Here is a process to which an analogy in a different setting is of interest.
Let $G = (V, E)$ be a graph with $F \subset V$. Let $G$ represent, for example, a road network, and $F$ are refueling stations. A random walk (random drive) on this network is limited to $k$ steps without refueling. After refueling (visiting a node in $F$), the driver can take another $k$ steps.
Apart from road networks, are there any other analogies for such a process? Maybe for example, in social networks rumor/gossip spreading, disease spreading? It seems like there should be one since "realistic" random walks should not be infinite in practice.
A lot of theoretical properties can be proved about this process and I am looking for any literature that has anything remotely connected.