Reliability Analysis of Embedded System with Different Modes of Failure Emphasizing Reboot Delay

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1 International Journal of Applied Science and Engineering 3., 4: Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay Deepak Kumar* and S. B. Singh Department of Mathematic, Statitic and Computer Science,. B. Pant Univerity of Agriculture and Technology, Pantnagar Abtract: Preent paper dicue the reliability analyi of a complex ytem which conit of two repairable ubytem namely A and B connected in erie. Subytem A i a k-out of -n: ytem and ubytem B i a circular conecutive -out of -3: F ytem. In thi tudy an inpection hop ha been taken into conideration along with a pecial type of delay viz. reboot delay. By employing upplementary variable technique, Laplace tranformation and umbel-hougaard family of copula variou tranition tate probabilitie, reliability, availability, M.T.T.F., cot analyi, enitivity analyi and teady tate behaviour of the ytem have been obtained. At the end ome pecial cae of the ytem have been taken and important reult have been derived a particular cae. Keyword: Reliability; availability; mean time to failure; enitivity; reboot delay; complex ytem; gumbel-hougaard family of copula.. Introduction Many reliability tudie aume tochatically independent component of a ytem, i.e. the condition of one component doe not affect the lifetime ditribution of other component. But in real world there are ytem having dependent component in which a component failure may have ome influence on the working component by cauing an increaed tre. The reliability characteritic of uch ytem depend not only on it component but alo on the interaction among the component. In thee ytem a ingle failure event can propagate and caue failure of other component. For example, in a load haring ytem if one component fail, the work load ha to be hared by remaining working component, which can caue an increae in the failure rate of every working component. In general a ytem pae through three phae of failure viz. initial failure (due to deigning problem), random failure (due to accident or any other reaon) and wear out failure (due to over age of the ytem). Each ytem may have different failure propertie. For intance, the failure rate of the ytem may be contant, increaing or decreaing and the type of the failure (partial failure, catatrophic failure, human failure etc.) may be different. Likewie it may be the cae that in the ytem chance of initial failure i maximum. In thi cae once the ytem pae the initial phae uccefully then it will work properly for a long period of time. Many ueful reult [3, 4, 5, 8, 9,, ] have been publihed by reearcher regarding variou type of failure in a ytem. But it eem that a ytem having maximum initial chance of failure ha not been tudied widely. Reearche [, ] carried out analyi of ytem * Correponding author; deepakdev6@gmail.com 3 Chaoyang Univerity of Technology, ISSN Received 8 April 3 Revied 3 Augut 3 Accepted 7 Augut 3 Int. J. Appl. Sci. Eng., 3., 4 449

