RELIABILITY ANALYSIS OF A COMPLEX REPAIRABLE SYSTEM COMPOSED OF TWO 2-OUT-OF-3: G SUBSYSTEMS CONNECTED IN PARALLEL

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1 Journal of Reliability and Statitical Studie; ISSN (Print: 97-8, (Online: Vol. 7, Iue Secial (: 89- RELIILITY NLYSIS OF COMPLEX REPIRLE SYSTEM COMPOSE OF TWO -OUT-OF-: G SUSYSTEMS CONNECTE IN PRLLEL lka Munjal and S.. Singh eartment of Mathematic, Statitic and Comuter Science G.. Pant Univerity of griculture and Technology, Pantnagar, INI E Mail: alkamunjal8@gmail.com, drurajbingh@yahoo.com Received July 9, Modified January 7, cceted March, btract Thi aer deal with the reliability analyi of a comlex reairable ytem. The conidered ytem conit of two reairable ubytem L and M connected in arallel configuration. Subytem L and M both are -out-of-: G conit of tye- and tye- comonent reectively which are in arallel configuration. hot are of tye- and tye- i connected to the L and M ubytem reectively. Sytem ha two tate: Good and Failed. Sulementary variable technique ha been ued for mathematical formation of model. With the hel of Gumbel-Hougaard family of coula reliability and cot analyi of the ytem i evaluated. The ytem i tudied by uing the ulementary variable technique to obtain variou related meaure uch a mean time to failure, teady tate robability, availability and cot analyi. t lat ome articular cae of the ytem are taken to highlight the different oibilitie. Key Word: Sytem, Reliability, vailability, MTTF, Senitivity, k-out-of-m: G, Gumbel- Hougaard Coula.. Introduction In the reent cometitive market cutting down roduction cot and imroving roductivity and delivery erformance of manufacturing ytem are the key objective of Indutrie. Mot of the indutrie and manufacturer are trying to introduce more comlex mechaniation and automation in their indutrial roce to comete in the rate race of manufacturing more ohiticated equiment which lead to more comlexitie in the ytem, ubequently chance of failure of uch ytem increae. Reliability and it related iue have therefore become very relevant for manufacturing ytem to meet the miion ucce. Variou reearcher have made their contribution in the develoment of different technique to find the reliability characteritic of comlex ytem. They focued themelve on the analyi of the variou reairable and nonreairable ytem. Some of them ued the concet of coula to find the joint ditribution of failure and reair rate. They alo alied reliability technique to analye the different indutrie. Reliability and availability analyi make it oible to evolve different alternative for the imrovement of the ytem deign and configuration. With the advent of modern age and develoment of the comlex engineering ytem, an imrovement in the deign to imrove the reliability of the ytem became a challenge for the engineer. Engineer are intereted in the develoment of uch ytem

2 9 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S whoe reliability and rofit are maximum with minimum cot. For thi, it i eential to develo uch ytem whoe erformance are failure free. lot of tudie dealing with the reliability and availability of k-out-of-n ytem (tandby ytem have been carried out. The ytem of n unit in which k unit are ufficient to erform the entire function of the ytem are called k-out-of-n redundant ytem. Thee ytem have wide alication in the real world. The communication ytem with three tranmitter can be ited a a good examle of -outof- redundant ytem. Many ueful reult [8, and 5] have been ublihed by reearcher (Moutafa; Haett et al.; and Sidarovky et al. regarding variou tye of failure in a ytem. In recent at ome good work have been done in reliability modelling with the alication of coula. Lindkog [6] alied the coula into modelling deendence. Nailwal et al. [9] have tudied erformance evaluation and reliability analyi of a comlex ytem with three oibilitie in reair with the alication of coula. Nailwal et al. [] have alied coula in reliability meaure and enitivity analyi of a comlex matrix ytem including ower failure. Kumar et al. [] delt with rofit analyi of two unit non-identical ytem with degradation and relacement. ngu [] ha tudied k-out-of-n: G ytem in different configuration. Malik et al. [7] have tudied the rofit analyi of a tochatic model of -out-of- redundant ytem with inection by a erver who aear and diaear randomly. Goel et al. [] have analyed tochatic behavior of a two unit arallel ytem with artial and catatrohic failure and reventive maintenance. Ram and Singh [, and ] have analyed the reliability characteritic of variou comlex reairable ytem by uing Gumbel-Hougaard family of coula. Pandey et al. [] ha analyed reliability and cot of a ytem with multile comonent uing coula. Many reearch reult have been reorted on reliability of -out-of- redundant ytem involving -out-of- ytem a main ytem, but a little attention ha been aid for the cae where -out-of- ytem i ued a a ubytem. Keeing thi fact in view, here we have focued on thi iue while develoing reliability model. In the reent model we have conidered a ytem coniting of two -out-of-: G ubytem L and M. Subytem L and M are connected in arallel configuration which conit of tye- and tye- comonent reectively. hot are of tye- and tye- i connected to the ubytem L and M reectively. S and S are two different tye of are that can relace only own tye comonent (S can relace only, S can relace only ued in model. In the tranition tate diagram (Figure of the ytem, we x y w denote S S by the joint tate that there x tye- comonent, y tye- comonent, tye- are comonent and w tye- are comonent are functional (x; y k, k,, m; ; w,. Each comonent of the ytem ha two mode- good and failed. Failure rate of comonent of tye- and tye- are contant. ll comonent of tye-/tye- are reairable and reair rate follow general ditribution in all the cae. We have ued Gumbel-Hougaard family of coula to find joint ditribution of reair whenever both the ubytem are being reaired imultaneouly with two different reair rate. The reair of the failed comonent i erfect. y the hel of Lalace tranform and ulementary variable technique the following reliability characteritic of the ytem have been analyed in thi model: (i Tranition tate robabilitie. (ii ymtotic behaviour of ytem.

