# Reliability of a multistate item

I have a question and appreciate your time and input to give me some hints. If a system could stay in several states ranked based on the performance of the system such as (state 8 - perfect state, state 7 - good state, 6 - fair state, 5- poor state, and state 4 - severe state). Data are available for the family of this system in each of the states. Assuming the system should pass and stay some time in each state sequentially, state 8, 7, 6, 5, and 4). Is it possible to calculate the probability of failure (POF)( failure here means the transition from state 8 to 4) by multiplying the probability of failure from each state? For example, the POF (the probability that the system would transition to state (i-1) until 10 years)) is 30% from state 8 to 7, 28% from state 7 to 6, 35% from state 6 to 5, 28% from state 5 to 4 is given. Is the POF for the system to transition from state 8 to state 4 equal to (0.3*0.28*0.35*0.28=0.008)? Please provide me with a reference that I could rely on during the discussions.

Yes. I would suggest looking for discrete state space Markov chains.

The following course covers the topic (with much more detail than needed for your question):

http://www.statslab.cam.ac.uk/~rrw1/markov/index.html

But essentially, you need to set up your transition matrix. You can then calculate one of several things: 1) Average time to hit failure state. 2) N-step transition probability to failure state.

• Thanks for your reply. Is it possible to put this in transition matrix and calculate the POF of reaching from state 8 to 4, maybe in 10 years or any new criteria you would like to set. Thanks, Jan 13, 2020 at 17:35

Based on the @conjectures suggestion I prepared the transition matrix and for example, in 7 years (n=7) the transition matrix is Does this matrix say that the probability for a system now in state 8 to reach state 4 in 7 years equal to 9.8% (1st row, column 5) of the matrix? Also, state 3 which is not defined in the original post has a probability of 2.9% (1st row, 6th column). Does it mean the system now in state 8 would reach state 3 in 7 years with a probability of 2.9%? Thanks,

• Yes. More generally these things get calculated as x' P^N, where x is a non-negative vector which sums to 1. x represents the initial state probability and N is the number of transitions. Jan 15, 2020 at 9:40
• Thank you @conjectures. Jan 20, 2020 at 1:10