2 Deepak Kumar and S. B. Singh having k-out of-n: ytem but they have not conidered the concept of inpection hop where it i to be decided whether the ytem hould go for repair or for replacement of the failed unit. Furthermore, the ytem configuration alo affect the failure propertie of a ytem. The ytem having different configuration poe different propertie. A circular conecutive -out of-3: F i a type of ytem configuration having a equence of 3 ordered component arranged along a circle uch that the ytem will be in failure condition if at leat conecutive component in the ytem will fail. In thee ytem the order in which the component are arranged i alo important. There are ome tudie [3] in which uch ytem with identical component have been conidered. But if the component are not identical the ytem will have different propertie. Thu, there i a cope for further work in thi direction. In real life it can be een that the toppage of a ytem due to any error take ome time to recover/retart, ofcoure with ome recovery rate. The time between the ytem failure and retart i aid to be reboot delay. The concept of reboot delay and it effect on the reliability and/or availability of a repairable ytem ha been propoed by []. The above ituation can be viualied in communication and computer cience application where there i a reboot delay. Recently, [6, 7] tudied reliability meaure of repairable ytem in which witching failure, detection delay and reboot delay are conidered. Reboot delay can be viewed a the unreliability of a recovery oftware. Further, the reliability of a ytem with inpection hop concept emphaizing reboot delay employ an increaingly important iue in power ytem, manufacturing ytem, and indutrial ytem. Therefore, keeping all the above fact under conideration here reearcher attempt to tudy the reliability and enitivity meaure of a complex ytem having different mode of failure incorporating reboot delay. Thi paper differ from previou work in the following ene: (i) it emphaize the reliability problem with inpection hop concept emphaizing reboot delay (ii) it perform enitivity analyi for the reliability with repect to different ytem parameter. Variou reliability meaure have alo been computed for the conidered reliability model for different cae. An attempt ha alo been made to provide performance analyi of the ytem by applying the reliability theoretic approache. Several reliability characteritic of complex ytem evaluated in thi tudy will be helpful to ytem deigner a well a operation manager. The preent contribution i tructured a follow: Section decribe the material and method of the model; ub-ection.,.,.3,.4 and.5 cover the aumption, notation, formulation of mathematical model and olution of the model repectively of the propoed problem. Section 3 cover reult and dicuion of the model; ub-ection 3., 3. and 3.3 cover aymptotic behaviour of the ytem, particular cae in the abence of reboot delay in the ytem and numerical computation repectively. Again ub-ection 3.3 ub divided into five ection 3.3., 3.3., 3.3.3, and which compute different reliability meaure namely availability analyi, reliability analyi, M. T. T. F. analyi, cot analyi and enitivity repectively. Section 4 preent the concluion of the propoed analyi. Block diagram, tranition diagram and tate pecification chart of the ytem are hown in Figure, and Table repectively.. Material and method In thi tudy, the complex ytem conidered conit of two repairable ubytem namely A and B. Subytem A i a k-out of -n: ytem, i.e if any k unit out of n unit work, ubytem A will be operational wherea ubytem B i a circular conecutive -out of -3: F ytem i.e, if 45 Int. J. Appl. Sci. Eng., 3., 4

3 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay conecutively two unit out of three unit fail, ubytem B will fail. The ubytem A and B are arranged in erie configuration. The ubytem A can fail only from fully operational tate. Once it paed through the initial phae it will be operational for ufficiently long period of time. During thi period if any failure occur, then it may lead to further failure in the ytem. Since ubytem A i a non repairable ytem o whenever there occur any failure in ubytem A we go for replacement. It i alo aumed that after failure in the ubytem B, the ytem goe to the inpection hop where it i to be decided whether the ytem hould go for repair or for replacement of the failed unit. We have aumed that after repair of the ubytem B it work at higher rik of failure. Further, the policy of the maintenance company providing the replacement or repair of the unit i to replace the unit if they further fail after repair. So any further failure in the ubytem B i being replaced by new ubytem without any inpection. The tate S6 in which the ubytem A and B both are in failed tate i a critical degraded tate. Any further failure in the ytem from a critical degraded tate will be replaced by the new one. A mentioned earlier, many time failure take ome time to recover/retart. For example, when the power goe, the generator take ome time to tart. Conequently power ytem take ome time to retart which i aid to be reboot delay for thi. Thi apect i alo incorporated in reliability modeling of the preent ytem. The ytem i tudied by uing the upplementary variable technique, Laplace tranform and umbel-hougaard family of Copula. When two type of repair occur in preent ytem, coupled repair rate evaluated by umbel-hougaard family of copula. At lat ome numerical example have been taken to highlight the reliability characteritic of the ytem. The following characteritic of the ytem have been analyzed: a) Tranition tate probabilitie of the ytem. b) Aymptotic behaviour of the ytem. c) Variou meaure uch a reliability, availability, M.T.T.F. and cot effectivene of the ytem. d) Senitivity analyi of the ytem with repect to different parameter. Subytem B Subytem A B B3 B Figure. Diagram of invetigated ytem Int. J. Appl. Sci. Eng., 3., 4 45

