I am working through the book "Big queues", and I do not know how to proceed with one of the exercises in this book.
Let $(A_t^{(i)}, t \in \mathbb{Z})$ be a two-state Markov chain representing a source which produces an amount of work $h$ in each timestep while in the on state and no work in the off state, and which flips from on to off with probability $p$ and from off to on with probability $q$. Show that $$\Lambda_t(\theta) = \log \left( \frac{q}{q+p}E_t + \frac{p}{q+p}F_t\right)$$ where $$\left( \begin{matrix} E_t\\ F_t \end{matrix} \right) = \left( \begin{matrix} (1-p)e^{\theta h} & p \\ qe^{\theta h} & 1-q \end{matrix} \right)^t \times \left( \begin{matrix} 1\\1 \end{matrix} \right)$$
I do not know how to proceed - I have written down the transition probability matrix, $$A = \left( \begin{matrix} p & 1-p \\ 1-q & q \end{matrix} \right)$$ and I know that the log mgf is supposed to be something like $$\Lambda_t (\theta) = \log E e^{\theta A_t}$$ but I don't know how to move on from here.
I would appreciate any hints and suggestions. Thank you.