REPAIRABLESYSTEMS
1
AVAILABILITYENGINEERING
A Single Repairable ComponentWith Failure Rate λ and Repair Rate μ
The component may exist in one of Two States SO (Working) and S1 (Failing).It transits from SO to S1 by Failure and back from S1 to SO by Repair
SO
S1
λμ
The Concept of AVAILABILITY
STATE GRAPH
After time interval ΔtAt Present
SO S1
SO 1 - λ Δt λ Δt
S1 μ Δt 1 - μ Δt
TRANSITION MATRIX
It is required to evaluate Probabilities PO and P1 Of the component being in states SO and S1
TRANSITION EQUATIONS
After time interval ΔtAt Present
SO S1
SO 1 - λ Δt λ Δt
S1 μ Δt 1 - μ Δt
OO
OOO
OOO
OOO
OO
O
PdtdP
Pt
tPttPttPtPttPPttPtttP
PttPtttPPP
)()(
)1()(1)(1
1
1
1
Multiply both sides byte )(
KeeP
dteePd
eePdtd
ePeedtdP
ttO
ttO
ttO
tO
ttO
)()(
)()(
)()(
)()()(
)(
KeeP
dteePd
eePdtd
ePeedtdP
ttO
ttO
ttO
tO
ttO
)()(
)()(
)()(
)()()(
)(
tO KeP )(
)(
Then
, ,)()(
1 then )(
1
, 1 0
ThereforeKK
ThereforePtAt O
tO eP )(
)()(
At Steady state as t tends to infinity, the component(system) will be AVAILABLE with probability
APO
)(
After time interval ΔtAt Present
SO S1
SO 1 - λ Δt λ Δt
S1 μ Δt 1 - μ Δt
OO
OOO
OOO
OOO
OO
O
PdtdP
Pt
tPttPttPtPttPPttPtttP
PttPtttPPP
)()(
)1()(1)(1
1
1
1
As t tends to infinity, Po = Availability = const and dPo / d t =0, then
A
A λ=1/MTBF μ=1/MTTR Therefore
TimeDownMeanTimeUpMeanTimeUpMean
MTTRMTTFMTBFA
Another Approach toAVAILABILITY
Availability of A System with One Unit Standby
This system could exist in one of three states So , S1 and S2:
So state of system operating with probability Po
S1 state of system with the main component failed and switched to the standby unit to operate ( probability of being in this state P1,S2 State of having both units main and standby failed with probability P2
SOS1 S2
λ λ
μ2 μ
SO S1 S2So 1 - λ Δt λ Δt 0
S1 μ Δt 1 - λ Δt - μ Δt λ Δt
S2 0 2 μ Δt 1 – 2 μ Δt
Two Repair gangs are available
STATE GRAPH
TRANSITION MATRIX
TRANSITION EQUATIONS
)........(3....................2
)(2)()(
)(2)()()(21)(
..(2)..............................
before,shown as Therefore)(1
.....(1)..............................1
122
2122
2122
212
1
1
21
PPdtdP
tPtPt
tPttPttPttPtPttP
tPtttPttP
PPdtdP
PttPtttPPPP
OO
OO
O
SO S1 S2SO 1 - λ Δt λ Δt 0
S1 μ Δt 1 - λ Δt - μ Δt λ Δt
S2 0 2 μ Δt 1 – 2 μ Δt
Taking the Limit as the time tends to infinity, we find
)5(..........212
)4........(..........
2
212
11
O
OO
PPPP
PPPλP
.
Substitute (4) and (5) in (1), we find
22
2
2
2
222
1211
121.
