IJPAM: Volume 120, No. 1 (2018)

Title

SOME NOTES ON THE EXTENDED BURR XII
SOFTWARE RELIABILITY MODEL

Authors

Vesselin Kyurkchiev$^1$, Anna Malinova$^2$,
Olga Rahneva$^3$ and Pavel Kyurkchiev$^4$
$^{1,2,4}$Faculty of Mathematics and Informatics
University of Plovdiv Paisii Hilendarski
24, Tzar Asen Str., 4000 Plovdiv, BULGARIA
$^{3}$Faculty of Economy and Social Sciences
University of Plovdiv Paisii Hilendarski
24, Tzar Asen Str., 4000 Plovdiv, BULGARIA

Abstract

In this paper we study the Hausdorff approximation of the Heaviside step function by extended Burr XII cumulative distribution function.

The results have independent significance in the study of issues related to debugging theory. Numerical examples, illustrating our results are presented using programming environment Mathematica.

We give also real example with data provided in Yamada and Tamura [8] for testing Apache HTTP Server Project which is developed and maintained an open-source Apache HTTP server for modern operating systems including UNIX and Windows.

History

Received: April 11, 2018
Revised: August 30, 2018
Published: September 9, 2018

AMS Classification, Key Words

AMS Subject Classification: 68N30, 41A46
Key Words and Phrases: extended Burr XII cumulative function (Bcdf), Hausdorff approximation, upper and lower bounds

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Bibliography

1
G. Cordeiro, M. Alizadeh, P. Marinho, The Type I Half-Logistic Family of Distributions, Journal of Statistical Computation and Simulation, 86 (2015), 707-728.

2
I. Ghosh, M. Bourguignon, A new extended Burr XII distribution, Australian J. of Statistics, 16 (2017), 33-39.

3
F. Hausdorff, Set Theory (2 ed.) (Chelsea Publ., New York, (1962 [1957]) (Republished by AMS-Chelsea 2005), ISBN: 978-0-821-83835-8.

4
B. Oluyende, B. Makubate, A. Fagbamigbe, P. Mdlongwa, A new Burr XII-Weibull-Logarithmic distribution for survival and lifetime data analysis: Model, Theory and Applications, Stats, 1(6) (2018), 15 pp.

5
W. Zimmer, J. Keats, F. Wang, The Burr XII distribution in reliability analysis, J. of Quality Technology, 30 (4) (1988), 386-394.

6
M. Daniyal, M. Aleem, On the mixture of Burr XII and Weibull distributions, J. Stat. Appl. Probability, 2 (2014), 251-267.

7
G. Mudholkar, D. Srivastava, Exponented Weibull Family Analyzing Bathtub Failure-Rate Data, IEEE Transaction and Reliability, 42 (1993), 299-302.

8
S. Yamada, Y. Tamura, OSS Reliability Measurement and Assessment, In: Springer Series in Reliability Engineering (H. Pham, Ed.), Springer International Publishing Switzerland (2016), 185 pp., ISBN 978-3-319-31817-2.

9
D.R. Jeske, X. Zhang, Some successful approaches to software reliability modeling in industry, J. Syst. Softw., 74 (2005), 85-99.

10
K. Song, H. Pham, A Software Reliability Model with a Weibull Fault Detection Rate Function Subject to Operating Environments, Appl. Sci., 7 (2017), doi:10.3390/app7100983, 16 pp.

11
S. Yamada, Software Reliability Modeling: Fundamentals and Applications, Springer (2014).

12
S. Yamada, Y. Tamura, OSS Reliability Measurement and Assessment, In: Springer Series in Reliability Engineering (H. Pham, Ed.), Springer International Publishing Switzerland (2016).

13
H. Pham, System Software Reliability, In: Springer Series in Reliability Engineering, Springer-Verlag London Limited (2006).

14
Advances in Degradation Modeling. Applications to reliability, survival analysis and finance (M. Nikulin, N. Limnios, N. Balakrishnan, W. Kahle and C. Huber-Carol, Editors), Birkhauser (2010).

15
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Some software reliability models: Approximation and modeling aspects, LAP LAMBERT Academic Publishing (2018), ISBN: 978-613-9-82805-0.

