Mathematical Theory and Modeling                                                                www.iiste.orgISSN 2224-580...
Mathematical Theory and Modeling                                                                  www.iiste.orgISSN 2224-5...
Mathematical Theory and Modeling                                                                              www.iiste.or...
Mathematical Theory and Modeling                                                   www.iiste.orgISSN 2224-5804 (Paper)    ...
Mathematical Theory and Modeling                                                                                    www.ii...
Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                                                       www.iiste.orgISSN 2...
Mathematical Theory and Modeling                                                                     www.iiste.orgISSN 222...
Mathematical Theory and Modeling                                                              www.iiste.orgISSN 2224-5804 ...
Mathematical Theory and Modeling                                                                www.iiste.orgISSN 2224-580...
Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                                 www.iiste.orgISSN 2224-5804 (Paper)    IS...
Upcoming SlideShare
Loading in …5
×

Behavioural analysis of a complex manufacturing system having queue in the maintenance section

307 views

Published on

International Journals Call for paper by IISTE: http://www.iiste.org

Published in: Technology, Business
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
307
On SlideShare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
3
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Behavioural analysis of a complex manufacturing system having queue in the maintenance section

  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 Behavioural Analysis of a Complex Manufacturing System Having Queue in the Maintenance Section Surabhi Sengar* S.B.Singh Department of Mathematics, Statistics and Computer Science, G. B. Pant University of Agriculture and Technology, Pantnagar (India) * E-mail of the corresponding author: sursengar@gmail.comAbstractIn this paper reliability characteristic of a complex manufacturing system incorporating queue in service isstudied. The considered system consists of three units namely A, B and C connected in series. Unit Aconsists of a main unit a1 and an active redundant unit a2, unit B consists of a main unit b1 and a coldredundant unit b2 and unit C consists of two units’ c1 and c2 connected in parallel configuration. Consideredsystem can completely fail due to failure of any of the subsystems. It is also assumed that the system canfail due to catastrophic failure. General repair facility is available for units c1 and c2 whereas there exits amaintenance section with one repairmen for repairing the units a1, a2, b1 and b2. Various reliabilitycharacteristics such as steady state behavior, availability, reliability, and MTTF and cost analysis have beenobtained using supplementary variable technique and Gumble-Hougaard copula methodology.Keywords: Supplementary variable technique, reliability, MTTF, asymptotic behavior, markov process,Gumble-Hougaard copula, profit function, queue.1. IntroductionIn modern engineering systems standby redundancy is used for improving the reliability and availability ofcomponents/units. Liebowitz (1966); Mine et al (1968) and Subba Rao (1970), while studying redundantsystem have assumed that a unit, immediately