Date post: | 02-Jan-2016 |
Category: |
Documents |
Upload: | lynn-freeman |
View: | 219 times |
Download: | 0 times |
1
Stracener_EMIS 7305/5305_Spr08_01.24.08
System Reliability Analysis- Concepts and Metrics
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
EMIS 7305/5305Systems Reliability, Supportability and Availability Analysis
Systems Engineering ProgramDepartment of Engineering Management, Information and Systems
2Stracener_EMIS 7305/5305_Spr08_01.24.08
Reliability Definitions and Concepts
• Figures of merit
• Failure densities and distributions
• The reliability function
• Failure rates
• The reliability functions in terms of the failure rate
• Mean time to failure (MTTF) and mean time between failures (MTBF)
3Stracener_EMIS 7305/5305_Spr08_01.24.08
Reliability Concepts, Principles and Methodology
• Hardware
• Software
• Operator
• Service
• Product
• Production/Manufacturing Processes and Equipment
• Product and Customer Support
• Systems
4Stracener_EMIS 7305/5305_Spr08_01.24.08
What is Reliability?
• To the user of a product, reliability is problem free operation
• Reliability is a function of stressTo understand reliability, understand stress on hardware
- where its going to be used- how its going to be used- what environment it is going to be used in
To efficiently achieve reliability, rely on analytical understanding of reliability and less on understanding reliability through testing
Field ProblemsStress/Design, Parts and Workmanship
5Stracener_EMIS 7305/5305_Spr08_01.24.08
Definitions of Reliability
• Reliability is a measure of the capability of a system, equipment or component to operate without failure when in service.
• Reliability provides a quantitative statement of the chance thatan item will operate without failure for a given period oftime in the environment for which it was designed.
• In its simplest and most general form, reliability is the probability of success.
• To perform reliability calculations, reliability must first bedefined explicitly. It is not enough to say that reliability is a probability. A probability of what?
6Stracener_EMIS 7305/5305_Spr08_01.24.08
More Definitions of Reliability
• Reliability is defined as the probability that an item will perform its intended unction for a specified interval under stated conditions. In the simplest sense, reliability means how long an item (such as a machine) will perform its intended function without a breakdown.
• Reliability: the capability to operate as intended, whenever used, for as long as needed.
Reliability is performance over time, probability that something will work when you want it to.
7Stracener_EMIS 7305/5305_Spr08_01.24.08
Definitions of Reliability
• Essential elements needed to define reliability are:– What does it do?
• System, subsystem, equipment or component functions
– What is satisfactory performance?• Figures of merit @ System• Allocations &/or derived @ subsystem, equipment & component
– How long does it need to function?Life: required number of operational units (time, sorties, cycles, etc)
– What are conditions under which it operates?• Environment• Operation• Maintenance• Support
8Stracener_EMIS 7305/5305_Spr08_01.24.08
Reliability Figures of Merit
• Basic or Logistic Reliability
MTBF - Mean Time Between Failures
measure of product support requirements
• Mission Reliability
Ps or R(t) - Probability of mission success
measure of product effectiveness
9Stracener_EMIS 7305/5305_Spr08_01.24.08
10Stracener_EMIS 7305/5305_Spr08_01.24.08
Basic Reliability
• Design and developmentBasic reliability is a measure of serial reliability or logistics reliability and reflects all elements in a system
• Measures
Air Force MFHBF - Mean Flight Hours Between FailuresMFHBUM - MFHB Unscheduled Maintenance
Army MFHBE - Mean Flight Hours Between Events
Navy MFHBF - Mean Flight Hours Between FailuresMFHBMA - MFHB Maintenance Actions
Automotive IndustryNumber of defects per 100 vehicles
11Stracener_EMIS 7305/5305_Spr08_01.24.08
12Stracener_EMIS 7305/5305_Spr08_01.24.08
Mission Reliability
• Mission Reliability is defined as the probability that a system will perform its mission essential functions during a specified mission, given that all elements of the systemare in an operational state at the start of the mission.
• Measure
Ps or R(t) - Probability of mission success based on:
Mission Essential FunctionsMission Essential EquipmentMission Operating EnvironmentMission Length
13Stracener_EMIS 7305/5305_Spr08_01.24.08
Basic Elements of Reliability Modeling & Analysis
• Reliability is a probability
• Therefore a working knowledge of probability, random variables and probability distributions is required for:
- Development of reliability models
- Performing reliability analyses
• An understanding of the concepts of probability is required for design and support decisions
14Stracener_EMIS 7305/5305_Spr08_01.24.08
Reliability Humor: Statistics
“If I had only one day left to live, I would live it in my statistics class --it would seem so much longer.”
From: Statistics A Fresh ApproachDonald H. SandersMcGraw Hill, 4th Edition, 1990
15Stracener_EMIS 7305/5305_Spr08_01.24.08
Failure Density Function
associated with a continuous random variable T, the time to failure of an item, is a function f, called the probability density function, or in reliability, the failure density. The function f has the following properties:
for all values of t
and
0)t(f
1dt)t(f0
16Stracener_EMIS 7305/5305_Spr08_01.24.08
Failure Distribution Function
The failure distribution function or, the probability distributionfunction is the cumulative proportion of the population failing in time t, i.e.,
dy)y(f)tT(PtFt
0
17Stracener_EMIS 7305/5305_Spr08_01.24.08
Failure Distribution Function
The failure distribution function, F, has the followingproperties:
1. F is nondecreasing, i.e., if 0 t1 < t2 < , thenF(t1) F(t2),
2. 0 F(t) 1 for all t
3. in general, but here F(0) = 0
4.
