There is a system which follows the equation: $X(n+1) = X(n) - 1 + A(n)$, where $A(n)$ is a random variable taking values $0,1,2$ with probabilities $p_0$, $p_1$ and $p_2$, respectively. Now, $X(n) = n$ and $X(n)\geq1$ are two provided conditions. I need to find the relation between $p_0$, $p_1$ and $p_2$ which ensures stability of the Markov Chain.
Answer is $p_0>p_2$, but I can't figure out the way to proceed.
Just for reference, this is actual a model of packet arrivals in a queueing network. Basically, $n$ is an index of discrete time, and $X(n) = n$ implies that the number of packets that arrived at time $n$ equals $n$ itself.
My progress: I tried to derive the steady state matrix, but am not getting $p_0>p_2$ as the answer. In fact, I don't understand how an inequality like this can be obtained.