應用於機場安全檢查之等候模型 - 政大學術集成

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(1)國立政治大學應用數學系碩士學位論文. 立. 政治大. ‧. ‧ 國. 學. A Tiered Security Screening System at Airport. n. al. er. io. sit. y. Nat. 應用於安全檢查之等候模型 Ch. engchi. i n U. v. 碩士班學生：黃鵬錕撰指導教授：陸行博士中華民國 105 年 5 月 24 日.

(2) Abstract This thesis proposes a tiered inspection system for airport security, wherein passengers are divided into three classes based on historical security records. A twodimensional Markov process and a Markov modulated Poisson process (MMPP) queue were used in the formulation of the security inspection system. Simulated annealing was then used to obtain near-optimum solution for the model. The efficacy of the proposed model was evaluated using the arrival data of passengers at Taoyuan International Airport and other two international airports. A comparison with two conventional queueing models with regard to the average waiting time demonstrated the effectiveness of the proposed security inspection system in enhancing service efficiency and boosting the level of security.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i. i n U. v.

(3) 中文本論文中，我們提出基於機場安全檢查的分層排隊理論模型，模型中的旅客基於歷史的安全數據被分成三組。我們運用二維馬可夫過程（two-. 政治大. dimensional Markov process）以及馬可夫調控卜瓦松過程（Markov modulated. 立. Poisson process）構建模型的排隊系統並加以分析。我們收集了台灣桃園國. ‧ 國. 學. 際機場和其它兩個機場的旅客數據以驗證我們提出的模型，並運用模擬退火法（simulated annealing）求得近似最佳解（near-optimum solution）。最後我. ‧. 們通過模型的旅客平均等候時間和另外兩種等候模型進行比較，之後得出我. sit. y. Nat. 們的模型確實可以在不增加成本，甚至提升安全性的同時能夠有效地減少平. io. n. al. er. 均等候時間。. Ch. engchi. ii. i n U. v.

(4) Contents. Abstract. 立. 中文. ‧ 國. iii. v. n. al. vii. er. io. sit. y. Nat. List of Tables. ii. ‧. List of Figures. 政治大. 學. Contents. i. Ch. i n U. v. 1. Introduction. 2. Model Formulation and Analysis. 4. 2.1. Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.1.1. A single-queue aggregated system without differentiation of passengers. 5. 2.1.2. A three-independent-lane without sharing system . . . . . . . . . . . .. 5. 2.1.3. A three-lane with sharing system . . . . . . . . . . . . . . . . . . . . .. 6. Model analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 2.2.1. H-lane and M-lane . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 2.2.2. L-lane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. 2.2. engchi. iii. 1.

(5) 2.2.3. 3. Stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 26. Simulation Study. 29. 3.1. Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29. 3.2. Simulation assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31. 3.2.1. Case A: A single-queue aggregated system . . . . . . . . . . . . . . .. 31. 3.2.2. Case B: A three-independent-lane without sharing system . . . . . . .. 32. 3.2.3. Case C: A three-lane with sharing system . . . . . . . . . . . . . . . .. 5. Conclusions. 立. ‧. A Approximation for GI/G/c Queue. 47. 48. sit. y. Nat. n. al. er. io. B Simulated Annealing Method. Bibliography. 33. 34. ‧ 國. Numerical Results. 學. 4. 政治大. Ch. engchi. iv. i n U. 52. v. 55.

(6) List of Figures. 2.1. Tiered security inspection lanes . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.2. The state transition diagram for two-dimensional Markov process . . . . . . .. 11. 2.3. Markov modulated Poisson process model . . . . . . . . . . . . . . . . . . . .. 22. 2.4. The state transition diagram for MMPP(2)/M/1 model . . . . . . . . . . . .. 4.1. Process of finding the optimal solution with D = 0.02 at Taoyuan airport . . .. 4.2. Comparison between proportion p and current average waiting time EWC with. 立. 政治大. ‧. ‧ 國. 學. 22. Nat. io. sit. y. 36. er. D = 0.02 at Taoyuan airport . . . . . . . . . . . . . . . . . . . . . . . . . . .. al. 36. . . . . . . . .. 38. 4.4. The thresholds h and m of different D at Taoyuan airport . . . . . . . . . . . .. 38. 4.5. The proportions p M and p L of different D at Taoyuan airport . . . . . . . . . .. 39. 4.6. The buffer size M of different D at Narita airport . . . . . . . . . . . . . . . .. 41. 4.7. Comparison between proportion p and current average waiting time EWC with. n. 4.3. v i n The average waiting time CEhW of different D Uat Taoyuan airport engchi C. D = 0.2 at Narita airport . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. 4.8. The average waiting time EWC of different D at Narita airport . . . . . . . . .. 42. 4.9. The thresholds h and m of different D at Narita airport . . . . . . . . . . . . .. 42. 4.10 The proportions p M and p L of different D at Narita airport . . . . . . . . . . .. 43. v.

(7) 4.11 Comparison between proportion p and current average waiting time EWC with D = 0.2 at Sydney airport . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 4.12 The average waiting time EWC of different D at Sydney airport . . . . . . . . .. 45. 4.13 The thresholds h and m of different D at Sydney airport . . . . . . . . . . . . .. 46. 4.14 The proportions p M and p L of different D at Sydney airport . . . . . . . . . .. 46. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. vi. i n U. v.

(8) List of Tables. 3.1. Arrival schedules at Taoyuan airport . . . . . . . . . . . . . . . . . . . . . . .. 30. 3.2. Parameters used in simulation at Taoyuan airport . . . . . . . . . . . . . . . .. 31. 4.1. Numerical results of Case A . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 4.2. Numerical results of Case B . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34. 4.3. The range of parameters used in simulated annealing at Taoyuan airport . . . .. 4.4. Optimal solutions by simulated annealing of Case C with fixed D at Taoyuan. 立. 政治大. ‧. ‧ 國. 學. io. sit. y. Nat. er. airport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. al. v i n Optimal solutions at Taoyuan annealing of Case C with C h airport by simulated engchi U n. 4.5. 35. 35. different D at Taoyuan airport . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. 4.6. Arrival schedules at Narita airport . . . . . . . . . . . . . . . . . . . . . . . .. 39. 4.7. Parameters used in simulation at Narita airport . . . . . . . . . . . . . . . . . .. 40. 4.8. Optimal solutions by simulated annealing with different D at Narita airport . .. 40. 4.9. Arrival schedules at Sydney airport . . . . . . . . . . . . . . . . . . . . . . . .. 43. 4.10 Parameters used in simulation at Sydney airport . . . . . . . . . . . . . . . . .. 44. 4.11 Optimal solutions by simulated annealing with different D at Sydney airport . .. 44. vii.

(9) Chapter 1 Introduction 立. 政治大. ‧ 國. 學. Since its launch in Salt Lack City by the Transportation Security Administration (TSA) in February 2008, the “Black Diamond” self-select program has been expanded to 51 airports. [2]. ‧. The self-selection process is meant to enable travelers familiar with TSA procedures to pass through checkpoints more quickly and efficiently, while giving families and others with special. y. Nat. sit. needs more time and assistance. Self-select lanes use familiar icons based on those used at ski. n. al. er. io. resorts to guide people along trails or lanes in accordance with their skill level. Green designates. i n U. v. a queue for families or beginners, blue is for casual travelers at an intermediate level, and the. Ch. engchi. black diamond is reserved for expert travelers who are familiar with TSA rules and arrive at the checkpoint fully prepared. Self-selection also helps to reduce stress and anxiety levels among passengers, and infuse a sense of calm into the checkpoint environment. Reduced stress is a winwin situation for the traveling public as well as officers involved in maintaining transportation security. From the perspective of controlling waiting lines, this program can be seen as the conversion of a security screening system into a tiered service system. Since September 11, 2001, considerable attention has been directed toward counter-terrorism. The TSA introduced the Computer Assisted Passenger Prescreening System II (CAPPS II) system, which is designed by the Office of National Risk Assessment (ONRA), a subsidiary office of the TSA. [1] On August 2009, TSA announced plans to replace CAPPS II with the Secure Flight program, another airline passenger prescreening program intended to partition passengers 1.

