部分服务台同步多重休假的排队库存系统

doi:10.15960/j.cnki.issn.1007-6093.2024.01.004

摘要/Abstract

摘要：

本文研究了$\left({s, S}\right)$ 库存策略的多服务台排队库存系统, 其中库存为空时有部分服务台同步多重休假, 休假时间服从指数分布。顾客到达为泊松过程, 每个服务台的服务时间和补货时间均服从指数分布。利用拟生灭过程和矩阵几何解法, 计算了系统稳态概率和一些性能指标, 并给出了系统单位时间的平均费用函数。最后, 通过数值算例分析了参数对费用函数的影响, 并得到最优库存策略和最优平均费用。

关键词: 排队库存系统, 部分服务台休假, (s, S) 库存策略, 拟生灭过程, 矩阵几何解

Abstract:

In this paper, we consider a Markovian $\left({s, S}\right)$ queueing-inventory system in which only partial servers take multiple synchronous vacations when the on-hand inventory level is zero. It is assumed that the vacation time follows an exponential distribution. The customers arrive according to a Poisson process, and the service time of the customers is distributed exponentially. The lead times for the orders are assumed to have independent and identical exponential distributions. Using the theory of quasi-birth-and-death process, the matrix-geometric solution of the steady-state probability is derived. On this basis, the steady-state performance measures and cost function of the system are obtained. Finally, the effect of the parameters on cost function is analyzed by numerical examples, and the optimal inventory policy and the optimal expected cost are also computed.

Key words: queueing-inventory system, vacations of partial servers, (s, S) policy, quasi-birth-and-death process, matrix-geometric solution

中图分类号:

O226

叶子钦, 岳德权. 部分服务台同步多重休假的排队库存系统[J]. 运筹学学报, 2024, 28(1): 40-56.

Ziqin YE, Dequan YUE. Queueing-inventory system with multiple synchronous vacations of partial servers[J]. Operations Research Transactions, 2024, 28(1): 40-56.

图/表 9

表1

表2

表3

表4

表5

服务台总数不同情况下, 休假服务台数对系统费用和相关性能指标的影响"

c	$ d $	$ F $	$ {L_q} $	$ I $	$ {E_l} $	$ {N_b} $	$ R $	$ {E_r} $	$ {E_v} $
6	1	254.565 7	5.534 8	12.105 7	0.578 3	3.430 0	0.168 8	14.978 8	0.045 4
	2	245.129 8	5.784 1	12.110 0	0.573 4	2.690 6	0.168 7	14.977 6	0.052 7
	3	242.046 6	6.041 3	12.131 8	0.565 7	2.366 5	0.168 3	14.954 8	0.059 9
	4	247.237 1	6.294 4	12.168 5	0.565 3	2.591 4	0.167 9	14.913 3	0.065 9
	5	262.020 3	6.534 6	12.220 5	0.561 5	3.458 1	0.167 3	14.852 5	0.070 0
8	1	319.633 2	3.586 5	11.669 0	0.521 8	8.000 0	0.171 1	15.513 1	0.066 9
	2	302.503 0	3.724 1	11.681 6	0.521 6	7.422 9	0.170 3	15.505 0	0.063 8
	3	290.779 2	3.872 4	11.702 0	0.5212	6.5642	0.1700	15.4865	0.062 1
	4	286.677 1	4.030 2	11.726 9	0.520 3	6.210 0	0.169 8	15.462 4	0.061 4
	5	292.009 1	4.197 1	11.754 7	0.519 2	6.481 2	0.169 5	15.434 2	0.061 5
	6	307.893 7	4.371 5	11.786 3	0.517 7	7.452 7	0.169 2	15.401 0	0.061 8
	7	317.449 5	4.551 7	11.822 3	0.515 9	8.000 0	0.168 9	15.361 9	0.062 1
10	1	344.919 2	2.921 3	10.641 6	0.629 0	10.000 0	0.196 8	16.756 5	0.133 6
	2	341.718 3	3.234 5	10.679 2	0.642 5	9.759 9	0.195 9	16.754 0	0.080 0
	3	327.034 8	3.544 6	10.715 0	0.649 8	8.879 4	0.195 4	16.706 8	0.076 6
	4	305.090 9	3.851 0	10.760 8	0.653 6	7.271 7	0.194 8	16.648 1	0.079 4
	5	293.631 5	4.129 9	10.800 1	0.654 3	6.270 5	0.194 3	16.599 8	0.084 7
	6	294.274 4	4.359 8	10.838 0	0.651 3	6.395 0	0.193 9	16.553 9	0.089 3
	7	308.409 8	4.534 3	10.883 1	0.644 7	7.139 7	0.193 4	16.499 0	0.092 8
	8	338.448 5	4.665 3	10.932 8	0.635 2	9.058 9	0.192 9	16.438 4	0.096 1
	9	353.768 7	4.781 4	10.979 9	0.623 7	10.000 0	0.192 5	16.380 6	0.100 0
12	1	371.130 3	1.052 3	18.819 6	0.034 4	11.017 0	0.103 9	5.066 4	0.242 2
	2	352.440 6	1.208 0	18.614 2	0.050 6	9.716 5	0.106 6	5.374 6	0.170 9
	3	332.259 6	1.322 6	18.617 9	0.052 3	8.365 9	0.106 3	5.363 1	0.079 2
	4	315.001 8	1.441 4	18.673 6	0.050 4	7.164 5	0.105 5	5.280 7	0.053 1
	5	304.435 4	1.558 2	18.735 7	0.047 8	6.388 1	0.104 7	5.194 0	0.048 6
	6	304.185 8	1.663 6	18.790 8	0.045 1	6.296 7	0.104 0	5.118 8	0.050 0
	7	316.099 0	1.748 8	18.829 4	0.042 7	7.025 6	0.103 5	5.066 4	0.052 8
	8	338.501 0	1.806 7	18.847 6	0.040 9	8.469 7	0.103 3	5.042 1	0.055 4
	9	366.455 6	1.835 3	18.845 7	0.039 6	10.301 7	0.103 3	5.046 4	0.057 4
	10	395.726 5	1.839 0	18.822 4	0.039 0	11.237 7	0.103 6	5.081 3	0.059 1
	11	427.025 1	1.829 0	18.774 0	0.038 2	12.000 0	0.104 3	5.151 1	0.060 7

