IJPAM: Volume 118, No. 4 (2018)
Title
ON THE EXTENDED CHEN'S AND PHAM'S SOFTWARERELIABILITY MODELS. SOME APPLICATIONS
Authors
Nikolay Pavlov, Anton Iliev,Asen Rahnev, Nikolay Kyurkchiev
Faculty of Mathematics and Informatics
University of Plovdiv Paisii Hilendarski
24, Tzar Asen Str., 4000 Plovdiv, BULGARIA
Abstract
The Hausdorff approximation of the impulse function by sigmoidal functions based on the extended Chen's and Pham's cumulative functions are studied and an expression for the error of the best approximation is found. The received results are of independent significance in the study of issues related to neural networks and impulse technics. Using programming environment Mathematica we give results of many numerical examples which confirm the theory presented here. We give also real examples with data provided in [21] using extended Chen's software reliability model and extended Pham's deterministic software reliability model. Dataset included [22] Year 2000 compatibility modifications, operating system upgrade, and signaling message processing. Some direct comparisons are made.History
Received: March 2, 2018
Revised: May 12, 2018
Published: May 23, 2018
AMS Classification, Key Words
AMS Subject Classification: 41A46
Key Words and Phrases: three-parameters Chen's cumulative function (3Ccdf), four parameters Pham's cumulative function (4Pcdf), impulse function
, Hausdorff approximation, upper and lower bounds
Download Section
Download paper from here.You will need Adobe Acrobat reader. For more information and free download of the reader, see the Adobe Acrobat website.
Bibliography
- 18
- S. Yamada, Software Reliability Modeling: Fundamentals and Applications, Springer (2014).
- 19
- S. Yamada, Y. Tamura, OSS Reliability Measurement and Assessment, In: Springer Series in Reliability Engineering (H. Pham, Ed.), Springer International Publishing Switzerland (2016).
- 20
- H. Pham, System Software Reliability, In: Springer Series in Reliability Engineering, Springer-Verlag London Limited (2006).
- 21
- D.R. Jeske, X. Zhang, Some successful approaches to software reliability modeling in industry, J. Syst. Softw., 74 (2005), 85-99.
- 22
- K. Song, H. Pham, A Software Reliability Model with a Weibull Fault Detection Rate Function Subject to Operating Environments, Appl. Sci., 7 (2017), 983; doi:10.3390/app7100983, 16 pp.
- 23
- 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).
- 24
- 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.
- 25
- K. Ohishi, H. Okamura, T. Dohi, Gompertz software reliability model: Estimation algorithm and empirical validation, J. of Systems and Software, 82 (2009), 535-543.
- 1
- D. Satoh, A discrete Gompertz equation and a software reliability growth model, IEICE Trans. Inform. Syst., E83-D (2000), 1508-1513.
- 2
- D. Satoh, S. Yamada, Discrete equations and software reliability growth models, in: Proc. 12th Int. Symp. on Software Reliab. and Eng., (2001), 176-184.
- 3
- S. Yamada, A stochastic software reliability growth model with Gompertz curve, Trans. IPSJ, 33 (1992), 964-969. (in Japanese)
- 4
- 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, 1 (2017).
- 5
- W. Shaw, I. Buckley, The alchemy of probability distributions: Beyond Gram-Charlier expansions and a skew-kurtotic-normal distribution from a rank transmutation map, (2009), Research report.
- 6
- M. Khan, Transmuted generalized inverted exponential distribution with application to reliability data, Thailand Statistician, 16 (2018), 14-25.
- 7
- A. Abouammd, A. Alshingiti, Reliability estimation of generalized inverted exponential distribution, J. Stat. Comput. Simul., 79 (2009), 1301-1315.
- 8
- I. Ellatal, Transmuted generalized inverted exponential distribution, Econom. Qual. Control, 28 (2014), 125-133.
- 9
- N. Kyurkchiev, S. Markov, Sigmoid functions: Some Approximation and Modelling Aspects, LAP LAMBERT Academic Publishing, Saarbrucken (2015), ISBN 978-3-659-76045-7.
