A hemoglobin molecule can carry one oxygen or one carbon monoxide molecule. Suppose that the two types of gases arrive at rates 1 and 2 and attach for an exponential amount of time with rates 3 and 4, respectively. Formulate a Markov chain model with state space {+, $0$, -} where + denotes an attached oxygen molecule, – denotes an attached carbon monoxide molecule, and $0$ a free hemoglobin molecule and find the long-run fraction of time the hemoglobin molecule is in each of its three states.
I'm not sure where to start with this problem and would greatly appreciate any guidance/hints. Thanks in advance!
Best Answer
I assume the following additional condition: upon arrival a oxygen or carbon monoxide molecule may only occupy the hemoglobin molecule if the hemoglobin molecule does not already carry any molecule. (correct me if this is wrong but seems logical)
You can write down the generator as a 3x3 matrix via $$Q = \begin{pmatrix} -3 & 3 & 0 \\ 1 & -3 & 2\\ 0 &4 & -4\end{pmatrix}$$ if the first row/column corresponds to $+$, the second row/column to $0$ and the last row/column to $-$. The transition graph is just a graph with 3 nodes and 4 directed edges (+,0), (0,+), (-,0), (0,-). The transitions between + and - are not possible by my additional assumption. If the hemoglobin is empty it can take one other molecule with one of the rates 1 or 2.
The stationary distribution can then be directly calculated via $\pi Q = 0$. The step from the stationary distribution to the mean time spend in each state is a formula which you should have seen in your course/lecture.