4 Deepak Kumar and S. B. Singh Figure. State tranition diagram 45 Int. J. Appl. Sci. Eng., 3., 4

5 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay Table. State pecification Number of good Sytem tate State State of ubytem A unit of ubytem B S 3 S S S3 3 S4 F S5 F S6 DR S7 Fr S8 FR S9 3 S S S FR S3 FR S4 F 3 FR S5 F FR : ood tate, F: Failed tate; Fr: Failed under repair; FR: Failed under replacement; DR: Critical degraded under replacement... Aumption a) b) c) d) e) f) g) h) i) Initially all component are functioning perfectly. The ytem conit of two ubytem namely A and B. Sytem ha three tate good, critical degraded and failed. Subytem A and B are arranged in erie configuration. The tate S6 in which both ubytem A and B are in failed tate, i a critical degraded tate. From a critical degraded tate we replace the failed unit directly without inpection. Subytem A can fail only from fully operational tate. Once it tart working properly then it will not fail for a long period of time. During thi period if any failure occur then it may lead to further failure. The policy of the maintenance company providing the replacement or repair of the unit i to replace the unit without inpection if thee unit further fail after repair. After repair of ubytem B it work at higher rik of failure, i.e. failure rate of ubytem B increae in thi tate. Due to the reboot delay the ytem top and retart after ome time with ome recovery rate. Int. J. Appl. Sci. Eng., 3., 4 453

6 Deepak Kumar and S. B. Singh.. Notation The following notation are aociated with thi model Failure rate of firt and third unit of B. a Failure rate of econd unit of B. b R Increaed failure rate of econd unit of B. Increaed failure rate of firt and third unit of B. Reboot delay rate. ( x) ( x) A eneral repair rate after detecting when individual repair i done. eneral repair rate after failure in critical degraded tate when individual repair i done. Failure rate of ubytem A. y Coupled repair rate i.e. repair rate when two type of repair occur. C (, ) exp( (( log ) ( log ) ) ), Recovery rate after reboot delay. Pi (t ) Probability that the ytem i in Si tate at intant t for i = to 5. Pi ( x, t ) P5 ( y, t ) Pi ( ) E p (t ) The probability ditribution function (ytem i in tate Si and i under repair, elaped repair time i (x, t) where i = 4, 5, 6, 7, 8,, 3, 4. The probability ditribution function (ytem i in tate S5 and i under repair, elaped repair time i (y, t)). Laplace tranformation of Pi (t). Expected profit during the interval (, )..3. Formulation of mathematical model By probability conideration and continuity argument, we obtain the following et of integro-differential equation governing the behavior of the ytem. d dt A R P ( t ) P6 (x, t ) ( x )dx P8 (x, t ) ( x )dx P ( x, t ) ( x )dx () P3 ( x, t ) ( x )dx P4 ( x, t ) ( x )dx P5 ( y, t ) ( y)dx d dt P (t ) P (t ) () d dt P (t ) P (t ) (3) 454 Int. J. Appl. Sci. Eng., 3., 4

7 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay d dt A P3 (t ) A P (t ) (4) t x d e P4 ( x, t ) (5) t x d e P5 ( x, t ) (6) t x ( x) P6 ( x, t ) (7) t x ( x ) P7 ( x, t ) (8) t x ( x ) P8 ( x, t ) (9) d dt a b P9 (t ) P7 ( x, t ) ( x) dx () d dt a P (t ) b P9 (t ) () d dt a b P (t ) a P9 (t ) () t x ( x ) P ( x, t ) (3) t x ( x ) P3 ( x, t ) (4) t x ( x ) P4 ( x, t ) (5) t y ( y) P5 ( y, t ) (6) Int. J. Appl. Sci. Eng., 3., 4 455

