CC BY
32
A Case Study to Analyze Ageing Phenomenon in
Reliability Theory
PULAK SWAIN1* SUBARNA BHATTACHARJEE2" SATYA Kr. MlSRA3
•
1 School of Basic Sciences, IIT Bhubaneswar, Argul-752050, Odisha, India
2 Department of Mathematics, Ravenshaw University, Cuttack-753003, Odisha, India
3 Department of Mathematics, KIIT University, Bhubaneswar-751024, Odisha, India [email protected], [email protected], [email protected]
Abstract
Hazard rate, and ageing intensity (AI) are measures or functions required for qualitative and quantitative analysis of ageing phenomena of a system with a well defined statistical distribution respectively. In this paper, we reiterate upon the fact that in a few cases hazard rate and ageing intensity do not depict the same pattern as far as monotonicity is concerned. So, a question naturally arises which among hazard rate, and ageing intensity is a preferable measure for characterizing ageing phenomena of a system. As a consequence, an example involving two design systems are analyzed and is illustrated to answer the aforementioned question.
Keywords: Ageing phenomenon, hazard rate, ageing intensity function. AMS 2020 Subject Classification: Primary 60E15, Secondary 62N05, 60E05
1. INTRODUCTION
The notion of ageing phenomena and its mathematical counterpart are established by Barlow and Proschan (1975), Shaked and Shanthikumar (2007), Deshpande and Purohit (2005), Nanda et al. (2010) to name a few. The measures (or functions) usually used in this context are many, namely, survival function, hazard rate function, reversed hazard rate function, mean residual function, reversed mean residual function (cf. Block et al. (1998), Nanda et al. (2003,2005)).
Jiang et al. (2003) came forward with ageing intensity function relevant in reliability analysis. He established that the quantitative analysis of ageing phenomena for a system can be done using ageing intensity (AI) function, whereas hazard rate does the qualitative analysis.
The ageing intensity function (AI), denoted by Lx(t) of a random variable X at time t > 0, with probability density function fX(t), survival function FX(t) and failure rate AX(t) =
*The work was jointly done with the first author when he was in Ravenshaw University, Cuttack-753003, Odisha, India "Corresponding author : E-mail: [email protected]
fX(t)/FX(t) is given by (cf. Jiang et al. (2003)),
LX(t) = = , [fx,, where defined, X() Fx(t) lnFx(t)' '
tXx(t) (1.1)
f0 (u)du
Nanda et al. (2007) and Bhattacharjee et al. (2013), Giri et al. (2021) derive the AI function of a few distributions. Sunojand Rasin (2017) introduce quantile-based ageing intensity function and study its various ageing properties. To learn more on ageing intensity function, one can refer to Misra and Bhattacharjee (2018), Szymkowiak (2018a,b) to name a few.
Stochastic orders play an important role in the theory of reliability as it helps in comparison of systems based on the functions, discussed in this section, namely survival function F(t), hazard rate function A(t), reversed hazard rate function f(t), mean residual function m(t), ageing intensity function L(t) etc. giving rise to usual stochastic order (ST order), hazard rate order (HR order ), reversed hazard rate order (RHR order), mean residual order (MRL order) and ageing intensity order(AI order) respectively. The stochastic orders are mathematically represented as given in the next definition.
Definition 1.1. A random variable X is said to be smaller than another random variable Y in
(i) usual stochastic order (denoted by X <ST Y) ifFX(t) < FY(t),for all t > 0.
(ii) hazard rate order (denoted by X <HR Y) if Ax(t) > AY(t),for all t > 0.
(iii) reversed hazard rate order (denoted by X <RHR Y) if px(t) < pY(t),for all t > 0.
(iv) mean residual life order (denoted by X <MRL Y) ifmX(t) < mY(t),for all t > 0.
(v) AI order (denoted by X <AI Y) ifLX(t) > LY(t),for all t > 0.
Based on the hazard rate function, an ageing class has been defined in the literature as follows.
Definition 1.2. A random variable X is said to have increasing (decreasing) hazard rate function, denoted by IFR(DFR), if AX(t) is increasing (decreasing) in t > 0.
