It depends how the PCM is programmed but on yours, the PCM activates the fuel pump relay for 2 to 3 seconds when the ignition is initially turn to the RUN position. If the PCM does not receive data from the crankshaft position sensor during that time, the PCM deactivates the fuel pump relay. The PCM activates the fuel pump relay continuously when the ignition is in the START position, regardless of crankshaft position sensor data. When the ignition is initially turned to the OFF position, the relay stays activated for 2 to 3 seconds, presumably to prime for the next start. Rapidly cycling the ignition between off and run, without giving the PCM a 5 second "time out" between cycles, may cause erratic relay activation.

 

Did you ever solve the problem regarding the fuel pump? I have a 2001 Blazer doing the same thing.