This paper presents a method for allocating transmission rates in a communication network that uses knowledge of the size of the file to be transmitted to seek to obtain the benefits of shortest remaining processing time (SRPT) scheduling. It provides fairness among file transmissions taking into account their sizes and the congestion on the links they use. The proposed approach is based on network utility maximisation (NUM), but with utility functions that vary in time depending on the remaining file size. A parameterized family of utility functions is proposed. For each value of the parameter, there is a threshold load such that an arbitrary network will be stable provided no link load exceeds that threshold. This is in contrast with SRPT, for which any non-zero load becomes unstable for topologies with sufficiently long paths. The method can be implemented by a distributed algorithm analogous to that of traditional NUM.