I second what dlaue wrote above.

Let me add that the LAX Airline Connections (Route A) runs on the inner roadway. Access to the inner roadway is currently limited to airport operated shuttles and the LAX FlyAway Bus. Thus it is not affected by the congestion on the outer roadway which the rest of the airport traffic - including hotel and car rental shuttles - uses (parking shuttles use the upper roadway).

Route A shuttles run every 10 minutes. I don't have first hand experience of how long it takes the bus to travel from TBIT to T1, however, here is a link to the Live Shuttle Tracker (the Red marker) which may give you an idea of how long it takes the shuttle to travel between these two terminals.