When may a train run in advance of its schedule?

A train may run in advance of its schedule in 251 and 261 territory because these specific areas have regulations that allow train movements based on operational needs and traffic conditions. The rules in these territories are designed to facilitate the efficient flow of trains and can permit adjustments to schedules under certain circumstances, enhancing overall service reliability and punctuality.

In contrast, the other options are not conditions under which a train may advance its schedule. The stipulation of weekends, peak hours, or holiday schedules does not inherently provide the logistical framework to authorize a train to run ahead of its scheduled time; instead, they focus on the timing of service rather than operational flexibility. Therefore, the correct understanding of how and where trains can run in advance of their schedules centers around the specific regulations applicable in 251 and 261 territory.