(10) into separate classes according to the level of risk they pose. [3] CAPPS II and Security Flight are both risk-based passenger prescreening programs aimed at enhancing security by profiling low- and high-risk passengers prior to their arrival at the airport by matching their names with lists of trusted travelers and watchlists of potentially dangerous individuals. There is growing support for tiered risk-based security system at airports. [4] Again, the passengers are divided into three classes: trusted, regular, and risky. Different screening techniques are then applied to passengers of each class, based on what is known about them. This approach is based on the notion of applying at security checkpoints passenger-related data that the government and the airlines are already collecting. Trusted passengers are individuals who have undergone a. 政治大. background check in order to gain access to an expedited security lane. Risky passengers, as. 立. identified by government intelligence systems, are subjected to more intensive scrutiny, using. ‧ 國. 學. body scanners and interviews with officers trained in behavioral analysis. Regular passengers in the middle group, who are neither vetted nor risky, would receive an intermediate level of. ‧. screening; however, ideally the process would be made quicker and more efficient than current. sit. y. Nat. procedures by prescreening suspicious passengers. The “self-select” three-tier system has al-. io. under consideration.. er. ready been implemented in many airports; however, the “risk-based” three-tier system is still. al. n. v i n arriving passengers into service groups to be Whitt [25] discussed when andChow h etonpartition gchi U. served separately. He provided a methodology by which to quantify the trade-off between economics and scale associated with larger systems and the benefits of having passengers requiring shorter service times separated from other passengers requiring longer service times. Poole and Passantino [19] demonstrated that dividing the processing of passengers into multiple levels can be more effective (from a security standpoint) than treating all passengers in the same way. They proposed a risk-based system capable of classifying passengers into two or more risk groups according to the level of risk they pose. When applying risk analysis, data mining, and applied probability to the analysis of prescreening systems, Barnett [5] concluded that Secure Flight program could be transformed from a security centerpiece to one of many components of aviation security systems. There has been tremendous interest in developing airport security screening. 2.

(11) systems using risk-based profiling methods due to the high false-alarm rates and undue pressure on security officers (see Ryu and Rhee [21]). Jacobson et al. [13] reported that a risk-based approach could greatly enhance security. Cavusoglu et al. [6] claimed that the deployment of a two-screening device architecture by TSA could blunt criticism that profiling is discriminatory while making procedures more convenient for normal passengers and reducing the economic burden of security systems. Nie et al. [17] investigated the means by which to assign passengers to queueing lanes in an airport screening system based on risk, using a steady-state nonlinear binary integer model. Song and Zhuang [23] provided a number of policy insights that are highly useful for security screening practices.. 政治大. The aim of this study investigated the benefits of implementing the proposed three-tiered se-. 立. curity screening system in terms of service quality and security screening. The proposed system. ‧ 國. 學. was developed using a two-dimensional Markov process in conjunction with a Markov modulated Poisson process. Near-optimum solutions are then obtained using the matrix geometric. ‧. method with simulated annealing. Actual data is used to evaluate the efficacy of the proposed. sit. y. Nat. model.. n. al. er. io. The remainder of the thesis is structured as follows: Chapter 2 outlines the formulation of a. i n U. v. three-lane with sharing system for airport security. We then compare three configurations: (a) a. Ch. engchi. single-queue aggregated system without differentiation of passengers; (b) a three-independentlane without sharing system, and (c) a three-lane with sharing system. In Chapter 3, we analyze the configurations using simulations and numerical examples. Chapter 4 presents specific computing results and graphics. In Chapter 5, we draw conclusions and propose directions for future research into airport inspection and waiting time models. Details of the methods used in this study can be found in the Appendix.. 3.

(12) Chapter 2 Model Formulation and Analysis 政治大立 Model formulation. ‧. ‧ 國. 學. 2.1. Passengers arriving at an airport security inspection can be classified into multiple classes.. sit. y. Nat. This classification can be made based on either risk level assessments or passenger service re-. io. er. quirements. Each class of passengers has its own service time distribution and service goal. For a given service capacity such as total number of servers, we examine different service config-. n. al. Ch. i n U. v. uration to minimize the waiting time while maintaining the security level. Namely, the goal. engchi. is to maximize the security inspection effectiveness which can be translated to minimize the probability of “false clear.” From the passenger’s viewpoint, the goal is to maximize the passenger service quality which can be achieved by minimizing the passenger waiting cost or the probability of “false alarm.” Assume that there are three classes of passengers according to the risk level which are labeled as H-, M-, and L- classes. The arrival process of passengers of class-i (i = H, M, L) is a Poisson process with rate λi . The three arriving processes are independent from each other, and λ M ≥ λ L ≥ λ H . Let Λ = λ H + λ M + λ L be the total arrival rate. The service time for class-i is independent and identically distributed (i.i.d). Denoted by Bi (t) the service time’s cumulative distribution function for class i is with service rate µi where µ L ≥ µ M ≥ µ H . In the following sections, we first study a single-queue aggregated system. 4.

(13) that does not discriminate the passengers in section 2.1.1, and study a three-independent-lane without sharing system in section 2.1.2. Finally, a three-lane with sharing system is developed in section 2.1.3.. 2.1.1 A single-queue aggregated system without differentiation of passengers The basic configuration is to combine arrivals of the three classes into a Poisson process with total arrival rate of Λ and serve these passengers with 3 inspection levels of homogeneous. 政治大. servers. Thus, the average waiting time can be estimated as M/G/3 in which the service time’s. 立. B (t) =. 學. λH λM λL BH (t) + BM (t) + B L ( t ). Λ Λ Λ. ‧. ‧ 國. c.d.f, denoted by B(t), is a mixture of c.d.f’s. i.e.. sit. y. Nat. We will adopt the well-known approximations to evaluate the performance of such an M/G/3. io. er. system. This basic configuration is used as a bench mark for other two security inspection mod-. al. els. In this setting, there is no significantly different inspection procedure for security among. n. v i n C h of approximationUmethod is described in Appendix A. all passengers. The specific procedure engchi 2.1.2 A three-independent-lane without sharing system. This is a configuration with three dedicated server systems serving three classes. It implies that we may estimate the waiting time by three independent M/M/1 systems. The i-class system has a Poisson arrival process with rate λi . The service rate is set with respect to each inspection level. In contrast to the single-queue aggregated system, this setting is constructed separately without sharing the resources when passengers are examined at the security inspection.. 5.

(14) 2.1.3 A three-lane with sharing system This is a configuration with the features of both the single-queue aggregated system and the three-independent-lane without sharing system. It can also be called a hybrid system. We need to develop the new procedures to compute the performance measures for this configuration. This is a more complex situation. Thus we first assume that the service times are all exponentially distributed and each queue is served by a single server. Due to the security inspection requirement, passengers in the lower class can share the service with the higher class but not the other way around. Such a sharing scheme will ensure that the required security level will not be hurt.. 政治大 of the system performance. Figure 2.1 below shows the idea described above. 立. Therefore, any improvement in passenger service measures indicates the overall improvement. ‧ 國. h. pM λ M m. n. Ch. sit. io. al. y. ‧. Nat. λM (1 − pM ) + pL λL if H-queue is shorter than h and M -queue is shorter than m; λM + pL λL if Hqueue is at least h and M -queue is shorter than m; λM (1 − pM ) if H-queue is shorter than h and M queue is at least m; otherwise λM .. engchi. er. otherwise λH .. 學. λH + pM λM if H-queue is shorter than h,. i n U. v. pL λ L. λL (1 − pL ) if M -queue is shorter than m, otherwise λL .. Figure 2.1: Tiered security inspection lanes The p M proportion of M-class passengers are sent to H-queue for security inspection, as long as the number of passenger in H-queue is below the threshold h. In the same way, the p L proportion of L-class passengers are sent to M-queue for security inspection, as long as the 6.

(15) number of passenger in M-queue is below its threshold value m. Obviously, this is a more general configuration as p M = p L = 0, reducing the three-lane with sharing system is to a three-independent-lane without sharing system. In contrast to the previous two configurations in section 2.1.1 and section 2.1.2, this configuration must be examined in detail. We use the queueing model to compute the expected waiting time and the simulation method to determine the optimal configuration of parameters. The computational approach provides a benchmark for numerically evaluating the performance of the system and the simulation approach elucidates the performance effects of the system and configuration of parameters such as M, h, m, p M , and p L .. System stability analysis. 立. 政治大. ‧ 國. 學. In this section, we will analyze the stability of the three-lane with sharing system using the. ‧. method in [26] to show that the proposed model is solvable. Firstly, we define the following events: A = {inspection system gives an alarm}; T = {the passenger is threat}; FI = {the. y. Nat. sit. passenger is selected for further inspection (to H-lane from M-lane)}; FI c is the complement. n. al. er. io. of FI. Then we set the proportion of further inspection is P( FI ) = p, which is the proportion. i n U. v. of passengers who are sent to H-lane from M-lane, where p can be two: (i) when the number. Ch. engchi. of passengers in H-lane is smaller than h; (ii) when the number of passengers in M-lane is full. Secondly, we need to define True Alarm (TA), which is the case that the system gives an alarm and a threat exists. Our task now is to prove that the tiered queueing model can indeed improve the security level via maximizing the P( TA), the probability of true alarm, and existence of p as the average waiting time of passengers is reduced. To achieve this goal, we define the following probabilities: the TA rate includes the threat from the further inspection of passengers in M-lane θ FI ( p) = P( A| T ∩ FI ), and the threat from the passengers in Mland θ FI c ( p) = P( A| T ∩ FI c ). We also need the information that the threat rate of further inspection passengers α( p) = P( T | FI ), and threat rate of passengers of inspection in M-lane β( p) = P( T | FI c ). Note that these probabilities have been denoted as functions of p, since they all can be controlled by p. Because the inspection procedure in H-lane is far more strict than in 7.

