On the Maximum Likelihood Estimation of Weibull Distribution with Lifetime Data of Hard Disk Drives

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1 314 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'17 On the Maxiu Likelihood Estiation of Weibull Distribution with Lifetie Data of Hard Disk Drives Daiki Koizui Departent of Inforation and Manageent Science, Otaru University of Coerce, Hokkaido, Japan Abstract The axiu likelihood estiations (MLEs) of the shape and scale paraeters under the two-paraeter Weibull distribution are considered with both coplete and randoly censored data. These estiation ethods are applied to real lifetie data of hard disk drives (HDDs) where the nuber of the is ore than 90,000 for alost 4 years (fro 2013 to 2016). Then the ean tie to failure (MTTF) of each HDD is estiated. It turned out that alost all estiated shape paraeters were ore than one that indicates the Increasing Failure Rate (IFR). Furtherore, the estiated MTTF fro coplete data can be interrupted as the epirical lower bound of MTTF. But soe lower bounds were uch saller than the official MTTF value guaranteed by a anufacturer. Finally, it also turned out that the ratio of the estiated MTTF fro randoly censored data over that fro coplete data can easure the efficiency of this epirical lower bound. Keywords: Reliability Engineering, Probability Model, Weibull Distribution, Maxiu Likelihood Estiation (MLE), Mean Tie to Failure (MTTF), and Hard Disk Drive (HDD) 1. Introduction 1.1 Background The lifetie data analysis is one of iportant research fields not only in the quality anageent of products but also in the reliability engineering [1], [2], [3]. In order to guarantee the certain lifetie for products, soe probabilistic approaches are often chosen. Typically, a probability distributions have been assued for the lifetie data and its paraeters are estiated by observed data fro reliability testing. For the probability distributions of lifetie data, the Weibull, exponential, noral, log noral, Poisson, and Gaa distributions etc. have been assued [1], [2], [3]. Aong those distributions, the Weibull distribution is one of faous probability density functions in the reliability engineering [1], [2], [3]. Especially the two-paraeter Weibull distribution [4], [5] has the shape and scale paraeters. If the shape paraeter is less than one, the target device has the decreasing failure rate (DFR) [3]. If it equals to one, the target device has the constant failure rate (CFR) and the Weibull distribution corresponds to the exponential distribution [3]. If it is ore than one, the target device has the increasing failure rate (IFR) [3]. These characteristics have been often referred as a part of Bathtub curve [2], [3]. On the other hand, there also exist soe types of observed data in the reliability engineering. If the exact lifetie data are observed, those are called coplete data [1], [2]. Otherwise the device is still alive and the coplete lifetie data cannot be observed during the reliability testing. In this case, the reliability testing would be then forcibly stopped. This action has been called censoring [1], [2] and the rest of data except for the coplete data would be recorded as censored data [1], [2]. There are several types of censoring such as Type I, Type II, and Rando censoring [1], [2]. The rando censoring is the ost general type aong those. The censored data are not the exact lifetie data but the lower bounds for the exact lifetie data. Therefore there is soe possibility of their inforation to copensate the inforation of sall nuber of coplete data. The axiu likelihood estiation (MLE) ethods of the Weibull distribution for the coplete and randoly censored data have been proposed [4]. Recently, its application for the rocket engine tests has been also reported [5]. 1.2 Previous Studies In lifetie data analysis, it often happens that nubers of the coplete or censored data are sall. In this case, soe graphical analysis techniques such as probability papers, probability plotting, and hazard plotting etc. have been developed and widely applied to estiate the paraeters of the probability density functions for the lifetie data [2]. One of probles for those ethods is that the objective functions for the estiations are undefined. However, the above situation has not been the case in soe engineering fields. For exaple, the cloud technology has been widely spread in the network server counities. Cloud technology requires a huge nuber of hard disk drives (HDDs) as storage devices. The set of the lifetie data is now big data and those have been analyzed [6], [7] in soe reliability engineering approaches. Especially, Schroeder et al. exained the lifetie data of HDDs [7]. They have been reported that the Weibull and Gaa distributions are accepted for the lifetie of HDDs but the exponential and log noral distributions are rejected according to their Chi- Square tests. They also tried to plot probability distribution curves including Weibull plot, however, they have not define any detail estiation ethods [7].

