Dependability state model

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

It consist 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 fail with an intensity of $\lambda$. When failed, they are repaired one at the time by a single repairman with negative exponential distributed repair times with expection $\mu ^{-1}$.

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

Availability

The asymptotic availability, ie 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.

Reliability

The reliability of the system is found by making the failure states absorbing, ie. remove all outgoing state transitions.

For this system the function is: $R(t)=e^{-(\lambda *t)}$

Critism

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 infeasable to solve or draw the model.