$\alpha$-Fairness Based Resource Allocation for Ambient Backscatter Communications

This paper presents a resource allocation strategy for ambient backscatter (AmBack) communication systems, with a focus on analyzing the impact of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex>-fairness on system efficiency (measured by throughput) and Jain's index (a metric for measuring fairness). We first formulate an optimization problem to balance these metrics by dynamically adjusting <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha$</tex>-utility function. To address the challenge of achieving this balance, we develop a gradient-based optimization algorithm that integrates the principles of the Robbins-Monro (RM) algorithm. This algorithm effectively allocates resources in a multi-user environment, ensuring an optimal balance between fairness and system efficiency. Simulation results validate the algorithm's ability to enhance system performance while maintaining fairness under various network scenarios.