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It also falls down on its not just time slots, but the relative speeds of the services, the number and location of stops and the time taken to embark/disembark at each station dependent upon the type of train and number of passengers on that service. It's also dependent upon the number of lines e.g. 2 pairs or lines (one fast one slow), just one or bi-directional, and the signalling layout (can only ever run one train per block).
Some people suggest that this is exactly what should happen. The model would be the same as airlines where (as I understand it) airlines bid for departure slots and it's then on them to make their service as competitive as possible to attract custom and make profit. The claim is that what you end up with is a price war and better and better deals for customers.
I guess where the analogy falls down is that trains are a necessary public service in a way that air connections are not. Therefore you need someone to run services on routes that aren't profitable, and to work out how to incentivise them to do that.