Modeling Lambda Tau Events
Descriptions follow for how Lambda Tau models calculate results.
• Constant probability. When availability equals the constant probability value that you enter for Probability.
• Lambda Tau. When selected, unavailability is approximated. This model is applicable for both repairable and non-repairable events. Under the Maintenance properties heading, for Mission Time/MTTR, you enter a mission time for a non-repairable event or the MTTR (mean time to repair) for a repairable event. You then select the appropriate units.
◦ If the event is non-repairable, τ represents the mission time, which in this situation is a short duration. When mission time is short, the approximate unreliability is equivalent to λτ because:
Unreliability = 1 - exp(−λτ) ≈ λτ
◦ If the event is repairable, τ represents the MTTR. If τ is small and the mission time is relatively large or unknown, the approximate unavailability is equivalent to the steady state unavailability. Also, because MTTR is small, in this case, the following approximation can be made:
Unreliability = MTTR / (MTTR + MTTF) ≈ MTTR / MTTF
OR:
Unreliability = λτ since MTTF = 1 / λ
• Average unavailability (approx). When selected, the average unavailability is approximated. This model is applicable when the event undergoes regular preventive maintenances, particularly if the event is tested regularly. The test interval is assumed to be very short in comparison to the failure time. The actual test time is assumed to be negligible. In this model, τ represents the test interval, which is equivalent to the time between consecutive tests. Under the Maintenance properties heading, you enter a value for Time Between Tests and select the appropriate units.
The event will be unavailable if it has failed prior to the test time. The unavailable duration may vary from 0 to τ, depending on the actual failure time. If you consider that on average the failure will occur at the middle of the test interval, then the failure time can be considered as τ /2. Then:
Unavailability = λτ / 2
◦ Probability of failure. When selected, the probability of failure is calculated. This model is applicable for non-repairable events. In this model, τ represents the mission time. Under the Mission properties heading, you enter a value for Mission Time and select the appropriate units.
The Lambda Tau calculations used in this model produce exact results. These calculations are applicable even for large mission times. Because the event is non-repairable, the unavailability is equivalent to unreliability. Therefore:
Unavailability = Unreliability = 1 − exp(-λτ)
• Asymptotic unavailability. When selected, the asymptotic unavailability is calculated. This model is applicable for repairable events, where the mission time is very large. In this model, τ represents MTTR. Under the Maintenance properties heading, you enter a MTTR value for Mission Time/MTTR and select the appropriate units.
For these events, the system is considered to be in a steady state, where the asymptotic unavailability is an appropriate measure. The asymptotic unavailability is equivalent to:
Unavailability = MTTR / (MTTR + MTTF) = λτ / (λτ + 1)
• Average Unavailability. When selected, average unavailability is exactly calculated. Like the Average unavailability (approximate results) model, this model is applicable when the event undergoes regular preventive maintenances, particularly if the event is tested regularly. The actual test time is assumed to be negligible. In this model, τ represents the test interval, which is equivalent to the time between consecutive tests. Under the Maintenance properties heading, you enter a value for Time Between Tests and select the appropriate units.
The event will be unavailable if it is failed prior to the test time. The unavailable duration can vary from 0 to τ, depending on the actual failure time. If the mission time is unknown or very large, the appropriate measure for the unavailability is the average unavailability of the event during the test interval. The Lambda Tau calculations used in this model produce exact results for the average unavailability of the event during a test interval, which is equivalent to:
Unavailability = 1 - (1 - exp(−λτ)] / λτ