8 RELIABILITY

1 18 RELIABILITYS ystems ReliabilityA system consists of components which determine whether or not itwill work. There are various types of configurations of the componentsin different systems. Series SystemThis is a system in which all the components are in series and theyall have to work for the system to work. If one component fails,the system fails. Parallel SystemThis is a system that will fail only if they all fail. Series-Parallel SystemThis is a system where some of the components in series are repli-cated in RELIABILITY of Series SystemExamples1. A simple computer consists of a processor, a bus and a computer will work only if all three are functioning probability that the processor is functioning is , that thebus is functioning , and that the memory is functioning is.

2 P rocessorBusM emoryinput outputThe probability that the computer will work is:Rel=.99 .95 .99 = even though all the components have above 95% or more reli-ability, the overall RELIABILITY of the computer is less that 90%.2. A system consists of 5 components in series each having a relia-bility of What is the RELIABILITY of the system ?C1C2C3C4C5input outputRel= series systems, RELIABILITY decreases as the number ofcompo-nents 6 components in series;Rel= 7 components;Rel= An electronic product contains 100 integrated prob-ability that any integrated circuit is defective is.

3 001 andthe in-tegrated circuits are independent. The product operates only ifall the integrated circuits are operational. What is the probabilitythat the product is operational?Solution:The probability that any component is functioning is .999. Sincethe product operates only if all 100 components are operational,the probability that the 100 components are functioning is:Rel=.999100obtained **100[1] RELIABILITY is just over 90% even though each component hasa RELIABILITY of in mind that computing and electrical systems have hundredsor thousands of components, a series formation on its own will neverbe sufficiently reliable, no matter how high the individual componentreliability is.

4 The components need to be backed up in RELIABILITY of a Parallel SystemSystems with parallel structure have built in redundancy. Componentsare backed up, and the system will work even if some of these cease A system consists of 5 components in parallel. If each componenthas a RELIABILITY of , what is the overall RELIABILITY ofthe system ?input outputSystem will function provided at least one of the 5 componentsworks:Rel= P(At least one component is functioning)Taking the complementary approach,P(at least one component functioning ) = 1-P(all componentsfail).

5 ThereforeRel= 1 ( )5 parallel systems the law of diminishing returns operates:With 2 componentsRel= 1 3 componentsRel= 1 4 componentsRel= 1 of a parallel system with increasing number ofcomponents: component RELIABILITY =. of components Reliabilityk<-1:10p <- .97plot (k, 1-(1-p)^k , xlab = "Number of components ",ylab = " RELIABILITY ")abline(1,0)8 RELIABILITY6 Series-Parallel SystemsExamples1. Consider a system with 5 kinds of component, with reliabilities component 1 : , component 2 : , component 3 : , component 4 : , component 5 : of the low RELIABILITY of the third and fourth components, theyare replicated; the system contains 3 of the third componentand 2 ofthe fourth C2 C3 C4 C5P(C1) P(C2) P( at least 1C3) P(at least 1C4) P(C5)Rel=.

6 95 .95 (1 .33) (1 .252) . *.95*( ^3) * (1- .25^2) *.9[1] The following system operates only if there is a path of functionaldevices from left to right. The probability that each devicefunctions isshown on the graph. Assume the devices fail outputReliability:P(at least 1 of 2C1) P(at least 1 of 3C2) P(C3)1 .12 1 .153 .99 InR> ( ^2)* ( ^3)*.99[1] The following diagram is a representation of a system thatcom-prises two subsystems operating in 1 PPPPPPPPPPqSubsystem ASubsystem B 1outputThe system operates if at least one of subsystem A (at least one subsystem works)EquivalentlyRel= 1-P(both subsystems fail)P(Subsystem A fails) =1.

7 (Subsystem B fails) =1 . system will operate if at least one of these subsystems (at least one subsystem works)EquivalentlyRel= 1-P(both subsystems fail)Rel= 1 (1 .92)2= withksubsystems, the RELIABILITY isRel= 1 (1 .92)kWe can examine this with different values ofkk<-1:10plot(k, 1-( ^2)^k, xlab = "Number of Subsystems", ylab =" RELIABILITY ")abline(.99, 0)calculates and plots the RELIABILITY for various levels of of SubsystemsReliability8 RELIABILITY10 SummaryThe RELIABILITY of a system , the probability that it is functioningproperly depends on the RELIABILITY of each of its the components the type of systemReliability with Series SystemsThe problem with series systems is that RELIABILITY quicklydecreases asthe number of components with Parallel SystemsThe problem with parallel systems is that the law of diminishing re-turns operates.

8 The rate of increase in RELIABILITY with each additionalcomponent decreases as the number of components systems are combinations of series and parallel systems