Question

4. The structural deterioration of a stormwater pipe is modeled using a Markov Chains based model. The transition probability matrix (T) for this model is given below. The row matrix below (s) provides the probabilities of being in different condition states at t=0 for a particular pipe Calculate the probability of this pipe to be in each one of these condition states after three transition periods. 0.90 0.07 0.03 0 0 1 0 0.85 0.10 0.05 0 T= 0 0 0.70 0.25 0.05 10 0 0 0.60 0.40 Lo 0 0 0 11 S = [0.85 0.15 0 0 0]

Answer 1

so this question is based upon the concept of Markov chain method of counting probability.

let us understand the complete process by the images that I have uploaded.

Ans. I Mankow's chain Based model deterioration of stormwater Pipe for structural The Row matrix (s) provides different state at to probability of being at for this problem is The transition probability matrix given as matrix (T) We have each one to calculate of these probability of state after 3 this pipe to be in transition. Soln Let us calculate Probability to be each one these state after I transition for that, we n ed to multiply matot (s) to matrix (T) 9.90 0.07 0.03 0 0 3= 10.85 0.15 o o o T = o 0.85 010 oos o 0 0 0.70 0.25 0.05 0 0 0 0.60 0.40 Lo o o o t 3 = Prob to be each one of these state after 1 transition = [3] x [I] (multiply Row of the column of T) 30, Si = 0.765 · 187 0.0405.0075 o

further for 2nd and 3rd transition

Similarly, Probability to be in each one of these state after and transition = [s2] will be [3.] x [T. S, = [31]*[1] = 365.187.040.0075 0] .90 .07.03 o o lo 85 10 05 o 10 o 70 .25 0.05 0 0 0 .60 0.40 S = 2:56.2125132615 0.02397 ·OOSO Similonly, Sz = [s2] x [ty] o Sg= 1:4192 21945.1062303084 Hence, probability of each one condition after three of to be in these transition are state [53] 53] = 0.6192 O B 0.2287 0.26945 0.2097 0.26945 0.106 29 0103084 Answer

See, it was truly conceptual and hence I have tried to be as short as possible so that I don't crowd the solution with words. Hope you will like it and it helps you. Don't forget to upvote if you liked it.