A Note on the Exponentiated Exponential–Poisson Software Reliability Model

Neural, Parallel, and Scientific Computations, 26, No. 3 (2018), 257-267 ISSN: 1061-5369 A NOTE ON THE EXPONENTIATED EXPONENTIAL–POISSON SOFTWARE RELIABILITY MODEL VESSELIN KYURKCHIEV1 , HRISTO KISKINOV2 , OLGA RAHNEVA3 , AND GEORGI SPASOV4 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 consider an application to the debugging theory of a class of cumulative exponentiated exponential–Poisson distribution functions introduced by Ramos, Dey, Louzada and Lachos. By this family of cumulative distribution functions we study the Hausdorff approximation of the shifted Heaviside step function. Numerical examples, illustrating our results using the programming environment Mathematica are presented. As application in the field of debugging and test theory are given examples with real data including compatibility modifications, operating system upgrade and signaling message processing from year 2000 using the new software reliability model. AMS Subject Classification: 68N30, 41A46 Key Words: exponentiated exponential–Poisson cumulative distribution function (EEPcdf), Heaviside function, Hausdorff approximation, upper and lower bounds Received: July 4, 2018 ; Accepted: September 26, 2018 ; Published: October 3, 2018. doi: 10.12732/npsc.v26i3.3 Dynamic Publishers, Inc., Acad. Publishers, Ltd. https://acadsol.eu/npsc 1. INTRODUCTION Some extensions of the well-known Poisson, Poisson–exponential, Chen, Exponentiated Chen, modified Weibull and Burr distributions can be found in [1]–[8]. 258 V. KYURKCHIEV, H. KISKINOV, O. RAHNEVA, AND G. SPASOV A software reliability model that uses the ”Gompertz–type correction” is the Exponentiated Exponential–Poisson cumulative distribution function (EEPcdf) introduced by Ramos, Dey, Louzada and Lachos in [9]:  M (t; λ; θ) =  −θt e λ 1−e −θ α − 1 eλ − 1 1−e (1) where θ > 0; λ > 0; α > 0. For other software reliability models see [10]– [29]. Without loss of generality we consider the following class of the family (1) with application to the debugging theory:  M1 (t) =  with −θt e λ 1−e −θ α − 1 eλ − 1 1−e    1 − e−θ eλ − 1 1 ; ln 1 + t0 = − ln 1 − 1 θ λ 2α (2) M1 (t0 ) = 1 . 2 (3) In this note we study the Hausdorff approximation of the shifted Heaviside step function  0, if t < t0 ,    ht0 (t) = [0, 1], if t = t0 ,    1, if t > t0 , by the family (2)–(3). Hausdorff approximation of some modeling functions can be found in [30]–[35]. Furthermore, we propose a software module (intellectual property) within the programming environment CAS Mathematica for the analysis. Numerical examples, illustrating our results are presented. As application in the field of debugging and test theory we give also real examples with data provided in [36] using the new software reliability model. The dataset includes [37] Year 2000 compatibility modifications, operating system upgrade and signaling message processing. 2. HAUSDORFF APPROXIMATION OF THE SHIFTED HEAVISIDE STEP FUNCTION Definition 1. [38] The Hausdorff distance (the H–distance) ρ(f, g) between two interval functions f, g on Ω ⊆ R, is the distance between their completed graphs F (f ) EXPONENTIAL–POISSON SOFTWARE RELIABILITY MODEL 259 and F (g) considered as closed subsets of Ω × R. More precisely, ρ(f, g) = max{ sup inf A∈F (f ) B∈F (g) ||A − B||, sup inf B∈F (g) A∈F (f ) ||A − B||}, wherein ||.|| is any norm in R2 , e. g. the maximum norm ||(t, x)|| = max{|t|, |x|}; hence the distance between the points A = (tA , xA ), B = (tB , xB ) in R2 is ||A − B|| = max{|tA − tB |, |xA − xB |}. The one–sided Hausdorff distance d between the function ht0 (t) and the function (2)–(3) satisfies the relation M1 (t0 + d) = 1 − d. (4) The following theorem gives upper and lower bounds for d. Theorem 2. Let 1 p=− , 2  αθλ 1 + eλ −1     1 − e−θ eλ − 1 1− , ln 1 + q = 1 + α−1 1 λ 2α 2 α (1 − e−θ )(eλ − 1) 1 2α r = 2.1q For the one–sided Hausdorff distance d between ht0 (t) and the function (2)–(3) for q> e1.05 2.1 the following inequalities hold: dl = ln r 1 <d< = dr . r r (5) Proof. Let us examine the function: F (d) = M1 (t0 + d) − 1 + d. (6) From F ′ (d) > 0 we conclude that the function F is increasing. Consider the function G(d) = p + qd. From the Taylor expansion we obtain G(d) − F (d) = O(d2 ). Hence G(d) approximates F (d) with d → 0 as O(d2 ) (see Fig. 1). In addition G′ (d) > 0. 