Evaluation of Instantaneous Reliability Measures for a Gradual Deteriorating System

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1 General Leers in Mahemaic, Vol. 3, No.3, Dec 27, pp e-issn , p-issn Available online a hp:\\ Evaluaion of Insananeous Reliabiliy Measures for a Gradual Deerioraing Sysem Tijjani A. Waziri, 2 Bashir M. Yakasai, and 3 Ibrahim Yusuf School of Coninuing Educaion, Bayero Universiy, Kano, Nigeria, ijjaniw@gmail.com 2 Deparmen of Mahemaical Sciences, Bayero Universiy, Kano, Nigeria, yakasaibashir9@gmail.com 3 Deparmen of Mahemaical Sciences, Bayero Universiy, Kano, Nigeria, iyusuf.mh@buk.edu.ng Absrac This paper sudies various reliabiliy measures. Reliabiliy measures are quaniies used for analyzing reliabiliy and mainenance models. Someimes when he mainenance cos rae is minimized, he sysem reliabiliy measures are so low ha hey are no accepable in pracice. The following insananeous reliabiliy measures are considered in his paper: insananeous failure rae, insananeous repair rae, insananeous mainainabiliy, insananeous reliabiliy and insananeous availabiliy. The explici expression and numerical resuls for he insananeous reliabiliy measures are shown. Also, heir relaionships are shown. Keywords: Reliabiliy, Gradual, Deerioraion, Insananeous, Sysem 2 MSC No: 9B25. Inroducion Mainenance involves prevenive (planned) and unplanned acions carried ou o reain a sysem or resore i o an accepable operaing condiion. Opimal mainenance policies aim o provide opimum sysem reliabiliy and safey performance a he lowes possible mainenance coss. Proper mainenance echniques have been emphasized due o increased safey and reliabiliy requiremens of sysems, increased complexiy, and rising coss of maerial and labor. One imporan research area in reliabiliy engineering is he sudy of various mainenance policies in order o improve sysem reliabiliy, o preven he occurrence of sysem failure, and o reduce mainenance coss. Reliabiliy heory has grown ou of he valuable experiences from many defecs of miliary sysems in World War II and wih he developmen of modern echnology. For he purpose of making good producs wih high qualiy and designing highly reliable sysems, he imporance of reliabiliy has been increasing grealy wih he innovaion of recen echnology. The heory has been acually applied o no only indusrial, mechanical, and elecronic engineering bu also o compuer, informaion, and communicaion engineering. Many researchers have invesigaed saisically and sochasically complex phenomena of real sysems o improve heir reliabiliy. During operaion he srenghs of sysems are gradually deerioraed, unil some poin of deerioraion failure, or oher ypes of failures. To proper deal wih repairable sysems ha deerioraes, we need o undersand he characerisics and he behavior of he sysem. And his will be done by invesigaing he properies of reliabiliy measures. Reliabiliy measures are quaniies for analyzing reliabiliy and mainenance models. I is imporan o noe ha, minimizing he sysem mainenance cos rae for a paricular sysem may no resul in opimal sysem reliabiliy measures. Someimes when he mainenance cos rae is minimized, he sysem reliabiliy measures are so low ha hey are no accepable in pracice. The insananeous reliabiliy measures ha are considered in his paper are: insananeous failure rae, insananeous repair rae, insananeous mainainabiliy, insananeous reliabiliy and insananeous availabiliy.

