IJPAM: Volume 119, No. 2 (2018)




Nikolay Pavlov$^1$, Anton Iliev$^2$,
Asen Rahnev$^3$, Nikolay Kyurkchiev$^4$
$^{1,2,3,4}$Faculty of Mathematics and Informatics
University of Plovdiv Paisii Hilendarski
24, Tzar Asen Str., 4000 Plovdiv, BULGARIA


The Hausdorff approximation of the shifted Heaviside function $h_{t_0}(t)$ by sigmoidal functions based on the Pham [1] and Song-Chang-Pham [2] cumulative functions is investigated and an expression for the error of the best approximation is obtained in this paper.

The results of numerical examples confirm theoretical conclusions and they are obtained using programming environment Mathematica.

We give real examples with data provided by IBM entry software package [3] using Song-Chang-Pham [2] software reliability model.


Received: May 10, 2018
Revised: June 20, 2018
Published: July 4, 2018

AMS Classification, Key Words

AMS Subject Classification: 68N30, 41A46
Key Words and Phrases: deterministic Pham's model, Song-Chang-Pham model, shifted Heaviside function $h_{t_0}(t)$, 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.


H. Pham, A new software reliability model with vtub-shaped fault-detection rate and the uncertainty of operating environments, Optimization, 63(10) (2014), 1481-1490.

K. Song, I. Chang, H. Pham, An NHPP Software Reliability Model with S-Shaped Growth Curve Subject to Random Operating Environments and Optimal Release Time, Appl. Sci., 7 (2017), 1304, doi:10.3390/app7121304

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

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

B. Sendov, Hausdorff Approximations, Boston, Kluwer (1990).

C. Stringfellow, A. A. Andrews, An empirical method for selecting software reliability growth models, Emp. Softw. Eng., 7 (2012), 319-343.

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

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

I. H. Chang, H. Pham, S. W. Lee, K. Y. Song, A testing coverage software reliability model with the uncertainty of operation environments, International Journal of Systems Science: Operations and Logistics, 1(4) (2014), 220-227.

K. Y. Song, I. H. Chang, H. Pham, A three-parameter fault-detection software reliability model with the uncertainty of operating environments, Journal of Syst. Sci. Syst. Eng., 26 (2017), 121-132.

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

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.

K. Y. Song, I. H. Chang, H. Pham, Optimal release time and sensitivity analysis using a new NHPP software reliability model with probability of fault removal subject to operating environments, Appl. Sci., (2018), pp. 26; doi: 10.3390/app8050714.

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.

D. Satoh, A discrete Gompertz equation and a software reliability growth model, IEICE Trans. Inform. Syst., E83-D(7) (20000, 1508-1513.

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

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

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).

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.

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

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

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

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

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.

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.

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

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

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

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

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.

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.

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.

M. Khan, A. Sharma, Generalized order statistics from Chen distribution and its characterization, J. of Stat. Appl. and Prob., 5(1) (2016), 123-128.

S. Dey, D. Kumar, P. Ramos, F. Louzada, Exponentiated Chen distribution: Properties and Estimations, Comm. in Stat.-Simulation and Computation, (2017), 1-22.

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.

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

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).

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).

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).

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

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

Q. Li, H. Pham, NHPP software reliability model considering the uncertainty of operating environments with imperfect debugging and testing coverage, Applied Mathematical Modelling, 51 (2017), 68-85.

J. Wang, An Imperfect Software Debugging Model Considering Irregular Fluctuation of Fault Introduction Rate, Quality Engineering, 29 (2017), 377-394.

Q. Li, H. Pham, A testing-coverage software reliability model considering fault removal efficiency and error generation, PLoS ONE, 12(7) (2017), doi:10.1371/journal.pone.0181524.

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.

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.

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.

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.

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.

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.

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)

N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, On Some Nonstandard Software Reliability Models, Compt. rend. Acad. bulg. Sci., 71 (2018). (to appear)

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.

V. Ivanov, A. Reznik, G. Succi, Comparing the reliability of software systems: A case study on mobile operating systems, Information Sciences, 423 (2018), 398-411.

How to Cite?

DOI: 10.12732/ijpam.v119i2.8 How to cite this paper?

International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 119
Issue: 2
Pages: 357 - 368

Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).