What do the blue/black and yellowish/black vacuum lines do?

[LemonHunter 583]

And could the deteriorated state of the yellowish one be to blame for slow rough starts?

2001 e39 540i

[aja540i 1903]

Supply vacuum to open the valve for the secondary air pump, which pumps extra air through the exhaust on start up to warm the cats up faster, would affect idle if they were the cause of a vacuum leak. Pretty easy to replace as a precaution.

[LemonHunter 583]

Replaced them from the solenoid to the plenum and boom, starts straight away no missing... i can't beleive i waited so long to look in to this, $11 fix.

[aja540i 1903]

Result!