  Assignment of one of the two possible directions to every edge of an
undirected graph $G=(V,E)$ is called an orientation of $G$. The resulting
directed graph is denoted by $\overrightarrow{G}$. A strong orientation is one
in which every vertex is reachable from every other vertex via a directed path.
The diameter of $\overrightarrow{G}$, i.e., the maximum distance from one
vertex to another, depends on the particular orientation. The minimum diameter
among all possible orientations is called the oriented diameter
$\overrightarrow{\text{diam}}(G)$ of $G$. Let $n,k$ be two integers with $1
\leq k < n$. In the realm of interconnection networks of processing elements,
an $(n,k)$-star graph $S_{n,k}$ offers a topology that circumvents the lack of
scalability of $n$-star graphs $S_n$. In this paper, we present a strong
orientation for $S_{n,k}$ that combines approaches suggested by Cheng and
Lipman [Journal of Interconnection Networks (2002)] for $S_{n,k}$ with the one
proposed by Fujita [The First International Symposium on Computing and
Networking (CANDAR 2013)] for $S_n$. Next, we propose a distributed routing
algorithm for $\overrightarrow{S_{n,k}}$ inspired by an algorithm proposed by
Kumar, Rajendraprasad and Sudeep [Discrete Applied Mathematics (2021)] for
$\overrightarrow{S_n}$. With the aid of both the orientation scheme and the
routing algorithm, we show that $\overrightarrow{\text{diam}}(S_{n,k}) \leq
\lfloor \frac{n+k}{2} \rfloor + 2k + 6 - \delta(n,k)$ where $\delta(n,k)$ is a
non-negative function. The function $\delta(n,k)$ takes on values $2k-n$, $0$,
and $\left\lfloor \frac{n-3k}{2} \right\rfloor$ respectively for three disjoint
intervals $k>\frac{n}{2}$, $\frac{n}{3} < k \leq \frac{n}{2}$ and $k\leq
\frac{n}{3}$. For every value of $n$, $k$, our upper bound performs better than
all known bounds in literature.