after failure, enters repair. Gupta et al (1983) and Pandey etal (2008) have assumed that the repair times of the failed units are independently distributed. This impliesthat there exist a fairly large number of independent repair facilities which would take up each unit as, andwhen, it fails. However, one can see in many practical situations, it is not feasible to have more than onerepair facilities, in which the units that fail queue up for repair. Queuing Theory plays a vital part in almost all investigations of service facilities. Queuing modelsprovide a useful tool for predicting the performance of many service systems including computer systems,telecommunication systems, computer/communication networks, and flexible manufacturing systems.Traditional queuing models predict system performance under the assumption that all service facilitiesprovide failure-free service. It must, however, be acknowledged that service facilities do experience failuresand that they get repaired. Trivedi (1982) argues that failure/repair behavior of such systems’ is commonlymodeled separately using techniques classified under reliability/availability modeling. In recent years, it hasbeen increasingly recognized that this separation of performance and reliability/ availability models is nolonger adequate. Also Altiok’s (1997) focused on the Performance Analysis of Manufacturing Systems.Mangey & Singh (2010) analyzed a complex system with common cause failures using Gumble-Hougaardcopula methodology. Barlow & Proschan (1975) give the statistical theory of reliability and life testing. Keeping above facts in view, the present paper deals with the reliability characteristics of a complexmanufacturing system having 3-units A, B and C, connected in series, incorporating queue in service. UnitA consists of a main unit a1 and an active redundant unit a2, Unit B consists of a main unit b1 and a coldredundant unit b2 whereas unit C consists of two units’ c1 and c2 connected in parallel configuration. Thesystem can completely fail due to failure of any of the subsystems. Initially when the system startsfunctioning, the main units of subsystem A and B and both units of the subsystem C are operational. When 15
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012the main units a1 and b1 of the subsystems A and B fail, The units in standby are switched onautomatically and failed units are taken up for repair to maintenance section. An unusual situation isdiscussed here that when main units and standby units of A and B fail and they are taken up for repair tomaintenance section, where repairmen is busy in repairing of other machine of the system. At this situationa queue is generated at the maintenance section. So here study is focused on the issue that all the four unitsafter failure are in the queue waiting for repair. It is also assumed that the system can fail due tocatastrophic failure. Once the system is failed due to catastrophic failure (CSF), two types of repairs i.e.constant and exponential are involved to repair the system. Hence the joint distribution is obtained with thehelp of Gumble-Hougaard copula. General repair facility is available for the repairing of units c1 and c2whereas there exists a maintenance section with one repairmen for repairing the units a1, a2, b1 and b2 .Forthe units c1 and c2 failures follows exponential time distribution while repairs follow general timedistribution and for the units a1, a2, b1 and b2 failure and repairs both follows exponential time distributionwith Poisson arrivals. System specification and transition diagram is shown by figures 1 and 2 respectively.Table 1 shows the state specification of the system.The following reliability characteristics of interest are obtained.