3 Reliability nalyi of Comlex Reairable Sytem Comoed... 9 (iii Variou reliability meaure uch a availability, reliability, mean time to failure and enitivity with reect to different arameter. t lat, ome ecial cae of the ytem are taken to highlight the reliability characteritic of the ytem. Thee are a follow:. Reairable and non-identical.. Reairable and identical. C. Non-reairable and non-identical.. Non-reairable and identical. The tate ecification chart of the conidered ytem i given in Table. lock diagram and tranition tate diagram of invetigated ytem are hown in Figure and Figure reectively.. umtion (i Initially the ytem i in erfectly good tate i.e. all the comonent are functioning erfectly. (ii t t all the comonent are erfectly well and at t > they tart oerating. (iii The ytem conit of two ubytem L and M connected in arallel. (iv Subytem L i -out-of-: G ytem of comonent of tye- and M i -outof-: G ytem of comonent of tye-. (v hot are of tye- and tye- i connected to the ubytem L and M reectively. When a comonent fail in ubytem, the hot are i witched into oeration. (vi Each comonent i either functional or failed. (vii Failure rate of tye- comonent and tye- comonent are aumed a contant. (viii Each ubytem on comlete failure goe for reair. (ix The reaired ubytem i a good a new and i immediately reconnected to the ytem. (x Tranition from the comletely failed tate S to the initial tate S follow two different ditribution. (xi Joint robability ditribution of reair rate from S to the initial tate S i comuted by Gumbel-Hougaard family of coula. (xii If both unit fail, the ytem fail comletely.. Nomenclature / : Failure rate of comonent of tye-/tye-. η ( x : Reair rate of tye- comonent. ψ ( y : Reair rate of tye- comonent. uv ( t : Probability that the ytem i in S tate at intant t for u; v to. uv uv : Lalace tranform of P uv ( t. uv ( j, t : The df (ytem i in tate S and i under reair; elaed reair time i j, uv t, where j x, y,. : Couled reair rate. (

4 9 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S Conidering u η( x and u ψ ( y, the exreion for joint robability (failed tate S to good tate S according to Gumbel-Hougaard family of coula i given by. State Secification G Good tate, F Failed tate State State of ubytem ] θ θ θ ( ex[(log( u (log( u State of ubytem S G G G S G G G S G G G S G G G S G G G S G G G S F G G S G G G S G G G S G F G S F G G S G G G S G F G S F G G S G F G S F F F Table : State Secification State of ytem 5. lock and State Tranition iagram Figure and rereent the block diagram and the tate tranition diagram of invetigated ytem reectively.

5 Reliability nalyi of Comlex Reairable Sytem Comoed... 9 Subytem L Figure : lock diagram of ytem Subytem M Figure : Tranition State iagram

6 9 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S 6. Formulation of Mathematical Model y robability conideration and continuity argument, we obtain the following et of integro-differential equation governing the behavior of the ytem. d ( t ( y t ψ( y dy ( x t η( x dx ( t ( d ( t ( d dt,,,, (, t ( d (, t ( d (, t ( d d ( t ( t dt ( d ( t ( t ( dt d ( t ( t dt ( d ( t ( t ( t dt (5 d ( t ( t dt (6 η( x ( x, t t x (7 d ( t ( t ( t dt (8 d ( t ( t ( t dt (9 ψ( y ( y, t t y ( ( (, t t ( d ( t ( t ( t dt ( ( (, t t ( ( (, t t ( ( (, t t (5 ( (, t t (6 oundary condition (, t ( t (7 (, t ( t (8, t t t (9 (