16
K. Ohishi, H. Okamura, T. Dohi, Gompertz software reliability model: Estimation algorithm and empirical validation, J. of Systems and Software, 82(3) (2009), 535-543.

17
D. Satoh, A discrete Gompertz equation and a software reliability growth model, IEICE Trans. Inform. Syst., E83-D(7) (2000), 1508-1513.

18
D. Satoh, S. Yamada, Discrete equations and software reliability growth models, in: Proc. 12th Int. Symp. on Software Reliab. and Eng., (2001), 176-184.

19
S. Yamada, A stochastic software reliability growth model with Gompertz curve, Trans. IPSJ, 33 (1992), 964-969. (in Japanese)

20
P. Oguntunde, A. Adejumo, E. Owoloko, On the flexibility of the transmuted inverse exponential distribution, Proc. of the World Congress on Engineering, July 5-7, 2017, London, U.K., 1 (2017).

21
W. Shaw, I. Buckley, The alchemy of probability distributions: Beyond Gram-Charlier expansions and a skew-kurtotic-normal distribution from a rank transmutation map, research report, (2009).

22
M. Khan, Transmuted generalized inverted exponential distribution with application to reliability data, Thailand Statistician, 16(1) (2018), 14-25.

23
A. Abouammd, A. Alshingiti, Reliability estimation of generalized inverted exponential distribution, J. Stat. Comput. Simul., 79(11) (2009), 1301-1315.

24
I. Ellatal, Transmuted generalized inverted exponential distribution, Econom. Qual. Control, 28(2) (2014), 125-133.

25
E. P. Virene, Reliability growth and its upper limit, in: Proc. 1968, Annual Symp. on Realib., (1968), 265-270.

26
S. Rafi, S. Akthar, Software Reliability Growth Model with Gompertz TEF and Optimal Release Time Determination by Improving the Test Efficiency, Int. J. of Comput. Applications, 7(11) (2010), 34-43.

27
F. Serdio, E. Lughofer, K. Pichler, T. Buchegger, H. Efendic, Residua-based fault detection using soft computing techniques for condition monitoring at rolling mills, Information Sciences, 259 (2014), 304-320.

28
S. Yamada, M. Ohba, S. Osaki, S-shaped reliability growth modeling for software error detection, IEEE Trans, Reliab., R-32 (1983), 475-478.

29
S. Yamada, S. Osaki, Software reliability growth modeling: Models and Applications, IEEE Transaction on Software Engineering, SE-11 (1985), 1431-1437.

30
A. Pandey, N. Goyal, Early Software Reliability Prediction. A Fuzzy Logic Approach, In: Studies in Fuzziness and Soft Computing (J. Kacprzyk, Ed.), 303, London, Springer (2013).

31
N. D. Singpurwalla, S. P. Wilson, Statistical Methods in Software Engineering. Reliability and Risk, In: Springer Series in Statistics (P. Bickel, Adv.), New York, Springer 1999.

32
M. Bisi, N. Goyal, Artificial Neural Network for Software Reliability Prediction, In: Performability Engineering Series (K. Misra and J. Andrews, Eds.), New Jersey, John Wiley & Sons Inc. (2017).

33
P. K. Kapur, H. Pham, A. Gupta, P. C. Jha, Software Reliability Assessment with OR Applications, In: Springer Series in Reliability Engineering, London, Springer-Verlag (2011).

34
P. Karup, R. Garg, S. Kumar, Contributions to Hardware and Software Reliability, London, World Scientific (1999).

35
M. Lyu (Ed. in Chief), Handbook of Software Reliability Engineering, IEEE Computer Society Press, Los Alamitos, The McGraw-Hill Companies (1996).

36
M. Ohba, Software reliability analysis models, IBM J. Research and Development, 21(4) (1984).

37
Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Stat. and Prob. Letters, 49(2) (2000), 155-161.

38
M. Xie, Tang, Y., Goh, T., A modified Weibull extension with bathtub-shaped failure rate function, Reliability Eng. and System Safety, 76(3) (2002), 279-285.

39
Y. Chaubey, R. Zhang, An extension of Chen's family of survival distributions with bathtub shape or increasing hazard rate function, Comm. in Stat.-Theory and Methods, 44(19) (2015), 4049-4069.

40
N. Pavlov, G. Spasov, A. Rahnev, N. Kyurkchiev, A new class of Gompertz-type software reliability models, International Electronic Journal of Pure and Applied Mathematics, 12(1) (2018), 43-57.

