Boundedness of expected reward Markov chain
It is not bounded sorry for the inconvenience. Here is the proof a friend gave me (thanks a lot to him): 

Let consider the set $W_1 = \{w = (s,w_1,\dots)| w_1 \leq n\}$. Then $\mathcal{E}^n_n$ is bigger than $\displaystyle\sum_{w\in W_1}O^n_n(w)*P(w)$.

\begin{align}
\sum_{w\in W_1}O^n_n(w)*P(w) &amp;= \sum_{k=0}^n \sum_{i=0}^k {n\choose i}(1-p)^i p^{n-i}q^{k-i}(1-q) * k \\
&amp;= (1-q) \sum_{i=0}^n \sum_{k=i}^n {n\choose i}(1-p)^i p^{n-i}q^{k-i}*k\\
&amp;= (1-q) \sum_{i=0}^n  {n\choose i}(1-p)^i p^{n-i}\sum_{k=i}^n q^{k-i}*k\\
&amp;\geq (1-q) \sum_{i=1}^n  {n\choose i}(1-p)^i p^{n-i}i \\
&amp;\geq (1-q) \sum_{i=0}^{n-1} {n\choose i+1}(1-p)^{i+1} p^{n-1-i}(i+1) \\
&amp;\geq (1-q) n (1-p) \sum_{i=0}^{n-1} {n-1 \choose i}(1-p)^i p^{n-1-i} \\
&amp;\geq (1-q) n (1-p)
\end{align}