Combining Reliabilities (Without Repair)
The general principles described in the foregoing sections apply only to systems that are not maintained (i.e., those which cannot be restored to a failure-free condition if they fail during any part of their operational duty cycle).
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Operational duty cycles are explained on Operational Duty Cycle.
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For a repairable system, the methods of computing the probability ( ) that the system functions as required must be modified to take into account the maintainability and hence the availability of the system.
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In this context, repairable means repair during an operational duty cycle. For more information, see Reliability Evaluation when Redundant Sub-systems can be Repaired Before System Failure.
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Maintainability and availability can be defined in quantitative terms as follows:
• Maintainability–The probability that an item can be restored to a serviceable condition within a specified period of time. A most useful measure of maintainability is the quantity Mean Down Time (MDT), which includes administrative and other logistic delays beyond the designer's control.
• Availability (steady state)–The proportion of an item's operational duty cycle during which the item is not engaged in any activity preventing its immediate use and is serviceable.
Thus, in the simplest terms, if reliability is identified in terms of MTTF and maintainability in terms of MDT, the intrinsic availability of a system (i.e., that which can be designed in) is given by:
| | This expression is based on many assumptions, including: • The system is operating continuously except when failed. • Any failures are detected immediately upon occurrence. • The operational duty cycle is large compared with the MTTF and MDT values so that the system can be considered to be in a steady state. • Further failures do not occur during the repairs. In practice, the relationships between Availability, MTTF and MDT may be complex if the operational duty cycle is complex. However, this simple expression does serve to illustrate the principles associated with repairable systems. When considering the probability that a repairable system functions as required, it is often convenient to introduce the concept of system success (SS) and to denote the associated probability as the product of two probabilities: • The probability that it is failure-free at some appropriate point within the operational duty cycle (steady state availability, ), and also • The probability that the system survives the remainder of the operational duty cycle, given that it was failure-free at the appropriate point within the cycle (Reliability,R(t)). |
Example
The missile launch and interception phase is a critical phase in the operation of a guided weapon system. Assume that the surveillance radar is a repairable component that operates continuously for some time prior to launch, and is required to operate throughout the launch and interception phase (time ). Assume also that the failure
rate of the radar is constant throughout the duty cycle so that MTTF = 1/λ. Then:
◦ From equation (2.13), the probability that the radar is failure-free at launch is denoted by (availability) and is given by:
◦ Because the radar is effectively non-repairable during the launch and interception phase (they are too short), the probability that it survives this period is denoted by (reliability) and is given by:
R(t)=e-λt
◦ The probability that the radar is not compromised by failure is denoted by (system success) and is thus the product of the above two expressions:
Because MDT will be specified for the equipment concerned, can be predicted
from this expression.
| | The foregoing is intended only to introduce the different philosophy that must be adopted when considering repairable systems. |