COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS

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1 Avalable onlne at J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS IBRAHIM YUSUF Department of Mathematcal Scences, Faculty of Scence, Bayero Unversty, Kano, Ngera Abstract: In ths paper, two dfferent systems both are requrng a supportng unt for ther operaton are studed. The frst system consst of 3-out-of-4 subsystem requrng ts support from -out-of-4 subsystem for ts operaton whle the other system s two unt cold standby system where each unt s attached to ts supportng unt for ts operaton. Each system s attended by two reparmen, one reparng the man unt and the other reparng the supportng unt. Explct expressons for mean tme to system falure (MTSF) and steady- state avalablty are developed. We analyzed the system by usng lnear dfferental equatons. Effect of falure and repar rates on mean tme to system falure and steady-state avalablty have also been dscussed graphcally. Comparsons are made graphcally for specfc values of parameters. Furthermore, we compare these relablty characterstcs for the two models and found that model I s more relable and effcent than the remanng models. Keywords: Redundant system, Relablty characterstcs, supportng unt AMS Subject Classfcaton: 9B5. Introducton Redundancy s a technque used to mprove system relablty and avalablty. Receved November, 6

2 7 COMPARISON OF SOME RELIABILITY CHARACTERISTICS Relablty s vtal for proper utlzaton and mantenance of any system. It nvolves technque for ncreasng system effectveness through reducng falure frequency and mantenance cost mnmzaton. Studes on redundant system are becomng more and rcher day by day due to the fact that numbers of researchers n the feld of relablty of redundant system are makng huge contrbutons. Models of redundant systems as well as methods of evaluatng system relablty ndces such as mean tme to system falure (MTSF), system avalablty, busy perod of reparman, proft analyss, etc have been studed n order to mprove the system effectveness (see for example [] - [4]). Example of such systems are -out-of-, -out-of-3,-out-of-4, or 3-out-of-4 redundant systems. These systems have wde applcaton n the real world. The communcaton system wth three transmtters can be sted as a good example of -out-of-3 redundant system. In ths paper, we construct two dfferent redundant systems both requrng supportng unts/subsystem for ther operaton. The frst system/confguraton conssts of 3-out-of-4 subsystem requrng ts support from -out-of-3 subsystem for ts operaton as can be seen n Fg.. In the second system/confguraton the system s a two unt cold standby where each unt s attached to ts supportng unt as can be seen from Fg.3.Each system s attended by two reparmen. One reparng the man unt and the other reparng the supportng unt. Example of such systems can be seen n satellte amplfer redundant system, computer systems, etc. In ths paper, we construct two dstnct redundant systems and derved ther correspondng mathematcal models. Furthermore, we study relablty characterstcs of each model usng lnear dfferental equaton method. The focus of our analyss s prmarly to capture the effect of both falure and repar rates on the measures of system effectveness lke MTSF, avalablty and proft. We also looked at the effect of the system desgn. The organzaton of the paper s as follows. In Secton, we gve the notatons and assumptons of the study. In Secton 3, we gve detaled descrpton of the state of the systems.some relablty characterstcs of model I and II are derved n Sectons 4. The results of our numercal smulatons are presented n Secton 5 and dscussed n

3 IBRAHIM YUSUF 8 Secton 6. Fnally, we make a concludng remark n Secton 7.. Prelmnares. Notatons and assumptons Notatons S : Transton states,,,,3, 4,5,6,7 for system I and,,,3,4,5,6 for system II : repar rate of the man unt for both system I and II : repar rate of the supportng unt for both system I and II : Falure rate of the man unt for both system I and II : Falure rate of the supportng unt for both system I and II Pt (): Probablty row vector Pt: () Probablty that the system s n state S A : Man operatonal unts,,,3 n system I and, n system II P : Supportng unts,,3 n system I and, n system II MTSF : Mean tme to system falure,, A ( ): System avalablty,, Assumptons. System I consst s 3-out-of-4 subsystem and a -out-of-3 supportng subsystem. System II consst of two cold standby wth the supportng attached to each uny 3. Falure and repar rates are constant 4. Intally two unts are n operable condton of full capacty 5. System I faled when the number of workng unt goes down beyond one or when two or three of the supportng unts faled 6. System II faled when all the unts faled 7. Falure and repar tme follow exponental dstrbuton

