IJQRM-Oct-2011-0143

International Journal of Quality & Reliability ManagementGeneralized q-Weibull model and the bathtub curveEdilson M. Assis Ernesto P. Borges Silvio A.B. Vieira de Melo

Article information:To cite this document:Edilson M. Assis Ernesto P. Borges Silvio A.B. Vieira de Melo, (2013),"Generalized q-Weibull model and thebathtub curve", International Journal of Quality & Reliability Management, Vol. 30 Iss 7 pp. 720 - 736Permanent link to this document:http://dx.doi.org/10.1108/IJQRM-Oct-2011-0143

Downloaded on: 05 October 2015, At: 20:07 (PT)References: this document contains references to 25 other documents.To copy this document: [email protected] fulltext of this document has been downloaded 233 times since 2013*

Users who downloaded this article also downloaded:S. Samar Ali, S. Kannan, (2011),"A diagnostic approach to Weibull-Weibull stress-strength model and itsgeneralization", International Journal of Quality & Reliability Management, Vol. 28 Iss 4 pp. 451-463http://dx.doi.org/10.1108/02656711111121834Yi-Hui Liang, (2013),"Applying the generalized autoregressive conditional Heteroskedastic model toanalyze and forecast the field failure data of repairable systems", International Journal of Quality &Reliability Management, Vol. 30 Iss 7 pp. 737-750 http://dx.doi.org/10.1108/IJQRM-Feb-2012-0027

Access to this document was granted through an Emerald subscription provided by emerald-srm:273599 []

For AuthorsIf you would like to write for this, or any other Emerald publication, then please use our Emerald forAuthors service information about how to choose which publication to write for and submission guidelinesare available for all. Please visit www.emeraldinsight.com/authors for more information.

About Emerald www.emeraldinsight.comEmerald is a global publisher linking research and practice to the benefit of society. The companymanages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well asproviding an extensive range of online products and additional customer resources and services.

Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committeeon Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archivepreservation.

*Related content and download information correct at time of download.

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

RELIABILITY PAPER

Generalized q-Weibull modeland the bathtub curve

Edilson M. Assis, Ernesto P. Borges and Silvio A.B. Vieira de MeloPolytechnic School Federal University of Bahia, Salvador, Brazil

Abstract

Purpose The purpose of this paper is to analyze mathematical aspects of the q-Weibull model andexplore the influence of the parameter q.

Design/methodology/approach The paper uses analytical developments with graphillustrations and an application to a practical example.

Findings The q-Weibull distribution function is able to reproduce the bathtub shape curve for thefailure rate function with a single set of parameters. Moments of the distribution are also presented.

Practical implications The generalized q-Weibull distribution unifies various possibledescriptions for the failure rate function: monotonically decreasing, monotonically increasing,unimodal and U-shaped (bathtub) curves. It recovers the usual Weibull distribution as a particularcase. It represents a unification of models usually found in reliability analysis. Q-Weibull model has itsinspiration in nonextensive statistics, used to describe complex systems with long-range interactionsand/or long-term memory. This theoretical background may help the understanding of the underlyingmechanisms for failure events in engineering problems.

Originality/value Q-Weibull model has already been introduced in the literature, but it was notrealized that it is able to reproduce a bathtub curve using a unique set of parameters. The paper brings amapping of the parameters, showing the range of the parameters that should be used for each typeof curve.

Keywords Failure rate, Generalized Weibull, Reliability, Bathtub curve, Reliability management,Distribution curves, Distribution functions

Paper type Research paper

1. IntroductionReliability analysis widely uses Weibull (1951) distribution, that is a simple andpowerful empirical model. Many branches of knowledge have applied this distribution.These are some recent examples: service operations (Hensley and Utley, 2011), theproblem of the strength of a manufactured item against stress (Ali and Kannan, 2011)and large-scale information systems supporting infrastructures deterioration processformulated by a Weibull hazard model (Kobayashi and Kaito, 2011). Weibull probabilitydensity function (pdf) at time t, where t , T and T is time to failure, is given by:

f t bh2 t0

t 2 t0h2 t0

b21exp 2

t 2 t0h2 t0

b" #1

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0265-671X.htm

Ernesto P. Borges and Edilson M. Assis are grateful to Giovana O. Silva for remarks and forsending to them some references. This work is partially supported by Brazilian agency CNPq Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (473925/2007-9).

