Reliability Measures of a Series System with Weibull Failure Laws

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Iteratioal Joural of Statistics ad Systems ISSN 973-2675 Volume, Number 2 (26), pp. 73-86 Research Idia Publicatios http://www.ripublicatio.com Reliability Measures of a Series System with Weibull Failure Laws S.K. Chauha ad S.C. Malik Departmet of Statistics, M.D. Uiversity, Rohtak 24 (Haryaa) Email: statskumar@gmail.com & sc_malik@rediffmail.com Abstract The Weibull distributio is widely used i reliability ad life data aalysis due to its versatility. Ad, this distributio has bee cosidered as a popular life time distributio which describes modelig pheomea with mootoic failure rates of compoets. Depedig o the values of the parameters it ca be used to model a variety of life behaviors. I this paper, reliability measures such as reliability ad mea time to system failure (MTSF) of a series system of idetical compoets by cosiderig Weibull failure laws are obtaied. The results for these measures are also evaluated for the special case of Weibull distributio i.e. by assumig Rayleigh failure laws. The behavior of MTSF ad reliability has bee observed graphically for arbitrary values of the parameters related to umber of compoets, failure rates ad operatig time. Keywords: Series System, Reliability, MTSF, Weibull Failure Laws. INTRODUCTION It has commoly kow that performace of operatig systems depeds etirely o the cofiguratios of their compoets. The system may have simple or complex structure of the compoets. Ad, accordigly several cofiguratios of the compoets have bee evolved as a result of research i the field of reliability egieerig. The series systems are oe of them beig used i may systems like wheat harvestig system where a tractor, wago ad combie are coected i series. I a series system, the compoets are arraged i such a way that the successive operatio of the system depeds o the proper operatio of all the compoets. Therefore, reliability of such systems has become a matter of cocer for the egieers ad researchers i order to idetify the factors which ca be used to improve their performace. There are several systems i which compoets have

74 S.K. Chauha ad S.C. Malik mootoic failure rates. For example, the hazard rate of rotatig shafts, valves ad cams are of o liear ature due to agig ad workig stress. I such systems, the compoet s life time distributed by cumulative damage ad thus they have icreasig failure rate with passage of time. Balagurusamy (984) ad Sriath(985) determie reliability measures of a series system for Expoetial distributio. Elsayed(22) developed reliability measure of some system cofiguratios usig Expoetial, Rayleigh ad Weibull distributios. Navarro ad Spizzichio(2) made a Compariso of series ad parallel systems with compoets sharig the same copula. Recetly, Nadal et al. (25) evaluated the Reliability ad Mea time to System Failure (MTSF) of a Series System with expoetial failure laws. The Weibull distributio is widely used i reliability ad life data aalysis due to its versatility. Ad, this distributio has bee cosidered as a popular life time distributio which describes modelig pheomea with mootoic failure rates of compoets. Depedig o the values of the parameters it ca be used to model a variety of life behaviors. I this paper, reliability measures such as reliability ad mea time to system failure (MTSF) of a series system of idetical compoets by cosiderig Weibull failure laws are obtaied. The results for these measures are also evaluated for the special case of Weibull distributio i.e. by assumig Rayleigh failure laws. The behavior of MTSF ad reliability has bee observed graphically for arbitrary values of the parameters related to umber of compoets, failure rates ad operatig time. 2. NOTATIONS R(t) = Reliability of the system, R i (t) = Reliability of the i th compoet h(t)= Istataeous failure rate of the system, h i (t) = Istataeous failure rate of i th compoet, λ = Costat failure rate T = Life time of the system, T i = Life time of the i th compoet. 3. SYSTEM DESCRIPTION Here, a series system of compoets is cosidered which ca fail at the failure of ay oe of the compoets. The state trasitio diagram is show i Fig. Fig: A series system of compoets. The reliability of the system is give by R(t) = Pr[T>t] = Pr[mi(T, T 2,.., T )>t] = Pr[T >t, T 2 >t,.,t >t] = i= Pr [Ti > t] = R i (t) i= ()

