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I don't know if they will "run out"; I do know there is a limited supply available. They are not meant to be placed on the floor, and it appears there are only two bolts for one bassinet in the bulkhead row.

I don't think you can place a travel bed on the floor; you aren't supposed to have anything on the floor if you are seated at the bulkhead. In any case, your child cannot be in a bassinet during take off or landing, or when the seat belt sign is on.

I did find a picture of a bassinet on UA - it's even on flight #895; you can see there's not much leg room left.
http://www.flickr.com/photos/17554153@N00/4084114951/

You must hold your baby during take off and landing. I personally would prefer a baby sling; not sure if a Baby Bjorn would be allowed.

I have a nine hour experience with a lap child and that was the end of that for me. I never did it again; children got their own seats with their car seats installed. Made life much easier.