IEEE TRANSACTIONS ON RELIABILITY, VOL. R-25, NO. 2, JUNE 1976 105 Steady-State Profit in a 2-unit Standby System Ashok Kumar Standard notation is given in Information for Readers & Authors, p. 128. Abstract-Three models are presented for a 2-unit standby system, Models I and 2 deal with a 2-unit warm-standby system with and with- Notation Common to Models 1 and 2 out preventive maintenance (pm) to standby unit. Model 3 is a 2-unit cold-standby system with pm to operative unit. An income structure has been superimposed on the semi-Markov processes generated by the systems to obtain s-expected profit. For models 2 and 3, s-expected mo, Ms mean time in state ro or rs for a unit profit is suggested as the measure of maintenance effectiveness. Numeri- MO, Ms Xnm, Xmns cal examples are included to illustrate the results. Xyi amount the system earns per unit time in Si Reader Aids: Ng s-expected net gain (earnings) of the system per Purpose: Report of derivation unit time. Special math needed for explanations: Semi-Markov processes Special math needed for results: None Notation for Model 1 Results useful to: Reliability theoreticians i4 1,2,3,4,5 p ratio of failure rate of standby to failure rate of operative unit INTRODUCTION pO, ps Laplace transforms of Cdf's of the time-to-repair (time in system states ro, rs respectively) evaluated Most of the earlier workers in the field of standby systems at X have evaluated preventive maintenance (pm) policies either by minimizing system steady-state unavailability [3] or by maxi- Notation for Model 2 mizing mean time to system failure [4, 7]. Mine and Kawai [5] have evaluated pm policies that maximize net profit rate i4 0, 1, 2, 3, 4, 5, 6 from the system over an infinite time span for a 2-unit system H(t) Cdf of pm time with degraded states. The s-expected profit is immensely im- Go(t), G7(t) Cdf's of time to repair (times in ro, rs re- portant in the maintenance of equipment in modern business- spectively) industrial systems. Three models are considered: Model 1 is m;Ml mean pm time; ?m that of Shah and Branson [1], model 2 is similar to that of mo, mS mean time to repair in system states ro, rs [7], and model 3 is that of Osaki [6]. Howard's [4] reward structure has been superimposed on Notation for Model 3 the semi-Markov processes generated by the systems to obtain s-expected profit, denoted by g for the 3 models. In models i, j 0, 1, 2, 3, 4, 5 2 and 3, g has been used as the measure of maintenance effec- L(t) Cdf of failure time of the operative unit tiveness. An optimal pm policy is defined to be one that maxi- yi amount the system earns per unit time for the mizes g. duration it stays in Si g s-expected net profit of the system per unit time. NOTATION COMMON TO ALL MODELS Assumptions common to all Models Si, S system states i4 subscripts which imply system state 1. Switching of units to change system state is perfect P1, 1-step transition probability from Si to S, (instantaneous, without errors, and without system damage). pi unconditional s-expected value of sojourn time in Sn 2. The system earns (loses) at a fixed rate, which can be P transition probability matrix, P--((p4)) different for each state. There is a transition reward (cost) I identity matrix of suitable order when the system changes state; it can be different for each D I-P transition. di subdeterminant of D formed by deleting row i and column i Assumptions common to Modelsl1and 2 rj reward (cost) for transition from Si to S - implies the complement e.g., lpI--- 1. Each unit can be in one of 5 states
Transcript
IEEE TRANSACTIONS ON RELIABILITY, VOL. R-25, NO. 2, JUNE 1976 105
Steady-State Profit in a 2-unit Standby System
Ashok Kumar Standard notation is given in Information for Readers &Authors, p. 128.