O
O
OOO
P
P
PPP
This Result could be obtained directly by a method called HARMONIC BALANCE Apply Harmonic Balance to state So
SO
Expected IN [ µ P1 ] = Expected out [ λ Po ]
μ
S1λ
Similarly, apply Harmonic Balance to state S2 S1 S2
λ
2μ.2 21 PP .....(1)..............................121 PPPO
The System is AVAILABLE in case of being in either states So and S1
22
2
1
2222
1
A
PPPPA OOO
. 1 OPP
22
2
222
OP
In a metal cutting workshop, there are two different machines:Machine A: a mechanical cutting with capacity of 10 tons per hour. The machine has a constant failure rate of 3 failures per month and mean repair time of 5 daysMachine B: a gas cutting machine with capacity of 20 tons per hour . The machine has a constant failure rate of 4 failures per month and mean repair time of 3 days.Evaluate the EFFECTIVE CAPACITY of the workshop._____________________________________________________________________The Workshop could exist in one of the following statesSo Both machines are working. S1 Mechanical Cutting failed and gas Cutting worksS2 Gas Cutting failed and Mechanical Cutting worksS3 Both failed
S1S2
S3
λ m
μ mλ gRequired to find the FOUR probabilities, Po, P1, P2, P3
The first equation:Po + P1 + P2 + P3 = 1 ………………………….(1)The other THREE equation will be obtained by applying HARMONIC BALANCE to the three statesSo, S1, S2 as follows:
SO
μ m
λ mλ g
μ g
μ g103/30 465/30 3
gg
mm
)2...(..........21 PPP gmOgm Apply to So
Apply to S1 )3...(..........31 PPP gOmmg
Apply to S2 )4...(..........32 PPP mOgmg Omitting P3 from the two equations (3) and (4), we get
)5......(21 Oggmmggmmmg PPP Consider (2) and (5) to omit P2, we find
)6.(..........01
1
PP
PP
m
m
Ogmgmmmgmgm
Substitute in (2)
)7.........(02
20
PP
PPP
g
g
gmOgm
Substitute in (3)
)8.(..........3
30
Og
g
m
m
gOmm
mmg
PP
PPP
Effective Capacity = (20+10)*Po + 20*P1 + 10 * P2
tons95.201905.0*102381.0*204762.0*30
095.0 1905.0 2381.0 321 PPP
Substitute in (1), we get
i
ii
gmgmO
Og
g
m
mO
g
gO
m
mO
P
PPPP
)9.(..........1
1
1
Po + P1 + P2 + P3 =1……….(1)..01 PP
m
m
02 PPg
g
..3 Og
g
m
m PP
10 46 3
gg
mm
4762.02.04.05.01
1 4.0 5.0 0
Pgg
gm
m
m
Evaluation of the RELIABILITY of the above considered REPAIRABLE SYSTEM
)........(3....................
..(2)..............................
before, shown as Therefore
.....(1)..............................
122
2122
2122
212
1
1
21
2
)(2)()()(2)()()(21)(
)(11
PPdtdP
tPtPt
tPttPttPttPtPttP
tPtttPttP
PPdtdP
PttPtttPPPP
OO
OO
O
...(3)..............................2
...(2)..............................
...(1).......... ......................1
122
1
21
PPdtdP
PPdtdP
PPP
OO
O
Consider the following three equations in the three unknowns:
Apply Laplace Transform on the above three equations
0)0( 1)0(2)0(
)0(
1
2
*1
*22
*2
*1
**
*2
*1
*
PPPPPsP
PPPsPs
PPP
O
OOO
O
0
* dtePP stmm
Where, Is the Laplace Transform of Pm. Therefore,
..(6)..........2
5)1........(
)4..(..........1
*1
*2
*2
*1
**
*2
*1
*
PPsP
PPsPs
PPP
OO
O
Solving the above system of algebraic equations
At t = 0, PO =1 :
)(1
)(1
K
K
Therefore,
A
AtRtAs
etRtP
eeP
tO
ttO
)(
)()(
)(
)(
)()(
A is the AVAILABILITY λ=1/MTTF μ=1/MTTR
Therefore
TimeDownMeanTimeUpMeanTimeUpMean
MTTRMTTFMTTFA
KeeP ttO
)()(
)(
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