A Study on Weibull Distribution for Estimating the Reliability Dr. P. K. Suri 1, Parul Raheja 2

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1 Iteratioal Joural Of Egieerig Ad Computer Sciece ISSN: Volume 4 Issue 7 July 205, Page No A Study o Weibull Distributio for Estimatig the Reliability Dr. P. K. Suri, Parul Raheja 2 Dea, Research ad Developmet, Chairma, MCSE/IT/MCA, HCTM Kaithal, Haryaa, Idia pksurikuk@gmail.com 2 Studet, HCTM Kaithal, Haryaa, Idia parulraheja296@gmail.com Correspodig Author Abstract: I this paper we estimate reliability of a system usig Weibull distributio. Reliability is defied as whe a product does ot fail i a expected lifetime i specified coditios. I reliability applicatios Weibull Distributio is commoly used distributio. The Weibull distributio is a geeralizatio of the expoetial distributio. The two-parameter Weibull distributio ca accommodate icreasig, decreasig, or costat failure rates because of the great flexibility expressed through the model's parameter. This distributio is recogized today as idustrial stadard. This paper presets the maximum likelihood method for estimatig the Weibull distributio parameters. Weibull distributio is widely used i egieerig applicatios. Weibull distributio is aalysis with the Weibull Data Plot depict failure aalysis with extremely small samples. The Data plot is very helpful for egieers ad maagers. Keywords: Reliability, Weibull distributio, Parameter estimatio, Maximum likelihood method, Weibull probability plot... Itroductio With the developmet of ew techologies ad techiques the demad of bug free software icreasig,the software udergo extesive testig ad debuggig which is very costly ad time cosumig, ad compaies demad accurate iformatio how software reliability grows. Thus Weibull distributio is widely used i theory of reliability. This distributio was amed by Swedish scietist Waloddi Weibull ( ), who developed it i 937 ad published it i 95. Much type of failure rate behaviors ca be represeted by Weibull distributio. It is very flexible ad reliable i may practical applicatios. Reliability ca be estimated usig two or three parameter (shape, scale ad locatio) Weibull distributios. I this we estimate the reliability usig two parameter Weibull distributios. Dolas et.al []describes Graphical methods ad aalytical methods for estimatig the values of these parameters. Parameters estimatig usig Graphical methods like Weibull probability plottig ad hazard plot are ot very accurate but they are fast. But aalytical methods like maximum likelihood method, least square method etc are cosidered more accurate. This distributio is popular amog egieers because it has the ability to assume the characteristics of may distributios. Ivaa et.al [2]itroduced the four methods for estimatig the Weibull distributio parameters: the least square method (LSM), the weighted least square method (WLSM), the maximum likelihood method (MLM) ad the method of momets (MOM) ad compare their performace usig the Mote Carlo simulatio. The maximum likelihood method is most commoly used method for estimatig the parameters of a distributio ad commoly used by researchers. Li et.al [7] used mea square error to compare the least square method, the weighted least square method, the maximum likelihood method ad the method of momets. Estimatio method used Mote Carlo simulatio to obtai their statistical properties. A Advaced algorithm for maximum likelihood estimatio is give by A Yag et.al. [3].Wid Eergy has bee expressed by Weibull distributio fuctio ad two aalytical methods for calculatio of parameters are give by Paritosh [4]. Distributio of wid speeds of wid power statios is most commoly applicatio of the Weibull distributio. I[8], three differet models are used for calculatig the shape ad scale parameters of the Weibull distributio. To determie which model is best suited to desig wid coversio machies, i the Nigeria locatios. The method of momets is a appropriate method for estimatig. The failure probability of certai compoets ca be derived from the Weibull plot [5]. There are may egieerig problems that ca be solved with Weibull aalysis. Examples of problem solved are: [6]to order spare parts ad schedule maiteace labor, how