7 Reliability nalyi of Comlex Reairable Sytem Comoed (, t ( t ( t ( (, t ( t ( t ( (, t ( t ( t ( (, t ( t ( t ( Initial condition ( and other robabilitie are ero at t. ( Solution of the model Solving equation (- with the hel of Lalace tranform and uing equation (, we get [ ] ( y, ψ( y dy ( x, η( x dx (, ( d (, ( d (, ( d (, ( d (, ( d [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] (6 (7 (8 (9 ( ( ( ( η( x ( x, x ( ψ( y ( y, y (5 ( (, (6 ( (, (7 ( (, (8 ( (, (9 ( (, ( oundary condition (, ( (, (, ( (5

8 96 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S (, ( (, (5 (, (6 (, t ( t ( t (7 The tranition tate robabilitie for the ytem can be viewed a a reult of olving the et of equation (5- with the hel of (- 7 (8 ( ( (5 ( (. (5. ( ( ( (5. (5.. (5.. (55 [ ].. [ Sη( ] ( x, ( ( (.. [ Sψ( ] ( x, ( ( ( (56 ( x (57 (58 E..., (59 ( x, (, (, (,... (6 F ( S ( C E ( S ( C F ( S ( ( S ( ( C E ( C F... (6... (6... (6

9 Reliability nalyi of Comlex Reairable Sytem Comoed where y y x x ψ ψ η η ( (6 (65 (66 C ( (67 S S E η (68 S S F η (69 lo x u, y,,,, (7 down, (7 7. ymtotic ehaviour of the Sytem Uing bel lemma in Lalace tranform, ( lim } ( { lim t F t F rovided the limit on the right hand ide exit, the time indeendent oerational robabilitie are obtained a follow: (7 (7 (7

10 98 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S ( ( (. (75. ( ( ( ( ( (76. (77.. (78 ( (.. (79 ( (... (8 ( C( ( ( ( ( η( x.. ( ( ( ( ψ( y ( E(..... (8... (8 ( F(... ( C E (85... ( C F (86... ( ( C E C F (87 8. Secial Cae When reair follow exonential ditribution. In thi cae the reult can be derived by utting θ { } θ S η( x ψ( y Sµ, Sψ(, ( ex logη( x logψ ( x θ (88 η( x ψ( y θ {( η ( ψ } θ ex log ( x log ( x θ. Reairable and Non Identical When ytem i reairable and all the unit of the ytem are non-identical then the tranition tate robabilitie correonding to ytem can be obtained a (89 (8 (8

11 Reliability nalyi of Comlex Reairable Sytem Comoed (9 ( ( (9. ( ( (9. (9 (. (9 ( (.. (95.. (96... (97 C.. (98 ( ( ( η( x ( ( ( ψ ( y.. (99... ( E... ( F ( C F ( C E ( ( ( C E C F... (... (... (. Reairable and Identical When the conidered ytem i taken to be reairable and unit are identical then tranition tate robabilitie in thi cae tranition tate robabilitie can be obtained by utting in equation (8-6, which are given by (5

12 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S (6 ( 7 (7 ( 7. (8 ( 7( 6.. ( 6( 7. ( 7( 6 (9 (.. (.. (.. ( ( 7( 6 ( η( x.. ( ( 7( 6 ( ψ ( y... (5 E... F (6... C ( ( C E ( ( C F ( ( ( ( E C F ( C. Non Reairable and Non Identical When the conidered ytem i taken to be non-reairable (i.e. η ( x ψ ( y ( and unit are non-identical then the tranition tate robabilitie correonding to the reent ytem are given by ( (

13 Reliability nalyi of Comlex Reairable Sytem Comoed... ( (. ( ( (. (5 (. (6 ( (.. (7.. (8 ( ( ( (9.. ( ( (.. (... ( E... ( C... ( F... ( C E... (5 C F... (6 C E C F. Non Reairable and Identical When the conidered ytem i taken to be non-reairable and unit are identical then the tranition tate robabilitie correonding to reent ytem are given by (7 (8 ( 7 (9 7

14 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S. ( ( 7( 6.. ( 6( 7 ( 7( 6 (. (.. (.. (.. (5 ( 7( 6(.. (6 ( 7( 6(... (7 E... (8 C... (9 F 5 C E 5 C F 6 E C F... ( (5 (5 9. Numerical Comutation The Male oftware ha been ued to analye reliability, availability, MTTF, cot effectivene and enitivity of the ytem. (I Reliability nalyi Let the failure rate of the comonent are. and., reair rate η ( x ψ ( y (, θ and x y. lo let the reair follow exonential ditribution, i.e. equation (88 hold. Putting all thee value in equation (7, taking invere Lalace tranformation and etting t,,,,, 5, 6, 7, 8, 9,, one can obtain Table and Figure which rereent how reliability varie with reect to the time.