(16) M-lane, we assume θ FI ( p) > θ FI c ( p). By the law of total probability, we write the probability of true alarm P( TA) as a function of p, P( TA) = f ( p) = θ FI ( p)α( p) p + θ FI c ( p) β( p)(1 − p).. We also need to make the following specific assumptions in developing the model. Assumption 2.1.1. By profiling assumption, we know that the true threat from H-lane is greater than that of M-lane, i.e., P( A ∩ T | FI ) > P( A ∩ T | FI c ). 政治大 Assumption 2.1.2. The sensitivity 立 to change of further inspection is higher than that of its com-. ‧ 國. 學. plement, which means. dP( A ∩ T | FI ) dP( A ∩ T | FI c ) > . dp dp. ‧. In other words, (2.1.1) is equal to. y. Nat. n. al. er. io. sit. | ( P( A| T ∩ FI ) P( T | FI ))′ p| > | ( P( A| T ∩ FI c ) P( T | FI c ))′ (1 − p)|, and. Ch. i. e. i n U. v. h | (θ FI ( p)α( p))′ p|n>g| (c θ FI c ( p) β( p))′ (1 − p)|.. Therefore, we get the following proposition of P( TA). Proposition 2.1.3. P( TA) is an increasing function of p, that is, ∂P( TA) > 0. ∂p. 8. (2.1.1).

(17) Proof: ( ′ ) ( ′ ) ∂P( TA) ′ = p θ FI ( p)α( p) + θ FI ( p)α′ ( p) + (1 − p) θ FI c ( p ) β ( p ) + θ FI c ( p ) β ( p ) ∂p. + θ FI ( p)α( p) − θ FI c ( p) β( p) = p (θ FI ( p)α( p))′ + (1 − p) (θ FI c ( p) β( p))′ + −. P( A ∩ T ∩ FI ) P( T ∩ FI ) P( T ∩ FI ) P( FI ). P( A ∩ T ∩ FI c ) P( T ∩ FI c ) P( T ∩ FI c ) P( FI c ). = p (θ FI ( p)α( p))′ + (1 − p) (θ FI c ( p) β( p))′ + P( A ∩ T | FI ) − P( A ∩ T | FI c ). 政治大. According to Assumption 2.1.1 and 2.1.2, we know. 立. ∂P( TA) ∂p. > 0 and complete the proof.. So far, we have shown that P( TA) is an increasing function of p, i.e., with an increase. ‧ 國. 學. in the number of passengers sent to H-lane from M-lane, the true alarm rate and the safety. ‧. performance of the proposed model will enhance stability. Herein, we modify the value of p to adjust the requirement of true alarm TA in the model. In addition to enhancing security, it was. y. Nat. sit. our intention that the model would decrease the average waiting time. Assume that EW ( p) is. n. al. er. io. the average waiting time for the model, which is obviously a function of p. The purpose of the. i n U. v. model can be summarized using the following mathematical programming.. Ch. engchi Min EW ( p). s.t. D ≤ p < 1,. where D is a predetermined threshold for p, which is presented as the expected minimum requirement of the true alarm. The use of simulation to obtain the optimal model configuration is outlined in a later chapter.. 9.

(18) 2.2. Model analysis. The H- and M-class queue can be modeled as a two dimensional Markov process. The Lclass queue can be treated as MMPP(2)/M/1 queue. We first solve for the stationary joint distribution for the H- and M-queues. Then we approximate the MMPP(2)/M/1 L-queue with the superimposition of the two M/M/1 queues.. 2.2.1. H-lane and M-lane. 政治大 sents the number of passengers in queue i at time t. We assume that the stability condition (to 立 Consider a two-dimensional Markov process {( X H (t), X M (t)), t ≥ 0} where Xi (t) repre-. ‧ 國. 學. be specified later) is satisfied and the steady state is reached. To numerically solve for the stationary distribution of the system, we assume that the buffer size of the passengers with middle. ‧. risk is set to M.. io. sit. y. Nat. Next, the state space of the Markov process {( X H (t), X M (t)), t ≥ 0} is. n. al. er. Ω = {(i, j)| i = 0, 1, 2, . . . , j = 0, 1, . . . , M}.. Ch. engchi. i n U. v. Let h and m be the thresholds at the H-lane and M-lane, respectively. The state transition diagram for a case with h = 4 and m = 3 is demonstrated in Figure 2.2 below. Denoted by n is the set of state vectors with common first component n (the number of passengers in H-lane), where n = 0, 1, 2, . . . n = {(n, 0), (n, 1), (n, 2), . . . , (n, M)}.. Let ( X H , X M ) be the limit of ( X H (t), X M (t)) as t → ∞. Define the stationary probabilities. 10.

(19) λM (1 − pM ) + pL λL. 0,0. µM. λM (1 − pM ) + pL λL. µM. λM (1 − pM ). µM. µM. 1,3. µM. µM. µH. λM (1 − pM ). λM (1 − pM ). µM. µM. 1,4. 政治大. 立. µM. λM (1 − pM ). λM (1 − pM ) + pL λL. 2,2. 2,3. µM. µM. ‧ 國. λM (1 − pM ). λM (1 − pM ) + pL λL. 3,2. µM. 學. µM. λM (1 − pM ) + pL λL. 3,1. 3,3. µH. µM. µM. µM. λM (1 − pM ). µM. µM. ‧. io. 5,0. λ M + pL λ L. µH. µM. λH µ H. µH. 5,1. λH. al. n. λH. µH. Ch. λ M + pL λ L. µH. µM. λH. µH. e n gµ c h i5,3U 5,2 λ M + pL λ L M. λH. λH. µH. µH. y. λM µM. sit. Nat. λ M + pL λ L λ M + pL λ L λ M + pL λ L 4,1 4,2 4,3 µM µM µM. iv. 4,4. nλ. M. µM. λH. µH. 3,M. µH λM µM. λM µM. 4,M. λ H + λM λM µM. λM µM. Figure 2.2: The state transition diagram for two-dimensional Markov process. µH. 5,M. λ H + λM. h = 4, m = 3. 11. µH. λ H + λM. µH. 5,4 λH. M. λ H + λM. λ H + pM λ M λ H + pM λ M λ H + pM λ M λ H + pM λ M µH µH µH µH µH. 4,0. λM (1 − pM ) 2,M µ. λM (1 − pM ). 3,4. 1,M. λ H + λM. λM (1 − pM ). 2,4. 0,M. λ H + λM. λ H + pM λ M λ H + pM λ M λ H + pM λ M λ H + pM λ M µH µH µH µH µH λM (1 − pM ) + pL λL. λH. µM. λM (1 − pM ). λM (1 − pM ) + pL λL. 1,2. λM (1 − pM ) + pL λL. 2,1. µM. 3,0. λH. λM (1 − pM ). 0,4. µM. λ H + pM λ M λ H + pM λ M λ H + pM λ M λ H + pM λ M µH µH µH µH µH. 2,0. λ H + pM λ M. λM (1 − pM ) + pL λL. 1,1. λM (1 − pM ) + pL λL. λ H + pM λ M. 0,3. µM. λ H + pM λ M λ H + pM λ M λ H + pM λ M λ H + pM λ M µH µH µH µH µH. 1,0 λ H + pM λ M. µM. λM (1 − pM ). λM (1 − pM ) + pL λL. 0,2. er. λ H + pM λ M. λM (1 − pM ) + pL λL. 0,1. µH.

(20) as follows: π (i, j) = lim P{ X H (t) = i, X M (t) = j} = P( X H = i, X M = j) t→∞. πi = (π (i, 0), π (i, 1), . . . , π (i, M)), i ≥ 0.. Because of the high complexity of this queueing system, matrix analytic approach is employed to establish the steady-state equations in matrix form. The infinitesimal generator Q is given by 0. A11 A12 .. .. . . A2. ... .. ‧. h−1. Nat. A(h−1)(h−1) A(h−1)h A1. sit. A2. A2. n. al. er. io. h+1 .. .. Notation. A2. y. 1 .. .. h. h+1 ··· ···. 學. Q=. 立. 治政 ··· h−1 大h. A00 A01. ‧ 國. 0. 1. Ch. n U i engch. iv. A0 A1 .... A0 .... M : the buffer size of M-lane for passengers with middle risk h : the threshold of queue length associated with passengers with high risk m : the threshold of queue length associated with passengers with middle risk p M : the percentage of passengers in M-lane who are sent to H-lane p L : the percentage of passengers in L-lane who are sent to M-lane λ H : the average arrival rate of passengers with high risk µ H : the average service rate per server for passengers with high risk 12. ....

