key: cord-0880166-ox8h0fu0
authors: nan
title: A Three-Level Health Inspection Queue Based on Risk Screening Management Mechanism for Post-COVID Global Economic Recovery
date: 2020-09-25
journal: IEEE Access
DOI: 10.1109/access.2020.3026786
sha: 652cfa41d449f9f2fc0d3840b63c7cbd67cedede
doc_id: 880166
cord_uid: ox8h0fu0

Due to the serious impact of border control measures during the ongoing novel coronavirus (COVID-19) epidemic, new difficulties and challenges have been brought to the border lines of most countries and regions. Focusing on the post-COVID global economic recovery, we develop a computing method for studying the health inspection process based on a three-color risk screening management mechanism at border crossing checkpoints. In this article, to manage the cross-border risk efficiently during the epidemic prevention and control, we formulate a queueing model with hierarchical health inspection channels. The structural characteristics and properties of this three-level queueing model are also analyzed by studying the health inspection process with risk classification. Furthermore, we conduct a series of sensitivity analysis on several performance measures for the studied queueing system. In the numerical results, we figure out the monotonicity, convexity and complicated patterns of the derived formulas.

The novel coronavirus disease, so-called COVID-19, has been declared a global pandemic by World Health Organization (WHO) since March 11, 2020 . In order to prevent and control the severe situation of the ongoing COVID-19 epidemic, most countries and regions in the world have carried out strict border control measures [1] . For example, the inspectors and quarantine officers at airports, ports and land borders implement the health inspection and quarantine procedures, including entry quarantine, inspection and quantity control of carry-on items, screening and guiding isolation of abnormal travelers. For those abnormal travelers, the quarantine officers will evacuation them to designated isolation or medical places.

However, those epidemic prevention and control tasks related to border control have brought new difficulties and challenges to the global economic development [2] . The restrictive measures for cross-border flow of travelers The associate editor coordinating the review of this manuscript and approving it for publication was Qi Zhou.

have hindered the activities of international supply chain, and affected the industrial production of various regions. Therefore, it has become an important problem for optimizing the health inspection process at border crossing points.

In this article, we investigate a hierarchical inspection queueing process with three levels of inspection channels, which is of great significance to efficiently execute the border control management for releasing COVID-19 travelling restriction. Based on a three-color risk screening management mechanism, we are going to formulate a queueing model with hierarchical inspection channels. In addition, we will analyze the risk index and system performance of the studied threelevel inspection queueing model. In the practical operation of border control and risk management, the efficiency of inspection services and the risk level are two dilemmatic goals to be achieved [3] - [5] . So, how to balance the risk and efficiency is a scientific problem worthy of study in the inspection process.

In general, there are two basic modes for inspection: uniform screening and selective screening. For uniform screening, we treat everyone with identical screening procedures; however, for selective screening, we selectively apply inspection technologies and procedures on a targeted subgroup of people [6] - [9] . The uniform screening for COVID-19 virus has inevitably put hospitals and health systems in vulnerable positions, due to the shortages in medical capacity and resources to screen the whole people [10] . According to different system scenarios, many analytical tools were developed to study how to allocate security resources and identify where emphasis should be placed, e.g., [11] - [15] . According to the relevant information about persons to be screened, each of them can be assessed and given a corresponding risk value. Based on the assessed risk value of persons to be screened, the inspection system will assign him/her to the appropriate type of inspection channels.

However, when we are forced to expend resources to protect ourselves from COVID-19, we are reducing the resources available for investment, production and consumption of goods and services. In order to increase the likelihood of successfully managing emerging pandemics, it is important to consider valid population-based management approaches, decision-making, and continuous assessment [16] . It is also found that resources devoted to defense activities of any kind do not directly increase economic welfare [17] , [18] . Besides, it is verified that long waiting time in the inspection queues has a negative impact on customer satisfaction [19] - [21] . Over the past decades, queueing theory has been applied extensively in the research works dedicated to the improvement in the security-check or inspection processes, such as [22] - [26] . Stochastic models can be used to estimate the performance of a queueing system in which the events are random [27] , [28] , and to identify the distribution of possible outcomes of a process (as in risk analysis) [29] , [30] .

