Serial port side of connection hangs

We are using a Digi ConnectPort TS 16 MEI to connect serial ports to ethernet using ssh. We have found that when we connect to a port, then disconnect, the serial device seems to hang, apparently due to a full serial port buffer. Once we reconnect, everything seems to be okay again.
Is there some way to configure the serial port to continue receiving data even though there is no connection on the ethernet side?