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A new and flexible solution to the problem of multiple users accessing a single resource, such as communication bandwidth or composite object in memory, is derived. The means of communication consists of sending and receiving messages in known locations (or equivalently, mailboxes without queueing). Any particular user is able to deposit, and hence destroy, previous messages in a mailbox. It is assumed that exclusive access to a mailbox is supplied by an underlying system. The major results of this paper are: 1) a simple tree-based algorithm that guarantees ¿¿ no user or group of users can conspire to prevent access by some other user to the resource; ¿¿ only one user accesses the resource at a time; ¿¿ if there are N users, an individual user is guaranteed access, when requested, to the resource in no more than N-1 turns; Knuth's solution  can delay a user up to 2** (N-1)-1 turns; 2) an extension of Dekker's algorithm (2 users)  that allows the relative rates of reservations for access to the resource to be proportional to a set of N integers. When a reservation is not being used by its ``owner,'' it will be assigned to another contending request. The assignment is optimal for periodic requests.