The innovation of this article lies in the formulation of a three-level inspection queueing model based on the threecolor risk screening management mechanism, which are a new phenomenon and an innovative method for border control under the current pandemic prevention and control. A flowchart for the present research is depicted in Fig. 1 . In the present work, we aim to investigate the stochastic process of inspecting cross-border travelers by formulating a three-level inspection queueing model based on a health risk screening mechanism. The presented mathematical model and analysis provide a scientific tool which is necessary when exploring the border control policies for releasing crossborder travelling restriction and carrying out the management on the smooth flow of cross-border travelers. Taking into account both risk and inspection efficiency, we analyze the proposed queueing model and evaluate its system performance. The originality of this work is to provide an analytical method for the inspection management at border checkpoints, and the queueing analysis conducted in this article can be applied to meet the demand of border control on improving the efficiency of inspection service while ensuring the overall safety level.

The main contribution of this work is to present an executable queueing management scheme for health inspection process of cross-border travelers in order to achieve the goal of effectively optimizing the epidemic prevention and control at border checkpoints. The mathematical model presented in this article is useful as quantitative prescriptions of stochastic behaviors for making border control policies of government departments. It aims to provide information about the relationships among and relative importance of several factors influencing the studied queueing system. Besides, we reveal the stochastic behavior of the studied inspection queueing system through a series of computational experiments. For the post-COVID economic recovery, our research results would be useful in dredging the blocking points of inspection queues under the implementation of reopening the border lines.

The structure of this article is organized as follows. In Section II, we will introduce the health information code and a three-color risk screening management mechanism for reopening the border control under the current epidemic prevention control. The problem definitions and a three-level health inspection queueing system based on the three-color risk screening mechanism will be introduced in Section III. Next, in Section IV, we will give an illustrative example to demonstrate the proposed queueing analysis, and conduct a series of numerical experiments to reveal the performance evaluation of the studied queueing system. Finally, the concluding remarks are summarized in Section V.

The global economy is facing great challenges due to the serious impact of border control measures on the cross-border flow of international travelers. Strict health and quarantine measures have been enforced at points of entry and exit across the borders of most countries and regions to prevent inbound and outbound spread of the virus. The strictest-ever measures were applied at border control to suspend nonurgent and nonessential outbound travel by their citizens. Due to COVID-19 travelling restriction, it has impeded the crossborder flow of people, and delivered a blow to both the supply chains and other economic activities, making a severe global economic recession unavoidable.

For the sake of our work and life gradually returning to normal, many countries/regions are looking for effective measures to lift inbound and outbound traffic restrictions during the battle against COVID-19. As of July 23, 2020, several countries have begun to gradually open the border for international travelers, such as the member states of European Union, United Kingdom, China, Thailand, Turkey, Ukraine, North Macedonia, Haiti, and so on.

Here, taking the measures implemented in China as an example, we introduce a three-color risk management mechanism to release the travelling restriction. In China, the government has launched ''the epidemic prevention health information code'' to implement the strategy of precise prevention and control at different risk levels [31] . As shown in Fig. 1 , the health information code is divided into three color codes: Green code (low level of risk), Yellow code (medium level of risk), and Red code (high level of risk). A database for epidemic prevention health information code has been set up in accordance with the law to provide data services for COVID-19 risk control, and precisely identify different groups at risk.

In specific, the three-color risk management mechanism is based on the real health data and connects to relevant databases, such as the COVID-19 risk management system and control lists for travelling personnel. Through the online declaration and background review of an individual, it generates a color QR code belonging to him/her, which can be used as an electronic voucher for him/her to pass in and out. Thus, it can predict risk factors in different areas, and facilitate the orderly flow of people and the resumption of business operations.

With authorization from the public, the health information codes and digital travel records have been employed in China as permits for making trips. For example, when the travelers to be inspected enter the health inspection checkpoints, the three-color health information code belonging to an individual should be generated to mark his/her risk value by the risk pre-screening system. The health inspection personnel at the checkpoint will check the electronic voucher data of the travelers to be inspected, and then assign him/her to the appropriate inspection channel according to his/her risk level.

In summary, the three-color health information code can manage the flow of travelers efficiently for the economic recovery. The results shown on the health information code and historical records provide a base for the COVID-19 risk identification of cross-border travelers in different areas. In addition, the three-color health information code has made it possible to implement differentiated border control measures at different risk levels during the post-COVID global economic recovery. Therefore, it is essential to develop an analytical model that takes into account risk management and inspection efficiency under this three-color risk management mechanism. 

