Time Control Chart Some IFR Models

Transcription

1 Time Control Chart Some IFR Models R.R.L.Kantam 1 and M.S.Ravi Kumar 2 Department of Statistics, Acharya Nagarjuna University, Gunturr , Andhra Pradesh, India. 1 kantam.rrl@gmail.com ; 2 msrk.raama@gmail.com Abstract: The time to failure of a product is considered as a quality characteristic to judge the quality of the product. Control limits are evaluated for the time to failure. Life time data falling below the lower control limits/ above the upper control limits/ between the control limits are made use of to conclude whether the life pattern of the product is performing satisfactorily/ favorably/ adversely. Comparative study of our limits with those existing in literature is made. Key Words: Control Charts, Gamma Distribution, Half Logistic Distribution, IFR Models. 1. Introduction Life time data generally contain the failure times of sample products or interfailure times or number of failures experienced in a given time. Assuming a suitable probability model the reliability of the product is computed and the quality with respect to reliability would be assessed. From a different point of view if the specific life time data contain, times between failures, also called inter failure times, probability limits for such a data can be constructed in a parametric approach. Taking central line at the median of the distribution of the data, the probability limits as usual control limits we can think of a control chart for the data. Points above the upper control limit of such a data would be an encouraging characteristic of the product because they lead to a large gap between successive failures so that the uptime of the product is large. Hence the product is preferable. That is detection of out of control above the UCL is desirable and its causes are to preserved or encouraged. Similarly detection of out of control below the LCL results in shorter gaps between successive failures. The assignable causes for this detection are to be minimized or eliminated. Points within the control limits indicate a smooth failure phenomenon. Thus such a set of control limits would be helpful in assessing the quality of the product based on inter failure time data. The control chart may be accordingly named as Time Control Chart. 1

2 Xie et al. (2002) have suggested time control charts for failure data modeled by the well known exponential distribution. In this context a Non Homogeneous Poisson Process with its mean value function as generated by the exponential distribution, Half Logistic Distribution and other standard life testing models available in the literature. The references relevant to the approach of distribution function for the mean value function are studied by a number of authors. These are Kantam and Sriram (2001), Kantam and Subba Rao (2009), Kantam et al.(2009), Satya Prasad and Kantam (2009a), Satya Prasad and Kantam (2009b), Naga Raju et al.(2010), Satya Prasad and Kantam (2010), Satya Prasad and Nagaraju et al. (2010), Kantam and Priya (2011a), Kantam and Priya (2011b), Krishna Mohan et al.(2011), Ramchand H Rao et al. (2011), Satya Prasad and Srinivasa Rao et al.(2011), Srinivasa Rao et al.(2011), Satya Prasad et al.(2011a), Satya Prasad et al.(2011b), Satya Prasad et al.(2011c), Kantam et al.(2012a), Kantam et al.(2012b). Based on this research article and the published paper- of that of are very much useful for supplementing to the literature. In our paper we adopt the principle of Xie et al. (2002) to develop time control chart in the cases of Gamma distribution, and Half Logistic distribution, which are Inter Failure Rate (IFR) models. The rest of the paper is organised as follows. Section-2 deals with general theory of Xie et al. (2002) and its application to the above two IFR models. Section-3 consists of evaluation of control limits of the time control chart for the above two models with a live data along with the respective goodness of fit criteria and a comparison of the two models with respect to early detection criteria. 2. Time Control Chart Let F(X) be the cumulative distribution function of a continuous positive valued random variable, f(x) be its probability density function. Further let this distribution be a scaled density. If the random variable is taken as representing inter failure time of a device (time lapse between successive failures), a control chart for such a data would be based on probability limits (on par with the probability content chosen by Schewart for the classical control charts) of the times between failure random variable say x. These limits and the central line are respectively the solutions of the following equations taking equitailed probabilities. 2

