两类具有N-策略和单重休假的M/G/1排队系统的最优控制策略

doi:10.15960/j.cnki.issn.1007-6093.2021.04.002

Abstract

Abstract:

This paper considers two types of M/G/1 queueing systems with N-strategy and single vacation, one of which is uninterruptible on vacation and the other is of interruptible on vacation. Using the stochastic decomposition of the steady-state queue length, the expressions of the probability generating function of the steady-state queue length are obtained. And the sensitivity of the system's idle rate and the additional average queue-length towards system parameters is discussed. Furthermore, the explicit expressions of the long-run expected cost per unit time are derived under a given cost structure. At the same time, numerical calculation examples are used to determine the control strategy N^* for minimizing the long-run expected cost per unit time as well as the combined control strategy (N^*, T^*) when the vacation time is fixed duration T.

Key words: N-strategy, single vacation, interruptible vacation, uninterruptible vacation, optimal control strategy

CLC Number:

O226

Beilei TANG, Yinghui TANG. Optimal control strategies for two types of M/G/1 queueing systems with N-strategy and single vacation[J]. Operations Research Transactions, 2021, 25(4): 15-30.

Figures/Tables 18

References 21

1	Yadin M , Naor P . Queueing systems with a removable service station[J]. Operational Research Quarterly, 1963, 14, 393- 405. doi: 10.1057/jors.1963.63
2	Balachandran K R . Control policies for a single server system[J]. Management Science, 1973, 19 (9): 1013- 1018. doi: 10.1287/mnsc.19.9.1013
3	Heyman D P . T-policy for the M/G/1 queue[J]. Management Science, 1977, 23 (7): 775- 778. doi: 10.1287/mnsc.23.7.775
4	Doshi B . Queueing systems with vacations—a survey[J]. Queueing Systems, 1986, 1, 29- 66. doi: 10.1007/BF01149327
5	田乃硕. 休假随机服务系统[M]. 北京: 北京大学出版社, 2001.
6	Agarwal R P , Dshalalow J H . New fluctuation analysis of $ D $-policy bulk queues with multiple vacation[J]. Mathematical and Computer Modelling, 2005, 41 (2-3): 253- 269. doi: 10.1016/j.mcm.2003.12.012
7	唐应辉, 唐小我. 排队论——基础与分析技术[M]. 北京: 科学出版社, 2006.
8	Wang K H , Wang T Y , Pearn W L . Optimal control of the $ N $-policy M/G/1 queueing system with server breakdowns and general startup times[J]. Applied Mathematical Modelling, 2006, 31 (10): 2199- 2212.
9	井彩霞, 崔颖, 田乃硕. Min($ N, V $)-策略休假的M/G/1排队系统分析[J]. 运筹与管理, 2006, 15 (3): 53- 58. doi: 10.3969/j.issn.1007-3221.2006.03.011
10	Wang K H , Kuo C C , Ke J C . Optimal control of the $ D $-policy M/G/1 queueing system with server breakdowns[J]. American Journal of Applied Sciences, 2008, 5 (5): 565- 573. doi: 10.3844/ajassp.2008.565.573
11	唐应辉, 刘名武. $ N $-策略单重休假M/G/1排队系统的队长分布[J]. 应用数学, 2008, 21 (1): 20- 26.
12	唐应辉, 吴文青, 刘云颇, 等. 基于多重休假的min($ N, V $)-策略M/G/1排队系统的队长分布[J]. 系统工程理论与实践, 2014, 34 (6): 1533- 1546.
13	唐应辉, 吴文青, 刘云颇. 基于单重休假的min($ N, V $)-策略M/G/1排队系统分析[J]. 应用数学学报, 2014, 37 (6): 976- 996.
14	蒋书丽, 唐应辉. 具有多级适应性休假和min($ N, V $)-策略控制的M/G/1排队系统[J]. 系统科学与数学, 2017, 37 (8): 1866- 1884.
15	Gu J X , Wei Y Y , Tang Y H , et al. Queue size distribution of Geo/G/1 queue under the min($ N, D $)-policy[J]. Journal of Systems Science and Complexity, 2016, 29 (3): 752- 771. doi: 10.1007/s11424-016-4180-y
16	蔡晓丽, 唐应辉. 具有温储备失效特征和单重休假min($ N, V $)-控制策略的M/G/1可修排队系统[J]. 应用数学学报, 2017, 40 (5): 702- 726.
17	魏瑛源, 唐应辉, 余玅妙. 基于min($ N, D $)-策略的M/G/1排队系统的队长分布及最优策略[J]. 系统科学与数学, 2015, 35 (6): 729- 744.
18	高丽君, 唐应辉. 具有min($ N, D $)-策略控制的M/G/1可修排队系统及最优控制策略[J]. 数学物理学报, 2017, 37A (2): 352- 365. doi: 10.3969/j.issn.1003-3998.2017.02.014
19	王敏, 唐应辉. 基于min($ N, D, V $)-策略和单重休假的M/G/1排队系统的最优控制策略[J]. 系统科学与数学, 2018, 38 (9): 1067- 1084.
20	罗乐, 唐应辉. 具有min($ N, D, V $)-策略控制的M/G/1排队系统[J]. 运筹学学报, 2019, 23 (2): 1- 16.
21	罗乐, 唐应辉. 具有$ p $-进入规则和min($ N, D, V $)-策略的M/G/1排队系统容量的优化设计及最优控制策略[J]. 数学物理学报, 2019, 39 (5): 1228- 1246. doi: 10.3969/j.issn.1003-3998.2019.05.024