The words 'failure rate' and 'hazard rate' have been synonymously used in this article. Throughout the article, the words increasing (decreasing) and non-decreasing (non-increasing) are used interchangeably.
Section 2 discuss the monotonic properties of failure rate and ageing intensity functions in a few statistical distributions. Section 3 simply highlights the estimator of functions appearing in this paper. Section 4 cites an example to illustrate the study of ageing phenomena through reliability function, hazard rate, reversed hazard rate and ageing intensity functions. Section 5 demonstrates the concluding remarks of the work.
2. MONOTONICITY OF FAILURE RATE AND AGEING INTENSITY FUNCTIONS
On the basis of the monotonicity of the AI function, Nanda et al. (2007) define ageing classes, namely increasing ageing intensity class (IAI) (decreasing ageing intensity class (DAI)) if
the corresponding AI function L(t) is increasing(decreasing) in t > 0. It was pointed out that the monotonic behavior of the failure rate function is not, in general, transmitted to the monotonicity of the AI function, which is established by the following examples.
Example 2.1. (cf. Nanda et al. (2007)) Let X has Erlang distribution, with density function with fX (t) = A2 te-At, t > 0. Clearly,its failure rate function is rX (t) = A21/(1 + At) which increases for t > 0, i.e., X has increasing failure rate (IFR). On the other hand, LX (t) = A2t2/(1 + At)(At - ln(1 + At)), decreases for t > 0, i.e., X is DAI. So, X is IFR but DAI.
Example 2.2. (cf. Nanda et al. (2007)) Let Xbe a random variable having uniform distribution over [a, b], 0 < a < b < ro, i.e., Then, its failure rate rX (t) = 1/ (b - t), a < t < b is increasing in t e (a, b), i.e., X is IFR. However, LX(t) = t/(b - t)/ ln (b/b - t), for a < t < b, is increasing in t,a < t < b. So, X is IFR and IAI.
In the next example, we find that a random variable is DFR and DAI.
Example 2.3. Let Xbe a random variable having Pareto distribution with density function or fX (t) = aka/ta+1, for t > k > 0, so that its failure rate rX (t) = a/1, is decreasing in t e (k, ro). i.e., X is DFR. However, LX(t) = 1/(ln t - ln k), is increasing in t e (k, ro). Thus, X is DFR and IAI.
Through these aforementioned examples, one concludes that an IFR random variable could be IAI or DAI. So, does a DFR random variable. The non-monotonic nature are also observed for some statistical distributions (cf. Nanda et al. (2007, 2013)).
Reliability analysts can obviously strive for a question, if a system (or a random variable) depicts different characteristics in terms of failure rate and ageing intensity function then which function should be used in the final conclusion of knowing the behavior of the system in terms of ageing phenomena. In this paper, we try to answer this question by giving a case study mentioned in Section 4 and analyzing it.
3. ESTIMATOR OF FUNCTIONS
Nanda et al. (2013) gives the logical estimates of survival function Fx (t), probability density function fx(t), hazard rate function AX(t), reversed hazard rate pX(t) and ageing intensity function LX(t). Let n units be put to test at t = 0. Further, let the number of units having survived at ordered times tj be ns(tj). Then logical estimates of FX(t), fX(t), AX(t), pX(t) and LX(t) for tj < t < tj + Atj, are respectively given by
Fx (t) =
fx(t) -
ns(tj)
ns (tj) - ns (tj + Atj) nAtj '
{ns (tj) - ns (tj + Atj)) AX (t) = --r^-
XW ns (tj)Atj '
|ns (tj) - ns (tj + Atj)}
p X (t)
(n - ns (tj ))Atj
n
Thus, logical estimate of LX(t) is
-t{ns(tj) - ns(tj + Atj)}
Lx (t) =
ns (tj )Atj ln "M
for tj < t < tj + Atj.
4. An Example to illustrate The study of ageing PhENomENA Through
RELIAbILITY FUNCTION, hAzARD RATE, REvERSED hAzARD RATE AND AGEING
INTENSITY functions
A good number of life testing data can be found for analysis in Shooman (1968), Ebeling (1997) and others.
