The online Dial-a-Ride problem is a fundamental online problem in a metric space, where transportation requests appear over time and may be served in any order by a single server with unit speed. Restricted to the real line, online Dial-a-Ride captures natural problems like controlling a personal elevator. Tight results in terms of competitive ratios are known for the general setting and for online TSP on the line (where the source and target of each request coincide). In contrast, online Dial-a-Ride on the line has resisted tight analysis so far, even though it is a very natural online problem. We conduct a tight competitive analysis of the Smartstart algorithm that gave the best known results for the general, metric case. In particular, our analysis yields a new upper bound of 2.94 for open, nonpreemptive online Dial-a-Ride on the line, which improves the previous bound of 3.41 [S. O. Krumke, “Online Optimization Competitive Analysis and Beyond,” Habilitation thesis, Technische Universität Berlin, 2001]. The best known lower bound remains 2.04 [A. Bjelde et al., in Proceedings of the 28th Annual Symposium on Discrete Algorithms (SODA), SIAM, 2017, pp. 994--1005]. We also show that the known upper bound of 2 [N. Ascheuer, S. O. Krumke, and J. Rambau, in Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS), Springer, 2000, pp. 639--650] regarding Smartstart's competitive ratio for closed, nonpreemptive online Dial-a-Ride is tight on the line.

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.