(21) λ M : the average arrival rate of passengers with middle risk µ M : the average service rate per server for passengers with middle risk λ L : the average arrival rate of passengers with low risk µ L : the average service rate per server for passengers with low risk The state equations are given by ΠQ = 0 in which Π denotes the steady-state probability vector and 0 is the zero row vector. The sub-matrices in Q are defined in the following. For 0 ≤ i ≤ h − 1, let θ = λ M (1 − pm ) + p L λ L , then we have. 政治大. 立. (i, 0) (i, 1) · · · (i, m − 1) (i, m). q1i .. .. (i, m − 1). .. qim−1. θ. µM. qim .. .. n. er. io. (i, M). al. ... .. µM. Nat. (i, m) .. .. θ ... Ch. engchi U. .. y. µM. sit. (i, 1) .. .. ‧ 國. θ. ‧. q0i. (i, M). 學. Aii =. (i, 0). ···. v ni. λ M (1 − p m ) .. .. ... µM. qiM. Let τ = λ H + p M λ M , then we have. (1, 0) · · · (1, M − 1). A01 =. (0, 0) .. .. (1, M). τ ... (0, M − 1). .. . τ λH + λM. (0, M) Notice that A01 = A12 = A23 = · · · = A(h−1)h .. Considering the repetitive portion A2 , A1 and A0 , we have A2 = µ H I. 13. ..

(22) For i ≥ h,. A1 =. (i, 0). (i, 1). (i, 0). q0i. λM + pL λL. (i, 1) .. .. µM. q1i .. .. ···. (i, m − 1). λM + pL λL .. .. ... µM. qim−1. λM + pL λL. µM. qim .... (i, m − 1) (i, m) .. . (i, M). 政治大 λH ... . λH. y. Nat. n. qij j=0. i=0. Ch. engchi. −λ H − λ M − p L λ L. µM. qiM. λH + λM. i n U. v. i≥1. −λ H − λ M − p L λ L − µ H. 1 ≤ j ≤ m − 1 −λ H − λ M − p L λ L − µ M m≤j≤M. .... er. io. sit. (i, M). al. λM .... (i + 1, 0) · · · (i + 1, M − 1) (i + 1, M). (i, M − 1). where qij is given as follows:. .. ‧. A0 =. (i, 0) .. .. · · · (i, M). 學. ‧ 國. 立. (i, m). −λ H − λ M − µ M. −λ H − λ M − p L λ L − µ H − µ M −λ H − λ M − µ H − µ M. 14.

(23) Before proceeding further, let us define the submatrix of Q in the initial portion as follows . . B00. A A01   00     A2 A11 A12       = A2 A22 A23      .. .. .. . . .     A2 A(h−1)(h−1). B01. 0.         h( M+1)×( M+1). h( M+1)×h( M+1). ]. B10 = 0 · · · 0 A2. . ( M+1)×h( M+1). 治政 Therefore, we can rewrite the infinitesimal generator大 Q 立. 0. A0. 0. 0 0. io. er. Nat. A2 A1 A0 .. .. .. . . .. ‧. A1 A0. · · ·  · · ·    · · ·   · · ·   ···. y. 0. sit. 0. . 學. ‧ 國. 0 .. ..     =   . A ( h −1) h. [.  B00 B01  B  10 A1   Q =  0 A2    0 0   . .. .. .. . . al. n. v i n Below we analyze this model in We now turn to using matrix geometric method Cmore h e detail. ngchi U. to analyze this model. After writing down the sub-matrices explicitly, we can obtain the steadystate probability by a computation recursively. Noting that for the repetitive portion, we have. πi A0 + πi+1 A1 + πi+2 A2 = 0,. i = h, h + 1, · · · .. (2.2.1). Therefore, πi is a function only of the transition rates between stationary probabilities with. (i − 1) queued passengers and stationary probabilities with i queued passengers in H-lane. Then. 15.

(24) there exists a matrix R such that πi+1 = πi R, or. i = h, h + 1, · · ·. πi = πh Ri−h ,. (2.2.2). i = h, h + 1, · · · .. Substituting (2.2.2) into (2.2.1), we get πh Ri−h A0 + πh Ri−h+1 A1 + πh Ri−h+2 A2 = 0,. i = h, h + 1, · · · .. 政治大. Since it is true for πh Ri−h ̸= 0, substituting i = h, we have. 立. (2.2.3). ‧. ‧ 國. 學. A0 + RA1 + R2 A2 = 0.. n. er. io. al. sit. Nat. R = −{ A0 + R2 A2 } A1−1 .. y. It is common practice to use the iterative procedure that is derived from the (2.2.3), namely. Ch. Therefore the recursive solution is given by. engchi. i n U. v. R(0) = 0, R(k + 1) = −{ A0 + R2 (k ) A2 } A1−1 ,. where R(k) is the value of R in the kth iteration. The iteration is repeated until the two successive iterations differs by less than a predefined parameter δ, that is. ∥ R(k + 1) − R(k)∥2 < δ.. 16.

(25) For the initial portion, we solve (π0 π1 · · · πh ) by using R, as we already know,. [π0 π1 · · · πh−1 ] B00 + πh B10 = 0, [π0 π1 · · · πh−1 ] B01 + πh A1 + πh+1 A2 = 0,. or [. ] π0 π1 · · · πh−1 πh. 立. Expanding the expression. 政治大 ]. . . ‧. ‧ 國. (2.2.4).  B00   = 0, π0 π1 · · · πh−1 πh  B10. . n. al. [ π0 π1 · · · πh−1. .  A00 A01    A   2 A11 A12     ]  A2 A22 A23   = 0. πh    .. .. ..   . . .      A2 A(h−1)(h−1)      A2. er. io. sit. y. Nat. which is. B01   B00  = 0.  B10 A1 + RA2. 學. [. . . Ch. engchi. 17. i n U. v.

(26) Then we have the following equations π0 A00 + π1 A2 = 0 π0 A01 +π1 A11 + π2 A2 = 0 π1 A12 +π2 A22 + π3 A2 = 0 .. . πh−3 A(h−3)(h−2) +πh−2 A(h−2)(h−2) + πh−1 A2 = 0 πh−1 A(h−1)(h−1) + πh A2 = 0.. 立. 學. ‧ 國. We can get. 政治大. π1 = −π0 A00 A2−1. ‧. π2 = − (π0 A01 + π1 A11 ) A2−1. y. .. .. io. sit. Nat. π3 = − (π1 A12 + π2 A22 ) A2−1 (. ). n. al. er. πh−1 = − πh−3 A(h−3)(h−2) + πh−2 A(h−2)(h−2) A2−1 ) ( πh = − πh−2 A(h−2)(h−1) + πh−1 A(h−1)(h−1) A2−1 .. Ch. engchi. i n U. v. Also, we rewrite the expressions above in the following form: π1 = π0. (. − A00 A2−1. ). π2 = − (π0 A01 + π1 A11 ) A2−1 ) ) ) ( ( ( = π0 − A01 + − A00 A2−1 A11 A2−1 π3 = − (π1 A12 + π2 A22 ) A2−1 ) ) ) ) ) ( ( ( ( ( = π0 − − A00 A2−1 A12 + − A01 + − A00 A2−1 A11 A2−1 A22 A2−1 .. .. 18.

(27) Therefore, (π0 π1 · · · πh ) can be expressed as [. ] π0 π1 · · · πh = π0 X1 ,. where [. (. − A00 A2−1. X1 = I. We also know that 1 = [π0 π1 · · · πh−1. (. − A01 +. − A00 A2−1. ). ) A11. 政治大 ]e + π ∑ R e = [π π · · · π 立 ∞. h. i −1. 0. 1. ] A2−1. ··· .. h − 1 ] e + πh ( I. − R)−1 e,. i =1. ∞. ∑ R i −1. = ( I − R ) −1 .. ‧. ‧ 國. 學. and. i =1. y. Nat. n. al.  π0 .  e. ∗ B01. er. io. . sit. Eventually, from (2.2.4), we get the system of linear equations in the matrix form [ ]  = 1 0 ,. i C ∗n − 1 ( I − R)h ee ( A1 + RA ) n g c h i 2U. v. ∗ and ( A + RA )∗ are B and A + RA with first column being eliminated, where B01 2 2 1 01 1. and e is the column vector of 1 with suitable size. Now, it is apparent that we can get the expected waiting time and the expected number of passengers in the H- and M-lanes. Firstly, the arrival rates at different state conditions of M-lane are given below. 0 ≤ j ≤ m−1. m≤j≤M. 0 ≤ i ≤ h − 1 λ M (1 − p M ) + p L λ L λ M (1 − p M ) i≥h. λM + pL λL. 19. λM.

(28) Because of limitation of capacity of inspection for passengers with middle risk, the probability of insufficient inspection is estimated by γ=. ∞. ∑ π (i, M).. i =0. So the effective arrival rate of passengers with middle risk is λ M (1 − γ ).. 政治大. We have the average queue length of M-lane. 立. ∑ j ∑ π (i, j).. 學. ‧ 國. ∞. M. SM =. j =1 i =0. ‧. Also, the average effective arrival rate of passengers in M-lane is given by. sit. y. Nat. { } h −1 { } h −1 m −1 λ Mave = λ M (1 − γ)(1 − p M ) + p L λ L ∑ ∑ π (i, j) + λ M (1 − γ)(1 − p M ) ∑. er. io. i =0 j =0. n. ∞ { } ∞ m −1 γ ) + λ M (1 − γ) + paL λl L ∑ ∑ π (i, j) + λ M (1 − v i ∑. Ch. i = h j =0. = λ M (1 − γ ) − p M λ M (1 − γ ). engchi. h −1 M. Un. M. ∑ π (i, j). ∞ m −1. π (i, j).. i =0 j =0. Thus we can compute the average waiting time of passengers in M-lane. WM =. 20. SM . λ Mave. ∑ π (i, j). i =0 j = m. i=h j=m. ∑ ∑ π (i, j) + p L λ L ∑ ∑. i =0 j =0. M.

