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Stochastic Resource Allocation via Dual Tail Waterfilling

Gokberk Yaylali, Dionysios S. Kalogerias

TL;DR

This work investigates a new risk-aware formulation of the classical stochastic resource allocation problem for point-to-point power-constrained communication networks over fading channels with no cross-interference, by leveraging the Conditional Value-at-Risk (CV@R) as a coherent measure of risk.

Abstract

Optimal resource allocation in wireless systems still stands as a rather challenging task due to the inherent statistical characteristics of channel fading. On the one hand, minimax/outage-optimal policies are often overconservative and analytically intractable, despite advertising maximally reliable system performance. On the other hand, ergodic-optimal resource allocation policies are often susceptible to the statistical dispersion of heavy-tailed fading channels, leading to relatively frequent drastic performance drops. We investigate a new risk-aware formulation of the classical stochastic resource allocation problem for point-to-point power-constrained communication networks over fading channels with no cross-interference, by leveraging the Conditional Value-at-Risk (CV@R) as a coherent measure of risk. We rigorously derive closed-form expressions for the CV@R-optimal risk-aware resource allocation policy, as well as the optimal associated quantiles of the corresponding user rate functions by capitalizing on the underlying fading distribution, parameterized by dual variables. We then develop a purely dual tail waterfilling scheme, achieving significantly more rapid and assured convergence of dual variables, as compared with the primal-dual tail waterfilling algorithm, recently proposed in the literature. The effectiveness of the proposed scheme is also readily confirmed via detailed numerical simulations.

Stochastic Resource Allocation via Dual Tail Waterfilling

TL;DR

This work investigates a new risk-aware formulation of the classical stochastic resource allocation problem for point-to-point power-constrained communication networks over fading channels with no cross-interference, by leveraging the Conditional Value-at-Risk (CV@R) as a coherent measure of risk.

Abstract

Optimal resource allocation in wireless systems still stands as a rather challenging task due to the inherent statistical characteristics of channel fading. On the one hand, minimax/outage-optimal policies are often overconservative and analytically intractable, despite advertising maximally reliable system performance. On the other hand, ergodic-optimal resource allocation policies are often susceptible to the statistical dispersion of heavy-tailed fading channels, leading to relatively frequent drastic performance drops. We investigate a new risk-aware formulation of the classical stochastic resource allocation problem for point-to-point power-constrained communication networks over fading channels with no cross-interference, by leveraging the Conditional Value-at-Risk (CV@R) as a coherent measure of risk. We rigorously derive closed-form expressions for the CV@R-optimal risk-aware resource allocation policy, as well as the optimal associated quantiles of the corresponding user rate functions by capitalizing on the underlying fading distribution, parameterized by dual variables. We then develop a purely dual tail waterfilling scheme, achieving significantly more rapid and assured convergence of dual variables, as compared with the primal-dual tail waterfilling algorithm, recently proposed in the literature. The effectiveness of the proposed scheme is also readily confirmed via detailed numerical simulations.
Paper Structure (13 sections, 2 theorems, 38 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 2 theorems, 38 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. The Dual Tail Waterfilling
  4. CV@R-Optimal Risk-Aware Resource Policy
  5. Optimal Value-at-Risk / Risk-Ergodic Rate
  6. Dual Descent
  7. Common Utilities & Fading Distributions
  8. Sumrate Utility
  9. Proportional Fairness Utility
  10. Weibull Fading
  11. Rayleigh Fading
  12. Performance Evaluation
  13. Conclusion

Key Result

Theorem 1

An optimal solution to the resource policy subproblem eqn:p_subproblem for terminal $i \in \{1, \dots, N_U\}$ is whenever $(\lambda_i, \mu) \neq \boldsymbol{ 0 }$, otherwise selecting $p_i^* = 0$ is optimal.

Figures (4)

  • Figure 1: CV@R-Optimal resource allocation policies for risk-aware (RA, $\alpha = 0.90$) and risk-neutral (RN, $\alpha = 1.00$) settings in a $3$-terminal network.
  • Figure 2: Outage probabilities for a $3$-terminal network with sumrate utility (top) and proportional fairness utility (bottom).
  • Figure 3: Achieved rates for the $3$-terminal network with sumrate utility (left), and proportional fairness utility (left). Top: risk-aware. Bottom: risk-neutral.
  • Figure 4: $\boldsymbol{ t }$-iterates for the $3$-terminal network with sumrate utility (left), and proportional fairness utility (right).

Theorems & Definitions (2)

  • Theorem 1: CV@R-Optimal Risk-aware Policy
  • Theorem 2: Optimal Value-at-Risk