Assessment of Reliability and Availability of Series-Parallel Sub- Systems

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1 ISSN -58 (Paer) ISSN 5-5 (Onlne) Vol., No., Assessment of Relablty and Avalablty of Seres-Parallel Sub- Systems Suleman K. Deartment of Mathematcs, Faculty of Scence, Faculty of Scence, Federal Unversty, Dutse, Ngera. U. A. Al Deartment of Mathematcs, Faculty of Scence, Faculty of Scence, Usmanu Danfodyo Unversty, Sokoto,Ngera. Ibrahm Yusuf Deartment of Mathematcal Scences, Faculty of Scence, Bayero Unversty, Kano, Ngera. Abstract The aer resents the exlct exressons for measurng relablty arameter of systems, Grahs were lotted to hghlght mortant results. Results have shown that measures of system effectveness such MTSF, system avalablty and roft ncreases wth rear rates and decreases wth falure rates.the develoed model hels n determnng the otmal mantenance strateges whch ensure the maxmum overall avalablty of the system. The otmum values of falure and rear rates for each subsystem were gven. It s observed that the frst subsystem s havng the maxmum avalablty wth (97%). The otmum values of falure and rear rates for maxmum avalablty level for each subsystem s also shown. Keywords: Mean tme to system falure, Avalablty, Relablty, Mantenance and Subsystem.. INTRODUCTION Stochastc models of redundant systems as well as methods of evaluatng system relablty ndces such as mean tme to system falure (MTSF), system avalablty, busy erod of rearman, roft analyss, etc have been nvestgated by many researchers n order to mrove the system effectveness. Some systems are seres-arallel desgn. These systems have wde alcaton n the real world esecally n ndustres. Furthermore, electrcal substaton, arcraft engne can be cted as a good examle of seres-arallel systems. Due to the mortance of seres-arallel nature of some comlex systems n varous ndustres, determnaton of ther avalablty has become an ncreasngly mortant ssue. System avalablty reresents the ercentage of tme the system s avalable to users. Falure s an unavodable henomenon whch can be dangerous and costly and brng about less roducton and roft. Proer mantenance lannng lays a role n achevng hgh system relablty, avalablty and roducton outut. It s therefore mortant to kee the equments/systems always avalable and to lay emhass on system avalablty at the hghest order. A large volume of lterature exsts on the ssue of redctng erformance evaluaton of varous systems. Kumar et al (988) dscussed the relablty analyss of the Feedng system n the aer ndustry, Kumar el al.(989) dscussed the avalablty analyss of the washng system n the aer ndustry, Kuren (988), develoed a smulaton model for analyzng the relablty and avalablty of an arcraft tranng faclty Kumar el al. (99) deal wth relablty, avalablty and oeratonal behavor analyss for dfferent systems n aer lant. Haggag (9), dscussed the behavor analyss of Urea decomoston n the fertlzer ndustry under the general rear olcy. Guta a et al (5), studed the desgn and cost analyss of a refnng system n a Sugar ndustry. Srnath (99), has exlaned a Markov model to determne the avalablty exresson for a smle system consstng of only one comonent. Guta b el al. (5) has evaluated the relablty arameters of butter manufacturng system n a dary lant consderng exonentally dstrbuted falure rates of varous comonents. Guta b et al. (5) studed the behavor of Cement manufacturng lant. Arora and Kumar (), studed the avalablty analyss of the cool handlng system n aer lant by dvdng t nto three subsystems. Sngh and Garg (5), erform the avalablty analyss of the core veneer manufacturng system n a lywood manufacturng system under the assumton of constant falure and rear rates. Guta and Twar (9), study smulaton modelng of comlex system usng thermal ower lant as. SYSTEM DESCRIPTION/ NOTATIONS/ASSUMPTIONS The system consdered n ths study consst of four subsystems arranged n seres connecton,.e. A, B, C and D are connected n seres to each other wth subsystem D havng two unts n arallel. 8

