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SRGM Using SDE

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Software Reliability Assessment with OR Applications

Part of the book series: Springer Series in Reliability Engineering ((RELIABILITY))

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Abstract

A number of NHPP based SRGM have been discussed in the previous chapters. These models treat the event of software fault detection/removal in the testing and operational phase as a counting process in discrete state space. If the size of software system is large, the number of software faults detected during the testing phase becomes large, and the change in the number of faults, which are detected and removed through debugging activities, becomes sufficiently small compared with the initial fault content at the beginning of the testing phase. Therefore, in such a situation, the software fault detection process can be well described by a stochastic process with a continuous state space. This chapter focuses on the development of stochastic differential equation based software reliability growth models to describe the stochastic process with continuous state space. Before developing any model we introduce the readers with the theoretical and mathematical background of stochastic differential equations.

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References

  1. Arnold L (1974) Stochastic differential equations. Wiley, New York

    MATH  Google Scholar 

  2. Wong E (1971) Stochastic processes in information and dynamical systems. McGraw-Hill, New York

    MATH  Google Scholar 

  3. Yamada S, Kimura M, Tanaka H, Osaki S (1994) Software reliability measurement and assessment with stochastic differential equations. IEICE Trans Fundam Electron Comput Sci E77-A(1):109–116

    Google Scholar 

  4. Yamada S, Nishigaki A, Kimura M (2003) A stochastic differential equation model for software reliability assessment and its goodness of fit. Int J Reliab Appl 4(1):1–11

    Google Scholar 

  5. Kapur PK, Anand S, Yadavalli VSS, Beichelt F (2007) A generalised software growth model using stochastic differential equation. Communication in Dependability and Quality Management Belgrade, Serbia, pp 82–96

    Google Scholar 

  6. Kapur PK, Anand S, Yamada S, Yadavalli VSS (2009) Stochastic differential equation-based flexible software reliability growth model. Math Prob Eng, Article ID 581383, 15 pages. doi:10.1155/2009/581383

  7. Tamura Y, Yamada S (2006) A flexible stochastic differential equation model in distributed development environment. Eur J Oper Res 168:143–152

    Article  MathSciNet  MATH  Google Scholar 

  8. Kapur PK, Singh VB, Anand S (2007) Effect of change-point on software reliability growth models using stochastic differential equation. In: 3rd international conference on reliability and safety engineering (INCRESE-2007), Udaipur, 7–19 Dec, 2007, pp 320–333

    Google Scholar 

  9. Kapur PK, Anand S, Yadav K (2008) Testing-domain based software reliability growth models using stochastic differential equation. In: Verma AK, Kapur PK, Ghadge SG (eds) Advances in performance and safety of complex systems. MacMillan India Ltd, New Delhi, pp 817–830

    Google Scholar 

  10. Jeske DR, Zhang X, Pham L (2005) Adjusting software failure rates that are estimated from test data. IEEE Trans Reliab 54(1):107–114

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Operational Research, University of Delhi, Delhi, 110007, India

    Prof. P. K. Kapur (Faculty of Mathematical Sciences), Dr. A. Gupta (Faculty of Mathematical Sciences) & Dr. P. C. Jha (Faculty of Mathematical Sciences)

  2. Department of Industrial and Systems Engineering, Rutgers University, Frelinghuysen Road 96, Piscataway, NJ, 08854-8018, USA

    Prof. H. Pham

Authors
  1. Prof. P. K. Kapur
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  2. Prof. H. Pham
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  3. Dr. A. Gupta
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  4. Dr. P. C. Jha
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Corresponding author

Correspondence to H. Pham .

Additional information

Exercises

  1. 1.

    Under what condition one should apply stochastic differential equation based SRGM.

  2. 2.

    Define the following

    1. a.

      Stochastic process

    2. b.

      Brownian motion

    3. c.

      \( It\hat{O} \) Integrals

  3. 3.

    Fault detection rate per remaining fault is known to have irregular fluctuations, i.e., it is represented as \( b\left( t \right) + \sigma \gamma \left( t \right) \), where σγ(t) represents a standardized Gaussian white noise. In such a case the differential equation for basic SDE model is given by

    $$ {\frac{{{\text{d}}N\left( t \right)}}{{{\text{d}}t}}} = \left\{ {b\left( t \right) + \sigma \gamma \left( t \right)} \right\}\left\{ {a - N\left( t \right)} \right\}. $$

    Derive the solution of the above differential equation. Here N(t) is a random variable which represents the number of software faults detected in the software system up to testing time t.

    1. 4.

      Derive the Exponential SDE based software reliability growth model. Give the expression of instantaneous and cumulative MTBF for the model.

    2. 5.

      Using the real life software project data given below

      1. a.

        Compute the estimates of unknown parameters of the models M1–M4, M7 and M8.

      2. b.

        Analyze and compare the results of estimation based on root mean square prediction error.

      3. c.

        Draw the graphs for the goodness of fit.

    Testing time (days)

    Cumulative failures

    Testing time (days)

    Cumulative failures

    1

    2

    12

    24

    2

    3

    13

    26

    3

    4

    14

    30

    4

    5

    15

    31

    5

    7

    16

    37

    6

    9

    17

    38

    7

    11

    18

    41

    8

    12

    19

    42

    9

    19

    20

    45

    10

    21

    21

    46

    11

    22

    		  

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© 2011 Springer-Verlag London Limited

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Kapur, P.K., Pham, H., Gupta, A., Jha, P.C. (2011). SRGM Using SDE. In: Software Reliability Assessment with OR Applications. Springer Series in Reliability Engineering. Springer, London. https://doi.org/10.1007/978-0-85729-204-9_8

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  • DOI: https://doi.org/10.1007/978-0-85729-204-9_8

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  • Publisher Name: Springer, London

  • Print ISBN: 978-0-85729-203-2

  • Online ISBN: 978-0-85729-204-9

  • eBook Packages: EngineeringEngineering (R0)

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