Received 30 October 2011Revised 26 April 2012Accepted 17 December 2012

International Journal of Quality &Reliability ManagementVol. 30 No. 7, 2013pp. 720-736q Emerald Group Publishing Limited0265-671XDOI 10.1108/IJQRM-Oct-2011-0143

IJQRM30,7

720

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

with b . 0, h . t0, t $ t0, andR1

0 f xdx 1. Equation (1) may be viewed as ageneralization of the exponential distribution, which is recovered if parameterb is takenas unity.

Various generalizations of Weibull model have been proposed: linear or nonlineartransformation of time, use of multiple distributions, time dependence of parameters,discrete, multivariate, stochastic models, etc. (Murthy et al., 2004) for a comprehensiveapproach). Xie et al. (2000) compares the approximated exponential distribution usingthe average failure rate with the Weibull reliability. Almost all proposals ofgeneralization of Weibull model share a common feature: they rely on the exponentialframework (single exponential, exponentials of a variety of functions and so forth).

In the following we briefly point out some theoretical remarks about the emergence ofexponential and non exponential distributions in statistical mechanics, which serve asmotivation for our approach to the problem. Exponentials are usually found innon-interacting or weakly interacting systems. Systems that exhibit long-range (spatial)interactions, long-term (temporal) memory, effects of competition/cooperation, amongothers, usually can be classified as complex (Bak, 1997) and power-laws dominate theirstatistical distributions, in contrast to simple systems, that are the realm of exponentiallaws. Failure of a component may have many (recent or not) multiple and interactingcauses, some of them acting on a cooperative and others on a conflictive basis, so it is notsurprising that complex behavior may appear. If this happens, power-law-likeexpressions are expected to substitute exponentials in the statistical description.

Statistical mechanics of simple systems has a well established theoretical framework,and probability distributions with exponentials (e.g. Boltzmann weight, Maxwelliandistribution among many others) are derived from Boltzmann-Gibbs-Shannon (BGS)entropy. On the other hand, theoretical basis of the statistical description of complexsystems is object of intense current research.

The definition of the nonextensive entropy (Tsallis, 1988), which is a generalization ofBGS entropy (by means of a parameter q, also known as entropic index), has introducedthe possibility to extend statistical mechanics to complex systems in a coherent andnatural way. The developments surpassed the bounds of physics and have lead toapplications in different areas, including topics in applied mathematics (Tsallis, 2009).We focus on the q-exponential function, which naturally appears in nonextensiveformalism, defined as:

expqx 1 12 qx1=12q; if 1 12 qx $ 00; otherwise;

8 1

Weibul, < 1

0.4

0.2

00 50 100 150

t

h (t)

Notes: The displayed q-Weibull distribution is a generalizationof ordinary Weibull and is able to represent the bathtub curve;the values of the parameters were chosen just to give a goodvisual representation

IJQRM30,7

722

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

Section 2 introduces the model and some of its features are shown in Section 3. Section 4brings an example and our conclusions and final remarks are developed in Section 5.

2. q-Weibull failure rate modelThe q-Weibull model is obtained from the classical Weibull model (equation (1)) by thesubstitution of the exponential function by a q-exponential (see details in Costa et al.(2006)):

f qt 22 q bh2 t0

t 2 t0h2 t0

b21expq 2

t 2 t0h2 t0

b" #: 3

The factor (2 2 q) and the constraint q , 2 are necessary due to normalizationrequirements. The ordinary Weibull pdf is recovered in the limit q! 1, and coherentlyequation (1) shall now be denoted as f1(t). h is the scale parameter and t0 is the locationparameter of q-Weibull model as well as in Weibull model. However, both theparameters b and q control the shape of the q-Weibull distribution, while in the Weibullmodel, only the parameter b affects its shape.

The q-Weibull distribution is also a generalization of Burr XII distribution function(Burr, 1942):

f t ck tc21

s c1 t

s

c 2k21k . 0; c . 0; s . 0; 4

if the parameters of q-Weibull are taken as b c, h s/(k 1)1/c and q (k 2)/(k 1) . 1. It is worth a mention that q-Weibull is a generalization of Burr XII,and not the opposite, as claimed by Nadarajah and Kotz (2006), once equation (4) demandsq . 1, while equation (3) is also defined for q # 1. Burr XII distribution can assumedifferent shapes, which allow it to be a good candidate to fit various lifetimes data.