4 9 COMPARISON OF SOME RELIABILITY CHARACTERISTICS 8. Repar s as good as new (Perfect repar).. Model descrptons.. Frst System / Confguraton P P P 3 A A A 3 A 4 Fg. Relablty block dagram of the System S 4 S S S S 3 S 6 S 7 S 5 Fg. transton dagram of the frst System

5 IBRAHIM YUSUF State of the System: State S : In subsystem I, two unts are operatonal, one unt s n standby. In subsystem II, three unts are operatonal, one unt s n standby. The system s operatonal State S : In subsystem I, two unts are operatonal, one s under repar. In subsystem II, three unts are operatonal, one unt s n standby. The system s operatonal State S : In subsystem I, two unts are operatonal, one unt s n standby. In subsystem II, three unts are operatonal, one unt s under repar. The system s operatonal State S 3 : In subsystem I, two unts are operatonal, one s under repar. In subsystem II, three unts are operatonal, one unt s under repar. The system s operatonal State S 4 : In subsystem I, one unt s under repar, one unt s watng for repar, one unt s good. In subsystem II, all the unts are good. The system faled State S 5 : In subsystem I, two unts are operatonal, one unt s n standby. In subsystem II, one unt s under repar, one s watng for repar and two unts are good. The system faled State S 6 : In subsystem I, one unt s under repar, one unt s watng for repar and one unt s good. In subsystem II, one unt s under repar, three unts are good. The system faled. State S 7 : In subsystem I, one unt s under repar, two unts are good. In subsystem II, one unt s under repar, one unt s watng for repar and two unts are good. The system faled... Second System / Confguraton

6 COMPARISON OF SOME RELIABILITY CHARACTERISTICS P A A P Fg. 3 relablty block dagram of the second system S S S S 3 S 4 S 5 S 6 Fg.4 Transton dagram of the second system States of the System: S : Unt I and supportng unt I are operatonal, unt II and supportng II are n standby. The system s operatonal S :Unt I s good, supportng unt I s under repar, unt II and supportng unt II are operatonal. The system s operatonal S : Unt I s under repar, supportng I s good, unt II and supportng unt II are operatonal. The system s operatonal S 3 : Unt I s good, supportng unt I s under repar, unt II s good, supportng unt II

7 IBRAHIM YUSUF s watng for repar. The system faled S 4 : Unt I s good, supportng unt I s under repar, unt II s under repar, supportng unt II s good. The system faled S 5 : Unt s under repar, supportng I s good, unt II s watng for repar, supportng unt II s good. The system faled S 6 : Unt s under repar, supportng I s good, unt II s good, supportng unt II s under repar. The system faled 3. Man results 3. Relablty Characterstcs of the frst System 3.. Mean tme to system falure analyss MTSF From Fg. above, defne Ptto () be the probablty that the system at tme tt, s n state S. Let Pt () be the probablty row vector at tme t, the ntal condton for ths paper are: P() [ P, P, P, P3, P4, P5, P6, P7()] =,,,,,,, we obtan the followng dfferental equatons: dp ( ) P P P dp ( ) P P P3 P4 dp ( ) P P P3 P5 dp3 ( ) P3 P P P6 P7

8 3 COMPARISON OF SOME RELIABILITY CHARACTERISTICS dp4 P4 P dp5 P5 P () dp6 P4 P3 dp7 P4 P3 Whch s n matrx form as : P A P () ( ) ( ) ( ) ( ) A It s dffcult to evaluate the transent solutons hence we delete the rows and columns of absorbng state of matrx A and take the transpose to produce a new matrx, say Q (see El sad and El hamd [, ], El sad [3], Haggag [4], Wang et al [5]). The expected tme to reach an absorbng state s obtaned from and Q P() P( absorbng ) () (3) E T P e t e Qt Q, snce Q (4) For system, explct expresson for the MTSF s gven by

