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[中文]二、问题假设
1、影响检查系统评价的因素是有限的，不包括一些特设或突发情况例如旅客蛮不讲理或遭遇灾难等；
2、对于不同影响因子之间的比例系数，经咨询了解是客观真实的，是有效的；
4、旅客的总体是无限的；
5、旅客一个一个地到来，不同旅客之间到达相互独立；
6、旅客相继到达的间隔时间是随机的，且其分布与时间无关；
8、旅客检查方式为一个一个地进行，采用先到先检查的原则；
9、每位旅客的检查时间长度是随机的，其分布对时间是平稳的
10、每个检查口单位时间的成本费均相同；
三、符号说明
：队长，即排队系统中的旅客数； 是其标准化后的值；
：一个旅客在系统中的等待时间； 为其标准化后的值；
：绝对通过能力，表示单位时间能被检查完旅客的均值；
：相对通过能力，表示单位时间能被检查完的旅客数与等待检查的旅客数之比值；
：检查系统中旅客的满意程度，可由队长 和等待时间 加权得到；
：检查系统评价的性能指标值；
：矩阵的特征值；
：矩阵的特征向量；
：表示在长为 的时间内到达 个旅客的概率；
：指损失概率，由于等待人数过多导致旅客放弃检查，选择改签或退票的概率； 为其改进后的值；
：进行检查的预检线路数；
λ：单位时间内有一个顾客到达的概率；
：系统的检查强度，或称检查机构的平均利用率， ；
：系统有 个旅客时有 个旅客被检查完的概率；
：系统中没有旅客到达的概率；
：单位时间平均进入系统的旅客数， ；
：平均正在进行检查的检查口个数；
：旅客在系统中的平均逗留时间；
：单位时间内每个检查台的成本费；
：旅客在系统中停留的时间的平均费用；
：系统单位时间内的总费用的期望值；
：系统中旅客总的等待时间；
un： 该检查系统的服务率；
vn： n个人时系统的平均移动速度；
v1：单个人在安检中的移动速度；
ρ：人员在该系统的密度
（注：本模型数据处理过程中相关变量时间单位为：秒，个体数量单位为：人）
四、Model Overview
首先，我们建立了关于分析机场安检的基础模型，它是找到安检瓶颈并进一步研究修正其他模型的基础，在该基础模型下，我们建立了层次分析、对比矩阵、泊松分布与负指数分布、多检查窗混合制排队的整体框架。
从基础模型出发，我们建立了检查线路数模型关于限制条件成本的修正，从而分析检查线路数的最优解问题，提供了对安检瓶颈的解决方案。
通过理解文化差异对安检流程和排队论的影响，借用社会动力学原理，我们分别提出排队间距、插队演化对安检效率模型的修正，并通过数据分析找到最优间距使得绝对通过能力最大，安检秩序最佳，从而完成对检查点的敏感性分析。
五、基础模型的建立
1）建立层次分析模型
我们把主要影响性能指标值 的五个因子：绝对通过能力 ，相对通过能力 ，队长 ，等待时间 ，旅客满意程度 分层，其中 由队长 和等待时间 加权得到，层次分析图如下：
由于队长L是越短越好，等待时间W也是越短越好，而总目标Z值，我们令其取值越大越好，那么这些值对于Z值的大小起负向作用，故我们在模型求解时要对它们进行标准化的改进；而绝对通过能力A是越大越好，相对通过能力Q是越大越好，旅客满意程度S是越大越好，因此只需直接将这三个因子直接无量纲化后与上面改进后的因子加权求和即得最终的性能指标值。
由图一及分析可得以下函数式：
综合以上两式，可得最终Z的函数式：
2）构造成对比较矩阵
经过上述建立的机场检查系统评价的层析分析，我们并不把所有的因子放在一起进行分析，而是根据通用的1—9判断矩阵尺度表及对以上各个因子重要性的判断，分别构造出检查系统层次结构中中间层对目标层，细则层对中间层的比较判断矩阵。
尺度 含义
1 与 的影响相同
3 比 的影响稍强
5 比 的影响强
7 比 的影响明显的强
9 比 的影响绝对的强
2,4,6,8 与 的影响之比在上述两个相邻等级之间
1,1/2,1/3，…，1/9 与 的影响之比为上面的互反数
表1:1-9尺度的含义
根据搜索到的资料并结合旅客自身的特点，我们认为对于细则层来说，等待时间 比队长 的影响稍强，故在相应位置赋值为3或1/3，用公式表示即为 ，得到以下矩阵D：
同理，把中间层对于目标层的影响分别赋值，根据对美国奥黑尔机场一些数据了解的情况和从网上查找到的资料，我们对以下五种因子的影响做如下排序：绝对通过能力A < 相对通过能力Q < 队长 < 等待时间 <顾客满意程度S 。我们把它们按照从小到大的顺序排序，逐一两两相比较，得到以下矩阵C.
