Evaluation of Mean Time to System Failure of a Repairable 3-out-of-4 System with Online Preventive Maintenance

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1 American Journal of Applied Mahemaics and Saisics, 0, Vol., No., 9- Available online a hp://pubs.sciepub.com/ajams/// Science and Educaion Publishing DOI:0.69/ajams--- Evaluaion of Mean Time o Sysem Failure of a Repairable -ou-of- Sysem wih Online Prevenive Mainenance U.A. Ali, Saminu I. Bala,, Ibrahim Yusuf,* Deparmen of Mahemaics, Usmanu Danfodio Universiy, Sokoo, Nigeria Deparmen of Mahemaical Sciences, Bayero Universiy, Kano, Nigeria Deparmen of Mahemaics, Federal Universiy, Duse, Nigeria *Corresponding auhor: Ibrahimyusuffagge@gmail.com Received Augus, 0; Revised November, 0; Acceped January 07, 0 Absrac Mos of he lieraure assumed ha sysems undergo prevenive mainenance. Lile lieraure is found on wheher he prevenive mainenance is online or offline. I is known ha mos of he engineering sysems undergo boh online and offline prevenive mainenance. In his paper, we sudied he mean ime o sysem failure of a repairable redundan -ou-of- sysem wih online prevenive mainenance involving four ypes of failures. We develop he explici expressions for mean ime o sysem failure for he sysem using Chapman-Kolmogorov equaions. Various cases are analyzed graphically o invesigae he impac of sysem parameers on mean ime o sysem failure. Resuls have shown ha sysem wih online prevenive mainenance is beer in erms of mean ime o sysem failure of sysem han sysem wihou prevenive mainenance. Keywords: Mean ime o sysem failure, online prevenive mainenance Cie This Aricle: U.A. Ali, Saminu I. Bala, and Ibrahim Yusuf, Evaluaion of Mean Time o Sysem Failure of a Repairable -ou-of- Sysem wih Online Prevenive Mainenance. American Journal of Applied Mahemaics and Saisics, no. (0): 9-. doi: 0.69/ajams---.. Inroducion Prevenive Mainenance schedules ha minimize resource consumpion or maximize availabiliy can be deermined hrough he use of quaniaive decisionmodels, based on facual informaion such as ime-ofailure disribuions, cos of inervenion (e.g. for inspecion, repair or replacemen) and consequences of failure. Mos of engineering sysems undergo eiher offline or online prevenive. Under online Prevenive mainenance is carried ou when he sysem is operaing and inends o slow down he wear process and reduce he frequency of occurrence of sysem or componens failures. There are sysems of hree/four unis in which wo/hree unis are sufficien o perform he enire funcion of he sysem. Example of such sysems are -ou-of-,-ou-of-, or -ou-of- redundan sysems which can be seen in modular communicaion amplifier sysem. Many research resuls have been repored on reliabiliy of redundan sysems. For example, Chander and Bhardwaj[], analyzed reliabiliy models for -ou-of- redundan sysem subjec o condiional arrival ime of he server. Chander and Bhardwaj [] presen reliabiliy and economic analysis of -ou-of- redundan sysem wih prioriy o repair. Bhardwaj and Malik [] sudied MTSF and cos effeciveness of -ou-of- cold sandby sysem wih probabiliy of repair and inspecion. Wang e al [] examined he cos benefi analysis of series sysems wih cold sandby componens and repairable service saion. El-Said [] and Haggag [6] examined he cos analysis of wo uni cold sandby sysem involving prevenive mainenance respecively. Haggag [7] analyzed cos analysis of repairable k-ou-of-n sysem wih dependable failures and sandby suppor. Wang and Kuo [8] sudied he cos and probabilisic analysis of series sysem wih mixed sandby componens. Wang e al [9] sudied cos