具有服务员工作休假的M/M/1生产库存系统分析

doi:10.15960/j.cnki.issn.1007-6093.2025.04.001

摘要/Abstract

摘要：

本文研究了具有服务员工作休假的M/M/1生产库存系统, 当系统中顾客数为零时服务员开始多重工作休假。基于(s, S) 库存策略, 建立系统中顾客数、库存水平、服务员状态和生产状态的四维Markov过程。利用拟生灭过程理论得出了系统稳态条件, 进而求出了系统稳态性能指标和系统费用函数, 分析了系统各项参数对性能指标和库存策略的影响, 对比分析了不同工作休假服务率下的最优库存策略和最优费用。

关键词: 生产库存系统, 工作休假, (s, S) 库存策略, 费用函数, 拟生灭过程

Abstract:

This paper studies the M/M/1 production inventory system with working vacations. The server takes multiple working vacations when there is no customers in the system. Based on the (s, S) inventory strategy, a four-dimensional Markov process of the number of customers in the system, the inventory level, the server state and the production state is established. We obtain the steady-state condition of the system by using quasi-birth-death process theory. Furthermore, we derive the steady-state performance indexes of the system and the system cost function. The effect of various parameters of the system on the performance indexes and inventory strategy is analyzed. Then the optimal inventory strategies and optimal costs with different service rates under working vacation are compared.

Key words: production inventory system, working vacation, (s, S) inventory strategy, cost function, quasi-birth-death process

中图分类号:

O226

刘文烨, 岳德权. 具有服务员工作休假的M/M/1生产库存系统分析[J]. 运筹学学报（中英文）, 2025, 29(4): 1-13.

Wenye LIU, Dequan YUE. Analysis of M/M/1 production-inventory system with working vacations[J]. Operations Research Transactions, 2025, 29(4): 1-13.

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参考文献 13

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$ \theta $	$ {E_q} $	$ {\overline E_q} $	$ {E_p} $	$ {\overline E_p} $	$ {E_{\rm inv}} $	$ \overline E_{\rm inv} $	$ E_{\rm loss} $	$ \overline E_{\rm loss} $	$ E_{\rm open} $	$ \overline E_{\rm open} $
1	5.721 8	2	2.521 3	2.991 5	3.583 3	2.876 1	1.034 5	1.008 5	0.003 80	0.000 81
10	2.39	2	2.715 8	2.991 5	2.907 4	2.876 1	1.008 2	1.008 5	0.000 84	0.000 81
100	2.004	2	2.991 2	2.991 5	2.880 3	2.876 1	1.008 4	1.008 5	0.000 82	0.000 81

	λ	1	1.5	2	2.5	3
µ_v = 0	(s, S)	(1, 3)	(1, 4)	(2, 6)	(3, 7)	(4, 9)
µ_v = 0	F(s, S)	207.540 0	297.490 7	391.520 5	489.020 9	591.700 8
µ_v = 2	(s, S)	(1, 4)	(1, 5)	(2, 6)	(2, 7)	(3, 8)
µ_v = 2	F(s, S)	159.678 3	223.674 6	298.292 4	382.614 0	475.241 1
µ_v = 4	(s, S)	(1, 5)	(1, 5)	(1, 6)	(2, 7)	(2, 8)
µ_v = 4	F(s, S)	150.020 3	194.401 5	251.594 0	315.189 7	391.918 3
µ_v = 6	(s, S)	(1, 5)	(1, 6)	(1, 6)	(2, 7)	(2, 8)
µ_v = 6	F(s, S)	148.008 1	185.119 9	230.967 9	286.133 0	349.134 8