3）泊松分布和负指数分布
1．泊松分布
在长为t的时间内到达n个旅客的概率为泊松分布：
，
数学期望和方差分别为 .
2．负指数分布
当旅客流为泊松流时，用 表示两个旅客相继到达的时间间隔，是一个随机变量，其分布函数 。由泊松分布的推导公式可知 ，于是 ， ；分布密度 ， .这里 表示单位时间内到达的旅客数。同时知T服从负指数分布，且 ， 。
设普通检查线路对一个旅客的检查时间为 （即在忙期内相继离开线路的两个旅客的间隔时间）服从负指数分布，分布函数为 ， ，分布密度为 ， ，其中 表示平均检查率，且期望值为 ，表示一个旅客的检查时间。
4）多检查窗混合制排队模型的建立（M/M/n/m）
1．系统的状态概率
等候人较多时，新来到系统的旅客考虑自己的乘机时间选择等待或是改签、退票。那么，此时系统有所损失，这样的系统为多服务窗混合制排队模型（M/M/n/m）。图示化如下：
2．排队论的公式推导
在M/M/n/m模型中，处于状态 时，由于每个线路的服务率为 ，故此时系统的总服务率为 ，而当 时， 个检查线路均为忙碌，故系统的总服务率为 。令 ，称为系统的服务强度。
已知旅客的到达规律服从参数为 的泊松分布，检查时间服从参数为 的负指数分布；若有 个旅客，只有 个接受检查，其余的顾客排队等待，有无限个位置可排队，于是在时间间隔 内有：
（1） 有一个旅客到达的概率为 ；
（2） 没有一个旅客到达的概率为 ；
（3） 若系统有 个旅客时有 个旅客被检查完的概率为
（4）多余一个旅客到达或多余 个旅客被检查完的概率为
设在 时刻系统中有 个旅客的概率为 ，我们首先计算 的情况：
计算得出
如果 ，则达到稳态解，故上式可化为：
，
即
特别地，当 时，有
同理，我们可得 的情形，这里求解过程同上，直接给出结果：
当 时，有
故有以下四式：
再由递推关系求得系统的状态概率为
其中， 由 可求得
3.模型指标和模型说明
系统的损失概率：
系统的相对通过能力：
系统的相对通过能力：
单位时间平均进入系统的旅客数：
绝对通过能力：
平均忙着的检查窗个数：
系统的服务率： un=vn/v1
队列长：当 时
；
当 时，
队长：
旅客逗留时间：
旅客等待时间：
六、模型的修正与求解
1.安检乘客流瓶颈分析
1）特征值，特征根的求解
以下定义一致阵：
如果一个对称阵A满足：
，i, j, k=1,2,…，n
则A称为一致性矩阵，简称一致阵，并且可知一致阵有下列性质：
1．A的秩为1，A的唯一非零特征根为n；
2．A的任一列向量都是对应于特征根n的特征向量。
我们得到的成对矩阵C和D均不属于一致阵，但在不一致的容许范围内，我们分别用C和D的最大特征根（记作V）得出最大特征根的特征向量（归一化后）作为权向量w，即w满足：
用MATLAB编程得到：
矩阵C的最大特征根
对应的特征向量
归一化后得到
因此有权重 ， ， ， ，
同理，得到权重 ，
2）对目标函数的求解
我们将求得的权重 （i=1,2,3,4,5）和 （j=1,2）代入之前求得的Z的函数式，得到以下式子：
接着计算以上式子的四个因子 ， ， ，
首先定义因子标准化的处理方法：
1．平移—标准差变换：原始数据之间有不同的量纲，采用下面的变换使每个变量的均值为0.，标准差为1，消除量纲差异的影响。令
其中 为原始变量的测量值， 和 分别为 的样本平均差与样本标准差，即
，
2．平移—极差变换：即经过上述平移—标准差变换后还有某些 ，则对其进行平移—极差变换，即把样本数据极值标准化，经过改进处理后，得到：
正向因子应得的分数：
负向因子应得的分数：
用上式处理后的所有 ，且不存在量纲因素影响。
根据题目中表格所给的数据，我们用MATLAB拟合出绝对通过能力 （即每秒办理的顾客数）的值，并经过标准化方法处理后得到 。
同理，我们可以得到所需各个参数值分别为： ， ， 。由于存在负向因子，需要根据平移—极差变换对其进行数据处理，处理后得到改进因子： ， ，
把这五个参数带入Z中求解得： （由于对数据都做了标准化处理，故Z的最大值为1）
3）瓶颈分析结果
综上，我们通过计算各个权重，得知顾客满意程度所占权重最大，瓶颈问题即为解决由队长和等待时间决定的顾客满意度方面，其次是等待时间方面，考虑到瓶颈的出现是由于前一个工作台的服务率高于后一个服务台时，后一个服务台无法承载相应数量的乘客所导致的，又由机场安检模拟图可知，瓶颈进一步体现在传送扫描阶段。
通过拟合题目数据表得到归一化的各个因子，由于算法可实现，故可视为准确值。所以得出的Z值较为精确，由 可知，该机场检查系统在我们建立的标准下大体令人满意，但还需进一步优化提高，解决瓶颈问题。