In this section, we present a three-level health inspection queueing model based on the three-color health information code, where the three-level inspection queues are studied based on a risk screening management mechanism. In the studied queueing system as shown in Fig. 2 , two risk thresholds are to be used to distinguish those cross-border travelers for going through three different risk level inspection channels. We will formulate the queueing model and determine the performance measures of the entire queueing system with these two risk thresholds.

As shown in Fig. 2 , the health information provided by the database of a risk pre-screening system can be used to mark a risk value α for every cross-border traveler to be inspected at the border checkpoints, where the risk value α is a real number and lies between 0% and 100%. We assume that the risk value α of cross-border travelers to be inspected follows a probability density function f (α), and the risk value of each traveler is independent of each other.

Two risk thresholds τ 1 and τ 2 are used to distinguish the cross-border travelers between high-risk, medium-risk and low-risk classes. The values of risk thresholds τ 1 and τ 2 are real numbers between zero and one, and the value range is 0 ≤ τ 2 < τ 1 ≤ 1. According to the assigned risk value α, the inspection officers will guide the travelers to be inspected to the corresponding queues of three level inspection channels.

It can be seen from Fig. 2 that the queueing process for inspecting cross-border travelers with three-level risk classification considered in this study is mainly divided into Red inspection channel (high level of risk), Yellow inspection channel (medium level of risk) and Green inspection channel (low level of risk). We represent s 1 as the number of inspectors (with equipment) set in Red inspection channel for high level of risk. In addition, the number of inspectors (with equipment) in the Yellow inspection channel for medium level of risk is expressed as s 2 , and the number of inspectors (with equipment) in the Green inspection channel for low level of risk is denoted as s 3 . Note that these three numbers s 1 , s 2 and s 3 are positive integers.

Suppose that the inter-arrival time of cross-border travelers to be inspected follows the exponential distribution and is independent of each other, with the expected value of 1 λ. The inspection time of three kinds of inspection channels follows an exponential distribution and is independent of each other. The expected inspection time of the Red inspection channel for high level of risk is 1 µ 1 . The expected inspection time of the Yellow inspection channel for medium level of risk is 1 µ 2 . The expected inspection time of the Green inspection channel for low level of risk is 1 µ 3 . Compared with the other two inspection channels, the Red inspection channel has the longest inspection time and strictest quarantine, but the highest recognition rate for dangerous travelers. The Green inspection channel has the shortest inspection time and the highest service efficiency, but the lowest recognition rate for dangerous travelers.

If two risk thresholds τ 1 and τ 2 are given, the probability that the travelers to be inspected are assigned to the Red inspection channel can be expressed as p 1 (τ 1 , τ 2 ) . Meanwhile, the probability that the travelers to be inspected are assigned to the Yellow inspection channel can be expressed as p 2 (τ 1 , τ 2 ), and the probability that the travelers to be inspected are assigned to the Green inspection channel can be expressed as p 1 (τ 1 , τ 2 ). Note that it is obvious that the following condition holds:

In the next subsection, we are going to formulate a queueing model based on a three-color risk management mechanism, which can be used to evaluate the system performance in the cross-border inspection process. The queue discipline is first come, first served, where travelers are inspected in the same order in which they arrive at the checkpoint. The mathematical notations and definitions of the queueing model studied in this article are summarized in Table 1 .

The independence of arrivals in disjoint time intervals together with the memoryless property of the exponential service time distribution implies that the counting number of cross-border travelers to be inspected at the three-level health inspection queueing system forms a time homogeneous Markov chain, in particular, a birth and death process.

Under the condition of risk thresholds 0 ≤ τ 2 < τ 1 ≤ 1, the probability that the travelers to be inspected are assigned to the Green inspection channel for low level of risk can be derived as:

and the probability that the travelers to be inspected are assigned to the Red inspection channel for medium level of risk can be derived as: Thus, we can determine the probability that the travelers to be inspected are assigned to the Yellow inspection channel VOLUME 8, 2020 for high level of risk:

For the Red inspection channel, we denote X (R) (t) as the number of cross-border travelers waiting in the Red inspection channel at time t. When a fixed number s 1 of inspectors (with equipment) are available in the Red inspection channel, then X (R) (t) forms a birth and death process with birth parameters

and death parameters

Then the number undergoing the inspection service is min X (R) (t) , s 1 , and the number waiting for service is max X (R) (t) − s 1 , 0 . Besides, we can denote the traffic intensity of the Red inspection channel as