41
N. Pavlov, G. Spasov, A. Rahnev, N. Kyurkchiev, Some deterministic reliability growth curves for software error detection: Approximation and modeling aspects, International Journal of Pure and Applied Mathematics, 118(3) (2018), 599-611.

42
N. Pavlov, A. Golev, A. Rahnev, N. Kyurkchiev, A note on the Yamada-exponential software reliability model, International Journal of Pure and Applied Mathematics, 118(4) (2018), 871-882.

43
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, A Note on The "Mean Value" Software Reliability Model, International Journal of Pure and Applied Mathematics, 118(4) (2018), 949-956.

44
N. Pavlov, A. Golev, A. Rahnev, N. Kyurkchiev, A note on the generalized inverted exponential software reliability model, International Journal of Advanced Research in Computer and Communication Engineering, 7(3) (2018), 484-487.

45
A. L. Goel, Software reliability models: Assumptions, limitations and applicability, IEEE Trans. Software Eng., SE-11 (1985), 1411-1423.

46
J. D. Musa, Software Reliability Data, DACS, RADC, New York (1980).

47
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Transmuted inverse exponential software reliability model, Int. J. of Latest Research in Engineering and Technology, 4(5) (2018), 1-6.

48
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Analysis of the Chen's and Pham's Software Reliability Models, Cybernetics and Information Technologies, 18(3) (2018). (to appear)

49
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, On Some Nonstandard Software Reliability Models, Dynamic Systems and Applications, 27(4) (2018). (to appear)

50
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, On the extended Chen's and Pham's software reliability models. Some applications, Int. J. of Pure and Appl. Math., 118(4) (2018), 1053-1067.

51
S. Markov, N. Kyurkchiev, A. Iliev, A. Rahnev, A note on the Log-logistic and transmuted Log-logistic Models. Some applications, Dynamic Systems and Applications, 27(3) (2018), 593-607.

52
M. Cinque, D. Cotroneo, A. Pecchia, R. Pietrantuono, S. Russo, Debugging-Workflow-aware Software Reliability Growth Analysis, Softw. Test. Verif. Reliab., (2017), 23 pp.

53
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Some deterministic growth curves with applications to software reliability analysis, Int. J. of Pure and Appl. Math., 119(2) (2018), 357-368.

54
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Investigations of the K-stage Erlangian software reliability growth model, Int. J. of Pure and Appl. Math., 119(3) (2018), 441-449.

55
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Some Transmuted Software Reliability Models, Journal of Mathematical Sciences and Modelling, 1(2) (2018). (to appear)

56
N. Pavlov, A. Golev, A. Iliev, A. Rahnev, N. Kyurkchiev, On the Kumaraswamy-Dagum log-logistic sigmoid function with applications to population dynamics, Biomath Communications, 5(1) (2018).

57
N. Kyurkchiev, S. Markov, A family of recurrence generated sigmoidal functions based on the Log-logistic function. Some approximation aspects, Biomath Communications, 5(1) (2018).

58
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, A Note on Ohbas Inflexion S-shaped Software Reliability Growth Model, Collection of scientific works from conference ”Mathematics. Informatics. Information Technologies. Application in Education”, Pamporovo, Bulgaria, October 10-12, 2018. (to appear)

59
V. Kyurkchiev, A. Malinova, O. Rahneva, P. Kyurkchiev, On the Burr XII-Weibull Software Reliability Model, Int. J. of Pure and Appl. Math., 119(4) (2018), 639-650.

60
N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Nonstandard Models in Debugging Theory (Part 2), LAP LAMBERT Academic Publishing (2018), ISBN: 978-613-9-87794-2.

61
N. Kyurkchiev, S. Markov, Sigmoid functions: Some Approximation and Modelling Aspects, LAP LAMBERT Academic Publishing (2015), ISBN: 978-3-659-76045-7.

62
N. Kyurkchiev, A. Iliev, S. Markov, Some techniques for recurrence generating of activation functions, LAP LAMBERT Academic Publishing (2017), ISBN: 978-3-330-33143-3.

How to Cite?

DOI: 10.12732/ijpam.v120i1.11 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 120
Issue: 1
Pages: 127 - 136


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