(i) Transition and steady state probabilities.(ii) Mean operating time between successive failures for different failures.(iii) Availability of the system.(iv) Profit analysis.2. Assumptions(1) Initially the system is in good state.(2) Subsystems A, B and C are connected in series.(3) System has two states namely good and failed.(4) For the units’ c1 and c2 failures follow exponential time distribution while repairs follow general timedistribution.(5) For the units’ a1, a2, b1 and b2 failure and repairs both follows exponential time distribution with Poissonarrivals.(6) There are two types of repairs from state S0 to S14 one is constant and other is exponential.(7) Subsystem a1, a2, b1 and b2 can be repaired only at maintenance section.(8) A special situation is discussed here that when all four machines a1, a2, b1 and b2 are in queue for repairat maintenance section because of repair of other machine.(9) Once the system is failed due to catastrophic failure two types of repairs are involved to repair thesystem i.e. constant and other is exponential.(10) Joint probability distribution of repair rates follows Gumbel-Hougaard copula.3. Notations Pr ProbabilityP0 (t ) Pr (at time t system is in good state S0)Pi (t ) Pr {the system is in failed state due to the failure of the ith subsystem at time t}, where i=2, 5, 7, 14. K Elapsed repair time, where k= x, y, z, u, q, g. λi Failure rates of subsystems, where i=a1, a2, b1, b2, c1, c2, CSF.ψ Arrival rate of unit’s a1, a2, b1, b2 to the maintenance section. 16
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012µ Repair rate of unit’s a1, a2, b1, b2.φ i (k ) General repair rate of ith system in the time interval (k, k+∆), where i= c1, c2, CSF and k=v, g, r, l.P3 (t ) Pr (at time t there is a queue (a1, a2, b1, b2) in the maintenance section due to servicing of some other unit and all four machines are waiting for repair.Pi ( j , k , t ) Pr (at time t system is in failed state due to the failure of jth unit when kth unit has been already failed, where i=9, 11. j=g, v. and k=v, g.K1, K2 Revenue cost per unit time and service cost per unit time respectively.Let u1 = e and u 2 = φCSF (l ) then the expression for joint probability according to Gumbel-Hougaard l θ θ 1/θfamily of copula is given as φ CSF (l ) = exp[l + (log φ CSF (l )) ) ] .4. Formulation of the Mathematical ModelUsing elementary probability considerations and limiting procedure, we obtain the following set ofdifference-differential equations governing the behavior of considered system, continuous in time anddiscrete in space: ∞d  dt + λ a1 + λ a2 + λb1 + λb2 + λc1 + λc2 + λCSF  P0 (t ) = ∫ µ (i) P3 (t )di + φ c1 P8 (t ) + φ c2 P10 (t ) +  0 ∞ ∫φ 0 CSF (l ) P14 (l , t )dl ... (1) ∞∂  ∂t + λ a2 + λc  P1 (t ) = λ a1 P0 (t ) + ∫ φC (r ) P12 (r , t )dr … (2)  0∂  ∂t + ψ  P2 (t ) = λ a2 P1 (t )  … (3)∂ ∂  (ψt ) 3 e −ψt + + ( µ + ψ ) P3 (t ) = ψ [ P2 (t ) + P5 (t ) + P6 (t ) + P7 (t )] + ∂t ∂i … (4)  6∂  ∂t + λa1  P4 (t ) = λa 2 P0 (t )  … (5)∂  ∂t + ψ  P5 (t ) = λa1 P4 (t )  … (6) ∞∂  + λb2 + ψ ∂t  P6 (t ) = λb1 P0 (t ) + ∫ φC (r ) P13 (r , t )dr  … (7) 0 17
