IF YOUR ECM or RELAY HAS FAILED AND YOU'RE ONLY RUNNING ON THE OIL PRESSURE SWITCH:
1 - You won't get the two second "prime" running of the fuel pump to pressurize the system so the engine starts right away.
2 - After some extended cranking, you will build up enough oil pressure to turn on the fuel pump via the oil pressure switch.
3 - Because when your oil is hot, the oil pressure drops - the amount of extended cranking will increase when the engine is at operating temperature.
4 - IF your engine is so worn as to make the HOT oil pressure drop to below the switch threshold at idle AND your fuel pump is running solely on the oil pressure switch, the engine will shut off at a hot idle. The engine can't be restarted while hot since the oil pressure at cranking speed is no longer enough to turn on the switch. Once the engine is cold you will be back to step 1 - which is the engine will start with an extended cranking.
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I am not saying this IS happening with your car, but I am saying it's likely possibility. A way to test this would be to insert a test light into the fuel pump test point in your ALDL connector (with the other end of your test light grounded). Turn on the key but don't start the engine. Watch to see if you get the two second "prime" from the ECM. No prime would indicate a bad ECM or Relay. Then start the engine. The light should come on sometime IF the engine starts. If the light comes on only after extended cranking that would also indicate a bad ECM or Relay. Then run the engine and wait till it's warm and see if the test light goes out before the engine stops turning. If the light goes out and THEN the engine stops this would also indicate that the reason your engine is stopping is caused by the fuel pump no longer getting voltage. One way to test re-start the engine would be to 'hotwire' power to the ALDL fuel pump test position.
ALDL Connector

Be careful - Yours might be upside down from this picture - Pin G is the fuel pump power test point. The ALDL connector is under the panel with the cigarette lighter in it.
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If however you get the light to light during the 2 second "prime" and the light doesn't turn off untill after the engine quits - then the ECM and Relay ARE doing their job and you don't need to look at them as a cause of this trouble. If the light stays lit untill after the engine dies then you also don't have to worry about the oil pressure switch as a cause.
[This message has been edited by phonedawgz (edited 06-20-2010).]