It was an age-old tradition at the Massachusetts Bay Transportation Authority: Four times a year, subway and bus drivers would arrive at their garages to find the walls plastered in sheets of paper, filled with the times and dates for thousands of shifts over the coming months.

Drivers, in order of seniority, had to fill each one of the time slots in. In all, the process could take up to eight weeks as driver after driver came to mark their shifts.

But now that the MBTA and its largest union, the Boston Carmen’s Union, have reached a new collective bargaining deal, the antiquated system is poised to come to an end. The union, which represents about 4,100 workers — many of them drivers who used the system — has agreed to try new scheduling software that cost the MBTA about $430,000.

“The electronic roster picking has been a huge priority for us, because we want to build bus service that works for our customers,’’ Transportation Secretary Stephanie Pollack said last week, as the agency prepared to brief the public on a new contract.

Brian Shortsleeve, the MBTA’s acting general manager, said making the system more efficient has been a leading priority.

“It’ll enable us to more rapidly change routes and modify routes in reaction to demand,’’ he said. “Most importantly, we’re going to save millions of dollars and drive productivity.’’

Computerizing the schedule sounds like an obvious move in this day and age, but the union had resisted the change for years. At one point, the union brought the issue to an independent arbitrator, who allowed the union to keep the system. (The Boston Carmen’s Union didn’t respond to questions for this story, but president James O’Brien has said the union is eager to “be part of the solution,’’ in general, for MBTA issues.)

For MBTA executives, who have been eager to cast Governor Charlie Baker and his administration as long-awaited reformers for an embattled system, the switch looks like a win. It not only deals a blow to a powerful union, but also saves money.

MBTA officials have pointed out that paper scheduling allowed some drivers to choose shifts that violated a federal rule prohibiting them from working unless they’ve had 10 hours between shifts.

In some cases, the MBTA had to pay some drivers to stay home during those shifts and pay another driver to fill in. Officials estimate the MBTA could save about $2.6 million over four years avoiding such problems.

The new system will also allow workers to select shifts through the software program, reducing the need for oversight at an expected savings of about $4.7 million over four years.

The change will take time, though. Implementing the system will take at least a year, although officials say they will be testing it in their garages starting in 2017.

Holiday schedules

Want to take advantage of the MBTA during your holiday revelry? Check the schedules first.

On Christmas Day, Dec. 25, and Monday, Dec. 26, everything will operate on a Sunday schedule. That means people who are going to work Monday will see fewer buses, trains, and trolleys.

But on New Year’s Eve, it’s a different story. First, the best news: You can ride free after 8 p.m.

That night, the MBTA will be running subway trains more frequently than on a typical Saturday, to accommodate the droves of people coming in and out of the city. Subway lines will start on a Saturday schedule, then switch to rush-hour levels of service between 3 p.m. and 2 a.m. Commuter rail lines will also run more often.

It’s tougher luck if you want to take the bus for your party plans, though. On New Year’s Eve, the buses will run on a regular Saturday schedule, which means several lines won’t run.

On New Year’s Day and Jan. 2, be prepared for less service again. They’ll be running a Sunday schedule on buses and trains.

You can find more details at http://www.mbta.com/events.

Nicole Dungca can be reached at nicole.dungca@globe.com. Follow her on Twitter @ndungca or on Facebook at facebook.com/startsandstops.