2 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA' Purpose of This Study This paper assues the two-paraeter Weibull distribution and estiates those paraeters by the axiu likelihood estiation (MLE) for the large nubers of lifetie data of HDDs. Fortunately, the lifetie data of HDDs in the cloud systes have been widely onitored and those data have been reported [8] fro 2013 to 2016 where the total nuber of lifetie data is close to hundred thousands. This paper regards those data as both coplete and randoly censored lifetie data where no outliers as well as no other operating factors such as teperatures, vibrations, and disk access error rates etc. are considered. Then, both shape and scale paraeters of Weibull distribution as well as the ean tie to failure (MTTF) of the HDDs would be estiated by MLE. Even if the nuber of coplete data is sall, it ay be possible for randoly censored data to provide the sufficient inforation for the estiations. As a result, the following three points would be at least observed. The first is that alost all estiated shape paraeters are ore then one and it indicates that each lifetie of objective HDDs has the Increasing Failure Rate (IFR). The second is that the estiated MTTF fro coplete data can be interrupted as the epirical lower bounds for MTTF. But soe lower bounds are uch saller than the official MTTF value guaranteed by a anufacturer. The last is that the ratio of the MTTF fro randoly censored data over that of coplete data can easure the efficiency of the epirical lower bound. The rest of this paper is organized as follows: the next section 2 gives definitions in ters of the reliability engineering under the Weibull distribution. Section 3 derives the axiu likelihood estiators under the Weibull distribution for both coplete and randoly censored data. Section 4 considers to estiate the shape paraeters, scale paraeters, and the MTTF fro the real lifetie data of hard disk drives. Section 5 discusses those results. The final section 6 gives concluding rearks. 2. Preliinaries Let T 0 denote the lifetie of the target device. Fro the probabilistic viewpoint, let T = t, i.e. t 0 denotes the observed value of a continuous rando variable T. Suppose that the probability density function of t is the following two paraeter Weibull distribution with the shape paraeter >0 and the scale paraeter >0. Definition 2.1 (Probability density function [1], [2]): f (t) = ( ) 1 [ ( ) ] t t exp, t 0, >0, >0. (1) Based on Definition 2.1, the following cuulative distribution function of F (t), the reliability function of R (t), and the ean tie to failure (MTTF) or the expectation of E (t) can be defined. Definition 2.2 (Cuulative distribution function [1], [2]): t F (t) = f(x)dx = 0 [ ( ) ] t 1 exp. (2) Definition 2.3 (Reliability function [1], [2]): R (t) = f(x)dx = t [ ( ) ] t exp. (3) Definition 2.4 (Mean Tie to Failure (MTTF) [1], [2]): E (t) = xf(x)dx 0 ( = Γ 1+ 1 ), (4) where Γ( ) is the following Gaa function, Γ(α) = 0 x α 1 exp ( x) dx, α > 0. (5) 3. Maxiu Likelihood Estiation (MLE) of Weibull Distribution Suppose that a reliability test about a device is executed. A sufficient nuber of products of the device have been prepared and a sufficient nuber of the lifetie data can be observed. The ain purpose of this test is to estiate and evaluate the ean tie to failure (MTTF) of the device. In order to estiate MTTF, the shape and scale paraeters of the Weibull distribution should be estiated. This paper considers the axiu likelihood estiation (MLE) and assues two types for the observed data, i.e. the coplete and the randoly censored data. The following subsection derives the axiu likelihood estiators for those two types. 3.1 MLE for Coplete Data Suppose that the coplete data sequence t 1,t 2,...,t n is observed where t i,i = 1, 2,...,n denotes the ith observed lifetie data of the device. Let L 1 (, ) denotes the likelihood function where, denote the shape and scale paraeters of the Weibull distribution, respectively. Then, the axiu likelihood estiators ˆ 1 and ˆ 1 can be forulated as follows:

