Grey-based approach for estimating software reliability under nonhomogeneous Poisson process

doi:10.23919/JSEE.2022.000038

Journal of Systems Engineering and Electronics ›› 2022, Vol. 33 ›› Issue (2): 360-369.doi: 10.23919/JSEE.2022.000038

收稿日期:2020-12-28 出版日期:2022-05-06 发布日期:2022-05-06

Grey-based approach for estimating software reliability under nonhomogeneous Poisson process

Xiaomei LIU^1,²(), Naiming XIE^1,*()

¹ College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China
² College of Science, Jiujiang University, Jiujiang 332005, China

Received:2020-12-28 Online:2022-05-06 Published:2022-05-06
Contact: Naiming XIE E-mail:liuxiaomeijju@126.com;xienaiming@nuaa.edu.cn
About author:|LIU Xiaomei was born in 1981. She received her B.S. and M.S. degrees from Nanchang University in 2003 and 2006, respectively. Now, she is a lecturer at Jiujiang University. She has been pursuing her Ph.D. degree in Nanjing University of Aeronautics and Astronautics since 2018. Her research interests include grey information modeling and reliability system modeling. E-mail: liuxiaomeijju@126.com||XIE Naiming was born in 1981. He received his B.S., M.S., and PH.D. degrees from Nanjing University of Aeronautics and Astronautics in 2002, 2005 and 2008, respectively. Now, he is a full professor from Institute of Grey System Studies, and serves as Vice-Dean of College of Economics and Management at Nanjing University of Aeronautics and Astronautics. His major research interests include grey information modelling, dynamic system modelling and data analytics in smart manufacturing. E-mail: xienaiming@nuaa.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (71671090), the Fundamental Research Funds for the Central Universities (NP2020022), the Qinglan Project of Excellent Youth or Middle-Aged Academic Leaders in Jiangsu Province.

摘要/Abstract

Abstract:

Due to the randomness and time dependence of the factors affecting software reliability, most software reliability models are treated as stochastic processes, and the non-homogeneous Poisson process (NHPP) is the most used one. However, the failure behavior of software does not follow the NHPP in a statistically rigorous manner, and the pure random method might be not enough to describe the software failure behavior. To solve these problems, this paper proposes a new integrated approach that combines stochastic process and grey system theory to describe the failure behavior of software. A grey NHPP software reliability model is put forward in a discrete form, and a grey-based approach for estimating software reliability under the NHPP is proposed as a nonlinear multi-objective programming problem. Finally, four grey NHPP software reliability models are applied to four real datasets, the dynamic R-square and predictive relative error are calculated. Comparing with the original single NHPP software reliability model, it is found that the modeling using the integrated approach has a higher prediction accuracy of software reliability. Therefore, there is the characteristics of grey uncertain information in the NHPP software reliability models, and exploiting the latent grey uncertain information might lead to more accurate software reliability estimation.

Key words: software reliability model, stochastic process, uncertainty system, non-homogeneous Poisson process, grey system theory

. [J]. Journal of Systems Engineering and Electronics, 2022, 33(2): 360-369.

Xiaomei LIU, Naiming XIE. Grey-based approach for estimating software reliability under nonhomogeneous Poisson process[J]. Journal of Systems Engineering and Electronics, 2022, 33(2): 360-369.

图/表 10

参考文献 50

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Test weeks	Defects found	Delayed S-shaped		Grey delayed S-shaped
Test weeks	Defects found	Predicted	$ {R^2} $	Predicted	$ {R^2} $
1	1	?	?	?	?
2	3	?	?	?	?
3	8	?	?	?	?
4	9	?	?	?	?
5	11	15.9835	0.9616	17.8443	0.9608
6	16	38.1267	0.9533	33.1945	0.9579
7	19	37.1073	0.9732	35.361 0	0.9740
8	25	61.9431	0.9717	44.2515	0.9720
9	27	47.486 0	0.9827	42.7439	0.9791
10	29	42.8805	0.9852	40.4875	0.9815
11	32	44.7539	0.9893	42.1101	0.9870
12	32	38.6972	0.9846	40.6727	0.9874
13	36	44.1493	0.9902	41.898 0	0.9898
14	38	44.480 0	0.9917	42.561 0	0.9913
15	39	43.3726	0.9929	42.1856	0.9918
16	39	41.4597	0.9917	41.1419	0.9907
17	41	42.7744	0.9939	41.9332	0.9930
18	42	42.8261	0.9945	42.069 0	0.9938
19	42	42.000 0	0.9941	41.6576	0.9934

