Supriya Maheshwari
Department of Mathematics, Noida Institute of Engineering &Technology, Greater Noida, Uttar Pradesh, India.

Masood Alam
Department of Math & IT, CPS, Sultan Qaboos University, Muscat, Oman.

Khursheed Alam
Department of Mathematics & Data Science, The A. H. Siddiqi Center for Advance Research in Applied Mathematics & Physics (CARAMP), Sharda University, India.

DOI https://doi.org/10.33889/IJMEMS.2026.11.1.018

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Abstract

This study provides a mathematical analysis of the steady-state probability distribution in a machine repair system characterized by threshold recovery, unreliable repair servers, and phase-type repairs. A series of steady-state equations shows the possibility of distinct failure states in the system. The number of broken machines can be ranging from 0 to N. The recovery policy is based on a threshold, which means that repairs only start when failures reach above a certain level T. This is the greatest way to use resources. Repair servers, on the other hand, aren't always trustworthy, which means that there is a potential of failure that takes longer to fix, making the system even less reliable. There are various processes in the repair process, and machines go through a number of random repair phases before they can be utilized again. You can find the steady-state distribution by utilizing probability balance equations. This tells us how well the system works, how likely it is to break down, and how reliable it is overall. The results help businesses figure out the best ways to keep their machines running smoothly, which helps them avoid breakdowns, make machines more available, and better plan repairs when things don't go as planned.

Keywords- Steady-state probability distribution, Machine repair system, Threshold recovery policy, Unreliable repair servers, Phase-type repairs.

Citation

Maheshwari, S., Alam, M., & Alam, K. (2026). Steady-State Probability Distribution in Machine Repair Systems with Unreliable Repair Servers and Multi-Phase Recovery. International Journal of Mathematical, Engineering and Management Sciences, 11(1), 417-438. https://doi.org/10.33889/IJMEMS.2026.11.1.018.