The q-Weibull reliability function is consistently given by Rqt R1t f qt 0dt 0, i.e.:

Rqt 12 12 q t2t0h2t0 b 22q=12q

expq 2 t2t0h2t0 b 22q

;

5

where we use the symbol [A ] (first line of equation (5)), that means that [A ] A ifA $ 0 and [A ] 0 if A , 0. This is already implicit in equation (2): we use it hereand also in some equations in the following just to remind the reader of the cut-offcondition of the q-exponential. To deduce equation (5) we have used the followingproperty of the q-exponential function:Z

expqaxdx 122 qa expqax22q: 6

Note that (expqx)a expq(ax) for q 1, but:

expqxa exp1212q=aax; 7

Generalizedq-Weibull model

723

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

So that equation (5) may be alternatively written as Rq(t) expq0[2 (2 2 q)((t 2 t0)/(h 2 t0))

b ] with q0 1/(22 q). The interested reader may find more propertiesof q-exponentials at Yamano (2002).

The cumulative distribution function Fq(t) is the complement to the reliabilityfunction, Fqt 12 Rqt, and the instantaneous failure rate, defined as hqt ;f qt=Rqt is generalized to:

hqt 22qbh2t0 t2t0h2t0 b21 12 12 q t2t0h2t0

b 21

22qbh2t0 t2t0h2t0 b21 expq 2 t2t0h2t0

b q21;

8

which is consistently reduced to the usual Weibull version as q! 1:

h1t bh2 t0

t 2 t0h2 t0

b21: 9

This is precisely the origin of the difference of behaviors between usual (q 1) andq-Weibull models: the integral of an ordinary exponential is an exponential (except froma multiplicative constant), and they cancel out in the expression for the failure rate withq 1. That does not happen with hq(t), due to the property given by equation (6).

Equation (8) is able to represent four different types of failure rate function, accordingto the values of the parameters, besides the constant type (with q 1 and b 1). hq(t) ismonotonically decreasing for 1 , q , 2 and 0 , b , 1, monotonically increasing for q, 1 andb. 1, unimodal for 1, q, 2 andb. 1 and U-shaped (bathtub curve) for q, 1and 0 , b , 1. The non-monotonic hazard function cited by Vuorenmaa (2006)corresponds to the unimodal type and the bathtub shape was not covered by that paper.Figure 2 shows the four possibilities (detailed analysis of the q parameter is performed inSection 3), and Figure 3 shows the corresponding four unreliability curves.

For q , 1, equation (8) presents a divergence that defines the maximum allowedtime (lifetime deadline) at:

tmax t0 h2 t012 q21=b: 10Finite tmax corresponds to a relaxation of the constraint usually imposed to a cumulativefailure rate functionHqt

R t0hqtdt (Pham and Lai, 2007): it is normally expected that

H1 ! 1 at t! 1. According to q-Weibull model, Hq,1! 1 at t! tmax , 1. That isto say that ordinary Weibull is unlimited, while q-Weibull (with q , 1) is limited to tmax.Coherently, limq!12tmax !1, as q approaches the unity from the left.

The time derivative of the q-failure rate is:

h0qt 22 qbb2 1

h2 t02t 2 t0h2 t0

b2212 12 q=12 bt 2 t0=h2 t0b12 12 qt 2 t0=h2 t0b2

:

11For the unimodal case (1 , q , 2 and b . 1) and for the U-shaped case (q , 1 and0 , b , 1), the root of equation (11) is located at:

IJQRM30,7

724

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

Figure 3.Unreliability curves of the

q-Weibull distribution

1

0.8

0.6

0.4

0.2

00 2 4 6 8 10 12

t

F q (t)

Notes: The parameters are the same of those used in Figure 2;the four types of failure rate associated are: (i) monotonicallydecreasing (solid line); (ii) monotonically increasing (dashedline); (iii) unimodal (dot-dashed line); (iv) U-shaped (bathtubcurve) (dot-dot-dashed line)

Figure 2.Types of failure ratecurves described by

q-Weibull

U-shaped

DecreasingUnimodal

Increasing

1.5

1

0.5

00 2 4 6 8 10

h q (t)

t

Notes: Values of the parameters were chosen to give a goodvisualization of the curves in the same figure; the four typesare: (i) monotonically decreasing function: (q = 1.5, = 0.5, = 1); (ii) monotonically increasing function: (q = 0.5, b = 2,h = 7.071, evaluated from equation (10) with tmax = 10);(iii) unimodal function: (q = 1.5, b = 2, h = 1); (iv) U-shaped(bathtub curve): (q = 0.5, b = 0.5, h = 2.5, evaluated fromequation (10) with tmax = 10


