I want to stream video of 1080p resolution to my consoles (2x Xbox 360 and 1x PlayStation 3) and I want to stream it simultaneously.
What is the minimum requirement of processor and RAM for that heavy work?
How exactly are you streaming it?
I would imagine that the biggest bottleneck would be your home network, rather than your computer hardware. Is it wireless, or is it connected up via ethernet?
Also, do you mean that you wish to stream simultaneously to 3 consoles? It's very difficult to say what exactly you need (and I have no experience in streaming 3 feeds all at once), but I think it might be fair to suggest at least a decent quad-core processor and at least 4GB of RAM.
Also, I would imagine that the video card installed might have the biggest impact. I think that most of today's GPU's have video acceleration built in to encode the images. I have no real idea of the minimum specs you should be looking for, but to be on the safe side I'd suggest looking at recent mid-range video cards.
Don't make any decision until someone else can contribute to this answer though, as all I've given you is an educated guess.
The bottleneck here would most likely be your home network, not your computer. Streaming video shouldn't use your processor much unless you're encoding it in real time.