A Minimum Capacity Optimization Scheme for Airport Terminals During Peak Periods

Air transportation as a public transport method is preferentially considered by passengers. However, this method will not only bring a congestion problem but also affect the passenger experiences in airport terminals during peak periods. Considering the fact that the capacities of airport terminals are proportional to their building costs, this paper proposes a scheme to calculate the minimum capacity for each airport terminal. The main purpose is to avoid the saturated congestion problems and reduce the waste of transition bus resources. For each airport terminal, its capacity is closely related with its inputs and outputs. The proposed scheme proves a relational expression for each airport terminal according to its inputs and outputs. Moreover, this work formally models the limited flow structures of flight process guidance systems by using finite capacity Petri nets and then verifies the correctness of the proposed relational expression. Finally, an example is presented to illustrate the proposed scheme.


I. INTRODUCTION
With the rapid development of social economy globalization, the demands of air transportation have increased sharply. Unfortunately, they may lead to a series of problems, such as the congestion at airport terminals during peak periods, the flight delays, and the serious declines in service quality. A core reason of these problems is that the current capacities of airport terminals are unreasonable for satisfying the demands of air transportation. These problems will greatly affect the developments of air transportation. In order to make sure the normal operations of air transportation and relieve passenger congestion in airport terminals, two urgent problems, i.e., how to improve the service quality and how to The associate editor coordinating the review of this manuscript and approving it for publication was Zhiwu Li . properly optimize the capacities of airport terminals, should be resolved more rationally.
As a long-term strategy, an excessively ideal approach is to build high capacity airports to resolve these urgent problems. However, it not only requires a lot of construction costs but also cannot be implemented in short periods. This approach has no advantage in alleviating the pressure of air transportation demands. Another practical approach is to optimize and expand the existing resources, such as airport terminals, runways, aprons and so on. The main purpose is to appropriately improve the capacities of these existing resources to improve the utilization rate and passenger satisfaction degree [1]. The prominent advantages of this approach are simple construction, rapid implementation, and relatively low cost. In recent years, various airport capacity optimization schemes are proposed to resolve these urgent problems, which can be roughly VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ summarized into two categories, i.e., queuing theory [2]- [4] and genetic algorithms [5]- [7]. Queuing theory as an excellent mathematical tool that can be used to simulate and analyze airport behavior [8], [9]. The authors of [10] observe the arrival time, queuing time, and queuing behavior of passengers in the terminals of Kerala airport. They establish an M/M/C queuing system to reduce passenger dissatisfactions and increase terminal capacities. The authors of [11] present a queuing model to simulate the departure processes of an airport. The model simulates airport departure processes with a current gate assignment and a robust gate assignment to assess the impact of gate assignments on departure metering. As a result, it improves the operations of crowded airports with limited gate resources and increases airport capacities. In [12]- [15], they focus on the resource allocation of airport security scanning channels. Some models are built based on the queuing theory to describe passenger flows in airport security processes. Finally, the optimization of airport security processes is solved and the utilization of airport resources is improved. In general, the queue theory can be used to establish the corresponding mathematical models and optimize the existing resources. It is easy to build a real model to improve the capacities of these existing resources based on queue theory. However, such a model requires a large amount of real time aviation data that is affected by numerous random factors.
Genetic algorithms are a class of search algorithms based on evolutionary theory [16], which greatly improve the adaptability of each individual and finally converge to the optimal solutions [17]. They are often used for scheduling airport resources to achieve the optimal utilization of airport resources. In [18], a genetic algorithm is provided to achieve flight scheduling optimization. This scheme uses genetic algorithms to solve a two-runway flight scheduling problem, which can not only efficiently determine the runways of aircraft and landing sequences, but also achieve a runway capacity optimization scheme. The authors of [19] present a model based on genetic algorithms. This model can reasonably solve the problem of airport capacity management. A simulation study shows that this scheme can improve runway capacities. In [20], an improved genetic algorithm is proposed to avoid the premature of genetic algorithms by considering structural characteristics. The experimental results show that the gate capacities are optimized by using genetic algorithms. The authors of [21] establish a multi-flight multi-service parallel timing optimization model based on genetic algorithms. The model can ensure the normal operations of flights, realize the coordinated task scheduling scheme among various businesses, and optimize the capacities of airport terminals.
We know that genetic algorithms are based on probability rules, which make the search algorithms more flexible and their parameters have as little impact as possible. Meanwhile, genetic algorithms can be well combined with other related algorithms. They can be used to optimize the capacities of the existing airport resources. However, it may involve a lot of complex calculations by using genetic algorithms to optimize complex systems. Unfortunately, airport systems coincidentally are complex systems and it is difficult to achieve a global optimal solution for a complex airport layout with genetic algorithms. Furthermore, the speeds of many search algorithms based on genetic algorithms are relatively slow if the feedback information of the networks cannot be used in time. Moreover, the efficiency of genetic algorithms depends on the selections of initial populations.