may uits will be retured to depot for overhaul for each failure mode moth-by-moth ext year? The program maager wats to be 95% cofidet that he will have eough spare parts ad labor available to support the overall program. Or a project egieer reports three failures of a compoet i service operatios durig a three-moth period. The Program Maager asks, "How may failures will we have i the ext quarter, six moths, ad year?" What will it cost? What is the best corrective actio to reduce the risk ad losses? Thus Weibull aalysis proactively maage spare parts ivetories, compare reliability or predictio about product life. Kuag Chu et.al [0] calculate sample meas ad the sample root mea square errors for simulate three types of Weibull distributio costat failure rate, icreasig failure rate ad decreasig failure rate. The performace of the maximum likelihood method ad least squares method is uderstad by compariso betwee these two computatio approaches. Dr. P. K. Suri, IJECS Volume 4 Issue 7 July, 205 Page No Page 3447

2 Muhammad et. al [] proves how Weibull distributio is flexible to differet distributio. They also study the life time of the electrical ad mechaical compoets with the help of Weibull Quatile modelig. Reliability ad Quatile aalysis of the Weibull distributio is itroduced. Quatile models are also preseted graphically ad mathematically. Existig i literature there are may computatio approaches to estimate the values of the parameters. Also compariso betwee differet approaches has bee itroduced. The mai purpose of Weibull distributio is used to defie reliability ad ureliability. Weibull Plot is very useful i determiig the life of the compoets or their failure rates. Weibull Plot allow the user to determie that the proposed distributio fits the data it do ot give cofidece limits 2. Weibull Distributio i Weibull Failure Class Models Model A large umber of models have bee derived from the twoparameter Weibull distributio ad Referred as Weibull Models. Weibull ad gamma are two types of failure time classes models. I this we take the per-fault failure distributio to be the traditioal Weibull ad gamma distributios, respectively rather tha the expoetial distributio. I short Weibull model is statistical rather tha expoetial. Weibull distributio has great flexibility give for failure modelig because of the shape ad scale parameters that defie them. Hardware failure used this distributio. Weibull model is oe of the most widely used models for hardware reliability modelig i the Weibull distributio. It ca accommodate icreasig, decreasig, or costat failure rates because of the great flexibility expressed through the model's parameter. Weibull model is based o the followig assumptios: the umber of faults detected i each iterval is idepedet for ay fiite collectio of times. There is fixed umber of faults(n) i the software at the begiig of the time i which the software is observed. The time to fault of failure is distributed as a Weibull distributio with parameters α ad β. A fault is corrected istataeously without itroducig ew faults i the software. The Data requiremets to implemet this fault cout model are: (a) The umber of fault o each testig iterval i.e., f, f 2,.., f. (b)the Completio time of each period that the software is uder observatio: i.e., X,,X 2,...,X. The Weibull distributio ca be used i Weibull Model. The probability desity fuctio f(x) of the Weibull distributio W (β, α) with parameters β > 0ad α > 0 is give as: f(x) = β α (x α )β e (x α )β () probability desity fuctio Figure : The Weibull probability desity fuctio Where x > 0,β is the shape parameter, ad α is the scale parameter or the characteristic life at which 63.2% of the populatio ca be expected to have failed. Probability desity Fuctio f(x) determies how may parts have failed or will fail up to a defied ruig time. Figure shows the shape of Weibull probability desity fuctio with value of α=0 ad β=(, 2, 2.5,3.4) The cumulative distributio fuctio F(x) of the Weibull distributio is give as: F(x) = e (x α )β (2) cumulative distributio Figure 2: The Weibull cumulative distributio fuctio Where x >0, the costat α is called the scale parameter [9], because it scales the x variable ad β is called the shape parameter, because it decides the shape of the rate fuctio. The costat x is called lifetime variable. Figure 2 shows the Weibull cumulative distributio fuctio with value of α=0 ad β=(, 2, 2.5,3.4). All curves itersect at oe poit called the characteristic poit. Reliability R(x)determies how may parts are still i use after a certai ruig time. The reliability fuctio is probability i which system performs its iteded fuctio over a specific period of time. The two-parameter Weibull distributio reliability is give as: R(x) = e (x α )β (3) Dr. P. K. Suri, IJECS Volume 4 Issue 7 July, 205 Page No Page 3448

