The sketch below is a 'phase-plane' plot, same as used in the solution to the Trail Braking puzzle.  We observe a train-in-trail approaching a station at cruise speed vC.  As a practical matter, trains cannot be operated in stations at full cruise speed vC.  Imagine the danger to passengers waiting on the platform when a train comes barreling into the station at, say, 80 miles per hour.

Accordingly, wayside signage or the train control system at Point 1 will command the train to slow down. At Point 2, the train has slowed to vP , "platform operating speed," let's call it.  Upon reaching Point 3 the train will be stabilized at vP, ready to enter the station platform.

Prior to the arrival of the train-in-trail at Point 3, the train control system will have routinely determined whether the station platform is occupied by the previous train.  The time-table, naturally, will be designed to assure that the platform will not be occupied by the train ahead, so the train-in-trail will generally be allowed to continue to into the platform at vP all the way to Point 4a.

Positioning a train with precision in a station platform is more technically demanding than full service braking between stations.  For one thing, passengers standing in the aisles preparing to disembark are susceptible to sudden changes in brake-rate ("jerk" is the technical name for it) at the instant a train comes to rest.  Limiting jerk calls for what is called "the feathered stop."  Beginning at Point 4a, deceleration must gradually be reduced as the train pulls into the platform.  The feathered stopping distance xF0 brings the train to a halt with its lead car at Point 5.

In the sketch above, we now shift our attention to Point 1', location of the rear-most extreme of a train as it is berthed in the platform.  When tD has elapsed, the doors are closed and the train is dispatched to the next station.  The train accelerates at a constant rate aP to vP, the platform operating speed, at Point 2'.  As the tail car of the train reaches the far end of the platform at Point 3', the train is released to continue accelerating from vP to cruise speed vC, which occurs well clear of the platform at Point 4'.

Prior to the arrival of the train-in-trail at Point 3, the train control system has made a decision: If the station platform is occupied by any part of the train ahead, full service braking will be commanded at Point 3.  The train-in-trail will begin decelerating -- not from vC but from vP.  That can be accomplished at Point 4b, which allows for a buffer distance xB.  Here we note that xB defines an allowance for variations in brake-rate.  After stopping, the train-in trail must wait with its lead car at Point 4b until the tail car of the train ahead reaches Point 3', after which the train-in-trail will be released to accelerate back up to vP for entry into the station platform.

Each of the distances shown in the sketch and their respective time intervals may be derived as follows:

For estimating minimum headway, then, we focus our analysis on Point 3 and Point 3', postulating their occurences at the same time -- at the same time, but for different trains.  Suppose that when the lead car of the train-in-trail reaches Point 3, the tail car of the train ahead is just reaching Point 3'.  The question then becomes, how much time after that does it take for...

[a] the lead car on the train-in-trail to proceed at vP beyond Point 3,
[b] to execute a feathered stop at Point 5,
[c] to wait out its dwell time tD, and finally for...
[d] the tail car of the train-in-trail to reach Point 3'?

Exclamation Point Alert: For the base-case values used in the Trail Braking puzzle, the minimum headway imposed by a Station Stop is more than 92 seconds, which is more than twice as long as the headway at crusing speed along trackways and tunnels -- actually independent of cruise speed!

1. Headway is most sensitive to Platform Operating Speed vP -- in this case, faster is faster -- and may be improved best by the application of platform doors, as seen on elevators and people-movers.
2. The feathered Platform Stop deserves design attention, particularly if platform doors are applied to enable higher platform operating speeds.
3. Acceleration has a significant impact on headway, which confirms an advantage of all-wheel, electric propulsion systems over locomotive-per-consist trains (typically limited to < 1 mphps).
4. Buffer Distance depends on precision in service braking.  Using a value larger than that in the base-case should be avoided.
5. Parameters such as Train Length and Dwell Time are not likely to be variable in a train control system design but are included for completeness in the sensitivity analysis.