  The $k$-Server Problem covers plenty of resource allocation scenarios, and
several variations have been studied extensively for decades. We present a
model generalizing the $k$-Server Problem by preferences of the requests, where
the servers are not identical and requests can express which specific servers
should serve them. In our model, requests can either be answered by any server
(general requests) or by a specific one (specific requests). If only general
requests appear, the instance is one of the original $k$-Server Problem, and a
lower bound for the competitive ratio of $k$ applies. If only specific requests
appear, a solution with a competitive ratio of $1$ becomes trivial. We show
that if both kinds of requests appear, the lower bound raises to $2k-1$.
  We study deterministic online algorithms and present two algorithms for
uniform metrics. The first one has a competitive ratio dependent on the
frequency of specific requests. It achieves a worst-case competitive ratio of
$3k-2$ while it is optimal when only general requests appear or when specific
requests dominate the input sequence. The second has a worst-case competitive
ratio of $2k+14$. For the first algorithm, we show a lower bound of $3k-2$,
while the second algorithm has a lower bound of $2k-1$ when only general
requests appear. The two algorithms differ in only one behavioral rule that
significantly influences the competitive ratio. We show that there is a
trade-off between performing well against instances of the $k$-Server Problem
and mixed instances based on the rule. Additionally, no deterministic online
algorithm can be optimal for both kinds of instances simultaneously.
  Regarding non-uniform metrics, we present an adaption of the Double Coverage
algorithm for $2$ servers on the line achieving a competitive ratio of $6$, and
an adaption of the Work-Function-Algorithm achieving a competitive ratio of
$4k$.