Example 4.1. (cf. Ebeling (1997)) Fifteen units each of two different deadbolt locking mechanisms were tested under accelerated conditions until 10 failures of each were observed. The following failure times in thousands of cycles were recorded as in Table 1. Which design appears to provide the best function?
Note that, estimator of probability density function for ti < t < ti+1 is
R(ti+1) - R(ti)
f (t)
(ti+1 - ti) 1
that of failure rate function is
X(t)
(ti+1 - ti)(n + 1) f (t)
(4.2)
R(t)
(ti+1 - ti)(n + 1 -i))' The estimator of reversed hazard rate is given by,
№ = (f (t))/(F(t))
= 1/(ti+1 - ti)(n + 1) i / (n + 1) 1
(4.3)
i(ti+1 - ti)
Now, for the ageing intensity, it is given by,
-tf(t)
(4.4)
L(t)
F(t) ln F(t)
_-t/(ti+1 - ti)(n + 1)_
{(n + 1 - i)/(n + 1)} ln{(n + 1 - i)/(n + 1)} -t
(ti+1 - ti)(n + 1 - i) ln(n + 1 - i)/(n + 1)
(4.5)
The detailed analysis of the example considered in this Section are given in Table 2, Table 3, Table 4, Table 5 and Table 6. The Plots are also displayed in Figure 1, Figure 2, Figure 3 and Figure 4.
1
5. CONCLUSION
According to the literature on stochastic orders, we know that any system, say, here, DesignA is said to be better than design-B, if design-A has less ageing intensity, less hazard rate and higher reliability than that of design-B. The concluding remarks as noted in Table 6 at a certain interval of time are summarized as follows:
(i) Design A is better than design B in terms of the function being doubly underlined in a time interval.
(ii) Design B is better than design A on the basis of the function being singly underlined during a certain time interval.
(iii) However, the function being starred in a time interval denotes the fact that we cannot specify which among A or B is of the better design.
(iv) For example, in the interval (56.8,77], design B is better in terms of ageing intensity, whereas according to hazard rate, design A is better during (56.8,63] and design B is better in the interval (63,77]. Also, the analyzing the systems in terms of reliability reveal that, both the designs A and B have equal reliabilities during (56.8,63] but design-A is better on (63,77].
(v) It is evident that Table 6 contains more singly underlined cells than than that of doubly underlined cells.
(vi) In a nutshell, design B is more efficient than that of design A.
(vii) We attempt to identify the function which should be preferred in determining the ageing behaviour of a system.
In Table 6, one can observe that if at some interval of time the ageing intensity, hazard rate and the reliability have the same nature (either single underlined or doubly underlined) or (doubly underlined with starred) or ( singly underlined with starred),then all the three measures give the same conclusion in choosing the best system design. But if one function is doubly underlined and another is singly underlined, then it gives different conclusion with regard to the performance of the systems.
(viii) For example, on the interval (56.8,63], the ageing intensity and the hazard rate show different behaviour, whereas on the interval (63,77] hazard rate and reliability show different behaviour. And on (897.8,1043.6], all the three measures show same behaviour.
(ix) Clearly, from Table 6 we can see that, hazard rate doesn't have opposite behaviour with the other two measures simultaneously. For example, on the interval (56.8,63], hazard rate shows opposite behaviour to ageing intensity function only, but not to reliability. Also, it shows opposite behaviour to reliability on (63,77], but not to the ageing intensity function in that interval. We note that, hazard rate doesn't have any doubtful situations (Ai = A2), which are in the case of ageing intensity or reliability at some intervals. (as, the equality sign doesn't say anything about which design is better, so these are the doubtful situations.)
Therefore, we conclude that, hazard rate should be preferred as a measure of ageing phenomena, while comparing the two systems in the problem concerned.
