Abstract: Mutual exclusion is a well-known problem that arise when multiple processes compete, in an uncoordinated way, for the acquisition of shared resources over a distributed system. In particular, k-mutual exclusion allows at most k processes to get one unit of the same resource simultaneously. These paradigms do not cover all the cases in which resource accesses must be serialized over a distributed system. There exist cases (e.g. the bandwidth of communication lines) where the amount of shared resource might differ from request to request (for example, audio and video communications). In this paper, we formalize this problem as the {h-out of-k mutual exclusion problem}, in which each request concerns some number h $(1 \le h \le k)$ of units of shared resource and no unit is allocated to multiple processes at the same time. We present a general scheme for a {\em quorum-based h-out of-k mutual exclusion algorithm} that relies on a collection of quorums called {\em $k$-arbiter}. Several examples of $k$-arbiters are discussed, two particular classes of k-arbiters are investigated and a metric to evaluate the resiliency with respect to failures of k-arbiters is also given.