A Reliability Calculations and Statistics

1 AReliability Calculations and StatisticsIn this appendix we will discuss basic probabilistic and statistical coher-ences and quantities. Our goal is to learn how to use them in real life. Thetargets of our discussion are hardware components and hardware config-urations. Software shows failures of different kinds which do not apply tothe findings in this might say that Statistics is about large numbers (of outages) itdoes not apply to predicting failures which happen very seldom. Even ifthis is true, we can use Statistics and probabilities to predict reliabilitywhen we configure a system. It also helps to identify anomalies duringsystem operations.

2 In particular we will address the following areas: In configuring a new system, there are many options. In theory, yourvendor of choice should offer thebestconfiguration for your needs; inpractice many of your hardware vendor s presales consultants fail todo this. And technicians who assemble the systems often do it in a waysuch that the system works, but does not show the best possible avail-ability. In the end, we want to achieve the best compromise betweenreliability, cost, and overcome this, we can do some basic computations and approxima-tions to understand the consequences of configuration decisions like what if we used a second power supply?

3 Or should we configure hot-spare disks? In the following, we will discuss the Reliability of disk configurationsand different redundant array of independent disks (Raid) levels indetail. There are many options to protect our data and the risk of aninadequate configuration is high. During the lifetime of your system, problems will occur. Some will becovered by redundancy (like a drive failure in a mirrored configura-tion), others will have an impact (like a system crash after a CPUfailure), up to real downtimes. If we monitor these occurrences, im-portant conclusions can be drawn. We can distinguish betweengood360 A Reliability Calculations and Statisticsfrombadsystems and can identify aging systems, which need to bereplaced.

4 Computer systems like all other technology differ in de-sign quality as well as in manufacturing quality. There are Mondaysystems similar to cars and we need to deal with them in an appro-priate way. Such Statistics are typically not published: they containtoo much political ammunition. Therefore the data of your outagesis an important information source, and you know that your numbersare Mathematical BasicsFor the following discussion, we use some basic statistical concepts. Inthis section we repeat the most important formulas if you are less in-terested in mathematics, skip this section and look only at the examplesgiven.

5 If you want more details, many undergraduate text books, suchas [9], present them. Empirical ProbabilityIf we execute an experimentatimes, and getbtimes a certain eventE, thenb/ais the empirical probability ofE. For largea, we speak of aprobabilityforEand can use it for future predictions:p=ba. Bernoulli ExperimentA Bernoulli experiment has exactly two possible outcomes: the probabil-ity for eventAisp, the probability for the opposite event isp=1 execute the experimentntimes and want to know the probability ofgetting exactlyktimes the resultE(andn ktimes the opposite resultE), the following formula applies:Bn,p(k)=pk(1 p)n k nk.

6 ( ) Distribution FunctionsWe interpret the Bernoulli formula as a distribution functionk Bn,p(k)with the independent variablek. It gives the probabilities that withnexperiments the number of resultsAwill be exactlyk=0,1,2,.., largenand smallpwe can use the following approximations to makereal Calculations easier. Equation ( ) is named thePoisson formula, andEq. ( ) is named theDe Moivre Laplace formula; Mathematical Basics 361Bn,p(k) (np)kk!e np,( )P(k1 x k2) (x2) (x1),( )withx1=k1 np np(1 p),x2=k2 np np(1 p).The De Moivre Laplace formula is used to calculate the probabilityfor an interval of outcomes (instead of a discrete number of outcomeslabeledkabove).

7 The term (x)istheGaussian function. It is an integralwhich has no elementary antiderivative; therefore, it can be calculatedonly numerically. Tests of Hypothesis and SignificanceIn reality, we execute only a limited number of experiments, , experi-ence a limited number of system failures during a limited time interval;therefore, the outcomes are not expected to show the given need to test thesignificanceof our result to accept a given probabil-ity ( , if we got the probability from our system vendor). On the otherhand, a given probability can be wrong as well, as its calculation is basedon wrong assumptions or without enough experiments to test order to evaluate the significance of a result, a so-called hypothe-sis test is done.

8 We first assume that the probabilityp(A) for resultAiscorrect and gives the distributionk Bn,p(k). We makenexperimentsto test the hypothesis. We expectnptimes the resultA, that is whereBn,p(k) has its maximum. If the measured result deviates very much fromnp, then we might need to disapprove the hypothesis the assumed prob-abilitypmight be wrong. In other words, we talk about the probabilitythat a measured result follows a given practice this is done as follows. We calculate the sumBn,p(0)+Bn,p(1)+ +Bn,p(i) for increasingiuntil the sum exceeds a given on the real numbers, the De Moivre Laplace formula can behelpful.

9 This limit describes the requested quality of our result. Typicalvalues for this sum limit are or This belongs to a level of sig-nificance of or , respectively. Using this method, we know thatthe probability of gettingk iis very small (according to our earlier def-inition). We reject the hypothesis, when the measured result belongs tothe interval [n i,n]. Confidence Interval for ProbabilitiesThis is a similar circumstance but from a different viewpoint: we madenexperiments and experiencedffailures. How close is our measured fail-ure probabilityf/nto the real probabilityp? This question can obviouslynot be answered, but we can calculate aninterval of probabilities[p1,p2]362 A Reliability Calculations and StatisticsTable levels and corresponding values ofc (%)c80 contains the real probabilitypwith a chosen confidence level.

10 Ifwe set very close to 1, this interval becomes very large. It depends onthe number of experiments and the value chosen for to achieve a smallenough ( , meaningful) interval of following formula is the criterion to calculate the interval [p1,p2].It denotes the probabilityQfor that interval with confidence :Q fn p c p(1 p)n =2 (c) 1.( )The value ofcis calculated from , using the Gaussian function. Ta-ble gives the values ofcfor typical values of .If we have done our experiments, measuredfandn, and have cho-sen the confidence level , we use Table to getc. Then we can calcu-late the interval of probability [p1,p2] by solving the following quadraticequation: fn p1,2 2=c2p1,2(1 p1,2)n.