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1 RELIABILITY AND SENSITIVITY ANALYSIS OF THE K-OUT-OF-N:G WARM STANDBY PARALLEL REPAIRABLE SYSTEM WITH REPLACEMENT AT COMMON-CAUSE FAILURE USING MARKOV MODEL M. A. El-Damcese 1 and N. H. El-Sodany 2 1 Mathematcs Department, Faculty of Scence, Tanta Unversty, Tanta, Egypt E-mal:meldamcese@yahoo.com 2 Natonal Accountant, Central Agency for Publc Moblzaton and Statstcs, Caro, Egypt E-mal: naglaa_hassan17@yahoo.com ABSTRACT Standby redundancy s a technque that has been wdely appled for mprovng system relablty and avalablty n system desgn. In ths paper, probablstc model for a redundant system wth replacement at each common-cause falure has been developed to analyze the relablty measures usng Markov models. We nvestgate the relablty and senstvty analyss of k-out-of-n:g warm standby parallel reparable system. All falure and repar tmes of the system are exponentally dstrbuted and when one of the operatng prmary unts fals then t s nstantaneously replaced by a warm standby unt f one s avalable. Comparatve analyss of relablty measures between two dssmlar confguratons has been developed. Confguraton Ι s a 2-out-of-4:G warm standby parallel reparable system, whle Confguraton ΙΙ s a 2-out-of-5:G warm standby parallel reparable system. We get a closed-form soluton of the relablty measures of the system for the two confguratons. Comparsons are performed for specfc values of system parameter. Senstvty analyss s also carred out to depct the effect of varous parameters on the relablty functon and mean tme to falure of the system. Numercal example s gven to llustrate the results obtaned. Keywords: k-out-of-n:g warm standby, relablty, parallel, common-cause falure, replacement, Markov model, senstvty analyss. 1. Introducton Standby redundancy s a technque used to mprove system relablty and avalablty. Standby redundancy represents a stuaton wth one unt operatng and a number of unts on standby. Gnedenko et al. (1969) classfed standby redundancy accordng to falure characterstcs; hot standby, cold standby, and warm standby. In hot standby redundancy, each unt has the same falure rate regardless of whether t s n standby or n operaton. In cold standby redundancy, only one component wll be workng at any gven tme, the others beng standbys and not workng. One of the standby components starts workng only when the currently workng component fals. In warm standby redundancy, the standbys may fal n standby state but wth a falure rate smaller than that of the prmary component but s greater than zero. Operatve and warm standby unts can be consdered to be reparable. Warm standby reparable systems have receved attenton by several authors. Guo et al. (2012) analyzed the dynamc behavor of a two unt parallel system wth warm standby and common-cause falure. The system s composed of three dentcal unts; two unts are operatng 53