8 Deepak Kumar and S. B. Singh Boundary condition P4 (, t ) P (t ) (7) P5 (, t ) P (t ) P (t ) (8) P6 (, t ) P3 (t ) (9) P7 (, t ) D P4 (t ) P5 (t ) () P8 (, t ) e P4 (t ) P5 (t ) () P (, t ) a P (t ) () P3 (, t ) b P (t ) a P (t ) (3) P4 (, t ) A P3 (t ) (4) P5 (, t ) R P (t ) (5) where exp( (( log ) ( log ) ) ), Initial condition P () and other probabilitie are zero at t=. (6).4. Solution of the model Taking Laplace tranform of equation ()-(5) and uing initial condition, we get A R P () P6 (x, ) (x)dx P8 (x, ) (x)dx P(x, ) (x)dx P3(x, ) (x)dx (7) P4(x, ) (x)dx P5(y, ) (y)dy P ( ) P ( ) (8) P ( ) P ( ) (9) A P3 ( ) A P ( ) (3) x d e P4 ( x, ) (3) 456 Int. J. Appl. Sci. Eng., 3., 4

9 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay x d e P5 ( x, ) (3) x ( x) P6 ( x, ) (33) x ( x) P7 ( x, ) (34) x ( x) P6 ( x, ) (35) a b P9 () ( x) P7 ( x, )dx (36) a P ( ) b P9 ( ) (37) a b P ( ) a P9 ( ) (38) x ( x) P ( x, ) (39) x ( x) P3 ( x, ) (4) x ( x) P4 ( x, ) (4) y ( y) P5 ( y, ) (4) Boundary condition P4 (, ) P ( ) (43) P5 (, ) P ( ) P ( ) (44) P6 (, ) P3 ( ) (45) P7 (, ) d P4 ( ) P5 ( ) (46) Int. J. Appl. Sci. Eng., 3., 4 457

10 Deepak Kumar and S. B. Singh P8 (, ) e P4 ( ) P5 ( ) (47) P (, ) a P ( ) (48) P3 (, ) b P ( ) a P ( ) (49) P4 (, ) A P3 ( ) (5) P5 (, ) R P ( ) (5) Solving equation (7-4) and uing equation (43-5), one can get the following tranition tate probabilitie: P ( ) P ( ) ( ) (5) P ( ) P ( ) ( ) (53) P3 ( ) A P ( ) ( A ) (54) P4 ( ) S ( ) P ( ) S ( ) P5 ( ) P ( ) ( 3 ) (55) (56) P6 ( ) S ( ) A P ( ) A (57) P7 () d S () P () ( d e ) ( ) (58) P8 () e S () P () ( d e ) ( ) (59) P9 ( ) S ( ) d S ( ) ( d e ) a b (6) P ( ) ( ) P ( ) S ( ) d b S ( ) ( d e ) a b a P ( ) ( ) 458 Int. J. Appl. Sci. Eng., 3., 4 (6)

11 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay P ( ) S ( ) 4 a d S ( ) ( d e ) a b a b (6) P ( ) ( ) S ( ) P ( ) a P ( ) (63) S ( ) P3 ( ) b P ( ) a P ( ) (64) S ( ) A P4 ( ) P ( ) A (65) S ( ) P5 ( ) R P ( ) (66) where S () ( x) ( x) ( y), S ( ), S ( ) ( x) ( y) ( x) Putting all thee value in equation (7) and implifying, we get tranition tate probability at S a P ( ) D( ) (67) where A D()= [ A R ] S ( ) A S ( ) ( d e ) ( ) S ( ) a b d S ( ) S ( ) a d S ( ) e ( a b ) a b a b A S ( ) R S ( ) a a b A (68) Tranition tate probability that the ytem i in the up tate i given by Int. J. Appl. Sci. Eng., 3., 4 459

12 Deepak Kumar and S. B. Singh P up ( ) P ( ) P ( ) P ( ) P 3 ( ) P 6 ( ) P 9 ( ) P ( ) P ( ) S ( ) A A D( ) ( ) ( ) ( A ) A d S ( ) ( d e ) a b ( ) (69) S ( ) b a a a b Tranition tate probability that the ytem i in the down tate i given by P down ( ) P 4 ( ) P 5 ( ) P 7 ( ) P 8 ( ) P ( ) P 3 ( ) P 4 ( ) P 5 ( ) S ( ) D( ) ( ) 3 d ( d e ) ( ) (7) S ( ) S ( ) e a b P ( ) a P ( ) d S ( ) S ( ) S ( ) A R A It i worth mentioning that P up ( ) P down ( ) (7) 3. Reult and dicuion 3.. Aymptotic behaviour of the ytem lim { F ()} lim F (t ) in equation (69) through (7), one can t obtain the following time independent probabilitie Uing Abel lemma, p up d S A A D A A d e a b b a a a b Tranition tate probability that the ytem i in the down tate i given by 46 Int. J. Appl. Sci. Eng., 3., 4 (7)

