No you are fine.
In order to be 100% sure about the psu failing, you really either need to test the 12V rail will a multimeter to check it is reading withing 12V +/-5% normally, and 12V +/-10% under load, so that you can confirm that the PSU is providing the correct output. The alternative is to borrow a powerful, quality psu, which also will make sure that the required output is being met.
If you cannot do this,
then there is an element of risk, that if you replace the PSU, it will not change anything, but i am pretty confident that this will not be the case.
The alienware website refers to your pc as having a "standard" psu, which isn't quite as specific as having a ATX PSU, which is a standard, specifying the outputs (as above) and the size etc. So in summary I would:
1. Purchase a multimeter (can be got cheap ie, 15 euro
http://www.maplin.co.uk/module.aspx?...=25414&doy=4m4)
2. Test the PSU as shown here
http://www.driverheaven.net/guides/testingPSU/ under load (ie when 3DMark06 is running) - please be careful!
.
OR
3. Borrow a quliaty single rail PSU 600W or greater.
If you find that in 2, the voltage on the 12V rail is less than 11.4V or 10.8 (under load) OR that a borrowed PSU (in 3) resolves the issue:
4.
Confirm with Alienware that your case will fit a
standard ATX PSU and then get a new PSU such as:
a. PC Power & Cooling 750 Watt:
http://www.newegg.com/Product/Produc...82E16817256017
b. Corsair TX750W :
http://www.corsair.com/products/tx.aspx