Abstract-Three models are presented for a 2-unit standby system,Models I and 2 deal with a 2-unit warm-standby system with and with- Notation Common to Models 1 and 2out preventive maintenance (pm) to standby unit. Model 3 is a 2-unitcold-standby system with pm to operative unit. An income structurehas been superimposed on the semi-Markov processes generated by thesystems to obtain s-expected profit. For models 2 and 3, s-expected mo, Ms mean time in state ro or rs for a unitprofit is suggested as the measure of maintenance effectiveness. Numeri- MO, Ms Xnm, Xmnscal examples are included to illustrate the results. Xyi amount the system earns per unit time in Si
Reader Aids: Ng s-expected net gain (earnings) of the system per
Purpose: Report of derivation unit time.Special math needed for explanations: Semi-Markov processesSpecial math needed for results: None Notation for Model 1Results useful to: Reliability theoreticians
i4 1,2,3,4,5p ratio of failure rate of standby to failure rate of
operative unitINTRODUCTION pO, ps Laplace transforms of Cdf's of the time-to-repair
(time in system states ro, rs respectively) evaluatedMost of the earlier workers in the field of standby systems at X
have evaluated preventive maintenance (pm) policies either byminimizing system steady-state unavailability [3] or by maxi- Notation for Model 2mizing mean time to system failure [4, 7]. Mine and Kawai[5] have evaluated pm policies that maximize net profit rate i4 0, 1, 2, 3, 4, 5, 6from the system over an infinite time span for a 2-unit system H(t) Cdf of pm timewith degraded states. The s-expected profit is immensely im- Go(t), G7(t) Cdf's of time to repair (times in ro, rs re-portant in the maintenance of equipment in modern business- spectively)industrial systems. Three models are considered: Model 1 is m;Ml mean pm time; ?mthat of Shah and Branson [1], model 2 is similar to that of mo, mS mean time to repair in system states ro, rs[7], and model 3 is that of Osaki [6].
Howard's [4] reward structure has been superimposed on Notation for Model 3the semi-Markov processes generated by the systems to obtains-expected profit, denoted by g for the 3 models. In models i, j 0, 1, 2, 3, 4, 52 and 3, g has been used as the measure of maintenance effec- L(t) Cdf of failure time of the operative unittiveness. An optimal pm policy is defined to be one that maxi- yi amount the system earns per unit time for themizes g. duration it stays in Si
g s-expected net profit of the system per unit time.NOTATION COMMON TO ALL MODELS
Assumptions common to all ModelsSi, S system statesi4 subscripts which imply system state 1. Switching of units to change system state is perfectP1, 1-step transition probability from Si to S, (instantaneous, without errors, and without system damage).
pi unconditional s-expected value of sojourn time inSn 2. The system earns (loses) at a fixed rate, which can beP transition probability matrix, P--((p4)) different for each state. There is a transition reward (cost)I identity matrix of suitable order when the system changes state; it can be different for eachD I-P transition.di subdeterminant of D formed by deleting row i and
column i Assumptions commonto Modelsl1and 2rj reward (cost) for transition from Si to S- implies the complement e.g., lpI--- 1. Each unit can be in one of 5 states
106 IEEE TRANSACTIONS ON RELIABILITY, VOL. R-25, NO. 2, JUNE 1976
0 operatings standby 6 0ro repair (from operating) I .75 .2 5rs repair (from standby) 5s0 II 7 25wo wait for repair (from operating) ivA 2'.25l
vi .05 .75Vii .05 .025
2. There is only one repairman. l 0
Model 1 30 -
1. The times in each state are mutually s-independent. The 20times to failure (times in o, s) are exponentially distributed.
2. The system state is the states of the 2 units.I.0
S5 (o, s), can go to (o, rs) or (o, ro) L54 (o, ro), can go to (wo, ro) or (o, s) 00-0. 05 1 5S3 (O, rs), can go to (wo, rs) or (o, s) - . eS2 (wo, rs) can go to (o, ro)SI (wo, ro) can go to (o, ro) Fig. 1. Variation of g with p for Model 1.
The system is up in S5, S4, S3; the system is down in S2, S1.
Model 2 For details and transition diagram see [6].
1. There is a 2-unit standby system; at t = 0, one unit is in INCOME STRUCTURE FOR ALL MODELSoperation and other begins as standby. Failure time distribu-tion of operative unit is exponential. Failure time of standby It has been shown in Howard [4] thatunit is F(t) with IFR property [8].
2. As soon as operative unit fails, the standby unit (if it has Xg = Xiwidgq./L,ud, (I)not failed by that time) becomes operative and its failure dis-tribution becomes exponential with rate X. q,= Ip.j r1, +Y.l.X, (la)
3. Time to begin the pm (measured from the beginning ofthe operation of unit in standby) follows general distribution s-Expected Profit for Model 1A(t).
4. A unit can be in pm state. System state is a pair of sym- When the solution of the model from [1] is substituted intobols with transitions as shown. (1), (2) is obtained.
