(FORTUNE Magazine) – At the heart of American Airlines is one of the world's largest privately owned computer networks. Powered by seven massive IBM mainframes in an underground concrete bunker outside Tulsa, the system has been a crucial factor in American's spectacular rise. It is called Sabre (Semi-Automated Business Research Environment), and it was conceived in 1959 as an internal link between the airline's offices and ticket counters to help track reservations. In 1975 talks between travel agents and the major carriers aimed at developing an industrywide reservation system broke down. Bob Crandall and Max D. Hopper, a data-processing manager, seized the moment to break away and begin marketing Sabre to outsiders. Though American has had difficulty cracking protected markets in Europe, today more than 85,500 Sabre terminals are in use at travel agencies in 47 countries. The system provides fares and schedules for 665 airlines, as well as information on prices and availability for more than 20,000 hotels and 52 rental car companies. Sabre is a huge, if hidden, moneymaker for parent AMR Corp. Analysts figure the company earns a higher return on investment by booking tickets than it does by flying airplanes. Along with its central role in reservations and ticketing, Sabre serves several key internal functions: -- Since deregulation opened up pricing competition in 1978, the big U.S. and foreign carriers continually adjust fares to match their rivals and ensure the highest yield possible for each seat sold. Sabre maintains a pricing profile on each American flight; as many as eight different fare classes may be available on a single trip. When fare wars break out, nearly 1.5 million new industry fares are loaded into Sabre every day. -- Among Crandall's shrewdest marketing innovations has been the American Advantage frequent-flier program. Largely because Sabre automatically calculates and records mileage, the program has attracted more than 11 million members and built up strong brand loyalty. While some other airlines now do the same for their frequent fliers, American was way out in front in offering this service. -- The system also calculates flight plans, each aircraft's weight and balance, fuel requirements, and takeoff power settings for about 2,300 American flights each day. -- Sabre maintains inventory control on nearly one billion spare parts distributed among American's maintenance centers, handles all aircraft and crew scheduling, and controls all baggage routing and freight tracking. The airline's frenetic growth is already testing the limits of Sabre's capacity. Says Hopper, 55, now senior vice president for information systems: ''As we expand, our problems do not grow in a linear way. They seem to explode sometimes.'' A slightly rumpled fellow, Hopper has a professorial air that contrasts sharply with the hard-chargers around headquarters. But if American continues to fulfill Bob Crandall's expansionist dreams, he knows that new hardware, such as a minisupercomputer, may be the only answer.