3 Reliability Fuctio l P(β,α) β = l α x i β l x i l α x β i= i= i + l x β α β i= i = 0 (9) O simplifyig the above equatios (8) ad (9) we obtai the equatios: α = ( l x i= i) β (0) Figure 3: The Weibull Reliability Fuctio Figure 3 shows the Weibull Reliability fuctio with value of α=0 ad β=(, 2, 2.5,3.4). The Weibull failure rate fuctio λ(x) is defied as the umber of failures per uit time that ca be expected to occur for the product. It is give as: x β i= i l x i β x β i= i + l x i= i = 0 () The estimate of the parameter α is obtaied i equatio (0). The estimate of the parameter β ca be obtaied by solvig the equatio () with respect to β. The equatio () must be solved umerically ot aalytical. This parameter is estimated usig Newto method. λ(x) = β α (x α )β (4) The mea life is the average time of failure-free operatio up to a failure evet. The MTTF of the Weibull Probability desity Fuctio is give as: MTTF = ( + ) (5) β 3. Estimatio of Parameter The most commoly method used method to estimate the parameters are the least square method (LSM), the weighted least square method (WLSM), the maximum likelihood method (MLM) ad the method of momets (MOM).All these methods are the aalytical methods. I this paper we used the maximum likelihood method which is used by researchers to estimate parameters of Weibull distributio. The estimatio of parameters ca be obtaied oly umerically[2]. Cosider Weibull radom variables X,,X 2,...,X are idepedet ad idetically distributed. Probability desity fuctio is give i equatio ().The likelihood fuctio of X,,X 2,...,X. ca be costructed from equatio () as: P(β, α) = f(x i, β, α) = β α β i= i= x i β e (x i α )β Now take atural logarithmic o both the sides, we l P (β, α) = l β β l α (6) x α β i= i β + (β ) i= l x i (7) Now differetiatig equatio (7) with respect to β ad α Equatig to zero we obtai followig equatios (8) ad (9): l P(β,α) α = β α + β x α β+ i= i β = 0 (8) 4.Weibull Aalysis ad Results Weibull aalysis helps to determie life of software by fittig a statistical distributio. Weibull aalysis is useful i graphical plot of the failure data. The data plot is extremely importat to the egieer ad to the maager. The Weibull data plot is particularly iformative as Weibull poited out i his 95 paper [6]. Figure 6 is a typical Weibull plot. Weibull probability plot is geerated to see how well radom Weibull data poits are actually fit by a Weibull distributio. The Weibull Plot is a plot of F(x) (cumulative distributio fuctio) vs. time. Note log scale used is base 0. Reliability fuctio R(x) is also kow as survivor fuctio []. Weibull parameter β is the slope of the Weibull plot determies which member of the family of Weibull failure distributios best fits or describes the data. [6] The slope, β, also idicates which class of failures is preset. Shape parameter α which is othig else tha the slope of the lie o the liearized Weibull graph 4. Effects of the Shape Parameter β 4.. Weibull Distributio with β Less Tha I early stage of software failures occur frequetly called ifat mortality (premature failures).the data that fit this distributio have iitial high umber of failures, which decrease over time as the defective items are elimiated from the sample. The probability decreases expoetially from ifiity Weibull Distributio with β Equal to Failure rate over the time remai cosistet called radom failures. This shape model the useful life of products. Figure shows the distributio decreases expoetially from / scale parameter Weibull Distributio with β betwee ad 2 Failure Rate shows whe β=icrease quickly, ad the decrease with time. This shape idicates early wear-out failures Weibull Distributio with β Equal to 2 Dr. P. K. Suri, IJECS Volume 4 Issue 7 July, 205 Page No Page 3449