- 10
- A. Iliev, N. Kyurkchiev, S. Markov, A note on the new activation function of Gompertz type, Biomath Communications, 4 (2017).
- 11
- 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.
- 12
- E. P. Virene, Reliability growth and its upper limit, in: Proc. 1968, Annual Symp. on Realib., (1968), 265-270.
- 13
- 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 (2010), 34-43.
- 14
- 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.
- 15
- S. Yamada, M. Ohba, S. Osaki, S-shaped reliability growth modeling for software error detection, IEEE Trans, Reliab., R-32 (1983), 475-478.
- 16
- S. Yamada, S. Osaki, Software reliability growth modeling: Models and Applications, IEEE Transaction on Software Engineering, SE-11 (1985), 1431-1437.
- 17
- 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 (2018), 43-57.
- 26
- 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 (2018), 599-611.
- 27
- 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 (2018), 871-882.
- 28
- 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 (2018), 949-956.
- 29
- 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 (2018), 484-487.
- 30
- A. L. Goel, Software reliability models: Assumptions, limitations and applicability, IEEE Trans. Software Eng. SE-11 (1985), 1411-1423.
- 31
- J. D. Musa, Software Reliability Data, DACS, RADC, New York (1980).
- 32
- N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Transmuted inverse exponential software reliability model, Int. J. of Latest Research in Engineering and Technology, 4 (2018) (accepted).
- 33
- N. Kyurkchiev, A. Iliev and S. Markov, Some techniques for recurrence generating of activation functions, LAP LAMBERT Academic Publishing (2017), ISBN 978-3-330-33143-3.
- 34
- V. Kyurkchiev, N. Kyurkchiev, A family of recurrence generated functions based on Half-hyperbolic tangent activation functions, Biomedical Statistics and Informatics, 2 (2017), 87-94.
- 35
- N. Guliyev, V. Ismailov, A single hidden layer feedforward network with only one neuron in the hidden layer san approximate any univariate function, Neural Computation, 28 (2016), 1289-–1304.
- 36
- D. Costarelli, R. Spigler, Approximation results for neural network operators activated by sigmoidal functions, Neural Networks, 44 (2013), 101-–106.
- 37
- D. Costarelli, G. Vinti, Pointwise and uniform approximation by multivariate neural network operators of the max-product type, Neural Networks, (2016), doi:10.1016/j.neunet.2016.06.002
- 38
- D. Costarelli, R. Spigler, Solving numerically nonlinear systems of balance laws by multivariate sigmoidal functions approximation, Computational and Applied Mathematics, (2016), doi:10.1007/s40314-016-0334-8
- 39
- D. Costarelli, G. Vinti, Convergence for a family of neural network operators in Orlicz spaces, Mathematische Nachrichten, (2016), doi: 10.1002/mana.20160006
- 40
- N. Kyurkchiev, A. Andreev, Approximation and antenna and filter synthesis: Some moduli in programming environment Mathematica, LAP LAMBERT Academic Publishing, Saarbrucken (2014), ISBN 978-3-659-53322-8.
- 41
- N. Kyurkchiev, Bl. Sendov, Approximation of a class of functions by algebraic polynomials with respect to Hausdorff distance, Ann. Univ. Sofia, Fac. Math., 67 (1975), 573-579. (in Bulgarian)
- 42
- N. Kyurkchiev, S. Markov, On the Hausdorff distance between the Heaviside step function and Verhulst logistic function, J. Math. Chem., 54 (2016), 109-119.
- 43
- A. Andreev, N. Kyurkchiev, Approximation of some impulse functions - implementation in programming environment MATHEMATICA, Proceedings of the 43 Spring Conference of the Union of Bulgarian Mathematicians, Borovetz, April 2-6, (2014), 111-117.
- 44
- N. Kyurkchiev, S. Markov, On the numerical approximation of the "cross" set, Ann. Univ. Sofia, Fac. Math., 66 (1974), 19-25. (in Bulgarian)
- 45
- N. Kyurkchiev, A. Andreev, Hausdorff approximation of functions different from zero at one point - implementation in programming environment MATHEMATICA, Serdica J. of Computing, 7 (2013), 135-142.