Reliability Measures of a Series System with Weibull Failure Laws 75 The mea time to system failure is give by MTSF= R i (t) dt i= (2) 4. RELIABILITY MEASURES OF A SERIES SYSTEM WITH WEIBULL DISTRIBUTION: Suppose failure rate of all compoets are govered by the Weibull failure law i.e. h i (t) = λ i t β i The, the compoets reliability is give by R i (t) = e t h i (u)du = e t λ iu β idu = e λ i t Therefore, the system reliability is give by R s (t) = R i (t) = e λ t i Ad, i= MTSF = e i= λ i t β i + β i + β i + β i + β i + β i + i= = e λ t β i + i= i β i + dt = For idetical compoets we ca have h i (t) = λt β Г β i + i= [λ i (β i +) β i] β i + The the system reliability is give by λtβ+ R s (t) = e β+ ad MTSF= e λtβ+ β+ dt = Г β+ [λ(β+) β ] β+ Illustratios. For a sigle compoet, the system reliability is give by R s (t) = i= R i (t) = e λ i t β i + β i + i= = e λ t β + For idetical compoets, we ca have h i (t) = λt β The the system reliability is give by R s (t) = e λtβ+ β+ ad MTSF= Г β+ [λ(β+) β ] β+ β+ ad MTSF= Г β + [λ (β +) β ] β+ 2. Suppose system has two compoets, the the system reliability is give by R s (t) = e λ t β i + i β i + = e 2 λ tβ i + i= t β + 2 i β i + = β+ +λ 2 tβ 2+ β2+ ] i= e [λ

76 S.K. Chauha ad S.C. Malik MTSF= Г β + Г β 2 + [λ (β +) β ] β+ [λ 2 (β 2 +) β 2] β2+ For idetical compoets, we ca have h i (t) = λt β The the system reliability is give by 2λtβ+ R s (t) = e β+ ad MTSF= R s (t)dt = e 2λtβ+ β+ dt = Г β+ [2λ(β+) β ] β+ I a similar way we ca obtai reliability ad MTSF of a system havig three or more compoets coected i series. 5. RELIABILITY MEASURES FOR ARBITRARY VALUES OF THE PARAMETERS Reliability ad mea time to system failure (MTSF) of the system has bee obtaied for arbitrary values of the parameters associated with umber of compoets(), failure rate (λ), operatig time of the compoet (t) ad shape parameter (β). The results are show umerically ad graphically as: No. of Compoets λ=., t=, β=. Table : Reliability Vs No. of Compoets () Reliability λ=.2, t=, β=. λ=.3, t=, β=. λ=.4, t=, β=. λ=.5, t=, β=..89859.79542.79395.6326797.56426 2.79542.63268.5324.42835.3839 3.79395.5324.356996.253253.79655 4.63268.4284.25325.62269.372 5.56426.3839.79655.3723.572 6.5324.25325.27446.64362.32276 7.44882.2439.94.45777.822 8.4284.6227.6436.256727.276 9.356996.27446.45498.62426.5799.3839.372.32276.2763.3272

Reliability Reliability Measures of a Series System with Weibull Failure Laws 77.9.8.7.6.5.4.3.2. 2 3 4 5 6 7 8 9 No. of Compoets λ=. λ=.2 λ=.3 λ=.4 λ=.5 Fig.2: Reliability Vs No. of Compoets () Table 2: MTSF Vs No. of Compoets () No. of MTSF Compo ets λ=., t=, β=. λ=.2, t=, β=. λ=.3, t=, β=. λ=.4, t=, β=. λ=.5, t=, β=. 69.235736 36.8667 25.5655 9.63228 6.2768 2 36.866728 9.63228 3.5796227.45459 8.53569 3 25.5655 3.57962 9.39393 7.23428 5.93697 4 9.6322766.45459 7.234286 5.567284 4.54599 5 6.276796 8.53569 5.9369698 4.54599 3.7593 6 3.5796227 7.23428 5.9728 3.85884 3.4384 7.839398 6.28584 4.3479843 3.347339 2.732749 8.454597 5.567284 3.85884 2.964693 2.42359 9 9.39393 5.97 3.45985696 2.663652 2.7459 8.535687 4.54599 3.4383993 2.42359.975967

MTSF 78 S.K. Chauha ad S.C. Malik 8 7 6 5 4 3 2 2 3 4 5 6 7 8 9 No. of Compoets λ=. λ=.2 λ=.3 λ=.4 λ=.5 Fig.3: MTSF Vs No. of Compoets () Table 3: Reliability Vs No. of Compoets () No. of Reliability Compoets β=., λ=.,t= β=.2, λ=.,t= β=.3, λ=.,t= β=.4, λ=.,t= β=.5, λ=.,t=.89859.876276.85776.8357544.8992 2.79542.767859.735678.6984855.655972 3.79395.672856.633.5837623.53286 4.63268.58968.5422.487889.43299 5.56426.56659.46424.477495.34859 6.5324.452736.39864.347784.282264 7.44882.39672.3452.28487.22862 8.4284.347637.29292.238288.8558 9.356996.34626.25243.989336.49963.3839.266937.25495.662596.2458

