Defines and handles segments in messages to place pauses and interruptions within the communication of a message between transmitted segments of the message. A common link switch is used in a network to connect links to all nodes, the segment structures in each message is preserved when packets of each message are passed within the switch to a switch transmitter connected to the destination node indicated in each packet of the message for transmitting each of the message segments. Each transmitter stores the source identifier of the first packet it transmits for a segment and then gives priority to transmitting packets which contain source and destination identifiers which match the current transmitter stored source identifier and match the destination node connected to the transmitter. This priority enables each switch transmitter to interleaves segments of concurrent messages while preserving the segmentation of transmitted packets to maintaining a maximum network communication rate for the messages. When an unexpected wait occurs within a transmitting segment, which exceeds a predetermined time-out period, a transmission of any other waiting segment is started, which improves the message transmission efficiency in the network.