Table 1: Failure Times
Design A 44 77 218 251 317 380 438 739 758 1115
Design B 32 63 211 248 327 404 476 877 903 1416
Table 2: Analysis of Design A
Table 3: Analysis of Design B
i t R1(t) Mi) F1 (*) L1(t)
0 0 1 0.002066
1 44 0.909 0.00303 0.022727 0.3179t
2 77 0.8182 0.000788 0.015152 0.00393i
3 218 0.7273 0.003788 0.002364 0.01189t
4 251 0.6364 0.002165 0.007576 0.004791
5 317 0.5455 0.002646 0.00303 0.004361
6 380 0.4546 0.003448 0.002646 0.004371
7 438 0.3636 0.000831 0.002463 0.000821
8 739 0.2727 0.017544 0.000415 0.01351
9 758 0.1818 0.001401 0.005848 0.000821
10 1115 0.0909 0.00028
i t Ri(t) A2 (t) Ml) L2(t)
0 0 1 0.002841
1 32 0.909 0.002933 0.03125 0.0339t
2 63 0.8182 0.000614 0.016129 0.00374t
3 211 0.7273 0.002457 0.002252 0.0106t
4 248 0.6364 0.001151 0.006757 0.004t
5 327 0.5455 0.001181 0.002532 0.00357t
6 404 0.4546 0.001263 0.002165 0.00352t
7 476 0.3636 0.000227 0.001984 0.00062t
8 877 0.2727 0.003497 0.000312 0.00987t
9 903 0.1818 0.000177 0.004274 0.00057t
10 1416 0.0909 0.000195
Table 4: Comparison ofR(t), A(t), f(t)
Time R1 (t) R2 (t Order R(t) A1() A2 (t Order A(t) n(t) n (t Order p(t)
(0,32] 1 1 R2 II R1 0.002066 0.002841 A1 < A2 0.022727 0.03125 F1 < F2
(32,44] 1 0.909 R1 > R2 0.002066 0.003226 A1 < A2 0.022727 0.016129 H > F2
(44,63] 0.909 0.909 R2 II R1 0.00303 0.003226 A1 < A2 0.015152 0.016129 F1 > F2
(63,77] 0.909 0.8182 R1 > R2 0.00303 0.000751 A1 < A2 0.015152 0.002252 F1 > F2
(77,211] 0.8182 0.8182 R2 II R1 0.000788 0.000751 A1 < A2 0.002364 0.002252 F1 > F2
(211,218] 0.8182 0.7273 R1 > R2 0.000788 0.003378 A1 < A2 0.002364 0.006757 F1 > F2
(218,248] 0.7273 0.7273 R2 II R1 0.003788 0.003378 A1 < A2 0.007576 0.006757 F1 > F2
(248,251] 0.7273 0.6364 R1 > R2 0.003788 0.001808 A1 < A2 0.007576 0.002532 F1 > F2
(251,317] 0.6364 0.6364 R2 II R1 0.002165 0.001808 A1 < A2 0.00303 0.002532 F1 > F2
(317,327] 0.5455 0.6364 R1 < R2 0.002646 0.001808 A1 < A2 0.002646 0.002532 F1 > F2
(327,380] 0.5455 0.5455 R2 II R1 0.002646 0.002165 A1 < A2 0.002646 0.002165 F1 > F2
(380,404] 0.4546 0.5455 R1 < R2 0.003448 0.002165 A1 < A2 0.002463 0.002165 F1 > F2
(404,438] 0.4546 0.4546 R2 II R1 0.003448 0.002778 A1 < A2 0.002463 0.001984 F1 > F2
(438,476] 0.3636 0.4546 R1 < R2 0.000831 0.002778 A1 < A2 0.000415 0.001984 F1 > F2
(476,739] 0.3636 0.3636 R2 II R1 0.000831 0.000623 A1 < A2 0.000415 0.000312 F1 > F2
(739,758] 0.2727 0.3636 R1 < R2 0.017544 0.000623 A1 < A2 0.005848 0.000312 F1 > F2
(758,877] 0.1818 0.3636 R1 < R2 0.001401 0.000623 A1 < A2 0.00028 0.000312 F1 > F2
(877,903] 0.1818 0.2727 R1 < R2 0.001401 0.012821 A1 < A2 0.00028 0.004274 F1 > F2
(903,1115] 0.1818 0.1818 R2 II R1 0.001401 0.000975 A1 < A2 0.00028 0.000195 F1 > F2
(1115,1416] 0.0909 0.1818 R1 < R2 0.000975
Figure 1: Plot of R1 and R2 versus time t.
