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Markov Analysis

Markov Analysis. Jørn Vatn NTNU. Introduction. Markov analysis is used to model systems which have many different states These states range from “perfect function” to a total fault state The migration between the different states may often be described by a so-called Markov-model

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Markov Analysis

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  1. Markov Analysis Jørn Vatn NTNU

  2. Introduction • Markov analysis is used to model systems which have many different states • These states range from “perfect function” to a total fault state • The migration between the different states may often be described by a so-called Markov-model • The possible transitions between the states may further be described by a Markov diagram

  3. Purpose • Markov analysis is well suited for deciding reliability characteristics of a system • Especially the method is well suited for small systems with complicated maintenance strategies • In a Markov analysis the following topics will be of interest • Estimating the average time the system is in each state. These numbers might further form a basis for economic considerations. • Estimating how frequent the system in average “visits” the various states. This information might further be used to estimate the need for spare parts, and maintenance personnel. • Estimate the mean time until the system enters one specific state, for example a critical state.

  4. Markov Analysis procedure • Make a sketch of the system • Define the system states • Draw the Markov diagram with the transition rates • Quantitative assessment • Compilation and presentation of the result from the analysis

  5. Make a sketch of the system • Pump system wit active pump and a spare pump in standby

  6. Definition of system states • x1 = state of active pump • x2 = state of standby pump

  7. State transitions • For this system we have assumed that if the active pump fails, the standby pump could always be started • Further we assume that if both pumps have failed, they will both be repaired before the system is put into service again • The following transition rates are defined 1 = failure rate of the active pump 2 = failure rate of the standby pump (while running, 2 = 0 in standby position) 1 = repair rate of the active pump (1/1 = Mean Down Time when the active pump has failed) B = repair rate when both pumps are in a fault state. I.e. we assume that if the active pump has failed, and a repair with repair rate 1 is started, one will ”start over again” with repair rate B, if the standby pump also fails, independent of “how much” have been repaired on the active pump.

  8. Markov state space diagram • The circles represent the system states, and the arrows represent the transition rates between the different system states • The Markov diagram and the description of states represent the total qualitative description of the system

  9. Quantitative assessment • We want to assess the following quantities • Average time the system remain in the various system states • The visiting frequencies to each system state

  10. Transition matrix • The indexing starts on 0, and moves to r, e.g. there are r +1 system states • Each cell in the matrix has two indexes,where the first (row index) represent the ”from” state, whereas the second (column index) represent the “to” state. • The cells represent transition rates from one state to another • aijis thus the transition rate from state i to state j • The diagonal elements are a kind of ”dummy”-elements, which are filled in at the end, and shall fulfil the condition that all cells in a row adds up to zero

  11. Example transition matrix: (From , To )

  12. State probabilities • Let Pi(t) represent the probability that the system is in state i at time t • Now introduce vector notation, i.e. • P(t) = [P0(t), P1(t),…,Pr(t)] • From the definition of the matrix diagram it might be shown that the Markov state equations are given by:P(t)A = dP(t)/dt • These equations may be used to establish both the steady state probabilities, and the time dependent solution

  13. Steady state probabilities • Let the vector P = [P0, P1,…,Pr] represent the average time the system is in the various system states in the long time run • For example, P0 is average fraction of the time the system is in state 0, P1 is average fraction of the time the system is in state 1 • The elements P = [P0, P1,…,Pr] are also denoted steady state probabilities to indicate that in the stationary situation Pi represents the probability that the system is in state i.

  14. The steady state solution • In the long run when the system has stabilized we must have that dP(t)/dt = 0, hence • PA = 0 • This system of equations is over-determined, hence we may delete one column, and replace it with the fact thatP0+ P1+…+Pr = 1 • Hence, we have

  15. The steady state solution PA1 = b where and b = [0,0, …,0,1]

  16. Example which gives

  17. Numerical solution • To solve the steady state equations PA1 = b is a tedious task • Often we therefore solve these equations by numerical methods • The Markov.xls program does this, where we have to: • Define the transition rates • Assign numerical values to the transition rates • Specify the Markov state space matrix

  18. Visiting frequencies • Often we are interested in evaluating how many times the system enters the various states, i.e. the visiting frequencies • The visiting frequency for state j is denoted j, and could be obtained by: • j = -Pjajj • From our example we obtain the “system failure rate”

  19. Time dependent solution • Up to now we have investigated the steady state situation • In some situations we also want to investigate the time dependent solution, i.e. the probability that the system is in e.g. state 0 at time t • We now let Pi(t) be the probability that the system is in state i at time t • The time dependent solution may be found by: • P(t)A = dP(t)/dt • Which could be solved by Laplace methods, or numerical methods • For numerical methods we apply Markov.xls

  20. Spare parts – Simple model • Assume a spare part regime where there is only one spare in the stock • Upon a demand (with demand rate ) a new spare is ordered • The intensity of arrival of a new spare is  = 1/ MTA (Mean Time to Arrival)

  21. Spare parts 2 spares in stock • Assume a spare part regime wheretherearetwo spares in thestock • Upon a demand (withdemand rate ) a new spare is ordered • The intensityofarrivalof a new spares is  = 1/ MTA (Mean Time to Arrival) independentofhowmany in order

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