13 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay P down ( ) D( ) ( ) 3 ( 3 ) (73) A R A d e a b P a P where S (), S (), S ( ) (74) D() lim D( ) (75) 3.. Particular cae When reboot delay doe not occur in the ytem then the tranition tate probabilitie can be obtained by putting λr = in (5) to (66) which are given below P up A A S D A A d S d e a b (76) S b a a a b P down ( ) S ( ) D ( ) ( ) 3 d ( d e ) ( ) (77) S ( ) S ( ) e a b P( ) a P ( ) d S ( ) S ( ) S ( ) A R A where P ( ) D ( ) (78) Int. J. Appl. Sci. Eng., 3., 4 46

14 Deepak Kumar and S. B. Singh A D [ A ] S ( ) A S ( ) ( d e ) ( ) S ( ) a b d S ( ) S ( ) a d S ( ) e ( a b ) a b a b A S a a b A (79) 3.3. Numerical computation Availability analyi Auming repair rate to be exponential then one can obtain where S () ( x) ( x) ( y), S ( ), S ( ) ( x) ( y) ( x) (8) Let the failure rate λ =., λ =., λa=.4, λb=.5, λd =.6, λe =.7, λr =., λa=.8, and taking repair rate µ =, µ =, θ =, x = and y=. Subtituting thee value in equation (69) and computing invere Laplace Tranform, we get Pup t e t e t co t e t in t e t co t.397 e t in t (8) e t e t co t e t in t e t Reliability analyi a) Let the failure rate of ytem be λ=., λ=., λa=.4, λb=.5, λd=.6, λe=.7, λr=., λa=.8 and repair rate be µ=, µ=, θ=, x= and y=. Alo auming repair rate to be exponential and putting thee value in equation (69) and uing equation (8), one can obtain Table 3. b) If the ytem cannot be recovered, i.e. λr= and failure rate be λ=., λ=., λa=.4, λb=.5, λd=.6, λe=.7, λa=.8 and repair rate be µ=, µ=, θ=, x= and y=. Putting thee value in equation (69) and uing equation (8), one can obtain Table 3. In both the cae (a) and (b), Figure 4 how how reliability varie with repect to time. 46 Int. J. Appl. Sci. Eng., 3., 4

15 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay Figure 3. Time v. availability Figure 4. Time v. reliability Table. Time v. availability Time Pup(t) Int. J. Appl. Sci. Eng., 3., 4 463

16 Deepak Kumar and S. B. Singh Time Table 3. Time v. reliability Pup(t) Pup(t)(λR=) M. T. T. F. analyi Let the repair follow exponential ditribution, i.e. (8) hold then M. T.T. F. of the ytem i given by M.T. T. F. lim P up ( ) S A A D A A d S d e a b (8) S b a a a b a) Setting λ=., λa=., λr=., x=, y=,θ= and varying the value of λ a.,.,.3,.4,.5,.6,.7,.8,.9,., in equation (8), we can obtain Table 4. b) Taking λ=., λa=., λr=., x=, y=, θ= and varying the value of λ a.,.,.3,.4,.5,.6,.7,.8,.9,., in equation (8), computational value obtained are given in Table 4. c) Keeping the value a λ=., λ=., λa=., x=, y=, θ= and varying the value of λr a.,.,.3,.4,.5,.6,.7,.8,.9,., in equation (8), one can get Table 4. Figure 5 how variation of M. T. T. F. with repect to λ, λ and λr Cot analyi Let the failure rate be λ=., λ=., λa=.4, λb=.5, λd=.6, λe=.7, λa=.8, λr=. and repair rate be µ=, µ=, θ= and x=,y=. Alo letting the repair follow exponential ditribution. Subtituting all thee value in equation (69) and uing equation (8) then taking invere Laplace tranform, one can obtain equation (8). Let the ervice facility be alway available, then expected profit during the interval (, t] i given by 464 Int. J. Appl. Sci. Eng., 3., 4

