Stochastic Analysis of a Compound Redundant System Involving Human Failure

Transcription

1 Journal of Matheatis an Statistis (3): 47-43, 6 ISSN Siene Publiations Stohasti nalysis of a Copoun Reunant Syste Involving uan Failure Ritu Gupta, S.. Mittal an 3 C. M. Batra,3 Departent of Matheatis, rishna Institute of Engineering an Tehnology, Ghaziaba, Inia Departent of Matheatis, M. M. (PG) College, Moinagar, Inia bstrat: This stuy eals with a opoun stanby reunant syste onsisting of three subsystes, B, an C onnete in series. The sub-syste B onsists of one ain unit an the other is its stanby reunant unit. These units further onsist of two sub-units onnete in series. The sub units of ain unit are onnete to sub-units of stanby unit through iperfet swithing over evie. Failure of all sub-systes an repair rate of swithing over evies are exponential while repairs of all sub-systes are istribute quite generally. The various reliability paraeters have been opute an analyze by tabular an graphial illustrations. ey wors: M.T.T.F., availability, reliability, profit funtion, stanby reunany INTRODUCTION Reliability is an iportant onept at the planning, esign an operation stages of various oplex systes. s long as an has built things, he has wante to ake the as reliable as possible. In pratie, we oe aross with a nuber of oplex systes onsisting of one or ore parts, failure of any of the parts results in the reution of effiieny of whole systes or the oplete failure of the syste an as a result of it, the reliability of the syste reues. The better aintenane of suh parts originate better reliability an then only we an ahieve the arkets eans of reliability, funtionality, prie an perforane of that syste. On the other han it ay not be eonoial to obtain higher orer of reliability always through any aount of aintenane. Thus introuing reunant parts an proviing aintenane an repair at the tie of nee ay ahieve high egree of reliability. In a reunant syste, soe aitional paths are reate for the proper funtioning of the syste. If all the reunant parts start working together at the tie of operation, then it is tere as parallel reunany. stanby reunant syste is the one in whih one operating unit is followe by spare units alle stanbys. On the failure of the operating unit, a stanby unit is swithe on by perfet or iperfet swithing evie. In the present isussion, the authors have onsiere a opoun syste onsisting of three sub-systes, B an C. The sub-syste B onsists of two units, one is ain an the other is its stanby. These units further onsist of two sub-units viz, (B, B) in ain unit an (B, B) in its stanby reunant syste. The sub syste B also has two iperfet swithing evies S an S. S onnets B an B an S onnets B an B. B an B sub-units are sae as B an B respetively. Initially, in sub-syste B, the sub-units B an B are assue to be in operation. If B fails then B an B ay begin to operate through a swithing evie S. If B fails then B an B ay begin to operate through a swithing evie S. The failure in sub-syste C requires waiting tie for repair. The syste will be in own state ue to failure of subsyste or C or ourrene of any huan error. lso, ue to failure of two sub-units of one unit an one subunit of other unit of sub-syste B, the syste suffers oplete break own. The Laplae transfors of the tie epenent probabilities of the syste being in various states have been obtaine by eploying the suppleentary variable tehnique. Various reliability paraeters have been opute an soe tabular an graphial illustrations are also given at the en of the paper. The state transition iagra of the syste is shown in Fig. a. ssuptions * Initially, the syste works perfetly. * The syste onsists of three subsystes, B an C onnete in series. * The Sub-syste B onsists of two units, one ain unit an other in stanby oe. * The ain unit of subsyste B further onsists of two sub-units B an B onnete in series. Siilarly, stanby unit onsists of two sub-units B & B onnete in series. * The Sub-unit B is onnete to sub-unit B by swith S an sub-unit B is onnete to subunit B by swith S. * Iperfet swithing evie is assue. * ll the repairs of sub-systes are istribute quite generally while the repair rate of swithing over Corresponing uthor: Ritu Gupta, Departent of Matheatis, rishna Institute of Engineering & Tehnology, 3 M Stone, Ghaziaba-Meerut Roa, P.O. Muranagar-6, Distt. Ghaziaba (U.P.), Inia, Tel: 67534,5 extn. (36) 47

