Hi, Does anyone know what is the purpose of this tiny hose linked to the upper intake manifold ?

I know the PVC and EGR, but don't know what's this. See red circle.
Thanks.

http://img16.imageshack.us/img16/4049/valvec.jpg

normally it goes To the tranmission:

The vacuum modulator system on the transaxle controls shift sensing changes in engine load,by engine vacuum.

but yours dont look right,,what make and model??

02 malibu with 3100 v6 engine.
I just broke this hose, see the duct tape.
It was bundled with electric wires, it's very fragile after years.
Not sure if I need to replace this.

normally it goes To the tranmission:

The vacuum modulator system on the transaxle controls shift sensing changes in engine load,by engine vacuum.

but yours dont look right,,what make and model??

ok then it goes to the air vacuum solenoid valve by the coil pack.

You can either replace it or use rubber tubing over the tube,careful not to break it any further,the duct tape will eventually breakdown and recreate a vacuum leak.