Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes
American Journal of Theoretical and Applied Statistics
Volume 4, Issue 4, July 2015, Pages: 277-290
Received: May 28, 2015;
Accepted: Jun. 10, 2015;
Published: Jun. 30, 2015
Views 4196 Downloads 124
Medhat Ahmed El-Damcese, Mathematics Department, Faculty of Science, Tanta University, Tanta, Egypt
Naglaa Hassan El-Sodany, National Accounts Department, Central Agency for Public Mobilization and Statistics, Cairo, Egypt
This paper presents the reliability and availability measures of a two non-identical unit cold standby redundant system (unit-1 is operating, and unit-2 is cold standby) using semi-Markov process under discrete parametric Markov-Chain i.e. failure and repair times of a unit and time to PM and PM time are taken as discrete random variables assuming three different modes (normal (N) mode, partial failure (P) mode and total failure (F) mode) of each unit. The unit-1 is sent for preventive maintenance (PM) after its working for a random period of time assuming that the failure and repair times of a unit and time to PM and PM time are taken as discrete random variables having geometric distributions with different parameters. A single repairman is available with the system for PM of unit-1 and repair of both units. The system is considered in up-state if only one or two units are operative or in partial failure (P) mode. After some basic definitions and notations, we obtain various measures of system effectiveness; reliability, availability, mean time to failure, busy period of repairman due to PM of unit-1, busy period of repairman due to repair of unit-1 and unit-2 from total failure, and the expected profit function using regenerative point technique. The mathematical problem thus developed has next been solved numerically and graphically represented by the aid of Maple program.
Medhat Ahmed El-Damcese,
Naglaa Hassan El-Sodany,
Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes, American Journal of Theoretical and Applied Statistics.
Vol. 4, No. 4,
2015, pp. 277-290.
Smith, W. L.: Regenerative Stochastic Processes. Proc. Roy. Soc (GB). Series A, pp. 232:6-31, 1955.
Limnios, N., Oprisan, G.: Semi-Markov Processes and Reliability.Boston. Birkhauser, 2001.
Limnios, N., Barbu, V. S.: Semi-Markov chains and hidden semi-Markov models toward applications: their use in reliability and DNA analysis. Lecture Notes in Statistics. New York. Vol. 191, 2008.
Grabski, F.: Semi-Markov models of reliability and operation. Warsaw: IBS PAN (in Polish), 2002.
Grabski, F.: Application of semi-Markov processes in reliability. Electronic Journal. Reliability: Theory & Applications. Vol. 2, pp. 60-75, 2007.
Grabski, F., Kołowrocki, K.: Asymptotic Reliability of Multistate System with semi-Markov states of components. Proc. of the European Conference on Safety and Reliability. Safety and Reliability. Pp. 317-322, 1999.
Grabski, F.: The reliability of an object with semi-Markov failure rate. Applied Mathematics and Computation. Vol. 135, pp. 1-16, 2003.
Limnios, N., Ouhbi, B.: Nonparametric estimation of some important indicators in reliability for semi-Markov processes.Statistical Methodology.Vol. 3, pp. 341-350, 2006.
Korolyuk, V.S., Turbin, A.F.: Semi-Markov processes and their applications. Naukova Dumka (in Russian), 1976.
Limnios, N., Barbu, V.: Nonparametric estimation for discrete time semi-Markov processes with applications in reliability. Journal of Nonparametric Statististics. Vol. 18, pp. 483-498, 2006.
Gupta, R., Bhardwaj, P.: A Discrete Parametric Markov-Chain Model of a Two Non-Identical Unit Cold Standby System with Preventive-Maintenance. International Journal of Research and Reviews in Applied Sciences. Vol. 17, pp. 295-305 2013.
El-Said, K. M., El-Sherbeny, M. S.: Profit Analysis of a Two-Unit Cold Standby System with Preventive Maintenance and Random Change in Units. Journal of Mathematics and Statistics. Vol.1, pp. 71-77, 2005.
El-Damcese, M. A., Helmy, A. N.: Study of Reliability with Mixed Standby Components. Applications and Applied Mathematics: An International Journal (AAM). Vol.7, pp. 672-690, 2012.
Mokaddis , G. S., Matta, C. H.: Cost Analysis of a Two Dissimilar-Unit Cold Standby Redundant System Subject To Inspection and Random Change in Units. Journal of Mathematics and Statistics. Vol. 6, pp. 306-315, 2010.
Gupta, R., Bhardwaj, P.: A Two-Unit Standby System with Regular Repairman and Waiting Time of Skilled Repairman. Journal of Mathematical and Computational Science. Vol. 3, pp. 1115-1130, 2013.
Gupta, R., Mumtaz, S. Z.: Stochastic Analysis of a Two-Unit Cold Standby System with Maximum Repair Time and Correlated Failures and Repairs. Journal of Quality in Maintenance Engineering. Vol. 2, pp. 66-76, 1996.
Singh, S. K., Srinivasu, B.: Stochastic Analysis of a Two-Unit Cold Standby System with Preparation Time for Repair. Microelectronics Reliability. Vol. 27, pp. 55-60, 1987.
GoelL. R. and Srivastva, P.: A Two-Unit Cold Standby System with Three Modes and Correlated Failures and Repairs. Microelectronics Reliability. Vol. 31, pp. 835-840, 1991.
Malik, A.: Probabilistic Analysis of a Two-Unit Cold Standby System with Partial Failure. Ordinary and Expert Repairman. International Journal of Innovative Research and Studies. Vol. 13, pp. 195-209, 2014.
El-Said, K. M., Abd El-Hamid, R.: Comparison of Reliability Characteristics of Two Systems with Preventive Maintenance and Different Modes. Information and Management Sciences. Vol. 19, pp. 107-118, 2008.
Gupta, R., Bhardwaj, P.: A Two-Unit Standby System with Two Operative Modes of the Units and Preparation Time for Repair. Journal of Reliability and Statistical Studies. Vol. 6, pp. 87-100, 2013.
Goel, L.R., Gupta, P.: Stochastic Analysis of a Deteriorating Standby System with Three Modes and Inspection. Microelectronics Reliability. Vol. 25, pp. 285-290, 1985.
Goel, L.R., Gupta, R.: Analysis of a Two-Unit Cold Standby System with Three Modes and Imperfect Switching Device. Microelectronics Reliability. Vol. 24, pp. 425-429, 1984.
Haggag, M. Y.: Cost Analysis of Two-Dissimilar-Unit Cold Standby System With Three States and Preventive Maintenance Using Linear First Order Differential Equations. Journal of Mathematics and Statistics. Vol. 5, pp. 395-40,2009.
Sharma , V., Singh , D. Availability Analysis of a Two-Unit Standby System with Delayed Replacement Under Perfect Switching, International Journal Of Scientific & Engineering Research, Vol. 4, pp. 1405-1409,2013.
Gertsbakh, I.: Reliability theory with applications to preventive maintenance. Springer. Berlin, 2000.
El-Damcese, M. A., Temraz, N. S.: Semi-Markov Model of a Series-Parallel System Subject to Preventive Maintenance. Journal of Statistics Applications and Probability. Vol.2, pp. 307-318, 2013.