  We consider the open, non-preemptive online Dial-a-Ride problem on the real
line, where transportation requests appear over time and need to be served by a
single server. We give a lower bound of 2.0585 on the competitive ratio, which
is the first bound that strictly separates online Dial-a-Ride on the line from
online TSP on the line in terms of competitive analysis, and is the best
currently known lower bound even for general metric spaces. On the other hand,
we present an algorithm that improves the best known upper bound from 2.9377 to
2.6662. The analysis of our algorithm is tight.
