THE IMMIGRATION and Naturalization Service is an agency in trouble. Public attention understandably has focused on the extensive and complex problem of illegal immigration and the horrendous headaches of the understaffed Border Patrol.But those are just the beginning of INS's current woes. Responsibilities that might seem easier, such as processing routine papers and answering public inquiries, have also overwhelmed the agency. According to Rep. Joshua Ellberg (D-Pa.), chairman of the House immigration subcommittee, a legal alien now has to wait more than a year for a decision on a change-of-status request. An applicant for naturalization has to wait more than six months. A person with a simple question has to spend hours standing in line at major offices or days trying to get through by phone.

How did the agency slide into such a state? It's hard to put down the suspicion that inadequate manangement has been partly to blame. That would help explain, for instance, why INS's records systems are so archaic and clogged that transferring a file within one office may consume three days and shipping a file to another office may take three weeks. INS officials naturally prefer to emphasize that their work-load has been outrunning their resources for 10 or 15 years. Last year, for example, applications for naturalization increase 17 per cent over 1975, while legislated changes in immigration policies for the Western Hemisphere piled new burdens on the agency. Congress did provide funds for 750 new positions in fiscal 1976 and 641 more last year. However, because of executive-branch constraints, only about one-third of last year's increase has materialized so far.

The new commissioner, Leonel J. Castillo, seems determined to shake up the agency. He has just launched a crash program to clean up the huge backlog to over 146,000 immigration applications in six major offices. INS says that, at a "normal" pace, some of that work might take four years - which suggests that the agency's notion of "normal" productivity could use an overhaul.

Even if Mr. Castillo can achieve management miracles, INS still needs more resources. The House this month approved the Carter administration's request for 144 more employees for INS - and added 600 on top of that. Around 500 of those 744, however, would be assigned to border inspections, deportations and patrols.The Senate might well consider changing that allocation a bit. The problems on the borders, however important, are so vast that 500 more agents can make only a modest difference. Assigning a few more to office work, on the other hand, could substantially improve service to legal aliens and citizens, and also bolster INS's morale. The effective control of millions of illegal immigrants may be too much to ask of an agency that can't even answer its phones.