This app models the remaining-life Markov chain corresponding to the geometric distribution with parameter \(p\). When the chain is in state 0, the next state is \( x \in \N \) with probability \( (1 - p) p^x \). When the chain is in state \( x \in \N_+ \), the next state is \( x - 1 \) deterministically. The given geometric distribution is also the limiting distribution of the chain.
The top graph shows the state space, with the current state colored red and the initial state colored blue. As the process runs, the chain moves from state to state. The time and the sate are recorded at each update in the record table. The limiting distribution and the proportion of time that the chain is in each state are shown in the distribution graph and the distribution table. The parameter \(p\) and the initial state \(x_0\) can be varied with the input controls.