Separation of Circulating Tokens

Self-stabilizing distributed control is often modeled by token abstractions. A system with a single token may implement mutual exclusion; a system with multiple tokens may ensure that immediate neighbors do not simultaneously enjoy a privilege. For a cyber-physical system, tokens may represent physical objects whose movement is controlled. The problem studied in this paper is to ensure that a synchronous system with m circulating tokens has at least d distance between tokens. This problem is first considered in a ring where d is given whilst m and the ring size n are unknown. The protocol solving this problem can be uniform, with all processes running the same program, or it can be non-uniform, with some processes acting only as token relays. The protocol for this first problem is simple, and can be expressed with Petri net formalism. A second problem is to maximize d when m is given, and n is unknown. For the second problem, the paper presents a non-uniform protocol with a single corrective process.