(29) Secondly, the arrival rates at different state conditions of H-lane are 0 ≤ j ≤ M−1. j=M. 0 ≤ i ≤ h−1. λH + pM λM. λH + λM. i≥h. λH. λH + λM. Then we have the average number of passengers in H-lane ∞. SH =. M. ∑ i ∑ π (i, j).. i =0 j =0. 政治大 Also, we can compute the average effective arrival rate of passengers in H-lane as 立. ‧ 國. h −1 M −1. ∑ ∑. i =0 j =0. ∑ ∑. i = h j =0. ∞. π (i, j) + (λ H + λ M ) ∑ π (i, M ) i =0. ∞. ‧. = λH + pM λM. i =0 j =0. ∞ M −1. 學. { } h −1 M −1 λ Have = λ H + p M λ M ∑ ∑ π (i, j) + λ H. π (i, j) + λ M ∑ π (i, M). i =0. sit. y. Nat. n. al. er. io. Therefore, the average waiting time of passengers in H-lane is given by. 2.2.2. Ch. i n U. SH = . λ Have. eWnH g c h i. v. L-lane. For L-lane, we consider an MMPP(2)/M/1 model. This model is a process that behaves like a Poisson process in phase 1 of which the duration is described by a random variable Y1 with parameter ω1 for a time that is exponentially distributed when the number of passengers in M-lane increase from (m − 1) to m with a mean. 1 α12 .. Then it switches to a Poisson process in. phase 2 having Y2 with parameter ω2 for a time period that is exponentially distributed when the number of passengers in M-lane decrease from m to (m − 1)with a mean. 1 α21 .. It then switches. back to the phase of Y1 back and force, again and again infinitely often. The Figure 2.3 shown. 21.

(30) below illustrates the model.. α12. Y1. ω1. ω2. α21 Figure 2.3: Markov modulated Poisson process model. 政治大. The process of the MMPP(2)/M/1 queue is shown in Figure 2.4. 立. ω1. 2,1. 1,1. µL. µL α12. α21. Nat. io. α21. y. ω2. 1,2. al. µL. Ch. ...... ω2. 2,2. n. µL. α12. sit. ω2. µL. er. α21. 0,2. .... ‧. α12. ω1. 學. 0,1. ‧ 國. ω1. engchi U. v ni. ... µL. Figure 2.4: The state transition diagram for MMPP(2)/M/1 model. To motivate the discussion on the MMPP(2)/M/1 queue, we need the following notations. Let ϕi = [ϕ(i, 1), ϕ(i, 2)] denote the vector of the steady-state probabilities that the process is in state with i passengers in L-lane, where ϕ(i, j) is the steady-state probability of being in state. (i, j), and p1 and p2 denote the probabilities that the process is in phase 1 and 2, respectively. Note that. ∞. pj =. ∑ ϕ(i, j). i =0. 22. j = 1, 2..

(31) From the MMPP(2)/M/1 model, we know α12 p1 = α21 p2 1 = p1 + p2 .. Thus the average arrival rate of L-lane is given by λ Lave = ω1 p1 + ω2 p2 = ω1. α21 α12 + ω2 . α12 + α21 α12 + α21. 政治大. From the discussion of H-lane and M-lane, we can compute the probabilities p1 and p2 as. 立. ∞ m −1. p1 =. ∑∑. 學. ‧ 國. the following equations:. π (i, j). i =0 j =0 M. ∑ ∑ π (i, j) = 1 − p1.. io. y. sit. Nat. i =0 j = m. ‧. ∞. p2 =. n. al. er. Changing between the phases only happens when the number of passengers in M-lane change. i n U. v. between m and (m − 1). In other words, the effect arrival rate of L-lane depends on the number. Ch. engchi. of passengers in M-lane, when it decreases from m to (m − 1), the MMPP(2) model switches to phase 1 from phase 2. Similarly, when the number of passengers in M-lane increases from. (m − 1) to m, the model switches back to phase 2. Because we already know the service rate of M-lane, we can compute the probability from phase 2 to phase 1 in MMPP(2) model, which is

(32) α21 = P{ X M = m

(33) X L is in phase 2}µ M P{ X M = m}µ M P{ X L is in phase 2} ∑∞ π (i, m)µ M . = i =0 p2. =. 23.

(34) We also know that α12 p1 = α21 p2 , then we have p2 α21 p1 p ∑∞ π (i, m)µ M = 2 i =0 p1 p2 ∞ ∑ π (i, m)µ M = i =0 . p1. α12 =. Therefore, we define the parameters of MMPP(2)/M/1 model of L-lane, ω1. ω2. λ L (1 − p L ). α21. 治政 λ 大 L. ∑i∞=0. π (i,m)µ M p1. We have the infinitesimal generator. ···. n. y.   F 0 0 · · ·   L F 0 · · ·    B L F · · ·  .. .. .. . . .. engchi. . where. 0. sit. io. Ch. 0. er. Nat. al. 0. . ‧.  D D1  0  B L   U= 0 B   0 0 . π (i,m)µ M p2. 學. ‧ 國. 立. α12. ∑i∞=0. i n U . v. . . α12 −(ω1 + α12 )   ω1 0  D0 =   , D1 =   α21 −(ω2 + α21 ) 0 ω2 . . . . . . α12   ω1 0  −(ω1 + α12 + µ L ) µ L 0  B= . , F =  , L =  0 ω2 α21 −(ω2 + α21 + µ L ) 0 µL. 24.

(35) Let ϕ = [ϕ0 , ϕ1 , ϕ2 , · · · ], because ϕU = 0, it produces     ϕ0 D0 + ϕ1 B     ϕ0 D1 + ϕ1 L + ϕ2 B       ϕk−1 F + ϕk L + ϕk+1 B. =0 =0 = 0 ( k ≥ 1).. Since ϕj is a function only of the transition rates between states with (k − 1) queued passengers and states with k queued passengers, we have. 政治大. ϕj = ϕj−1 T j = 2, 3, · · ·. 立or ϕ = ϕ T 1. j.   ϕ0 D1 + ϕ1 ( L + T B). =0. io. y. ]. n. a l[. Ch. ϕ0 ϕ1. . . sit. =0. er. ‧ 國.    ϕ0 D0 + ϕ1 B. ‧. Nat. Or. j = 2, 3, · · · .. 學. It means that. j −1. i n U. v. D1   D0  = 0.  B L + TB. engchi. We also need ∞. ∞. k =0. k =0. ∑ ϕk e = ϕ0 e + ϕ1 ∑ T k e = {ϕ0 + ϕ1 [ I − R]−1 }e = 1.. The expected number of passengers and the expected waiting time in the L-lane are given by 2. SL =. ∑. ∞. ∑ iϕ(i, j) =. j =1 i =0. WL =. SL λ Lave 25. 2. ∞. ∑ ∑ iϕ0 ( Ri ) j. j =1 i =0.

(36) where ( Ri ) j is the jth column of the matrix Ri .. 2.2.3 Stability conditions Before approaching simulation study, we first pose the problem of finding what is the stability conditions in the model. We use the following two lemmas to give these conditions. Then we will rewrite the mathematical programming of the proposed model to find the optimal solution. Lemma 2.2.1. To reach a stable state, we must make sure that the following inequalities are. 政治大. satisfied.. 立. 學. ‧ 國. λ Have < µ H λ Lave < µ L .. ‧. In other words, the inequalities can be simplified as. io. sit. y. µ H − λ H − λ M ∑i∞=0 π (i, M) 1 λ M ∑ih=−01 ∑ jM=− 0 π (i, j ). er. Nat. 0 ≤ pM <. λL − µL < pL ≤ 1 λ L p1. n. al. Ch. engchi. i n U. v. where p1 = ∑im=−01 ∑ jH=0 π (i, j) as defined before. Proof:. From definition of the average arrival rate in H-lane and L-lane, it follows that { } h −1 M −1 λ Have = λ H + p M λ M ∑ ∑ π (i, j) + λ H i =0 j =0. = λH + pM λM. h −1 M −1. ∑ ∑. i =0 j =0. ∞ M −1. ∑ ∑. i = h j =0. ∞. π (i, j) + λ M ∑ π (i, M). i =0. 26. ∞. π (i, j) + (λ H + λ M ) ∑ π (i, M ) i =0.

