Research Scholar, Department of Statistics, Shri Jagdish Prasad Jhabarmal Tibrewala University Dist Jhunjhunu, Rajasthan , INDIA **

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1 [Ahmad, 4 (5): May, 05] ISSN: (IOR), Publicaion Imac Facor: (ISRA), Imac Facor:.4 IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY AVAILABILITY ANALYSIS OF A TWO-UNIT CENTRIFUGE SYSTEM CONSIDERING THE HALT STATE ON OCCURRENCE OF MINOR/MAJOR FAULT Shakeel Ahmad and Vinod Kumar Research Scholar, Dearmen of Saisics, Shri Jagdish Prasad Jhabarmal Tibrewala Universiy Dis Jhunjhunu, Rajashan-33300, INDIA Dearmen of Mahemaics, S.B.M.N. Engg. College, Ashal Bohar, Rohak-400(Haryana) INDIA ABSTRACT This aer deals wih a cenrifuge sysem consising wo idenical uni cold sandby considering major/minor faul. I is assumed ha sysem leads o arial failed sae on occurrence of a minor faul whereas on occurrence of a major faul i leads o comlee failure. Some ime he sysem is need o be brough a hal sae for reair/ relacemen (off-line) and reairs/ relacemens of he sysem/ comonens is done. In general on comlee failure of he sysem, he reairman firs insec wheher he faul is reairable or non reairable and accordingly carry ou he reair or relacemen of he comonens involved. Various measures of sysem effeciveness are obained by using Markov rocesses and regeneraive oin echnique. The analysis of he sysem is carried ou on he basis of he grahical sudies and conclusions are drawn regarding he availabiliy of he sysem. KEYWORDS: Cenrifuge Sysem, MTSF, Execed Uime, Markov Process, Regeneraive Poin Technique INTRODUCTION In he resen scenario filraion and urificaion lays a very imoran role in he modern sociey eraining o he healh of he human being and he qualiies of he roducs used by hem. A large number of equimens or sysems of equimens are involved in he indusries o mee ou he requiremens of such roducs. One such sysem is a cenrifuge sysem used for searaion of wo objecs having differen ye of densiy. Cenrifuge sysem is being used in Refineries for oil urificaion, in milk lans o exrac he fas, in laboraories for blood fracionaion and wine clarificaion ec. Thus he reliabiliy and cos of he cenrifuge sysem lays a very significan role in such ye of indusries and hence need o be analyzed. In fac a large number of researchers in he field of reliabiliy modeling including Gua and Kumar (983), Goalan and Murlidhar (99), Tueja e al (00), Taneja e al (004), Taneja and Parashar (007), Gua e al (008), Kumar e al (00), ec. analyzed various one-uni/ wo-uni sysems. Kumar e al. (00) invesigaed a wo-uni redundan sysem wih he degradaion afer firs failure and relacemen afer second failure. Tueja e.al (00) sudied reliabiliy and rofi analysis of wo-uni cold sandby sysem wih arial failure and wo yes of reairman. Taneja and Nanda (003) sudied robabilisic analysis of a wo-uni cold sandby sysem wih resume and reea reair olicies. Singh and Chander (005) analyzed reliabiliy of wo sysems each of which conains non-idenical unis-an elecric ransformer and a generaor. Kumar and Bhaia (0, 0, 03) discussed he behaviour of he single uni cenrifuge sysem considering he conces of insecions, hal of sysem, degradaion, minor/major fauls, negleced fauls, online/offline mainenances, reairs of he fauls ec. Recenly, Kumar V. e al. (04) discussed he reliabiliy and rofi analysis of a wo-uni cold sandby cenrifuge sysem considering reair and relacemen wih insecion. As far as we concern wih he research work on reliabiliy modeling, none of he researchers have analyzed such a wo-uni cold sandby cenrifuge sysem considering such a siuaion wih occurrence of various fauls. To fill u his ga, he resen aer discussed an analysis of a sochasic model for wo-uni cenrifuge sysem considering hal of he sysem on occurrence of minor/major faul. On comlee failure of he sysem, he reairman firs insec h: // Inernaional Journal of Engineering Sciences & Research Technology [00]

