# Expected time of distinguishability of a series of Poisson processes bounded by each other

Consider a system of $n$ "bounded" Poisson processes over the integers, $X_1, \ldots X_n$, all incrementing at rate $\lambda$. Initially all the processes begin at $0$. The process $X_i$ is inactive until $X_{i+1} - X_i > 1$, at which point it begins incrementing itself at rate $\lambda$. Whenever $X_{i+1} - X_i \leq 1$, the process $X_i$ stops, waiting for $X_{i+1}$ to increment itself, before becoming active again. ($X_n$ is always active.)

We can see that eventually every $X_i$ will have a unique value. My question is regarding the expected time $E[\mathcal{T}]$ of the first time this event occurs: what is the first time all processes have a unique value?

A simple bound seems to be $E[\mathcal{T}] = O(n^2)$. However, my simulations indicate that this takes linear time in $n$. I'd love to see an analysis that shows something along the lines of $E[\mathcal{T}] = O(n)$, or any other asymptotic analysis for this system.

The process you are looking at is called TASEP (totally asymmetric simple exclusion process); though your initial conditions are unusual. A more conventional version of your question would be to consider the TASEP with step initial conditions (one particle at every negative integer) and ask for a typical time that the first $n$ particles are at distances at least, say, 2 from each other.
in particular, suggest that the time of interest should indeed scale linearly with $n$; e. g. Theorem 1 say that the typical time before the $n$-th particle moves scales linearly, and the intuition (corroborated by Theorem 2) is that once they all have started moving, they will spread out quickly.