725

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

t* t0 h2 t0 12 b12 q

1=b; 12

which corresponds to the extreme value (maximum for unimodal case, minimum forbathtub case):

hqt* 22 qh2 t0

12 b

12 q

b21=b: 13

Figure 4 shows the change of sign in time derivative of hq(t).The time derivative of the usual (q 1) Weibull failure rate is a monotonic

power-law:

h01t bb2 1h2 t02

t 2 t0h2 t0

b22; 14

Hence it is unable to represent the whole bathtub curve. h01t , 0 for 0 , b , 1, andthis situation can just describe the warm in phase. Wear out phase needs h01t . 0, andthis happens in usual Weibull for b . 1. Description of intermediary random failurephase happens by imposing b 1. q-Weibull failure rate reproduces the whole curveby a continuous function with the same set of parameters.

3. Influence of the parameter qIn order to exhibit the effect of the parameter q , 1 on the q-Weibull model, let us considerthe instance b 0.5. First we keep parameter h constant (let us assume h 1 forsimplicity). The usual (q 1) Weibull does not present a limiting lifetime (i.e. tmax 1).As q departs from unity (from below), lifetime deadline gets smaller values, as Figure 5shows. Second, let us keep tmax constant (we choose the instanceb 0.5 and tmax 100),

Figure 4.Time derivative h0qt,given by equation (11),with q 0.5, b 0.5 andh 2.5 (corresponding tothe U-shaped curve) andq 1.5, b 2 and h 1(corresponding to theunimodal curve)

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0 2 4 6 8 10t

h'q

(t)

U-shaped

Unimodal

Notes: Parameters are the same of those in Figure 2 (monotoniccases are not shown); the change of sign in h'q(t) is responsiblefor the proper description of the whole bathtub curve

IJQRM30,7

726

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

soh is obtained according to equation (10). Figure 6 shows curves for different values of q.As q approaches unity (from below), intermediate random failure phase decreases andminimum of failure rate (equation (13)) increases. Particularly limq!12hqt* !1.Minimum value of hq is found at limb!1limq! 2 1 hq(t*) 1/tmax.

Influence of q on unimodal case (1 , q , 2 and b . 1) can be viewed in Figure 7.There is a displacement of the maximum failure rate as q approaches the value 2.

For 1 , q , 2 and 0 , b , 1, q-Weibull failure rate is a monotonically decreasingfunction and Figure 8 shows examples.

4. ExamplesWe illustrate the flexibility and the reach of the q-Weibull distribution, incomparison to the usual Weibull model, with three examples, extracted fromcomponents of oil wells.

We maximize the coefficient of determination R 2 of the estimated unreliabilityFî i2 0:3=n 0:4 (Bernards approximation), for each sample i (n is the totalnumber of samples), properly linearized as yi ln2lnq0 12 Fî vs xi lnti 2 t0,with q0 1=22 q. Note that the usual procedure must be changed, as we are dealingwith the q-Weibull model, and thus there is a q-logarithm within the expression of yi .As an additional criterion to evaluate the goodness of the fittings, we also evaluate themean squared error (MSE).

When the censoring of the data is simple type-I, type-II or multiply censored data Fîis corrected as done with Weibull distribution (Rinne, 2008).

Tables I-III present times to failure data (in days) of oil pumps, pumping rods, andproduction tubings, respectively. In Table II, 1,448 times to failure of pumping rods weregrouped within 20 time intervals, and the relative frequency of occurrence was used toestimate the unreliability, for each interval. Table III shows 115 different values of timeto failure (repetitions were excluded from 438 samples in order to reduce the size of

Figure 5.q-Weibull failure rate

curve as a function of timefor different values ofq , 1 in log-log scale

104

103

102

101

100

104

102

h q (t)

104 103 102 101 100 101 102 103

t

q = 1

= 0.5 = 1

q = 100.50.80.9

Notes: All curves are calculated with b = 0.5 and h = 1;limiting lifetime comes from t = , for q = 1 to closerand finite values, as the parameter q departs from unityfrom below


727

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

the table). All 438 sample values were used to estimate the unreliability, according to themedian rank.