Note that, as the traffic intensity ρ 1 approaches one, the mean queue length in the Red inspection channel becomes unbounded. Let π (R) n = lim t→∞ Pr{X (R) (t) = n}, for n = 0, 1, . . . , (8) be the equilibrium distribution of queue length in the Red inspection channel. When it holds the stability condition that ρ 1 < 1, we can determine

and

For the Yellow inspection channel, we denote X (Y ) (t) as the number of cross-border travelers waiting in the Yellow inspection channel at time t. When a fixed number s 2 of inspectors (with equipment) are available in the Yellow inspection channel, then the appropriate birth and death parameters are λ (Y ) n = p 2 (τ 1 , τ 2 ) · λ, for n = 0, 1, . . . ,

and µ (Y ) n = n·µ 2 , for n = 0, 1, . . . , s 2 , s 2 · µ 2 , for n > s 2 .

In addition, the number undergoing the inspection service of the Yellow inspection channel is min X (Y ) (t) , s 2 , and the number waiting for service is max X (Y ) (t) − s 2 , 0 . Next, we denote the traffic intensity of the Yellow inspection channel as

Under the stability condition that ρ 2 < 1, we can derive the equilibrium distribution

and

Similarly, for the Green inspection channel, we denote X (G) (t) as the number of cross-border travelers waiting in the Green inspection channel at time t. When a fixed number s 3 of inspectors (with equipment) are available in the Green inspection channel, then the appropriate birth and death parameters are

and µ (G) n = n·µ 3 , for n = 0, 1, . . . , s 3 , s 3 · µ 3 , for n > s 3 .

Furthermore, the number undergoing the inspection service of the Green inspection channel is min X (G) (t) , s 3 , and the number waiting for service is max X (G) (t) − s 3 , 0 . Next, we denote the traffic intensity of the Green inspection channel as

If it holds the stability condition that ρ 3 < 1, we can derive the equilibrium distribution

and

The equilibrium distributions (9), (10), (14) , (15) , (19) and (20) can demonstrate the limiting behavior of the studied queueing system. Besides, these equilibrium distributions enable us to calculate the major system performance measures, such as the expected number of travelers in the studied queueing system and the average time spent by each crossborder traveler.

Here, we are going to derive three major performance measures for the proposed three-level inspection queueing model, including the average system size, the average time spent in the system, and the safety level of the overall system. When the studied queueing system operates sufficiently long to have reached an appropriate steady state, we can evaluate the average queue size and average waiting time with the equilibrium distributions. With the equilibrium distributions (9) and (10), we can calculate L (R) q , the average number of travelers in the system waiting for (and not undergoing) the inspection service at the Red inspection channel as follows:

Then, with the help of Little's formula, we can determine the average time spent by a traveler with high level of risk in the queue line

and the average time spent for passing through the Red inspection channel

Thus, we determine the average number of travelers with high level of risk in the whole queueing system

With the equilibrium distributions (14) and (15), we determine L (Y ) q , the average number of travelers in the system waiting for (and not undergoing) the inspection service at the Yellow inspection channel as follows:

Then, with the help of Little's formula, we can determine the average time spent by a traveler with medium level of risk in the queue line

and the average time spent for passing through the Yellow inspection channel

Thus, we obtain the average number of travelers with medium level of risk in the whole queueing system

With the equilibrium distributions (19) and (20), we can evaluate L (G) q , the average number of travelers in the system waiting for (and not undergoing) the inspection service at the Green inspection channel as follows:

Then, with the help of Little's formula, we can determine the average time spent by a traveler with low level of risk in the queue line

and the average time spent for passing through the Green inspection channel

Thus, we obtain the average number of travelers with low level of risk in the whole queueing system

After determining the values of W (R) , W (Y ) and W (G) , we can obtain an important performance measure for the health inspection queueing system, that is, the average time spent by a cross-border traveler at the health inspection queueing system, W (τ 1 , τ 2 , s 1 , s 2 , s 3 ). Given the risk thresholds τ 1 , τ 2 and the numbers of inspectors s 1 , s 2 , s 3 , we have the following equation:

In addition, with the values of L (R) , L (Y ) and L (G) , we can also obtain the average number of cross-border travelers at the health inspection queueing system, L(τ 1 , τ 2 , s 1 , s 2 , s 3 ). Given the risk thresholds τ 1 , τ 2 and the numbers of inspectors s 1 , s 2 , s 3 , we have the second performance measure for the health inspection queueing system, L (τ 1 , τ 2 , s 1 , s 2 , s 3 ), which is derived in the following equation:

Furthermore, if it is given that 0 ≤ τ 2 < τ 1 ≤ 1, we can derive the risk ratio of travelers assigned to the Green inspection channel is:

where f (α) is a probability density function of the risk value α for those cross-border travelers to be inspected. In addition, VOLUME 8, 2020 the risk ratio of travelers assigned to the Red inspection channel is:

Therefore, from the condition (1), we can determine the risk ratio of travelers assigned to the Yellow inspection channel as follows:

Finally, we denote SL(τ 1 , τ 2 ) as the safety level of the overall health inspection queueing system given the risk thresholds τ 1 and τ 2 . With the values of three risk ratios R 1 (τ 1 , τ 2 ), R 2 (τ 1 , τ 2 ) and R 3 (τ 1 , τ 2 ), we can determine the safety level of the overall health inspection queueing system as follows:

where d i is the recognition rate for inspection and quarantine at the corresponding inspection channel, which is a constant number between 0% and 100%. The safety level, SL (τ 1 , τ 2 ), is the third performance measure for the overall health inspection queueing system.

In this section, we demonstrate the proposed approach through solving an illustrative example, and conduct a sensitivity analysis on the studied queueing system through a series of numerical experiments to figure out the effects of varying two risk thresholds on the presented three performance measures. Besides, a special case when two risk thresholds are identical is also presented as an application of our model.

The parameter settings given here are used to provide a numerical example of our calculation process in Section III, and the different setting of model parameters will not change the correctness of those derived formulas in the present paper. The parameter settings of the illustrative example are given as follows. The average arrival rate is λ = 5 travelers per minute. The numbers of inspectors (with equipment) set in three levels of inspection channels are given as s 1 = 5, s 2 = 3, and s 3 = 2. Meanwhile, the recognition rates at those three levels of inspection channels are given as d 1 = 99%, d 2 = 80%, and d 3 = 75%, and the average service rate for each level of inspection channel is set as µ 1 = 1.0 (traveler/minute), µ 2 = 1.5 (traveler/minute), and µ 3 = 2.6 (traveler/minute), individually. Besides, two risk thresholds τ 1 and τ 2 are varied within the interval [0%, 100%], i.e., 0 ≤ τ 2 < τ 1 ≤ 1. Moreover, as an illustrative example taken from [20] , it is given that the risk value α of the travelers to be inspected follows a truncated exponential distribution, and each risk value is independent of each other. The probability density function of a truncated exponential distribution is given as follows:

where θ is the parameter of the truncated exponential distribution. Therefore, the average risk value E[α] of travelers to be inspected can be expressed as follows:

That is, the average risk value E[α] is approximately the parameter θ of the truncated exponential distribution. According to the statistics of McLay et al. [32] , we take θ = 0.0625 as the parameter of truncated exponential distribution in the numerical experiments.

Our computational environment is run by means of MATLAB 2017a on Windows 10 Professional 64-bit with the processor Intel (R) Core (TM) i7-6500U CPU@2.50GHz dual-core and 8 GB memory.

Given 0 ≤ τ 2 < τ 1 ≤ 1, we can derive the probability that those cross-border travelers are to be assigned to the Green inspection channel

and the probability that those cross-border travelers are to be assigned to the Red inspection channel

With the help of condition (1), we can easily determine the probability that those cross-border travelers are to be assigned to the Yellow inspection channel as follows:

After obtaining the above three probabilities, we can evaluate the traffic intensities ρ 1 = p 1 (τ 1 , τ 2 ), ρ 2 = 10 9 · p 2 (τ 1 , τ 2 ) and ρ 3 = 25 26 · p 3 (τ 1 , τ 2 ) via the equations (7), (13) and (18) . Moreover, when it holds the stability conditions, the equilibrium distributions π (G) n , π (Y) n , and π (R) n can be determined by using equations (9), (10), (14) , (15) , (19) and (20) . Therefore, two important performance measures can be easily obtained via the equations (33) and (34) , that is, the average time W (τ 1 , τ 2 , s 1 , s 2 , s 3 ) spent by a cross-border traveler and the average number L(τ 1 , τ 2 , s 1 , s 2 , s 3 ) at the health inspection queueing system. Furthermore, from the equations (35) and (36), we can derive the risk ratio of travelers assigned to the Green inspection channel and the risk ratio of travelers assigned to the Red inspection channel

where it is given that 0 ≤ τ 2 < τ 1 ≤ 1. So, with the help of condition (1), we can easily determine the risk ratio of travelers assigned to the Yellow inspection channel as follows:

After obtaining the above three risk ratios, we can determine the last performance measure via the equation (38), i.e., the safety level SL (τ 1 , τ 2 ) for the overall health inspection queueing system. Next, we observe the effects of varying two risk thresholds τ 1 and τ 2 on the studied three performance measures under the parameter settings s 1 = 5, s 2 = 3 and s 3 = 2. From Fig. 3 to Fig. 5 , it illustrates the numerical results of the average waiting time W (τ 1 , τ 2 , 5, 3, 2) , the average number L (τ 1 , τ 2 , 5, 3, 2) , and the safety level SL (τ 1 , τ 2 ) when varying two risk thresholds τ 1 and τ 2 in the domain range 0 ≤ τ 2 < τ 1 ≤ 1.

In Fig. 3 , it shows that the safety level SL (τ 1 , τ 2 ) is decreasing when we enlarge two risk thresholds. In Fig. 4 and Fig. 5 , we find that the surface of average number L(τ 1 , τ 2 , 5, 3, 2) and average time W (τ 1 , τ 2 , 5, 3, 2) are decreasing when two risk thresholds lies in a small region near the origin point (0,0), while the function values will become increasing as enlarging two risk thresholds outside this small region. 

In this subsection, we demonstrate that our three-level health inspection queueing model can be simplified to an inspection queueing model with two types of inspection channels and only one risk threshold. When two risk thresholds τ 1 and τ 2 are identical, it is obvious that the probability that those crossborder travelers are to be assigned to the Yellow inspection channel will become zero, i.e., p 2 (τ 1 , τ 2 ) = 0. In such a special case, the proposed three-level inspection queueing model can be reduced to a two-level inspection queueing model consists of Red inspection channel (for higher level of risk) and Green inspection channel (for lower level of risk). That is, in the case when τ 1 = τ 2 , there is only two-level risk classification for cross-border travelers, and there is only one risk threshold used in this simpler inspection queueing system.

Next, we investigate the effects of changing a risk threshold on such a simplified inspection queueing system with only two types of inspection channels. For the simplification of notation, we take τ = τ 1 = τ 2 in the following derivation. Given 0 ≤ τ ≤ 1, we can derive the probability that the crossborder travelers are to be assigned to the Green inspection channel

and the probability that those cross-border travelers are to be assigned to the Red inspection channel

Therefore, under the stability conditions ρ 1 = p 1 (τ ) < 1 and ρ 3 = 25 26 · p 3 (τ ) < 1, we can determine the average time spent for passing through the Red inspection channel, W (R) , and the average time spent for passing through the Green inspection channel, W (G) , via equations (23) and (31) . In addition, we can also calculate the average number of travelers with lower level of risk in the two-level inspection queueing system, L (G) , and the average number of travelers with higher level of risk in the system, L (R) , via equations (24) and (32) . Furthermore, from the equations (35) and (36), we can derive the risk ratio of travelers assigned to the Green inspection channel, R 3 (τ ), and the risk ratio of travelers assigned to the Red inspection channel, R 1 (τ ). Finally, we obtain three major performance measures for the two-level inspection queueing, such as the average time W (τ, s 1 , s 3 ), the average number L(τ, s 1 , s 3 ), and the safety level SL (τ ). The numerical results of the derived performance measures are illustrated in Figs. 6-9 under the parameter settings λ = 5, s 1 = 5, and s 3 = 2.

When varying the risk threshold τ between 0% and 100%, Fig. 6 illustrates four performance indices for two types of inspection channels, including the probability of travelers assigned to each type of inspection channel, the risk ratio, the average time, and the average number of travelers. For a closer insight into the curves of changes, Fig. 7 shows those numerical results especially when the risk threshold lies in the interval [0%, 25%] . It can be found that the probability p 3 (τ ) increases and the probability p 1 (τ ) decreases when increasing the risk threshold τ . Besides, if we increase the risk threshold τ , then the risk ratio, the average time and the average number are increasing for the Green inspection channel while those three indices are decreasing for the Red inspection channel.