(37) Since λ Have < µ H , simplifying the expression, we get h −1 M −1. ∑ ∑. λH + pM λM. i =0 j =0. ∞. π (i, j) + λ M ∑ π (i, M ) < µ H . i =0. Therefore, we have the following inequality. pM <. µ H − λ H − λ M ∑i∞=0 π (i, M ) 1 λ M ∑ih=−01 ∑ jM=− 0 π (i, j ). .. By the same method, as λ Lave < µ L , we know. 立λ. L ave. = λ L (1 − p L ) p1 + λ L p2 ,. ‧. ‧ 國. 學. and. 政治大. p1 + p2 = 0.. y. Nat. n. Ch. λL − µL . λ L p1. engchi. er. io. al. pL >. sit. Simplify the equation, we get. i n U. v. p M and p L are proportions, so this completes the proof of Lemma 2.2.1. Next we will discuss the range of p by Lemma 2.2.2. First, we have h −1 M −1. p=. ∑ ∑. i =0 j =0. ∞. π (i, j) + ∑ π (i, M ).. (2.2.5). i =0. Lemma 2.2.2. To reach a stable state, the proportion p sent from M-lane to H-lane need to satisfy the inequality below. 0<p<. µ H − λ H − (1 − p M )λ M ∑i∞=0 π (i, M ) . pMλM. The proof of this result is quite similar to that given for Lemma 2.2.1 and so is omitted. After 27.

(38) we have Lemma 2.2.2, let D be the pre-given minimum requirement to adjust true alarm rate of the model. As a consequence, we rewrite the mathematical programming in section 2.1.3 as follows:. Min EW ( p). (2.2.6). h −1 M −1. s.t. p =. ∑ ∑. i =0 j =0. p≥D. ∞. π (i, j) + ∑ π (i, M ) i =0. 政治大. λL − µL < pL ≤ 1 λ L p1. ‧. ‧ 國. 立. 學. µ H − λ H − (1 − p M )λ M ∑i∞=0 π (i, M) 0≤p< pMλM µ H − λ H − λ M ∑i∞=0 π (i, M ) 0 ≤ pM < 1 λ M ∑ih=−01 ∑ jM=− 0 π (i, j ). n. er. io. sit. y. Nat. al. Ch. engchi. 28. i n U. v.

(39) Chapter 3 Simulation Study 政治大立 ‧ 國. 學. Simulations were conducted using the actual data pertaining to Taiwan Taoyuan International Airport and other two international airports in order to determine the extent to which the tiered. ‧. security system enhances airport security and reduces the expected waiting time.. sit. y. Nat. The simulation results demonstrate that the proposed tiered security system is able to achieve. io. er. both of these goals in (2.2.6).. n. al. 3.1. C Simulation setup h e n g c h i. i n U. v. In the following, we consider three cases: (A) a single-queue aggregated system without differentiation of different type of passengers; (B) a three-independent-lane without sharing system; (C) a three-lane with sharing system. The purpose of simulation is to evaluate the efficacy of the model by comparing simulation results with data collected from three international airports. Finally, the simulation results are analyzed to find optimal queueing policy of the model for Case C. Assume that when passengers join the security system, historical records of the passengers are used to categorized them H-, M-, L-classes in the formulation of the model. Data included the scheduled times of departure on the website of Taiwan Taoyuan Interna29.

(40) tional Airport(http://www.taoyuan-airport.com/english/flight_depart/) for February 10th, 2015. The following fixed key parameters were used in the study: (1) arrival rate of passengers; (2) service rate of passengers. In the Taoyuan International Airport website, we collect the scheduled fight between 10:30 and 14:30 in Terminal 1. According to the aircraft type of the flight, we can estimate the total number of passengers arriving the airport. Table 3.1 gives information about the arrival process. Table 3.1: Arrival schedules at Taoyuan airport Time Period. 8 9. 1675 1057 2100 2255. 政9 治大 4. ‧. ‧ 國. 立. Number of Passengers. 學. 10:30 - 11:30 11:30 - 12:30 12:30 - 13:30 13:30 - 14:30. Number of Flights. The average arrival rate of passengers for the period 10:30-14:30 was 1771.75 per hour. We. sit. y. Nat. assume that 60% of them are assigned to M-class and 10% of them are assigned to H-class.. io. according to this proportion.. er. On contrary, 30% of arrival passengers are allocated to L-class. We divide it into three classes. al. n. v i n C It was also assumed that securityhlanes security level have the different service e ningdifferent chi U. rate. We assume that the average service rates of each corresponding lane in H-lane, M-lane and L-lane are 185, 220 and 270 per hour respectively. Thus, the total average service rates of H-lane, M-lane and L-lane are 185, 1100 and 540 per hour respectively. Specific data is listed in Table 3.2. In addition to the data collected in Taoyuan International Airport, we also have collected the scheduled times of departure on the website of Narita International Airport(http://www. narita-airport.or.jp/ais/flight/today/e_inter_dep.html) for July 4th, 2015 and on the website of Sydney Airport(http://www.sydneyairport.com.au/flights/ flight-arrivals-and-departures/international-departures.aspx) for July 13th, 2015. Also, we use the data to validate the proposed model as well. The detailed data 30.

(41) and numerical results will be presented in the next Chapter. Table 3.2: Parameters used in simulation at Taoyuan airport. 3.2. Classes. Proportion. Arrival Rate. Total Service Rate. Service rate per lane × Number of lanes. H-class M-class L-class. 10% 60% 30%. 177.175 1063.05 531.525. 185 1100 540. 185 × 1 220 × 5 270 × 2. 政 Simulation assumptions. 治. 大. 立. ‧ 國. 學. Investigation of the queueing system, requires a number of assumptions based on findings in the existing literature. The simulations do not account for flight delays, mechanical problems,. ‧. balking, reneging, or the effects of dealing with families or groups.. sit. y. Nat. io. n. al. er. 3.2.1 Case A: A single-queue aggregated system. Ch. i n U. v. A single-queue aggregated system is an M/G/3 queueing system. Passengers arrive at the. engchi. airport according to a Poisson process with mean arrival rate Λ and service time’s cdf B(t) is a mixture of cdf’s. According to Choi et al. [7], the M/G/3 queueing model can be approximated by GI/G/c/c + r, where c = 3 and r is the length of the queue, which can be set sufficient large. The specific method of the approximation is outlined in Appendix A. In the study, arrivals occur at rate Λ according to a Poisson process and service time has a hyper-exponential distribution. Then we obtain c2A = 1 and c2S =. Var [S]. ( E[S])2. 31. ,.

(42) where 3. E[S] = E [ S2 ] =. p. ∑ µii. i =1 3. 2. ∑ µ2 pi. i =1. λH 1 λ 1 λ 1 + M + L Λ µH Λ µM Λ µL. =. =. i. λH 2 λ 2 λ 2 + M 2 + L 2 2 Λ µH Λ µM Λ µL. Var [S] = E[S2 ] − E[S]2 [ ]2 )2 ( 3 3 3 p 1 1 = ∑ i + ∑ ∑ pi p j − µ µi µ j i =1 i i =1 j =1 ( ) ( ) ( ) ( ) λH 1 2 λM 1 2 λL 1 2 λH λM 1 1 2 = + + +2 − Λ µH Λ µM Λ µL Λ Λ µH µM ( )2 ( )2 λ λ 1 1 λ λ 1 1 +2 H L − +2 M L − . Λ Λ µH µL Λ Λ µM µL. 立. 政治大. ‧ 國. 學. Therefore, the estimated value of the average waiting time in Case A is given by the follow-. y. 1 ∞ ˜ i Pn . Λ n∑ =1. io. sit. Nat. EWA =. ‧. ing. n. al. er. 3.2.2 Case B: A three-independent-lane without sharing system. Ch. engchi. i n U. v. As described in Chapter 2, we designate a three-independent-lane without sharing system is a three independent M/M/1 system. This simulation includes a novel measure for the estimation the average waiting time of Case B, which is given as follows:. EWB =. SH + S M + SL , Λ. where Si is the estimated average queue length of the M/M/1 queue for H, M, L classes, respectively. EWB is set up to enable a comparison of the overall differences between Case B and Case C. This type of comparison provides a more direct indication of the performance of the proposed model as the three lanes are considered as a whole. Using this kind of objective function also 32.

(43) makes it possible to determine the average waiting time for each lane in the numerical results. Keeping EWB as an average upper bound of Case C, the same method is used to assess the model performance in Case C.. 3.2.3 Case C: A three-lane with sharing system An optimization model provides an alternative means by which to obtain an optimal solution to the proposed tiered model. In this thesis, we employed simulated annealing to find the optimal solution for Case C. This begins with the establishment of an objective function of for. 政治大. the simulated annealing model, as follows:. 立. ‧ 國. SH + S M + SL . Λ. 學. EWC =. ‧. Simulated annealing method is based on the pre-given domain of these controllable param-. y. Nat. eters to determine the range to find the optimal feasible solution. Begin with initial point x0 ,. io. sit. simulated annealing algorithm looks for the best neighbor point of x0 and make it be xk . After. n. al. er. found every possible solution xk+1 , the algorithm will make sure xk+1 satisfy the conditions in. Ch. i n U. v. mathematical programming, only coincident solution will be outputted. Otherwise it will use. engchi. some specific approaches to deal with, which include the physics concepts of annealing, more material of simulated annealing method may be found in Appendix B.. 33.

