System Reliability and Availability
Estimation Under Uncertainty
Tongdan Jin, Ph.D.
Ingram School of Engineering
Texas State University, San Marcos, TX
tj17@txstate.edu
4/11/2012
1
Contents
 System Reliability Estimation
* Variance of reliability estimate
* Series, and parallel systems
 Operational Availability
* Performance based maintenance/logistics/contracting
* Reliability growth or spare parts stocking ?
* A unified availability model
 Conclusion
2
3
Topic One
Modeling System Reliability
With
Uncertain Estimates
Two Components having Same Reliability?
Component
Test Plan 1
Testing 100 hours
Sample n=10, survivals=9
rˆ1 
9
 0 .9
10
Test Plan 2
Testing 100 hours
Sample n=20, survivals=18
rˆ2 
18
 0.9
20
Which component is more reliable?
4
Risk-Averse vs. Risk-Neutral Design
system 2
f (Rˆ )
E[ Rˆ1 ]  E[ Rˆ 2 ]
Var ( Rˆ1 )  Var ( Rˆ 2 )
system 1
Rˆ
• f (Rˆ ) = probability density function for reliability estimate Rˆ
• risk-neutral design would always choose system 1
• risk-adverse design might choose system 2
5
Variance of Reliability Estimate
Test Plan 1
Testing 100 hours
Sample n=10, survivals=9
9
rˆ1 
 0 .9
10
vaˆ r(rˆ) 
0.9(1  0.9)
 0.01
10  1
Which component is
more reliable?
Test Plan 2
Testing 100 hours
Sample n=20, survivals=18
18
rˆ1 
 0. 9
20
vaˆr(rˆ) 
0.9(1  0.9)
 0.0047
20  1
rˆ(1  rˆ)
vaˆ r( rˆ) 
n 1
6
Variance vs. Sample Size
rˆ1 
x
n
n=sample size
rˆ(1  rˆ)
vaˆ r(rˆ) 
n 1
x=survivals
Variance of Component Reliability Estimate
Variance
0.200
r=0.8
0.150
0.100
r=0.9
0.050
0.000
0
10
20
Sample Size
30
40
7
Reliability Variance of Series Systems
Component 1
Component 2
Rˆ s  rˆ1rˆ2
vaˆr(Rˆ s )  rˆ12 rˆ22  rˆ12  vaˆr(rˆ1 ) rˆ22  vaˆr(rˆ2 )



k Components in Series
k
Rˆ s   rˆi
i 1
k
k
i 1
i 1
vaˆr(Rˆ s )   rˆi 2   (rˆi 2  vaˆr(rˆi ))
8
Numerical Example
Component 1
Test Plan 1
Testing 100 hours
n1=10, x1=9
n2=20, x2=17
rˆ1  0.9
rˆ2  0.85
vaˆr(rˆ1 )  0.01
vaˆr(rˆ2 )  0.0067
Component 2
k
Rˆ s   rˆi
i 1
k
k
i 1
i 1
vaˆr(Rˆ s )   rˆi 2   (rˆi 2  vaˆr(rˆi ))
Rˆ s  (0.9)(0.85)  0.765
vaˆr(Rˆ )  (0.9 2  0.852 )
s
 (0.9 2  0.01)  (0.852  0.0067)
 0.0126
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Reliability Confidence Estimate
Assuming Rˆ s is normally distributed, the lower bound
Rˆ s  E[ Rˆ s ]  Z vaˆr(Rˆ s )
E[ Rˆ s ]  0.765
vaˆr(Rˆ )  0.0126
s
Rˆ s  0.765 (1.28) 0.0126  0.62 With 90% confidence
Rˆ s  0.765 (1.64) 0.0126  0.58 With 95% confidence
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Reliability Variance of Parallel System
Component 1
Component 2
Rˆ p  1  (1  rˆ1 )(1  rˆ2 )  1  qˆ1qˆ2
Where
qˆi  1  rˆi
for i=1, and 2
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Parallel System
Series System
Estimates for Reliability and Unreliability
rˆ 
x
n
vaˆ r(rˆ) 
qˆ 
n=sample size
rˆ(1  rˆ)
n 1
x=survivals
nx
n
vaˆr(qˆ ) 
(1  qˆ )qˆ rˆ(1  rˆ)

 vaˆr(rˆ)
n 1
n 1