5. P(a < T b) = F(b) - F(a)
0tFlim0t
1tFlimt
18Stracener_EMIS 7305/5305_Spr08_01.24.08
Remark
The time to failure distribution has a special nameand symbol in reliability. It is called the unreliabilityand is denoted by Q, i.e.
Q(t) = F(t) = P(T t)
19Stracener_EMIS 7305/5305_Spr08_01.24.08
Failure Densities and Distributions
Failure Density
Failure Distribution
f(t)
t
Area = P(t1 < T <t2)
F(t)
t
F(t2)
F(t1)
t2t1
P(t1 < T < t2) = F(t2) - F(t1)
1
0
0
20Stracener_EMIS 7305/5305_Spr08_01.24.08
Percentile
The 100pth percentile, 0 < p < 1, of the time to failure probability distribution function, F, is the time, say tp, within which aproportion, p, of the items has failed, i.e., tp is the value of t such that
F(tp) = P(T tp) = p
or tp = F-1(p)F(t)
p
tp
21Stracener_EMIS 7305/5305_Spr08_01.24.08
Reliability
In terms of the failure density, f, of an item, the100pth percentile, tp, is
pdttfpt
0
t
p
f(t)
0 tp
22Stracener_EMIS 7305/5305_Spr08_01.24.08
The Reliability Function
The Reliability of an item is the probability that the item willsurvive time t, given that it had not failed at time zero, when used within specified conditions, i.e.,
)tT(PtR
t
)t(F1dt)t(f
23Stracener_EMIS 7305/5305_Spr08_01.24.08
Properties of the Reliability Function
1. R is a non-increasing function, i.e., if 0 t1 < t2 < , then
R(t1) R(t2)
2. 0 R(t) 1 for all t
3. R(t) = 1 at t = 0
4. 0tRlimt
24Stracener_EMIS 7305/5305_Spr08_01.24.08
Properties of the Reliability Function
The probability of failure in a given time interval, t1
to t2, can be expressed in terms of either reliabilityor unreliability functions, i.e.,
P(t1 < T < t2) = R(t1) - R(t2)
= Q(t2) - Q(t1)
25Stracener_EMIS 7305/5305_Spr08_01.24.08
Reliability
Relationship between failure density and reliability
tRdt
dtf
26Stracener_EMIS 7305/5305_Spr08_01.24.08
Relationship Between h(t), f(t), F(t) and R(t)
Remark: The failure rate h(t) is a measure of proneness to failure as a function of age, t.
tF-1
tf
tR
tfth
27Stracener_EMIS 7305/5305_Spr08_01.24.08
Properties of the Failure Rate
The (instantaneous) failure rate, h, has the followingproperties:
1. h(t) 0 , t 0
and
2.
t
0t
dyyhlim
28Stracener_EMIS 7305/5305_Spr08_01.24.08
The Reliability Function
The reliability of an item at time t may be expressed in termsof its failure rate at time t as follows:
where h(y) is the failure rate
t
0dy)y(ht
0
edy)y(hexp)t(R
29Stracener_EMIS 7305/5305_Spr08_01.24.08
Cumulative Failure Rate
The cumulative failure rate at time t, H(t), is the cumulative number of failures at time t, divided by the cumulative time, t, i.e.,
The average failure rate of an item over an interval of time fromt1 to t2, where t1 < t2, is the number of failures occurring in the interval (t1, t2), divided by the interval length, t2 - t1
t
0
dy)y(ht
1)t(H
12
1221 tt
)t(H)t(H)t,t(H
30Stracener_EMIS 7305/5305_Spr08_01.24.08
Mean Time to Failure and Mean Time Between Failures
Mean Time to Failure (or Between Failures) MTTF (or MTBF)is the expected Time to Failure (or Between Failures)
Remarks:MTBF provides a reliability figure of merit for expected failure
free operationMTBF provides the basis for estimating the number of failures in
a given period of timeEven though an item may be discarded after failure and its mean
life characterized by MTTF, it may be meaningful tocharacterize the system reliability in terms of MTBF if thesystem is restored after item failure.
31Stracener_EMIS 7305/5305_Spr08_01.24.08
MTTF
MTTF (Mean Time to Failure) or MTBF (Mean TimeBetween Failures) may be determined from thetime to failure probability density function by useof three equivalent methods:
1. definition of MTBF2. moment generating functions3. characteristic function
32Stracener_EMIS 7305/5305_Spr08_01.24.08
Relationship Between MTTF and Failure Density
If T is the random time to failure of an item, themean time to failure, MTTF, of the item is
where f is the probability density function of timeto failure, iff this integral exists (as an improperintegral).
0
dtttfMTTFTE
33Stracener_EMIS 7305/5305_Spr08_01.24.08
Relationship Between MTTF and Reliability
0
dttRMTTFMTBF
34Stracener_EMIS 7305/5305_Spr08_01.24.08
Reliability “Bathtub Curve”
35Stracener_EMIS 7305/5305_Spr08_01.24.08
Reliability Humor