17 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay t (83) E P (t) c Pup (t)dt ct where c and c are revenue rate per unit time and ervice cot per unit time, repectively. Uing (8) and (83) for the above mentioned parameter, we get E P (t) = c e t e t co t e t in t.6636 e t co t e t in t e t (84) e t co t e t in t e t t t c Taking C= and C=.,.,.3,.4,.5 and uing equation (8) one can obtain the variation of EP(t) with repect to time. The computational value obtained are given in Table 5 and hown in Figure 6. λ M. T. T. F Table 4. Failure rate v. M. T. T. F. λ M. T. T. F λr M. T. T. F Int. J. Appl. Sci. Eng., 3., 4 465

18 Deepak Kumar and S. B. Singh Figure 5. Failure rate v. M. T. T. F. Time C = Table 5. Time v. expected profit Ep(t) C =. C =.3 C = Figure 6. Time v. expected profit 466 Int. J. Appl. Sci. Eng., 3., 4 C =

19 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay Senitivity Table 6 and 7 are correponding to the enitivity analyi of the ytem reliability with repect to change in λ and λ repectively. The ame i hown in Figure 7 and 8 repectively. One can eaily conclude by thee figure that enitivity of the ytem reliability decreae with the increae in the value of λ and λ. Alo one can viualize that the ytem reliability i more enitive with repect to λ than λ. Time Time Table 6. Senitivity of the ytem reliability w. r. t. λ Value of R t / E E E E E-4 -.E E E- -.3E E E-3 -.E-3-3.7E E-4-5.4E E-5 Table 7. Senitivity of the ytem reliability w. r. t. λ Value of R t / E E+ -8.5E- -6.3E E- -.6E- -.63E E- -6.5E E- -.53E- -.6E E E-3-3.6E E E-3-4.9E E- -8.4E-3-3.3E-3 -.E-3-4.E-4 -.5E E-5 -.4E-5-7.5E-6 -.5E E E- -.6E- -3.6E- -.8E- -3.E E E E-5 Int. J. Appl. Sci. Eng., 3., 4 467

20 Deepak Kumar and S. B. Singh Figure 7. Senitivity of the ytem reliability w. r. t. λ Figure 8. Senitivity of the ytem reliability w. r. t. λ 4. Concluion In the preent tudy variou reliability meaure have been computed for the reliability model with reboot delay for different cae. In thi model different reliability meaure uch a tranition tate probabilitie, availability, reliability, M.T.T.F., expected profit and enitivity with repect to different parameter have been obtained. From Figure 3 we can ee the variation of availability with repect to time when failure rate are fixed at different value. When failure rate are aumed a λ=., λ=., λa=.4, λb=.5, λd=.6, λe=.7, λa=.8 and λr=., one can oberve that initially availability of the ytem decreae rapidly with repect to time but a the time increae it tend to tabilize at The Figure 4 provide the variation of reliability with repect to time. It how the change in reliability with repect to time in both condition: () when Reboot Delay preent in the ytem and () when Reboot Delay doe not occur in the ytem. By oberving the figure one can viualize that it decreae harply during initial tage and uniformly in the later. One can alo eaily conclude that the reliability of the ytem i better in the abence of Reboot Delay epecially during initial tage though later on they are tending toward the ame value a time pae away. Figure 5 and correponding Table 4 how the variation in M. T. T. F. of the ytem w.r.t. failure rate λ, λ and λr keeping other parameter contant. A the value of failure rate λ, λ 468 Int. J. Appl. Sci. Eng., 3., 4