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Variance of Parallel System
qˆ1
ˆ2
q
ˆk
q
k components in parallel
k
k
i 1
i 1
Rˆ p  1   (1  rˆi )  1   qˆi
k
k
vaˆr(Rˆ p )   qˆ   (qˆi2  vaˆr(qˆi ))
2
i
i 1
Where
i 1
qˆi  1  rˆi
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Numerical Example
Component 1
Component 2
Test Plan 1
Testing 100 hours
n1=10, x1=9
n2=20, x2=17
rˆ1  0.9; qˆ1  1  rˆ1  0.1
rˆ2  0.85; qˆ 2  1  rˆ2  0.15
vaˆr(qˆ1 )  0.01
vaˆr(qˆ 2 )  0.0067
k
k
i 1
i 1
Rˆ p  1   (1  rˆi )  1   qˆi
k
k
vaˆr(Rˆ p )   qˆ   (qˆi2  vaˆr(qˆi ))
2
i
i 1
i 1
Rˆ p  1  (0.1)(0.15)  0.985
vaˆr(Rˆ p )  (0.12  0.152 )
 (0.12  0.01)  (0.152  0.0067)
 0.000225
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Reliability Confidence Estimate
Assuming Rˆ p is normally distributed, then
Rˆ p  E[ Rˆ p ]  Z vaˆr(Rˆ p )
E[ Rˆ p ]  0.985
vaˆr(Rˆ p )  0.000225
Rˆ p  0.985 (1.28) 0.000225 0.966 With 90% confidence
Rˆ  0.985 (1.64) 0.000225 0.960 With 95% confidence
p
15
Series-Parallel Systems
5
4
1
2
6
3
Variance Estimation
1
7
5’
4
1’
2’
7
4’
1’’
7
1’’
7’
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Compute r and var(r) over Time
time (hours)
Sample
Size
Failures
Cum
Failures
1
20
0
0
1
0
2
20
0
0
1
0
3
20
0
0
1
0
4
20
1
1
0.95
0.0025
5
20
0
1
0.95
0.0025
6
20
0
1
0.95
0.0025
7
20
1
2
0.9
0.0047
8
20
1
3
0.85
0.0067
9
20
2
5
0.75
0.0099
10
20
1
6
0.7
0.0111
Reliability Variance
17
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Topic Two
Operational Availability
under
Performance Based Contract (PBC)
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Service Parts Logistics Business
• Representing 8-10% of GDP in the US.
• US airline industry is $45B on MRO in 2008.
• US auto industry is $190B and $73B for parts in 2010.
• US DoD maintenance budget $125B and $70B
inventory with 6,000 suppliers.
• Joint Strike Fighter (F-35): $350B for R/D/P, and
$600B for after-production O/M for 30 years.
• EU Wind turbine service revenue €3B in 2011
• IBM computing/network servers, etc.
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Cost ($)
Total Ownership Cost Distribution
30-40%
50-60%
10-20%
Research
Manufacturing
Development
Operation and Support
5%
Retirement
PBC aims to lower the cost of ownership
while ensuring system performance goals
Reference DoD 5000, University of Tennessee
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Reliability Allocation and Spare Parts Logistics
Reliability Allocation
Spare Parts Logistics
r5 (t)
r1(t)
r6(t)
r2(t)
r4(t)
s21
r8(t)
s
s22
r7(t)
min varRsys (r(t ), n) 
s32
Fleet 2
s3,n-1 Fleet n-1
Fleet n
max Ap (s,x)
min Cost r(t ), n 
Tillman et al. (1977)
Kuo et al. (1987)
Chen (1992)
Jin & Coit (2001)
Levitin & Lisnianski (2001)
Coit et al. (2004)
Ramirez-Marquez et al. (2004)
Marseguerra, Zio (2005)
Jin & Ozalp (2009)
Ramirez-Marquez & Rocco (2010)
More .....