3 316 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'17 Definition 3.1 (Likelihood function [4]): n L 1 (, ) = f (t i ) = i=1 n i=1 ( ) 1 [ ti exp Definition 3.2 (Maxiu likelihood estiators): ˆ 1 =argaxl 1 (, ) ; ˆ 1 =argaxl 1 (, ). ( ) ] ti. (6) Then, the log likelihood equations with respect to Definition 3.1 and 3.2 becoe, ln L 1(,) ln L 1(,) =0; =ˆ1 =0. =ˆ1 The above equations (8) give the following conditions for the axiu likelihood estiators of both ˆ 1 and ˆ 1 : Lea 3.1 (Conditions for estiators [4]): [ n ] i=1 (ti) ˆ 1 ln t i n 1ˆ i=1 (ti) ˆ n n i=1 ln t i =0; ˆ 1 = [ n i=1 (ti) ˆ 1 n (7) (8) ] 1 ˆ 1. (9) 3.2 MLE for Randoly Censored Data Suppose that the observation consists of both coplete and randoly censored data. Assue that the forer sequence is t 1,t 2,...,t n and the latter sequence is u n+1,u n+2,...,u n+r. Note that the exact lifetie data t 1,t 2,...,t n is observed and t n+1,t n+2,...,t n+r is not observed but u n+j < t n+j,j = 1, 2,...,r is observed in this case. This eans that the censored data express the lower bounds for unknown coplete data. In order to siplify the notation, u j would be used instead of u n+j for the rest of this paper. Then, the ML estiation proble for the shape and scale paraeters of and can be forulated as follows: Definition 3.3 (Likelihood function [4]): (n + r)! n r L 2 (, ) = f (t i ) R (u j ) r! i=1 j=1 (n + r)! n ( ) 1 [ ( ) ] ti ti = exp r! i=1 r [ ( uj ) ] exp. (10) j=1 Definition 3.4 (Maxiu likelihood estiators): ˆ 2 =argaxl 2 (, ) ; ˆ 2 =argaxl 2 (, ). (11) The above equations (11) give the following conditions for the axiu likelihood estiators of both ˆ 2 and ˆ 2 : Lea 3.2 (Conditions for estiators [4]): [ n ˆ 2 = i=1 (ti) ˆ 2 ln t i+ r j=1 (uj) ˆ 2 ln u j n i=1 (ti) ˆ 2 + r j=1 (uj) ˆ 2 1ˆ 2 1 n n i=1 ln t i =0; (12) [ n i=1 (ti) ˆ 2 + ] 1 r j=1 (uj) ˆ 2 ˆ 2 n. 4. Lifetie Data Analysis for Hard Disk Drives (HDDs) 4.1 Data Specifications For data analysis, the lifetie data of hard disk drives (HDDs) in data center of Backblaze, Inc. [8] is used. The data contain daily status of HDDs based on the Self-onitoring, Analysis and Reporting Technology (S.M.A.R.T.). As of Apr. 2017, the total nuber of both coplete and randoly censored data is 92,369 for alost 4 years (1362 days). Note that each unit of lifetie data is [hrs] according to the specification of S.M.A.R.T. Each nuber of both n and r corresponds to single physical device of HDD with single serial nuber. Furtherore, a single HDD odel consists of plural serial nubers. Additionally no outliers as well as no other operating factors such as teperatures, vibrations, and disk access error rates etc. are considered. The basic data specifications are suarized in Table 1 and 2. Table 1: HDDs Lifetie Data Specifications [1/2] Year Fro To Days n r 2016 Jan. 1st. Dec. 31st ,422 78, Jan. 1st. Dec. 31st ,427 4, Jan. 1st. Dec. 31st ,200 3, Apr. 10th Dec. 31st ,062 Total 1,362 5,789 85, Estiation Results for Coplete Data The coplete lifetie data of t i,i = 1, 2,...,n are fro the serial nubers of single odel of HDD. For each odel of HDDs, the shape and scale paraeters of Weibull distribution have been estiated as ˆ 1, ˆ 1 by Eq. (9). Fro ]

