In order to fire a bow you need to Ready the bow with a Simple Action, Ready the arrow with another Simple Action, and then Fire Weapon with a Simple Action - generally this is enough to turn people off to archery altogether (or walk around with a bow out and arrow nocked at all times). Enter Arsenal's Martial Arts.
For 5 BP you can pick up one rank in Krav Maga and take the Advantage that allows you to change Ready Weapon to a Free Action (Arsenal, page 157). Additionally, this allows you to take Maneuvers.
QUOTE (Arsenal, page 160, Iaijutsu)
A character with this maneuver may use the Quick Draw rules (p. 127, SR4) to draw any weapon, not just pistols. The character may draw and use a single melee weapon, missile weapon, throwing weapon or firearm of Reach 1 or less with a Simple Action. If attacking with the weapon requires a Complex Action, he may still draw and attack with a Complex Action. He must succeed in an appropriate Weapon Skill + Reaction (3) Test to Quick Draw.
This is what makes it all come together. Now we can have a character with a bow stashed, use Krav Maga to Ready the bow as a Free Action, and then use Quick Draw (either the arrows or the bow - it is unclear which one you need here but it is clear that it is possible) to get the same rate of fire with a bow that can be achieved with a semi-auto firearm. When you further take into account that a Strength 8 bow is a base DV of 10 which can be increased to 15 DV with net successes (and possibly 16 with Exlosive Arrowheads from Arsenal - the bow rules, including prices for arrows, were tweaked in SR4A but did not include any information about tweaking those found in Arsenal) this allows the bow retake its place among its more contemporary ranged weapons.
This means an archer can now fire up to eight times in a Combat Turn, pumping out up to 15P DV per shot. Add to this MRSI software and you can pump out up to 30 DV attacks in one Initiative Pass. At another extreme is a humanoid drone with a rigger that has both Sinsense Accelerator and Simsense Booster getting these same results at 5 IP.
tl:dr = bows are still very competitive.