9 IBRAHIM YUSUF 4 where N E T MTSF P()( Q ) P() P( absorbng ) D (5) Q ( ) ( ) ( ) ( ) N ( 3 ) (3 ) ( ) D Steady-State Avalablty analyss A ( ) For the analyss of avalablty case of Fg. usng the same ntal condtons for ths problem as =,,,,,,, P() [ P, P, P, P, P, P, P, P ()] The dfferental equatons n () above can be expressed as P P ( ) P ( ) P P ( ) P P 3 ( ) P3 P P4 4 P5 P 5 P 6 P 6 P7 P 7

10 5 COMPARISON OF SOME RELIABILITY CHARACTERISTICS The steady-state avalablty s gven by A ( ) P ( ) P( ) P ( ) P ( ) (6) 3 In the steady state, the dervatves of the state probabltes become zero so that whch n matrx form AP (7) ( ) P ( ) ( ) P 4 P5 P P P ( ) P3 6 P7 Usng the followng normalzng condton P ( ) P ( ) P ( ) P ( ) P ( ) P ( ) P ( ) P ( ) (8) To obtan P ( ), P ( ), P ( ), P3 ( ), we substtute (8) n one of the redundant rows of (7) to gve the followng system of lnear equatons n matrx form whch solved usng MATLAB to gve A ( ) ( ) P ( ) ( ) P ( ) ( ) P ( ) ( ) P3 ( ) P4 ( ) P5 ( ) P ( ) 6 P7 ( ) Explct expresson for steady state avalablty A ( ) s:

11 IBRAHIM YUSUF 6 A ( ) P ( ) P( ) P ( ) P ( ) (9) 3 3. Relablty Characterstcs of the second system 3.. Mean tme to System falure analyss MTSF From Fg. above, defne Ptto () be the probablty that the system at tme tt, s n state S. Let Pt () be the probablty row vector at tme t, the ntal condton for ths paper are: P() P[ P, P, P3, P4, 5 P, P6 =,,,,,,, ] we obtan the followng dfferental equatons: dp ( ) P P P dp ( ) P P P3 P4 dp ( ) P P P5 P6 dp3 P3 P dp4 P4 P dp5 P5 P dp6 P6 P () Whch s n matrx form as :

12 7 COMPARISON OF SOME RELIABILITY CHARACTERISTICS P A P () A ( ) ( ) ( ) It s dffcult to evaluate the transent solutons hence we delete the rows and columns of absorbng state of matrx A and take the transpose to produce a new matrx, say Q (see El sad and El hamd [, ], El sad [3], Haggag [4], Wang et al [5]). The expected tme to reach an absorbng state s obtaned from N E T MTSF P()( Q ) 3 P() P( absorbng ) D3 () Q P() P( absorbng ) () (3) E T P e and t e Qt Q, snce Q (4) Q ( ) ( ) ( ) For system, explct expresson for the MTSF s gven by

13 IBRAHIM YUSUF 8 MTSF P()( Q ) ( )( ) ( ) ( ) Steady-State Avalablty Analyss A ( ) For the analyss of avalablty case of Fg. usng the same ntal condtons for ths problem as =,,,,,, P() [ P, P, P, P, P, P, P ] The dfferental equatons n () above can be expressed as P ( ) P ( ) P ( ) P 3 P 4 P 5 P6 The steady-state avalablty s gven by A ( ) P ( ) P ( ) P ( ) (5) In the steady state, the dervatves of the state probabltes become zero so that whch n matrx form AP (6)