(44) Chapter 4 Numerical Results 政治大立 ‧ 國. 學. In this chapter, we outline the numerical results of the three cases. First of all, there are extensive discussions on these cases by using the date collected in Taoyuan International Airport.. ‧. We will get some useful conclusions of the proposed model. In the final part of this Chapter, we will use the data collected in the other two international airports to strengthen our conclusions.. sit. y. Nat. io. al. v i n TableC 4.1: h Numerical i Uof Case A e n g c hresults. n. mation method.. er. To begin with Case A, Table 4.1 presents the parameters computed using the proposed esti-. a. b. aR. 1 1771.75. 0.0016. 1 1771.75. c2S. bR. 0.0022 1.6336. EWA 1.434. Thus, we obtain the numerical results of Case B, as shown in Table 4.2. Table 4.2: Numerical results of Case B SH. SM. SL. 22.64. 28.77. 62.72. WH. WM. 7.668 1.626. WL. EWB. 7.080. 3.86. It should be noted that Case C achieves an average waiting time shorter than that of Case B, 34.

(45) as anticipated. In the following, we present the optimal numerical results obtained via simulated annealing with a given fixed minimum requirement D. Table 4.3 below presents the range of controllable parameters used in simulated annealing. Table 4.3: The range of parameters used in simulated annealing at Taoyuan airport M. h. m. pM. pL. [70, 100] [1, 20] [1, 20] [0, 1] [0, 1]. Then, the optimal solutions can be found in Table 4.4, we assume D = 0.02 in this part,. 政治大 this figure shows that the changing 立 of feasible solutions varies with iterations, the lower part and the process of finding the optimal solutions is presented in Figure 4.1, the upper part of. ‧ 國. 學. demonstrates the changing of optimal solution EWC . In the search for an optimal solution, we also record the changes in proportion p of all passengers transferred from M-lane to H-lane.. ‧. Figure 4.2 presents a comparison between p and the current feasible solution EWC .. Nat. a l4. sit. 100. 2. m. pM. pL. EWC. 0.28. 0.99. 2.359. Ch. engchi. WH. er. h. n. 0.02. M. io. D. y. Table 4.4: Optimal solutions by simulated annealing of Case C with fixed D at Taoyuan airport. i n U. v. WM. 10.419 1.403. WL 1.474. As indicated above, the relationship between EWC and p is unpredictable rather than linear. Nonetheless, the simulation has an optimal solution, which can be found via simulated annealing. Compared to Case B, Case C shortens the average waiting time in L-lane and M-lane but extends the waiting time in H-lane, resulting in an overall decrease. A lack of passengers in higher level lanes will tend to attract passengers from lower level lanes. However, it is necessary to ensure that p is not less than D at any point during this period. Thus, in the following, we discuss the changes to the configuration of parameters and the average waiting time with different values for D. Table 4.5 lists the optimal parameter configurations with various values for D. An increase in D was shown to increase EWC ; however, the effect is not significant. Figure 4.3 can reflect 35.

(46) 立. 政治大. ‧ 國. 學 ‧. Figure 4.1: Process of finding the optimal solution with D = 0.02 at Taoyuan airport. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 4.2: Comparison between proportion p and current average waiting time EWC with D = 0.02 at Taoyuan airport 36.

(47) this trend more directly. The thresholds h and m can be adjusted in accordance with the value for D. As shown in Figure 4.4, an increase in D makes the requirements for the security inspection system more stringent, which leads to an increase in h, while the value of m remains unchanged. Table 4.5: Optimal solutions at Taoyuan airport by simulated annealing of Case C with different D at Taoyuan airport D. h. m. pM. pL. EWC. 0 100 0.01 100 0.02 100 0.03 100 0.04 100 0.05 100 0.06 100 0.07 100 0.08 100 0.09 100 0.10 100. 1 1 2 2 3 3 3 4 4 5 5. 4 4 4 4 4 4 4 4 4 4 4. 1 0.53 0.28 0.15 0.13 0.09 0.06 0.06 0.04 0.05 0.04. 1 0.99 0.99 1 0.99 0.99 0.99 0.99 1 1 1. 2.339 2.351 2.359 2.370 2.379 2.384 2.394 2.399 2.406 2.410 2.412. ‧. ‧ 國. 立. 政治大. 學. M. y. Nat. sit. In the next part, we discuss the issue of proportions p M and PL , which send passengers to. n. al. er. io. a higher level lane from lower lane. Figure 4.5 presents the results of this situation. The result. i n U. v. is somewhat non-intuitive: an increase in D results in the transfer of a smaller proportion of. Ch. engchi. arriving passengers from M-lane to H-lane, such that the value of p M drops steadily, as shown in Figure 4.5. Nonetheless, we know that the threshold h gradually increases, which drives up the overall number of passengers being transferred. At the same time, the threshold of M-lane remains constant; therefore, the value of p L remains equal to 1. Following the statements above, we will change some of the model configurations to check that the conclusions are still holds. Table 4.6 shows the arrival schedules at Narita Airport which are collected from 10:30 to 14:30. So we have the average total arrival rate during the period at Narita Airport is 1770.5 per hour, which is very close the data we collected at Taoyuan airport. But we use different proportions to assign the number of passengers to the three lanes. In this time, we assume that 70% of arrival passengers, in which 65% of them are assigned to M-lane and 5% of them are assigned to H-lane. Thus 30% of arrival passengers are allocated to L-lane. 37.

(48) 立. 政治大. ‧. ‧ 國. 學. Figure 4.3: The average waiting time EWC of different D at Taoyuan airport. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 4.4: The thresholds h and m of different D at Taoyuan airport 38.

(49) 立. 政治大. ‧ 國. 學 ‧. Figure 4.5: The proportions p M and p L of different D at Taoyuan airport. Nat. sit. y. Since the number of arrival passenger is close to that at Taoyuan airport, we use the same service. al. er. io. rates of three lanes. Table 4.7 demonstrates this assignment. The range of parameters we used. v. n. in simulated annealing method is still the same as we did at Taoyuan airport.. Ch. engchi. i n U. Table 4.6: Arrival schedules at Narita airport Time Period. Number of Flights. Number of Passengers. 10:30 - 11:30 11:30 - 12:30 12:30 - 13:30 13:30 - 14:30. 16 5 3 6. 4075 986 630 1391. We directly obtain the optimal solutions via simulated annealing method with different D in the Table 4.8. In this configuration, we can still get the similar conclusions. Nevertheless, the only difference is that the optimal configurations of M at Taoyuan airport occur at the upper bound of its range, while at Narita airport, the optimal configurations of M become varied with the increase of the minimum requirement D but there is no strict rules. It can be intuitively 39.

(50) Table 4.7: Parameters used in simulation at Narita airport Classes. Proportion. Arrival Rate. Total Service Rate. H-class M-class L-class. 5% 65% 30%. 88.525 1150.825 531.15. 185 1100 540. observed from Figure 4.6. As the same methods we dealing with at Taoyuan airport, the comparison between proportion p and current average waiting time EWC at Narita airport is given in Figure 4.7, the average waiting time EWC of different D is given in Figure 4.8, and the thresholds. 政治大. h and m of different D are given in Figure 4.9. Also, Figure 4.10 demonstrates the proportions p M and p L of different D.. 立. ‧ 國. 學. Table 4.8: Optimal solutions by simulated annealing with different D at Narita airport h. m. pM. pL. EWC. 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50. 70 70 75 71 75 70 70 70 79 90 82. 2 2 3 3 4 4 5 5 6 6 7. 4 4 4 4 4 4 4 4 4 4 4. 0.99 0.99 0.82 0.52 0.39 0.30 0.25 0.21 0.19 0.17 0.15. 0.99 0.99 0.99 0.99 1 1 0.99 1 0.99 1 0.99. 1.516 1.516 1.522 1.532 1.549 1.560 1.577 1.590 1.606 1.625 1.641. n. al. Ch. engchi. y. sit. er. io. M. ‧. Nat. D. i n U. v. Since the average arrival rates collected at Taoyuan airport and Narita airport are almost equal, we have collected data from different time periods at Sydney International Airport. Table 4.9 gives the arrival schedules. From Table 4.9, the average arrival rate at Sydney airport during 18:00-22:00 is 1140.5 per hour, which is quite smaller than that at Taoyuan airport. We allocate the passengers according to the same proportions at Taoyuan airport. Therefore, 60% of them and 10% of them are assigned to M-lane and H-lane. On contrary, 30% of arrival passengers are allocated to L-lane. Similarly, 40.

(51) 立. 政治大. ‧ 國. 學 ‧. Figure 4.6: The buffer size M of different D at Narita airport. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 4.7: Comparison between proportion p and current average waiting time EWC with D = 0.2 at Narita airport 41.

(52) 立. 政治大. ‧. ‧ 國. 學. Figure 4.8: The average waiting time EWC of different D at Narita airport. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 4.9: The thresholds h and m of different D at Narita airport 42.

