This work is motivated by the following observation: modern wireless applications such as content prefetching, allow different extent of delay tolerance, thus in practice users may have different time-averaged data rates during a certain period of time. In this study, the authors consider resource allocation in small cell networks under time-varying channels while each user has an individual time-averaged rate requirement. Stochastic optimisation model is employed to minimise the time-averaged power consumption of small base stations subject to individual user's time-averaged rate constraint. They develop an online power optimal resource allocation (PORA) algorithm to achieve the optimal power allocation and subcarrier assignment decisions without prior knowledge of channel statistics. Furthermore, considering that the power allocation and subcarrier assignment problem is a nonconvex combinatorial problem, they further develop an iterative heuristic algorithm with polynomial complexity. Simulation results show the effectiveness of PORA and verify the theoretical analysis on the network performance.