Fleet 1
s3,n
max E[ Rsys (r(t ), n)]
•
•
•
•
•
•
•
•
•
•
•
s32
min Cost(s,x), EBO(s, x)
•
•
•
•
•
•
•
•
•
•
•
Scherbrooke (1968, 1992)
Muckstadt (1973)
Graves (1985)
Lee (1987)
Cohen et al. (1990)
Diaz & Fu (1996)
Alfredsson (1997)
Zamperini & Freimer (2005)
Lau & Song (2008)
Kutanoglu et al. (2009)
More .....
A 4-Step Performance-Based Contracting
Step 1
Performance
Outcome
Step 2
Performance
Measures
System
readiness,
operational
reliability,
assurance of
spare parts
supply
System
availability,
MTBF,
MTTR, Mean
downtime,
logistics
response time
Step 3
Performance
Criteria
Step 4
Performance
Compensation
Mini
availability,
max failure
rate, max repair
waiting time,
max cost per
unit time
Cost plus
incentive fee,
cost plus award
fee, linear
reward,
exponential
reward
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Five Performance Measures by US DoD
• Operational availability (OA)
• Inherent reliability or mission reliability (MR)
• Logistics response time (e.g. MTTR, LDT)
• Cost per unit usage (CUU)
• Logistics footprint
23
Interactions of Five Performance Measures
MTBF
Ao 
MTBF  MTTR  MLDT
MTBF=Mean Time Between Failures
MTTR=Mean Time to Repair
MLDT=Mean Logistics Delay Time
Mission
Reliability (MR)
Operational
Availability(OA)
Logistics
Footprint (LF)
Cost Per Unit
Usage (CUU)
Logistics Response
Time (LRT)
24
Total Ownership Cost
Evolution of Sustainment/Maintenane Solution
CM=>{Warranty, MBC}
PM=>{MBC}
CBM=>{Warranty, MBC}
PBM/PBL=>{PBC}
CM
PM
CBM
PBM
PBC aims to lower the cost of ownership while ensuring system
performance (e.g. reliability and availability).
Note: PBM=performance-based maintenance
25
Integrating Manufacturing with Service
Emergency Repair
OEM for design
and
manufacturing
Repair
Center
Local
spares
stocking
Supplier or OEM
System
fleet
N(t)
Customer
Emergency Repair
OEM for design
and
manufacturing
Repair
Center
Supplier or OEM
Local
spares
stocking
System
fleet
N(t)
Customer
26
Availability and Variable Fleet Size
Variable Fleet Size
• Availability
MTBF(hours)
1,200
800
600
System
Population
800
400
400
200
135
120
105
90
75
60
45
30
15
0
1
0
Weeks
Cumulative Installed WT (1998 to 2030)
150,000
160,000
1.5 MW (2010-2014 )
2.0 MW (2015-2020)
2.5 MW (2020-2025)
3.0 MW (2025-2030)
140,000
120,000
100,000
80,000
60,000
40,000
22,500
20,000
2030
2028
2026
2024
2022
2020
2018
2016
2014
2012
2010
2008
2006
0
02-03
Installed WT Population
Wind Power Industry
200
A
 0.95
200  10
MTBF
1,600
100
A
 0.95
100  5
• MTBF=200 hours, MDT=10 hours
1000
Cumulative Fleet Size
2,000
98-99
• MTBF=100 hours, MDT=5 hours
Figure 1: System Reliability and Fleet Size
Semiconductor Industry
MTBF
A
MTBF  MDT
27
Performance Measures and Drivers
Inherent
Reliability ()
MTBF
OEM
Controlled
Maintenance
Schedule ()
MTTR
Operational
Availability (Ao)
Logistics Support
(s, ts, tr)
MLDT
Customer
Controlled
System Fleet
(n, )
28
29
A Unified Operational Availability Model
Ao ( , s,  , n, tr , t s ) 
1
s

(ntr ) x e ntr
1  t s  tr 1  
x!