Comparison of Reliability
L
0 200 400 600 800 1000 1200 1400 1600
time (t)
Figure 2: Plot of H R1 and HR2 versus time t
Comparison of Hazard rate
time (t)
Table 5: L1 (t) and L2(t)
Design A Design B
t L1 (0 t L2(t)
32 1.0848 44 1.39876
38.2 1.29498 50.6 1.608574
44.4 1.50516 57.2 1.818388
50.6 1.71534 63.8 2.028202
56.8 1.92552 70.4 2.238016
63 0.23562 77 0.30261
92.6 0.346324 105.2 0.413436
122.2 0.457028 133.4 0.524262
151.8 0.567732 161.6 0.635088
181.4 0.678436 189.8 0.745914
211 2.2366 218 2.59202
218.4 2.31504 224.6 2.670494
225.8 2.39348 231.2 2.748968
233.2 2.47192 237.8 2.827442
240.6 2.55036 244.4 2.905916
248 0.992 251 1.20229
263.8 1.0552 264.2 1.265518
279.6 1.1184 277.4 1.328746
295.4 1.1816 290.6 1.391974
311.2 1.2448 303.8 1.455202
327 1.16739 317 1.38212
342.4 1.222368 329.6 1.437056
357.8 1.277346 342.2 1.491992
373.2 1.332324 354.8 1.546928
388.6 1.387302 367.4 1.601864
404 1.42208 380 1.6606
418.4 1.472768 391.6 1.711292
432.8 1.523456 403.2 1.761984
447.2 1.574144 414.8 1.812676
461.6 1.624832 426.4 1.863368
476 0.29512 438 0.35916
556.2 0.344844 498.2 0.408524
636.4 0.394568 558.4 0.457888
716.6 0.444292 618.6 0.507252
796.8 0.494016 678.8 0.556616
877 8.65599 739 9.9765
882.2 8.707314 742.8 10.0278
887.4 8.758638 746.6 10.0791
892.6 8.809962 750.4 10.1304
897.8 8.861286 754.2 10.1817
903 0.51471 758 0.62156
1005.6 0.573192 829.4 0.680108
1108.2 0.631674 900.8 0.738656
1210.8 0.690156 972.2 0.797204
1313.4 0.748638 1043.6 0.855752
1416 1115
Table 6: Interval-wise Study
Interval Compare L(t) Interval Compare m Interval Compare R(t)
(56.8,77] L1 > l2 (56.8,63] Ä1 < A2 (56.8,63] R1 = R2
(63,77] A1 > A2 (63,77] R1 > R2
(77,211] L1 = L2 (77,211] A1 > A2 (77,211] R1 = R2
(211,240.6] L1 > l2 (211,218] A1 < A2 (211,218] R1 > R2
(218,240.6] A1 > A2 (218,240.6] R1 = R2
(240.6,248] L1 = L2 (240.6,248] A1 > A2 (240.6,248] R1 = R2
(248,418.4] L1 > l2 (248,418.4] A1 > A2 (248,251] R1 > R2
(251,317] R1 = R2
(317,327] R1 < R2
(327,380] R1 = R2
(380,404] R1 < R2
(404,418.4] R1 = R2
(418.4,476] L1 < l2 (418.4,438] A1 > A2 (418.4,438] R1 = R2
(438,476] A1 < A2 (438,476] R1 = R2
(476,636.4] L1 = L2 (476,636.4] A1 > A2 (476,636.4] R1 = R2
(636.4,796.8] L1 > l2 (636.4,796.8] A1 > A2 (636.4,739] R1 = R2
(739,758] R1 < R2
(758,796.8] R1 < R2
(796.8,897.8] L1 < l2 (796.8,877] A1 > A2 (796.8,877] R1 < R2
(877,897.8] A1 < A2 (877,897.8] R1 < R2
(897.8,1043.6] L1 > l2 (897.8,1043.6] A1 > A2 (897.8,903] R1 < R2
(903,1043.6] R1 = R2
Figure 3: Plot of RHRi and RHR2 versus time t
Figure 4: Plot of AI1 and AI2
Acknowledgements
The authors would like to thank the editor and the anonymous reviewers for their useful comments.