4 Whe Risk of wear-out failure icreases steadily over the product lifetime this form of distributio is also called Rayleigh distributio. Figure shows β = Weibull Distributio with β betwee 3 ad 4 Whe software is i fial period of product life most failure happe. The bell-shaped is formed like ormal curve. 4.2 Case Study We calculate ow probability desity fuctio f(x),cumulative distributio fuctio F(x) ad reliability R(x)of the compoet for a operatig time of 000 hours. The failure time of certai Compoet has a Weibull distributio with Weibull parameters are: x = 000 β =.5 α = 5000 Reliability = R(x) = e (x α )β = e ( ).5 = or, I( I( Q(x)) = β I( β α ) I (I ( Q(x) y = I (I ( )) = β I(x) βi(α) )) Q(x) y = βt βi (α) The X-axis simply shows I(x).The Y-axis is βt βi (α) which is complicated. Now X-axis ad Y-axis of the Weibull Probability Plot show i Figure.4 is costructed. Weibull probability is oe of the method to calculatig the parameters of the Weibull distributio. Probability plot shows how well data fit by a Weibull distributio. Figure 4 show probability plot time to failure data. Today there are may software programs that are available i the market to aalyze reliability data. If poits makes a straight lie the data follows Weibull distributio. Otherwise, we cosider data is followig aother lifetime distributio. Cumulative desity fuctio = F(x)= e (x α )β = e ( ).5 = Failure rate = λ(x) = β α (x α )β = = Probability desity fuctio = R(x) λ(x) ( ).5 = = Coclusio This work has proposed a method of estimatig the reliability usig Weibull distributio. We have also study differet distributios fit ito Weibull distributio whe shape parameter chages. The reliability of the compoet is Estimatio of parameter is doe by the maximum likelihood method which is mostly used by researchers. The approximatio of fuctios was take with Weibull parameter are β =.5 ad α = 5000.Weibull aalysis tells which specific failure distributio is followig by the failure data. As the software compoet icreases day by day complexity of the system also icreases which leads a icrease i failure mechaism. This aalysis also help us to determie how items are failig. I (/ (-F(x)) I(x) Figure 4:Weibull probability plot[0] The cumulative distributio fuctio or reliability fuctio of the two parameter Weibull distributio is : F(x) = Q(x) = e (x α )β liearize this fuctio ito y = mt + c Q(x) = e (x α )β I( Q(x)) = I( e (x α )β ) I( Q(x)) = ( x α ) β Refereces [] D.R.Dolas, M.D. Jaybhaye, S.D. Deshmukh, Estimatio the System Reliability usig Weibull Distributio, IPEDR V [2] Ivaa Pobočíková ad Zuzaa Sedliačková, Compariso of Four Methods for Estimatig the Weibull Distributio Parameters, Applied Mathematical Scieces, Vol. 8, 204, o. 83, HIKARI Ltd, [3] J. Dai, A. Yag, ad Z. Yag., A Advaced Algorithm for Equatios of Maximum Likelihood Estimatio, Applied Mathematics A Joural of Chiese Uiversities (Ser. A),03, 2009 [4] Paritosh Bhattacharya., A Study o Weibull Distributio for Estimatig the Parameters, Joural applied quatitative methods,200, Vol.5 No.2, pp [5] K. Weibull Curt Roiger 202, "Reliability aalysis with Weibull" [6] Dr. Robert B. Aberethy., The New Weibull Hadbook, Chapter A Overview Of Weibull Aalysis, 536 Oyster Road, North Palm Beach, FL Dr. P. K. Suri, IJECS Volume 4 Issue 7 July, 205 Page No Page 3450

5 [7] D. Wu, J. Zhou, Y. Li., Methods for estimatig Weibull parameters for brittle materials, Joural of Material Sciece, 4(2006), [8] F.C.Odo ad G.U.Akubue, "Comparative Assessmet of Three Models for Estimatig Weibull Parameters for Wid Eergy Applicatios i a Nigeria Locatio," Iteratioal Joural of Eergy Sciece202 Vol.2 No.,PP [9] Yu-Kuag Chu ad Jau-Chua Ke., Computatio Approaches For Parameter Estimatio Of Weibull Distributio,,Mathematical ad Computatioal Applicatios, Vol. 7, No., pp , 202 [0] [Olie].Available: ok/apr/sectio/apr62.htm [] Muhammad Sahib Kha, Ghulam Rasul Pasha ad Ahmed Hesham Pasha., Reliability ad Quatile Aalysis of the Weibull distributio ISSN Joural of Statistics Volume 4, 2007, pp Dr. P. K. Suri, IJECS Volume 4 Issue 7 July, 205 Page No Page 345