14 9 COMPARISON OF SOME RELIABILITY CHARACTERISTICS ( ) P ( ) P ( ) P P3 P 4 P5 P 6 Usng the followng normalzng condton P ( ) P ( ) P ( ) P ( ) P ( ) P ( ) P ( ) (7) to obtan P ( ), P ( ), P ( ), we substtute (7) n one of the redundant rows of (6) to gve the followng system of lnear equatons n matrx form whch solved usng MATLAB to gve A ( ) ( ) P ( ) ( ) P ( ) ( ) P ( ) P3 ( ) P4 ( ) P5 ( ) P6 ( ) A ( ) P ( ) P ( ) P ( ) 3.3. Numercal smulatons of Systems behavor For the study of system behavor, we plot graphs n Fg. 5 to 8 for MTSF and system avalablty wth respect to and.

15 IBRAHIM YUSUF 3 effect of on Avalablty.9 AV AV.8.7 Avalablty Fg. 5 effect of on systems avalablty effect of on Avalablty Avalablty AV AV Fg. 6 effect of on systems avalablty 6 5 effect of on MTSF MTSF MTSF MTSF Fg. 7 effect of on MTSF

16 3 COMPARISON OF SOME RELIABILITY CHARACTERISTICS 6 5 effect of on MTSF MTSF MTSF MTSF Fg. 8 effect of on MTSF 3.4 Dscusson For Fg. 5 we fxed =.69,. and vary. From Fg. 5 t s clear that avalablty ncreases wth ncrease n the value of whch reflects the effect of repar on avalablty. Ths reflects the effect repar rate on man unt (3-out-of-4 subsystem). It s clear that system I tend to ncrease wth ncrease n than system II. Thus, A( ) A( ).From Fg. 6, we fxed.5 and vary. It s evdent from Fg. 6 that system avalablty decrease wth ncrease n value of. Ths depcts the effect falure rate on man unt (3-out-of-4 subsystem). The graph n Fg. 6 below reveals that system II decreases more that system I. From the result n Fg. 6, t s clear that A( ) A( ). Fg. 7 shows the relatonshp between and MTSF. In ths fgure we fxed.,.5,.5 and vary where MTSF decrease as ncrease. The result have ndcated that MTSF MTSF. From Fg. 8, t s evdent that MTSF ncrease as ncreases. We fxed =.69,. and vary. From Fg. 8 t s clear that avalablty ncreases wth ncrease n the value of whch reflects the effect of repar on avalablty. It s clear that system I tend to

17 IBRAHIM YUSUF 3 ncrease wth ncrease n more than system II. Thus, MTSF MTSF 3.5 Concluson In ths paper, two dfferent reparable systems both wth standby unt and requrng a supportng unt for ther operatons are consdered. Explct expressons for mean tme to system falure MTSF and steady-state avalablty A ( ),, are developed for the two systems and comparatve analyss are performed to determne the optmal confguraton between the two systems. Graphcal studes of the systems behavor have shown that system I s better than system II. REFERENCES [] El sad, K.M. and El hamd, R. A. Comparson between two dfferent systems by usng lnear frst order dfferental equatons, Informaton and management Scences, Vol. 7, No.4, pp 83-94, 6 [] El sad, K.M. and El hamd, R. A. Comparson of Relablty characterstcs of two systems wth preventve mantenance and dfferent modes, Informaton and management Scences, Vol. 9, No., pp 7-8, 8 [3] El-Sad, K.M. Cost analyss of a system wth preventve mantenance by usng Kolmogorov s forward equatons method. Amercan Journal of Appled Scences 5(4), 45-4, 8 [4] Haggag, M.Y., (9). Cost analyss of a system nvolvng common cause falures and preventve mantenance. Journal of Mathematcs and Statstcs 5(4), 35-3, 9 [5] Wang., k. Hseh., C. and Lou, C. Cost beneft analyss of seres systems wth cold standby components and a reparable servce staton. Journal of qualty technology and quanttatve management, 3(), 77-9, (6)