Nevertheless, it can be observed in Fig. 8 and Fig. 9 that the major performance measures for the whole queueing system would be neither monotonically increasing nor monotonically decreasing when varying the risk threshold. It can be seen from Fig. 8 that the safety level SL (τ ) is a decreasing function of risk threshold τ . In Fig. 8 , we also observe that the curves of average number L(τ, s 1 , s 3 ) and average time W (τ, s 1 , s 3 ) are decreasing and concave upward when the risk threshold is smaller than an inflection point, while these two curves become increasing and concave downward if the risk threshold is larger than an inflection point. In Fig. 9 , it shows a closer insight into the curves of those three major performance measures when the risk threshold lies in the interval [0%, 25%] .

For more detailed discussion and application on a two-level inspection queueing model with only one risk FIGURE 9. Three performance indices for a two-level inspection queueing system when varying the risk threshold between 0% and 25%.

threshold, interested readers may refer to those research works in [33] - [36] and references therein.

In this article, we study the queueing management on health inspection process based on a three-color risk screening mechanism at border checkpoints. We develop an analytical method for a queueing system with three levels of inspection channels to meet the need of border control during the post-COVID economic recovery. The mathematical model derived in this article is useful as a quantitative description of cross-border travelers' stochastic behaviors in order to make border control policies of government departments, and it aims to provide information about the relationships among relative factors influencing the studied queueing system.

The originality of this article lies in the formulation of a three-level inspection queueing model based on the threecolor risk screening management mechanism. It is a new phenomenon and an innovative method for border control under the current prevention and control situation of the ongoing COVID-19 pandemic. The main contribution of the present work is to get a more detailed understanding of the trade-off between safety and efficiency in the epidemic prevention and control management at the border crossing checkpoints.

In the presented analysis, we formulated a three-level queueing model taking both risk and efficiency into account, and derived the equilibrium distributions of the studied queueing system under the stability conditions. Moreover, three major performance measures were derived theoretically to evaluate the proposed queueing model, including the average number of travelers to be inspected, the average time spent by each cross-border traveler, and the safety level of the overall queueing system. Furthermore, by studying a special case (when two risk thresholds were assumed to be identical), we also demonstrated that our three-level health inspection queue could be simplified to a model with two types of inspection channels and only one risk threshold, which had been widely applied in the related works on inspection process.

In the numerical results, we illustrated the monotonicity, convexity and sensitivity analysis of the studied queueing system. Through the subsequent computational experiments, it revealed some managerial insight into the three-level inspection queueing system, which could reconcile a conflict between efficiency and safety in the inspection tasks. Besides, our findings could also provide a helpful reference for implementing the prevention and control strategy more accurately classified and differentiated, especially when dredging the bottleneck points for the flow of travelers at border customs.

The application of our research work can be used in formulating feasible inspection measures for hierarchical diversion in the cross-border flow of travelers. For example, the queueing analysis conducted in this article can be applied to meet the demand of border control on improving the efficiency of inspection service while ensuring the overall safety level. Furthermore, the system performance of the border crossing points can be quantitatively evaluated and analyzed while taking into both risk and efficiency account.

In a COVID-19 world, it is necessary to scientifically carry out their border control management and accurately maintain the smooth flow of cross-border travelers. Meanwhile, it is essential to take the initiative on an orderly return to normal cross-border activities for post-COVID global economic recovery. Consequently, the mathematical analysis and performance evaluation in this article can be a scientific tool of exploring the measures for releasing cross-border travelling restriction during the current epidemic prevention and control.

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Optimizing airline passenger prescreening systems with Bayesian decision models

The State Council of the People's Republic of China

Risk-based policies for airport security checkpoint screening

A modelling framework for managing risk-based checkpoint screening systems with two-type inspection queues

Arena simulation for aviation passenger security-check systems

A queueing analysis of a security-check system with two types of inspection channels

China, where he currently holds an Associate Professor position. Meanwhile, he also serves as a Research Fellow with the Fujian Provincial Key Laboratory of Big Data Mining and Applications, China. He is also an Expert in operations research, applied mathematics, information systems, and data science. He has been involved in several collaborative projects with international professionals, including the modeling and analysis of security-check queues, the management mechanisms for service systems, and the design of management schemes on communication networks