2 and one unt s n warm standby state. Yun and Cha (2006) proposed a general method for modelng a warm standby system wth three unts and derved the system performance measures (system relablty and mean lfe); one unt s operatng n an actve state and two unts wat n warm standby state. Hajeeh (2011) studed relablty and avalablty for four seres confguratons wth both warm and cold standby and common-cause falure. Dhllon and Yang (1992) and Dhllon (1993) analyzed the relablty and avalablty of warm standby systems wth common-cause falures and human errors. Labb (1991) proposed the stochastc analyss of a two-unt warm standby system wth two swtchng devces. Sngh (1989) consdered a warm standby redundant system wth (M+N) dentcal unts, r repar facltes. The system s under common-cause falure and repar tmes are arbtrary dstrbuted. Srnvasan and Subramanan (2006) developed relablty and avalablty functons of a three unt warm standby system wth dentcal components. In ths model, one unt s workng at the begnnng and the other two are n standby. The k-out-of-n:g reparable system s one of the most popular and wdely used systems n practce. The k-out-of-n:g systems have been studed n certan stuatons where redundancy s of mportance. Redundancy s requred not only to extend the functonng of the system but also to acheve a certan relablty of the system. The k-out-of-n:g systems can be classfed nto: Actve redundant systems (k-out-of-n:g system): n whch all the n unts are actve even though only k unts are requred for the proper functonng of the system; Cold standby systems: n whch the n-k cold standby unts wll not be actve and upon falure of one of the k actve unts, cold standby unt wll nstantaneously replace the faled unt; Warm standby systems: n whch the n-k warm standby unts wll have a smaller falure rate compared to the k actve ones; Hot standby systems: n whch the n-k hot standby unts and the k actve ones wll have the same falure rate. Due to ther mportance n ndustres and desgn, the k-out-of-n:g systems have receved attenton from dfferent researchers. El-Damcese and El-Sodany (2014) analyzed the relablty and avalablty of a k-out-of-n:g system wth three falures usng Markov model. El-Damcese (2009) presented the relablty and avalablty analyss of a k-out-of-(m+s):g warm standby system wth tme varyng falure and repar rates n presence of common-cause falure. El- Damcese (2010) presented contnuous-tme homogeneous Markov process to evaluate avalablty, relablty and MTTF for crcular consecutve k-out-of-n:g system wth reparman. El-Damcese (2009) analyzed the k-out-of-n:g system model wth crtcal human errors, common-cause falures and tme dependent system repar-rate. Zhang et al. (2006) analyzed the k-out-of-(m+n):g warm standby system. In the system, not all components n standby can be treated as dentcal as they have dfferent falure and repar rates. Kumar and Bajaj (2014) analyzed the vague relablty of k-out-of-n:g system (partcularly, seres and parallel system) wth ndependent and non-dentcally dstrbuted components, where the relablty of the components are unknown. The relablty of each component has been estmated usng statstcal confdence nterval approach. Then converted these statstcal confdence ntervals nto trangular fuzzy numbers. Based on these trangular fuzzy numbers, the relablty of the k- out-of-n:g system has been calculated. Moustafa (2008) presented a contnuous tme Markov chan (CTMC) model to obtan closed form expressons of the mean tme between system falures (MTBF) for k-out-of-n:g systems subject to M exponental falure modes and repars. She and Pecht (1992) made a bref revew on standby redundancy technques. In ther research, a general closed form equaton was developed for system relablty of a k-out-of-n warm 54

3 standby system. Chryssaphnou et al. (1997) consdered a 1-out-of-(m+1) warm standby system wth non-dentcal unts. Goel et al. (1989) analyzed a 1-out-of-3 warm standby system wth two types of spare unts a warm and a cold standby unt, and nspecton. A general closed form equaton was developed for system relablty of a k-out-of-n warm standby system where components n k-out-of-n:g standby systems were assumed to be statstcally dentcal. An analyss on 1-out-of-2:G warm standby system has been presented by Henley and Kumamoto (1992). Yusuf and Bala (2013) analyzed the mean tme to system falure (MTSF) of a reparable 2-out-of-4 warm standby system. Yusuf and Gmba (2013) analyzed the MTSF of 2-out-of-5 warm standby reparable system wth replacement at the occurrence of each common-cause falure usng Kolmogorov s forward equatons method. In recent years, t has been realzed that n order to predct realstc relablty and avalablty of standby systems, the occurrence of common-cause falures must be consdered. Common-cause falures can only occur n the system wth more than one good unt. A common-cause falure s defned as any nstance where multple unts or components fal due to a sngle cause. The concept of common-cause falure and ts mpact on relablty measures of system effectveness has been ntroduced by several authors. Dhllon and Anude (1993) studed common-cause falure analyss of a non-dentcal unt parallel reparable system wth arbtrary dstrbuted repar tmes. Haggag (2009) studed cost analyss of a system nvolvng commoncause falures and preventve mantenance. Vashsth (2011) have analyzed the relablty of redundant system wth common-cause falure. Mantanng a system wth common-cause falure s often an essental requste. Seres system s a confguraton n whch all components are n seres and all components have to work for the system to work. If any one of the system components fals, the system fals. Whereas the parallel system fals only when all the system components fals. Parallel confguraton s used to ncrease the relablty of a system wthout any change n the relablty of the ndvdual components that form the system. The problem of evaluatng the avalablty and relablty of the parallel system has been subject of many studes throughout the lterature (Kolowrock (1994), Pan and Nonaka (1995), Ebelng (2000) and Kwatuszewska (2001)). In ths paper, we analyze the relablty measures of the k-out-of-n:g warm standby parallel reparable system wth replacement at each common-cause falure wth constant falure and repar rates of the operatng prmary unts and warm standby unts usng Markov model. The followng relablty measures of the system are obtaned usng Markov model for two confguratons. Confguraton Ι s a 2-out-of-4:G warm standby parallel reparable system, whle Confguraton ΙΙ s a 2-out-of-5:G warm standby parallel reparable system.. Avalablty and steady state avalablty of the system.. Relablty and mean tme to system falure. We also perform senstvty analyss for changes n the relablty characterstcs along wth changes n specfc values of the system parameters. Ths paper s organzed as follows: Secton 2 s devoted to the descrpton and basc assumptons of the system. Secton 3 s devoted to the relablty and avalablty analyss of the k-out-of-n:g warm standby parallel reparable system. In Secton 4 we make a comparatve analyss of the relablty measures of two dssmlar confguratons of the k-out-of-n:g warm 55

