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

Abstract

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

CLC Number:

O226

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

Figures/Tables 13

References 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 $

Analysis of M/M/1 production-inventory system with working vacations

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