benefi analysis of series sysems wih warm sandby componens involving general repair ime where he server is no subjec o breakdowns. Yusuf [0] sudied he availabiliy and profi of -ou-of- sysem. Reference [0] sudied he availabiliy and profi of - ou-of- sysem wih prevenive mainenance. In his sudy, i is no cerain wheher he prevenive mainenance is offline or online. In fac mos of sudies on sysem reliabiliy evaluaion, did no menion which caegory of prevenive mainenance he sysem will undergo. In our sudy, we assume ha he sysem undergo online prevenive mainenance where he sysem is operaion while he prevenive mainenance aking place. Also availabiliy and profi canno be use o judge he effeciveness of he sysem. The ime inerval from he ime sysem is new o he ime when he sysem experience is firs failure (Mean ime o sysem failure) is equally imporan in deermining he effeciveness which has no been sudied in Reference [0]. This paper is coninuaion of he work of Yusuf [0], a repairable -ou-of- sysem under online prevenive

2 0 American Journal of Applied Mahemaics and Saisics mainenance is considered and derived is corresponding mahemaical model. The main conribuion of his paper is hree fold. Firs, is o develop he explici expressions mean ime o sysem failure. The second is o perform a parameric invesigaion of sysem parameers on mean ime o sysem failure and capure heir effec. The hird o compare he mean ime o sysem failure wih and wihou online prevenive mainenance o highligh he impac of online prevenive mainenance on mean ime o sysem failure. The res of he paper is organized as follows. Secion gives he noaions and assumpions of he sudy. Secion is he descripion of he sysem. Secion deals wih model formulaion. The resuls of our numerical simulaions are presened and discussed in Secion. The paper is concluded in Secion 6.. Noaions and Assumpions α i : Consan repair raes for ype i =,,, β i : Consan failure raes for ypes i =,,, µ : Consan rae end of prevenive mainenance λ : Consan rae of aking he uni ino prevenive mainenance A : Sysem ransiion rae marix MTSF : Mean ime o sysem failure wih online prevenive mainenance MTSF : Mean ime o sysem failure wihou online prevenive mainenance Table. Transiion rae able. The sysem is -ou-of- sysem. The sysem can be in Operaion, Fail sae or online prevenive mainenance. The sysem suffer four ypes of failures. The sysem is down when number of unis failure goes beyond one. Failure and repair ime follow exponenial 6. Failure raes and repair raes are consan 7. The sysem is aended by one repairman. Descripion of he Sysem In his secion, he -ou-of- redundan sysem is considered. Through Markov assumpion, he Chapman- Kolmogorov s equaions are obained for he analysis of sae probabiliies. The sysem comprise of four unis in which a leas hree unis mos be in operaional for he sysem o work. Malfuncioning of wo unis lead he sysem o go down. The unis can work consecuively or randomly as can be seen in he saes of he sysem given below. The sysem ransi o prevenive mainenance before failure a he rae λ wih corresponding rae of prevenive mainenance µ. Uni i fails wih rae β i and is under minimal repair wih raeα i and he sandby uni is swich on. I is assumed ha he swich from sandby o operaion is perfec. The sysem when wo unis. The saes of he sysem according Markov chain is shown in Table below. S 0 S S S S S S 6 S 7 S 8 S 9 S 0 S 0 0 β β β λ S α β β S α β β β 0 S α β β 0 β S 0 0 α α S 0 α 0 α S 6 0 α S α S α S α S 0 µ Sae of he Sysem Sae 0: Unis, and are working, uni in sandby, he sysem is working Sae : unis,,and are working; uni is down and