Petri nets [22] as a graphical tool are widely used to describe and analyze discrete event systems [23]. They have established applications to discrete event systems from modeling [24], optimization control [25], and deadlock analysis [26]. It is possible to create mathematical models, state equations, and algebraic equations to analyze and verify the behavior of discrete event systems by using Petri nets [27]. With the help of formal modeling and analysis methods of Petri nets, researchers can analyze the static models and the operational dynamic behavior of discrete event systems. The obtained results are more similar to the exact results for the real systems. For example, a Petri net model of an automated manufacturing system is established in [28]. In this model, genetic algorithms are used to solve multi-objective scheduling problems of automated manufacturing systems with limited resource capacities and high deadlocks. The studies in [29] use Petri nets to establish a model and propose a method to avoid deadlock in flexible manufacturing systems. In [30], the fault diagnosis of power systems is realized by building a model with Petri nets. The authors of [31] propose a scheme to ease the traffic congestion problems based on Petri nets in traffic management systems. The authors of [32] present two heuristic algorithms to solve a marking optimization problem and a cycle time optimization problem based on a subclass of timed Petri nets called deterministic timed weighted marked graphs. The simulation studies show that their presented algorithms are significantly more efficient. Moreover, they also establish a formal model for resource allocation systems by deterministic timed weighted marked graphs in [33]. The performance optimization problems are considered with the aim of maximizing their throughput for resource allocation systems under a given budget for acquiring resources.
Considering the above characteristics of Petri nets, many researchers also use Petri nets to optimize the capacities for airport resources. It can not only avoid the inputs of a large amount of real time aviation data, but also establish formal models with intuitive graphics to represent complex airport systems. The authors in [34] use Petri nets to establish a situational awareness method. This method introduces the queuing theory and perceptual parameters into the existing Petri nets. Moreover, the perceptual Petri net model of a general service system is constructed, which can quickly establish models by different scene service systems. In [35], the authors study the model and simulation of baggage handling systems based on colored Petri nets. The simulation results show that the model can not only realize fast baggage sorting, but also improve baggage efficiency and airport capacities. The authors in [36] propose a new Petri net model to dynamically analyze aircraft movements on the runways with given input and predetermined exit parameters. Their main purpose is to improve the runway capacities. In [37], a stochastic Petri net model is used to optimize the airport security processes and finally the security processes of terminals are optimized. In [38], a dynamic modeling method is proposed for airport aprons based on Petri nets, which realizes the optimization of apron capacities.
Airport terminals as a critical part of airports, their role is to provide waiting services for passengers. The capacity of an airport terminal largely determines the congestion levels during peak periods. For example, Beijing Capital International Airport as one of the three complex airport hubs in China, it has three main airport terminals, i.e., T 1 , T 2 and T 3 [39], where terminal T 3 is the largest single terminal in Asia and it is composed of three terminals, i.e., T 3 -C, T 3 -D, and T 3 -E [40]. Terminal T 3 -C is a main building of T 3 . Its role is to perform ticket purchasing, security checking, and waiting for boarding. Passengers should take transition buses from T 3 -C to T 3 -D or T 3 -E, where they are waiting for boarding if T 3 -C is extremely congestion during peak periods. There are many runways around terminals T 3 -D and T 3 -E, which can improve airplane utilization and reduce passenger congestion during peak periods. Similarly, passengers should take transition buses from T 3 -D or T 3 -E to T 3 -C to leave the airports. Therefore, the capacity of T 3 -D will affect the congestion levels in Beijing Capital International Airport. A whole process is shown in Fig. 1. In Beijing Capital International Airport, passengers should go through a series of complex passenger handing processes from entering the airport to returning and leaving the airport [41]. It is difficult to completely analyze the whole processes. In order to simplify these processes, this paper divides the passenger handling processes into six core steps, as shown in Fig. 2. The six core steps form a flight process guidance system that is a typical discrete event system. Therefore, a flight process guidance system can be modeled and analyzed by using Petri nets.
In a flight process guidance system, it usually suffers from the saturated congestion problems but also exists the waste of transition bus resources if many passengers are waiting in terminals and the capacities of the terminals are unsuitable, which are illustrated by the following examples.
To facilitate the dispatch and control of flights and transition buses and enhance the statistical and analysis of passengers, all passengers who are entering and exiting a terminal will be divided into different groups. Let x = 200 and y = 150 be the input and output of T 3 -D, respectively, C = 250 be the capacity of T 3 -D, and there be 250 passengers who are waiting to leave T 3 -D, where x and y represent the numbers of passengers who are entering and exiting T 3 -D, respectively. Specially, x is closely controlled by the arrived flights and y is closely related with the number of transition bus seats at T 3 -D. If the flight process guidance system starts to run during the peak period, it may suffer a failure state after a few system running periods. At the failure state, there are 100 passengers who still are waiting to leave terminal T 3 -D by transition buses. Certainly, they can leave T 3 -D by a transition bus at this state. However, this transition bus cannot work with a full load and 50 seats of this transition bus are wasted since y = 150 > 100. Moreover, a more noteworthy problem is that the other passengers cannot enter T 3 -D at this failure state since x + 100 = 300 > C = 250. Therefore, the flight process guidance system suffers a saturated congestion problem in T 3 -D at this failure state. To avoid this problem, three ways can be considered as follows.