2 ISSN -58 (Paer) ISSN 5-5 (Onlne) Vol., No., The lfe tme of the unts ( A, B, C, D;,) s exonentally dstrbuted random varables wth arameter,,,,. The rear tme of the unts s exonentally dstrbuted random varables wth arameter j; j,,,. th th, Denote falure rate of andj unts resectvely. All falures are rearable ones. In state 8 D has mantenance rorty over D The system s attended by one rearman. Each unt s as good as new after rear. The rear s done at down tme or at the tme of falure There s no smultaneous falure among the four subsystems State and ndcate the system s n fully oeratonal states. Swtchng devce s erfect for the unts n subsystem D. Subsystems D andd are n cold standby.. SYSTEM STRUCTURE A tycal system conssts of a number of subsystems connected to each other n seres-arallel. The erformance of the system deends on the confguraton and erformance of ts subsystems. Before analyzng the falure data, t s better to descrbe the confguraton of the system and classfy t nto varous subsystems so that the falure can be categorzed. The system under study conssts of the followng four subsystems: Fg. Relablty block dagram and transton dagram of the system. METHODOLOGY. MEAN TIME TO SYSTEM FAILURE FOR THE SYSTEM Let P( t ) be the robablty row vector at tme t and ( ) tme t,then the ntal condtons for ths roblem are as follows: P t s the robablty that the system s n state [ P P P P P P P P P ] [,,,,,,,, ] P() (), (), (), (), (), (), (), (), () P P + P P P + P P P + P P ( ) P + P + P5 + P6 + P7 + P8 P P + P 5 5 P 6 P6 + P P 7 P + P () P P + P 8 8. MODEL VALIDATION In ths secton the range of values of the ar (, ) were use for analyss, the range values used by Guta and Twar (9) n the modelng of ar system of a thermal ower lant to valdate the model develoed n fg.. S at 85

3 ISSN -58 (Paer) ISSN 5-5 (Onlne) Vol., No., The followng transton robabltes were obtaned by settng the left hand sdes of equatons () equal to zero and solve them recursvely. as t. Equatons () are solved Therefore, followng Kumar et al (989), P,,,,...,8 recursvely to obtan the followng: P ( ) 8 (9) Now by solvng to 8 recursvely, followng Guta et al 5, The sum of states robabltes equal to zero.. NORMALISING CONDITION The robablty of full workng caacty, namely, and determned by usng normalzng condton: (sum of the robabltes of all workng states s equal to zero) 8, therefore uttng the values of 8 and solvng, one gets ( ) [ ( ) ] Therefore Where D D ( ) Therefore snce normalzng condton s use to obtan, then steady state avalablty would be comuted n terms of. STEADY STATE AVAILABILITY Now, the steady state avalablty of the system may be obtaned as summaton of all workng states robabltes as: Avalablty the sum of robabltes of oeratonal states. () () () (5) (6) (7) (8) () 86

4 ISSN -58 (Paer) ISSN 5-5 (Onlne) Vol., No., Thus, + P( ) And Where Therefore AV AV P + P P + P P ( ) D D ( ) + D () Now, followng Guta and Twar (9) to comute avalablty table through. Takng the range of values of both falure and rear rates of sub-system A,B,C and D..5 ANALYSIS OF SYSTEM MODEL The smulaton model s used to redct the avalablty/erformance of the system for known nut values of falure and rear rates of ts subsystems. The erformance of the system s manly affected by the falure and rear rates of each subsystem. Arorate falure and rear rates of all subsystems are taken and decson matrces (avalablty values) are reared accordngly by uttng these falure and rear rates values n avalablty exresson, the avalablty smulaton model (Av.). Ths model forms the foundaton for all other erformance mrovement actvtes (e.g. soluton desgn and develoment, mlementaton and analyss). These unt arameters ensure the hgh avalablty/erformance of the system. Ths model ncludes all ossble states of nature, that s, falure events ( ) and the dentfcaton of all the courses of acton,.e, rear rortes ( ). Tables - reresent the avalablty matrces for varous subsystems of the system. These matrces smly reveal the varous avalablty levels for dfferent combnatons of falure and rear rates/rortes. On the bass of analyss made, the best ossble combnatons (, ) may be selected. These avalablty values n avalablty matrces further hel n dentfyng the subsystem whch ensures the maxmum avalablty, as shown n Table 5. The otmum vales of falure/rear rates of each subsystem of concerned system can easly be taken from Table 5. PARAMETER DESCRIPTION SOURCE Falure rate of subsystem A Guta and Tewar (9) Falure rate of subsystem B Guta and Tewar (9) Falure rate of subsystem C Guta and Tewar (9) Falure rate of subsystem D Guta and Tewar (9) Rear rate of subsystem A Guta and Tewar (9) Rear rate of subsystem B Guta and Tewar (9) Rear rate of subsystem C Guta and Tewar (9) Rear rate of subsystem D Guta and Tewar (9) 6. RESULTS AND DISCUSSION The result of the fndngs could be dscussed n two ways as can be observed. Case the system s consdered as a whole and some arbtrary values were use to see the effect of falure/rear on the system effectveness, meantme to system falure, and the effect of falure/rear on the system avalablty and lastly the otmum roft ncurred when the system s recevng roer mantenance and vse-versa. Whle n the second case the four subsystems were analyzed, comutng the avalablty level of each subsystem usng range of values of () 87