2.最优检查线路数的分析
1）检查线路数模型修正
由上述分析可知，机场目前需要解决的瓶颈问题为顾客的等待时间，如何设计检查线路的数量使得既能满足旅客尽量缩短检查等待的时间，又能使机场的花费尽可能小是研究的重点。
假设每位接受预检查的旅客所花费用为 ，预检查的旅客比例为 ,检查系统单位时间内每个检查线路的平均费用（其中包括检查人员的工资，设备维修、磨损的费用总和）为 ，平均每位旅客在系统中的等待成本（包括等待时间成本和检查时间成本）为 ，则单位时间内机场检查系统的利润为
在这里我们要确定检查线路数量的一个系统容量m，由于C不连续，这里采用边际分析法来求解最优的检查线路数量 ，使 。要使 达到最优，必须同时满足
代入上式，化简得
2）检查线路数模型求解
以题目所给数据及文献资料来分析论证，以下是芝加哥奥黑尔机场2000年1月份到12月份旅客吞吐量统计情况：
MONTH DOMESTIC INTERNATIONAL TOTAL
1 4711330 839718 5551000
2 4417410 690000 5100000
3 5433360 930000 6363490
4 5435850 879934 6315790
5 5857170 1000390 6893560
6 6178420 1150060 7329000
7 6169980 1217630 7387620
8 6213780 1171710 7385490
9 5839000 1000280 6846360
10 6100420 958940 7000360
11 5400000 828868 6229000
12 4941880 915535 5857410
根据表中数据可知，奥黑尔机场每月平均旅客吞吐量为6527938位，假设进站和出站人数各占50%，旅客到达率为100799位/天，根据资料显示，每个检查线路的服务率为2177.28位/天，每位接受预检查的旅客支付费用为85美元，接受预检查的旅客所占比例为45%，查找资料得知每张机票机场可获得平均利润为30美元，则 ，开放一个检查线路每天花费为 =1002美元，每位旅客等待消耗成本为 美元,用matlab进行处理，得到检查线路数目和机场总利润间的关系图
有图像可以得出，当机场共设置51个检查线路时，达到的利益最高。
再单独分析预检查线路的设置个数，由题目知，预检查的旅客人数占总人数的45%，则预检查旅客到达率为48960位/天， ，开放一个检查线路每天花费为 =1002美元，每位旅客等待消耗成本为 美元。用matlab进行分析得到下面图像
由图像知，设置预检查线路的数目为17条时，机场获得的收益最高。
综合以上分析，机场设置17条预检查线路，设置34条普通检查线路时可以获得最高的收益。
3.文化差异对检查点的敏感性分析
由题目可以知道，不同国家的人之间存在一定的文化差异，比如美国人特别尊重个人空间，希望人与人之间能保持一定的距离，部分国家的人则更喜欢和其他人“紧密接触”；瑞典人更注重集体利益，而中国人更注重个人利益，这会影响他们在时间紧急的情况下会不会选择插队。所以文化差异会对安检服务率产生影响，进而导致绝对通过能力和相对通过能力发生变化。下面就根据不同的文化差异对机场安检流程做出优化改进。
1）排队的间距分析
根据社会动力学模型可以得到
又
化简得到公式
用软件绘制出 关于 和 的三维图形如下图
分析图可知，当 时，可以使 取得最小值48.277，即机场应建议旅客排队等候时前后间保持约0.5米的距离，可以使安检绝对通过能力达到最优值。
2）插队的演化博弈分析
旅客安检时有2种策略选择,一种是排队，另一种是插队。假设有2个人甲和乙,他们会做出经验决策。
S为可能的策略集合, , 表示排队, 表示插队;
A为甲和乙都选择 时双方各自获得的盈利。
B为一方排队而另一方插队时,排队方获得的盈利;
C为一方排队而另一方插队时,插队方获得的盈利;
D为甲和乙都选择 时双方各自获得的盈利;
盈利指的是排队时间与冒险插队可能被工作人员要求重新排队从而使时间耽误的总效用，甲和乙的决策收益矩阵为
在决策过程中,旅客可能选择不同的策略并获得相应的盈利，在每次决策后积累经验，动态调整自己的行为策略,从而尽可能提高自己的效用。
设选择策略 的比例为x,则选择策略 的比例为1-x，x应为时间t的函数。采用策略 和 的旅客的期望收益 、 以及群体的平均期望收益 分别为