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012∂  ∂t + ψ  P7 (t ) = λb2 P6 (t )  … (8) ∞∂  ∂t + φ c1 (v ) + λc2  P8 (t ) = λc1 P0 (t ) + ∫ φ c2 ( g ) P11 ( g , v, t )dg  … (9) 0∂ ∂  ∂t + ∂g + φ c2 ( g ) P9 ( g , v, t ) = 0 … (10)  ∞∂  ∂t + φ c2 ( g ) + λc1  P10 (t ) = λc2 P0 (t ) + ∫ φ c1 (v) P11 (v, g , t )dv  … (11) 0∂ ∂  ∂t + ∂v + φ c1 (v) P11 (v, g , t ) = 0  … (12)∂ ∂  ∂t + ∂r + φC (r ) P12 (r , t ) = 0 … (13) ∂ ∂  ∂t + ∂r + φC (r ) P13 (r , t ) = 0  … (14)∂ ∂  ∂t + ∂l + φCSF (l ) P14 (l , t ) = 0 … (15) Boundary Conditions:P3 (i = 0, t ) = ψ [ P2 (t ) + P3 (t ) + P6 (t ) + P7 (t )] … (16)P8 (0, t ) = λc1 P0 (t ) … (17)P9 (0, v, t ) = λc2 P8 (t ) … (18)P10 (0, t ) = λc2 P0 (t ) … (19)P11 (0, g , t ) = λc1 P10 (t ) … (20)P12 (0, t ) = λC P1 (t ) … (21)P13 (0, t ) = λC P6 (t ) … (22) 18
  5. 5. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012P14 (0, t ) = λCSF P0 (t ) … (23)Initial condition:P0 (0) = 1 , otherwise zero. ... (24)5. Solution of the ModelTaking Laplace transforms of equation (1) through (23) subject to initial and boundary conditions and thenon solving them one by one; we obtain the following:Pup = P 0 ( s ) + P1 ( S ) + P 4 ( s ) + P 6 ( s ) + P 8 ( s ) + P10 ( s ) 1 λa1 λa2 λb1 = [ 1+ + + + K ( s) [ s + λa2 + λC − λC S φC ( s )] [ s + λa1 ] [ s + λb2 + ψ − λC S φ C ( s )] λc 1 λc 2 + ] … (25) [ s + λc 2 + φc1 (v) − λc 2 S φ C2 ( s )] [ s + λc1 + φc2 ( g ) − λc1 S φc1 ( s )]P Down = P 2 ( s ) + P 5 ( s ) + P 7 ( s ) + P 9 ( s ) + P 11 ( s ) + P 12 ( s ) + P 13 ( s ) + P 14 ( s ) λa λa 1 λ a1 λ a 2 1 = 1 2 + + [ s + ψ ][ s + λa 2 + λC − λC S φ C ( s )] K ( s ) [ s + ψ ][ s + λa1 ] K ( s ) λb λb 1 λc1 λc 2 Dφ c ( s ) 1 1 2 + 2 + [ s + ψ ][ s + λb2 + ψ − λC S φ C ( s )] K ( s ) [ s + λc 2 + φc1 (v) − λc 2 S φ c2 ( s )] K ( s ) λc λc Dφ ( s ) 1 λc λa1 Dφc ( s ) 1 1 2 c1 + + [ s + λc1 + φc 2 ( g ) − λc1 S φ c1 ( s )] K ( s ) [ s + λa 2 + λC − λC S φ C ( s )] K ( s ) λc λb Dφ ( s ) 1 λCSF DφCSF ( s) 1 c + … (26) [ s + λb2 + ψ − λC S φ C ( s )] K ( s ) K ( s)Where, λa λaK ( s ) = s + λa1 + λa2 + λb1 + λb2 + λc1 + λc2 + λCSF −ψ {[ 1 2 [ s + ψ ][ s + λa2 + λC − λC S φC ( s )] λa λa λb λb λb+ 1 2 + 1 + 1 2 ]Dµ ( s ) [ s + ψ ][ s + λa1 ] [ s + λb2 +ψ − λC S φC ( s )] [ s +ψ ][ s + λb2 +ψ − λC S φC ( s )] 19
  6. 6. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 ψ3 λc φc (v) λc φc ( g )+ }− 1 − 1 2 2 (s + ψ ) 4 [ s + λc + φc (v) − λc S φ ( s )] [ s + λc + φc ( g ) − λc S φ ( s )] 2 1 2 C2 1 2 1 c1− λCSF S φ CSF ( s ) ... (27) λ a λa λa λa λbM ( s ) = ψ {[ 1 2 + 1 2 + 1 [ s + ψ ][ s + λa 2 + λC − λC S φ C ( s )] [ s + ψ ][ s + λa1 ] [ s + λb2 + ψ − λC S φ C ( s )] λb λb ψ3 + 1 2 ]Dµ ( s ) + + } … (28) [ s + ψ ][ s + λb2 + ψ − λC S φ C ( s )] (s + ψ )4 1 − S µ (s)Dµ ( s ) = … (29) s +ψφCSF (l ) = exp[l θ + (log φCSF (l ))θ )1 / θ ] … (30)6. Asymptotic Behaviour of the SystemUsing Abel’s lemma in Laplace transforms, viz;lim sf (s) = lim f ( t ) = f (say)s→ 0 t →∞ ... (31)provided the limit on the right hand side exists, the time independent operational probabilities for up anddown states are obtained as follows: 1 λa λa λb1 λc1 λc2P up ( s ) = [1 + 1 + 2 + + + ] … (32) K (0) ψλ a2 λa1 λb2 + ψ − λc φ c1 (v) φ c2 ( g ) 1 λa1 λa λb1 λb2 λc λc M φ c λc λc M φ cP down = [ + M (0) + 2 + + 1 2 2 + 1 2 1 + K ( 0) ψ ψ ψ (λb2 + ψ − λC ) φc1 (v) φc 2 ( g ) λc λa M φ λC λb M φ 1 C + 1 C + λCSF M φCSF ] … (33) λa 2 λb +ψ − λC 2where,M (0) = lim M ( s ) s→0 … (34) 1 − S φi ( s )M φ i = lim s→ 0 s … (35) 20
