Dependability state model
# 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 $\lambda$. When failed, they are repaired one at the time by a single repairman with negative exponential distributed repair times with expectation $\mu ^{-1}$.

* 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 $2\lambda$, 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 represent 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, 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.

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

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

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

In addition the equality clause must be taken into account:

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

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:

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

## 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.