2 [Ahmad, 4 (5): May, 05] ISSN: (IOR), Publicaion Imac Facor: (ISRA), Imac Facor:.4 wheher he faul is reairable or non reairable and accordingly carry ou he reair or relacemen of he comonens involved. In general all he insecions, reairs and relacemens have done on-line as well as off-line during he uni oeraive/ inoeraive, bu someimes in emergency he oeraive uni of he sysem may be brough o hal for he reair or relacemen. Various measures of sysem effeciveness such as mean sojourn ime, MTSF, execed u ime and execed down ime of he sysem are obained using Markov rocesses and regeneraive oin echnique. The conclusions regarding reliabiliy and availabiliy of he sysem are given on he basis of grahical sudies. SYSTEM DESCRIPTION AND OTHER ASSUMPTIONS AND METHODS a. Fauls are self- announcing on occurring in he sysem. b. There is a single reairman faciliy wih he sysem o reair he faul. c. Afer each reair he sysem is as good as new. d. Insecion is carried ou only on he occurrence of major fauls. e. During online reair/waiing for reair here may be occurrence of major faul. f. On occurrence of minor/major faul wheher i is reairable or irrearable on-line, he sysem is need o be brough a hal sae for reair/ relacemen (off-line). g. The failure ime disribuions are exonenial while oher ime disribuions are general. h. Swiching is erfecly done on occurrence of major faul. i. All he random variables are muually indeenden. Noaions λ / λ λ 3 a / b η / η :Rae of occurrence of major/ minor failure :Rae of occurrence of failure due o delay in reair :Probabiliy ha a faul is reairable/ non-reairable :Rae a which he sysem brough o be a hal sae i ()/I ():.d.f./ c.d.f. of ime o insecion of he uni a failed sae i ()/ I ():.d.f./ c.d.f. of ime o insecion of he uni a haled sae g ()/G ():.d.f./ c.d.f. of imes o reair of minor faul a down sae g ()/G ():.d.f./ c.d.f. of imes o reair he uni a failed sae h ()/H ():.d.f./ c.d.f. of imes o relacemen of he uni a failed sae O r /O w /O cs :Oeraive uni under reair/ waiing/ cold sandby F i /F r /F r /F w :Failed uni under insecion/ reair/ relacemen/ waiing F R /F RP :Failed uni under reair/ relacemen coninue from he revious sae TRANSITION PROBABILITIES AND MEAN SOJOURN TIMES A sae-ransiion diagram in fig. shows various saes of ransiion of he sysem. The eochs of enry ino saes 0,,, 3, 4, 5 and 7 are regeneraion oins and hus hese are regeneraive saes. The sae s 6 and 8 are failed sae and 9 and 0 are hal sae. h: // Inernaional Journal of Engineering Sciences & Research Technology [0]

3 [Ahmad, 4 (5): May, 05] ISSN: (IOR), Publicaion Imac Facor: (ISRA), Imac Facor:.4 Figure: The ransiion robabiliies are given by dq e d 0 0 dq g d dq ai d 3 dq e d 0 dq bi d dq e g d dq e g d Sae Transiion Diagram dq e G d dq e G d 3 dq e h d dq e h d dq 59 e G d 3 dq 7 e h d dq e h d dq e H d dq e H d dq e g d dq e g d dq e g d dq e G d dq g d dq e h d dq e H d 0 7 dq e h d dq h d 8 dq g d 9 dq e g d h: // Inernaional Journal of Engineering Sciences & Research Technology [0] 3 7,0 dq e H d

