Consider a process that is evolving through a timeline, starting at $t=0$. The process can be described by two types of events (Type-I and Type-II) occurring one after the another (Type-I Type-II Type-I Type-II Type-I Type-II ...). The process starts with Type-I event.
Time duration of the events Type-I and Type-II follow the continuous distributions $f(t)$ and $g(t)$ respectively. Let us divide the timeline into unit time intervals. At the beginning of each time-slot, the process is observed. It may be assumed, if required, that both events have memoryless property.
I need to find the probability of finding the process in Type-II event at the beginning of a slot, given that the process was in Type-I event at the start of the previous slot.
I came up with this idea when I was trying to formulate a problem for a recreational math competition. My first approach was to use Laplace transform, and turn convolution to product form. I cannot quite solve it. Any help will be much appreciated. Thanks.
If the two types of events always alternate, and can be memoryless, this can be modeled as a two-state continuous-time Markov chain with transition rate matrix
$$Q=\left[ \begin{array}{c c} -\lambda & \lambda\\ \mu & -\mu\\ \end{array} \right],$$
where $\lambda$ and $\mu$ are the parameters of the two respective distributions. For such a process it is possible to work out the transition probability function in closed form by calculating $P\left(t\right) = e^{tQ}$ where
$$e^{tQ} = \sum^{\infty}_{n=0} \frac{1}{n!}\left(tQ\right)^n$$
is the matrix exponential (the computation can be done by diagonalizing $Q$). Then, $P_{ij}\left(t\right) = P\left(X_t = j\mid X_0 =i\right)$ for any $t > 0$.