2 J. Math. & Stat., (3): 47-43, 6 λ λ S( ) P ( x, t ) µ ( z ) λ S (3) P 3 ( t ) ( a) µ ( x) λ µ ( x ) η ( z) S () P ( t ) w S () P ( z, t ) R a S () P ( t ) ν ( x) b S (5) P 5 ( x, t ) µ ( y) ν ( z) S (9) S (8) P P 8 ( y, t ) 9 ( z, t ) ν ( y) ( b) S (4) P 4 ( t ) R P 6 ( x, t ) β ( x) S (6) λ µ ( z) S () P ( y, t ) λ S () S (7) λ P ( z, t) β ( x) λ P 7 ( x, t ) S () P ( z, t ) λ λ Fig. a: State transition iagra evies an the failures of sub-systes are exponentially istribute. * ll the units reover their funtioning perfetly after repair. * ll the repairs are perfet i.e., the repair faility never oes any aage to the units. * faile subsyste is repaire at a single servie hannel. * t tie t, B an B are operating an B an B are in stanby oe. * s soon as the operating unit fails, it is replae by it s stanby unit. * The syste eases to funtion ue to failure of subsyste or C or ue to ourrene of any huan error. 48

3 J. Math. & Stat., (3): 47-43, 6 * The syste suffers oplete break own ue to failure of sub-systes B, B or B, B or any three sub-units of sub-syste B at a single state. * The sub-syste C requires waiting tie for repair. * In state S(7) of the syste no priority is given to any sub-unit i.e., B, B an repair rate of these sub-units is assue to be iential. * Stanby sub-units in sub-syste B are assue to be perfet as long as they are in stanby oe. Notations : Constant failure rate of subsyste C. η ( z) : Repair rate of sub-syste C when the syste is in the state S(). w : Constant waiting tie to repair subsyste C. : Failure rate of sub-unit B or B of sub-syste B. : Failure rate of sub-unit B or B of sub-syste B. λ, µ ( ) : Failure an repair rate of the syste ue to huan failure. λ, µ ( ) : Failure an repair rate of sub-syste. z µ ( x) : Repair rate of sub-unit B when the syste is in state S(5). µ ( y) : Repair rate of sub-unit B an B when the syste is in state S(8). µ ( z) : Repair rate of sub-unit B, B an B when the syste is in state S(). β ( x) : Repair rate of sub-unit Bor B when the syste is in state S(7). ν ( x) : Repair rate of sub-unit B when the syste is in state S(6). ν ( y) : Repair rate of sub-unit B an B when the syste is in state S(). ν ( z) : Repair rate of sub-unit B, B an B when the syste is in state S(9) x, y, z : Elapse repair tie. a, b R, R : Probability of suessful operation of swithing over evies S an S respetively. : Constant repair rate of swithing over evies S an S respetively. : Operable : Iperfet swith : Faile : Waiting State probability esription P ( t ) : Probability that the syste is in operable state S() at tie t. 49 P ( t ) : Probability that the syste is in faile state S() at tie t. P ( z, t ) : Probability that the syste is in waiting state S() with elapse repair tie lying in the interval ( z, z ). P 3 ( t ) P 4 ( t ) : Probability that the syste is in faile state S(3) at any tie t : Probability that the syste is in faile state S(4) at any tie t. P5 ( x, t ) : Probability that the syste is in operable state S(5) with elapse repair tie lying in the interval ( x, x ). P6 ( x, t ) : Probability that the syste is in operable state S(6) with elapse repair tie lying in the interval ( x, x ). P7 ( x, t ) : Probability that the syste is in operable state S(7) with elapse repair tie lying in the interval ( x, x ). P8 ( y, t ) : Probability that the syste is in faile state S(8) with elapse repair tie lying in the interval ( y, y ). P9 ( z, t ) : Probability that the syste is in faile state S(9) with elapse repair tie lying in the interval ( z, z ). P ( z, t ) : Probability that the syste is in faile state S() with elapse repair tie lying in the interval ( z, z ). P ( y, t ) : Probability that the syste is in faile state S() with elapse repair tie lying in the interval ( y, y ). P ( z, t ) : Probability that the syste is in faile state S() with elapse repair tie lying in the interval ( z, z ). P ( x, t ) : Probability that the syste is in faile state S() with elapse repair tie lying in the interval ( x, x ). Forulation of atheatial oel: By eleentary probability onsieration an ontinuity arguents, the following ifferene-ifferential equations governing the behaviour of the syste ay be hol goo: a b ( b) λ λ ( a). P ( t) µ ( x). P5 ( x, t) ν( x). ν ( y). P ( y, t) y µ ( z). P ( z, t) z µ ( z). 6 η ( z). P ( z, t) z µ ( z). P ( z, t) z ν ( z). P9 ( z, t) z µ ( y 8 ). P ( y, t) y ()

