Reliability and LifetimesIn the previous sections a high level of abstraction was used that ignored thetime dependence of reliability. Now we look into some fundamental quantitiesthat describe the temporal behavior of component reliability.Basic Probabilistic ModelLet T be the failure time when the considered component brakes down. Nat-urally, T is a random variable.A fundamental characteristics of any random variable is its probability dis-tribution function (sometimes also called cumulative distribution function).For the failure time it is defined asF (t) = Pr(T < t).In other words, F (t) is the probability that the breakdown occurs before agiven time t. Naturally, the value of this probability depends on what is thisgiven time t, this dependence is described by F (t).Let us define now further important functions that characterize how thereliability of a component behaves in time.Lifetime measures• Survivor function (also called reliability function)The survivor function is defined byS(t) = 1 − F (t)1where F (t) is the probability distribution function of the failure time.The meaning of the survivor function directly follows from the defini-tion of F (t):S(t) = 1 − Pr(T < t) = Pr(T ≥ t).In other words, S(t) is the probability that the component is still op-erational at time t, since T ≥ t means the component has not failed upto time t.• Probability density function (pdf)If F(t ) is differentiable, then its derivative is called probability densityfunction:f(t) = F0(t)With the pdf f(t) we can easily express the probability that the failureoccurs in a time interval [t1, t2]:Pr(t1≤ T ≤ t2) =Zt2t1f(t)dt.• Hazard function (also called failure rate or hazard rate)The goal of the hazard function h(t) is to express the risk that thecomponent fails at time t.How can we capture this risk with the already known quantities?Take a very small time ∆t. Then the fact that the failure occurs at timet can be approximately expressed with the condition t ≤ T ≤ t + ∆t.Of course, this can only happen if the component still has not brokendown before t. Thus, the risk that the component brakes down at timet can be approximated by the conditional probabilityPr(t ≤ T ≤ t + ∆t | T ≥ t). (1)Note that the risk may be quite different from the probability of failure.As an example, one can say that the probability that a person dies at2age 120 is very small, since most people do not live that long. On theother hand, the risk that a person of age 120 dies is quite high, sincethe concept of risk assumes that the person is still alive (otherwise itwould be meaningless to talk about risk).To make the risk expression (1) exact and independent of ∆ t, we con-sider the limit ∆t → 0. Then, however, the probability would alwaystend to 0. To avoid this, we divide it by ∆t, thus obtaining a quan-tity that is similar in spirit to the pdf. This gives the definition of thehazard function:h(t) = lim∆t→0Pr(t ≤ T ≤ t + ∆t | T ≥ t)∆tThus, the hazard function gives the risk density for the failure to occurat time t.Relationships between different lifetime measuresAs we have seen there is a direct relationship between the survivor functionand the probability density function:S(t) = 1 − F (t)F (t) = 1 − S(t)By taking derivatives we can get a relationship between the survivior functionand the pdf:f(t) = −S0(t).We can also express the survivior function with the pdf from the definitionS(t) = 1 − F (t). Using that 1-F(t)=R∞tf(t)dt, we obtainS(t) =Z∞tf(t)dt.3Relating the hazard function with the others is slightly more complex. Onecan prove the following formula:h(t) =−S0(t)S(t).Using the previous relationships, this impliesh(t) =f(t)R∞tf(t)dt.It is kown from basic calculus that the formula h(t) = −S0(t)/S(t) is equiv-alent toh(t) = −ddtln S(t).Using this we can express S(t) by the hazard function asS(t) = e−Rt0h(t)dt.Typical hazard functionsIf we look at the formulah(t) =f(t)R∞tf(t)dtthen we can see that there are two possible reasons for having high hazard(risk) at time t:• either the numerator is large, that is, the probability of a failure is higharound time t, and/or• the denominator is small. The denominatorR∞tf(t)dt gives the prob-ability that the failure has not occured before t. If this is small, thatmeans the probability that the component is still alive is low. In otherwords, the component is old from the reliability point of view.4Since a typical pdf sooner or later starts decreasing with time, this effecttends to diminish the hazard as time advances. On the other hand, thedenumerator decreases as the component gets older, which will increase thehazard. These two opposite effects can balance each other in different ways.A special case when they precisely balance each other is the exponentialdistribution, where the pdf is of the formf(t) = λe−λt.In this case, if we compute the hazard function, we obtain h(t) = λ. Thatis, in this case the hazard function is constant.In many practical cases the hazard function has a bathtub shape that consiststhree typical parts:1. Initial “burn-in” period, when the hazard is relatively large, due topotential manufacturing defects that result in early failure.2. Steady part with approximately constant hazard function.3. Aging with increasing
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