 x 0
 =system or subsystem inherent failure rate
s =base stock level
β =usage rate, and 0β1
n =installed base size
tr =repair turn-around time
ts =time for repair-by-replacement
Ref: Jin & Wang (2011)



Trading Reliability with Spares Stocking (II)
Reliability vs. Spare Parts Inventory
spare parts number
10
=0.5, n=50, tr=60 days
8
6
Ao=0.8
4
2
0
2000
Ao=0.95
4000
6000 8000 10000 12000 14000 16000
MTBF (1/lambda) in hours
Note: here lambda=alpha in previous slide
30
Trading Reliability with Spares Stocking (I)
Reliability vs. Spare Parts Inventory Level
number of spare parts
10
=0.5, n=50, tr=30 days
8
Ao=0.8
6
4
Ao=0.95
2
0
1000
2000
3000
4000
5000
6000
7000
8000
MTBF (1/lambda) in hours
31
Trading Reliability and Spares Stocking (III)
Reliability vs. Spare Parts Inventory
spare parts number
20
=0.8, n=50, tr=30 days
15
10
Ao=0.8
5
Ao=0.95
0
0
2000
4000
6000
8000 10000
MTBF (1/lambda) in hours
12000
32
Key Terminologies
1. Variance of reliability estimate
2. Variance propagation
3. Series/parallel reduction
4. Unbiased estimate
5. Operational availability
6. Mean downtime
7. Mean time to repair
8. Mean logistics delay time
9. Mean time between failures
10. Mean time to failure
11. Performance based logistics/contracting/maintenance
12. Performance measure
13. Performance criteria
14. Material based contracting
33
Conclusion
1. Variance is a simple, yet accurate metric to gauge
the reliability uncertainty
2. Estimating the reliability variance for series,
parallel and mixed series-parallel systems
3. PBC aims to guarantee the system performance
while lowering the cost of ownership
4. PBC incentivizes the OEM/3PL to maximize the
profit by optimizing the development, production
and logistics delivery.
34
References
Reliability Estimation
1.
2.
3.
4.
5.
D. W. Coit, “System reliability confidence intervals for complex systems with estimated component
reliability,” IEEE Transactions on Reliability, vol. 46, no. 4, 1997, pp. 487-493.
J. E. Ramirez-Marquez, and W. Jiang, “An improved confidence bounds for system reliability,” IEEE
Transactions on Reliability, vol. 55, no. 1, 2006, pp. 26-36.
E. Borgonov, “A new uncertainty measure”, Reliability Engineering and System Safety, vo;. 92, pp. 771784, 2007.
T. Jin, D. Coit, "Unbiased variance estimates for system reliability estimate using block
decompositions," IEEE Transactions on Reliability , vol. 57, 2008, pp.458-464.
H. Guo, T. Jin, A. Mettas, “Designing reliability demonstration test for one-shot systems under zero
component failures," IEEE Transactions on Reliability , vol. 60, no. 1, 2011, pp. 286-294
Availability Estimation
1.
2.
3.
4.
5.
6.
7.
Huang, H.-Z., H.J. Liu, D.N.P. Murthy. 2007. Optimal reliability, warranty and price for new products.
IIE Transactions, vol. 39, no. 8, pp. 819-827.
Kang, K., M. McDonald. 2010. Impact of logistics on readiness and life cycle cost: a design of
experiments approach, Proceedings of Winter Simulation Conference. pp. 1336-1346.
Kim, S.H., M.A. Cohen, S. Netessine. 2007. Performance contracting in after-sales service supply chains.
Management Science, vol. 53, pp. 1843-1858.
Nowicki, D., U.D. Kumar, H.J. Steudel, D. Verma. 2008. Spares provisioning under performance-based
logistics contract: profit-centric approach. The Journal of the Operational Research Society. vol. 59, no.
3, 2008, pp. 342-352.
Öner, K.B., G.P. Kiesmüller, G.J. van Houtum. 2010. Optimization of component reliability in the design
phase of capital goods. European Journal of Operational Research, vol. 205, no. 3, pp. 615-624.
T. Jin, P. Wang, “Planning performance based contracts considering reliability and uncertaint system
usage,” Journal of the Operational Research Society , 2012 (forthcoming)
Jin, T., Y. Tian, “Optimizing reliability and service parts logistics for a time-varying installed base,”
European Journal of Operational Research, vol. 218, no. 1, 2012, pp. 152-162
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For Questions
E-mail to
tj17@txstate.edu
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