arXiv is running out of IDs -- is their solution known?
The format presumed in the question is incorrect. The format is YYMM.NNNN, rather than YYYY.NNNN. Shortening the year to YY and including the month as MM gives a factor of 12 increase in the number of available IDs. Version numbers are appended to the ID, as v1, v2, and so on. That said, it is still a problem, but arxiv have thought about it.


  NNNN is a zero-padded sequence number starting at 0001 and permitting up to 9999 submissions per month. If current growth rates
  continue, we expect to change the sequence number to 5-digits NNNNN in
  10 to 15 years. We will do this in a uniform fashion so that, likely
  starting on some year boundary, all subsequent identifiers are
  zero-padded to 5-digits. We cannot currently anticipate extension
  beyond that although extension to 6-digits would be possible.


This was posted in 2007 so we are 6 or so years into their 10-15 year window. Submissions of multiple versions might grant some extra headroom, however it does appear that we could run into problems in about two years. It looks like they underestimated their growth a little bit!



Status change in 2015. arXiv.org switched to 5-digit IDs in January 2015. The format is now YYMM.NNNNN or YYMM.NNNNNvV.