  This paper presents an extension of Naor's analysis on the join-or-balk
problem in observable M/M/1 queues. While all other Markovian assumptions still
hold, we explore this problem assuming uncertain arrival rates under the
distributionally robust settings. We first study the problem with the classical
moment ambiguity set, where the support, mean, and mean-absolute deviation of
the underlying distribution are known. Next, we extend the model to the
data-driven setting, where decision makers only have access to a finite set of
samples. We develop three optimal joining threshold strategies from the
perspective of an individual customer, a social optimizer, and a revenue
maximizer, such that their respective worst-case expected benefit rates are
maximized. Finally, we compare our findings with Naor's original results and
the traditional sample average approximation scheme.