Table IV shows the fitting parameters for each example, and also the coefficient ofdetermination and the MSE. There are two examples with b , 1 and q , 1, and one

Figure 7.q-Weibull failure ratefor unimodal case, withb 2 and h 1, anddifferent values of q . 1(indicated)

1.2

1.5

1.71.9

= 2

2

1.5

1

0.5

00 10 20 30 40

t

1 1.5 2q

102

101

100

101

102

h q (t)

h q (t*

)

Note: Inset shows maximum of failure rate as a function of q(equation (13))

Figure 6.q-Weibull failure ratecurve as a function of timefor different values ofq , 1

0 20 40 60 80 100

1

0.8

0.6

0.4

0.2

0

h q (t)

t

01

0.5

0.9

0.95

= 0.5tmax = 100

q = 0.97

Notes: All curves are calculated with b = 0.5 and tmax = 100,so h is taken from equation (10): h = 0.09, 0.25, 1, 25, 100,400 corresponds to q = 0.97, 0.95, 0.9, 0.5, 0, 1, respectively;as q approaches unity, intermediate random failure phasedecreases, and minimum value of failure rate hq(t*) increases(hq(t*) for q 1). As q , curves tend to a lowerbound (this particular instance, hq(t*) = 0.02, from equation(9) with b = 0.5 and tmax =100)

IJQRM30,7

728

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

with b . 1 and q . 1, thus covering different behaviors of the failure rate. In all thecases, the q-Weibull presented a greater coefficient of determination and a smallermean square error.

Figures 9-11 shows the reliability and failure rate curves for the three examples. It isto be noted that the usual Weibull model (with q 1) systematically departures fromthe experimental data (circles) for large times, in the three examples considered, whilethe q-Weibull model is able to fit the whole range of the data.

i Time interval Failures i Time interval Failures

1 1 # t # 93.1 625 11 922 , t # 1,014.1 82 93.1 , t # 185.2 320 12 1,014.1 , t # 1,106.2 43 185.2 , t # 277.3 170 13 1,106.2 , t # 1,198.3 44 277.3 , t # 369.4 65 14 1,198.3 , t # 1,290.4 45 369.4 , t # 461.5 65 15 1,290.4 , t # 1,382.5 86 461.5 , t # 553.6 67 16 1,382.5 , t # 1,474.6 47 553.6 , t # 645.7 12 17 1,474.6 , t # 1,566.7 18 645.7 , t # 737.8 33 18 1,566.7 , t # 1,658.8 19 737.8 , t # 829.9 33 19 1,658.8 , t # 1,750.9 1

10 829.9 , t # 922 22 20 1,750.9 , t # 1,843 1

Table II.Time to failure of

pumping rods in days

Pump time to failure in ascending order (days)

8 38 42 59 71 146 184185 199 204 214 379 457 457494 515 568 680 684 808 964

Table I.Pump time to failure in

days in ascending order(days)

Figure 8.q-Weibull failure rate,

given by equation (8), withb 0.5, h 1 and

different values of q . 1

3

2

1

0

h q (t)

0 0.5 1 1.5 2t

q = 1.0

q = 1.2

q = 1.5

q = 1.9

= 0.5 = 0.5

Note: hq(t) is a monotonically decreasing function for b < 1and 1 q < 2


729

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

Tim

eto

fail

ure

ofp

rod

uct

ion

tub

ing

(in

day

s)

1(2)

4(2)

6(10

)7(

8)8(

7)9(

2)10

(3)

12(2

)14

(4)

15(2

)17

(9)

19(8

)20

(3)

22(3

)23

(3)

24(5

)25

(1)

26(1

0)27

(5)

29(6

)30

(5)

31(2

)32

(1)

34(3

)35

(5)

36(1

2)38

(16)

41(2

)42

(3)

43(1

1)44

(2)

46(2

)47

(5)

48(3

)51

(2)

53(6

)55

(6)

56(3

)60

(2)

61(4

)63

(9)

64(8

)65

(6)

68(2

)69

(3)

70(4

)73

(6)

74(3

)75

(2)

78(3

)79

(3)

80(2

)81

(4)

82(2

)83

(4)

86(4

)87

(4)

88(2

)89

(6)

91(5

)92

(3)

93(4

)94

(3)

101(

4)10

6(3)

107(

2)10

8(3)

111(

4)11

7(3)

119(

3)12

1(3)

123(

3)12

4(3)

126(

3)13

3(4)

136(

2)14

2(3)

143(

1)14

8(2)

150(

2)15

4(2)

157(

3)16

1(4)

