Vietnammese

1. REPORTING QUALITY CONTROL
SUPERVISOR: Ms. Nguyen Nhu Phong

2. Monitoring Software Reliability using Statistical
Process Control: An Ordered Statistics Approach
Hoàng Minh
Công
21200393
Nguyễn Văn
Bình
21200267
Nguyễn
Trường Minh
Members:

3. Contents
4.CONLUCSION
3.STATISTICAL PROCESS CONTROL
2.ORDERED STATISTICS
1.INTRODUCTION

4. 1. INTRODUCTION
Software Operation T/F = Reliable
Control
ORDERED STATISTICS
Goel-Okumoto model
SPC
𝑋𝐶𝐶 𝑎𝑛𝑑 𝑀𝑅𝐶𝐶

5. M(t) : the random process representing the number of
failures experienced by time t
- mean value function (the expected number of failures at
time t): μ(t)=E[m(t)]
- failure intensity function (the instantaneous rate of
change of the expected number of failures with respect to
time) : λ(t)=
𝑑𝜇(𝑡)
𝑑𝑡
- software reliability function : R(t) = e-m(t)
1. INTRODUCTION

6. X denote a continuous random variable with:
- Probability Density Function (PDF) f(x)
- Cumulative Distribution Function (CDF) F(x)
(X1 , X2 , …, Xn) denote a random sample of size n
drawn on X
- X1 < X 2 <... <Xn
=> (X1, X2, …, Xn) are collectively known as the order
statistics derived from the parent X.
2.ORDERED STATISTICS

7. The inter-failure time data
- the time lapse between every two consecutive
failures.
We can group the inter-failure time data into non
overlapping successive sub groups
- size 4 or 5
- add the failure times within each sub group.
The probability distribution of such a time lapse
= the rth ordered statistics in a subgroup of size r
= rth power of the distribution m (t)
2.ORDERED STATISTICS

8. 2.ORDERED STATISTICS
Model
Description
MODEL
Parameter
estimation
and Control
limits

9. 2.1. Model Description
Considering failure detection:
- a non homogenous Poisson process
- have an exponentially decaying rate function
- the expected number of failures observed by time t
m(t)=a(1-e-bt)
- the failure rate : λ(t)=m’(t)
2.ORDERED STATISTICS

10. Group the inter-failure time data into non overlapping
successive sub groups of size r.
The mean value function can be written as
The likelihood function L
2.ORDERED STATISTICS

11. For the given observations using equations
2.ORDERED STATISTICS

12. 2.2. Parameter estimation and Control limits
The parameters “a” and “b”
- computed by using the popular Newton Rapson
method
- obtained from Goel-Okumoto model
Table 1: Parameter estimates and their control limits of
4 and 5 order Statistics
2.ORDERED STATISTICS

13. Control limits
These limits are convert at m(tu), m(tC) and m(tL)
2.ORDERED STATISTICS

14. 3. STATISTICAL PROCESS CONTROL
Statistical process control
- the application of statistical methods
- provide the information necessary to continuously control
- or improve processes throughout the entire lifecycle of a
product
 SPC techniques
- help to locate trends, cycles,
- irregularities within the development process and
- provide clues about how well the process meets
specifications or requirements.

15. 3. STATISTICAL PROCESS CONTROL
3.1. Control Chart
 Control chart
- displays sequential process measurements relative to the
overall process average and control limits.
 The upper and lower control limits (UCL & LCL)
- establish the boundaries of normal variation for the
process being measured.
- give the boundaries within which observed fluctuations
are typical and acceptable

16. 3. STATISTICAL PROCESS CONTROL
3.2. Illustration
Table 2: Software failure data reported by Musa (1975)

17. 3. STATISTICAL PROCESS CONTROL
Table 3: 4th order cumulative faults and their m(t) successive
difference.

18. 3. STATISTICAL PROCESS CONTROL

19. 3. STATISTICAL PROCESS CONTROL
Table: 4: 5th order cumulative faults and their m(t) successive
difference

20. 3. STATISTICAL PROCESS CONTROL

21. 4.CONCLUSION
The Mean value charts of Fig 1 and 2
- out of control signals i.e. below LCL.
- we identified that failures situation is detected at an early
stages.
The early detection of software failure will improve the
software reliability.
When the control signals are below LCL
- have causes leading to significant process deterioration and
- it should be investigated.