In this paper we study how to design a scheduling strategy aimed at minimizing the average holding cost for flows with general size distribution when the feasible transmission rate of each user varies randomly over time. We employ a Whittle-index-based approach in order to achieve an opportunistic and non-anticipating size-aware scheduling index rule proposal. When the flow size distribution belongs to the Decreasing Hazard Rate class, we propose the so-called Attained Service Potential Improvement index rule, which consists in giving priority to the flows with the highest ratio between the current attained-service-dependent completion probability and the expected potential improvement of this completion probability. We further analyze the performance of the proposed scheduler, concluding that it outperforms well-known opportunistic disciplines.