In response to Albot's reply:
After the first week or so of each month there's generally not much of a "pile" remaining; and, of course, in winter it's about wide-open.
If the month is already "open" (ie. Feb. - June right now) they will generally tell you, if asked, if the day(s)/zone(s) that you're wanting is currently available or not. Then, I've found, it's generally a slam dunk if you fax in the permit request right away.
The first week or so of every month they are inundated with faxed requests that they make their way through. By about the 8th-9th the've mostly finished the "rush" and things calm down til the next month.
The BCO guys/gals are some of the nicest, hard working folks at the Park IMO.