4 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA' Table 2: HDDs Lifetie Data Specifications [2/2] Coplete Data Randoly Censored Data t i u j Total Manufacturers 6 6 Total Models Total Serial Nubers n =5, 789 r =85, 044 Max Data Capacity 8 [TB] 8 [TB] Min Data Capacity 80 [GB] 80 [GB] Eq. (4), the estiated MTTF [hrs] of Ê 1 (x) can be calculated as the following: Ê 1 (t) =ˆ 1 Γ (1+ ) 1ˆ1 [hrs]. (13) In order to obtain those estiators, nuerical calculations have been executed by the statistical coputing software R version [9]. The results for the ain ten odels have been suarized in Table 3. Table 3: Estiation Results for Coplete Data Models n ˆ 1 ˆ1 Ê 1 (t) ST4000DM000 1, ,188 11,339 ST3000DM001 1, ,891 17,417 ST AS ,261 35,912 Hitachi HDS722020ALA ,359 35,093 ST AS ,604 34,684 WDC WD30EFRX ,292 8,227 Hitachi HDS5C3030ALA ,689 25,408 HGST HMS5C4040ALE ,198 7,902 ST1500DL ,236 9,202 ST320LT ,384 24, Estiation Results for Randoly Censored Data For randoly censored data, a sequence of both t i,i= 1, 2,...,nand u j,j=1, 2,...,r has been used to estiate the shape and scale paraeters of Weibull distribution as ˆ 2, ˆ 2 by Eq. (12). The estiated MTTF [hrs] of Ê2 (x) has been also calculated by Eq. (13). The results for the ain ten odels have been suarized in Table Discussions 5.1 Estiated Shape Paraeters As entioned in Section 1.1, the value of the shape paraeter in the Weibull distribution classifies at least three types of failure rates: DFR for ˆ <1, CFR for ˆ =1, and IFR for ˆ >1. For coplete data, Table 3 shows that all of the estiated shape paraeters are ˆ 1 > 1.00 and have classified into IFR. The only exception can be WDC WD30EFRX. This odel has ˆ 1 =1.019 and it is alost CFR. Note that a anufacturer has been estiated ˆ = 0.55 [10] for the first year MTTF and Schroeder et al. have been estiated 0.71 ˆ 0.76 [7]. Thus there exist rearkable differences between our result and previous results. For randoly censored data, Table 4 shows that two odels i.e. WDC WD30EFRX and HGST HMS5C4040ALE640 have the shape paraeters of ˆ 2 =0.638, 0.666, respectively and those ean DFRs. The rest of odels have the estiated shape paraeters ˆ 2 > 1.00 that eans IFRs. Furtherore, Table 5 shows that the ratios of estiated shape paraeters for both coplete and randoly censored data i.e. ˆ 2 / ˆ 1. Those ratios indicated that ˆ 2 / ˆ 1 < 1.00 except for ST3000DM001. Thus it can be concluded that there is a strong tendency to be ˆ 2 ˆ Estiated Scale Paraeters For coplete data, the estiated scale paraeters of ˆ 1 range fro 8,198 to 39,261 according to Table 3. A anufacturer obtained the ˆ =3,787,073 for the first year [10]. The corresponding HDD odels fro this anufacturer are ST4000DM000, ST3000DM001, ST AS, ST AS, ST1500DL003, and ST320LT007. Those estiated ˆ 1 range fro 10,236 to 39,261 and each of the is uch saller than the above ˆ. For randoly censored data, the estiated scale paraeters of ˆ 2 range fro 11,873 to 12,288,903 according to Table 4. Furtherore, Table 5 shows the ratios of ˆ 2 /ˆ 1. All ratios in Table 5 becoe ˆ 2 /ˆ 1 > Iteans that all estiated scale paraeters fro randoly censored data is larger than those fro coplete data. 5.3 Estiated MTTFs For coplete data, the estiated MTTFs of Ê1 (t) range fro 7,902[hrs] to 35,912[hrs] according to Table 3. A anufacturer obtained the ˆ =0.55, ˆ =3,787,073 [10]. Fro Eq. (4), these estiators give the following estiated MTTF for this anufacturer: ( Ê (t) = ˆ Γ 1+ 1ˆ ) ( = 3,787,073 Γ 1+ 1 ) 0.55 = 6,447,223[hrs]. On the other hand, Schroeder et al. reports that The MTTF for today s highest quality disks range fro 1,000,000 hours to 1,500,000 hours [7]. Our result fro the coplete lifetie data is extreely saller than the above estiations. For randoly censored data, the estiated MTTFs of Ê 2 (t) range fro 10,709[hrs] to 16,366,251[hrs] according to Table 4. Table 7 shows the ratios of Ê2 (t) /Ê1 (t). According to Table 7, all ratios satisfy Ê2 (t) /Ê1 (t) > It eans that all MTTFs of Ê2 (t) for randoly censored data are larger than those of Ê1 (t) for coplete data. But each degree varies depending on the HDD odel. For exaple, the ratios of ST3000DM001 and ST1500DL003 are 1.14