163(

2)16

7(3)

168(

2)17

0(5)

172(

3)17

7(2)

178(

4)18

5(4)

189(

3)19

4(3)

203(

3)20

7(6)

210(

4)21

9(3)

220(

3)22

2(3)

226(

3)22

7(2)

233(

3)23

4(3)

238(

3)24

5(3)

248(

4)26

5(2)

277(

3)34

7(4)

393(

2)42

5(3)

432(

4)48

8(2)

688(

2)69

1(3)

Note:

Nu

mb

erof

rep

etit

ion

sof

val

ues

inp

aren

thes

es

Table III.Time to failure ofproduction tubing in days

IJQRM30,7

730

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

Two examples (oil pumps and pumping rods) present failure rates with a bathtub shape,and the last example (production tubings) exhibits the failure rate as a unimodal function.Of course the usual Weibull model (with q 1) is unable to represent these cases.

5. Final remarksSeveral models for failure rate function are found in the literature, many of them useWeibull (or Weibull-like) as a basis. These distributions share in common theexponential nature. The q-Weibull generalization uses a function that is exponential

Pump Pumping rod Production tubingWeibull q-Weibull Weibull q-Weibull Weibull q-Weibull

b 1.05 0.82 0.95 0.42 0.12 1.31h (day) 383 1,277 195 520 105 65t0 (day) 27.66 20.51 256 59 0.61 0.24Q 1.00 0.00 1.00 0.46 1.00 1.30R 2 0.9761 0.9815 0.9904 0.9981 0.9818 0.9900MSE 2.16 1023 1.59 1023 2.05 1024 2.74 1025 5.81 1024 2.52 1024

Table IV.Fitting results

Figure 10.Weibull (dotted lines) and

q-Weibull (solid lines)models, and experimental

data (circles)

weibull q-Weibull

101

102

103

weibull

q-Weibull

102 103

t (102day)t (day)5 10 15

3

4

5

6

7

h q (10

3 d

ay

1 )

R q

Notes: Left panel: log-log plot of reliability curves; right panel: failure rate curves; both plotsshow abscissas in time to failure of pumping rods

Figure 9.Weibull (dotted lines) and

q-Weibull (solid lines)models, and experimental

data (circles)

3.2

3

2.8

2.6

2.4

h q (1

03 d

ay

1 ) q-WeibullWeibull

Weibull

0 1 2 3 4 5 6 7 8 9t (102day)

101

100

101

102 103

t (day)

R q

q-Weibull

Notes: Left panel: log-log plot of reliability curves; right panel: failure rate curves; abscissasshow time to failure of pumps


731

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

only as a limiting case, and may yield asymptotic power-laws. The q-Weibull model isable to describe four types of failure rate function, namely monotonically decreasing,monotonically increasing, unimodal and U-shaped curves, with a single parsimoniousset of three parameters, representing a unification of various models, including theversatile Burr XII distribution. Table V summarizes the possibilities with thecorresponding ranges of parameters.

Usual (q 1) Weibull model is unable to represent the whole bathtub curve, once h1(t)is monotonically decreasing or monotonically increasing, depending on the value ofparameter b. Modeling of U-shaped bathtub curve with Weibull requires a piecewisedescription, with b , 1 for the warm in phase, then b 1 for the intermediary randomfailure phase and finally b . 1 for the wear out phase. In the present work we haveshown the q-Weibull capacity of continuously reproducing the whole bathtub curve withthe same set of constant parameters and without need of introducing ad hoc hypotheses.

We compare the usual Weibull with the q-Weibull models by means of threeexamples which present different behaviors: failure rate as a bathtub shape and as aunimodal curve. In all cases, the performance of the q-Weibull was superior to that of theusual Weibull. Of course such a result was to be expected, due to the extra parameter q,but it is important to remark that the improvements of the fittings are not merelyquantitative (as it should be, due to the additional parameter), but also qualitative, oncethe q-Weibull model can describe behaviors (bathtub shape, and unimodal shape in thefailure rate curve) that are impossible to be described by the usual Weibull model.

q-Weibull is a natural extension of usual Weibull, and it has the advantage of beingoriginated from a theoretical background rooted in nonextensive statistical physics. Ofcourse the introduction of additional (empirically or theoretically based) generalizations,like the use of linear or nonlinear transformation of time, use of multiple distributions,time dependence of parameters, etc. as it was done with Weibull, will further enhanceflexibility and accuracy of q-Weibull model.