$ N $	$ {p}_{\mbox{ 0, 模 1}} $	$ {p}_{\mbox{ 0, 模 2}} $	$ N $	$ {p}_{\mbox{ 0, 模 1}} $	$ {p}_{\mbox{ 0, 模 2}} $
1	0.226 2	0.700 0	11	0.063 1	0.233 9
2	0.197 3	0.452 7	12	0.058 0	0.231 9
3	0.168 9	0.359 0	13	0.053 6	0.230 4
4	0.144 5	0.311 3	14	0.049 9	0.229 3
5	0.124 4	0.283 5	15	0.046 6	0.228 5
6	0.108 3	0.265 9	16	0.043 7	0.227 9
7	0.095 3	0.254 3	17	0.041 1	0.227 5
8	0.084 8	0.246 3	18	0.038 9	0.227 1
9	0.076 2	0.240 7	19	0.036 8	0.226 9
10	0.069 0	0.236 8	20	0.035 0	0.226 7

$ N $	$ {p}_{\mbox{ 0, 模 1}} $	$ {p}_{\mbox{ 0, 模 2}} $	$ N $	$ {p}_{\mbox{ 0, 模 1}} $	$ {p}_{\mbox{ 0, 模 2}} $
1	0.032 3	0.100 0	11	0.009 0	0.033 4
2	0.028 2	0.064 7	12	0.008 3	0.033 1
3	0.024 1	0.051 3	13	0.007 7	0.032 9
4	0.020 6	0.044 5	14	0.007 1	0.032 8
5	0.017 8	0.040 5	15	0.006 7	0.032 6
6	0.015 5	0.038 0	16	0.006 2	0.032 6
7	0.013 6	0.036 3	17	0.005 9	0.032 5
8	0.012 1	0.035 2	18	0.005 6	0.032 4
9	0.010 9	0.034 4	19	0.005 3	0.032 4
10	0.009 9	0.033 8	20	0.005 0	0.032 4

$ N $	$ {\overline{L}}_{\mbox{ d, 模 1}} $	$ {\overline{L}}_{\mbox{ d, 模 2}} $	$ N $	$ {\overline{L}}_{\mbox{ d, 模 1}} $	$ {\overline{L}}_{\mbox{ d, 模 2}} $
1	2.594 5	0.000 0	11	5.081 1	2.211 7
2	2.390 6	0.353 3	12	5.558 2	2.289 1
3	2.334 4	0.694 3	13	6.041 9	2.351 9
4	2.430 7	1.000 3	14	6.530 3	2.402 5
5	2.648 2	1.268 3	15	7.021 9	2.443 2
6	2.952 8	1.499 5	16	7.515 9	2.475 6
7	3.317 8	1.696 5	17	8.011 6	2.501 4
8	3.724 3	1.862 7	18	8.508 4	2.521 8
9	4.159 1	2.001 7	19	9.006 1	2.537 8
10	4.613 3	2.116 9	20	9.504 5	2.550 5

