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{\bf Question}
A simple random walk has the infinite set $(a,\ a-1,\ a-2,\
\cdots)$ as possible states. State $a$ is an upper reflecting
barrier, for which reflection is certain, i.e., if the random walk
is in state $a$ at step $n$ then it will be in state $a-1$ at step
$n+1$. For all other states, transitions of $+1,\ -1,\ 0$ take
place with probabilities $p,\ q,\ 1-p-q$ respectively.
Let $p_{j,\ k}^{(n)}$ denote the probability that the random walk
is in state $k$ at step $n$, having starter in state $j$. Obtain
difference equations relating to these probabilities, for $k=a,\
k=a-1$ and $k<1-1$.
Assuming that there is a long-term equilibrium distribution
$\pi_k)$, where
$$\pi_k=\lim_{n\to \infty}p_{j,\ k}^{(n)}\ \rm{for}\ j=a,\ a-1,\
a-2,\ \cdots\ ,$$
use the difference equations for $p_{j,\ k}^{(n)}$ to obtain a set
of difference equations for $\pi_k$ for $k