4 standby parallel reparable system. Senstvty analyss s carred out to depct the effect of varous parameters on the relablty functon and mean tme to falure of the system. In Secton 5, a numercal example s gven. In Secton 6, some concludng remarks are gven. 2. Model Descrpton and Assumptons The followng assumptons are assocated wth the system: 1. The system under consderaton s a k-out-of-n:g warm standby parallel reparable system. At least k unts of the system are requred for the system to work. 2. The system conssts of k prmary unts and n k warm standby unts and all the unts are dentcal. 3. The system s subject to falure of a sngle unt and common-cause falure of more than one unt. 4. All prmary unts and warm standby unts are consdered to be reparable. 5. Each of the prmary unts fals ndependent of the state of the others, accordng to an exponental falure tme dstrbuton wth parameter, and the avalable warm standby unts can also fal accordng to an exponental falure tme dstrbuton wth parameter, 0. s s 6. When one of the prmary unts fals, t s nstantaneously replaced by a warm standby unt f one s avalable. Swtchng from warm standby to operatve unt s perfect and nstantaneous. 7. When a standby unt swtches nto the operatng prmary unt successfully, ts falure characterstcs wll be the same as that of the operatng prmary unts. 8. Whenever one of the operatng unts or warm standby unts fal, t s mmedately sent to a repar. After reparng, the faled unt works lke a new one. 9. There s a sngle reparman who attends to the faled unts. 10. The reparmen can repar only one faled unt at a tme. 11. The faled system repar tmes are exponentally dstrbuted. The unts are repared accordng to an exponental repar tme dstrbuton wth parameter. 12. Common-cause falure and falure of a sngle unt are statstcally ndependent. 13. The common-cause falure affects only the unts n operaton and the affected unts are replaced nstantaneously. 14. The system at any workng state can completely fal due to common-cause falure wth constant common-cause falure rate. 15. When the system fals, no falure wll occur for other workng components. Notatons: S : state of the system, 0,1,2,..., n k, n s cj j : falure rate of a sngle prmary unt : falure rate of a sngle warm standby unt : repar rate of a sngle unt : common-cause falure rate/replacement rate of j unts, j 2,3,4,..., n 56

5 * P s : Laplace transformaton of P t : probablty that the system s n state at tme t, 0,1,2,..., n k, n 1 2 P t, 0,1,2,..., n k, n A t A t : avalablty of Confguraton Ι / Confguraton ΙΙ A1 A 2 : steady state avalablty of the Confguraton Ι / Confguraton ΙΙ 1 2 R t R t : relablty of Confguraton Ι / Confguraton ΙΙ * * 1 2 R s R s : Laplace transformaton of the relablty functon of Confguraton Ι / Confguraton ΙΙ MTTF1 MTTF 2: mean tme to falure of Confguraton Ι / Confguraton ΙΙ 3. Relablty and Avalablty Analyss of the System Wth the help of the above notatons and possble states of the system; the state transton dagram of the k-out-of-n:g warm standby parallel reparable system wth replacement at each common-cause falure s shown n Fgure 1. Fgure 1: State transton dagram of the k-out-of-n:g warm standby parallel reparable system 57

6 Probablty consderatons gves the followng set of dfferental dfference equatons assocated wth the state transton dagram of the k-out-of-n:g warm standby parallel reparable system: d P t 0 k n k s cn P 0 t P 1 t P n n t (1) 1 d P t k n k s c n P t k n k s P 1 t 1 P t P t,0 n k 1 n n (2) d P t n k P t k n k ck n k s P n k 1 t P k n t, n k 2 (3) n k n k d P t P t P t, n k 2 A t (4) n n n c n 0 0 The system avalablty s gven by: nk 0 P t (5) The ntal condtons of the system are P P 0 0, 1,2..., n k, n To obtan the relablty functon of the system, we assume that the faled states are absorbng states and set all transton rates from these states equal to zero. Now let P t P t, 0,1,..., n k, n n Eqs.(1-4). The system relablty s gven by: R t nk 0 P t (7) 4. Comparatve Analyss of Relablty Measures On the bass of the above descrpton and assumptons, we nvestgate the relablty measures of two dssmlar confguratons. Confguraton Ι s a 2-out-of-4:G warm standby parallel reparable system, whle Confguraton ΙΙ s a 2-out-of-5:G warm standby parallel reparable system. 4.1 Confguraton Ι Confguraton Ι conssts of 2 operatng prmary unts and 2 warm standby unts and all the unts are dentcal and at least 2 unts are requred for the system to work.. Avalablty Analyss of the System The state transton dagram of the 2-out-of-4:G warm standby parallel reparable system s shown n Fgure 2. (6) 58

