lecture 2: combinatorial modeling cs 7040 trustworthy system design, implementation, and analysis...
TRANSCRIPT
Lecture 2: Combinatorial Modeling
CS 7040
Trustworthy System Design, Implementation, and Analysis
Spring 2015, Dr. Rozier
Adapted from slides by WHS at UIUC
Introduction
Introduction to Combinatorial Methods
• One of the simplest validation methods utilizing analytical/numerical techniques that can be used for reliability and availability modeling.
• Requires certain assumptions…
Combinatorial Assumptions
• Component failures are independent
• For availability, repairs are independent
When these assumptions hold, simple formulas for reliability and availability exist!
Defining Reliability
Reliability
• A key to trustworthy systems is the use of reliable components and systems.– Leads to high availability!
• Reliability: The reliability of a system at time t, R(t), is the probability that system operation is proper throughout the interval [0,t].
Reliability
• Reliability: The reliability of a system at time t, R(t), is the probability that system operation is proper throughout the interval [0,t].
• Probability theory and combinatorics can be applied directly to reliability models.
Reliability
• Reliability: The reliability of a system at time t, R(t), is the probability that system operation is proper throughout the interval [0,t].
• Let X be a random variable representing the time to failure of a component. The reliability at time t is given by:
Reliability
• Reliability: The reliability of a system at time t, R(t), is the probability that system operation is proper throughout the interval [0,t].
• Unreliability can be defined similarly as:
Probability Refresher
• A random variable X is unique determined by its set of possible values, , and and the associated probability distribution (or density) function (pdf), a real-valued functiondefined for each possible value as a probability that X has the value x
Probability Refresher
• The cumulative distribution function (cdf) of the discrete random variable X is the real valued function defined for each as
Probability Refresher
• The cumulative distribution function (cdf) of the continuous random variable X is the real valued function defined for each as
PDFs and CDFs
Reliability and Unreliability
• Reliability:
• Unreliability:
Failure Rates
Failure Rate
• What is the rate at which a component fails at time t?– The probability that a component that has not yet
failed, fails in the interval
Note: We are not looking at
We are seeking
Failure Rate
Failure Rate
Failure Rate
Failure Rate
• is called the failure rate or hazard rate
Survival Function
• In addition to the reliability/hazard function we have the survival function
Survival Function
• In addition to the reliability/hazard function we have the survival function
Survival Function
• In addition to the reliability/hazard function we have the survival function
Survival and Hazard
• Hazard (or Failure) function – instantaneous failure rate at some time t.
• Survival function – the probability that the time of failure is later than some time t.
Typical Failure Rate
System Reliability
System Reliability
• While can give the reliability of a component, how do you compute the reliability of a system?
System Reliability
System failure can occur when one, all, or some of the components fail. If one makes the independent failure assumption, system failures can be computed quite simply.
The independent failure assumption states that all component failures of a system are independent, i.e., the failure of one component does not cause another component to be more or less likely to fail.
System Reliability
• Given this assumption, we can determine:– Minimum failure time of a set of components– Maximum failure time of a set of components– Probability that k of N components have failed at a
particular time t.
Maximum of n Independent Failure Times
• Let be independent component failure times. Suppose the system fails at time S if all the components fail.
• Thus,
• What is ?
Maximum of n Independent Failure Times
Maximum of n Independent Failure Times
By independence
Maximum of n Independent Failure Times
By definition
Maximum of n Independent Failure Times
Minimum of n Independent Failure Times
• Let be independent component failure times. A system fails at time S if any of the components fail.
• Thus,
• What is ?
Minimum of n Independent Failure Times
• What is ?
Minimum of n Independent Failure Times
• What is ?
• Trick: If is an event, and is the set complement such that and , then
Minimum of n Independent Failure Times
Minimum of n Independent Failure Times
By trick
Minimum of n Independent Failure Times
By independece
Minimum of n Independent Failure Times
By LOTP
Minimum of n Independent Failure Times
k of N
• Let be component failure times that have identical distributions (i.e., ). The system has failed by time S if k or more of the N components have failed by S. P[at least k components failed by time t] = P[exactly k failed OR exactly k+1 failed …] = P[exactly k failed] + P[exactly k+1 failed] …
k of N
• What is P[exactly k failed]? = P[k failed and (N – k) have not]
where is the failure distribution of each component
k of N in General
• For non-identical failure distributions, we must sum over all combinations of at least k failures.
• Let be the set of all subsets ofsuch that each element in is a set of size at least k, i.e.,
k of N in General
• The set represents all the possible failure scenarios.
• Now is given by
Component Building Blocks
• Complex systems can be analyzed hierarchically.
Example: A computer fails if both power supplies fail, or both memories fail, or if the CPU fails.
System problem is one of a minimum: the system fails when the first of three subsystems fails…
Component Building Blocks
• Power supply subsystem is a maximum: both must fail
• Memory supply subsystem is a maximum: both must fail
Summary
A system comprises N components, where the component failure times are given by the random variables . The system fails at time S with distribution if:
Condition Distribution
All components fail
One component fails
k components fail, identical distributions
k components fail, general case
Reliability Formalisms
Reliability Formalisms
• There are several popular graphical formalisms to express system reliability. The core of the solvers for these formalisms are the methods we have just examined. We will discuss a subset of these formalisms:– Reliability Block Diagrams– Fault Trees– Reliability Graphs
Reliability Formalisms
• There are several popular graphical formalisms to express system reliability. The core of the solvers for these formalisms are the methods we have just examined. We will discuss a subset of these formalisms:– Reliability Block Diagrams– Fault Trees– Reliability Graphs
There is nothing special about these formalisms except their popularity.
What is a Graphical Formalism
• A way to draw visual diagrams with formal underlying mathematical meanings.
Reliability Block Diagrams
• Blocks represent components• A system failure occurs if there is no path from
source to sink.
Reliability Block Diagrams
• Series:– System fails if any component fails
Reliability Block Diagrams
• Parallel:– System fails if all components fail
Reliability Block Diagrams
• k of N:– System fails if at least k of N components fail.
Example
A NASA satellite architecture under study is designed for high reliability. The major computer system components include the CPU system, the high-speed network for data collection and transmission, and the low-speed network for engineering and control. The satellite fails if any of the major systems fail.
There are 3 computers, and the computer system fails if 2 or more of the computers fail. Failure distribution of a computer is given by
There is a redundant (2) high-speed network, and the high-speed network system fails if both networks fail. The distribution of a high-speed network failure is given by
The low-speed network is arranged similarly, with a failure distribution of
Example
Example
Example
Example
Background: Series-Parallel Graphs
Series-Parallel Decomposition of NASA Example
Fault Trees
Fault Tree Example
Reliability Graphs
Reliability Graph Example
Solve by Conditioning
Solve by Conditioning
Conditioning Fault Trees
Reliability/Availability Point Estimates
Reliability/Availability Tables
Reliability Modeling Process
Reliability Modeling Process
Reliability Modeling Process
For next time
• Homework 1!• Due next Tuesday
• Review combinatorics