2 78 Tijjani A. Waziri e al. There is an exensive lieraure on reliabiliy measures. See Barlow and Proschan [2], Elsayed [3], Nakagawa[4], Pham [5] and Wang and Pham [8]. Also, many researchers have sudied availabiliy for various ypes of sysems. For example, see Yusuf and Hussain [9], Yusuf and Yusuf [], Yusuf e al. [], Sun e al. [7]. Moreover, Suleiman e al. [6] and Aliyu e al. [], analyzed he characerisics of reliabiliy and availabiliy of series parallel subsysems in differen configuraions. 2 Noaion and Descripion of he Sysem 2. Descripion of he Sysem: Consider a sysem, which gradually deerioraes wih ime and usage. The sysem is subjec o a paricular failure, which is recify by minimal repair. 2.2 Noaions: : Insananeous failure rae of he sysem a ime : Insananeous repair rae of he sysem a ime : Failure disribuion of he sysem a ime f(): Time o failure (pdf) he sysem a ime g(): Time o repair (pdf) he sysem a ime : Insananeous reliabiliy of he sysem a ime : Insananeous availabiliy of he sysem a ime : Insananeous mainainabiliy of he sysem a ime 2.3 Insananeous Reliabiliy Measures Insananeous failure rae of he sysem a ime is : () Insananeous reliabiliy of he sysem a ime is :. (2) Insananeous availabiliy ofhe sysem a ime is:. (3) Insananeous mainainabiliy of he sysem a ime is:. (4) Insananeous repair rae of he sysem a ime is:. (5) 3 Resuls and Discussions 3. Example (Supposing he failure and repair raes following he same disribuion): Assumed ha he ime o failure and ime o repair he sysem follows Weibull disribuion:, (6)

3 r() Evaluaion of Insananeous Reliabiliy Measures (7) Le he se of parameers be used for his paricular example:,, and. Observaions made from resuls obained for example :. From able, figure and figure 2, observed ha he insananeous repair rae h() and insananeous failure rae r() boh increases as ime increases. Furhermore, from figure 3, observed ha, he insananeous repair rae h() increases as he insananeous failure rae r() increases. 2. From able and figure 4, observed ha, he insananeous reliabiliy and insananeous availabiliy decreases as ime increases. While, he insananeous mainainabiliy increases as ime increases. 3. Also, observed from able and figure 5, he insananeous reliabiliy and insananeous availabiliy decreases as he insananeous failure rae increases. While, he insananeous mainainabiliy increases as insananeous failure rae increases. 4. Also, observed from able and figure 6, he insananeous reliabiliy and insananeous availabiliy decreases as insananeous repair rae increases. While, he insananeous mainainabiliy increases as insananeous repair rae increases. Observaions made from 2, 3, and 4 above, showed ha mainainabiliy is equivalen o un reliabiliy Table. Resuls obained from he evaluaion of insananeous reliabiliy measures when he failure rae and repair rae boh followed Weibull disribuion r() h() IA() IR() IM() Figure. The plo of insananeous failure rae r() agains ime.

4 h() h() 8 Tijjani A. Waziri e al. 4 x Figure 2. The plo of insananeous repair rae h() agains ime. 4 x r() Figure 3. The plo of insananeous repair rae h() agains insananeous failure rae r()..9 IA() / IR() IM() Figure 4.The plo of insananeous reliabiliy/availabiliy and insananeous mainainabiliy agains ime

5 Evaluaion of Insananeous Reliabiliy Measures IA() / IR() IM r() Figure 5. The plo of insananeous reliabiliy/availabiliy and insananeous mainainabiliy agains insananeous failure rae r().9 IA() / IR() IM() h() x 4 Figure 6. The plo of insananeous reliabiliy/availabiliy and insananeous mainainabiliy agains insananeous repair rae h() 3.2 Example 2 (failure and repair raes follows differen disribuions): Assumed ha he ime o failure of he sysem follows Weibull disribuion: While he repair ime of he sysem follows power law model:. (8). (7) Le he se of parameers be used for his paricular example:,, and.

