Recent publications prove that runtime systems oriented to the Bulk Synchronous Parallel Model usually achieve remarkable accuracy in their predictions. That accuracy can be seen in the capacity of the software for packing the messages generated during the superstep and their capability to find a rearrangement of the messages sent at the end of the superstep. Unfortunately, barrier synchronisation imposes some limits both in the range of available algorithms and in their performance. The asynchronous nature of many MPI/PVM programs makes their expression difficult or infeasible using a BSP oriented library. Through the generalisation of the concept of superstep we propose two extensions of the BSP model: the BSP Without Barriers (BSPWB) and the Message Passing Machine (MPM) models. These new models are oriented to MPI/PVM parallel programming. The parameters of the models and their quality are evaluated on four standard parallel platforms. The use of these BSP extensions is illustrated using the Fast Fourier Transform and the Parallel Sorting by Regular Sampling algorithms. Copyright © 1999 John Wiley & Sons, Ltd.