Figure 11.Weibull (dotted lines) andq-Weibull (solid lines)models, and experimentaldata (circles)

100

100 101 102

101

102

103

t (day) t (102day)0 1 2 3 4 5 6 7

4

6

8

10

12

h q (1

03 d

ay1 )

Weibull

q-WeibullWeibull

q-WeibullR q

Notes: Left panel: log-log plot of reliability curves; right panel: failure rate curves; theindependent variable is time to failure of production tubing for both plots

0 , b , 1 b 1 b . 1q , 1 Bathtub curve Monotonically increasing Monotonically increasingq 1 Monotonically decreasing Constant Monotonically increasing1 , q , 2 Monotonically decreasing Monotonically decreasing Unimodal

Table V.Behavior of q-Weibullfailure rate according tothe range of parameters qand b

IJQRM30,7

732

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

References

Ali, S.S. and Kannan, S. (2011), A diagnostic approach to Weibull-Weibull stress-strength modeland its generalization, International Journal of Quality & Reliability Management, Vol. 28No. 4, pp. 451-463.

Bak, P. (1997), How Nature Works, Oxford University Press, Oxford.

Burr, I.W. (1942), Cumulative frequency functions, Ann. Math. Statist., Vol. 13, pp. 215-232.

Costa, U.M.S., Freire, V.N., Malacarne, L.C., Mendes, R.S., Picoli, S., de Vasconcelos, E.A. andda Silva, E.F. (2006), An improved description of the dielectric breakdown in oxides basedon a generalized Weibull distribution, Physica A Statistical Mechanics and itsApplications, Vol. 361 No. 1, pp. 209-215.

Hensley, R.L. and Utley, J.S. (2011), Using reliability tools in service operations, InternationalJournal of Quality & Reliability Management, Vol. 28 No. 5, pp. 587-598.

Kobayashi, K. and Kaito, K. (2011), Random proportional Weibull hazard model for large-scaleinformation systems, International Journal of Quality&ReliabilityManagement (unpublished)

Lenzi, E.K., Mendes, R.S. and Rajagopal, A.K. (1999), Quantum statistical mechanics fornonextensive systems, Phys. Rev. E, Vol. 59, pp. 1398-1407.

Murthy, D.N.P., Xie, M. and Jiang, R. (2004), Weibull Models, Wiley, New York, NY.

Nadarajah, S. and Kotz, S. (2006), q-exponential is a Burr distribution, Phys. Lett. A, Vol. 359,pp. 577-579.

Pham, H. and Lai, C.D. (2007), On recent generalizations of the Weibull distribution, IEEETrans. Reliability, Vol. 56 No. 3, pp. 454-458.

Picoli, S., Mendes, R.S. and Malacarne, L.C. (2003), q-exponential, Weibull, and q-Weibulldistributions: an empirical analysis, Physica A Statistical Mechanics and its Applications,Vol. 324, pp. 678-688.

Prato, D. and Tsallis, C. (1999), Nonextensive foundation of Levy distributions, PhysicalReview E, Vol. 60 No. 2, pp. 2398-2401.

Rinne, H. (2008), The Weibull Distribution: A Handbook, McGraw-Hill, New York, NY.

Sartori, I., Assis, E.M., da Silva, A.L., Vieira de Melo, R.L.F., Borges, E.P. and Vieira de Melo, S.A.B.(2009), Reliability modeling of a natural gas recovery plant using q-Weibull distribution,in Alves, R.M.B., Nascimento, C.A.O. and Biscaia, E.C. (Eds), Computer Aided ChemicalEngineering, Vol. 27, Elsevier, Amsterdam, pp. 1797-1802.

Tsallis, C. (1988), Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., Vol. 52,pp. 479-487.

Tsallis, C. (1994), Extensive versus nonextensive physics, in Moran-Lopez, J.L. andSanchez, J.M. (Eds), New Trends in Magnetism, Magnetic Materials and TheirApplications, Plenum Press, New York, NY, pp. 451-463.

Tsallis, C. (2005), Nonextensive statistical mechanics, anomalous diffusion and central limittheorems, Milan J. Math., Vol. 73, pp. 145-176.

Tsallis, C. (2009), Introduction to Nonextensive Statistical Mechanics: Approaching a ComplexWorld, Springer, New York, NY.

