I am trying to better understand the steps in constructing a transition
probabililty matrix from a worked example in a study text. The example actually makes use of a software program to do the calculations, I would like to build a matrix to make sure I understand what is happening.

I am particularly interested in converting the transistion data into the
probabilities to populate the matrix.

The problem is as follows;

A system comprises two pumps, A & B arranged in parallel. Pump A has
a MTBF of 900hr and a MTTR of 100hr. Pump B has a MTBF of 800hr and a
MTTR of 200hr.
There are 4 potential states:
State 1 - A & B Operating.
State 2 - A failed and being repaired, B running.
State 3 - A running, B failed and being repaired.
State 4 - Both A & B failed.
It is assumed there are no transitions between states 2 & 3.

Can anyone help?

What is the end result that you want to achieve?

Are you trying to calculate Probability (Event X) = P(X) = ? Where X is a combination of Pump A & B in running and/or failed state.

P(A) = 0.????
P(B) = 0.????
P(C) = 0.????
P(D) = 0.????
P(A) + P(B) + P(C) + P(D) = 1.0000

The bottom line is that I wish to determine the probabilities of the system being in each of the 4 possible states after the first time interval. If I construct a transition probability matrix correctly and do a matrix multiplication of P^n I should then be able to determine the probabilities after n time intervals and ultimately the limiting probability of each state.

This all depends on correctly filling in each row of the 4 x 4 matrix (I have four possible states). Each row and column represents a state with the sum of each row equalling unity. I have to calculate the 16 possible transistion probabilities from the MTTR and MTBF data. Some probabilities can equal zero as the system conditions prevent transistions between states 2 & 3.

..1 2 3 4
1
2
3
4

I hope this makes some kind of sense.

This message has been edited. Last edited by: Ron Churchill,

Unless there is some time step here that works, this doesn’t seem to have the Markov property of depending upon only the most recent time step.

If the pump is in maintenance it depends upon how long it has been in maintenance for the transition probabilities and not just that it is in maintenance.

Have you tried reliaSofts Blocksim6 programme. You can evaluate acrive and inactive systems incl warm standby and include differnt repair times. AND its simple to use and low cost. You can download the programme for evaluation. Let me know if you want more info.

Hello Ron,

Recently made a calculation myself.
See the attachment for a example.

1) Calculate availbility and unavailability per pump
2) make a thruth tabel from all possible system states (think binary).
3 Fill in for every pump in case of up the availability in and in case of down the unavailability
4)Multiply the probabilitys per row and you have the probability the system can be in.

Regards Rogier

Event_probability_example.xls (14 Kb, 27 downloads) Calculation example Probability Event

Why system probability of both pumps down is the highest?

MTBF MTTR Availability Unavailability
Pump A 900 100 0.9 0.1
Pump B 800 200 0.8 0.2

0 = pump up
1 = pump down


Pump A Prob. 1 PumpB Prob. 2 Sys prob
0 0.1 0 0.2 2.0%
0 0.1 1 0.8 8.0%
1 0.9 0 0.2 18.0%
1 0.9 1 0.8 72.0%
100.0%

Josh, I don't understand your question.
The system probability of having both pumps down is the lowest i.e. 2%

Are referring to this line:
0 0.1 0 0.2 2.0%

In that case, your notation should be 0 = pump down and 1 = pump up. Typo error?

I still don't see any error.

system states are:
A B
0 0
0 1
1 0
1 1

probability of both pump A & B Down = Uavail. Pump A * unavail. Pump B
Unavail. pump A = 1 -(900/900+100) = 0,1
Unavail. pump B = 1 -(800/800+200) = 0,2

Probability A & B down = 0.1 * 0.2 = 0.02 = 2%

Ok, what does 0 and 1 stand for? 0 = pump up or down?

Finaly I see the typo error.
Must be indeed be
0 = pump down
1 = pump up
Thanks for beeing sharp.