$ N $	$ {\overline{L}}_{\mbox{ d, 模 1}} $	$ {\overline{L}}_{\mbox{ d, 模 2}} $	$ N $	$ {\overline{L}}_{\mbox{ d, 模 1}} $	$ {\overline{L}}_{\mbox{ d, 模 2}} $
1	0.387 8	0.000 0	11	5.000 0	0.266 4
2	0.626 7	0.115 7	12	5.500 0	0.266 5
3	1.044 8	0.191 4	13	6.000 0	0.266 5
4	1.516 7	0.231 6	14	6.500 0	0.266 5
5	2.006 4	0.251 0	15	7.000 0	0.266 5
6	2.502 5	0.259 8	16	7.500 0	0.266 5
7	3.001 0	0.263 7	17	8.000 0	0.266 5
8	3.500 4	0.265 3	18	8.500 0	0.266 5
9	4.000 2	0.266 0	19	9.000 0	0.266 5
10	4.500 1	0.266 3	20	9.500 0	0.266 5

$ N $	$ {{F}}_{\mbox{ 模 1}} $	$ {{F}}_{\mbox{ 模 2}} $	$ N $	$ {{F}}_{\mbox{ 模 1}} $	$ {{F}}_{\mbox{ 模 2}} $
1	34.344 7	61.642 9	11	32.907 7	33.085 7
2	30.866 2	42.388 0	12	34.861 4	33.297 5
3	28.172 5	36.125 3	13	36.909 9	33.484 1
4	26.575 7	33.605 7	14	39.031 8	33.644 2
5	$\bf{\fbox{25.961 5}}$	32.581 6	15	41.211 1	33.779 1
6	26.114 5	32.244 7	16	43.435 9	33.891 1
7	26.836 7	$\bf{\fbox{32.240 2}}$	17	45.697 2	33.983 1
8	27.974 3	32.393 9	18	47.988 2	34.057 9
9	29.415 1	32.614 7	19	50.303 6	34.118 2
10	31.078 6	32.853 8	20	52.639 2	34.166 5

Optimal control strategies for two types of M/G/1 queueing systems with N-strategy and single vacation

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Figures/Tables 18

References 21

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	1	2	3	4	5	6	7	8
0	61.642 9	34.392 9	26.976 2	24.517 9	$\bf{\fbox{24.042 9}}$	24.559 5	25.642 9	27.080 4
0.000 1	61.642 9	34.392 9	26.976 2	24.517 9	24.042 9	24.559 5	25.642 9	27.080 4
0.000 2	61.642 9	34.392 9	26.976 2	24.517 9	24.042 9	24.559 5	25.642 9	27.080 4
0.000 3	61.642 9	34.392 9	26.976 2	24.517 9	24.042 9	24.559 5	25.642 9	27.080 4
0.000 9	61.642 8	34.392 9	26.976 2	24.517 9	24.042 9	24.559 5	25.642 9	27.080 4
0.001	61.642 8	34.392 9	26.976 2	24.517 9	24.042 9	24.559 5	25.642 9	27.080 4
0.1	61.452 6	34.391 8	26.976 2	24.517 9	24.042 9	24.559 5	25.642 9	27.080 4
1	50.135 3	33.697 8	26.937 0	24.517 0	24.043 1	24.559 6	25.642 9	27.080 4
2	37.584 3	31.264 6	26.643 2	24.510 8	24.052 2	24.563 0	25.643 7	27.080 5
4	27.890 1	27.039 2	25.748 1	24.675 2	24.314 4	24.718 5	25.712 8	27.106 4
8	27.888 6	27.862 2	27.783 1	27.640 7	27.492 6	27.477 3	27.759 4	28.448 0