5 ISSN -58 (Paer) ISSN 5-5 (Onlne) Vol., No., falure and rear rates [see Guta and Twar (9)]. Lastly the maxmum avalablty of each subsystem s comuted to dentfy the subsystem to be gven mantenance rorty. SECTION The erformance of each subsystem s analyzed usng the develoed model. On the bass of avalablty values, as gven n Table -5 and lotted n Fgure -9, the followng observatons are made usng the ar values (, ) as n Guta and Twar (9), whch reveals the effect of falure and rear rates of varous subsystems on the avalablty of the system. Table Avalablty matrx of the subsystem A of seres-arallel system Table and Fgures and reveal the effect of falure and rear rates of subsystem A on the avalablty of the system. It s observed that for some known values of falure / rear rates of other three subsystems, as falure rate of frst subsystem ncreases from.5 to., the subsystem avalablty decreases.smlarly as rear rate of frst subsystem ncreases from. to., the subsystem avalablty ncreases. Table Avalablty matrx of the subsystem A of seres-arallel system

6 ISSN -58 (Paer) ISSN 5-5 (Onlne) Vol., No., Fg. Plot of avalablty vs Fg. Plot of avalablty vs Fg. 5 Plot avalablty vs Table and Fgure and 5 reveal the effect of falure and rear rates of subsystem B on the avalablty of the system. It s observed that for some known values of falure / rear rates of other three subsystems, as falure rate of Subsystem B ncreases from. to. the subsystem avalablty decreases. Smlarly as rear rate of subsystem B ncreases from.8 to., the subsystem avalablty ncreases. Table Avalablty matrx of the subsystem C of seres-arallel system Fg. 6 Plot of avalablty vs Fg. 7 Plot of avalablty Vs Table and Fgures 6 and 7 reveal the effect of falure and rear rates of subsystem C on the avalablty of the system. It s observed that for some known values of falure/rear rates of other three subsystems, as falure rate of subsystem C ncreases from.67 to.5, the subsystem avalablty decreases. Smlarly, as rear rate of subsystem C ncreases from.5 to.5, the subsystem avalablty ncreases. 89

7 ISSN -58 (Paer) ISSN 5-5 (Onlne) Vol., No., Table Avalablty matrx of the subsystem D of seres-arallel system Fg. 8 Plot of avalablty vs Fg. 9 Plot of avalablty vs Table and Fgures 8 and 9 reveal the effect of falure and rear rates of subsystem D on the avalablty of the system. It s observed that for some known values of falure/rear rates of other three subsystems, as falure rate of subsystem.5 ncreases from. to.5, the subsystem avalablty decreases. Smlarly, as rear rate of subsystem C ncreases from. to.5, the subsystem avalablty ncreases. Table 5 Otmum values of Falure/Rear rates of Subsystems of Seres-Parallel system S/N Subsystem Falure rate Rear rate Maxmun Avalablty Level A 97% B C D Table 5 hels n dentfyng the subsystem wth maxmum avalablty. It s observed that the frst subsystem s havng the maxmum avalablty wth (97%). The otmum values of falure and rear rates for maxmum avalablty level for each subsystem s also shown n Table CONCLUSIONS In ths resentaton,the exlct exressons for measurng the effectveness of system such as steady state avalablty, MTSF, and roft functon. Grahs were lotted to hghlght mortant results. Results have shown that measures of system effectveness ess such MTSF, system avalablty and roft ncreases wth rear rates and decreases wth falure rates. Secondly, t can be concluded from tables through, that as falure rate ncreases, the avalablty goes on decreasng and as rear rate ncreases, the avalablty goes on ncreasng. The develoed model hels n determnng the otmal mantenance nce strateges, whch wll ensure the maxmum overall avalablty of the system. The otmum values of falure and rear rates for each subsystem are gven n table 5.It s also 89% 96% 89% 9