4 J. Math. & Stat., (3): 47-43, 6 w. P ( t). P ( t). P5 ( x, t) η( z). P ( z, t) R. P3 ( t) ( a). P ( t) R. P4 ( t) ( b). P ( t) λ λ µ ( x). P5 ( x, t) λ λ ν ( x). P6 ( x, t) λ λ β ( x). P7 ( x, t) µ ( y). P8 ( y, t) y ν ( z). P9 ( z, t) µ ( z). P ( z, t) ν ( y). P ( y, t) y ( z µ ). P ( z, t ) µ ( x). P ( x, t) () (3) (4) (5) (6) (7) (8) (9) () () () (3) (4) Bounary onitions P (, t) w. P ( t) (5) P (, t) R. P ( t) a.. P ( t) β ( x) (6) P (, t) b.. P ( t) R. P ( t) β ( x) (7) P (, t) (8) P (, t) (9) P (, t) () P (, t) () P (, t) () 6 P (, t) λ λ. P ( t) 6 λ λ 5 7 P (, t) λ. P ( t) λ λ 5 λ 6 7 (3) (4) Initial onitions: P () an other state probabilities are zero. Solution of the oel: Taking Laplae transfor of () to (4) an on further siplifiation one ay obtain: P (5) ( ) (6) P 5 s P 6. (7) P3 7. (8) P4 8. (9) P5. (3) P6 3. (3) P7 4. (3) P8 5. (33) P9 6. (34) P 7. (35) 4

5 J. Math. & Stat., (3): 47-43, 6 P 8. (36) P. (37) P., (38) Operational availability an non-availibility: The Laplae transfor of the probabilities that the syste is in operable an own state at tie t an be evaluate as follows: Pup { 3 4 (39) Pown 8 It is worth notiing that Pup Pown. s Where, ( s λ λ ) ( s λ λ µ ( x)) ( s λ λ ν ( x)) 3 ( s λ λ β ( x)) 4 β ( x) s w w 6. 5 ( s η( z)) ( a) 7 ( s R ) ( b) 8 ( s R ) { {. b.. R. 8. R.. a. 7 9 { {. R a R b (4) R a.. 4 ( s µ ( y)) ( ). 3( ) s s ( s ν ( z)). ( ). 3( ) s s. 3( ) s ( s ν ( y)) { λ. 9 λ. 9 ( s µ ( x)) µ ( x) ( s λ λ µ ( x)) {. R a ν ( x) ( s λ λ ν( x)) {. R b ν ( y) 4. 3 ( s ν ( y )) ( µ ( z) s) λ. 5 9 µ ( x) 6 λ. 9 ( s µ ( x)) w.. η( z) 7 ( s w)( s η( z)) µ ( z) ν ( z) 9. ( s). 3( s) ( s ν ( z)).. µ ( y) 3 ( s µ ( y)) {. R a Ergoi behaviour: Using bel s lea in Laplae transfor, viz li( s. f ) li f ( t) f ( say), s t provie that the liit on the right han sie exists, the tie inepenent up an own state probabilities are obtaine as follows: Pup { () 3 () 4 () ' (4) () P own Where P (4) up