1.05 Further, for q > e2.1 we have G(dl ) < 0 and G(dr ) > 0. This completes the proof of the theorem. (7) 260 V. KYURKCHIEV, H. KISKINOV, O. RAHNEVA, AND G. SPASOV Figure 1: The functions F (d) and G(d) for θ = 10; λ = 0.5; α = 3.5. The model (2)–(3) for θ = 10, λ = 0.5, α = 3.5, t0 = 0.191957 is visualized on Fig. 2. From the nonlinear equation (4) and inequalities (5) we have: d = 0.14436, dl = 0.101423, dr = 0.232102. The model (2)–(3) for θ = 10, λ = 0.05, α = 0.5, t0 = 0.0293988 is visualized on Fig. 3. From the nonlinear equation (4) and inequalities (5) we have: d = 0.120798, dl = 0.0569498, dr = 0.163195. From the above examples, it can be seen that the proven estimates in Theorem 2 for the value of the Hausdorff approximation are reliable when assessing the important characteristic - ”saturation”. This characteristic has its equal participation together with the other two characteristics - ”confidence intervals” and ”confidence bounds” in the area of the software reliability theory. We propose a software module (intellectual property) within the programming environment CAS Mathematica for the analysis of the considered family M1 (t). The module offers the following possibilities: - generation of the function under user defined values of the parameters λ, α and θ; - calculation of the H-distance d between the function ht0 (t) and the function M1 (t); - software tools for animation and visualization. EXPONENTIAL–POISSON SOFTWARE RELIABILITY MODEL Figure 2: The model (2)–(3) for θ = 8, λ = 0.5, t0 = 0.103098; H–distance d = 0.155378, dl = 0.103958, dr = 0.235337. Figure 3: The model (2)–(3) for β = 15, λ = 0.01, t0 = 0.0463767; H–distance d = 0.104496, dl = 0.0561458, dr = 0.161689. 261 262 V. KYURKCHIEV, H. KISKINOV, O. RAHNEVA, AND G. SPASOV 3. APPLICATION IN THE FIELD OF DEBUGGING AND TEST THEORY We give real examples with data provided in [36]. The operating time of the software is 167,900 days. 115 failures are detected for these days which contain 71 unique failures. Table 1 shows the failures data which are united for each of the 13 months. The dataset includes [37] Year 2000 compatibility modifications, operating system upgrade and signaling message processing. Month In- System Days System Days (Cu- Failures Cumulative dex (Days) mulative) Failures 1 961 961 7 7 2 4170 5131 3 10 3 8789 13,920 14 24 4 11,858 25,778 8 32 5 13,110 38,888 11 43 6 14,198 53,086 8 51 7 14,265 67,351 7 58 8 15,175 82,526 19 77 9 15,376 97,902 17 94 10 15,704 113,606 6 100 11 18,182 131,788 11 111 12 17,760 149,548 4 115 13 18,352 167,900 0 115 Table 1. Field failure data [36]. The fitted model M1 (t) = ω  α − 1 eλ − 1 λ 1−e −θ −θt e 1−e based on the data of Table 1 for the estimated parameters: ω = 402; θ = 0.23822; λ = 0.830748; α = 19.0736 is plotted on Fig. 4. We hope that the results will be useful for a lot of specialists in this scientific area. Remark 3. model In [29] the authors developed the following new software reliability  M (t) = ω  λ 1−e −θ −θt e  − 1 eλ − 1 1−e EXPONENTIAL–POISSON SOFTWARE RELIABILITY MODEL 263 Figure 4: The fitted model M1 (t). The model for ω = 7; θ = 0.142302; λ = 0.317432 is plotted on Fig. 5. We will explicitly note that in some cases the software reliability model M1 (t) provides better results than other much more sophistical models. ACKNOWLEDGMENTS This work has been supported by project FP17-FMI-008 of Department for Scientific Research, Paisii Hilendarski University of Plovdiv. 264 V. KYURKCHIEV, H. KISKINOV, O. RAHNEVA, AND G. SPASOV Figure 5: The fitted model M (t). REFERENCES [1] Z. Chen, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Stat. and Prob. Letters, 49 (2000), 155-161. [2] M. Xie, Y. Tang, T. 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