lecture 2: combinatorial modeling cs 7040 trustworthy system design, implementation, and analysis...

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