(53) 立. 政治大. ‧. ‧ 國. 學 sit. y. Nat. io. n. al. er. Figure 4.10: The proportions p M and p L of different D at Narita airport. Ch. engchi. i n U. v. Table 4.9: Arrival schedules at Sydney airport Time Period. Number of Flights. Number of Passengers. 18:00 - 19:00 19:00 - 20:00 20:00 - 21:00 21:00 - 22:00. 8 3 0 6. 1956 595 0 2311. 43.

(54) we give the appropriate service rate in this case, Table 4.10 summarizes the information about parameters used in simulation at Sydney airport. Table 4.10: Parameters used in simulation at Sydney airport Classes. Proportion. Arrival Rate. Total Service Rate. H-class M-class L-class. 10% 60% 30%. 114.05 684.30 342.15. 125 720 380. After optimization using the simulated annealing method, we get the optimal solutions at. 政治大. Sydney airport. Table 4.11 demonstrates the configuration of parameters with different D.. 立. Therefore we can find that, although the number of passenger is declined, the parameters with. ‧ 國. 學. two sets of data have the same variation with increase of D. Figure 4.11 to Figure 4.14 are showing the changing of various parameters, which all have maintained the same trend with the. ‧. previous case at Taoyuan airport.. y. Nat. n. a 0 l. 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50. M. h. m. pM. pL. 100 100 100 100 100 100 100 100. 3 4 5 6 7 7 9 10. 2 2 2 2 2 2 2 2. 0.05 0.04 0.03 0.02 0.02 0.01 0.01 0.01. 0.99 0.99 1 0.99 1 0.99 0.99 1. EWC. er. io. D. sit. Table 4.11: Optimal solutions by simulated annealing with different D at Sydney airport 100 1 2 0.99 0.99i v 1.752 n 1.782 C U 100h e2 2 0.26 1 i h 0.99 1.809 n 2 c 0.12 100 3 g. 44. 1.824 1.836 1.849 1.864 1.873 1.879 1.890 1.896.

(55) 立. 政治大. ‧. ‧ 國. 學. n. al. er. io. sit. y. Nat. Figure 4.11: Comparison between proportion p and current average waiting time EWC with D = 0.2 at Sydney airport. Ch. engchi. i n U. v. Figure 4.12: The average waiting time EWC of different D at Sydney airport 45.

(56) 立. 政治大. ‧. ‧ 國. 學. Figure 4.13: The thresholds h and m of different D at Sydney airport. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v. Figure 4.14: The proportions p M and p L of different D at Sydney airport 46.

(57) Chapter 5 Conclusions 立. 政治大. ‧ 國. 學. This thesis reports a tiered security screening system for airports based on a two-dimensional Markov process and a Markov modulated Poisson process. The proposed model was evaluated. ‧. using the matrix geometric method, wherein the optimal configuration of parameters is determined using simulated annealing. The proposed security screening system was shown to reduce. y. Nat. sit. the overall average waiting time, even as security was improved. We also constructed a compre-. n. al. er. io. hensive queueing strategy and a novel approach to calculating optimal model parameters. This. i n U. v. makes it possible to adjust the configuration of the model according to the number of arriving. Ch. engchi. passengers or the specific requirements of the system security. The efficacy of the proposed methodology depends on the validity of profiling, in the form of a background pre-check by the TSA and the airline company. This makes it possible to send passengers to different queueing lanes according the risk they pose. We present the following recommendations for future study. Despite reductions in the computation time of the proposed model, it still lags behind real-time calculations. Enabling calculations in real time will require adjustment of the queueing strategy according to the number of passengers arriving in real time. Secondly, in the determination of optimal solutions, it will be necessary to take into account not only the overall average waiting time but also the inspection cost and inconvenience cost of passengers.. 47.

(58) Appendix A Approximation for GI/G/c Queue 政治大立 ‧ 國. 學. According to Choi et al. [7], they can approximate the M/G/3 queueing model by the GI/G/c/c + r model. Let N denote the number of passengers in the system at an arbitrary. ‧. time, and N A (N D ) denotes the number of passengers that an arriving passenger finds (that a departing passenger leaves behind) in steady state. In particular, passengers who arrive to. y. Nat. sit. find c + r passengers in the system are assumed to depart immediately from the system with. n. al. er. io. those c + r passengers left behind (note that we take into account those rejected passengers who. i n U. v. find and leave behind c + r passengers). Now let Pn , PnA , and PnD denote the probabilities that. Ch. engchi. N = n, N A = n, and N D = n, respectively, for 0 ≤ n ≤ c + r. These probabilities will be expressed in terms of the following quantities:. a = E[ A] =. 1 , λ. b = E [ S ], anD = E[ AnD ], 0 ≤ n ≤ c + r, bnA = E[SnA ], 0 ≤ n ≤ c + r, bnD = E[SnD ], 0 ≤ n ≤ c + r, where AnD , 0 ≤ n ≤ c + r, is the residual inter-arrival time at the departure instant of a customer who leaves behind n passengers in the system, and SnA (SnD ), 1 ≤ n ≤ c + r, is the residual 48.

(59) service time of a randomly chosen busy server at the arrival instant (the departure instant) of a customer who finds (leaves behind) n passengers in the system. From these definitions, we have acD+r = a and bcD+r = bcA+r (recall that passengers who leave behind c + r passengers in the system are those who are rejected from the system immediately after their arrivals). Finally, we set b0A = b0D = 0 for convenience. We assume that all the above quantities are well defined and finite. Also, we have the steady-state queue-length distributions of the GI/G/c/c + r queue as the following theorem. Theorem. The steady-state queue-length distributions of the GI/G/c/c + r queue are given by PnA. =. PnD. 立. n −1 i =0. i. i +1. ‧ 國. Pn = PnA γn , 0 ≤ n ≤ c + r,. ‧. where. A 0. 學. and. 治政 λ 大n ≤ c + r, =P ∏ , 0≤ µ. y. Nat.    b − i ( a − a D ) + (i − 1)(b A − b D ), i −1 i −1 i −1. n. al. sit. er. io. 1 ≤ i ≤ c, 1 =  µi  −c( a − a D ) + b A + (c − 1)(b A − b D ), c + 1 ≤ i ≤ c + r, i −1 i −1 i −1 i −1. Ch. engchi. i n U. v.     (i + 1)( aiD+1 + biA+1 − biD+1 ), 0 ≤ i ≤ c − 2,     1 = ca D + b A − b + (c − 1)(b A − b D ), c − 1 ≤ i ≤ c + r − 2, i +1 i +1 i +1 i +1  λi      ca, i = c + r − 1,    λa0D , i = 0, [ ] γi =  µ ( a− a D )  λ i λ i−1 + aiD , 1 ≤ i ≤ c + r, i −1. and by normalization, ∑cn+=r0 PnA = 1, [ P0A = 1 +. c +r n −1. ∑∏. n =1 i =0. 49. λi µ i +1. ] −1 ..

(60) Since equations in the theorem above involve quantities a D , bnA , and bnD , which are not easy to compute in general, except for some special cases such as Poisson arrivals or exponential service times. The approximations of steady-state queue-length distributions are obtained by replacing (unknown) arrival and departure-average quantities anD , bnA , and bnD of the above theorem by their corresponding (well-known) time-average counterparts. That is. (1 + c2A ) a E [ A ]2 = , 0 ≤ n ≤ c + r − 1, and 2E[ A] 2 (1 + c2S )b E [ S ]2 bnA (bnD ) ≈bR = = , 1 ≤ n ≤ c + r − 1, 2E[S] 2 anD ≈ a R =. 政治大. where c2X = Var [ X ]/( E[ X ])2 is the squared coefficient of variation (SCV) of a r.v. X with [ ] distribution function F. Note that E X 2 /2E [ X ] is the mean of the equilibrium distribution of. 立. ‧ 國. 學. F.. Finally, we obtain a two-moment approximation for the steady-sate queue-length distribution. λ˜. ∏ µ˜ i+i 1 , 0 ≤ n ≤ c + r, and i =0. n. er. io. al. n −1. Pñ = PñA γ˜ n , 0 ≤ n ≤ c + r,. where. Ch. sit. Nat. PñA = PñD = P˜0A. y. ‧. as follows:. engchi. i n U. v.     b − i ( a − a R ),. 1 ≤ i ≤ c, 1 =  µ˜ i  −c( a − a R ) + bR , c + 1 ≤ i ≤ c + r,     ( i + 1) a R , 0 ≤ i ≤ c − 2,     1 = ca R + bR − b, c − 1 ≤ i ≤ c + r − 2, and  λ˜ i      ca, i = c + r − 1,. 50.

(61)     λa R ,     [ ] γ˜ i = λ µ˜ i ( a − a R ) /λ˜ i−1 + a R ,     [ ]   λ µ˜ i ( a − a R ) /λ˜ i−1 + a ,. 立. i = 0, 1 ≤ i ≤ c + r − 1, and i = c + r.. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 51. i n U. v.