  We prove a sample path large deviation principle (LDP) with sub-linear speed
for unbounded functionals of certain Markov chains induced by the Lindley
recursion. The LDP holds in the Skorokhod space $\mathbb{D}[0,T]$ equipped with
the $M_1'$ topology. Our technique hinges on a suitable decomposition of the
Markov chain in terms of regeneration cycles. Each regeneration cycle denotes
the area accumulated during the busy period of the reflected random walk. We
prove a large deviation principle for the area under the busy period of the
MRW, and we show that it exhibits a heavy-tailed behavior.
