在服务启动N-策略控制下具有检修策略和不同到达率的M/G/1排队分析

doi:10.15960/j.cnki.issn.1007-6093.2025.02.009

Abstract

Abstract:

Based on the background of the manufacturing system, this paper proposes an M/G/1 queuing model with maintenance strategy and different arrival rates, in which the server begins its service under the control of $N$-policy. Firstly, using the renewal process theory, total probability decomposition technique and Laplace transform, we study the transient properties of the queue length at any time $t$, and obtain the expressions of the Laplace transform of the transient queue size distribution with respect to time $t$. Then, on the basis of the transient analysis, the recursive formulas of the steady-state queue-length distribution are presented by employing L'Hospital's rule. Finally, applying the renewal reward theory, the explicit expression of the long-run expected cost per unit time of the system is obtained under a given cost structure, and numerical examples are provided to discuss the optimal control policy that the server begins its service and optimal maintenance strategy.

Key words: M/G/1 queue, maintenance strategy, service startup N-policy, queue length distribution, optimal policy

CLC Number:

O226

Fengrui LI, Yinghui TANG. Analysis of M/G/1 queue with maintenance strategy and different arrival rates under service startup N-policy control[J]. Operations Research Transactions, 2025, 29(2): 113-127.

Figures/Tables 4

References 25

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$ N $	$ \rho=0.2 $	$ \rho=0.6 $	$ \rho =0.9 $
1	76.132 47	88.749 37	75.691 30
2	48.758 01	68.096 89	69.530 15
3	37.846 70	55.992 24	66.243 28
4	32.938 70	49.176 65	$ \boxed{{\bf{64.993\ 83}}}$
5	30.769 62	45.416 99	64.994 91
6	$ \boxed{{\bf{30.061\ 81}}}$	43.483 75	65.758 61
7	30.222 71	42.699 93	67.001 17
8	30.941 61	$ \boxed{{\bf{42.676\ 70}}}$	68.556 02
9	32.040 11	43.179 49	70.322 41
10	33.408 55	44.060 01	72.237 31

$ N $	$ T_N^* $	$ F(T_N^*, N) $	$ {\left. {\frac{{{\rm \partial} F(T, N)}}{{{\rm \partial} T}}} \right\|_{T = T_N^ * }} $	$ N $	$ T_N^* $	$ F(T_N^*, N) $	$ {\left. {\frac{{{\rm \partial} F(T, N)}}{{{\rm \partial} T}}} \right\|_{T = T_N^ * }} $
1	12.47	47.18	0	9	3.98	42.89	0
2	12.29	47.11	0	10	4.34	43.75	0
3	11.72	46.88	0	11	4.80	44.87	0
4	10.06	46.24	0	12	5.35	46.18	0
5	5.54	44.65	0	13	5.96	47.63	0
6	4.09	43.05	0	14	6.61	49.21	0
7	3.76	$ \boxed{{\bf{42.37}}}$	0	15	7.31	50.87	0
8	3.77	42.39	0	16	8.03	52.61	0

[1]	Yanjun LUO, Yinghui TANG. Analysis of M/G/1 queue with single vacation and modified (p, N)-policy [J]. Operations Research Transactions, 2024, 28(4): 1-17.
[2]	LUO Le, TANG Yinghui. M/G/1 queueing system with Min(N,D,V)- policy control [J]. Operations Research Transactions, 2019, 23(2): 1-16.
[3]	WEI Yingyuan, TANG Yinghui, YU Miaomiao. Queue length distribution and numerical calculation of queueing system with delay Min(N,D)-policy [J]. Operations Research Transactions, 2016, 20(2): 23-37.

Analysis of M/G/1 queue with maintenance strategy and different arrival rates under service startup N-policy control

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