	1	2	3	4	5	6	7	8
1	61.642 9	52.210 9	50.471 5	50.180 6	50.140 3	50.135 8	50.135 3	50.135 3
4	61.642 9	36.552 1	30.139 9	28.197 5	27.715 1	27.692 9	27.768 7	27.831 9
10	61.642 9	34.419 2	27.035 1	24.611 5	24.143 6	24.588 4	25.455 7	26.468 1
16	61.642 9	34.393 1	26.977 0	24.519 6	24.045 1	24.558 5	25.626 7	27.021 5
19	61.642 9	34.392 9	26.976 3	24.518 1	24.043 1	24.559 3	25.639 5	27.066 9
20	61.642 9	34.392 9	26.976 2	24.518	24.043 0	24.559 4	25.641 0	27.072 4
20.6	61.642 9	34.392 9	26.976 2	24.517 9	24.043 0	24.559 4	25.641 5	27.074 6
20.635	61.642 9	34.392 9	26.976 2	24.517 9	24.043 0	24.559 4	25.641 5	27.074 7
20.638 5	61.642 9	34.392 9	26.976 2	24.517 9	24.043 0	24.559 4	25.641 5	27.074 7
20.638 6	61.642 9	34.392 9	26.976 2	24.517 9	24.043 0	24.559 4	25.641 5	27.074 7
20.638 7	61.642 9	34.392 9	26.976 2	24.517 9	$\bf{\fbox{24.042 9}}$	24.559 4	25.641 5	27.074 7
20.64	61.642 9	34.392 9	26.976 2	24.517 9	24.042 9	24.559 4	25.641 5	27.074 7
20.7	61.642 9	34.392 9	26.976 2	24.517 9	24.042 9	24.559 4	25.641 6	27.074 9
21	61.642 9	34.392 9	26.976 2	24.517 9	24.042 9	24.559 4	25.641 8	27.075 7

$ N $	$ {{F}}_{\mbox{ 模 1}} $	$ {{F}}_{\mbox{ 模 2}} $	$ N $	$ {{F}}_{\mbox{ 模 1}} $	$ {{F}}_{\mbox{ 模 2}} $
1	31.460 6	41.500 0	11	35.968 0	29.920 9
2	29.036 0	31.254 3	12	38.106 8	30.207 8
3	27.376 3	28.406 3	13	40.313 9	30.449 0
4	$\bf{\fbox{26.670 3}}$	27.622 3	14	42.572 9	30.649 1
5	26.785 3	$\bf{\fbox{27.611 4}}$	15	44.871 7	30.813 2
6	27.525 5	27.914 2	16	47.201 5	30.946 7
7	28.720 4	28.333 9	17	49.555 8	31.054 5
8	30.241 4	28.777 9	18	51.929 7	31.140 9
9	31.996 4	29.201 9	19	54.319 3	31.209 9
10	33.920 5	29.585 4	20	56.721 8	31.264 5

$ N $	$ {{F}}_{\mbox{ 模 1}} $	$ {{F}}_{\mbox{ 模 2}} $	$ N $	$ {{F}}_{\mbox{ 模 1}} $	$ {{F}}_{\mbox{ 模 2}} $
1	60.719 4	53.500 0	11	71.171 1	58.899 0
2	59.348 6	$\bf{\fbox{52.263 4}}$	12	73.495 0	59.261 0
3	$\bf{\fbox{58.723 0}}$	52.830 1	13	75.860 7	59.556 8
4	58.907 9	53.781 8	14	78.256 8	59.796 8
5	59.752 0	54.784 0	15	80.675 1	59.990 2
6	61.079 3	55.726 6	16	83.110 0	60.145 2
7	62.747 0	56.570 4	17	85.557 3	60.268 8
8	64.651 4	57.304 8	18	88.013 9	60.366 9
9	66.720 6	57.931 9	19	90.477 8	60.444 4
10	68.905 1	58.459 8	20	92.947 2	60.505 4