6 r() 82 Tijjani A. Waziri e al. Table 2. Resuls obained for evaluaing insananeous reliabiliy measures when he failure rae follows Weibull disribuion and he repair rae follows power law r() h() IA() IR() IM() E E Observaions made from resuls obained for example 2:. From able 2, figure 7 and figure 8, observed ha he insananeous repair rae h() and insananeous failure rae r() boh increases as ime increases. Furhermore, from figure, observed ha, he insananeous repair rae h() increases as he insananeous failure rae r() increases. 2. From able 2 and figure 9, observed ha, he insananeous reliabiliy and insananeous availabiliy decreases as ime increases. While, he insananeous mainainabiliy increases as ime increases. 3. Also, observed from able 2 and figure, he insananeous reliabiliy and insananeous availabiliy decreases as he insananeous failure rae increases. While, he insananeous mainainabiliy increases as insananeous failure rae increases. 4. Also, observed from able 2 and figure 2, he insananeous reliabiliy and insananeous availabiliy decreases as insananeous repair rae increases. While, he insananeous mainainabiliy increases as insananeous repair rae increases. 5. Observaions made from 2, 3, and 4 above, showed ha mainainabiliy is equivalen o un reliabiliy Figure 7.The plo of insananeous failure rae r() agains ime

7 h() h() Evaluaion of Insananeous Reliabiliy Measures Figure 8. The plo of insananeous repair rae h() agains ime.9 IR / IA IM Figure 9.The plo of insananeous avail./ relaib. and insananeous mainainabiliy agains ime r() Figure. The plo of insananeous repair rae agains insananeous failure rae

8 84 Tijjani A. Waziri e al..9 IA() / IR() IM r() Figure. The plo of insananeous avail./reliab. and insananeous agains insananeous failure rae.9 IA() / IR() IM() h() Figure 2. The plo of insananeous availabiliy / reliabiliy and insananeous mainainabiliy agains insananeous repair rae 4 Conclusion In aemping o undersand he sysem behavior, he characerisics of insananeous reliabiliy measures (which are: insananeous failure rae, insananeous repair rae, insananeous mainainabiliy, insananeous reliabiliy and insananeous availabiliy) wih ime and heir relaionships are need o be fully invesigaed. Which provide full informaion on deerioraingsysems, and make decisions on which ype of mainenance policy need o be applied in mainaining he sysems. Because sysems, have differen paern and manner of deerioraing. References [] S. M.Aliyu, I. Yusuf,U.A. Ali.,25. Availabiliy and Profi Opimizaion of Series- Parallel Sysem Consising wih Linear Consecuive Cold Sandby Unis. Journal of Applied Mahemaics 6, [2] R.E.Barlow, F.Proschan, 965. Mahemaical Theory of Reliabiliy. Wiley, New York [3] E. Elsayed,, 996. Reliabiliy Engineering, Addison Wesley, Reading, MA. [4] T. Nakagawa, 25.Mainenance Theory of Reliabiliy. Springer-Verlag, London Limied. [5] H. Pham, 23. Handbook of Engineering. Springer- Verlag, London Limied.

9 Evaluaion of Insananeous Reliabiliy Measures.. 85 [6] K. Suleiman, U. Ali, I. Yusuf, 23. Assessmen of Reliabiliy and Availabiliy of Series- Parallel Subsysems. 3(9), [7] Sun e al., 28. Reliabiliy Modeling and Analysis of Series- Parallel Hybrid Muli Operaional Manufacuring Sysem Considering Dimensional Qualiy, Tool Degradaion and Sysem Configuraion. Inernaional Journal of Producion Economics, 4, [8] H. Wang, and H. Pham, (26). Reliabiliy and Opimal Mainenance. Springer-Verlag London Limied. [9] I. Yusuf, N. Hussaini, 22. Evaluaion of Reliabiliy and Availabiliy Characerisics of 2- ou- 3 Sandby Sysem Under a Perfec Repair Condiion. American Journal of Mahemaics. 2(5),4-9. []B.Yusuf, Yusuf, I.,23. Evaluaion Some Reliabiliy Characerisics of a Sysem Under Three Types of Failures wih Repair- Replacemen a Failure. American Journal of Operaional Research, 3(3), [] I.Yusuf, N. Hussaini, B. M.Yakasai, 24. Some Reliabiliy Measures of a Deerioraing Sysem. Inernaional Journal of Applied Mahemaical Research, 3(),