  7. 7. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 φiSφ i ( s ) = s + φi… (36)6.1Particular CaseA particular case is also discussed as given below:When all repairs follow exponential time distribution: In this case setting,S φi ( s ) = φi , ∀i , S µ ( s ) = µ and S φ ( s ) = [ exp l θ + [log φCSF (l )] θ ]1/θ [ ] in equation (25) s + φi s+µ s + exp l θ + [log φ CSF (l )] 1/θ CSF θand (26), we can obtain the Laplace transforms of various state probabilities of the system.7. Numerical Computation7.1 Availability of the considered systemFor the numerical computation let us consider the values:λa = 0.3, λa = .5, λb = .4, λb = .9, λc = .5, λc = .6, λC = .8, λCSF = .7,ψ = .5 and φC = φc = φc 1 2 1 2 1 2 1 2= µ = 1 and x=y=z=l=v=g=u=1.Putting all these values in equation (32) and taking inverse Laplace transform, we getPup=.1686680814e(-4.323965466t)-.008768799073e(-2.117902295t)-.04845342456e(-2.071912570t)+.09122411851e(-2.062984318t) +.4158882739e(-1.735709603t)-.02422312588e(-1.114879943t)+.005458900298e(-.8020535367t)+.2063822881e(-.5004630522t) cos(.5063484832t)-.02881439167e(-.5004630522t)sin(.5063484832t)+.1082411055*10(-89)I.1331028150*1 (89) (-.5004630522t)0 e cos(.5063484832t)-.9533452524*10(89)e(-.5004630522t)sin(.5063484832t))+.1082411055*10(-89)*I(.1331028150*10(89)e(-.5004630522t)cos(.5063484832t)+.9533452524*10(89)e(-.500463052t)sin(.5063484832t))-.1143200992*10(-5)e(-.4693387571t)-.001713357894e(-.3432408717t)+.0005397902571e(-.2917232784t)-.0002082937963e(-.2487154115t) +.1949572589e(.08335215466t) …(37)Now in equation (37) setting t=0, 1, 2….10 one can obtain the Figure 3.7.2 Reliability AnalysisLet λ a = 0.3, λ a = .5, λb = .4, λb = .9, λ c = .5, λ c = .6, λC = .8, λCSF = .7,ψ = .5 and φ C = φ c 1 2 1 2 1 2 1= φc = µ = 0 and x=y=z=l=v=g=u=1. 2Putting all these values in equation (32) and taking inverse Laplace transform, we get,Pup=-.0003992289179e(-.6000000000t)+.1301800390e(-1.400000000t)+.2561750275e(-3.885958190t)+.2579813141e(-1.229778949t) -.0237396891e(-1.034271878t)+.1939947786e(-.5178813385t)cos(.4067318666t)-.07626612243e(-.5178813385t)sin(.4067318666t)-.4458387683*10(-35)I(.8553105723*1034)e(-.5178813385t)cos(.4067318666t)+.217561855*1035e(-.5178813385t) sin(.4067318666t)-.4458387683*10(-35)I(.8553105723*1034)e(-.5178813385t)cos(.4067318666t)-.21756 35 (-.5178813385t)1855*10 e sin(.4067318666t)-.002547494817e(.3188919519t)+.1883552536e(.004663645435t) …(38)Now varying time in equation (38), one can obtain the Figure 4.7.3 MTTF AnalysisMTTF of the system can be obtained as MTTF = lim P up ( s) (as s tends to 0) ... (39) s→0 21
  8. 8. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 20127.3.1 With respect to λ a1 :Suppose λa = .5, λb1 = .4, λb2 = .9, λc1 = .5, λc2 = .6, λC = .8, λCSF = .7,ψ = .5 in equation (39) and 2putting λ a1 = 0.1,.5,1,1.5,2... one can obtain the MTTF for different values of λa1 as shown in Figure5.7.3.2 With respect to λc1 :Consider λa = 0.3, λa = .5, λb = .4, λb = .9, λc = .6, λC = .8, λCSF = .7,ψ = .5 in equation (39) 1 2 1 2 2and putting λc = .1,.5,1,1.5,2.... one can obtain the MTTF for different values of λc as depicted in 1 1Figure 6.7.3.3 With respect to λ c2 :Assume λa = 0.3, λa2 = .5, λb1 = .4, λb2 = .9, λc1 = .5, λC = .8, λCSF = .7,ψ = .5, in equation (39) 1and taking λc = .1,.5,1,1.5,2.... we have Figure 6 which shows the variation of MTTF for a range of 2values of λc2 .7.3.4 With respect to λCSF :Setting λa = 0.3, λa = .5, λb = .4, λb = .9, λc = .5, λc = .6, λC = .8,ψ = .5 in equation (39) and 1 2 1 2 1 2putting λCSF = .1,.5,1,1.5,... one can get Figure 7 which exhibits the variation of MTTF for differentvalues of λCSF .7.4 Cost AnalysisLetting λ a1 = 0 .3, λ a 2 = .5, λ b1 = .4, λ b2 = .9, λ c1 = .5, λ c 2 = .6, λ C = .8, λ CSF = .7,ψ = .5 and repairrates are φC = φ c = φ c = µ = 1 1 2 and x=y=z=l=v=g=u=1. Furthermore, if the repair follows exponentialdistribution then, on putting all these values and taking inverse Laplace transform one can obtain equations(37). If the service facility is always available, then expected profit during the interval (0, t] is given byG(t) = K1[.03900773092e(-4.323965466t)+.00414032759e (-2.117902295 t)-.02338584420 e(-2.071912570 t)-.04421949198e(-2.062984318t)-.2396070594e(-1.735709603t)+.02172711681e(-1.114879943t)-.006806154350 e(-.8020535367 t)-.1749950223e(-.5004630522t)cos(.5063484832t)+.2346284213e(-.5004630522t)sin(.5063484832t)+.1082411055*10(-89)I(.1083823101*10(90)e(-.5004630522t)cos(.5063484832t)+.8083575158*10(89)e(-.5004630522t)sin(.5063484832t))+.1082411055*10(-89)I(-.1083823101*10(90)e(-.5004630522t)cos(.5063484832t)-.8083575158*10(89)e(-.5004630522t)sin(.5063484832t))+.2435764807*10(-5)e(.4693387571t)+.004991707093e(-.3432408717t)- .001850350306e(-.2917232784t)+.0008374784427e(-.2487154115t)+2.338958839e(.08335215466t)-1.840786251]-K2t … (40)Keeping K1 = 1 and varying K2 at 0.1, 0.2, 0.3 in equations (40), one can obtain Figure 8.8. Results and DiscussionIn this paper, we have evaluated availability, reliability, MTTF and cost function for the considered systemby employing Supplementary variables technique and copula methodology. Also, we have computedasymptotic behavior and a particular case to improve practical utility of the system. M. Jain & CharuBhargava (2008) have done the analysis of bulk arrival retrial queue. Indra & Sweety Bansal (2010) havegiven the concept of vacations to unreliable M/G/1 queue. But we have analysed a common phenomenon ofqueue, which usually occurs in the manufacturing system. Analysis of the Figure 3 gives us the idea of theavailability of the system with respect to time t. Critical examination of Figure 3 yields that the values of 22
  9. 9. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012the availability decreases approximately in a constant manner with the increment in time.The Figure 4 shows the trends of reliability of the system with respect to time when all the failures andrepair rates have some fixed values. From the graph we conclude that the reliability of the system decreasesrapidly with passage of time when all failures follows exponential time distribution. The reason for therapid decrement is queue in the maintenance section due to which the system is in down state for a largeperiod of time.Next, we study the effect of various parameters on the MTTF. A critical examination of Figures.5, 6, 7shows that MTTF decreases with increment in λ a1 , λc1 , λc 2 and λCSF . An unusual phenomenon can beseen by observing the graph that for all the parameter, initially MTTF is negative due to queue inmaintenance section and later it becomes positive.Finally, Figure 8 represents the graph of the “Cost function vs. time. In this Figure we plotted a costfunction G (t) for different values of cost K1 and K2. One can easily observe that increasing service costleads decrement in expected profit.ReferencesAltiok, T. (1997), Performance Analysis of Manufacturing Systems, New York: Springer-Verlag.Barlow, R. E. & Proschan, F. (1975), Statistical Theory of Reliability and Life Testing: Probability models,New York: Holt, Rinehart and Winston.Gupta, S.M., Jaiswal, N. K. and Goel, L. R. (1983), “Stochastic behavior of a standby redundant systemwith three modes”, Microelectron Reliability 23, 329-331.Indra & Bansal, Sweety (2010), “The Transient Solution of an Unreliable M/G/1 Queue with Vacations”,International Journal of Information and Management Sciences 21, 391-406.Jain, M. & Bhargava Charu (2008), “Bulk Arrival Retrial Queue with Unreliable Server and PrioritySubscribers”, International Journal of Operations Research 5(4), 242−259.Liebowitz, B. R. (1966), “Reliability considerations for a two element redundant system with generalizedrepair times”, Operation Research 14, 233-241.Mangey, Ram & Singh, S.B. (2010), “Analysis of a complex system with common causes failure and twotypes of repair facilities with different distributions in failure”, International Journal of Reliability andsafety 4(4), 381-392.Mine, H., Osaki, S. & Asakura, T. (1968),”Some considerations for multiple units redundant systems withgeneralized repair time distributions”, IEEE Transaction on Reliability R-17, 170-174.Pandey, S. B., Singh, S. B. & Sharma, S. (2008), “Reliability and cost analysis of a system with multiplecomponents using copula”, Journal of Reliability and Statistical Studies 1(1), 25-32.Subba Rao, S. & Natrajan, R. (1970), “Reliability with standby” Opsearch, 7, 23-36.Trivedi, K.S. (1982), Probability and Statistics with Reliability, Queuing and Computer Scienceapplications, Englewood Cliffs, NJ: Prentice-Hall. Table 1 State specification chart States Description System State S0 When the system is in fully operational condition. G S1 When the system is in operating state when unit a1 is failed. G S2 When the system is in failed state due to the failure of unit a2. F S3 When all four units a1, a2, b1, b2 are in queue at maintenance section due to F repairing of some other unit of the manufacturing system. The system is in 23
  10. 10. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 failed state at this time. S4 When the system is in operable state when unit a2 is failed. G S5 When the system is in failed state due to the failure of unit a1. F S6 When the system is in operable condition when unit b1 is failed. G S7 When the system is in failed state due to failure of unit b2. F S8 When the system is in operable condition when unit c1 is failed. G S9 When the system is in failed state from the state S8 due to failure of unit c2. FR S10 When the system is in operable condition when unit c2 is failed. G S11 When the system is in failed state from the state S10 due to failure of unit c1. FR S12 When the system is in failed state from the state S1 due to failure of unit C. FR S13 When the system is in failed state from the state S6 due to failure of unit C. FR S14 When the system is in failed state due to catastrophic failure. FRG: Good state; F: Failed State; FR= Failed state and under repair. Subsystem A Subsystem B Subsystem C A B C c1 a1 a2 b1 b2 c2 Figure 1: System Configuration 24
  11. 11. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 µ P9 (g, v, t) S9 P7 (t) S7 ψ λ b2 φc (g) λc ψ 2 2 λb 1 P6 (t) S6 S φC (r) P8 (t) S8 P13 (r, z, t) S13 λC φc 1 λc 1 λa 1 λa 2 ψ P0 (t) S0 P1 (t) S1 P2 (t) S2 P3 (t) S3 φ (r) λc 2 C φc2 (g) λa 2 λC P12(r, y, t) S12 P10 (t) S10 λa 1 ψ P4 (t) S4 P5 (t) S5 µ λc φc (v) 1 1 λCSF µ P14 (l, t) P11(v, g, t) S11 φCSF(l) = Operational, == Maintenance section, = Failed Figure 2: State transition diagram 25
  12. 12. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.3, 2012 Figure 3. Availability Vs Time Figure 4. Reliability Vs Time Figure 5. MTTF Vs λa1 Figure 6. MTTF Vs λc 1 and λc 2 Figure 7. MTTF Vs λCSF Figure 8. Cost Vs time 26

×