The city has come up with a plan to clear the massive backlog of repair orders in public housing buildings. Mayor Michael Bloomberg and New York City Housing Authority Chairman John Rhea announced a new system today aimed at clearing a backlog of 420,000 repair requests by the end of the year. In some NYCHA buildings, residents have waited years for repairs, but agency officials hope to only have about 50,000 remaining repair orders by January 2014. The new system would speed up repairs by making sure the correct type of repair person gets assigned to jobs in the right order. City officials said they are paying for the program with $40 million in administrative savings.