7 Fgure 2: state transton dagram of 2-out-of-4:G warm standby parallel reparable system The system of dfferental dfference equatons assocated wth the state transton dagram of the system are gven by: d P t s c 4 P 0 t P 1 t P 4 4 t (8) d P t 1 2 s c 3 P 1 t 2 2s P 0 t 2 P 2 t P 3 4 t (9) d P t 2 c2 2 P 2 t 2 s P 1 t P 2 4 t (10) d P t P 4 t P c 4 0 t P c 3 1 t P c 2 2 t (11) The system avalablty s gven by A t P t P t P t (12) The ntal condtons of the system are gven by P P 0 P 0 P (13) The steady state equatons of the system are then 59

8 0 2 2 s c P P P (14) P 2 2 P 2P P (15) s c 3 1 s P 2 P P (16) c2 2 s P4 c 4P0 c 3P1 c 2P2 (17) Solvng the system of lnear Eqs.(14-17) usng Maple program, we get the state probabltes determnng the steady state avalablty of the system: The steady state avalablty of the system s gven by A1 P0 P1 P2 (18). System Relablty and Mean Tme to Falure We assume that the faled states are absorbng states and set all transton rates from these states equal to zero. Now let P t P t, 0,1,2,4 n Eqs.(8-11). The set of dfferental equatons assocated wth the system are gven by: d P t P t P t s c (19) d P t 1 2 s c 3 P 1 t 2 2s P 0 t 2 P 2 t (20) d P t 2 c2 2 P 2 t 2 s P 1 t (21) d P t 4 P 4 0 t c P c 3 1 t P c 2 2 t (22) The system relablty s gven by R t P t P t P t (23) Takng Laplace transformaton of Eqs.(19-22) usng the ntal condtons Eq.(13), we obtan: * * s P s P s (24) s c4 0 1 * * * s P s P s P s (25) s c 3 1 s 0 2 * * s P s P s (26) c2 2 s 1 sp s P s P s P s (27) * * * * 4 c 4 0 c 3 1 c On solvng Eqs.(24-27), we obtan the Laplace transformatons * P s, 0,1,2,4 60

9 The Laplace transformaton of the relablty functon of the system s gven by (28) R s P s P s P s * * * * The mean tme to system falure ( MTTF 1 ) s obtaned usng: lm * * 0 MTTF R t R s R (29) s 0 0. Senstvty Analyss of the Relablty and Mean Tme to Falure of the System The objectve of relablty senstvty analyss s to determne nput varables that mostly contrbute to the varablty of the falure probablty. The results whch can be obtaned from any model are senstve to many factors. In ths paper, we concentrate our attenton on parametrc senstvty analyss. Parametrc senstvty analyss helps n dentfyng the model parameters that could produce sgnfcant modelng errors. One approach to parametrc senstvty analyss s to use upper and lower bounds on each parameter n the model to compute optmstc and conservatve bounds on system relablty (Smotherman et al. (1986)). Our approach s to compute the dervatve of the measures of nterest wth respect to the model parameters (Goyal et al. (1987) and Smotherman (1984)). We frst perform senstvty analyss for changes n the system relablty R1 t resultng from changes n parameters, c 2, c 3, c 4 and. We obtan the dervatve of Eq.(23) wth respect to the parameters, c 2, c 3, c 4 and. Now we perform senstvty analyss for changes n the mean tme to falure MTTF 1 of the system resultng from changes n parameters, s, c 2, c 3, c 4 and. We obtan the dervatve of Eq.(29) wth respect to the parameters, s, c 2, c 3, c 4 and. 4.2 Confguraton ΙΙ Confguraton ΙΙ conssts of 2 operatng prmary unts and 3 warm standby unts and all the unts are dentcal and at least 2 unts are requred for the system to work.. Avalablty Analyss of the System The state transton dagram of the 2-out-of-5:G warm standby parallel reparable system s shown n Fgure 3. 61

