In recent years, the numbers of requests to transfer huge files via large high-speed computer networks have been increasing rapidly. Typically, these requests are handled in a ``best effort'' manner, resulting in unpredictable completion times. To overcome this problem, some studies have introduced a model where a file transfer request must either be completed by a user-specified deadline or be rejected if the deadline cannot be satisfied. In this model, the request is rejected if the request's deadline cannot be satisfied strictly. However, a little expiration deadline is often acceptable. In this study, I consider a model where the user utility varies toward the transfer time for time constrained file transfer. In this model, the request which cannot meet the deadline is not always rejected, but accepted if the total user utility increases. In this thesis, I assume a utility function describing user's satisfaction with respect to transfer time and propose a bandwidth assignment method for maximizing users' utility over the network. Finally, I show the effectiveness of the proposed method by simulation experiments.