Run related probability function and their application to start-up demonstration tests

- Journal title : Journal of the Korean Data and Information Science Society
- Volume 27, Issue 4, 2016, pp.1067-1074
- Publisher : Korean Data and Information Science Society
- DOI : 10.7465/jkdi.2016.27.4.1067

Title & Authors

Run related probability function and their application to start-up demonstration tests

Bi, Yi-Ming; Oh, Jung-Taek; Cho, Gyo-Young;

Bi, Yi-Ming; Oh, Jung-Taek; Cho, Gyo-Young;

Abstract

A start-up demonstration test is a mechanism that is usually used to determine the reliability of equipment, for example water pumps, car batteries and power generators. The simplest and oldest start-up demonstration tests are called CS (consecutive successes) which have been studied by Hahn and Gage (1983), Viveros and Balakrishnan (1993).At first Hahn and Gage (1983) discussed the start-up demonstration test. I was based on i.i.d (independently and identically distributed) binary outcomes with the specified number of consecutive successful start-ups. Oh (2016) studied CSNCF (consecutive successful, but not consecutive failures). In this paper, we investigated the CS and CSNCF models, also their applications to start-up demonstration tests. The numerical results showed that the expectations and variances of the total number of attempted start-ups until the acceptance of the unit are gradually increasing in all of the specified number of successes as the p (probability of a successful start-up in an single trial) decreases from 0.99 to 0.90. The difference between means of the CS mode and CSNCF model is small, but variances of the CS and CSNCF are big.

Keywords

CS;CSNCF;geometric distribution of order k;runs;start-updemonstration tests;

Language

English

References

1.

Balakrishnan, N., Balasubramanian, K. and Viveros, R. (1995). Start-up demonstration tests under corre-lation and corrective action. Naval Research Logistics, 42, 1271-1276.

2.

Balakrishnan, N., Mohanty, S. G. and Aki, S. (1997). Start-up demonstration tests under Markov dependence model with corrective actions. Annals of the Institute of Statistical Mathematics, 49, 155-169.

3.

Feller, W. (1968). An introduction to probability theory and Its applications, 3rd Ed., John Wiley & Sons, New York.

4.

Hahn, G. J. and Gage, J. B. (1983). Evaluation of a start-up demonstration test. Journal of Quality Technology, 15, 103-106.

5.

Kolev, N. W. and Minkova, L. D. (1997). Discrete distributions related to success runs of length K in a multi-state Markov-chain. Communications in Statistics-Theory and Methods, 26, 1031-1049.

6.

Muselli, M. (1996). Simple expressions for success run distributions in Bernoulli trials. Statistics & Proba-bility Letters, 31, 121-128.

7.

Oh, J. T. (2016). Modified geometric distribution of order k. Preprint.

8.

Philippou, A. N. and Muwafi, A. A. (1982). Waiting for the k-th consecutive success and the Fibonacci sequence of order k. The Fibonacci Quarterly, 20, 28-32.

9.

Todhunter, I. (1865). A history of the mathematical theory of probability from the time of Pascal to that of laplace, Macmillan, London, Reprinted by Chelsea Publishing Company, New York, 1949.

10.

Uppuluri, V. R. R. and Patil, S. A. (1983).Waiting times and generalized Fibonacci sequences. The Fibonacci Quarterly, 21, 242-249.