Beware: Availability can drop below steady state!

Most system availability analyses relate to steady state i.e. the availability that the system converges to as operation time increases. Standard equations for calculating availability relate to the steady state (e.g. EN 61078).
Point availability is the expected system availability at a specific time point. While steady state is a good indicator for the system behavior, it is important to ask:

Are there cases where the system point availability drops below the steady state availability?
The surprising answer is YES!

This means that a steady state analysis may provide an over optimistic result!

We used BQR’s RBD Monte Carlo simulation software to simulate various systems and identify cases where point availability drops below steady state:

For simplicity a constant repair rate is assumed for all the following examples.

Block with non exponential failure distribution:

A single block with a Weibull failure distribution can have a point availability that drops below the steady state:

K out of N with cold repair:

Cold repair is a case where a failed block cannot begin restoration while the system operates. K or more block out of N are required for the system to operate. Restoration begins only when the system fails i.e. N-K+1 blocks fail. During restoration the system is stopped, the K-1 blocks that did not fail, are in standby mode. When all failed blocks are repaired the system goes back to operation.
In this case blocks’ operation and restoration is not independent. A block operation may be stopped due to failure of a neighbor block.
The correlation between the blocks can create a situation where the point availability drops below steady state even when the blocks have a constant failure rate.
The following figure presents a 2 out of 3 model where the child blocks have a constant failure rate:

Each child block in the K out of N model can be found in one of four states:

• Operating
• Standby (block is not failed but is stopped due to failure of neighboring blocks)
• Failed (block failed but restoration did not yet begin)
• Restoration

The following figure presents the probability of being in each state for a child block of the K out of N system from Fig. 2:

Standby with cold repair:

Standby means that a primary sub-system operates while a backup is not operating. When the primary unit fails, the backup kicks into operation.

Standby with cold repair means that repair only begins when both the primary and backup failed.

The primary and backup units are correlated, and this can create a situation where the point availability drops below steady state even when the units have a constant failure rate.
The following figure presents a standby model where the child blocks have a constant failure rate:

Each child block in the standby model can be found in one of four states (similar to the K out of N model with cold repair).

The following figures presents the probability of being in each state for the primary and backup units:

Figures 5 and 6 show the different behavior of the primary and standby units.

The primary unit begins in operation mode whereas the standby begins in standby mode.

Conclusions

There are cases where point availability drops below the steady state.

Monte Carlo simulation can be used to identify these cases and prevent an over optimistic availability analysis.