10 Fgure 3: state transton dagram of 2-out-of-5:G warm standby parallel reparable system The system of dfferental dfference equatons assocated wth the state transton dagram of the system are gven by: d P t s c 5 P 0 t P 1 t P 5 5 t (30) d P t 1 2 2s c 4 P 1 t 2 3s P 0 t 2 P 2 t P 4 5 t (31) d P t 2 2 s c 3 2 P 2 t 2 2s P 1 t 3 P 3 t P 3 5 t (32) d P t 3 c2 3 P 3 t 2 s P 2 t P 2 5 t (33) d P t P 5 t P c 5 0 t P c 4 1 t P c 3 2 t P c 2 3 t (34) The system avalablty s gven by A t P t P t P t P t (35)

11 The ntal condtons of the system are gven by P P 0 P 0 P t P t (36) Usually we are manly concerned wth systems runnng for a long tme. The steady state avalablty of the system s the avalablty functon as tme approaches nfnty. Ths can be d obtaned mathematcally by takng 0 as t n the system of Eqs.(30-34) therefore, the system of Eqs.(30-34) reduces to the followng system of lnear equatons: s c P P P (37) P 2 3 P 2P P (38) s c 4 1 s P 2 2 P 3P P (39) s c 3 2 s P 2 P P (40) c2 3 s P5 c 5P0 c 4P1 c 3P2 c 2P3 (41) Solvng the system of lnear Eqs.(37-41) usng Maple program, we get the state probabltes determnng the steady state avalablty of the system: The steady state avalablty of the system s gven by A2 P0 P1 P2 P3 (42). System Relablty and Mean Tme to Falure To obtan the relablty functon of the system, we assume that the set of faled states are absorbng states and set all transton rates from these states equal to zero. Now let P t P t, 0,1,2,3,5 n Eqs.(37-41). The set of dfferental equatons assocated wth the system are gven by: d P t s c 5 P 0 t P 1 t (43) d P t 1 2 2s c 4 P 1 t 2 3s P 0 t 2 P 2 t (44) d P t 2 2 s c 3 2 P 2 t 2 2s P 1 t 3 P 3 t (45) d P t 3 c2 3 P 3 t 2 s P 2 t (46) d P t 5 P t c 5 0 P c 4 1 t P c 3 2 t P c 2 3 t (47) 63

12 The system relablty s gven by R t P t P t P t P t (48) Takng Laplace transformaton of Eqs.(43-47) usng the ntal condtons Eq.(36), we obtan: * * s P s P s (49) s c5 0 1 * * * s P s P s P s (50) s c 4 1 s 0 2 * * * s P s P s P s (51) s c 3 2 s 1 3 * * s P s P s (52) c2 3 s 2 sp s P s P s P s P s (53) * * * * * 5 c 5 0 c 4 1 c 3 2 c On solvng Eqs.(49-53), we obtan the Laplace transformatons * P s, 0,1,2,3,5. The Laplace transformaton of the relablty functon of the system s gven by R s P s P s P s P s (54) * * * * * The mean tme to system falure ( MTTF 2 ) s obtaned usng: lm * * 0 MTTF R t R s R (55) s 0 0. Senstvty Analyss of the Relablty and Mean Tme to Falure of the System We frst perform senstvty analyss for changes n the relablty of the system resultng from changes n parameters, c 2, c 3, c 4, c 5 and. We obtan the dervatve of Eq.(48) wth respect to the parameters, c 2, c 3, c 4, c 5 and. Now we perform senstvty analyss for changes n the mean tme to falure MTTF 2 of the system resultng from changes n parameters, s, c 2, c 3, c 4, c 5 and. We obtan the dervatve of Eq.(55) wth respect to the parameters, s, c 2, c 3, c 4, c 5 and. R2 t 64

