Railway Interlocking Crossing

What is modelled

Where two lines cross on the level, an interlocking makes sure that only one of the two crossing routes is locked at a time. A train that arrives while the other route is locked waits at a red signal. Each train holds the crossing from the moment its route is locked until it has cleared the crossing and the route is released, so the crossing behaves like a single server with a fixed service time and random arrivals. The driver must also be able to stop at a red signal within the distance from which it can be seen.

Equations

• Service time s = (L + d) / v + tl, with L the train length, d the clearing distance and tl the route setting and release time.
• Load ρ = λ s; capacity 3600 / s trains per hour.
• Mean wait (M/D/1) Wq = ρ s / (2 (1 − ρ)); Little's law Lq = λ Wq.
• Stopping distance v tr + v2 / (2a).

Checks run in the page

• A simulation of 400,000 random arrivals against the M/D/1 mean wait (or growth of the wait when the load is above 1).
• The same run shows no overlap of two locked routes.
• Little's law from the integrated queue length.
• The braking distance stepped in time against the formula, compared with the sighting distance.

Honest simplifications

Trains arrive at random at a steady average rate, are served strictly in the order of arrival, cross at a constant speed and queue without blocking other lines, the whole crossing is one lock, and only one conflict is modelled. Real interlockings also check points, flank protection and overlaps. Use it for the trends, not for signalling design.

• Press Busy crossing and watch the queue build and the load approach 1.
• Press Slow release, unstable queue, then shorten the release time.
• Press Sighting too short to see the braking distance exceed the sighting distance.