under repair, he sysem is working Sae : unis, and are working, uni is down and under repair, he sysem is working Sae : unis, and are working, uni is down and under repair, he sysem is working Sae : unis is down, under repair, unis is down and waiing for repair, unis and are idle, he sysem Sae : uni is down, under repair, uni is down, waiing for repair, unis and are idle, he sysem Sae 6: uni and are idle, uni is down, and waiing for repair, uni is down, under repair, he sysem Sae 7: uni is down, waiing for repair, unis and are idle, uni is down, under repair, he sysem Sae 8: unis and are idle, uni is down, and waiing for repair, uni is down, under repair, he sysem Sae 9: unis and are idle, uni is down, and waiing for repair, uni is down, under repair, he sysem Sae 0: all unis are under online prevenive mainenance, he sysem is working

3 American Journal of Applied Mahemaics and Saisics. Model Formulaion Le Pi () be he probabiliy ha he sysem is in sae i a ime. Le P () be he probabiliy row vecor a ime, hen he iniial condiions for his problem are as follows: [ 0 6 ] [,0,0,0,0,0,0] P(0) = P (0), P(0), P (0), P (0), P (0), P (0), P (0) = The corresponding se of kolmogorov s differenial equaions obained from Table is as follows: dp0 () = ( β+ β+ β+ λ) P0() + αp() d + αp () + α P () + µ P0() dp () = ( α + β+ β) P() d + βp0() + αp() + αp6() dp () = ( α + β+ β + β) P() + βp0() d + αp() + αp8() + αp9() dp () = ( α+ β + β + β) P() + βp0() d + αp() + αp() + αp7() dp () = ( α+ α) P() + βp() + βp() d dp () = ( α+ α) P() + β P() + βp() d dp6 () = αp6() + βp() d dp7 () = αp7() + βp() d dp8 () = αp8() + βp() d dp9 () = αp9() + βp() d dp0() = µ P0() + λp0 () () d The differenial equaions in () above is ransformed ino marix as where P = TP () y α α α µ β y α α β 0 y 0 α α α 0 β 0 0 y α α 0 α β β y T = 0 β 0 β 0 y β α β α β α β α 0 λ µ say Q. The expeced ime o reach an absorbing sae is y = ( λ+ β+ β + β), y = ( α + β+ β), obained from y = ( α + β+ β + β), y = ( α+ β + β + β) y = ( α+ α), y6 = ( α+ α) I is difficul o evaluae he ransien soluions, hence N E TP(0) P( absorbing) we follow El-said [], Haggag [6] and Wang [9], he = MTSF = P(0)( Q ) = () D procedure o develop he explici expression for MTSF is o delee he rows and columns of an absorbing sae in marix T and ake he ranspose o produce a new marix, Where ( λ+ β+ β + β) β β β λ α ( α β β) Q = α 0 ( α + β+ β + β) 0 0 α 0 0 ( α + β + β + β) 0 µ µ

4 American Journal of Applied Mahemaics and Saisics αββ + βββ + βββ + αββ + βββ + ββ + ααβ + αββ + αββ + αββ + β β + αββ + αββ + αββ + αββ + αββ + βββ + ααβ + αββ + β β N = µ + ββ + ββ + ββ + ββ + β β + αβ + αβ + αα β+ αββ + αββ + β αα β αβ β αβ αβ αβ αα α ααβ αα β αα β αα β αβ + ββ + ββ + ββ + ββ + β + β + αα + βµ αβ ββ αβ αβ ββ αβ αβ αβ + αβ + αβ + αα + ββ + ββ αβ + ββ + ββ + αα + αβ + β µ + β µ αβ ββ ββ β αβ ββ αβ αβ β ββ β αβ λµ αββ + βββ + βββ + αββ + βββ + ββ + ααβ + αββ + αββ + αββ + β β + αββ + αββ + αββ + αββ + αββ + βββ + ααβ + αββ + β β + ββ + ββ + ββ + ββ + β β + αβ + αβ + αα β+ αββ + αββ + β + αα β + αββ + αβ + αβ + αβ + ααα + ααβ + ααβ + ααβ + ααβ β ββ + αβ β + αββ + αβββ + 6ββββ + αβββ + αβββ + αβββ + αββ + αββ + βββ + βββ + βββ + βββ + βββ + ααββ + αβββ + αβββ + αβ β + αββ + αββ + αβ β + β ββ + ββ β + αββ + βββ + ββ + β β + ββ D = µ + ββ + ββ + β β + ββ + βββ + αββ + αββ ααββ + αβββ + ααββ + ααββ + αβββ + ααββ ααββ + β ββ + βββ + αβ + ββ + β β + ββ + ββ + ββ + ββ + αβ β + αβ β + αα β + αβ β + αββ + αββ + αββ + αββ + αββ + αβ β + αβ β + αββ + ααβ + αβ β αααβ + ααββ + αβββ + αααβ + ααββ + ααββ + α αββ. Resuls and Discussions In his