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| 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.<ref>{{cite book
| | Im Ima and was born on 22 September 1988. My hobbies are Golfing and Auto racing.<br><br>Also visit my webpage [http://besiki.mx/profile/phferk.html Women mountain bike sizing.] |
| | author = Bjarne E. Helvik
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| | title = Dependable Computing Systems and Communication Networks
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| | quote =
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| | publisher = Gnist Tapir
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| | year = 2007
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| | pages =
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| | url =
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| | doi =
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| }}</ref>
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| [[File:Dep-state-model.png|thumb|A simple state model with two states]]
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| It consists of creating a [[finite state machine]] which represent the different
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| states a system may be in. Transitions between states happen as a result of events from underlying Poisson processes with different intensities.
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| == Example ==
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| [[File:Dep-state-model-example.png|thumb|Example FSM with two working states and one failed]]
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| A redundant computer system consist of identical two compute nodes, which each
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| fail with an intensity of <math>\lambda</math>. When failed, they are repaired one
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| at the time by a single repairman with negative exponential distributed
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| repair times with expection <math>\mu^{-1}</math>.
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| * state 0: 0 failed units, normal state of the system.
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| * state 1: 1 failed unit, system operational.
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| * state 2: 2 failed units. system not operational.
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| Intensities from state 0 and state 1 are <math>2\lambda</math>, since each
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| compute node has a failure intensity of <math>\lambda</math>. Intensity from
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| state 1 to state 2 is <math>\lambda</math>.
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| Transitions from state 2 to state 1 and state 1 to state 0 represents
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| the repairs of the compute nodes and have the intensity <math>\mu</math> since only a single unit is repaired at the time.
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| === Availability ===
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| The asymptotic [[availability]], i.e. availability over a long time
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| period, of the system is equal to the probability that the model
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| is in state 1 or state 2.
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| This is calculated by making a set of linear equations of the
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| state transition and solving the linear system.
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| The matrix is constructed with a row for each state. In a row the
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| intensity into the state is set in the column with the same index, with
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| a negative term.
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| : <math>\mathbf{A_0} = \begin{bmatrix} | |
| 0 & -\mu & 0 \\
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| -\lambda & 0 & -\mu \\
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| 0 & \lambda & 0
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| \end{bmatrix}.</math>
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| The identities cells balance the sum of their column to 0:
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| : <math>\mathbf{A_1} = \begin{bmatrix}
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| (\lambda) & -\mu & 0 \\
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| -\lambda & (\lambda+\mu) & -\mu \\
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| 0 & -\lambda & (\mu) \\
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| \end{bmatrix}.</math>
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| In addition the equality clause must be taken into account:
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| :<math> \sum_n P_n = 1.</math>
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| By solving this equation the probability of being in state 1 or state 2 can be found, which
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| is equal to the long term availability of the service.
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| === Reliability ===
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| The reliability of the system is found by making the failure states absorbing, i.e. remove all outgoing state transitions.
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| For this system the function is:
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| : <math>
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| R(t) = e^{-\lambda t} \,
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| </math>
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| == Criticism ==
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| Finite state models of systems are subject to [[State explosion problem|state explosion]]. To create
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| 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.
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| == References ==
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| <references/>
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| [[Category:Reliability engineering]]
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| [[Category:Markov models]]
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| [[Category:Graphical models]]
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Im Ima and was born on 22 September 1988. My hobbies are Golfing and Auto racing.
Also visit my webpage Women mountain bike sizing.