5 318 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'17 Table 4: Estiation Results for Randoly Censored Data Models n + r n r ˆ 2 ˆ2 Ê 2 (t) ST4000DM000 36,532 1,807 34, , ,260 ST3000DM001 4,537 1,720 2, ,362 19,787 ST AS 2, , ,530 61,737 Hitachi HDS722020ALA330 4, , , ,546 ST AS ,735 50,795 WDC WD30EFRX 1, , , ,660 Hitachi HDS5C3030ALA630 4, , , ,993 HGST HMS5C4040ALE640 7, , ,288,903 16,366,251 ST1500DL ,873 10,709 ST320LT ,296 24,814 Table 5: The Ratios of the Estiated Shape Paraeters Models ˆ 2 / ˆ 1 ST4000DM ST3000DM ST AS 0.77 Hitachi HDS722020ALA ST AS 0.93 WDC WD30EFRX 0.63 Hitachi HDS5C3030ALA HGST HMS5C4040ALE ST1500DL ST320LT Table 6: The Ratios of the Estiated Scale Paraeters Models ˆ2 /ˆ 1 ST4000DM ST3000DM ST AS 1.75 Hitachi HDS722020ALA ST AS 1.47 WDC WD30EFRX Hitachi HDS5C3030ALA HGST HMS5C4040ALE ST1500DL ST320LT and 1.16, respectively on Table 7. For these odels, Ê 1 (t) is slightly saller than Ê2 (t). Therefore Ê1 (t) is a superior epirical lower bound of the MTTF of each HDD odel. On the other hand, the ratio of HGST HMS5C4040ALE640 is on Table 5. Iteans that Ê 1 (t) << Ê2 (t). In this case, Ê 1 (t) is an inferior epirical lower bound of the MTTF for this odel. 5.4 Epirical Lower Bounds for MTTFs and their Efficiencies Fro the observations so far, it turns out that the each Ê 1 (t) fro coplete data gives the epirical lower bound of the MTTF for the respective HDD odel. But those efficiencies ust vary depending on those HDD odels. The typical exaples for two HDD odels would be considered in the following three figures. Fig. 1 depicts the relative frequency histogras and estiated Weibull probability density functions (p.d.f.) for ST3000DM001. In Fig. 1, the grey and dashed bars represent the relative frequencies of randoly censored and coplete data, respectively. Moreover, solid and dashed lines represent the estiated Weibull p.d.f. fro randoly censored and coplete data, respectively. In Fig. 1, two p.d.f. lines are located near each other since they are estiated fro two histogras of copleted and randoly censored data and those histogras are close together. In this HDD odel, the MTTF fro dashed line is slightly saller than that fro solid line. In fact, the ratio of Ê2 (t) /Ê1 (t) fro Table 3 and 4 becoes 19,787/17,417 = 1.14 which eans that Ê 1 (t) is superior epirical lower bound for MTTF of this HDD odel. Fig. 2 also depicts the relative frequency histogras and estiated Weibull p.d.f. for HGST HMS5C4040ALE640 and Fig. 3 is enlarged version of Fig. 2 with the agnified vertical axis. Additionally the legends in Fig. 2 and Fig. 3 are exactly sae as Fig. 1. In this HDD odel, two estiated p.d.f. lines are located far each other since two histogras are far each other. In this HDD odel, the MTTF fro dashed line is uch saller than that fro solid line. The ratio becoes Ê2 (t) /Ê1 (t) = 12,288,903/8,198 = which eans that Ê1 (t) is inferior epirical lower bound for MTTF of this HDD odel. Thus the efficiency of Ê1 (t) as the lower bound of the MTTF can be easured by the ratio of Ê2 (t) /Ê1 (t). The closer this ratio gets to 1.00 (fro larger side), the superior the efficiency of Ê1 as the lower bound of MTTF is. Table 7 shows all ratios and those efficiencies take various values depending on HDD odels. 6. Concluding Rearks This paper assues the two-paraeter Weibull distribution and considers the axiu likelihood estiation

