AEM 336: Reliability & Sampling
Prediction & Modeling
Outline
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System Reliability
Series System Reliability
Parallel System Reliability
Series-Parallel System Reliability
Review
Static Systems: Systems where failure of one
component has NO effect on the probability of
any other component failing
Dynamic Systems: Components are dependent;
failure of one component will affect the probability
of failure of another component
Review
Series System: A complex system of
independent units connected together
(interrelated) such that the entire system will fail if
any one the units fail.
Review
Parallel System: components are connected in
such a way that a redundant, or standby, part can
take over the function of a failed part to save the
system.
Review
The Product Rule: if a system has n
components, each with a reliability P1, P2,…,Pn,
the reliability of the system (Rs) is
Rs = P1 * P2 * … * Pn, where,
Rs = Prob. Of system functioning as intended
Pn = prob. Components functioning as intended
Example: 3 components – A(.92), B(.95), & C(.96)
Rs = A * B * C = (.92) * (.95) * (.96) = .839
Review
Equivalent Component Reliability:
Rs = Pc * Pc * Pc = (Pc)n
Establishing Equivalency for non-equivalent
reliabilities:
A(.92), B(.95), & C(.96)
𝑃𝑐 =
𝑛
𝑃1 ∗ 𝑃2 ∗ ⋯ ∗ 𝑃𝑛 =
3
.92 ∗ .95 ∗ .96 = .9432
Review
Unreliability (U): defined as 1-Reliability
U = 1 - Pc for a component
U = 1 - (P1 * P2 * … * Pn)
Or
U = 1 – (Pc)n
Series
Systems
Review
Series System Reliability using Failure Rate (λ):
Rs = P1 * P2 * …* Pn
Rs = e-λ1T * e-λ2T *… e-λnT
Rs = e-T(λ1 + λ2 + … + λn)
Where:
λ = failure rate of component
T = x-hour reliability of the system
Review
Example: Failure Rates:
λ1 = .002
λ2 = .001
λ3 = .0025
λ4 = .0005
∑ = .0060/ T = 100
Rs = e-T(λ1 + λ2 + … + λn) = e-100(.006) = .5488 or
Rs = e-100(.002) * e-100(.001) * e-100(.0025) * e-100(.0005) =
Rs = .8187 * .9048 * .7788 * .9512 = .5488
Calculator Tip: eˆ((-100).002)
Parallel Reliability
The reliability of a parallel (or redundant) system
MUST be determined by 1st calculating the
probability that the system or part WILL fail
(unreliability).
Rs = 1 – (U1 * U2 * …* Un)
Where: Ux is the unreliability of a component AND
Rs = 1 – (Uc)n = 1 – (1 – Pc)n
Parallel Reliability
Example: RA = .92; UA = 1 – PA = .08
RB = .95; UB = 1 – PB = .05
RC = .96; UC = 1 – PC = .04
Rs
= 1 – (1 – Pc) = 1 – (UA * UB * UC) =
= 1 – (.08 * .05 * .04) = .9998
SERIES
RA*RB*RC
83.9%
vs.
vs.
vs.
PARALLEL
1-(UA*UB*UC)
99.98%
Parallel vs. Series
2 vs. 3 vs. 4 components at Pc = .70
Parallel
Series
2
.70
.70
= .91
2 (.70)2 = .49
3
.70
.70
.70
= .973
3 (.70)3 = .34
.70
.70
4 .70
.70
= .992
4 (.70)4 = .2401
Parallel vs. Series
1.20
0.97
1.00
0.99
0.91
0.80
Parallel
0.70
0.60
Series
0.49
0.40
0.34
0.20
0.24
0.00
1
2
3
4
Series-Parallel Systems
(B) RB = .234
.9320
(A) RA = .358
UB = .766
.9520
(C) RC = .086
.9660
UC = .914
Series Part =>
Parallel Part =>
(RA)(RBC)
(RBC)
***Find unreliability of B & C ***
Must find this 1st!
Series-Parallel Systems
.107
1) RBC = 1-UBC = 1-UBUC = 1-(.766)(.914)
= 1-.700 = .300
2) RA = .358
3) RS = (RA)(RBC) = (.358)(.300) = .107
High vs. Low Level Redundancy
Parallel Systems
Parallel Components
High Level – Entire System in Parallel
Series
.7
.7
.7
.7
.7
.7
RS = 1 – {[1 – (.7*.7*.7)] * [1- (.7*.7*.7)]} = .5684
High vs. Low Level Redundancy
Low Level – Component Level CAN BE REPLACED
.7
.7
.7
.7
.7
.7
RS = [1 – (1 - .70)(1 - .70)3 = .7536
The Unreliability of each of the 3 Parallel
Parts of the System
Dynamic Systems
Series Dynamic Systems: Calculate the failure
rate of system by summing the reciprocals of the
means (MTBF) of each component
Example – 100-hr Reliability
Subsystem
MTBF
Reciprocals
1
5000
1/5000 = .0004
2
6000
1/6000 = .00016
3
4500
1/4500 = .0002
4
2200
1/2200 = .00045
5
8650
1/8650 = .000115
.00116
failures/hr
Dynamic Systems
Subsystem
MTBF
Reciprocals
1
5000
1/5000 = .0004
2
6000
1/6000 = .00016
3
4500
1/4500 = .0002
4
2200
1/2200 = .00045
5
8650
1/8650 = .000115
.00116
failures/hr
RS = e-λT = e-.00116(100) = .8905
Parallel Dynamic Systems
2 Types
1) Manual Switching
2) Electronic Switching
Assignment
1) Exam on Chapter VI Modeling & Prediction,
Tuesday, November 2
2) Assignment: Worksheet on Prediction &
Modeling Due November 2