8 ISSN -58 (Paer) ISSN 5-5 (Onlne) Vol., No., concluded that frst subsystem (A) s havng maxmum avalablty. Such results are found hghly benefcal to the lant management for the avalablty analyss of ther system. Lastly, we have consdered the roblem so as to fnd out the otmum combnaton of falure/rear rates, whch maxmzes the system avalablty subject to the cost and system weght. In many stuatons, roblem arameters are more cometent to take, for real lfe examles to valdate the model develoed. Hence ths work gves more sgnfcant contrbuton for relablty engneers for decson makng. For ractcal stuaton, based on decson maker s choce, several combnatons of dfferent falure and rear rate values may be consdered n the relablty model. REFERENCES. Arora, N., and Kumar, D. System analyss and mantenance management for coal handlng system n a aer lant. Internatonal Journal of Management and Systems,. El-Sad, K.M.,. ( 8).Cost analyss of a system wth reventve mantenance by usng Kolmogorov s forward equatons method. Amercan Journal of Aled Scences 5(), 5- [7] Guta and Twar (9). Smulaton modelng and analyss of comlex system of thermal ower lant. Internatonal journal of ndustral engneerng and management, vol no., Guta, S., Tewar, P.C. and Sharma, A.K. Relablty and avalablty analyss of handlng unt of a steam thermal lant. Internatonal Journal of Engneerng Research and Industral Alcaton, vol., no.5, 8, Guta, A. Lal, R. Sharma and J. Sngh. Numercal analyss of relablty and avalablty of seres rocesses n butter ol rocessng lant. Internatonal Journal of Qualty and Relablty Management, vol., no. 5, Guta, A. Lal, R. Sharma and J. Sngh. Behavoral study of the Cement Manufacturng lant. A numercal aroach. Journal of Mathematcs and systems scences. Vol., no., 5, Guta, A. Lal, R. Sharma and J. Sngh a. Numercal analyss of relablty and avalablty of seres rocesses n butter ol rocessng lant. Internatonal Journal of Qualty and Relablty Management, vol., no. 5, Haggag, M.Y., (9). Cost analyss of a system nvolvng common cause falures and reventve mantenance.journal of Mathematcs and Statstcs 5(), 5-8. Kuren, K. C. (988). Relablty and avalablty analyss of rearable system usng dscrete event smulaton. Ph. D. thess, IIT, New Delh. 9. Kumar, D., Sngh Ja and Pandey PC. Relablty analyss of the feedng system n the aer ndustry. Mcroelectron Relablty, vol. 8, no., [] Kumar, D., Sngh Ja and Pandey PC. Avalablty analyss of the washng system n the aer ndustry. Mcroelectron Relablty, vol.9, 989, Kumar, D., Pandey, P. C. and Sngh, J. (99) Process Desgn for a Crystallzaton System n the Urea Fertlzer Industry, Mcroelectroncs and Relablty, Vol., Kumar, D., Sngh Ja and Pandey PC. Oeratonal behavor and roft functon for a bleachng and screenng n the aer ndustry. Mcroelectron Relablty, vol., 99, -5.. Kumar, S., Kumar, D., & Mehta, N. P. (996). Behavoral analyss of shell gasfcaton and carbon recovery rocess n urea fertlzer lant. McroelectronRelablty, 6 (5), Lm, T. J., & Chang, H. K. (). Analyss of system relablty wth deendent rear models. IEEE Trans Relab, 9(), Srnath, L.S. Relablty Engneerng rd edton, East west ress Pvt Ltd. New Delh, Inda, Sngh, J., and Garg, S. Avalablty analyss of core veneer manufacturng system n lywood ndustry. Internatonal Conference on Relablty and safety engneerng, Inda nsttute of Technology, Kharagur, 5, Wang., k. Hseh., C. and Lou, C (6). Cost beneft analyss of seres systems wth cold standby comonents and a rearable servce staton. Journal of qualty technology and quanttatve management, (), Wolstenholme, L. C. (999). Relablty modelng a statstcal aroach. Chaman &Hall; CRC. 8. Wolstenholme, L. C. (999). Relablty modelng a statstcal aroach. Chaman & Hall; CRC. 9. Yusuf, I. and Bala, S.I. (). Stochastc modelng of a two unt arallel system under two tyes of falures. Internatonal Journal of Latest trends n Mathematcs, Vol. (), -5. 9

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