修理设备可更换且修理工多重休假的机器维修模型等待时间分布函数研究

doi:10.15960/j.cnki.issn.1007-6093.2025.04.015

Abstract

Abstract:

This paper considers the waiting time distribution of an arbitrary failed machine in a machine repair system with a replaceable repair facility and repairman's multiple vacations where the operating time and the repair time of machines all follow exponential distributions. By employing the Markov Chain with absorbing time, the probabilistic properties of the phase-type distribution and the Laplace-Stieltjes transform, the specific expressions of the waiting time distribution of failed machines and its mean waiting time are all derived. Based on this, we discuss the time-dependent behavior of the system performance measures and provide numerical results of the waiting time.

Key words: machine repair system, replaceable repair facility, multiple vacations, phase type distribution, the waiting time distribution of failed machine

CLC Number:

O226
O213.2

Wenqing WU, Haiwen XU, Miaomiao YU, Kelong ZHENG. Research on the waiting time distribution of a machine maintenance model with replaceable repair facility and repairman's multiple vacations[J]. Operations Research Transactions, 2025, 29(4): 191-204.

Figures/Tables 6

References 14

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$ (i, j) $	$ P_{i, j} $	$ (i, j) $	$ P_{i, j} $	$ (i, j) $	$ P_{i, j} $	$ (i, j) $	$ P_{i, j} $
(0, 0)	1.70$ \times 10^{-4} $	(4, 2)	3.44$ \times 10^{-4} $	(8, 2)	5.35$ \times 10^{-3} $	(12, 2)	1.12$ \times 10^{-2} $
(1, 0)	1.38$ \times 10^{-4} $	(5, 0)	5.04$ \times 10^{-5} $	(9, 0)	1.23$ \times 10^{-5} $	(13, 0)	1.10$ \times 10^{-6} $
(1, 1)	4.68$ \times 10^{-4} $	(5, 1)	2.02$ \times 10^{-2} $	(9, 1)	1.38$ \times 10^{-1} $	(13, 1)	7.43$ \times 10^{-2} $
(1, 2)	1.05$ \times 10^{-5} $	(5, 2)	7.92$ \times 10^{-4} $	(9, 2)	8.09$ \times 10^{-3} $	(13, 2)	8.04$ \times 10^{-3} $
(2, 0)	1.10$ \times 10^{-4} $	(6, 0)	3.72$ \times 10^{-5} $	(10, 0)	7.72$ \times 10^{-6} $	(14, 0)	3.96$ \times 10^{-7} $
(2, 1)	1.58$ \times 10^{-3} $	(6, 1)	3.85$ \times 10^{-2} $	(10, 1)	1.60$ \times 10^{-1} $	(14, 1)	3.02$ \times 10^{-2} $
(2, 2)	4.47$ \times 10^{-5} $	(6, 2)	1.66$ \times 10^{-3} $	(10, 2)	1.07$ \times 10^{-2} $	(14, 2)	4.05$ \times 10^{-3} $
(3, 0)	8.64$ \times 10^{-5} $	(7, 0)	2.67$ \times 10^{-5} $	(11, 0)	4.52$ \times 10^{-6} $	(15, 0)	8.71$ \times 10^{-8} $
(3, 1)	4.14$ \times 10^{-3} $	(7, 1)	6.64$ \times 10^{-2} $	(11, 1)	1.57$ \times 10^{-1} $	(15, 1)	6.28$ \times 10^{-3} $
(3, 2)	1.34$ \times 10^{-4} $	(7, 2)	3.14$ \times 10^{-3} $	(11, 2)	1.20$ \times 10^{-2} $	(15, 2)	1.08$ \times 10^{-3} $
(4, 0)	6.67$ \times 10^{-5} $	(8, 0)	1.85$ \times 10^{-5} $	(12, 0)	2.39$ \times 10^{-6} $
(4, 1)	9.59$ \times 10^{-3} $	(8, 1)	1.02$ \times 10^{-1} $	(12, 1)	1.24$ \times 10^{-1} $

$ (i, j) $	$ P^-_{i, j} $	$ (i, j) $	$ P^-_{i, j} $	$ (i, j) $	$ P^-_{i, j} $	$ (i, j) $	$ P^-_{i, j} $
(0, 0)	5.02$ \times 10^{-4} $	(4, 1)	2.07$ \times 10^{-2} $	(8, 0)	2.55$ \times 10^{-5} $	(11, 2)	9.47$ \times 10^{-3} $
(1, 0)	3.79$ \times 10^{-4} $	(4, 2)	7.44$ \times 10^{-4} $	(8, 1)	1.40$ \times 10^{-1} $	(12, 0)	1.41$ \times 10^{-6} $
(1, 1)	1.29$ \times 10^{-3} $	(5, 0)	9.91$ \times 10^{-5} $	(8, 2)	7.36$ \times 10^{-3} $	(12, 1)	7.31$ \times 10^{-2} $
(1, 2)	2.88$ \times 10^{-5} $	(5, 1)	3.97$ \times 10^{-2} $	(9, 0)	1.45$ \times 10^{-5} $	(12, 2)	6.58$ \times 10^{-3} $
(2, 0)	2.81$ \times 10^{-4} $	(5, 2)	1.56$ \times 10^{-3} $	(9, 1)	1.62$ \times 10^{-1} $	(13, 0)	4.32$ \times 10^{-7} $
(2, 1)	4.04$ \times 10^{-3} $	(6, 0)	6.59$ \times 10^{-5} $	(9, 2)	9.55$ \times 10^{-3} $	(13, 1)	2.92$ \times 10^{-2} $
(2, 2)	1.14$ \times 10^{-4} $	(6, 1)	6.82$ \times 10^{-2} $	(10, 0)	7.59$ \times 10^{-6} $	(13, 2)	3.16$ \times 10^{-3} $
(3, 0)	2.04$ \times 10^{-4} $	(6, 2)	2.93$ \times 10^{-3} $	(10, 1)	1.58$ \times 10^{-1} $	(14, 0)	7.78$ \times 10^{-8} $
(3, 1)	9.77$ \times 10^{-3} $	(7, 0)	4.20$ \times 10^{-5} $	(10, 2)	1.05$ \times 10^{-2} $	(14, 1)	5.93$ \times 10^{-3} $
(3, 2)	3.16$ \times 10^{-4} $	(7, 1)	1.04$ \times 10^{-1} $	(11, 0)	3.55$ \times 10^{-6} $	(14, 2)	7.95$ \times 10^{-4} $
(4, 0)	1.44$ \times 10^{-4} $	(7, 2)	4.93$ \times 10^{-3} $	(11, 1)	1.23$ \times 10^{-1} $

[1]	Wenqing WU, Qilin KE, Yinghui TANG, Lin CHEN. Analysis of waiting time of customers in an M/M/c/m + c queueing system with customer interjections [J]. Operations Research Transactions, 2023, 27(3): 109-120.
[2]	ZHANG Yuanyuan, WU Wenqing. Analysis of a k/n(G) system with expert repairman's multiple vacations and replaceable repair facility [J]. Operations Research Transactions, 2019, 23(4): 111-123.

Research on the waiting time distribution of a machine maintenance model with replaceable repair facility and repairman's multiple vacations

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