- 46
- N. Kyurkchiev, A. Andreev, Synthesis of slot aerial grids with Hausdorff-type directive patterns - implementation in programming environment Mathematica, C.R. Acad. Bulgare Sci., 66 (2013), 1521-1528.
- 47
- N. Kyurkchiev, Synthesis of slot aerial grids with Hausdorff type directive patterns, PhD Thesis, Department of Radio-Electronics, VMEI, Sofia (1979). (in Bulgarian)
- 48
- Bl. Sendov, H. Schinev, N. Kjurkchiev, Hausdorff-synthesis of aerial grids in scanning the directive diagram, Electropromishlenost i Priboroostroene, 16 (1981), 203-205. (in Bulgarian)
- 49
- H. Schinev, N. Kjurkchiev, M. Gachev, Experimental investigations of slot aerial grids with Hausdorff type directive patterns, Electropromishlenost i Priboroostroene, 14 (1979), 223-224. (in Bulgarian)
- 50
- H. Shinev, N. Kyurkchiev, M. Gachev, S. Markov, Application of a class of polynomials of best approximation to linear antenna array synthesis, Izv. VMEI, Sofia, 34 (1975), 1-6. (in Bulgarian)
- 51
- A. Golev, T. Djamiykov, N. Kyurkchiev, Sigmoidal functions in antenna-feeder technique, Int. J. of Pure and Appl. Math., 116 (2017), 1081-1092.
- 52
- N. Kyurkchiev, A new class activation functions with application in the theory of impulse technics, Journal of Mathematical Sciences and Modelling (2018), ID 421392. (accepted)
- 53
- Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Stat. and Prob. Letters, 49 (2000), 155-161.
- 54
- M. Xie, Y. Tang, T. Goh, A modified Weibull extension with bathtub-shaped failure rate function, Reliability Eng. and System Safety, 76 (2002), 279-285.
- 55
- M. Khan, A. Sharma, Generalized order statistics from Chen distribution and its characterization, J. of Stat. Appl. and Prob., 5 (2016), 123-128.
- 56
- S. Dey, D. Kumar, P. Ramos, F. Louzada, Exponentiated Chen distribution: Properties and Estimations, Comm. in Stat.-Simulation and Computation, (2017), 1-22.
- 57
- 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 (2015), 4049-4069.
- 58
- B. Sendov, Hausdorff Approximations, Kluwer, Boston (1990).
- 59
- A. Pandey, N. Goyal, Early Software Reliability Prediction. A Fuzzy Logic Approach, In: Studies in Fuzziness and Soft Computing (J. Kacprzyk, Ed.), vol. 303, Springer, London (2013).
- 60
- N. D. Singpurwalla, S. P. Wilson, Statistical Methods in Software Engineering. Reliability and Risk, In: Springer Series in Statistics (P. Bickel, Adv.), Springer, New York (1999).
- 61
- M. Bisi, N. Goyal, Artificial Neural Network for Software Reliability Prediction, In: Performability Engineering Series (K. Misra and J. Andrews, Eds.), John Wiley & Sons, Inc., New Jersey (2017).
- 62
- P. K. Kapur, H. Pham, A. Gupta, P. C. Jha, Software Reliability Assessment with OR Applications, In: Springer Series in Reliability Engineering, Springer-Verlag, London (2011).
- 63
- P. Karup, R. Garg, S. Kumar, Contributions to Hardware and Software Reliability, World Scientific, London (1999).
- 64
- M. Lyu (Ed. in Chief), Handbook of Software Reliability Engineering, IEEE Computer Society Press, The McGraw-Hill Companies, Los Alamitos (1996).
- 65
- M. Ohba, Software reliability analysis models, IBM J. Research and Development, 21 (1984).
How to Cite?
DOI: 10.12732/ijpam.v118i4.18 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: 118
Issue: 4
Pages: 1053 -
Google Scholar; DOI (International DOI Foundation); WorldCAT.
This work is licensed under the Creative Commons Attribution International License (CC BY).