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Let p is the probability of failure of switching

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Letpis the probability of failure of switching device on demand that switches the standby system to operation.
The resulting equations are given by:Solving these simultaneous differential equations:Reliability of the systemR(t) =P1(t) +P2(t) +P3(t)
Ifp= 1 observe that the standby system has no effect and the overall system reliability is that of the primary unit only.Three components Standby systemConsidering a system with one active and two standby units. Lets us assume that no units fail while in standby and that allthe three units have same constant failure rate when on-line. The associated states are summarized in the table below:With initial conditionsP1(0) = 1,P2(0) = 0,P3(0) = 0. Solving these differential equations:
Reliability of the systemSince the system is functioning while in any of the first three states,R(t) =P1(t) +P2(t) +P3(t)As may be evident from the results that it is the same case as discussed in the identical standby unit section.System having three identical units with one online and two standby will have MTTF three times higher than singleunit system.In general, if there arekidentical and independent units withk- 1 on standby, then the system MTTF isk/λ.Degraded SystemIn many situation system may continue to operate in a degraded mode (less than required level) following certain types offailures. The system may continue to perform its function but not at a specified operating level. Examples are, a computersystem may not be able to access all its direct access storage devices; a copying machine may not be able toautomatically feed originals and may thereby require a manual operation; a multi-engine aircraft may experience aproblem in one of its engines etc. Whether the degraded mode is considered as a failure or not must be determined as apart of the reliability specification. If it is important to distinguish the degraded state from that of a complete failure, Markovanalysis is used if CFR is assumed.
Such A situation may be represented by three states:State 1: Fully operational, State 2: degraded state; State 3: Failed state. The rate diagram and differential equations aregiven below:The solution of above equation isReliability of the systemThe reliability of the system is the probability in state 1 and state 2:Three State Devices
Components having three states (operating state, failed open and failed short state) may be analyzed using Markovanalysis since there is a dependency between the two failure states. In this analysis, the components are assumed tohave the constant failure rates.The solution is straight forward and is given byReliability of the system
Physical Reliability ModelsSo far the study was focused on development of reliability models in which the system or component reliability wasconsidered as a function of time only. In many applications, other factors may be equally important. For example, theelectronic components failures may depend on the applied voltage or on operating temperature of the equipment. Thestrength (and therefore, the reliability) of a pre-cast concrete beam may depend on the impurities found in water and inother materials used in the mixture. A more accurate reliability model is one in which the inherent characteristics orexternal operating conditions of a component are included. Covariate models incorporate these additional factors in failure

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Term
Spring
Professor
NoProfessor
Tags
Probability distribution, Probability theory, Failure rate

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