Suppose you have a distributed system of processes $N$ = $\{p_1, p_2, \dots, p_n\}$, where $|N| = n$. The system progresses in rounds $r_0, r_1,\dots$, where at each round a set of processes a. receives some messages, b. sends some messages, and/or c. performs some local computation.
At $r_0$, each process is initialized with some initial state $\{s_1^0, s_2^0, \dots, s_n^0\}$, where $s_i^j \in \{0, 1\}$. Let $S_0$ denote the collection of states from all nodes at $r_0$ (and, in general, let $S_i$ denote the system-wide state at $r_i$). An process may change its state at a round $r_i$ by collecting information about the state of a random subset of processes, and applying some function $f$ to this subset. The collection of state is performed via the message passing primitive.
For example, at $r_i$, process $p_k$ may change its state like so: $s_k^i = f(s_1^3, s_2^{i-2},s_{3}^{20}, \dots)$. If we were restricted to only selecting from the prior round, this would be a fairly easy Markovian process to model. However, since state can be coming from any past round, I've had quite a bit of trouble pinning down exactly how to approach this. Are there any papers that model this problem definition closely? I have spent a day looking at literature, but the vast array of papers in stochastic modelling doesn't seem to capture this notion of 'transition to the next state could include information from states arbitrarily in the past'.