Does not make sense

Ok here is the information my girlfriend gave me on her car issue! Its a 01 impala 3.8 ltr with 140K. She states at time it will not start, by not start i mean no cranking of any kind [/U]or any noise. This can last for up to 30 mins before it will start. She states if she pushes the gas pedal while trying to start it when it wont start it will start. This baffles me since it wont do anything until she gives it gas. It has a new battery, ignition switch and I checked the starter connections.

Please give me ideas

Pushing on the pedal is coincidental, does the security lite stay on when this happens, double check the battery cables, if you can, see if there is 12v to the starter solenoid when this happens.