A study on analysis of reliability distributions

When we buy a mobile, laptop, scooter, car, or even a small item like bulb or battery, we can expect function satisfactory for a specific period of time. For this purpose one life testing experiments are wanted. These experiments are designed to find average life of an item or to ensure the questions like is the probability. The item will fail in time interval (t0, t0+t) such thing we will estimate reliability of the item through some continuous probability distributions.

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Reliability of R(t): 1 vs 2 vs 3 vs 4 vs 5Close ×

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Introduction

The reliability distributions are used to model random events for which the outcome is uncertain such as one time of failure for a component Artin E 1964 [1]. Before placing a demand on that component, the aim it will fail is unknown. The distribution of the probability of failure at different times is modelled by reliability distributions Dekking FM et al 2007 [2]. The reliability Distributions Analysis allows us to describe one time to failure (TTF) as a statistical distribution which is usually characterised by a specific pattern.

Four different “A” types are supported: Weibull, Normal, Long Normal, and Exponential Hastings NAJ et al. 2000 [3]. 2. Measures of reliability The most commonly used methods to measure the reliability are. a. Failure rate b. Probability of survival c.

Mean Time Between Failure d. Mean Time To Failure. 3. Failure rate The failure rate is defined as where in (iii) the ratio of number of failures during a particular unit interval to average population in that interval Meeker WQ et al. 1998 [4]. It is given by z(i) = ( 1) ( ) 2 s i si ni   The failure rate is commonly expressed in terms of failures per hour or 100 hours (or) 1000hrs (or) percentage failures per 1000 hrs Meeker WQ et al. 1998 [5].

Sometimes the failure rate is represented by a Greek letter ‘  ’ and it is calculated as a simple ratio of total no. of failures during a specified test interval to total test hours. And it is given by f TotalNo.of failures t totaltesthours   1. Probability of survival: It indicates the probability or expected no. of items that will operate for a required period of time. and Maths 2022; 7(1): 118-121 G Vijaya Lakshmi Research Scholar, Department of Statistics, S.K. University, Anantapur, Andhra Pradesh, India K Pusphanjali Professor, Department of Statistics, S.K.

University, Anantapur, Andhra Pradesh, India G Vijaya Lakshmi Research Scholar, Department of Statistics, S.K. University, Anantapur, Andhra Pradesh, India http://www.mathsjournal.com 2. Mean Time between Failures: It is applicable to Then MTBF =   T 1 1 f f / T    repairable items and is expressed in hrs. It is a ratio of total survival time to no. of failures during that period and it is given by Modarres M et al. [6].

MTBF = T Totaltest hours f No.of failuresduring that period  We can say that MTBF is a reciprocal of constant failure rate. 4. Mean time to failure This measures is applicable to non – repair items. The MTTF is expressed as average time of item. That is expected to function before failure WQ et al. 1998 [7].

Table 1: Types of numbers survivals and failure Time in Number of No. of Failure Reliability of Failure Density F hours Fails(f) Number of Survivals (S) (F) R(t) Failure Rate F(t) d(t) Hazard Function Z(t) 1 130 1184 0 1.00 0.00 0.11 0.04 2 83 1054 130 0.89 0.11 0.07 0.04 3 75 971 213 0.82 0.18 0.06 0.04 4 68 896 288 0.76 0.24 0.06 0.05 5 62 828 356 0.70 0.30 0.05 0.05 6 51 766 418 0.65 0.35 0.04 0.06 7 46 715 469 0.60 0.40 0.04 0.06 8 41 669 515 0.57 0.43 0.03 0.06 9 37 628 556 0.53 0.47 0.03 0.07 10 34 591 593 0.50 0.50 0.03 0.07 11 31 557 627 0.47 0.53 0.03 0.08 12 28 526 658 0.44 0.56 0.02 0.08 13 64 498 686 0.42 0.58 0.05 0.09 14 72 434 750 0.37 0.63 0.06 0.10 15 62 362 822 0.31 0.69 0.05 0.13 16 40 300 884 0.25 0.75 0.03 0.15 17 39 260 924 0.22 0.78 0.03 0.17 18 38 221 963 0.19 0.81 0.03 0.21 19 35 183 1001 0.15 0.85 0.03 0.25 20 30 148 1036 0.13 0.88 0.03 0.31 21 25 118 1066 0.10 0.90 0.02 0.39 22 27 93 1091 0.08 0.92 0.02 0.52 23 25 66 1118 0.06 0.94 0.02 0.78 24 21 41 1143 0.03 0.97 0.02 1.36 25 20 20 1164 0.02 0.98 0.02 4.15 1184 1.00 Fig 1: Shows the time linear Table 2: Analysis of Descriptive statistics that the product will works Average 47 hrs, Minimum 20 hrs and Maximum 130 hrs Mean 47.36 Standard Error 5.02 Median 39.00 Mode 62.00 SD 25.09 Sample Variance 629.57 Kurtosis 3.56 Skewness 1.64 Range 110.00 Minimum 20.00 Maximum 130.00 Sum 1184.00 http://www.mathsjournal.com 1 R(t) = 1- 1 log( )t           The lognormal distribution is similar to the Weibull Distribution (W.D) yet parameters are the Mean and S.D. of the transformed data we have to consider natural log value of each data point Kolowrocki K 1994 [10] 3. The Reliability for Gamma Distribution This distribution has been used as a model for industrial reliability problems. This distribution is characterized by two parameters and  Roy B et al. 2008 [11] Its P.D.F. is given by 5. Reliability Estimation through Distributions 1.