Reliability Reliability Measures of a Series System with Weibull Failure Laws 79.9.8.7.6.5.4.3.2. 2 3 4 5 6 7 8 9 No. of Compoets β=. β=.2 β=.3 β=.4 β=.5 Fig.4: Reliability Vs No. of Compoets () Table 4: MTSF Vs No. of Compoets () No. of Compo ets MTSF β=.5, λ=.,t= β=., λ=.,t= β=.2, λ=.,t= β=.3, λ=.,t= β=.4, λ=.,t= 69.235736 5.82565 39.4668 3.9362 25.4854 2 36.866728 28.52493 22.99827 8.9577 6.5463 3 25.5655 2.3463 6.7733 4.8634 2.2598 4 9.6322766 6.98 3.44888.5523.378 5 6.276796 3.29254.327643 9.849336 8.75795 6 3.5796227.4888 9.8423475 8.64667 7.7826 7.839398.4233 8.73992948 7.74549 6.964474 8.454597 8.98479 7.8867626 7.4556 6.37284 9 9.39393 8.4488 7.23675 6.47246 5.8936 8.535687 7.4685 6.6428238 6.3237 5.4966

MTSF 8 S.K. Chauha ad S.C. Malik 8 7 6 5 4 3 2 2 3 4 5 6 7 8 9 No. of Compoets β=. β=.2 β=.3 β=.4 β=.5 Fig.5: MTSF Vs No. of Compoets () Table 5: Reliability Vs No. of Compoets () No. of Reliability Compoe ts t=5, λ=., β=. t=, λ=., β=. t=5, λ=., β=. t=2, λ=., β=. t=25, λ=., β=..9489.89859.836294.782459.7383 2.89872.79542.699387.622294.5342 3.85994.79395.584893.479394.39345 4.87698.63268.48942.3748248.285276 5.76575.56426.4967.293282.28488 6.725894.5324.342.2294787.52369 7.68854.44882.28696.795558.356 8.652376.4284.23926.44936.8382 9.68458.356996.292.99294.59476.58634.3839.67336.8643.43467

Reliability Reliability Measures of a Series System with Weibull Failure Laws 8.9.8.7.6.5.4.3.2. 2 3 4 5 6 7 8 9 No. of Compoets t=5 t= t=5 t=2 t=25 Fig.6: Reliability Vs No. of Compoets ad Time 6. RELIABILITY MEASURES FOR A SPECIAL CASE (RAYLEIGH DISTRIBUTION) OF WEIBULL DISTRIBUTION: The Rayleigh distributio has extesively bee used i life testig experimets, reliability aalysis, commuicatio egieerig, cliical studies ad applied statistics. This distributio is a special case of Weibull distributio with the shape parameter β=. Whe compoets are govered by Rayleigh failure laws, the compoet reliability is give by R i (t) = e t h i (u)du = e t λ iudu = e λ i t2 2, where h i (t) = λ i t Therefore, the system reliability is give by R s (t) = R i (t) = e λ it 2 2 = e λ it 2 i= 2 i= Ad, MTSF = t= R(t)dt i= = e λ i t2 i= t= For idetical compoets we ca have λ i t = λt The system reliability is give by 2 dt = Π 2 i= λ i R s (t) = e λt2 2 ad MTSF= R(t)dt = e λt2 2 dt t= t= = Π 2λ

82 S.K. Chauha ad S.C. Malik Illustratios:. For a sigle compoet the system reliability is give by R s (t) = Ad MTSF= t= e λ i t2 λ it2 2 i= = e i= 2 R(t)dt = e λ i t2 i= t= For idetical compoets, we ca have λ i t = λt The the system reliability is give by 2 dt R s (t) = e λt2 2 ad MTSF= e λt2 2 dt = Π t= 2λ = Π 2 i= λ i = Π 2λ 2. Suppose system has two compoets, the the system reliability is give by 2 R s (t) = i= R i (t) = e Ad MTSF= t= R(t)dt λ 2 it2 i= 2 = e 2 λ i t2 i= t= For idetical compoet, we ca have λ i t = λt The the system reliability is give by R s (t) = e λt2 ad MTSF= e λt2 dt t= 2 dt = 2 Π λ = Π 2 2 i= λ i = Π 2(λ +λ 2 ) I a similar way we ca obtai reliability ad MTSF of a system havig three or more compoets coected i series. 7. RELIABILITY MEASURES FOR ARBITRARY VALUES OF THE PARAMETERS Reliability ad mea time to system failure (MTSF) of the system has bee obtaied for arbitrary values of the parameters associated with umber of compoets(), failure rate (λ) ad operatig time of the compoet (t) The results are show umerically ad graphically as:

Reliability Reliability Measures of a Series System with Weibull Failure Laws 83 Number of Compoets Table 6: Reliability Vs No. of Compoets () Reliability λ=.,t= λ=.2,t= λ=.3,t= λ=.4, t= λ=.5, t=.6653.367879.22332.35335283.828499862 2.367879.35335.49787.835639.6737947 3.2233.49787.9.2478752.5538437 4.35335.836.24788.335463.4539993 5.8285.6738.553.454.372665 6.49787.2479.234.644.359 7.397.92.275.832.25 8.836.335.6.3.26 9.9.23.4.5.7.6738.45.3.2..7.6.5.4.3.2. λ=. λ=.2 λ=.3 λ=.4 λ=.5 2 3 4 5 6 7 8 9 No. of Compoets Fig.7: Reliability Vs No. of Compoets ()

