Asymptotic Analysis of Proportionally Fair Scheduling in Rayleigh Fading

This paper is concerned with the analysis of proportionally fair scheduling (PFS), and we provide an analytical approximation for the PFS throughput over Rayleigh fading channels. Though quite accurate, the ordinary differential equation (ODE) analysis, typically used to analyze the PFS throughput, is highly time-consuming when there are lots of users. On the other hand, due to the intricate interplay among these ODE equations, the ODE analysis generally fails to provide a closed-form approximation for estimating the PFS throughput unless with simplified models such as the linear rate model to characterize channel capacity. Our aim is to provide a novel framework to evaluate PFS in Rayleigh fading without the above-mentioned limitations. To put our work on a firm base, we use results of stochastic approximation in the analysis and take the Gaussian approximation for capacity modeling for fading channels. Simulations validate this approach and show that our analytic result provides highly accurate estimate of the PFS throughput. Compared to existing studies, our work advances the state of the art in three ways. First, it goes beyond the linear rate model and applies to the commonly used Shannon rate model. Second, it provides accurate estimate of the PFS throughput without the need for the time-consuming ODE analysis. Third, it provides a unified closed-form expression for estimating the PFS throughput for both the linear rate model and the Shannon rate model. It is interesting to note that our analysis provides the same result as existing studies when assuming the linear rate model. More importantly, our formula is intuitive yet easy to evaluate numerically.