根据演化博弈论可知,采用策略s1的驾驶员的比例x的动态变化速度
将期望收益公式与平均期望收益公式带入得到
令 得到演化模型的可能稳定状态
， ， 这3个可能的稳定状态仅表示博弈方采用特定策略的比例达到该水平后将不再发生变化,根据演化博弈的基本理论,某策略是演化稳定均衡策略，必须满足主导策略对于变异策略入侵的稳定性,即演化稳定策略对微小扰动的抗干扰性。
令 ，当 时， 为演化稳定策略（ESS）
由（1）式得到：
1 ) 若A>C且B>D
无论其他人是否排队,参加排队旅客的收益 总大于其不排队的收益，此时 ， ， 。
这种情况下, 是唯一演化稳定策略，其博弈结果为人趋向于采用排队的策略，此时旅客都在排队,不会插队,使安检处的秩序良好。
2 ) 若A
无论其他人是否排队,参加插队旅客的收益总大于其排队的收益
此时 ， ， 。
这种情况下，
是唯一演化稳定策略，其结果为人趋向于插队，造成的恶性循环会使安检处的秩序恶化。
3 ) 若A>C且B
无论其他人是否排队,某个旅客排队的收益大于插队的收益;同时,其他人插队,某个旅客排队的收益小于插队的收益。
此时 ， ，
和 都是演化稳定策略，其博弈结果取决于x的初始水平.当初始的 时，人趋向于采用插队策略;当初始的 时，人趋向于采用排队策略。当所有人都插队时,排队旅客的盈利反而减少，这样会使本来排队的旅客也去插队,使得秩序混乱。
4 ) 若AD
无论其他人是否排队,某个旅客排队的收益小于插队的收益;同时,其他人插队,某个旅客排队的收益大于插队的收益。
此时， ， ，
这种情况下
是唯一演化稳定策略.博弈结果为有 比例的人趋向于排队，有 比例的人趋向于插队。
综上分析结果表明,当安检部门能对插队的旅客及时发现并进行纠正时,排队的盈利大于插队的盈利,而当以优先考虑个人利益的旅客比例增大时应适当增加安检督导工作人员的人数，这样群体就会向采用排队策略这一趋势演化并成为演化稳定状态。
4.对安全管理提出建议
根据前几问对性能指标，机场利益，不同文化差异等的分析，得出了影响机场安检通过效率的因素，针对这些因素提出以下建议：
1、机场最优开设安检线路和检查人员
经过对2000年整年客流量的分析和对开设不同数目检查线路的模拟，奥黑尔机场开设51个检查线路，且每条线路配备一名X光检查人员，一名毫米以检查人员时，可以使机场受益最大，同时使安检检查效率最高。
2、每个机场入口设置一名维护人员
因为不同文化的差异，不同文化的人有不同的习惯，美国人习惯保持一定距离，而中国人更习惯人与人之间间距较近，设置一名维护人员维护队伍之间的人均距离大约在0.5米左右，这样就可以消除不同文化之间关于人与人距离之间的差异，提高通行效率。
3、增加X光检查仪前端传送带长度，使旅客有更充分的准备时间和空间；增加后端传送带长度，使缓冲长度增加，减少旅客取行李时的拥挤程度。查阅资料并分析得知，传送带前端长度为1.2米，后端长度为2.2米时为最优长度，能尽可能的增加安检效率。
4、可以增设一个显示屏实时显示各个机场安检口的拥挤程度，使旅客选择拥挤程度最小的安检口，避免某个安检口超负荷，某个安检口却人很少的情况，增加整体效率。
5、放置物品的托盘设计为彩色，使旅客能够快速的识别并取走自己的物品。
七、Further Discussion
1）模型在实际中的应用
1.研究最佳线路数模型中的吞吐量数据来自芝加哥奥黑尔机场，应用模型时可以灵活采用不同时间不同机场的即时数据，这样可以让模型更具实用性。
2.在分析文化差异对安检效率的影响中，修正模型是在单人的移动速度一定的情况下研究最佳安检系统人员密度的，在将模型应用到实际中时，可以不断改变该移动速度的值，从而取更加贴合现实的值使模型更准确。
3.在研究文化差异方面的插队问题时，我们分析发现核心问题是对其进行演化博弈分析。在实际应用中，插队对整体安检效率的影响很小，但其可能引发的秩序混乱会对机场安全产生危害。
2）基于模型的仿真模拟
在我们建立好模型的基础上可以对机场安检过程进行仿真模拟，模拟得到的数据进一步验证模型具有优良准确性，仿真数据为模型应用于实际生活提供了保障。
八、优劣性总结
优点
1、我们的模型根据奥黑尔机场2000年全年客流量数据进行分析，所以模型有比较强的可靠性。
2、Our model is fairly robust due to our careful corrections in consideration of real-life situations and detailed sensitivity analysis.