MTSF 84 S.K. Chauha ad S.C. Malik Table 7: MTSF Vs No. of Compoets () No. of MTSF Compoets λ=.,t= λ=.2,t= λ=.3,t= λ=.4,t= λ=.5,t= 2.533 8.8622 7.236 6.2645 5.649 2 8.8622 6.2665 5.66 4.43 3.9633 3 7.236 5.66 4.777 3.686 3.2364 4 6.2665 4.43 3.68 3.332 2.824 5 5.649 3.9633 3.236 2.824 2.566 6 5.66 3.68 2.954 2.5583 2.2882 7 4.737 3.3496 2.7349 2.3685 2.84 8 4.43 3.332 2.5583 2.255.987 9 4.777 2.9548 2.42 2.888.8683 3.9633 2.824 2.2882.986.7724 4 2 8 6 4 2 λ=. λ=.2 λ=.3 λ=.4 λ=.5 2 3 4 5 6 7 8 9 No. of Compoets Fig.8: MTSF Vs No. of Compoets ()

Reliability Reliability Measures of a Series System with Weibull Failure Laws 85 Table 8: Reliability Vs No. of Compoets () No. of Compoets t=5, λ=. t=, λ=. Reliability t=5, λ=. t=2, λ=. t=25, λ=..882497.6653.3246525.35335283.43936933623 2.7788.367879.53992.835639.9345436 3.687289.2233.3428.2478752.8488235 4.6653.35335.9.335463.3726653 5.53526.8285.3666.454.63738 6.472367.49787.79.644.794 7.46862.397.38.832.36 8.367879.836.234.3.4 9.324652.9.4.5..28655.6738.3.2..9.8.7.6.5.4.3.2. 2 3 4 5 6 7 8 9 No. of Compoets t=5 t= t=5 t=2 t=25 Fig.9: Reliability Vs No. of Compoets () 8. DISCUSSION OF THE RESULTS The results obtaied for arbitrary values of the parameters idicate that reliability ad mea time to system failure of a series system of idetical compoets keep o decreasig with the icrease of the umber of compoets ad their failure rates. However, the effect of umber of compoets ad their failure rates o reliability of the system is much more i case compoets govered by Rayleigh failure laws the

86 S.K. Chauha ad S.C. Malik that of Weibull failure laws. I case of mea time to system failure, the effect is much more whe compoets follow Weibull failure laws rather tha Rayleigh failure laws. The reliability of the system goes o decreasig with the icrease of operatig time irrespective of distributios govered by failure time of the compoets. The effect of operatig time o reliability is much more i case compoets follow Rayleigh failure laws as compare to Weibull failure laws. However, there is o effect of operatig time o mea time to system failure (MTSF). The results obtaied for some more particular values of the shape parameter β (.,.2,.3,.4 ad.5) idicate that reliability ad mea time to system failure of a series system of idetical compoets declie with the icrease of the value of β. The results are show umerically ad graphically i respective tables ad figures. CONCLUSION I preset study, we coclude that the reliability ad MTSF keep o decreasig with the icrease the umber of compoets, failure rates ad operatig time of the compoet. It is suggested that least umber of compoet should be used i a series system for better performace. However, the performace of such systems ca be improved by utilizig compoets which follow Weibull failure laws. REFERENCES [] Balagurusamy, E. (984): Reliability Egieerig, Tata McGraw Hill Publishig Co. Ltd., Idia. [2] Sriath, L.S. (985): Cocept i Reliability Egieerig, Affiliated East-West Press (P) Ltd. [3] Rausad, M. ad Hsylad, A. (24): System Reliability Theory, Joh Wiley & Sos, Ic., Hoboke, New Jersey, [4] Navarro, J. ad Spizzichio, F. (2): Comparisos of series ad parallel systems with compoets sharig the same copula, Applied Stochastic Models I Busiess ad Idustry, Vol. 26(6), pp. 775-79. [5] Elsayed, A. (22): Reliability Egieerig, Wiley Series i Systems Egieerig ad Maagemet. [6] Nadal, J., Chauha, S.K. & Malik, S.C. (25): Reliability ad MTSF of a Series ad Parallel systems, Iteratioal Joural of Statistics ad Reliability Egieerig, Vol. 2(), pp. 74-8. [7] Chauha, S.K. ad Malik, S.C. (26): Reliability Evaluatio of a Series ad Parallel systems for Arbitrary Values of the Parameters, Iteratioal Joural of Statistics ad Reliability Egieerig, Vol. 3(), pp.-9,

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