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Shows the time linear

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Mean	47.36
Standard Error	5.02
Median	39.00
Mode	62.00
SD	25.09
Sample Variance	629.57
Kurtosis	3.56
Skewness	1.64
Range	110.00
Minimum	20.00
Maximum	130.00
Sum	1184.00

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Analysis of Descriptive statistics that the product will works Average 47 hrs, Minimum 20 hrs and Maximum 130 hrs

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Time in hours	Number of Fails(f)	Number of Survivals (S)	No. of Failure (F)	Reliability of R(t)	Failure Rate F(t)	Failure Density F d(t)	Hazard Function Z(t)
1	130	1184	0	1.00	0.00	0.11	0.04
2	83	1054	130	0.89	0.11	0.07	0.04
3	75	971	213	0.82	0.18	0.06	0.04
4	68	896	288	0.76	0.24	0.06	0.05
5	62	828	356	0.70	0.30	0.05	0.05
6	51	766	418	0.65	0.35	0.04	0.06
7	46	715	469	0.60	0.40	0.04	0.06
8	41	669	515	0.57	0.43	0.03	0.06
9	37	628	556	0.53	0.47	0.03	0.07
10	34	591	593	0.50	0.50	0.03	0.07
11	31	557	627	0.47	0.53	0.03	0.08
12	28	526	658	0.44	0.56	0.02	0.08
13	64	498	686	0.42	0.58	0.05	0.09
14	72	434	750	0.37	0.63	0.06	0.10
15	62	362	822	0.31	0.69	0.05	0.13
16	40	300	884	0.25	0.75	0.03	0.15
17	39	260	924	0.22	0.78	0.03	0.17
18	38	221	963	0.19	0.81	0.03	0.21
19	35	183	1001	0.15	0.85	0.03	0.25
20	30	148	1036	0.13	0.88	0.03	0.31
21	25	118	1066	0.10	0.90	0.02	0.39
22	27	93	1091	0.08	0.92	0.02	0.52
23	25	66	1118	0.06	0.94	0.02	0.78
24	21	41	1143	0.03	0.97	0.02	1.36
25	20	20	1164	0.02	0.98	0.02	4.15
	1184					1.00

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Types of numbers survivals and failure

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Exponential distribution; 2. Weibull distribution; 3. Gamma distribution; and 4. Lognormal distribution.

We will calculate reliability for given to three different distributions. On the event, we want estimate the reliability of an item at a specific time. Whatever specific situation, you have the life distributions parameters Singpurwalla ND 2006 [8]. We just need to calculate reliability at a specific time.

We can do that and let’s try it with three distributions using their respective reliability functions: Exponential Distribution, Weibull Distribution, and lognormal Distributions Meeker Pabla AS [9] 1. The Reliability Function for the Exponential Distribution (E.D) is f(t) = r t t e r      1 ( ) ( )  =        t / t R(t) e e       2. The Reliability Function for the Weibull Distribution(W.D) is 1 0, 0, 0 ( ) r t t e fort r r t R t e   / ( )     Here  = shape of the parameter    t e = scale parameter; for = 1, it reduced as f(t) = Where the shape parameter and n is the scale parameter. which is a p.d.f. of E.D. Remark: If time equals to n it does not matter what the  value is?

