Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 38 studies the combination of various methods of designing for reliability, availability, maintainability and safety, as well as the latest techniques in probability and possibility modelling, mathematical algorithmic modelling, evolutionary algorithmic modelling, symbolic logic modelling, artificial intelligence modelling, and object-oriented computer modelling, in a logically structured approach to determining the integrity of engineering design. . | Theoretical Overview of Availability and Maintainability in Engineering Design 353 The following assumptions are associated with this multi-state model System failures are statistically independent. A partially or fully failed system is restored to a good as new state. System failure rates are constant. System component failure times are random. The partially failed system repair rate is constant. Failed system repair times are arbitrarily distributed. As with the two-state Markov model the mathematical expressions for the multistate Markov model including supplementary variables indicating partial operation or a reduced efficiency of the system are given in the following Markov multi-state model equations according to Fig. Po t At Po t 1 - AiAt 1 - foAt Pi t ppAt yP2 x t fx dx At 0 Pi t At Pi t 1 - . . 1 - lip. A Po t AiAt P2 x At t At P2 x t 1 - pf x At lj is the jth constant failure rate of the system with j 1 normal-partial transition j 2 normal to failed j 3 partial to failed pp is the system constant repair rate from the partial operating state 1 to the normal operating state 0 Pi x is the repair rate when the system is in the failed state and has the elapsed repair time of x Po t A Pi t At is the probability that the system is in an operating state 0 at time t At is the probability that the system is in a partially failed state 1 at time t At P2 x At t At is the probability that at time t the system is in a failed state 2 Po t Pi t P2 x t and the elapsed repair time lies in the interval x x Ax is the probability that the system is in an operating state 0 at time t is the probability that the system is in a partially failed state 1 at time t is the probability that the system is in a failed state 2 after an elapsed 1 - k At repair time of x is the probability of no failure in time interval At when the system is in state i 1 - ppAt is the probability of no repair in time interval At when the system is in state 1 1 pfAt is the probability