21 Reliability Analyi of Embedded Sytem with Different Mode of Failure Emphaizing Reboot Delay and λr increae, M. T. T. F. of the ytem decreae. From Figure 5 one can eaily interpret that initially the value of M. T. T. F. decreae harply then it decline gradually with repect to λ, λ and λr. Critical examination of the Figure 5 reveal an intereting fact that at one point where λ, λ and λr become., the value of M. T. T. F. correponding to λ, λ and λr i the exactly ame i.e. 35 Critical examination of thi figure reveal that M. T. T. F. of the ytem i in the order: M. T. T. F. w. r. t. λ > M. T. T. F. w. r. t. λ > M. T. T. F. w. r. t. λr. From the Table 5 one can oberve the variation in effective profit with repect to time. The correponding Figure 6 ha been drawn by keeping the revenue per unit time C et at., ervice cot C varied and failure rate are kept at contant value. By the obervation of Figure 6 one can ee that expected profit decreae a ervice cot increae with repect to time wherea with repect to time expected profit increae continuouly. The enitivitie of variou value of λ and λ on the ytem reliability are hown in Figure 7 and 8 repectively. We oberve that influence of λ and λ on ytem reliability increae a they decreae. Several reliability meaure of complex ytem obtained in thi tudy will be helpful to ytem deigner a well a operation manager. Reference [ ] Chung, W. K Reliability analyi of a K-out-of-N redundant ytem with the preence of chance with multiple critical error. Microelectronic and Reliability, 33, 3: [ ] Coit, D. W. and Liu, J.. Sytem reliability optimization with k-out-of-n ubytem. International Journal of Reliability, Quality and Safety Engineering, 7, : 9-4. [ 3] oel, L. R. and upta, P Stochatic analyi of a two unit parallel ytem with partial and catatrophic failure and preventive maintenance. Microelectronic and Reliability, 4, 3: [ 4] upta, R. and Bhardwaj, P. 3. A two-unit tandby ytem with two operative mode of the unit and preparation time for repair. Journal of Reliability and Statitical Studie, 6, : 87-. [ 5] Haett, T. F., Dietrich, D. L., and Szidarovzky, F Time varying failure rate in the availability and reliability analyi of repairable ytem. IEEE Tranaction on Reliability, 44, : [ 6] Ke, J. C., Lee, S. L., and Ko, M. Y.. Two-unit redundant ytem with detection delay and imperfect coverage: cofidence interval etimation. Quality Technology & Quantitative Management, 8, : -4. [ 7] Ke, J. B., Chen, J. W., and Wang, K. H.. Reliability meaure of a repairable ytem with tandby witching failure and reboot delay. Quality Technology & Quantitative Management, 8, : 5-6. [ 8] Moutafa, M. S.. Reliability analyi of ytem with time varying failure rate. Economic Quality Control, 6, : [ 9] Ram, M. and Singh, S. B. 8. Availability and cot analyi of a parallel redundant complex ytem with two type of failure under preemptive-reume repair dicipline uing umbel-hougaard family of copula in repair. International Journal of Reliability, Quality & Safety Engineering, 5, 4: [] Ram, M. and Singh, S. B.. Analyi of a complex ytem with common caue failure and two type of repair facilitie with different ditribution in failure. International Journal of Reliability and Safety, 4, 4: Int. J. Appl. Sci. Eng., 3., 4 469

22 Deepak Kumar and S. B. Singh [] Ram, M. and Singh, S. B.. Availability, MTTF and cot analyi of complex ytem under preemptive-repeat repair dicipline uing umbel-hougaard family copula. International Journal of Quality & Reliability Management, 7, 5: [] Trivedi, K. S.. Probability and Statitic with Reliability, Queuing and Computer Science Application. nd Edition. John Wiley & Son. New York. [3] Yam, R. C. M., Zuo, M. J., and Zhang, Y. L. 3. A method for evaluation of reliability indice for repairable circular conecutive-k-out-of-n: F ytem. Reliability Engineering and Sytem Safety, 79: Int. J. Appl. Sci. Eng., 3., 4