When traffic of connectionless best effort protocols such as IP is carried over connection oriented protocols with guaranteed bandwidth, such as CBR connection in ATM, the interface layer between the protocols (i.e., AAL-the ATM adaption layer) needs to specify the bandwidth requirement and the duration of the bandwidth reservation. The purpose of this paper is to develop policies for deciding and for adjusting the amount of bandwidth requested for a best effort connection over such networks. Our aim is to develop such policies that achieve a good trade off between latency and utilization. The performances of the different policies are compared by an empirical evaluation.