1. 1 Software Testing and Quality Assurance Lecture 36 – Software Quality Assurance

2. 2 Lecture Objectives Markov Models Software Reliability Growth

3. 3 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.

4. 4 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.

5. 5 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)

6. 6 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.

7. 7 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

8. 8 Markov Models – Example

9. 9 Transition Matrix for the two State Model We can represent the probabilities for the simple two model as a matrix.

10. 10 Transition Matrix for the two State Model Transition Matrix for the Markov Chain

11. 11 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.

12. 12 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.

13. 13 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.

14. 14 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

15. 15 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.

16. 16 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.

17. 17 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.

18. 18 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.

19. 19 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.

20. 20 Key points Markov Models – stochastic process Transition Matrix for the two State Model Software Reliability Growth Basic Execution Time Model