3、除了一般情况，我们还考虑了不同文化差异下的旅客习惯，模型有较好的灵敏性。
缺点
1、因为时间限制，我们并未考虑区域D的影响，还需要进一步的完善模型。
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[外文]2 Assumptions and Justification
To make it convenient for us to simulate practical conditions, we make the following basic assumptions, each of which is properly justified.
1. The factors that influence the service system evaluation are limited. It does not include particular situations or emergencies such as passengers being unreasonable or catastrophes as these are small probability events.
2. The proportionality coefficient between different factors is objective, real and affective according to consultancy.
3. The ensemble of passengers is limitless. The passenger flow is consistent and shows infinity in the process of time.
4. The passengers come one after another. The arrival of different individuals is independent.
5. The intervals in the arrivals are random and not related to time.
6. Passengers are checked one by one in the principal of first come, first serve as social protocol demands.
7. The security check time span of each passenger is random and its distribution towards time is stable according to the database of O′Hare.
8. The cost per checkpoint per hour is the same.
3 Notations
SYMBOLS DEFINITION
the length of the queue, aka the number of passengers in the queue
the value of standardized
the time a passenger spends waiting in the system
value of standardized
absolute passing capability, the average number of passengers passing the security checkpoint per second
relative passing capability, the number of passengers passing the security checkpoint per second to the number of passengers waiting in line ratio
the satisfaction of passengers in the system, the weight value of and
the index used to evaluate the system
the eigenvalue of the matrix
the eigenvector of the matrix
the probability of passengers arrive in a time span of
Loss probability, the probability of passengers giving up waiting in line because there are too many people in line
The value of standardized
total number of pre-check lines
λ the probability of 1 passenger′s arrival per second
the intensity of the system service, or the average utilization of the checking system,
the probability of passengers having been checked when there are passengers in the system
the probability that there is no arrival into the system
the average number of passengers entering the system per second,
UN the service efficiency of the system, un= vn/ v1
VN the average speed when there are n passengers in the line
V1 the speed of one person in security check
Ρ the linear density of passengers in line
Note: the time unit of the related variables in the models is second, and the individual unit is person.
4 Model Overview
First, we establish a basic model to analyze the airport security check, which is the basis to locate and modify other models. Based on our basic model, we set up the general frame including level analysis, comparing matrices, Poisson distribution, negative exponential distribution and multi-checkpoints mixed queue.
Starting from the basic model, we made the modification on the model of the number of lines about limited cost so that we can analyze the optimal number of lines, providing the solution to the security check bottleneck.
Via understanding the influence cultural differences have on security check and queuing theory, with the help of social dynamics, we made modification to the influence that linear density of the queue and cutting have on the security check model. Also we maximized the absolute passing capability and the order of security check through data analysis in order to finish the analysis of the sensitivity of checkpoints.
5 Sub-model Ⅰ: bottleneck
5.1 Establishment of the sub-model
1) Level analysis model
We make the five main factors that influence the performance index Z: absolute passing capability A, relative passing capability Q, the length of the queue Ls, waiting time Wq and the passengers′ satisfaction S, into hierarchy, in which S equals the weight of Ls and Wq. The hierarchy is as follows:
Considering that it is better that the length of the queue is shorter; the waiting time is shorter and we define that it is better that the performance index is bigger, L and W have a negative effect on Z, so we need to make standard modification improvement when we are solving the models. On the contrary, it is better that the absolute passing capability is bigger; the relative passing capability is bigger and the passengers′ satisfaction is bigger. Therefore, we only need to nondimensionalize these three factors and get the weighted sum of these three factors and the former three after modification to get the final performance index.
The following equation can be drawn based on Fig.1 and analysis:
The final function Z can be made from the two equations above:
2) Construction of pairing comparison matrices
After the level analysis of the evaluation of airport security checking system established above, we didn′t put the factors altogether to analyze, instead, based on the one-nine method of AHP, we created two comparative judgement matrixes between the middle layer to the target layer and the detail layer to the middle layer, which examine the level system.
Measure Indication
1 The influence of and is the same
3 The influence of is slightly higher
5 The influence of is higher
7 The influence of is significantly higher
9 The influence of is absolutely higher
2,4,6,8 The influence ratio is between the two level above
1,1/2,1/3，…，1/9 The influence ratio is reciprocity of the one above
Table 1: One-nine method of AHP
According to the data collected and the feature of the passengers themselves, we consider that to the detail layer, the waiting time has a slightly higher influence than the length of the queue, so we give an assignment 3 or 1/3 in the corresponding position. The formula is , so we get the matrix D below:
Similarly, we give an assignment to the influence that the middle layer has on the target layer. According to the data of the O′Hare international airport and the information collected online, we put these five factors in the following order: A
3) Derivation of Poisson Distribution and negative exponential distribution
Based on the multi-checkpoint queueing model, we brought in the Poisson Distribution and negative exponential distribution and gave the derivation formula.
