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Some Discrete Parametric Markov–Chain System Models to Analyze Reliability

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Advances in Reliability, Failure and Risk Analysis

Part of the book series: Industrial and Applied Mathematics ((INAMA))

Abstract

The chapter deals with the development of various reliability characteristics and the inter-relations between them when the random variable denoting the lifetime of a device follows a discrete distribution. It analyzes the reliability characteristics for an n-unit series, parallel and standby system models when the failure times of the units are discrete random variables. The results are also drawn in the case when the lifetimes of the units follow geometric distributions. Many important conclusions are drawn regarding the lower/upper bounds of the number of units in the system and the failure rate of a unit. This chapter also presents the cost–benefit analysis of two identical unit warm standby repairable system models, assuming the geometric distributions of failure and repair times. A single repairman is always considered with the system to repair a failed unit. The results may also be obtained for the following two particular cases—(i) Two identical unit cold standby systems with geometric failure and repair time distributions and (ii) Two identical unit parallel systems with geometric failure and repair time distributions. The curves for MTSF and net-expected profit per unit of time in steady state are drawn to study the system behavior in respect of different parameters, and various important conclusions from the curves of these characteristics are drawn.

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Author information

Authors and Affiliations

  1. Department of Statistics, Chaudhary Charan Singh University, Meerut, Uttar Pradesh, 250005, India

    Rakesh Gupta & Shubham Gupta

  2. Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, Uttar Pradesh, 202002, India

    Irfan Ali

  3. Department of Electronics and Communication Engineering, Manipal University Jaipur, Rajasthan, India

    Shubham Gupta

Authors
  1. Rakesh Gupta
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  2. Shubham Gupta
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  3. Irfan Ali
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Corresponding author

Correspondence to Irfan Ali .

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Editors and Affiliations

  1. School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, Punjab, India

    Harish Garg

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Gupta, R., Gupta, S., Ali, I. (2023). Some Discrete Parametric Markov–Chain System Models to Analyze Reliability. In: Garg, H. (eds) Advances in Reliability, Failure and Risk Analysis. Industrial and Applied Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-19-9909-3_14

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