Dependability state model

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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. [1]

Markov chain mathematical system

A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event.

Reliability engineering is a sub-discipline of systems engineering that emphasizes dependability in the lifecycle management of a product. Dependability, or reliability, describes the ability of a system or component to function under stated conditions for a specified period of time. Reliability is closely related to availability, which is typically described as the ability of a component or system to function at a specified moment or interval of time.

Contents

A simple state model with two states Dep-state-model.png
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

Example FSM with two working states and one failed Dep-state-model-example.png
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 . When failed, they are repaired one at the time by a single repairman with negative exponential distributed repair times with expectation .

Intensities from state 0 and state 1 are , since each compute node has a failure intensity of . Intensity from state 1 to state 2 is . Transitions from state 2 to state 1 and state 1 to state 0 represent the repairs of the compute nodes and have the intensity , 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.

In reliability theory and reliability engineering, the term availability has the following meanings:

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.

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

In addition the equality clause must be taken into account:

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:

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.

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References

  1. Bjarne E. Helvik (2007). Dependable Computing Systems and Communication Networks. Gnist Tapir.