Mutual exclusion in arbitrary networks when process identity numbers are not distinct

A study of the problem of the mutual exclusion of critical sections of several distributed processes is presented. Each process is given with a particular section of code called its critical section. The problem is to design protocol to ensure that, at a given time, no more than one process is executing its own critical section. A distributed algorithm for mutual exclusion working in arbitrary networks is proposed. This algorithm does not use the logical clock, and the identities of nodes are not necessarily distinct. Initially, each process knows only its adjacent channels and knows them by local identities. The network is organized as a logical rooted tree where the root holds the token. When a process wants to enter the critical section, it sends a request message over an outgoing channel toward the root and waits for the token. The number of messages required for every request is bounded by 2d where d is the diameter of the network