Last week, Jonathan and Jennifer Hart boarded an Amtrak "luxury" train in Chicago for a trip across the country on the television show "Hart to Hart."

One piece of information missing during the hour mystery show was how they got their reservations. Did they get through on the phone? Did they go to the station and stand in line?

Despite the installation of a new $56 million computerized reservations system on Nov. 1, travel agents and would-be Amtrak passengers continue to report that they are having difficulty reaching Amtrak on the phone to book reservations. Bookings are needed on a number of Amtrak's trains, including the Washington-New York Metroliner and long-distance trains.

Amtrak officials admit there has been a "rough start" with the new system; they acknowledge that busy signals and long periods on "hold" are still common. But they contend that there are operational improvements each day as the reservations agents become more familiar with the new system. The trends are up, Amtrak officials say:

* The percentage of calls "lost" because of busy signals is decreasing. Amtrak was losing between 45 percent and 50 percent of attempted calls in September and October under the old system, a level that rose to 63 percent during the first few days the new system was in use. It dropped to 46 percent early this week.

* Calls are taking less time. Once a lucky caller got through to a reservations agent, each call used to average five or six minutes. During the first few days of the new system, an average call took 10 1/2 minutes; early this week, it was down to five minutes.

* There has been a dramatic change in equipment reliability. It was not unusual for the old computer system to be "down," or unavail able to the reservations agents, for two to three hours a day. The first day the new system was in operation, the outtages were down to a total of an hour; now they are down to about 10 minutes a day.

* The number of questions asked the computer -- and answered -- has gone up significantly, from between 300,000 and 350,000 a day to more than 500,000 a day and "still climbing."

"It's brand new technology... and we've got to do on-the-job training," an Amtrak spokesman said "We knew we could have two to three weeks of pretty tough going, he said. He promised that the reservations system should be in top running order by Thanksgiving.

It better be. Describing the new reservations system to the American Council of Railroad Women just before it was put in, Amtrak President Alan Boyd said, "We have a vice president of computer services who has bet his job that it will work."