Tsallis, C., Plastino, A.R. and Alvarez-Estrada, R.F. (2009), Escort mean values and thecharacterization of power-law-decaying probability densities, Journal of MathematicalPhysics, Vol. 50 No. 4, p. 043303.

Tsallis, C., Levy, S.V.F., Souza, A.M.C. and Maynard, R. (1996), Statistical-mechanicalfoundation of the ubiquity of Levy distributions in nature, Physical Review Letters, Vol. 75No. 20, pp. 3589-3593, 1995, Vol. 77, p. 5442 (erratum).


733

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

Umarov, S., Tsallis, C. and Steinberg, S. (2008), On a q-central limit theorem consistent withnonextensive statistical mechanics,Milan Journal ofMathematics, Vol. 76 No. 1, pp. 307-328.

Vuorenmaa, T. (2006), A q-Weibull autoregressive conditional duration model and thresholddependence, Discussion Paper No. 117, University of Helsinki, Helsinki.

Weibull, W. (1951), A statistical distribution function of wide applicability, Journal of AppliedMechanics, pp. 293-297.

Xie, M., Kong, H. and Goh, T.N. (2000), Exponential approximation for maintained Weibulldistributed component, Journal of Quality in Maintenance Engineering, Vol. 6 No. 4,pp. 260-268.

Yamano, T. (2002), Some properties of q-logarithm and q-exponential functions in Tsallisstatistics, Physica A, Vol. 305, pp. 486-496.

Appendix. Some mathematical properties of the q-Weibull distributionA probability distribution is better characterized when its moments are known, and here weadvance them. We consider the usual raw moments (moments about zero), and the usual centralmoments (moments about the mean). For a detailed analysis of the generalized moments ofq-distributions, see Tsallis et al. (2009).

To evaluate the raw moments (moments about zero) of equation (3), m0n R1

0 tnf qtdt, we

shall consider separately the cases q , 1 and q . 1. It is not necessary to set h 1 as shown byVuorenmaa (2006). For the case q , 1, it is useful to consider the integral representation of theq-exponential given by Lenzi et al. (1999). For the case q . 1, it is necessary to use the integralrepresentation proposed by Tsallis (1994). Straightforward calculations lead to the raw moments.For q , 1:

m0n h nG 1 n

b

G32 2q=12 q

12 qn=bG32 2q=12 q n=b ; if t0 0; A1

or:

m0n Xnj0

n

j

!tn2j0 h2 t0jG 1

j

b

G32 2q=12 q12 qj=bG32 2q=12 q j=b

( );

with t0 0

A2

and for q . 1:

m0n h nG 1 n

b

G22 q=q2 12 n=bq2 1n=bG22 q=q2 1 ; if t 0 0; A3

or:

m0n Xnj0

n

j

!tn2j0 h2 t0jG 1

j

b

G22 q=q2 12 j=bq2 1j=bG22 q=q2 1

( );

witht0 0

A4

with 1 , q , qupper and qupper 1 b/(n b). Note that q ! 1 recovers themoments of usual Weibull pdf, m0n h nG1 n=b; for t0 0e m0n

Pnj0

n

j

!tn2j0 h2 t0jG1 j=b

( ); for t0 0. The upper limit qupper attains

IJQRM30,7

734

Dow

nloa

ded

by U

nive

rsita

s Gad

jah M

ada A

t 20:0

7 05 O

ctobe

r 201

5 (PT

)

the values limb!0qupper 1, limb!1qupper 2, and limn!1qupper 1. The latter limitingbehavior means that it is not possible that q-Weibull pdf has all its moments for q . 1(all moments are defined for q # 1). As q departs from unity from above (for constant b),q-Weibull loses its higher moments (normalizability, that is m00 1, is preserved ;q , 2). Notethat tmax is the mean time between failures (MTBF). We remind the reader that there are manydistributions that do not have all its moments. The Cauchy-Lorentz distribution, for instance, hasno mean, variance or higher moments. Usual Weibull pdf has all moments, which is typical fordistributions with exponential decay.

Central moments (moments about the mean) are found using the binomial transformation ofthe raw moments, as usual:

mn Xnk0

n

k

!21n2km0km01n2k; if t0 0; A5

or, for t0 0:

mn Xnj0

n

j

0@

1At0 2 m01n2jh2 t0j G 1 jb

G22 q=q2 12 j=bq2 1j=bG22 q=q2 1

8

IJQRM-Oct-2011-0143

Documents

Transcript of IJQRM-Oct-2011-0143