secion, we numerically obained he resuls for mean ime o sysem failure. For he model analysis, he following se of parameers values are fixed hroughou he simulaions for consisency: = 0., = 0., = 0., = 0., = 0.7, = 0.8, = 0.7, = 0., = 0.8, = 0.98 β β β β α α α α µ λ The simulaions in Figure and Figure have shown ha he mean ime o sysem failure for boh sysems wih and wihou prevenive mainenance increase wih increase in repair raeα. I is eviden from Figure ha he mean ime o sysem failure of sysem wih prevenive mainenance increases more wih respec o repair rae α han he mean ime o sysem failure of sysem of sysem wihou prevenive mainenance. In Figure and Figure i is clear ha he mean ime o sysem failure of sysem decreases for boh sysems wih and wihou prevenive mainenance. Here also he mean ime o sysem failure of sysem of sysem wih prevenive mainenance decreases more han he mean ime o sysem failure of sysem of sysem wihou prevenive mainenance. Mean Time o Sysem Failure α Figure. Effec of α on MTSF

5 American Journal of Applied Mahemaics and Saisics Mean Time o Sysem Failure Conclusion In his paper, we developed he explici expression for mean ime o sysem failure of repairable -ou-of- sysem in he presence of online prevenive mainenance. In order o deermine he effeciveness of he sysem under sudy, we performed numerical invesigaion o see he effec of failure and repair raes on mean ime o sysem failure. I is eviden from he resuls obained ha failure and repair raes decrease and increase he mean ime o sysem failure of he sysem respecively. Through he analysis, we conclude ha sysem wih online prevenive mainenance is more effecive han sysem wihou online prevenive mainenance. Mean Time o Sysem Failure i, i=, β Figure. Effec of β on MTSF MTSF MTSF α Figure. Effec of α on MTSF MTSF MTSF References [] Bhardwj, R.K. and Chander, S. (007). Reliabiliy and cos benefi analysis of -ou-of- redundan sysem wih general disribuion of repair and waiing ime. DIAS- Technology review- An In. J. of business and IT. (), 8-. [] Chander, S. and Bhardwaj, R.K. (009). Reliabiliy and economic analysis of -ou-of- redundan sysem wih prioriy o repair. African J. of Mahs and comp. sci, (), 0-6. [] Bhardwj,R.K., and S.C. Malik. (00). MTSF and Cos effeciveness of -ou-of- cold sandby sysem wih probabiliy of repair and inspecion. In. J. of Eng. Sci. and Tech. (), [] Wang, k. Hsieh, C. and Liou, C (006). Cos benefi analysis of series sysems wih cold sandby componens and a repairable service saion. Journal of qualiy echnology and quaniaive managemen, (), [] El-Said, K.M., (008). Cos analysis of a sysem wih prevenive mainenance by using Kolmogorov s forward equaions mehod. American Journal of Applied Sciences (), 0-0. [6] Haggag, M.Y., (009). Cos analysis of a sysem involving common cause failures and prevenive mainenance, Journal of Mahemaics and Saisics (), 0-0. [7] Haggag, M.Y., (009). Cos analysis of k-ou-of-n repairable sysem wih dependen failure and sandby suppor using Kolmogorov s forward equaions mehod.journal of Mahemaics and Saisics (), [8] Wang, K.H and Kuo, C.C. (000). Cos and probabilisic analysis of series sysems wih mixed sandby componens. Applied Mahemaical Modelling,, [9] Wang, K..C., Liou, Y.C, and Pearn W. L. (00). Cos benefi analysis of series sysems wih warm sandby componens and general repair ime. Mahemaical Mehods of operaion Research, 6, 9-. [0] Yusuf, I. Availabiliy and profi analysis of -ou-of- repairable sysem wih prevenive mainenance, Inernaional Journal of Applied Mahemaics Research, (), 0, 0-9. Mean Time o Sysem failure i, i=, β Figure. Effec of β on MTSF