S0: (o, s), can go to (o, rs) or (o, pm) or (o, ro) g = N/D (2)SI: (o, rs), can go to (o, s) or (rs, wr)S2: (o, pm), can go to (o, s) N=(l +pp5) [pO(rI4 + r4l +Y4 -yl) +por45 + Moy1] +53: (o, ro), can go to (o, s) or (r, wr)S4: (rs, wr), can go to (o, ro) + °po [p<s (r24 + r32 + Y3 -y2 ) + r54 + Y5 +55: (pm, wr), can go to (o, ro)S6: (r, wr), can go to (o, ro) +p(r53 +r3M ps +MVy2)] (2a)
Model 3 D=M0 +%+ p(Msu,o0 + MO0 (2b)
1. There is a 2-unit cold-standby system. At time t = 0, one s-Expected Profit for Model 2unit begins to operate and the other is on standby. Failuretime distribution of operative unit has IFR property to make g IN)Ds (3)pm effective.
2. Time to begin thepm (measured from the beginning of N5 =P30[P0I{(rol +r10)+plO(rl4 +r43 -rio +YI -y4)+M5.y4}operation) follows general distribution unless the operative+rOpy+P3(r +Y)+p (pr2+My4+y)+unit fails before that time.
3. Repair t4me and pm time of each unit is exponential+P2 r +r Y2-s)]+(1 o Pl-P0P ) 0(3+4. Repair/pm are made at most for 2 units simultaneously. +2(5 r5+2Y) ( oPo-2p2[3(35. System fails when both units are under repair/pm for + r63 + Y3 - Y6) + r30p30 + Moy6] (3a)
the first time.
KUMAR: STEADY-STATE PROFIT IN A 2-UNIT STANDBY SYSTEM 107
K-8
0 20 40 60 8-0 100 120 140 °
-to ~~ ~ ~ ~~~-
Fig. 2. Variation of g with t4for Model 2. Fig. 3. Variation of g with to for Model 3.
D120 C03+ poiM5 +PoM)p30 Model 3:
+ (1 -PoP5o - p20 ) (3b) The pm policy is same as for model 2.
See [9] for details and several particular cases. FO0, if t K to,B(t) = IL(t)=gamf(6;K)
NUMERICAL EXAMvPLES L1, otherwise
Model 1: Yi = 4*O,Y2 =5S.O,y3 = 3.O,y4 =-1,O0,y5 =-2.0, rij=O°,= .10. Effect of 6, 5 and 3 ong is shown in Fig. 3 for
Let repair time distribution be exponential; then ep = 1/ K = 6.(1 +Mo) and fs = 1/(1 +Ms). Choose the numerical valuesYi =Y2 =-5*°,3 = = 5*0,5 = 100,i4 =r24 =r3I= ACKNOWLEDGMENTS
=45=-1.0, r41 = r = =32=-3.0, t-53 =0-2.0. The behaviourof g is shown in Fig. 1 for various values of Mow Ms and p. The author thanks the Director of the D.S.L., for permnission
to publish this paper, and Dr. N.K. Jaiswal, Dr. D. Ray and Dr.Model 2: J.L. Jamn for their help. The author is indebted to the Editor
for improving this paper. The author did part of the workPerform pm at a fixed time from the instant a unit begins while at the Defence Science Laboratory, Ministry of Defence,
as standby. Delhi.
0, if t < to, ~~~~~~REFERENCESA(t)=
1,otherwise -[1] Bharat Shah, M.H. Branson, "Reliability analysis of systems com-prised of units with arbitrary repair time distributions," IEEE
= r63 = r20 = r30 = r43 = &-12.5,M= .2,Mo = .4, = .8, [3] 5. Osaki, Tatsuyuki Askura, "A two-unit standby redundantp = .02. system with repair and preventive maintenance," Jr. ofAppi.