15 Reliability nalyi of Comlex Reairable Sytem Comoed... (II vailability nalyi Let the reair rate η ( x ψ ( y (, failure rate.,., and x y. Putting all value in equation and taking invere Lalace tranformation, we get P u (t8.8598/ex(.6t.69865/ex( t /ex(.t /ex(.t /ex(.9t -.756/ex(.8t-5.75/ex(.7t /ex(.85t.985/ex(.66t /ex(.8788t.8875/ex(.86t -.798/ex(.5t /ex(.5 t (5 Now etting t,,,,, 5, 6, 7, 8, 9,, one can get Table and Figure which how the variation of availability with reect to time. (III MTTF nalyi Let u uoe that reair follow exonential ditribution then uing equation (88 and MTTF of the ytem i given by MTTF lim P (5 u We have the following three cae when reair rate η ( x ψ ( y (, θ and x y : (a uming failure rate.6 and varying the value of a.,.,.,.,.5,.6,.7,.8,.9,., we obtain the change of MTTF with reect to. (b Varying a.,.,.,.,.5,.6,.7,.8,.9,., one can find change of MTTF with reect to. (c Let u increae the value of and from. to., we obtain manner in which MTTF varie with reect to and imultaneouly. Table and Figure 5 how how MTTF varie with reect to time. (IV Cot nalyi Let the failure rate.,., reair rateη ψ and x y. Putting all thee value and taking invere Lalace tranform, one can obtain equation (56. If the reair facility i alway available, then exected rofit during the interval (, ] i given by E t ( t c P ( tdt t (55 P u c where c and c are revenue rate er unit time and ervice cot er unit time reectively, then E (t c (-69.69/ex(.6e-t /ex( t9.99/ ex(.t /ex(.t /ex(.9e-t /ex(.8e-t 78.96/ex(.7e-t.5688e-/ex(.85t-.6e-/ex(.66t.6558e-/ex(.8788t e-/ex(.86t

16 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S.7e-/ex(.5t e -8/ ex(.5t8.97-c t (56 Taking c and c.,.,.,.,.5 and uing equation (88 one can comute the variation of E P (t with reect to time. The comutational value obtained are given in Table 5 and hown in Figure 6. (V Senitivity nalyi uming that the equation (88 hold. We firt erform a enitivity analyi for change in R(t reulting from change in ytem arameter and which yield R( t -96tex(-(yxt9tex(-(xyt-96tex(-(xyt -tex(-(yxt6tex(-(xyt7tex(-(yxt-tex(-xt -tex(-xt 7tex(-(x yt tex(-xt ((-8xt-ex((xyt (57 (9xt ex(-(x yt inh(/xtex((-7/x-yt/x ((-8ty-ex(-(y xt (9ty ex(-(x yt R( t inh(/tyex((-x-7/yt/y9t ex((xyt-tex(-(xyt (58-96tex(-(yxt7tex(-(xyt-96tex(-(xyt7tex(-(yxt tex(-ty 6tex(-(x yt-tex(-ty-tex(-ty Table 6 and 7 are correonding to the enitivity analyi of the ytem reliability with reect to change in and reectively. The nature of enitivity ha been hown in Figure 7 and 8. One can ee that enitivity of the ytem reliability decreae with the increae in the value of and. Table : Time v. Reliability Figure : Time v. Reliability

17 Reliability nalyi of Comlex Reairable Sytem Comoed... 5 Table : Time v. vailability Figure : Time v. vailability MTTF MTTF and MTTF Table : Failure rate v. MTTF

18 6 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S Figure 5: Failure rate v. MTTF Time E(t C. C. C. C. C E E E E E E E E E Table 5: Time v. exected rofit

19 Reliability nalyi of Comlex Reairable Sytem Comoed... 7 Figure 6: Time v. exected rofit Time Value of R( t / Table 6: Senitivity analyi of the ytem MTTF w. r. t.

20 8 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S Figure 7: Senitivity of ytem MTTF with reect to different value of Time Value of R( t / Table 7: Senitivity analyi of the ytem MTTF w. r. t.

