Simulation of a queuing model with backup waiters and impatient customers
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摘要: 影响排队系统性能的因素有很多，在某些假设条件下，服务员的数量决定服务效率，而不耐烦顾客的存在会影响服务收益.传统的排队理论主要是针对不同时间分布类型的排队模型分别进行分析的，而蒙特卡洛仿真模型可以同时适应多种时间分布类型的排队过程.本文构建了几种常见条件下的排队模型，并利用蒙特卡洛仿真方法对其进行了模拟，特别是分析了若干常用指标.通过对这些模型的仿真结果比较分析，表明：若根据顾客排队的情况及时调整服务员数量，则既可以提高服务效率，又可避免过多资源闲置浪费以及顾客流失；同时，仿真结果的各项指标可以作为设置排队类型及其模型参数的依据，为有关决策提供参考.Abstract: There are many factors that affect the performance of a queuing system. Under certain assumptions, the number of waiters determines the service efficiency while the presence of impatient customers will affect the service earnings. Traditional queuing theory primarily analyzes queuing models with different time distribution types, while a Monte Carlo simulation model can adapt to the queuing processes of different time distribution types simultaneously. In this paper, several queuing models under common conditions were constructed and simulated by a Monte Carlo simulation; in particular, some commonly used indicators were analyzed. Through a comparison and analysis of the simulation results from these models, it is shown that if the number of waiters is adjusted based on the customer queuing circumstances, it can not only improve service efficiency, but also avoid excessive waste of idle resources and loss of customers. Meanwhile, the simulation results can be used as the basis for setting up queue types and model parameters, as well as a reference for decision making.
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Key words:
- queuing theory /
- Monte Carlo simulation /
- backup waiters /
- impatient customers /
- customer loss
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图 2 到达时间间隔和服务时间的概率密度函数
Fig. 2 The probability density function of the interval of arrival time and service time
图 3 到达时间间隔和服务时间分布对比
Fig. 3 The distribution contrast of the interval of arrival time and service time
表 1 各组随机数与原始数据对比
Tab. 1 A comparison of random numbers and raw data in each group
数据性质 原始数据 随机1 随机2 随机3 数据 相对偏差/% 数据 相对偏差/% 数据 相对偏差/% 到达间隔 平均值 1.600 000 0 1.609 863 1 0.616 4 1.591 562 8 0.527 3 1.613 077 8 0.817 4 标准差 1.602 475 283 1.495 093 897 6.701 0 1.604 870 369 0.149 5 1.622 452 515 1.246 6 服务时间 平均值 1.157 004 831 1.160 791 1 0.327 2 1.149 028 3 0.689 4 1.166 264 98 0.800 4 标准差 0.975 740 902 0.835 083 888 14.415 4 0.810 161 006 16.969 7 0.844 692 605 13.430 6 
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表 2 不考虑不耐烦顾客的排队系统指标
Tab. 2 Indicators for a queuing system without considering impatient customers
$L_{q}$ $L_{q1}$ $L_{q2}$ $W_{q}$ $W_{q1}$ $W_{q2}$ $E_{Tp1}$ $E_{Tp2}$ $R_{Tp1}$/% $R_{Tp2}$/% $R_{v}$/% $q=s=1$ 1.302 85 2.217 7 0.447 85 27.745 $q=1, s=2$ 0.177 05 0.138 5 1.410 78 1.396 05 87.475 86.562 q=s=2 rnd 0.654 65 0.325 34 0.329 31 0.528 82 0.538 11 0.519 6 1.033 09 1.027 13 64.045 63.675 q=s=2 short 0.523 22 0.258 76 0.264 46 0.188 7 0.188 16 0.189 24 1.030 3 1.029 68 63.872 63.833 $q=1, Q_{\max}=2$ 0.497 19 0.355 37 0.721 34 0.020 68 44.719 2.916 82.061 $q=1, Q_{\max}=3$ 0.624 64 0.622 57 0.637 54 0.015 32 39.523 2.005 85.948 $q=1, Q_{\max}=4$ 0.736 48 0.881 58 0.580 65 0.011 93 35.997 1.464 89.326 $q=s, Q_{\max}=2$ 0.604 68 0.466 44 0.138 25 0.394 12 0.402 37 0.367 85 0.717 96 0.112 35 44.509 14.434 84.173 $q=s, Q_{\max}=3$ 0.708 83 0.608 9 0.099 93 0.667 44 0.702 05 0.480 51 0.620 94 0.175 39 38.494 19.883 88.637 $q=s, Q_{\max}=4$ 0.808 76 0.734 12 0.074 64 0.936 56 0.972 88 0.627 61 0.569 37 0.195 33 35.297 21.000 92.296
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表 3 具有不耐烦顾客的排队系统指标
Tab. 3 Indicators for a queuing system with impatient customers
$L_{q}$ $L_{q1}$ $L_{q2}$ $W_{q}$ $W_{q1}$ $W_{q2}$ $E_{Tp1}$ $E_{Tp2}$ $R_{Tp1}$/% $R_{Tp2}$/% $R_{v}$/% $R_{l}$/% $q=s=1$ 0.812 73 1.401 11 0.495 32 30.694 4.26 $q=1, s=2$ 0.171 35 0.135 11 1.411 8 1.396 65 87.538 86.599 0.14 q=s=2 rnd 0.596 13 0.300 42 0.295 71 0.460 02 0.465 97 0.454 01 1.035 07 1.034 26 64.167 64.117 0.90 q=s=2 short 0.503 26 0.252 44 0.250 82 0.180 97 0.180 55 0.181 39 1.025 65 1.036 61 63.583 64.263 0.46 $q=1, Q_{\max}=2$ 0.484 04 0.348 01 0.723 52 0.020 85 44.862 0.528 82.033 0.30 $q=1, Q_{\max}=3$ 0.593 77 0.607 09 0.639 29 0.015 59 39.639 0.286 86.052 0.68 $q=1, Q_{\max}=4$ 0.685 27 0.857 39 0.584 17 0.011 16 36.221 0.145 89.655 1.16 $q=s, Q_{\max}=2$ 0.582 74 0.456 27 0.126 47 0.383 49 0.396 22 0.342 47 0.716 4 0.111 48 44.412 14.447 84.445 0.48 $q=s, Q_{\max}=3$ 0.661 1 0.583 11 0.077 99 0.639 19 0.685 43 0.376 47 0.624 93 0.173 75 38.742 20.103 89.038 0.98 $q=s, Q_{\max}=4$ 0.733 63 0.693 95 0.039 68 0.911 6 0.963 93 0.389 48 0.570 31 0.245 16 35.355 27.116 92.904 1.64
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