6 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA' Probability Density or Relative Frequency Randoly Censored Coplete p.d.f. (Randoly Censored) p.d.f. (Coplete) Lifetie [hrs] Fig. 1: Relative Frequency Histogra and Estiated Weibull p.d.f for ST3000DM001 (n =1,720,r =2,817). Probability Density or Relative Frequency 0e+00 1e 04 2e 04 3e 04 4e 04 5e 04 6e 04 Randoly Censored Coplete p.d.f. (Randoly Censored) p.d.f. (Coplete) Lifetie [hrs] Fig. 2: Relative Frequency Histogra and Estiated Weibull p.d.f for HGST HMS5C4040ALE640 (n = 103,r = 7,059). Probability Density or Relative Frequency 0e+00 2e 05 4e 05 6e 05 8e 05 1e 04 Randoly Censored Coplete p.d.f. (Randoly Censored) p.d.f. (Coplete) Lifetie [hrs] Fig. 3: Enlarged Figure of Fig. 2 with Magnified Vertical Axis. Table 7: The Ratios of the Estiated MTTFs Models Ê 2 (t) /Ê1 (t) ST4000DM ST3000DM ST AS 1.72 Hitachi HDS722020ALA ST AS 1.46 WDC WD30EFRX Hitachi HDS5C3030ALA HGST HMS5C4040ALE ST1500DL ST320LT (MLE) for the shape and scale paraeters to estiate the ean tie to failure (MTTF) of the hard disk drive (HDDs). For large nuber of real lifetie data of HDDs, the coplete and randoly censored data have been considered where the nuber of the is ore than 90,000 for alost 4 years. The estiation results showed that alost all estiated shape paraeters were ore than one that eans the Increasing Failure Rate (IFR). Furtherore, each estiated MTTF fro the coplete data can be interrupted as the epirical lower bound of the MTTF. But soe of those bounds were uch saller than the official MTTF that was guaranteed by a anufacturer. Finally, the ratio of the estiated MTTF fro the randoly censored data over that fro coplete data can easure the efficiency of this epirical lower bound. One of open probles would be to evaluate the validity of the Weibull distribution to express the lifetie data of HDD. It would be able to be evaluated by easuring and

7 320 Int'l Conf. Par. and Dist. Proc. Tech. and Appl. PDPTA'17 coparing the estiation errors aong several probability density functions. References [1] Jerald F. Lawless, Statistical Models and Methods for Lifetie Data, John Wiley & Sons, [2] Wayne B. Nelson, Applied Life Data Analysis, John Wiley & Sons, [3] Michael S. Haada, Alyson G. Wilson, C. Shane Reese, and Harry F. Martz, Bayesian Reliability, Springer, [4] A. Clifford Cohen, Maxiu likelihood estiation in the Weibull distribution based on coplete and on censored saples, Technoetrics, vol.7, no.4, pp , Nov [5] Haibo Li, Zhengping Zhang, Yanping Hu, and Deqiang Zheng, Maxiu likelihood estiation of Weibull distribution based on rando censored data and its application, Proceeding of the 8th International Conference on Reliability, Maintainability and Safety, pp , Jul [6] Eduardo Pinheiro, Wolf-Dietrich Weber and Luiz Andre Barroso, Failure Trends in a Large Disk Drive Population, Proceeding of the 5th USENIX Conference on File and Storage Technologies (FAST 07), pp , Feb [7] Bianca Schroeder and Garth A. Gibson, Disk failures in the real world: What does an MTTF of 1,000,000 hours ean to you?, Proceeding of the 5th USENIX Conference on File and Storage Technologies (FAST 07), Article no. 1, Feb [8] Backblaze, Hard Drive Data and Stats [Online]. Available: https: // [9] The R Foundation, The R Project for Statistical Coputing [Online]. Available: [10] Gerry Cole, Estiating Drive Reliability in Desktop Coputers and Consuer Electoronics Systes, Technology Paper fro Seagate, TP 338.1, Nov