Polarized light microscopy - Revision history

en>NewEnglandYankee: Reverted edits by 139.80.98.130 (talk) to last version by ClueBot NG

2014-06-03T14:42:52Z

Reverted edits by 139.80.98.130 (talk) to last version by ClueBot NG

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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~~
	~~\| title = Dependable Computing Systems and Communication Networks~~
	~~\| quote =~~
	~~\| publisher = Gnist Tapir~~
	~~\| year = 2007~~
	~~\| pages =~~
	~~\| url =~~
	~~\| doi =~~
	~~}}</ref>~~
	~~[[File:Dep-state-model.png\|thumb\|A simple state model with two states]]~~

	~~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 ==~~

	~~[[File:Dep-state-model-example.png\|thumb\|Example FSM with two working states and one failed]]~~

	~~A redundant computer system consist of identical two compute nodes, which each~~
	~~fail with an intensity of <math>\lambda</math>~~. ~~When failed, they~~ are ~~repaired one~~
	~~at the time by a single repairman with negative exponential distributed~~
	~~repair times with expection <math>\mu^{-1}</math>.~~

	* 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 <math>2\lambda</math>, since each~~
	~~compute node has a failure intensity of <math>\lambda</math>. Intensity from~~
	~~state 1 to state 2 is <math>\lambda</math>~~.
	~~Transitions from state 2 to state 1 and state 1 to state 0 represents~~
	~~the repairs of the compute nodes and have the intensity~~ <~~math~~>~~\mu~~<~~/math~~> ~~since only a single unit is repaired at the time.~~

	~~=== Availability ===~~

	~~The asymptotic~~ [~~[availability]], i.e. availability over a long time~~
	~~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.~~

	: ~~<math>\mathbf{A_0} = \begin{bmatrix}~~
	~~0 & -\mu & 0 \\~~
	~~-\lambda & 0 & -\mu \\~~
	~~0 & \lambda & 0~~
	~~\end{bmatrix}.<~~/~~math>~~

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

	~~: <math>\mathbf{A_1} = \begin{bmatrix}~~
	~~(\lambda) & -\mu & 0 \\~~
	~~-\lambda & (\lambda+\mu) & -\mu \\~~
	~~0 & -\lambda & (\mu) \\~~
	~~\end{bmatrix}.<~~/~~math>~~

	~~In addition the equality clause must be taken into account:~~

	~~:<math> \sum_n P_n = 1~~.</~~math>~~

	~~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. remove all outgoing state transitions.~~

	~~For this system the function is:~~

	~~: <math>~~
	~~R(t) = e^{-\lambda t} \,~~
	</~~math>~~

	~~== Criticism ==~~

	~~Finite state models of systems are subject to [[State explosion problem\|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 ==~~
	~~<references/>~~

	~~[[Category:Reliability engineering]]~~
	~~[[Category:Markov models]]~~
	~~[[Category:Graphical models]~~]

en>ZZninepluralZalpha: /* The Michel-Levy Chart */ Corrected multiplication symbols; not commenting (now) on correctness

2013-04-04T23:10:30Z

The Michel-Levy Chart: Corrected multiplication symbols; not commenting (now) on correctness

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en>SchreiberBike: Repairing links to disambiguation pages - You can help! - Interference

2012-02-10T04:38:19Z

Repairing links to disambiguation pages - You can help! - Interference

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