References
[1] Barlow, R.E. and Proschan, F. (1975), Statistical Theory of Reliability and Life Testing, Probability Models. Holt Rinehart and Winston Inc. New York.
[2] Bhattacharjee, S., Nanda A.K. and Misra, S.K. (2013), Reliability Analysis using ageing intensity function. Statistics and Probability Letters, 83,1364-1371.
[3] Block, H.W., Savits, T. and Singh, H. (1998), Reversed hazard rate functions. Probability in the Engineering and Informational Sciences, 12, 69-90.
[4] Ebeling C.E. (1997), An Introduction to Reliability and Maintainability Engineering Michigan: Mcgraw Hill.
[5] Deshpande, J.V. and Purohit, S.G. (2005), Life Time Data: Statistical Models and Methods. Series in Quality, Reliability and Engineering Statistics, Vol 11, World scientific Publishing Co. Pte Ltd.
[6] Giri, R. L., Nanda, A. K., Dasgupta, M., Misra, S. Kr. and Bhattacharjee, S., On ageing intensity function of some Weibull models, Communications in Statistics-Theory and Methods, DOI: 10.1080/03610926.2021.1910845, Published online on 26 April 2021.
[7] Jiang, R., Ji, P., and Xiao, X. (2003), Ageing property of univariate failure rate models, Reliability Engineering and System Safety, 79,113-116.
[8] Misra, S.K. and Bhattacharjee, S. (2018), A case study of ageing intensity funtion on censored data, Alexendria Engineering Journal, 57, 3931-3952.
[9] Nanda, A. K., Bhattacharjee, S. and Balakrishnan, N. (2010), Mean residual life fuction, associated orederings and properties, IEEE Transactions on Reliability, 59(1), 55-65.
[10] Nanda, A.K., Singh, H., Misra, N. and Paul, P. (2003), Reliability properties of reversed residual lifetime, Communications in Statistics-Theory and Methods, 32(10), 2031-2042. [Addendum: Communications in Statistics-Theory and Methods, (2004), 33(4), 991-992].
[11] Nanda, A.K. and Sengupta, D. (2005), Discrete life distributions with decreasing reversed hazard. Sankhya 67, Part 1,1-19.
[12] Nanda, A. K., Bhattacharjee, S., Alam, S.S. (2007), Properties of ageing intensity function, Statistics and Probability Letters, 77, 365-373.
[13] Nanda, A.K., Bhattacharjee, S. and Balakrishnan,N. (2010), Mean Residual Life Function, Associated Orderings and Properties, IEEE Transactions on Reliability, 59(1),55-65.
[14] Nanda, A.K., Bhattacharjee, S. and Alam, S.S. (2006), On Up Shifted Reversed Mean Residual Life Order, Communications in Statistics: Theory and Methods, 35(8), 1513-1523.
[15] Shaked, M., and Shanthikumar, J.G. (2007), Stochastic Orders. New York: Springer.
[16] Shooman, M. L. (1968), Probabilistic Reliability: An Engineering Approach. McGraw-Hill, Inc.
[17] Sunoj, S. M., and Rasin, R. S. (2018). A quantile-based study on ageing intensity function,
Communications in Statistics: Theory and Methods, 47 (22), 5474-5484.
[18] Szymkowiak, M. (2018a), Characterizations of Distributions Through Ageing Intensity, IEEE Transactions on Reliability, 67, 446-458.
[19] Szymkowiak, M. (2018 b), Lifetime Analysis by Ageing Intensity Functions Studies in Systems, Decision and Control Volume 196 Series editor, Springer.