Poisson Distribution
represents the number of arrivals within the time , represents the probability of passengers arriving within the time , aka,
When meet the three conditions below, the arrival forms a Poisson Flow.
The posterior effect: the probability of k passengers arriving within the time and the number of arrival before t are irrelevant
Stability: when a is small enough, the probability of one passenger arriving within is only related to the length of , so
Universality: when a is small enough, the probability of 2 or more passengers arriving within is small enough to be ignored, so
The following is the probability distribution when the system state is
If we define the starting point of time is , then .
Within , because
The probability of no passengers arriving within is
.
We divided into and , so the probability of n passengers arriving within should be
It shows the probability of n passengers arriving within a time span of t, which accords with the Poisson Distribution, so the expectation and variance is
Negative exponential distribution
When the passenger flow is a Poisson flow, we use T to represent the interval between the arrival of two passengers, which is a random variable whose distribution function is
From the derivation of Poisson Distribution we know , so , ; the distribution density , . In here stands for the number of arrival per second. At the same time, T accords with negative exponential distribution and ,
Assuming the time it takes for a regular check-in line to examine a passenger is ( the interval between two passengers who leave the line in busy period ), accords with negative exponential distribution and the function is , , the density is , ; in here stands for average examine rate whose expectation is , representing the time it takes to examine one passenger.
4) Multi-checkpoints mixed queuing model
The state probability of the system
When there are too many people waiting in line, the new arrival will choose to wait, change ticket or refund considering his flight schedule. Therefore, there is a loss in the system, which should be classified as multi-checkpoint mixed queueing model （M/M/n/m）.
A derivation of the formula of the queuing
In the M/M/n/m model, when in the state of , because the operation rate of each line is , the total operation rate of the system is ; when , n lines are busy, so the total operation rate is . Define , the operation intensity of the system.
It is known that the rule of the arrivals accords with Poisson Distribution whose parameter is and the time of checking accords with negative exponential distribution whose parameter is . If there are k passengers, only n of which get examined while the rest wait in line and there are limitless number of positions in the line, then within the interval of , the following equations will be true:
(1) The probability of the arrival of one passenger is
(2) The probability of no arrival is
(3)
The probability of passengers having been checked when there are passengers in the system is
(4) The probability of one extra passenger arriving or n passengers having been checked is
Assuming the probability of there being k passengers in the system at is , we analyze the circumstance when at first
Through calculation we get that
If , then the solution of the equation reaches its steady state. Therefore,
So
Especially when ,
Similarly, we can solve the circumstance when
When ,
Therefore, the following four equations are true.
Then we can calculate the state probability of the system via recurrence.
In here, And because , we get that
Model index and model description
The loss probability of the system
The relative passing capability of the system
The relative passing capability of the system
The absolute passing capability
The average of engaged checkpoints
The length of the line: when
；
When ,
The length of the queue
Time passengers spend staying
Time passengers spend waiting
5.2 Solution to the model
1) Solution to the eigenvalue and the characteristic root
Define consistent matrix
If a symmetric matrix A meets the following condition
，i, j, k=1,2,…，n
A is called a consistent matrix and it has the following properties:
1.The sequence of A is 1; the sole nonzero characteristic root is n
2.Any column vector of A is an eigenvector to the characteristic root n
The matrix C and D we found are not consistent matrixes, but within the permissible range of inconsistency, we use the max root of C and D (noted as V) to identify the eigenvector of the max root as weight vector w
Via MATLAB programming we reached that
The max characteristic root of C is
Its corresponding eigenvector is
So
Therefore the weight ， ， ， ，
Similarly, the weight ，
2) Solution to the target function
We substitute the weight （i=1,2,3,4,5）and （j=1,2）into Z and get the following equation:
Then we calculate the four factors ， ， ， in the equation above
First we define the method with standardization of the factors
Translation-standard deviation transform
The primary data has different dimension. Using the following transform to make the mean of each variable 0 and the standard deviation 1 can eliminate the influence of dimension. Define
In here is the observed value of the primary variables; and are the average deviation and sample standard deviation of
，
According to the data given by the chart of the question, we use MATLAB to fit the value of absolute passing capability after standardization.
Similarly, we calculate the parameter needed: ， ， . Because of the existence of the negative factors, it is needed to make data processing via translation-range transform and get the perfected factors: ，
Having substituted the five parameters into Z, we get
To sum up, we find that the satisfaction of passengers weights the most via calculation; the bottleneck is the aspect of solving the problem of satisfaction and next is the time of waiting, which corresponds to the reality. Each factor is, considering the arithmetic is achievable, accurate. Therefore, the value of Z is accurate. Knowing the fact that , the checking system of the airline is generally satisfying in the standard we established but it needs improvement and optimization to solve the problem of the bottleneck.
