Abstract: To ensure delivery of inter-process messages at the same priority level at which they were sent in a multiprocessor (20), each message (200) includes a field (201) that indicates the priority of the sending process. When a message is received on a processor, a message-delivery function (22) determines whether the priority indicated by the message's field is lower than the priority of a process presently running on the processor. If not, the message is immediately delivered to the destination process; if so, the message is stored in the one queue of a plurality of queues (301-303) on the processor that corresponds to priority indicated by the message's field. Upon change of running process on the processor, the message-delivery function causes delivery to destination processes of messages stored in queues of the processor that correspond to priorities at least as high as the priority of the about-to-be-run process.