表5

图1

图2

图3

图4

参考文献 17

1	Arivarignan G , Yadavalli V S S , Sivakumar B .A perishable inventory system with multi-server service facility and retrial customers[J].Management Science and Practice,2008,3-27.
2	Yadavalli V S S , Sivakumar B , Arivarignan G , et al.A multi-server perishable inventory system with negative customer[J].Computers and Industrial Engineering,2011,61(2):254-273. doi: 10.1016/j.cie.2010.07.032
3	Nair A N , Jacob M J , Krishnamoorthy A .The multi-server M/M/$(s, S)$ queueing inventory system[J].Advances in Operations Research,2013,33(1):321-333.
4	Krishnamoorthy A , Manikandan R , Dhanya Shajin .Analysis of a multiserver queueing-inventory system[J].Advances in Operations Research,2015,1-16.
5	赵国喜. 供需与服务影响下的排队-库存系统性能分析及控制研究[D]. 秦皇岛: 燕山大学, 2017.
6	Chakravarthy S R , Dhanya S , Krishnamoorthy A .Infinite server queueing-inventory models[J].The Indian Society for Probability and Statistics,2019,21(2):3-68.
7	Tian N , Zhang Z G .Vacation Queueing Models: Theory and Applications[M].New York: Springer-Verlag,2006.
8	Daniel J K , Ramanarayanan R .An $(s, S)$ inventory system with rest periods to the server[J].Naval Research Logistics,1988,35(1):119-123. doi: 10.1002/1520-6750(198802)35:1<119::AID-NAV3220350111>3.0.CO;2-P
9	Suganya C , Sivakumar B , Arivarignan G .Numerical investigation on MAP/PH$(1)$, PH$(2)$/2 inventory system with multiple server vacations[J].International Journal of Operational Research,2017,29(1):1-33. doi: 10.1504/IJOR.2017.083173
10	Suganya C , Shophia Lawrence A , Sivakumar B .A finite-source inventory system with service facility, multiple vacations of two heterogeneous servers[J].International Journal of Information and Management Sciences,2018,29(3):257-277.
11	Suganya C , Sivakumar B .MAP/PH$(1)$, PH$(2)$/2 finite retrial inventory system with service facility, multiple vacations for servers[J].International Journal of Mathematics in Operational Research,2019,15(3):265-294. doi: 10.1504/IJMOR.2019.102075
12	Jeganathana K , Abdul Reiyas M .Two parallel heterogeneous servers Markovian inventory system with modified and delayed working vacations[J].Mathematics and Computers in Simulation,2020,172,273-304. doi: 10.1016/j.matcom.2019.12.002
13	Yadavalli V S S , Jeganathan K .Perishable inventory model with two heterogeneous servers including one with multiple vacations and retrial customers[J].Control and Systems Engineering,2015,3(1):10-34.
14	Jeganathan K , Abdul Reiyas M , Prasanna Lakshmi K , et al.Two server Markovian inventory systems with server interruptions: Heterogeneous VS. homogeneous servers[J].Mathematics and Computers in Simulation,2019,155,177-200.
15	Neuts M F .Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach[M].Baltimore: Johns Hopkins University Press,1981.
16	田乃硕, 岳德权.拟生灭过程与矩阵几何解[M].北京: 科学出版社,2002.
17	Latouche G , Ramaswami V .A logarithmic reduction algorithm for quasi-birth-death process[J].Journal of Applied Probability,1993,30(5):650-674.

