FIGURE 98.7 State diagram for two-unit parallel system The reliability of the voter must be included when calculating the overall reliability of such a system. As the voter appears in every path from input to output, it can be included as a series element in a series-parallel model. This leads to (98.18) where r, is the reliability of the voter. More information on methods of using redundancy to improve system reliability can be found in Chapter 93 98.10 Markov Modeling Another approach to determining the probability of system failure is to use a Markov model of the system, rather than the combinatorial methods outlined previously. Markov models involve the defining of system states and state transitions. The mathematics of Markov modeling are well beyond the scope of this brief introduction but most engineering mathematics textbooks will cover the technique. To model the reliability of any system it is necessary to define the various fault-free and faulty states that ould exist. For example, a system consisting of two identical units(A and B), either of which has to work for he system to work, would have four possible states. They would be(1)A and B working;(2)A working, B failed; (3)B working, A failed; and (4)A and B failed. The system designer must assign to each state a series of probabilities that determine whether it will remain in the same state or change to another after a given time period. This is usually shown in a state diagram, as in Fig. 98.7. This model does not allow for the possibility of repair, but this could easily be added. 98.11 Software Reliability One of the major components in any computer system is its software. Although software is unlikely to wear out in a physical sense, it is still impossible to prove that anything other than the simplest of programs is totally free from bugs. Hence, any piece of software will follow the first and second parts of the normal bathtub curve ( Fig. 98.1). The burn-in phase for hardware corresponds to the early release of a complex program, where bugs are commonly found and have to be fixed. The useful life phase for hardware corresponds to the time when the software can be described as stable, even though bugs may still be found. In this phase, where the failure rate can be characterized as constant(even if it is very low), the hardware performance criteria, such as MTTF TR can be estimated. They must be included in any estimation of the overall availability for the computer as a whole. Just as with hardware, techniques using redundancy can be used to improve the availability erance. e 2000 by CRC Press LLC© 2000 by CRC Press LLC The reliability of the voter must be included when calculating the overall reliability of such a system. As the voter appears in every path from input to output, it can be included as a series element in a series-parallel model. This leads to (98.18) where rv is the reliability of the voter. More information on methods of using redundancy to improve system reliability can be found in Chapter 93. 98.10 Markov Modeling Another approach to determining the probability of system failure is to use a Markov model of the system, rather than the combinatorial methods outlined previously. Markov models involve the defining of system states and state transitions. The mathematics of Markov modeling are well beyond the scope of this brief introduction, but most engineering mathematics textbooks will cover the technique. To model the reliability of any system it is necessary to define the various fault-free and faulty states that could exist. For example, a system consisting of two identical units (A and B), either of which has to work for the system to work, would have four possible states. They would be (1) A and B working; (2) A working, B failed; (3) B working, A failed; and (4) A and B failed. The system designer must assign to each state a series of probabilities that determine whether it will remain in the same state or change to another after a given time period. This is usually shown in a state diagram, as in Fig. 98.7. This model does not allow for the possibility of repair, but this could easily be added. 98.11 Software Reliability One of the major components in any computer system is its software. Although software is unlikely to wear out in a physical sense, it is still impossible to prove that anything other than the simplest of programs is totally free from bugs. Hence, any piece of software will follow the first and second parts of the normal bathtub curve (Fig. 98.1). The burn-in phase for hardware corresponds to the early release of a complex program, where bugs are commonly found and have to be fixed. The useful life phase for hardware corresponds to the time when the software can be described as stable, even though bugs may still be found. In this phase, where the failure rate can be characterized as constant (even if it is very low), the hardware performance criteria, such as MTTF and MTTR can be estimated. They must be included in any estimation of the overall availability for the computer system as a whole. Just as with hardware, techniques using redundancy can be used to improve the availability through fault tolerance. FIGURE 98.7 State diagram for two-unit parallel system. r r r r tmr v = - [ ] 3 2 2 3