Dependability state model

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

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

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.

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

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

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

In addition the equality clause must be taken into account:

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

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.

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:

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

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.