Test weeks	Defects found	Inflection S-shaped		Grey inflection S-shaped
Test weeks	Defects found	Predicted	$ {R^2} $	Predicted	$ {R^2} $
1	16	?	?	?	?
2	24	?	?	?	?
3	27	?	?	?	?
4	33	?	?	?	?
5	41	66.3894	0.9159	55.1628	0.9141
6	49	85.7874	0.9220	71.1501	0.9304
7	54	87.0057	0.9501	75.5131	0.9518
8	58	86.0565	0.9650	77.0651	0.9644
9	69	114.1563	0.9561	98.8765	0.9653
10	75	116.6969	0.9674	102.6451	0.9686
11	81	119.2928	0.9747	100.8727	0.9718
12	86	118.8128	0.9803	102.5417	0.9741
13	90	116.7822	0.9841	102.5982	0.9773
14	93	113.7481	0.9859	101.2632	0.9783
15	96	112.097 0	0.9870	101.4722	0.9809
16	98	108.6041	0.9872	101.1054	0.9824
17	99	104.7258	0.9869	99.7897	0.9815
18	100	103.0323	0.9869	100.0392	0.9836
19	100	101.0997	0.9861	101.1259	0.9877
20	100	100.000 0	0.9856	100.980 0	0.9878

Test weeks	Defects found	Yamada exponential		Grey Yamada exponential
Test weeks	Defects found	Predicted	$ {R^2} $	Predicted	$ {R^2} $
1	13	?	?	?	?
2	18	?	?	?	?
3	26	?	?	?	?
4	34	?	?	?	?
5	40	78.2824	0.9792	64.1298	0.7320
6	48	100.7217	0.9837	86.3526	0.6750
7	61	138.7355	0.9625	92.0219	0.7907
8	75	154.2108	0.9310	138.5518	0.8463
9	84	159.0036	0.9534	159.2504	0.8833
10	89	153.6955	0.9760	125.8609	0.9551
11	95	147.0843	0.9798	121.6978	0.9720
12	100	142.6860	0.9824	119.1437	0.9807
13	104	136.9456	0.9818	117.2671	0.9857
14	110	136.8509	0.9845	122.8409	0.9763
15	112	129.7455	0.9809	120.7351	0.9895
16	114	125.2981	0.9777	119.5513	0.9912
17	117	123.8234	0.9778	120.8430	0.9915
18	118	120.6101	0.9742	120.1751	0.9927
19	120	119.5196	0.9743	120.8021	0.9928

Test weeks	Defects found	Yamada Rayleigh		Grey Yamada Rayleigh
Test weeks	Defects found	Predicted	$ {R^2} $	Predicted	$ {R^2} $
1	6	?	?	?	?
2	9	?	?	?	?
3	13	?	?	?	?
4	20	?	?	?	?
5	28	44.0892	0.9253	39.4451	0.6206
6	40	83.8606	0.9556	66.1143	0.7546
7	48	76.2715	0.9767	61.0024	0.8990
8	54	70.4148	0.9841	63.3178	0.9516
9	57	63.9641	0.9850	61.6199	0.9739
10	59	62.0392	0.9847	61.4499	0.9816
11	60	60.8038	0.9838	61.1707	0.9856
12	61	60.4684	0.9857	61.5068	0.9872

Grey-based approach for estimating software reliability under nonhomogeneous Poisson process

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