6 Sub-model Ⅱ: optimum check line
According to the analysis above, the bottleneck that needs to be solved by the airport is the time passengers spend waiting. It is the emphasis of the research to design the checking lines and their number so that not only the time it takes the passengers to wait in line can be shortened, but also the cost of the airport can be minimized while reducing the expense of pre-check, getting more people involved in pre-check which lowers the time of security check.
Assuming the expense of each passengers taking pre-check is , the proportion of pre-check passengers is , the average cost, including salaries, repairing fees and abrasion loss, of each line in the system per second is and the average cost of waiting per capita in the system is , the profit of the security check system per second is:
Here we need to identify the capacity of the number of checking lines m, and because of the inconsistency of C, we use marginal analysis to solve the optimal number , which makes . To make optimal,
Therefore,
MONTH DOMESTIC INTERNATIONAL TOTAL
1 4711330 839718 5551000
2 4417410 690000 5100000
3 5433360 930000 6363490
4 5435850 879934 6315790
5 5857170 1000390 6893560
6 6178420 1150060 7329000
7 6169980 1217630 7387620
8 6213780 1171710 7385490
9 5839000 1000280 6846360
10 6100420 958940 7000360
11 5400000 828868 6229000
12 4941880 915535 5857410
Table 2: the cargo handling capacity of the O′Hare International Airport in 2000
According to the data in the table, the average of monthly cargo handling capacity in O′Hare is 6527938. Assuming the people pulling in and the people heading out each takes up 50%, the number of arrivals is 100799 per day. Datum shows that the handling capacity of each checkpoint is 2177.28 per day; the proportion of passengers taking pre-check is 45%, whose expense is 85$ per capita; the average profit of each ticket is 30$. Therefore ; the daily cost of one checkpoint is =1002$; the cost of each passenger waiting is $. Via MATLAB process, we get the relationship diagram between the number of checkpoints and the total profit of the airport.
Fig.3: Relation schema between the profit and the number of lines
It can be seen from the figure that when the number of checkpoints reaches 51, the profit reaches its peak.
Then we analyze the number of pre-check lines separately. Based on the chart given, passengers who take pre-check take up 45% of the total number, so the frequency of pre-check arrival is 48960 per day; ; the daily cost of each line =1002$; the waiting cost of each passenger is $. Via MATLAB we got the following diagram.
Fig.4: Relation schema between the profit and the number of pre-check lines
From the diagram we know that when the number of pre-check lines is 17, the profit of the airport reaches its peak.
To sum up, the airport can set up 17 pre-check lines and 34 regular lines to gain its highest profit.
7 Sub-model Ⅲ: cultural influence
According to the question, there are some cultural differences among countries, for example, Americans respect personal space and like to keep a certain distance between other people while people in some other countries prefer to stay close to each other; Swedish people lay more importance on collective interests while Chinese think of personal benefit first, which to a certain degree decides whether they will cut in line when they are in a hurry. Those differences in cultural norms will have an influence on the efficiency of security check and further causes changes in the passing capability. We will analyze the impact they have and make modification below.
1) Analysis of the linear density of the queue
Based on social dynamics model, we get that
Also,
Therefore,
Using software, we drew the figure of the function, in which x represents ; y represents and z represents .
Fig.5: Relation Schema of time to linear density and speed
The empirical value of is 0.000 in an airport line, so we assign as 0.000 and get the following figure describing the relation between and .
Fig.6: Diagram of time and linear density
According to the figure, when , reaches its minimum 48.277.
Therefore, the airport should suggest passengers keeping a distance of 0.5 meter waiting in line in order to maximize the passing capability.
2) Evolutionary game analysis of cutting in line
When passengers going through security check, they have to choices: waiting in line and cutting in line. Assuming there are two passengers A and B, they will make decisions.
S is the set of the possible strategies. , in which represents waiting in line and represents cutting.
A is the profit of either passenger when A and B both choose
B is the profit of the one waiting in line when A choose while B choose .
C is the profit of the one cutting in line when A choose while B choose .
D is the profit of either passenger when A and B both choose
The profit refers to the total utility of wait time and the delay caused by being demanded to wait in line by the officers when risking cutting in line. The profit matrix of A and B is
In the decision-making process, the passengers may choose different strategies and get the corresponding profit. Also they will gain experience after each decision and adjust their strategy to improve their profit.
Assuming the proportion of is x; the proportion of is 1-x, x is a function of t. is the expected revenue of ; is the expected revenue of and is the average.