	5	15	25	35	45
10	$ (7, 13, 3) $	$ (7, 13, 3) $	$ (7, 14, 3) $	$ (7, 14, 3) $	$ (7, 14, 2) $
10	225.675 1	231.490 9	237.270 1	242.945 9	248.553 4
15	$ (7, 14, 2) $	$ (7, 14, 2) $	$ (7, 15, 2) $	$ (7, 15, 2) $	$ (7, 15, 2) $
15	239.461 5	244.937 3	250.396 9	255.710 7	261.024 5
20	$ (7, 15, 2) $	$ (7, 15, 2) $	$ (7, 16, 2) $	$ (7, 16, 2) $	$ (7, 16, 2) $
20	251.851 4	257.165 2	262.422 3	267.580 1	272.737 9
25	$ (7, 16, 2) $	$ (7, 16, 2) $	$ (7, 17, 2) $	$ (7, 19, 1) $	$ (7, 19, 1) $
25	263.587 2	268.745 0	273.870 5	278.546 0	282.717 8
30	$ (7, 19, 1) $	$ (7, 19, 1) $	$ (7, 19, 1) $	$ (7, 20, 1) $	$ (7, 20, 1) $
30	274.369 7	278.541 5	282.713 4	286.877 8	290.913 3

	5	10	15	20	25
10	$ (7, 13, 3) $	$ (7, 13, 3) $	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 12, 3) $
10	182.050 5	200.488 3	218.709 2	236.797 2	254.885 2
15	$ (7, 11, 3) $	$ (7, 11, 3) $	$ (7, 11, 3) $	$ (7, 11, 3) $	$ (7, 10, 3) $
15	206.594 4	224.257 9	241.921 5	259.585 0	277.225 0
20	$ (7, 10, 2) $	$ (7, 10, 2) $	$ (7, 10, 2) $	$ (7, 10, 2) $	$ (7, 9, 2) $
20	228.182 3	245.550 3	262.918 3	280.286 3	297.567 2
25	$ (7, 10, 2) $	$ (7, 9, 2) $	$ (7, 9, 2) $	$ (7, 9, 2) $	$ (7, 9, 2) $
25	248.596 7	265.969 7	282.755 5	299.541 3	316.327 1
30	$ (7, 9, 2) $	$ (7, 9, 2) $	$ (7, 9, 2) $	$ (7, 9, 2) $	$ (7, 9, 2) $
30	267.943 8	284.729 6	301.515 4	318.301 2	335.087 0

	5	10	15	20	25
5	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 11, 3) $	$ (7, 11, 3) $
5	168.850 2	186.938 2	205.026 3	222.928 1	240.591 6
10	$ (7, 13, 3) $	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 11, 3) $	$ (7, 11, 3) $
10	171.554 7	189.674 8	207.762 8	225.796 2	243.459 8
15	$ (7, 13, 3) $	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 11, 3) $
15	174.178 7	192.411 4	210.499 4	228.587 5	246.327 9
20	$ (7, 13, 3) $	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 11, 3) $
20	176.802 6	195.148 0	213.236 0	231.324 0	249.196 0
25	$ (7, 13, 3) $	$ (7, 13, 3) $	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 11, 3) $
25	179.426 6	197.864 4	215.972 6	234.060 6	252.064 1

	5	10	15	20	25
5	$ (7, 10, 2) $	$ (7, 10, 2) $	$ (7, 10, 2) $	$ (7, 11, 1) $	$ (7, 11, 1) $
5	170.214 4	171.895 7	173.576 9	175.138 5	175.694 4
10	$ (7, 11, 2) $	$ (7, 11, 2) $	$ (7, 11, 2) $	$ (7, 11, 2) $	$ (7, 11, 2) $
10	195.464 3	197.086 6	198.708 9	200.331 1	201.953 4
15	$ (7, 12, 3) $	$ (7, 12, 3) $	$ (7, 12, 2) $	$ (7, 12, 2) $	$ (7, 12, 2) $
15	218.709 2	221.489 9	223.356 1	224.879 0	226.402 0
20	$ (7, 13, 3) $	$ (7, 13, 3) $	$ (7, 13, 3) $	$ (7, 13, 2) $	$ (7, 13, 2) $
20	240.840 0	243.543 9	246.247 8	248.771 6	250.169 7
25	$ (7, 14, 3) $	$ (7, 14, 3) $	$ (7, 14, 3) $	$ (7, 15, 3) $	$ (7, 15, 3) $
25	262.326 4	264.927 1	267.527 8	270.103 2	272.584 6