4 [Ahmad, 4 (5): May, 05] ISSN: (IOR), Publicaion Imac Facor: (ISRA), Imac Facor:.4 0, dq h d Taking L.S.T Q ij s and ij s0 ij limq s ai 0 g 0 30 g g bi h 40, he non-zero elemens ij, are obained as under: g h 5 g 3 g h g 3 3 g 6 3 g 0 h h 3 3 h h 0 0, h g 0 7,0 By hese ransiion robabiliies, i can be verified ha = = = = = + = = , = = = = = = , = = h: // Inernaional Journal of Engineering Sciences & Research Technology [03] h 3 3 The uncondiional mean ime aken by he sysem o ransi for any regeneraive sae j, when i is couned from eoch of enrance ino ha sae i, is mahemaically saed as-

5 [Ahmad, 4 (5): May, 05] ISSN: (IOR), Publicaion Imac Facor: (ISRA), Imac Facor:.4 Thusm m ij ij ij 0 m = dq q m0 m3 m4 m30 m35 m36 3 m40 m47 m48 4 m5 m56 m59 5 m7 m78 m7,0 7 m 6 6 m8 8 m9 9 m 0, 0 m m m k m m m k where m m m k m m m k k g 0 k 3 k h 0 k 4 The mean sojourn ime in he regeneraive sae i (µ i ) is defined as he ime of say in ha sae before ransiion o any oher sae hen we have 0 g 0 i 0 h 4 6 g 0 h 0 8 h 0 0 g 3 g 5 h g 0 MEASURES OF THE SYSTEM EFFECTIVENESS Various measures of he sysem effeciveness obained in seady sae using he argumens of he heory of regeneraive rocess are as under: The Mean Time o Sysem Failure (MTSF) = N/D Execed U-Time of he Sysem wih Full Caaciy (AF 0 ) = N /D Execed U-Time of he Sysem wih Reduced Caaciy (AR 0 ) = N /D where h: // Inernaional Journal of Engineering Sciences & Research Technology [04]

6 [Ahmad, 4 (5): May, 05] ISSN: (IOR), Publicaion Imac Facor: (ISRA), Imac Facor: N N = µ µ + µ + µ µ µ k k 4 k 47k µ + µ D D = µ 0+ 0µ N = µ µ GRAPHICAL INTERPRETATION AND CONCLUSION For grahical analysis following aricular cases are considered- g e g e i e h e Therefore, we have a b , h: // Inernaional Journal of Engineering Sciences & Research Technology [05]

7 [Ahmad, 4 (5): May, 05] ISSN: (IOR), Publicaion Imac Facor: (ISRA), Imac Facor: Various grahs are loed for MTSF, Execed u ime and Execed down ime and Profi of he sysem by aking differen values of failure raes (λ, λ & λ 3 ), insecion rae ( ), reair raes (β & β ), relacemen rae ( ), hal rae (η & η ) and robabiliies of reairable & non-reairable (a & b). Fig. Fig. gives he grah beween MTSF (T 0 ) and he rae of failure (λ ) due o minor fauls for differen values of he rae of failure (λ ) due o major fauls. The grah reveals ha he MTSF decreases wih increase in he values of he failure raes. Fig. 3 The curves in Fig. 3 give he grah beween MTSF (T 0 ) and rae of failure due o delay in reair (λ 3 ) for differen values of rae of major failure (λ ) of he sysem. The grah reveals ha he MTSF decreases wih increase in he values of he failure raes. h: // Inernaional Journal of Engineering Sciences & Research Technology [06]