According to the Evolutionary Game Theory, the changing speed of x
Substitute the equations above and we get
Let and we get the possible stable state of the evolution model
， ， only represent that when the proportion reaches these levels, it stops changing. According to the basic theory of the Evolutionary Game, the strategy is evolutionary stable harmonious strategy, which must meet the stability of the leading strategy against the invasion of variant strategy, also known as the anti-interference performance.
Define . When , is evolutionary stable strategy (ESS).
Then
①If A>C and B>D
The profit of passengers waiting in line is always higher than the other one whether other people stay in line or not. And
， ， 。
Under this circumstance, is the sole evolutionary stable strategy, the result of which is that people intend to stay in line.
②If A
The profit of passengers cutting in line is always higher than the other one whether other people wait in line or not. And
Under this circumstance, is the sole evolutionary stable strategy, the result of which is people tend to cut in line.
③If A>C and B
The profit of one passenger waiting in line is higher than cutting in line whether other people waiting in line or not. At the same time, other people cut in line and the profit of a certain passenger waiting in line is lower than cutting. And ， ，
and are both evolutionary stable strategy, the game result of which depends on the primary level of x. When the primary , people tend to use cutting strategy; when the primary , people tend to use waiting strategy. When everyone cuts in line, the profit of the queue decline, which causes the passengers who wait in line at the first place go cutting in line, too, leading to chaos.
④If AD
The profit of one passenger waiting in line is lower than cutting whether others wait in line or not. Also, the profit of one passenger waiting in line is higher when others cut in line. And ， ， ,
Under this circumstance, is the only evolutionary stable strategy. The result of the game is that of people tend to wait in line while of people tend to cut in line.
To sum up, when the officers can catch the passengers cutting in line in time and make correction, the profit of waiting in line is higher. When the proportion of passengers taking priority in personal interest rises, the airport need to increase the number of officers and then the group will tend to wait in line and the tendency will become an evolutionary stability.
8 Strengths and Weakness
1) Strengths
Our model is established based on the passenger flow of the O′Hare airport in 2000, which increases the reliability.
Our model is fairly robust due to our careful corrections in consideration of real-life situations and detailed sensitivity analysis.
We consider the custom of different cultures, increasing the sensitivity of the model.
2) Weakness
Because of the limit of time, we haven′t considered the influence of Section D.
9 Conclusion
Our team use the multi-checkpoint mixed queuing theory to analyze the security check of the airport and we get the performance index Z is 0.8716, which is generally satisfying but has a relatively big space for improvement. So targeting on the bottleneck, based on the situation given in the question and the passenger flow of O′Hare airport in 2000, we find the optimal number of check lines and officers and the ratio between pre-check lines and regular lines which fits to the one given by the question. At last we analyzed the linear density of the waiting line and cutting in line in different cultures. Using social dynamics and MATHMATIC we found the best distance that passengers should keep to maximize the efficiency. Using evolutionary game analysis, we studied the influence that cutting has on general efficiency and personal efficiency. To sum up, our model will increase the quality and efficiency of the security check.
10 Further discussion
1) Application of the model on reality
The data we use to study the optimal lines model is from O′Hare International airport. When putting the model into real use, different up-to-date data of other airports can be used to make the model more applicable.
In the analysis of the influence cultural differences have on security check efficiency, the modification model studies the optimal density in the circumstance that the moving speed of one person is fixed. When putting into use, the data can be changed to fit reality and increase accuracy.
When we are analyzing the problem of cutting in line, we found that the main problem is to study the evolutionary game. In application, cutting only has a little impact on the efficiency of the whole security check, but the disorder it causes may do harm to the security of the airport.
2) Simulation based on the model
On the basis of the models we established, simulation of the process of security check can be made and the data from the simulation proved that the models have high accuracy, guaranteeing the application into reality.
11 Suggestions
Based on the models simulating performance index, profit and cultural influence we established above, we confirmed the factors that influence the security check efficiency and now we propose the following suggestions.
① Optimize the check lines and officers.
Based on the simulation of the passenger flow and the number of lines, opening 51 lines with 1 X-ray officer and 1 millimeter wave officer in each line can maximize the profit of O′Hare and the efficiency of security check.
② Add one maintainer in each entrance.
The job of the maintainer is to keep passengers in line and make them behave themselves. Make sure nobody cut in line except for emergencies and keep the distance between people about 0.5 meter.
③ Lengthen the front of the X-ray conveyor to give passengers more time to get prepared and enough space. Lengthen the end of the X-ray conveyor to reduce the crowdedness when getting baggage.
Based on the data we find and analysis, the optimal length is 1.2m in the front and 2.2m in the end, which will improve the efficiency.
④ Set up screens that show the crowdedness of each checkpoint.
This will help the passengers to choose the least crowded line to avoid an overload at one checkpoint.
⑤ Colorize the plates that carry the baggage to make passengers easier to find their items.
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