6 J. Math. & Stat., (3): 47-43, 6 '() ' () ' () ' () ' () ' () ' () ' () ' () ' () Reliability: The reliability of the syste is given as follows: a t R( t) e ( a. b. ). t. e. t (. b... a).. e Where a. t a. t a λ λ (43) Table : Variation of availability with tie Tie (t) vailability [P up (t)] M.T.T.F: The Mean Tie To Failure of the syste is given by ( a. b. ) M. T. T. F. R( t) a ( a.. b.. ) 3 a a (44) Profit funtion i. For Non repairable syste, the Profit Funtion (t) in the interval (,t) is given as follows: t ( t) C. R( t) C. t C (45) 3 ii. For repairable syste, the profit funtion (t) in the interval (,t) is given by t ( t) C. P ( t) C. t C (46) up 3 where C,C an C3 are revenue per unit tie, servie ost per unit tie an syste establishent ost respetively. Nuerial oputation. vailability analysis: Setting the values.,., λ., λ.,., a.95, b.97, R R µ ( x).9, ν ( x).8, β ( x), an taking the inverse laplae transfor of (39) the availability of the syste is obtaine as follows: t.93t.3t.77e.88e Pup ( t) e.95.3t.88e t.83t.3t ( e e e ) The values of (t) P up (47) for ifferent values of t are alulate fro (47) an have been shown in the Table. Table : Variation of reliability with tie Tie (t) Reliability[R(t)] Table 3: tie Tie(t) Variation of profit funtion(non repairable syste) with Profit Funtion[(t)] (Non Repairable Syste) C. C.5 C Table 4: Variation of profit funtion(repairable syste) with tie Tie (t) Profit Funtion[(t)] (Repairable Syste) C. C 5. C Reliability analysis: Setting the values.,., λ., λ.,., a.95, b.97 an taking ifferent values of t in (43) one ay obtain the reliability of the syste as shown in the Table. 4

7 J. Math. & Stat., (3): 47-43, 6 Table 5: Variation of M. T. T. F. with failure rate of B(or B) unit M.T.T.F λ. λ. λ Table 6: Variation of M. T. T. F. with huan failure rate M.T.T.F λ Table 7: Variation of M. T. T. F. with huan failure rate M.T.T.F λ λ. λ. λ obtain the variations of M.T.T.F. of the syste against the failure rate of unit B (or B), shown in Table 5. Setting the values.,., λ., a.95, b.97 an taking ifferent values of in (44), one ay obtain the variations of M.T.T.F. of the syste against the huan failure rate λ shown in Table 6. Setting the values.,.,., a.95, b.97 an taking ifferent values of λ in (44), one ay obtain the variations of M.T.T.F. of the syste against the huan failure rate λ shown in Table 7. RESULTS Table exhibits that the operational availability of the syste ereases with inrease in tie perio. Table shows that the reliability of the syste ereases as the tie perio inreases. Table 3 an 4 show the variation in profit funtion with the inrease in tie perio for non repairable an repairable syste respetively. The series of urve in both the figures exhibits that the profit funtion ereases with the inrease in servie ost of the syste. Table 5 epits the variation of M.T.T.F. with the failure rate of subsyste B (or B). The series of urve represents that M.T.T.F. ereases as the failure rate of sub syste inreases. Table 6 epits the variation of M.T.T.F. with huan failure rate. The series of urve represents that M.T.T.F. ereases as the failure rate of subsyste C inreases. Table 7 epits the variation of M.T.T.F. with the huan failure rate. The series of urve represents that M.T.T.F. ereases as the failure rate of subsyste inreases. CNOWLEDGEMENT 3. Profit funtion Non-repairable syste: Setting the values., λ.,.,., λ., a.95, b.97 C 5., C3 5., using (43), taking ifferent values of C an t in (45), one ay obtain the variation of profit funtion of the syste as shown in the Table 3. Repairable syste: Using (47), takingc 5., C3 5., ifferent values of C an t in (46), one ay obtain the variation of profit funtion of the syste as shown in the Table M.T.T.F analysis: Setting the values., λ.,., a.95, b.97 an taking ifferent values of λ in (44), one ay 43 uthors are grateful to Rear iral Dr. jay Shara (Diretor General) an Prof. G. S. Sanhu (Diretor), of rishna Institute of Engineering & Tehnology, Ghaziaba (Inia) for their onstant enourageent, an proviing neessary failities (inluing finanial support) to arry out the present work. The authors o not have any other inustrial links. REFERENCES. Masanori,., J. Fukata an I. Sawa, 985. Reliability onsieration on a opoun reunant syste with repair. Miroeletronis Reliability, 5: Calabro, S.R. Reliability Priniples an Pratie. MGraw ill Book Copany, In. New York. 3. Chung, W.., 987. Reliability analysis of a repairable parallel syste with stanby involving huan error an oon ause failure. Miroeletronis Reliability, 7: