  In their recent work, C. Doerr and Krejca (Transactions on Evolutionary
Computation, 2023) proved upper bounds on the expected runtime of the
randomized local search heuristic on generalized Needle functions. Based on
these upper bounds, they deduce in a not fully rigorous manner a drastic
influence of the needle radius $k$ on the runtime.
  In this short article, we add the missing lower bound necessary to determine
the influence of parameter $k$ on the runtime. To this aim, we derive an exact
description of the expected runtime, which also significantly improves the
upper bound given by C. Doerr and Krejca. We also describe asymptotic estimates
of the expected runtime.