Fig. 2 shows g as a function of to for different values of Prob., vol. 7, pp. 641-648, 1970.K [
108 IEEE TRANSACTIONS ON RELIABILITY, VOL. R-25, NO. 2, JUNE 1976
structures," Jr. of Ops. Res. Soc. ofJapan, vol. 6, pp. 163-199, Ashok Kumar//System Planning Study Project//W.R.D.T.C.//UniversityAugust 1964. of Roorkee//Roorkee-247667, U.P. INDIA
[51 H. Mine, H. Kawai, "An optimal maintenance policy for a 2-unitparallel system with degraded states," IEEE Trans. Rel., vol. Ashok Kumar was born on December 3, 1944 at Hasan Abdel now inR-23, pp. 81-86, June 1974. Pakistan. He received M. Stat. degree in 1967 from Institute of Social
[61 S. Osaki, "Reliability analysis of a two-unit standby redundant Sciences, Agra University.system with preventive maintenance," IEEE Trans. Rel., vol. From 1967 to 1969 he was lecturer in Statistics at Institute ofR-21, pp. 24-29, Feb. 1972. Social Sciences, Agra. He joined Research & Development Orgn.,
[7] R. Subramanian, K.S. Venkatakrishnan, "Reliability of a 2-unit Ministry of Defence in 1969. During his career with R&D, he acquiredstandby redundant system with repair, maintenance and standby a deep insight into the problems involving the application of statisticalfailure," IEEE Trans. Rel., vol. R-24, pp. 139-141, June 1975. and OR techniques to aeronautics, psychology and medicine. He held
[8] R. Barlow, F. Proschan, Mathematical Theory ofReliability, position of Sr. Scientiflc Officer when he left. He is now SeniorNew York: Wiley, 1962. Statistician with the System Planning Study Project, Water Resources
[9] Supplement: NAPS document No. 02688-C; 5 pages in this Sup- Devel. Training Center of the University of Roorkee. He is also regis-plement. For current ordering information, see inside rear cover tered with the University of Delhi for his Ph.D. degree in Operationof a current issue. Order NAPS document No. 02688, 15 pages. Research. His present interests are operations research, reliability andASIS-NAPS; Microfiche Publications; 440 Park Avenue South; applied statistics.New York, NY 10016 USA.
Manuscript received December 18, 1974; revised May 27, 1975, July 18,1975, and October 7, 1975 n n r
Stochastic Behaviour of a Special Complex System
Arun Kumar 5. Switching is perfect; it takes no time, is always correct,and never does any damage.
6. Each subsystem goes for repair when it fails. After repair,Abstract-The system has three subsystems: A, B, C. A has several the subsystem is like new-everything works properly, and
non-identical units in series. B has 2 identical units in parallel. C has time is set to zero for that subsystem. The system then,2 identical units in standby (cold) redundancy. Switching is perfect. operates again.Unit failure rates are constant, repair rates need not be constant. The 7. The repair discipline is Head of Line, that is, first comesystem fails if A or B fails; C is useful, but not essential. The supple- 7 .The afc the situad ofly when1S, first ,mentary variable technique is used to find Laplace transforms of sys- first served. ThiS affects the situation only when C fails first,tem state probabilities. otherwise only one subsystem can be failed at a time.
8. The repair times for A units, and for B and C subsystemsKey Words-Redundancy, Repair discipline, Head-of-line. (not units) have general distributions. The time is set to zero
Reader Aids: for each subsystem when it begins repair.Purpose: Report of derivationSpecial math needed for explanations: Markov processes, Laplace
transformsSpecial math needed for results: SameResults useful to: Reliability theoreticians
NOTATION
(Standard notation is given in Information for Readers &MODEL DESCRIPTION Authors)
1. There are 3 subsystems: A, B, C. The system fails if A m number of units in Aor B fails. The system is merely less useful if C fails, but it xi Constant failure rate for unit i from A, 1,..., m.continues to operate. When the system is failed no further mfailure can occur. Xa . Xi
2. A is a 1l-out-of-M:F, 'series' subsystem. The life of each 1unit is exponentially distributed; all units are operating at Xb, Xc Constant failure rate for units in B or C, respectively.Once. ui~(z) repair rate of unit i from A.
3. B is a 1 -out-of-9: G, 'parallel' subsystem. The life of each Ipb (Z), Pc (Z) repair rates for B or C (not units thereof), re-unit is exponentially i.i.d. Both units operate at once. spectively.
4. C is l-out-of-2: G, a cold-standby subsystem. The life of (x State of A: 0 implies no failure; Ri implies unit i haseach unit is exponentially i.i.d., while operating, but only 1 failed and is being repaired; W1 imnplies unit i has failedunit operates at a time. and is waiting for repair.