3 F(X)= (2.1) F(X)=0.5 (2.2) F(X)= (2.3) Let X U, X C, X L be respectively the solutions of equations (2.1), (2.2) and (2.3) in the standard form. i.e., X U =F -1 ( ) (2.4) X C =F -1 (0.5) (2.5) X L =F -1 ( ) (2.6) The graph between the serial number of the failure and corresponding inter failure time together with three parallel lines to the horizontal axis at X U, X C, X L is the time control chart. In our study we considered two IFR life testing models namely Gamma distribution, and Half Logistic distribution whose respective CDF s in standard form are given below. Gamma Distribution: F 1 (X)= Half Logistic Distribution: x 0 e x x 1 ( ) dx, x>0, è>0 (2.7) 1 e F 2 (X) = 1 e x x, x>0 (2.8) The parameter è in the above model (2.7) is called its shape parameter. We take è= 2 in the present paper. For Gamma distribution the percentiles X U,X C, X L are obtained from the specially evaluated tables of Kantam and Sriram(2001). For HLD these are simple inversion of its distribution function. The following are the evaluated percentiles. X U X C X L Gamma(2) Half Logistic

4 In order to use the above percentiles for scaled versions (containing the scale parameter) of the two models, the respective density functions would be as follows. Gamma(2): 1 f 1 (X)= xe x / (2.9) t / 2e HLD: f 2 (X)= t / 2 (1 e ) (2.10) We know that the expected values of these two densities are E 1 (X)=2ó (2.11) E 2 (X)= ó ln(4) (2.12) Therefore in these two models the population mean (process average) is a constant multiplier of the scale parameter. Accordingly an unknown scale parameter requires its estimation as given from the following respective unbiased estimators, which are highly efficient. Gamma (2): x the UMVUE 2 Half Logistic Distribution: x the moment estimator with an efficiency of ln(4) With the estimated scale parameters, our control limits can be demonstrated for an example in Section Illustration We use the notation t-chart for the time control chart, and apply our results of Section-2 through an example. The following Table-1 shows that time between failures in hours of a device that experienced 30 failures (Xie et al. (2002)). 4

5 Table-1:- Time between failures of a component Failure Number Time between failure(h) Failure Number Time between failure(h) Failure Number Time between Failure(h) Taking the confidence coefficient as the control limits under the two models of Section-2 are as follows: Gamma (2): T U =X U T C =X C T L =X L Where is given by 2 x from the data and X U,X C,X L are the table values of Section-2. We have = X U =8.9 X C = X L =

6 Therefore T U = T C = and T L = HLD: T U =X U T C =X C T L =X L where is given by Section-2. We have Therefore = X U = X C = X L = T U = T C = T L = x ln(4) from the data and X U, X C,X L are the table values of Xie et al. (2002) have used the well known exponential model for the above data and got the control limits as T U = ln( / 2) =66.1 T C = ln(0.5) =6.9 and T L = ln(1 / 2) =

7 Thus we have control limits for the t-chart adopting three different models- two suggested by us and one suggested by Xie et al. (2002) presented in the following table. t-chart constants T U T C T L r Models Exponential Gamma HLD The last column in the above table is the correlation coefficient between the sample quantiles and population quantiles and can be used as a measure of closeness of the sample data to the respective distributions. These indicate all the three models are significantly equally closer to their respective distributions. t-charts drawn using the limits of the three models reveal the following results. Chart charac- Count between Count above Count below Failure number -teristics the limits the UCL the LCL with the 1 st out of control Models signal Exponential (above UCL) Gamma (below LCL) HLD (above UCL) The nature of a Time Control Chart is such that an out of control signal below LCL is to be cautiously investigated because such a signal indicates less inter failure time inturn more frequent failures. Where as on the other hand an out of control signal above the UCL is also to be investigated for maintaining the assignable causes for that signal, because such a signal results in more uptime of the device and hence less frequent failures the points within the control limits indicate a smooth life pattern of the device. 7

