Software Testing and Quality
Assurance
Lecture 36 – Software Quality
Assurance
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Lecture Objectives

Markov Models

Software Reliability Growth
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Markov Models

Reliability block diagrams are

Useful for analyzing failure probabilities and reliability over fixed time intervals

We also need to understand

How reliability changes over time;

Especially in the presence of hardware.
Markov Chain can provide extra information about how a system’s reliability changes over time.
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Markov Models – stochastic process

Random variable are functions that

Assign a number to each outcome in the sample space of an experiment.

In Reliability Models

The number of failures at time T which is discrete, or an integer-valued; OR

The time at which the first failure occurs, which is a continuous, real-valued, random variable.
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Markov Models – Example

We are interested in Markov systems

We want to explore how the reliability of systems evolves over time.

Finite State Automaton (FSA)
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Markov Models – Example

There are two states of interest in the automation

Functioning state; and

Failed state.

The transitions in the automation are not labeled with events

Labeled with probabilities.
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Markov Models – Example

Each λ is the probability of making a transition from one state to the next

Within a specified time interval.

For the simple model of failure processes, we can interpret the states and probabilities over a time interval T
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Markov Models – Example
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Transition Matrix for the two State
Model

We can represent the probabilities for the simple two model as a matrix.
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Transition Matrix for the two State
Model

Transition Matrix for the Markov Chain
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Software Reliability Growth

Collection of techniques for estimating reliability as a program is being developed and tested.

Components/modules are tested and their failure rates are measured.
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Software Reliability Growth

Failure rate of a component is the number of failures observed per unit time.

Failure rates are plotted against reliability models to determine

How much more development time is required to reach acceptable levels of reliability.
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Software Reliability Growth

Measure Reliability

Through random testing.

Not specifically aimed at uncovering faults in a program, although it will certainly help to uncover any faults.

It aims at providing a random sample of inputs for the purpose of estimating a program ’s reliability.
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Software Reliability Growth

Reliability Growth Models

Assumes that as development and testing continues

The failure rates experienced should decrease.

If failure rates are decreasing then the reliability should be increasing.
If the failure rates are not decreasing then there may
Be something wrong with our testing and development process
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Software Reliability Growth –
Basic Execution Time Model

Reliability depends on:

The failure intensity λ (T) at time T and

The execution time itself.

Failure intensity is defined as

The failures experienced per unit time.
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Software Reliability Growth –
Basic Execution Time Model

Relation between failure rates and reliability is given by:

Where e is Euler ’s number (the base of the natural logarithm).
Reliability is approximated by an inverse exponential
Function of time and failure intensity.
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Software Reliability Growth –
Basic Execution Time Model

As the failure intensity λ approaches 0


Then R( T ) approaches 1; and
As the failure intensity approaches ∞

Then R( T ) approaches 0.
Even if we reached a low reliability estimate, the longer that a system is required toe execute without
Failure, the lower the reliability R ( T ) becomes.
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Software Reliability Growth –
Basic Execution Time Model

Obtaining Failure Data

Test cases are selected randomly

We can not predict what the next test case selected will be.

In reliability measurement, to get meaningful results, test cases are selected according to the patterns of usage of the program.
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Software Reliability Growth –
Basic Execution Time Model

Four ways of estimating failure intensity

The time of the failure;

The failures experienced in a specified time interval;

The time interval between failures;

The cumulative failures experienced up to a specified time.
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Key points

Markov Models – stochastic process

Transition Matrix for the two State Model

Software Reliability Growth

Basic Execution Time Model
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