21 Reliability nalyi of Comlex Reairable Sytem Comoed Figure 8: Senitivity of ytem MTTF with reect to different value of. Interretation of the Reult and Concluion For a more concrete tudy of the ytem behaviour, curve for reliability, availability, MTTF, exected rofit with reect to time and enitivity with reect to and have been lotted. The Table give the variation of reliability with reect to the time. y critical examination of the Figure we conclude that the reliability of the ytem decreae a time increae and attain a value.5756 at t. Figure deict the behaviour of availability with reect to time and it value ha been given in Table. It i clear from the Figure that availability decreae a the time increae from to and attain a value.578 at t. Figure 5 how the variation of ytem MTTF with reect to, and (. The correonding value have been given in Table. Obervation of thi figure reveal that in each cae MTTF of conidered ytem decreae a failure rate increae from. to.. It varie from , and.-. with reect to, and reectively. lo, MTTF of the ytem i greater with reect to than and. One of the intereting fact i that at failure rate.5, MTTF with reect to and are ame. Prior to failure rate.5, MTTF i higher with reect to than and afterward ituation got revered. We alo oberve that rior to failure rate.6, value of the MTTF i higher with reect to than and after thi the value of MTTF got revered. It i worth mentioning that the value of MTTF with reect to and are the ame at failure rate.6. For the cot analyi of the ytem we kee revenue cot er unit time at and vary ervice cot from. to.5. The behaviour of exected rofit can be oberved from Figure 6. We can ee from the figure that the rofit goe on decreaing with the

22 Journal of Reliability and Statitical Studie, ril, Vol. 7 (S increae in ervice cot. One can draw an imortant concluion from Figure 6 that ytem become more rofitable when ervice cot i.. The highet value of exected rofit i Figure 7 and 8 rereent how enitivity of the ytem reliability varie with reect to arameter and. It i clear from thee figure that enitivity of ytem reliability initially decreae and then increae a time ae with reect to and and increae with the increae in failure rate and from. to.. It i intereting to note that enitivity of the ytem reliability i ame with reect to and. y tudying the grah we can conclude that the ytem can be made le enitive by decreaing it failure rate. Reference. ngu, J. E. (988. On comuting MTF for a k-out-of-n: G reairable ytem, IEEE Tranaction on Reliability, 7(,. -.. Goel L. R. and Guta, P. (98. Stochatic analyi of a two unit arallel ytem with artial and catatrohic failure and reventive maintenance, Microelectron. Reliab., (, Haett, T. F., ietrich,. L. and Sidarovky, F. (995. Time varying failure rate in the availability and reliability analyi of reairable ytem, IEEE Tranaction on Reliability, (, Kumar, J. and Kadyan, M. S. (. Profit analyi of a ytem of non identical unit with degradation and relacement, International Journal of Comuter lication, (, Lim, Tae-Jin and Lie, C. H. ( nalyi of ytem reliability with deendent reair mode, IEEE Tranaction on Reliability, 9 (, Lindkog, F. (. Modelling deendence with coula, ETH, Zurich, Riklab reort. 7. Malik, S. C. and hardwaj, R. K. (8. Profit analyi of a tochatic model of -out-of- redundant ytem with inection by a erver who aear and diaear randomly, International Journal of uine and Engineering Reearch, (, Moutafa, M. S. (. Reliability analyi of ytem with time varying failure rate, Economic Quality Control, 6(, Nailwal,. and Singh, S.. (. Performance evaluation and reliability analyi of a comlex ytem with three oibilitie in reair with the alication of coula, International Journal of Reliability and lication. (, Nailwal,. and Singh, S.. (. Reliability meaure and enitivity analyi of a Comlex matrix ytem including ower failure, IJE Tranaction : aic, 5(,. 5-.

23 Reliability nalyi of Comlex Reairable Sytem Comoed.... Pandey, S.., Singh, S.. and Sharma, S. (8. Reliability and cot analyi of a ytem with multile comonent uing coula, Journal of Reliability and Statitical Studie, (, 5-.. Ram, M and Singh, S.. (9. nalyi of reliability characteritic of a comlex engineering ytem under coula, Journal of Reliability and Statitical Studie, (, Ram, M and Singh, S.. (. Cot benefit analyi of a ytem under head ofline reair aroach uing Gumbel-Hougaard Family coula, Journal of Reliability and Statitical Studie, 5(, Ram, M. and Singh, S.. (8. vailability and cot analyi of a arallel redundant comlex ytem with two tye of failure under reemtive-reume reair diciline uing Gumbel-Hougaard family of coula in reair, International Journal of Reliability, Quality & Safety Engineering, 5 (,. -65.