8 Among these, any point below the LCL is a real alarm asking for rectification in the process. Hence signals with points below LCL are more useful for an improvement in the quality of the product. Also the earlier such a signal the better for the designer for the rectification. In the present example one of our models namely Gamma distribution and the time control chart based on it has detected an alarm signal where such a signal is not identified by exponential nor by Half Logistic time control charts. This is an indication that the gamma based time control chart may have less average run length compared to the other two charts. We therefore conclude that one of our proposed time control charts is likely to exhibit preferable ARL. The work in this direction is in progress by the authors. References: [1] Kantam, R.R.L., and Priya, M.Ch. (2011a). Time Control Charts Using Order Statistics, Inter Stat, September. [2] Kantam, R.R.L., and Priya M.Ch. (2011b). Software Reliability Growth Model Based on Truncated Distributions, Calcutta Statistical Association Bulletin, 63, Nos , [3] Kantam, R.R.L., Ramakrishna, V., and Ravikumar, M.S. (2012a). Software Reliability Growth Model- Type-I Generalized Half Logistic Distribution, Proceedings of National Conference on Mathematical and Computational Sciences, [4] Kantam, R.R.L., Ramakrishna, V., and Ravikumar, M.S. (2012b). A New Software Reliability Growth Model, Mathematical Sciences International Research Journal, 1, No.1, [5] Kantam, R.R.L., Satya Prasad, R., and Ramakrishna, V. (2009). Software Reliability-Generalized Half Logistic Growth Model, International Journal of Systems and Technology, 2, No.2, [6] Kantam, R.R.L., and Sriram, B. (2001). Variable Control Charts Based on Gamma Distribution, IAPQR Transactions, 26, No.2, [7] Kantam, R.R.L., and Subba Rao, R. (2009). Pareto Distribution: A Software Reliability Growth Model, International Journal of Performability Engineering, 8

9 5, No.3, [8] Krishna Mohan, G., Satya Prasad, R., and Kantam, R.R.L. (2011). Software Reliability Using SPC and Weibull Order Statistics, Internatona1 Journal of Engineering Research and Applications (IJERA), 1, No.4, [9] Naga Raju, O., Satya Prasad, R., and Kantam, R.R.L. (2010). Predictability of Software Reliability with Imperfect Debugging Based on Multiple Change Point, International Journal of Advanced Research in Computer Science, 1, No.2, [10] Ramchand H Rao, K., Satya Prasad, R., and Kantam, R.R.L. (2011). International Journal of Computer Science, Engineering and Applications, 1, No.4, [11] Satya Prasad, R., and Kantam, R.R.L. (2009a). Detection of Reliable Software A Comparative Study of three Distinct Failure Situations, ANU Journal of Engineering and Technology, 1, No.1, [12] Satya Prasad, R., and Kantam, R.R.L. (2009b). A Comparative Evaluation of HL SRGM- Failure Count Data, ANU Journal of Engineering and Technology, 1, No.2, [13] Satya Prasad, R., and Kantam, R.R.L. (2010). Half Logistic Software Reliability Growth Model, International Journal of Advanced Research in Computer Science,1, No.2, [14] Satya Prasad, R., Naga Raju, O., and Kantam, R.R.L. (2010). SRGM with Imperfect Debugging by Genetic Algorithms, International Journal of Software Engineering and Applications (IJSEA), 1, No.2, [15] Satya Prasad, R., Ramchand H Rao, K., and Kantam, R.R.L. (2011a). Software Reliability with SPC, International Journal of Computer Science and Emerging Technologies, 2, No.2, [16] Satya Prasad, R., Ramchand H Rao, K., and Kantam, R.R.L. (2011b). Time Domain based Software Process Control using Weibull Mean Value Function, International Journal of Computer Applications, 18, No.3, [17] Satya Prasad, R., Ramchand H Rao, K., and Kantam, R.R.L. (2011c). Software Reliability Measuring Using Modified Maximum Likelihood Estimation and SPC, International Journal of Computer Applications, 21, No.7. [18] Satya Prasad, R., Srinivasa Rao, B., and Kantam, R.R.L. (2011). Assessing 9

10 Software Reliability using Inter Failures Time Data, International Journal of Computer Applications, 18, No.7. [19] Srinivasa Rao, B., Vara Prasad Rao, B., and Kantam, R.R.L. (2011). Software Reliability Growth Model Based on Half Logistic Distribution, Journal of Testing and Evaluation, 39, No.6, [20] Xie, M., Goh, T.N., and Ranjan, P. (2002). Some Effective Control Chart Procedures for Reliability Monitoring, Reliability Engineering and System Safety, 77,