13 5. Numercal Example: For comparatve analyss of relablty measures between confguraton Ι and confguraton ΙΙ; the falure, repar, common-cause falure and replacement rates are gven by: 0.2, 0.1, 0.4, 0.05, 0.1, 0.15, s c 5 c 4 c 3 0.2, 0.25, 0.5, 0.6, 0.7 c Fgures 4 5 show the avalablty and relablty for confguraton I and II versus tme. We conclude that the avalablty and relablty of confguraton ΙΙ s greater than the avalablty and relablty of confguraton Ι. Fgures 6 7 show the steady state avalablty of confguraton I and II versus falure and repar rates. It can be observed that the steady state avalablty confguraton ΙΙ s greater than that of confguraton Ι and the steady state avalablty of the two confguratons decreases wth the ncrease n the falure rate and ncreases wth the ncrease n the repar rate. Senstvty analyss for changes n the relablty functons R t and 2 R t resultng from changes n system parameters, c 2, c 3, c 4, c 5 and are shown n Fgures 8 9. We can easly observe that the system parameters, c 2, c 3, c 4, c 5 has bg mpact on the relablty functons R t of confguraton I and II at the same tme. The R1 t and 2 numercal results of the senstvty analyss of the mean tme to falure of confguraton Ι and II resultng from changes n system parameters, s, c 2, c 3, c 4, c 5 and are shown n Tables 1 2. It can be seen from Table 1 that the order of mpacts of the system parameters on MTTF 1 are: c 4 c 3 c 2 s. From Table 2 the order of mpacts of the system parameters on MTTF 2 are: c 4 c 5 c 3 s c 2 and the mean tme to falure of the two confguratons are not senstve to the replacement rates. It should be noted that these conclusons are only vald for the gven values of system parameters. We may reach other conclusons for other values of the system parameters. 1 (t) Avalablty A 1 (t) Avalablty A 2 Fgure 4: Avalablty A (t) versus tme, =1,2 65

14 (t) Relablty R 1 (t) Relablty R 2 Fgure 5: Relablty R (t) versus tme, =1,2 Steady state avalablty A 1 Steady state avalablty A 2 Fgure 6: steady state avalablty A versus falure rate, =1,2 Steady state avalablty A 1 Steady state avalablty A 2 Fgure 7: steady state avalablty A versus repar rate, =1,2 66

15 Now we perform senstvty analyss for changes n the relablty functons R t, 1,2 along wth changes n specfc values of the system parameters, c 2, c 3, c 4, c 5 and. Fgure 8: Senstvty of the relablty of confguraton Ι wth respect to system parameters Fgure 9: Senstvty of the relablty of confguraton ΙΙ wth respect to system parameters 67

16 Fnally we perform senstvty analyss for changes n the mean tme to falure MTTF, 1,2 along wth changes n specfc values of the system parameters, s, c 2, c 3, c 4, c 5 and. Table 1: Senstvty analyss for MTTF 1 s c 2 c 3 c 4 MTTF Table 2: Senstvty analyss for MTTF 2 s c 2 c 3 c 4 c 5 MTTF Concluson In ths paper we have utlzed the Markov model to develop the relablty measures of the k-out-of-n:g warm standby parallel reparable system. All falure and repar rates of the system are constant. Comparatve analyss of relablty measures between two dssmlar confguratons has been developed. Confguraton Ι s a 2-out-of-4:G warm standby parallel reparable system, whle Confguraton ΙΙ s a 2-out-of-5:G warm standby parallel reparable system. The system of dfferental equatons wth the ntal condtons has been solved numercally usng Laplace transformaton by the ad of Maple program. Graphcal representaton of the relablty and avalablty of the two confguratons versus tme are made. Senstvty analyss s also carred out to depct the effect of varous parameters on the relablty functon and mean tme to falure of the system. Numercal example s gven to llustrate the results obtaned, and the results were shown graphcally by the ad of Maple program. Results ndcate that the relablty and avalablty of the system ncrease by ncreasng of the number of warm standby unts. And the relablty and mean tme to falure of the two confguratons are senstve to the falure and repar rates of the system and are not senstve to the replacement rates. References: 1. Chryssaphnou, O., Papastavrds, S., Tsapelas, T.: A generalzed mult-standby multfalure mode system wth varous repar facltes. Mcroelectroncs and Relablty., Vol. 37, No. 5, pp , Dhllon, B.S.: Relablty and avalablty analyss of a system wth warm standby and common-cause falures. Mcroelectroncs and Relablty, Vol. 33, No. 9, pp , Dhllon, B.S. and Anude, O.C.: Common-cause falure analyss of a non-dentcal two unt parallel system wth arbtrary dstrbuted tmes. Mcroelectroncs and Relablty, Vol. 33, No. 1, pp ,

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