	$ \theta $	$ 1 $	$ 3 $	$ 5 $	$ 7 $	$ 9 $
$ {\mu_v} = 0 $	$ (s, S) $	$ (4, 9) $	$ (3, 8) $	$ (3, 8) $	$ (2, 8) $	$ (3, 5) $
$ {\mu_v} = 0 $	$ {F}(s, S) $	$ 591.700\, 8 $	$ 427.272\, 5 $	$ 389.098\, 6 $	$ 371.520\, 4 $	$ 360.709\, 3 $
$ {\mu_v} = 2 $	$ (s, S) $	$ (3, 8) $	$ (3, 8) $	$ (2, 8) $	$ (2, 8) $	$ (3, 8) $
$ {\mu_v} = 2 $	$ {F}(s, S) $	$ 475.241\, 1 $	$ 390.028\, 3 $	$ 367.035\, 4 $	$ 355.862\, 5 $	$ 349.240\, 3 $
$ {\mu_v} = 4 $	$ (s, S) $	$ (2, 8) $	$ (2, 8) $	$ (2, 8) $	$ (2, 8) $	$ (2, 8) $
$ {\mu_v} = 4 $	$ {F}(s, S) $	$ 391.918\, 3 $	$ 360.309\, 0 $	$ 349.404\, 8 $	$ 343.693\, 9 $	$ 340.132\, 8 $
$ {\mu_v} = 6 $	$ (s, S) $	$ (2, 8) $	$ (2, 9) $	$ (2, 9) $	$ (2, 9) $	$ (2, 9) $
$ {\mu_v} = 6 $	$ {F}(s, S) $	$ 349.134\, 8 $	$ 340.403\, 6 $	$ 336.238\, 0 $	$ 333.833\, 5 $	$ 332.250\, 4 $

	η	$3$	$4$	$5$	$6$	$7$
${\mu_v} = 0$	$(s, S)$	$(4, 12)$	$(4, 10)$	$(3, 8)$	$(3, 8)$	$(2, 7)$
${\mu_v} = 0$	${F}(s, S)$	$558.026\, 0$	$467.029\, 5$	$427.272\, 5$	$404.808\, 6$	$388.114\, 3$
${\mu_v} = 2$	$(s, S)$	$(2, 12)$	$(3, 10)$	$(3, 8)$	$(2, 7)$	$(2, 7)$
${\mu_v} = 2$	${F}(s, S)$	$522.272\, 9$	$429.688\, 3$	$390.028\, 3$	$366.549\, 8$	$350.994\, 9$
${\mu_v} = 4$	$(s, S)$	$(1, 12)$	$(3, 10)$	$(2, 8)$	$(2, 8)$	$(2, 7)$
${\mu_v} = 4$	${F}(s, S)$	$490.130\, 5$	$400.524\, 8$	$360.309\, 0$	$338.227\, 1$	$323.644\, 8$
${\mu_v} = 6$	$(s, S)$	$(1, 13)$	$(2, 10)$	$(2, 9)$	$(2, 8)$	$(2, 7)$
${\mu_v} = 6$	${F}(s, S)$	$464.504\, 3$	$379.849\, 5$	$340.403\, 6$	$318.345\, 3$	$306.287\, 1$

	$ {\mu_b} $	$ 3 $	$ 4 $	$ 5 $	$ 6 $	$ 7 $
$ {\mu_v} = 0 $	$ (s, S) $	$ (3, 8) $	$ (3, 8) $	$ (3, 9) $	$ (3, 9) $	$ (3, 9) $
$ {\mu_v} = 0 $	$ {F}(s, S) $	$ 435.245\, 1 $	$ 421.751\, 7 $	$ 413.848\, 5 $	$ 408.361\, 9 $	$ 404.507\, 6 $
$ {\mu_v} = 2 $	$ (s, S) $	$ (3, 9) $	$ (3, 9) $	$ (3, 9) $	$ (3, 9) $	$ (3, 9) $
$ {\mu_v} = 2 $	$ {F}(s, S) $	$ 398.007\, 3 $	$ 384.106\, 4 $	$ 376.170\, 8 $	$ 371.227\, 5 $	$ 367.844\, 1 $
$ {\mu_v} = 4 $	$ (s, S) $	$ (2, 8) $	$ (2, 8) $	$ (2, 8) $	$ (2, 9) $	$ (2, 9) $
$ {\mu_v} = 4 $	$ {F}(s, S) $	$ 490.130\, 5 $	$ 400.524\, 8 $	$ 360.309\, 0 $	$ 338.227\, 1 $	$ 323.644\, 8 $
$ {\mu_v} = 6 $	$ (s, S) $	$ (2, 8) $	$ (2, 9) $	$ (2, 9) $	$ (2, 9) $	$ (2, 9) $
$ {\mu_v} = 6 $	$ {F}(s, S) $	$ 347.801\, 0 $	$ 334.946\, 8 $	$ 328.101\, 7 $	$ 323.976\, 4 $	$ 321.217\, 2 $