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They are there to facilitate the payment of fares before a passenger without a ticket reaches the exit barriers. They are also there to sell tickets arising from a wide range of other situations. I'm sorry that I can't quite see the significance of this question, so I may be missing your point, but I cannot see how it is relevant to your incidentof travelling without a valid ticket and after you had not attempted to buy another ticket (or an excess) from that window.
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