INTRODUCTION
- Simultaneous wireless power and signal transfer is presented as promising for energy-constrained networks without natural light or wind –[3].Prior work considered multiple-antenna systems and imperfect transmitter CSI; energy harvested from incoming signals can power cooperative relays [6].Source: §I · p. 1
- The paper characterizes user outage while accounting for spatially random locations in a network with multiple source–destination pairs and an energy-harvesting relay.The paper also models user cooperation as a canonical coalitional game and reports that the grand coalition is preferred in the high-SNR regime.Source: §I · p. 1
II. COOPERATIVE ENERGY HARVESTING TRANSMISSIONS
The section specifies a spatially random cooperative network and describes how its users share transmission resources and relay energy. It defines outage-based rates and uses them to value coalitions.
II. COOPERATIVE ENERGY HARVESTING TRANSMISSIONS
- The model places 2N nodes uniformly in a disc of radius D, with the energy-harvesting relay at the origin, and assumes Rayleigh fading on source-to-relay channels.Channel gains and distances are unavailable as CSI before transmission; α denotes the path-loss factor.Source: §II · p. 1
- A coalition is a disjoint group of source–destination pairs, and each source transmits to the relay over an orthogonal channel with no direct source–destination link.Sources in the same coalition share power under a total power constraint.Source: §II · p. 1
- In Phase II, the relay decodes each received message, harvests any power left after decoding, and forwards correctly decoded messages to destinations.Coalition members share relaying power subject to the total harvested-energy constraint; harvested energy depends on channel gain and source–relay distance.Source: §II · p. 1
- For user pair i, outage probability P_i determines its achieved rate R(1 − P_i), and the section uses achieved rates to define coalition value.The section also frames coalition formation and user choice as an optimization problem.Source: §II · pp. 1–2
III. CANONICAL COALITIONAL GAME
The optimization is simplified using equal power allocation when CSI is unavailable, yielding an equivalent formulation for evaluating coalition choices.
III. CANONICAL COALITIONAL GAME
- With no CSI before transmission, the section treats user pairs as equally important and makes equal source-power allocation optimal.It likewise states that equal relay-power allocation is optimal among users whose information was successfully delivered to the relay.Source: §III · p. 2
- Applying the stated allocation steps produces an equivalent form of the optimization problem.Evaluating outage probability is identified as key to solving that formulation.Source: §III · p. 2
- The path-loss factor α is set to 2 to obtain closed-form expressions.Source: §III · p. 2
A. Characterizing Pi
This section gives high-SNR outage approximations and motivates more explicit asymptotic expressions for coalition analysis. The theorem’s analytical results are reported as accurate in moderate-SNR simulations.
A. Characterizing Pi
- Theorem 1 provides a high-SNR approximation for the outage probability of user pair i in coalition S_k.The approximation involves the modified Bessel function of the second kind.Source: A. Characterizing Pi · p. 2
- The theorem’s analytical outage results are reported as accurate even in a moderate-SNR regime, based on simulations in the next section.Source: A. Characterizing Pi · p. 2
- Because the theorem’s expressions are too involved for direct coalition analysis, the section introduces more explicit asymptotic expressions as SNR approaches infinity.The expressions distinguish coalitions of size at least two from singleton sets.Source: A. Characterizing Pi · p. 2
B. Characterizing the addressed game
In the high-SNR regime, the addressed game favors the grand coalition and has a stable grand-coalition outcome. The proposed payoff vector lies in a non-empty core.
B. Characterizing the addressed game
- The grand coalition is always preferred because outage probability depends only on coalition size and is lower for the grand set.The paper uses this comparison to show that the grand coalition yields greater value than a smaller coalition.Source: B. Characterizing the addressed game · p. 2
- At high SNR, the game is a canonical transferable-utility coalitional game, and the grand coalition is stable.The paper argues that singleton sets are not preferred, so their members try to join existing coalitions.Source: B. Characterizing the addressed game · p. 2
- For coalitions of size at least two, the game is superadditive; singleton users' outage probability decays more slowly than the 1/SNR rate achievable by larger coalitions.The proof uses these asymptotic properties to explain why singleton sets are not preferred.Source: B. Characterizing the addressed game · p. 2
- As SNR approaches infinity, the core is non-empty and contains the proposed payoff vector x*.The text introduces x* as the player payoff vector before characterizing the core.Source: B. Characterizing the addressed game · p. 2
IV. NUMERICAL RESULTS
The numerical results compare analytical outage expressions with simulations and examine outage across coalition sizes. They report that larger coalitions perform better, with the grand coalition achieving the best performance.
IV. NUMERICAL RESULTS
- The simulation comparison evaluates the analytical results from Theorem 1 and the asymptotic results from Corollary 1.The paper presents these comparisons as a demonstration of the analytical results' accuracy.Source: §IV · pp. 2–3
- Larger coalitions experience fewer errors and higher payoff, and the grand coalition achieves the best performance.The coalition-size comparison is reported to confirm Corollary 2.Source: §IV · p. 3
- For small D, the developed analytical results match simulations exactly; for a larger D, a gap remains, but further increasing SNR can reduce the gap.The simulation comparison covers analytical results developed in Theorem 1 and asymptotic results shown in Corollary 1.Source: §IV · pp. 2–3
V. CONCLUSION
The paper develops outage probabilities for users in a cooperative energy-harvesting relay network and models their cooperation as a coalitional game. In the addressed scenario, the grand coalition is shown to be stable.
V. CONCLUSION
- The paper develops outage probabilities for users communicating through an energy-harvesting relay.The network has multiple source-destination pairs.Source: §V · p. 3
- User cooperation is modeled as a canonical coalitional game, and the grand coalition is shown to be stable in the addressed scenario.Source: §V · p. 3
- The accompanying figure captions identify a simulation-versus-analytical comparison and outage probabilities for coalitions of different sizes.Source: Figure 1 · p. 3, Figure 2 · p. 3
APPENDIX
The appendix derives outage probabilities under spatially random node locations, using distributional approximations to handle a difficult conditional density. It also gives asymptotic outage behavior for coalitions of at least two users.
APPENDIX
- The derivation uses the spatial distribution of nodes and equal power allocation to characterize each pair’s outage probability.Node locations are uniform in D, and the source-relay channels are modeled as i.i.d.Source: APPENDIX · p. 3
- Because the conditional density of z is difficult to derive directly, the appendix approximates it where the integrand has its greatest effect.The approximation is matched near z = 0; differences at large z have insignificant impact because F_xi(γ(z, P_ow)) approaches zero.Source: APPENDIX · p. 3
- The exponential approximation is reported to match simulation results closely, including in the moderate-SNR regime.Source: APPENDIX · p. 4
- The passages state that for n ≥ 3, Q2,n decays at rate 1/P_ow, while Q2,2 has a much slower decay rate, described only as log P in the available text.Source: APPENDIX · p. 4
- For |S_k| ≥ 2, the per-user outage probability is asymptotically equivalent to the expression derived in the corollary.The supplied passage does not include the expression itself.Source: APPENDIX · p. 4