8 [Ahmad, 4 (5): May, 05] ISSN: (IOR), Publicaion Imac Facor: (ISRA), Imac Facor:.4 Fig. 4 The curves in Fig. 4 give he grah beween MTSF (T 0 ) and rae a which sysem haled (η ) for differen values of rae of major failure (λ ) of he sysem. The grah reveals ha he MTSF decreases wih increase in he values of he rae a which sysem haled as well as he failure rae. Fig. 5 Fig. 5 gives he grah beween Execed uime wih full caaciy (AF 0 ) and he rae of occurrence of minor fauls (λ ) for differen values of rae of occurrence of major fauls (λ ). The grah reveals ha he Execed uime wih full caaciy decreases wih increase in he values of he failure raes. REFERENCES [] R.C. Garg and A. Kumar (977) A comlex sysem wih wo yes of failure and reair, IEEE Trans. Reliabiliy, 6, [] L.R. Goel, G.C. Sharma and R. Gua (986) Reliabiliy analysis of a sysem wih revenion mainenance and wo yes of reairing, Micro elecron Reliabiliy, 6, [3] M.N. Goalan and N.N. Murlidhar (99) Cos analysis of a one uni reairable sysem subjec o on-line revenion mainenance and/or reair, Micro elecron Reliabiliy, 3(/3), [4] A. Goyel, G. Taneja and D.V. Singh (009) Analysis of a -uni cold sandby sysem working in a sugar mill wih oeraing and res eriods, Caledon. J. Eng., 5: -5. [5] A. Goyel, G. Taneja and D.V. Singh (00) Reliabiliy modeling and analysis of a sulhaed juice um sysem comrising hree idenical unis wih wo yes of working caaciy and res eriod, Pure and Alied Mahemaical Sciences,7,(-), [6] M. L. Gua and A. Kumar (983) On rofi consideraion of a mainenance sysem wih minor reair, Microelecron Reliabiliy, 3, [7] K. Murari and V. Goyal (983) Reliabiliy sysem wih wo yes of reair faciliies, Microelecron Reliabiliy, 3(6), [8] G. Taneja, V.K Tayagi and P. Bhardwaj (004) Programmable logic conroller, Pure and Alied Mahemaical Sciences, LX (-), [9] R. Kumar and P. Bhaia (0) Imac of Ignored fauls on reliabiliy and availabiliy of cenrifuge sysem ha undergoes eriodic res, Journal of Inernaional Academy of Physical Sciences, Vol.5, [0] R. Kumar and P. Bhaia (0) Reliabiliy and cos analysis of one uni cenrifuge sysem wih single reairman and insecion, Pure and Alied Mahemaika Sciences, 74 (-), 3-. h: // Inernaional Journal of Engineering Sciences & Research Technology [07]

9 [Ahmad, 4 (5): May, 05] ISSN: (IOR), Publicaion Imac Facor: (ISRA), Imac Facor:.4 [] Kumar, R. and Rani, S. (03) Cos-benefi analysis of a reliabiliy model on waer rocess sysem having wo yes of redundan subsysems, Inernaional Journal of Alied Mahemaical Research,, No., [] U.Sharma,Rekha, G Taneja.( 0) Analysis of a wo uni sandby oil delivering sysem wih a rovision of swiching over o anoher sysem a need o increase he availabiliy, Journal of Mahemaics and Saisics Vol 7 no [3] V. Kumar, P. Bhaia and S. Ahamd (04) Profi Analysis of a Two-Uni Cold Sandby Cenrifuge Sysem wih Single Reairman, Inernaional Journal of Scienific & Engineering Research, Volume 5, Issue 6, June-04, AUTHOR BIBLOGRAPHY Mr. Shakeel Ahmad Research Scholar, Dearmen of Saisics, Shri Jagdish Prasad Jhabarmal Tibrewala Universiy Dis Jhunjhunu, Rajashan-33300, INDIA affuzaa64@gmail.com Dr. Vinod Kumar Professor and Head De. of Mahemaics, S.B.M.N. Engg. College, Rohak, HARYANA, (INDIA) Associaed as Aroved Suervisor wih Dearmen of Saisics, Shri Jagdish Prasad Jhabarmal Tibrewala Universiy Dis Jhunjhunu, Rajashan-33300, INDIA kakoriavinod@gmail.com h: // Inernaional Journal of Engineering Sciences & Research Technology [08]