# Introduction to Continuous Time Markov Chains

- Let \(\{X(t),t\ge0\}\) be a continuous time Markov chain with state space \(S=\{1,2,3,4\}\) and transition rate matrix \(Q\):

\[ \mathbf{Q} = \begin{bmatrix} & 1 & 0 & 1\\ 2 & & 2 & 0\\ 0 & 3 & & 3\\ 4 & 0 & 4 & \\ \end{bmatrix} \]

- Find the diagonal entries of \(\mathbf{Q}\).
- Explain in full detail how you could simulate the process for a long time using only (1) a coin, and (2) an Exponential(1) spinner.

- A system is composed of \(5\) machines. A machine operates for an Exponentially distributed amount of time with rate \(\mu=1\) and then fails. When a machine fails it undergoes repair; repair times are Exponential distributed with rate \(\lambda=2\). Let \(X(t)\) represent the number of machines operating at time \(t\); then \(\{X(t)\}\) is a CTMC. Find the rate matrix of the CTMC.