  We consider the online traveling salesman problem on the real line (OLTSPL)
in which a salesman begins at the origin, traveling at no faster than unit
speed along the real line, and wants to serve a sequence of requests, arriving
online over time on the real line and return to the origin as quickly as
possible. The problem has been widely investigated for more than two decades,
but was just optimally solved by a deterministic algorithm with a competitive
ratio of $(9+\sqrt{17})/8$, reported in~[Bjelde A. et al., in Proc. SODA 2017,
pp.994--1005].
  In this study we present lower bounds and upper bounds for randomized
algorithms in the OLTSPL. Precisely, we show, for the first time, that a simple
randomized \emph{zealous} algorithm can improve the optimal deterministic
algorithm. Here an algorithm is called zealous if waiting strategies are not
allowed to use for the salesman as long as there are unserved requests.
Moreover, we incorporate a natural waiting scheme into the randomized
algorithm, which can even achieve the lower bound we propose for any randomized
algorithms, and thus it is optimal. We also consider randomized algorithms
against a \emph{fair} adversary, i.e. an adversary with restricted power that
requires the salesman to move within the convex hull of the origin and the
requests released so far. The randomized non-zealous algorithm can outperform
the optimal deterministic algorithm against the fair adversary as well.
