There exists an absorbing $(M \times M)$Markov chain with the following transition matrix:
$ \begin{array}{ccccccc} p_{11} & p_{12}&0&\cdots&\cdots&\cdots&0\\ p_{21} & p_{22}&p_{23}&0&\cdots&\cdots&0\\ 0 & p_{32}&p_{33}&p_{34}&0&\cdots&0\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&0\\ 0&0&0&0&p_{M-1,M-2}&p_{M-1,M-1}&p_{M-1,M}\\ 0&0&0&0&0&0&1 \end{array} $
I'd like to find $m_{1M}$, the expected first hitting time of the absorbing state given that the first state is $1$. In a simple case I'd use the system of linear equations, of course: $ \begin{eqnarray} &m_{1M}=1+p_{11}m_{1M}+p_{12}m_{2M} \\ m_{2M}&=1+p_{21}m_{1M}+p_{22}m_{2M} +p_{33}m_{3M} \end{eqnarray} $ and so on. Of course, this case is more complicated, and this approach is intractable.
I'll be grateful for the suggestions on how to find the approximation/upper bound on $m_{1M}$, becuase I'm quite sure the exact expression is probably impossible to find.