On-rail competition is perhaps the most far-reaching form of deregulation of the railways, giving travellers several options on a single line. It aims to lower fairs and raise quality of service, thereby boosting demand and social welfare. Concerns have been raised, however, regarding if effective competition is possible on such a market, allowing two or more operators to be profitable and eliminating through incentives or regulation the purchase by one operator of the others' access rights, thus restoring monopoly. In addition, the effect of competition on total welfare is unclear. The issue of how to regulate the market and conduct capacity allocation in order to maximise welfare is also as yet unanswered. Addressing these issues, the present paper studies a duopoly market through simulations. It builds on the hypothesis that competition occurs between trains with close departure times. Results indicate that total welfare increases significantly when going from profit-maximising monopoly to competition, as consumers make large gains while operators' profits fall. The way the regulator allocates departure slots has significant importance for market outcomes, including prices, frequencies and total welfare. In particular, it is possible to improve welfare by regulating the succession of departures. If trading in access rights is allowed, a would-be monopolist has incentives to buy its competitors' slots for a price they would accept. A monopolist that uses high frequency of departures as a deterrence strategy against competition increases frequency a lot compared to the profit-maximising level.