Markov Diagrams

We estimate the following probabilities per month to move between states:

• 1% chance to degrade from 100% to 80% capacity
• 10% chance to be restored from 80% to 100% capacity
• 3% chance to degrade from 80% to 60% capacity
• 8% chance to be restored from 60% to 100% capacity
• 6% chance to degrade from 60% to 40% capacity
• 5% chance to be restored from 40% to 100% capacity
• 8% chance to degrade from 40% capacity to salvage
  
  

Based on these percentages, the final diagram that is ready for analysis looks like this:

Since our estimated probabilities are on a month scale, we will take each step of the analysis to be the equivalent of one month. This means that we will run our calculation for 120 steps. After we calculate the diagram, we can see that the transition probability matrix between the states looks like this (which we can easily use to verify our inputs):

Full Diagram
FROM -> TO	100% capacity	80% capacity	60% capacity	40% capacity	Salvage
100% capacity	0.99	0.01	0	0	0
80% capacity	0.1	0.87	0.03	0	0
60% capacity	0.08	0	0.86	0.06	0
40% capacity	0.05	0	0	0.87	0.08
Salvage	0	0	0	0	1

We can use the state point probability plot to see if our system has reached steady state within our time frame.

In this case study example, because we have a "sink" state, we do not reach steady state, where all the probabilities have reached a constant value, but rather a pseudo-steady state where the probabilities are changing at a roughly constant rate.

 

Afterwards, we can check the results summary to determine the mean probabilities in each state and the point probabilities after 120 steps (10 years).

Results After 120 Steps
State name	Initial probability	Mean probability	Point probability	Steps spent in state
100% capacity	1	0.894127	0.859252	107.295203
80% capacity	0	0.064845	0.066382	7.781451
60% capacity	0	0.013046	0.014282	1.565469
40% capacity	0	0.005597	0.00662	0.671615
Salvage	0	0.022386	0.053464	2.686261