The Reliability Function for the Log N.D. is Table 3: Shows the Number of Fails (f) and Number of Survivals (S) Time Number of Fails(f) Number of Survivals (S) R(t)=e-λ t R(t)=λ e-λ t R(t)=1-ϕ((log(t)-μꞌ)/σꞌ) 1 130 1184 2702.797 128004.46 0.6784 2 83 1054 730.511 34597.0009 0.532 3 75 971 197.442 9350.853 0.6784 4 68 896 53.3646 2527.347 0.6784 5 62 828 14.423 683.073 0.6784 6 51 766 3.898 184.609 7 46 715 1.0534 49.8890 8 41 669 2847.78 134870.861 9 37 628 769.697 36452.8499 10 34 591 208.033 9852.443 11 31 557 56.2272 2662.92 12 28 526 15.197 719.73 13 64 498 4.1074 194.5264 14 72 434 1.1101 52.5743 15 62 362 3000.544 142105.76 16 40 300 0 0 17 39 260 0 0 18 38 221 0 0 19 35 183 0 0 20 30 148 0 0 21 25 118 0 0 22 27 93 0 0 23 25 66 0 0 24 21 41 0 0 25 20 20 0 0 1184

Time	Number of Fails(f)	Number of Survivals (S)	R(t)=e-λ t	R(t)=λ e-λ t	R(t)=1-ϕ((log(t)-μꞌ)/σꞌ)
1	130	1184	2702.797	128004.46	0.6784
2	83	1054	730.511	34597.0009	0.532
3	75	971	197.442	9350.853	0.6784
4	68	896	53.3646	2527.347	0.6784
5	62	828	14.423	683.073	0.6784
6	51	766	3.898	184.609
7	46	715	1.0534	49.8890
8	41	669	2847.78	134870.861
9	37	628	769.697	36452.8499
10	34	591	208.033	9852.443
11	31	557	56.2272	2662.92
12	28	526	15.197	719.73
13	64	498	4.1074	194.5264
14	72	434	1.1101	52.5743
15	62	362	3000.544	142105.76
16	40	300	0	0
17	39	260	0	0
18	38	221	0	0
19	35	183	0	0
20	30	148	0	0
21	25	118	0	0
22	27	93	0	0
23	25	66	0	0
24	21	41	0	0
25	20	20	0	0
	1184

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Shows the Number of Fails (f) and Number of Survivals (S)

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Conclusions

Reliability analysis is determined by based on the above Tables (1) and (2); this research has outlined the status of the extruder studied in terms of functions and typical graphical representations of reliability analysis. Thus, it is possible to state that the objective was successfully achieved, since all the reliability representations of interest were defined based on time to failure. It has been observed that the above calculations, Reliability starts with 1end with 0, total failure density is one. http://www.mathsjournal.com

1. Artin E. The Gamma Function, New York: Holt, Rinehart & Winston; c1964.
2. Dekking FM, et al. A Modern Introduction to Probability and Statistics: Understanding why and How, Springer; c2007.
3. Hastings NAJ, Peacock B, Evans M. Statistical Distributions, 3rd Edition 3rd ed., John Wiley & Sons Inc; c2000.
4. Meeker WQ, Escobar LA. Statistical Methods for Reliability Data 1st ed., Wiley-Inter science; c1998.
5. Meeker WQ, Escobar LA. Statistical Methods for Reliability Data 1st ed., Wiley-Interscience; c1998.
6. Modarres M, Kaminskiy M, Krivtsov V. Reliability engineering and risk analysis, CRC Press; c1999.
7. Meeker WQ, Escobar LA. Statistical Methods for Reliability Data 1st ed., Wiley-Interscience; c1998.
8. Singpurwalla ND. Reliability and Risk: A Bayesian Perspective 1st ed., Wiley; c2006.
9. Pabla AS. Electric Power Distribution, Tata Mc G raw Hill Publishing Company Limited, 5th Ed., New Delhi, India; c2008.
10. Kolowrocki K. Limit reliability functions of some series- parallel and parallel series systems, Journal of Applied Mathematics and Computation. 1994 May 1;62(2-3):129- 51.
11. Roy B, Allan RN. Reliability Evaluation of power systems, 2nd Ed. Springer, New Delhi; c2008. p. 220-221.