Artikel: Dependability state model GudangMovies21 Rebahinxxi

    • Source: Dependability state model

      • A dependability state diagram is a method for modelling a system as a Markov chain. It is used in reliability engineering for availability and reliability analysis.

      It consists of creating a finite-state machine which represent the different
      states a system may be in. Transitions between states happen as a result of events from underlying Poisson processes with different intensities.


      Example



      A redundant computer system consist of identical two-compute nodes, which each fail with an intensity of



      λ


      {\displaystyle \lambda }

      . When failed, they are repaired one at the time by a single repairman with negative exponential distributed repair times with expectation




      μ

      −
      1




      {\displaystyle \mu ^{-1}}

      .

      state 0: 0 failed units, normal state of the system.
      state 1: 1 failed unit, system operational.
      state 2: 2 failed units. system not operational.
      Intensities from state 0 and state 1 are



      2
      λ


      {\displaystyle 2\lambda }

      , since each compute node has a failure intensity of



      λ


      {\displaystyle \lambda }

      . Intensity from state 1 to state 2 is



      λ


      {\displaystyle \lambda }

      .
      Transitions from state 2 to state 1 and state 1 to state 0 represent the repairs of the compute nodes and have the intensity



      μ


      {\displaystyle \mu }

      , since only a single unit is repaired at the time.


      = Availability

      =
      The asymptotic availability, i.e. availability over a long period, of the system is equal to the probability that the model is in state 1 or state 2.
      This is calculated by making a set of linear equations of the state transition and solving the linear system.
      The matrix is constructed with a row for each state. In a row, the intensity into the state is set in the column with the same index, with a negative term.






      A

      0



      =


      [



      0


      −
      μ


      0




      −
      λ


      0


      −
      μ




      0


      λ


      0



      ]


      .


      {\displaystyle \mathbf {A_{0}} ={\begin{bmatrix}0&-\mu &0\\-\lambda &0&-\mu \\0&\lambda &0\end{bmatrix}}.}


      The identities cells balance the sum of their column to 0:






      A

      1



      =


      [



      (
      λ
      )


      −
      μ


      0




      −
      λ


      (
      λ
      +
      μ
      )


      −
      μ




      0


      −
      λ


      (
      μ
      )



      ]


      .


      {\displaystyle \mathbf {A_{1}} ={\begin{bmatrix}(\lambda )&-\mu &0\\-\lambda &(\lambda +\mu )&-\mu \\0&-\lambda &(\mu )\\\end{bmatrix}}.}


      In addition the equality clause must be taken into account:





      ∑

      n



      P

      n


      =
      1.


      {\displaystyle \sum _{n}P_{n}=1.}


      By solving this equation, the probability of being in state 1 or state 2 can be found, which
      is equal to the long-term availability of the service.


      = Reliability

      =
      The reliability of the system is found by making the failure states absorbing, i.e. removing all outgoing state transitions.
      For this system the function is:




      R
      (
      t
      )
      =

      e

      −
      λ
      t





      {\displaystyle R(t)=e^{-\lambda t}\,}



      Criticism


      Finite state models of systems are subject to state explosion. To create
      a realistic model of a system one ends up with a model with so many states that it is infeasible to solve or draw the model.


      References

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