Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes

Medhat Ahmed El-Damcese; Naglaa Hassan El-Sodany

doi:doi:10.11648/j.ajtas.20150404.18

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Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes

Medhat Ahmed El-Damcese, Naglaa Hassan El-Sodany

Published in American Journal of Theoretical and Applied Statistics (Volume 4, Issue 4)

Received: 28 May 2015 Accepted: 10 June 2015 Published: 30 June 2015

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Abstract

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.

Published in	American Journal of Theoretical and Applied Statistics (Volume 4, Issue 4)
DOI	10.11648/j.ajtas.20150404.18
Page(s)	277-290
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Semi-Markov, Discrete-Time, Cold Standby, Reliability, Mean Sojourn Time, Regenerative Point Technique

References

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[6]	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.
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[11]	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.
[12]	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.
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[17]	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.
[18]	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.
[19]	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.
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APA Style

Medhat Ahmed El-Damcese, Naglaa Hassan El-Sodany. (2015). 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, 4(4), 277-290. https://doi.org/10.11648/j.ajtas.20150404.18

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ACS Style

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. Am. J. Theor. Appl. Stat. 2015, 4(4), 277-290. doi: 10.11648/j.ajtas.20150404.18

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AMA Style

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. Am J Theor Appl Stat. 2015;4(4):277-290. doi: 10.11648/j.ajtas.20150404.18

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@article{10.11648/j.ajtas.20150404.18,
  author = {Medhat Ahmed El-Damcese and Naglaa Hassan El-Sodany},
  title = {Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {4},
  number = {4},
  pages = {277-290},
  doi = {10.11648/j.ajtas.20150404.18},
  url = {https://doi.org/10.11648/j.ajtas.20150404.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20150404.18},
  abstract = {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.},
 year = {2015}
}

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TY  - JOUR
T1  - Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes
AU  - Medhat Ahmed El-Damcese
AU  - Naglaa Hassan El-Sodany
Y1  - 2015/06/30
PY  - 2015
N1  - https://doi.org/10.11648/j.ajtas.20150404.18
DO  - 10.11648/j.ajtas.20150404.18
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 277
EP  - 290
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20150404.18
AB  - 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.
VL  - 4
IS  - 4
ER  -

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Author Information

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

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APA Style

Medhat Ahmed El-Damcese, Naglaa Hassan El-Sodany. (2015). 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, 4(4), 277-290. https://doi.org/10.11648/j.ajtas.20150404.18

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ACS Style

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. Am. J. Theor. Appl. Stat. 2015, 4(4), 277-290. doi: 10.11648/j.ajtas.20150404.18

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AMA Style

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. Am J Theor Appl Stat. 2015;4(4):277-290. doi: 10.11648/j.ajtas.20150404.18

Copy | Download

@article{10.11648/j.ajtas.20150404.18,
  author = {Medhat Ahmed El-Damcese and Naglaa Hassan El-Sodany},
  title = {Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {4},
  number = {4},
  pages = {277-290},
  doi = {10.11648/j.ajtas.20150404.18},
  url = {https://doi.org/10.11648/j.ajtas.20150404.18},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20150404.18},
  abstract = {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.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Discrete Time Semi-Markov Model of a Two Non-Identical Unit Cold Standby System with Preventive Maintenance with Three Modes
AU  - Medhat Ahmed El-Damcese
AU  - Naglaa Hassan El-Sodany
Y1  - 2015/06/30
PY  - 2015
N1  - https://doi.org/10.11648/j.ajtas.20150404.18
DO  - 10.11648/j.ajtas.20150404.18
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 277
EP  - 290
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20150404.18
AB  - 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.
VL  - 4
IS  - 4
ER  -

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