1) Decreasing the value of x, such as x = 150. This means that the number of passengers that are entering T 3 -D will be reduced to 150 and the remaining 50 passengers cannot enter T 3 -D timely. It will further aggravate the congestion in the entrance of T 3 -D. 2) Decreasing the value of y, such as y = 100. This means that the number of passengers that are exiting T 3 -D will be reduced to 100. In other words, the number of passengers that will leave T 3 -D by transition buses will be reduced, where the seats of transition buses are fixed. Therefore, the transition bus resources will be wasted unless more passengers are exiting T 3 -D rapidly. 3) Expanding the value of C, such as C = 400. The saturated congestion problems can be avoided if the flights and transition bus resources are difficult to be controlled during the peak periods.
The first way tries to decrease the value of x, but the value of x is determined by the number of arrived flights during the peak periods. It is obviously inadvisable to restrict the arrived flights. Conversely, it will further aggravate the congestion in the entrance of T 3 -D. The second way tries to decrease the value of y, but the value of y is determined by the number of VOLUME 8, 2020 transition bus seats that are usually constant. Comparing with the first two ways, we know that the third way is a reasonable method to solve the saturated congestion problems. But it also exists the waste of transition bus resources for different capacities. For example, assume that the capacity of T 3 -D is expanded to C = 400 and there are 400 passengers who are waiting to leave T 3 -D. If the flight process guidance system starts to run during the peak periods, it may also suffer another failure state after a few system running periods. At the failure state, 100 passengers also are waiting to leave terminal T 3 -D by transition buses. Similarly, the transition bus cannot work with a full load and 50 seats of the transition bus will be wasted if the 100 passengers leave T 3 -D by a transition bus. There exists the waste of transition bus resources in the flight process guidance system. Fortunately, the other passengers can enter T 3 -D at this failure state. If we assume that the capacity of T 3 -D is expanded to C = 450 and 450 passengers are waiting to leave T 3 -D at the initial state, the above two problems can be avoided. Therefore, a suitable capacity for T 3 -D is very important to avoid the saturated congestion problem and reduce the waste of transition bus resources. The existing studies ignore these problems.
In this paper, a scheme that is used to calculate the minimum capacity for each terminal is proposed to solve the above two problems. For each terminal, its capacity is closely related to its inputs and outputs. Therefore, a Petri net model is created for the terminal. Moreover, a relational expression among the capacity, inputs, and outputs of the terminal is derived based on the mathematical methods of Petri nets to relieve the congestion in the entrances of the terminal and reduce the waste of transition bus resources in its exits. The main contributions of this paper are concluded as follows.
1) The formal model of each terminal is proposed based on finite capacity Petri nets. The main purpose is to facilitate the analysis and verification for the behavior of terminals in a flight process guidance system by using the mathematical methods of Petri nets. 2) A relational expression is derived to calculate the minimum capacity for each terminal by analyzing the properties of the proposed Petri net model. It not only can avoid the saturated congestion problems in the entrances of terminals but also can reduce the waste of transition bus resources during the peak periods. Moreover, it also can reduce the building costs by minimizing the capacities of the terminals.
3) The proposed relational expression can also be expanded to control the dispatch of passengers among multiple infrastructures in aviation systems if the capacities of these infrastructures are difficult to be extended. The main purpose is also to avoid the congestion and reduce the waste of transition bus resources by controlling the inputs and outputs of the infrastructures.
This paper is organized as follows. Section II introduces the basics of Petri nets. Section III introduces a saturated congestion problem in a terminal. Section IV presents a capacity optimization scheme to solve the saturated congestion problem for each terminal. Moreover, a simulated study is given in Section V and Section VI concludes this paper.

II. BASICS OF PETRI NETS
This paper only provides some fundamental concepts of finite capacity Petri nets. More details about Petri nets can be found in [42]- [44]. A finite capacity Petri net [22] is a five-tuple N = (P, T , F, W , C), where P and T are finite, P = ∅ and T = ∅. P is a set of places and T is a set of transitions. F ⊆ (P×T )∪(T ×P) is called a flow relationship from places to transitions or transitions to places. W : F → N is a weight function that gives weights to arcs. C : P → N is a capacity function that gives capacities to places. Besides, a finite capacity Petri net can also be expressed as an input matrix In a finite capacity Petri net, t ∈ T is enabled at marking where σ : T → N is the Parikh vector of σ . The σ (t) represents the sum of all occurrences of t in σ . The set of reachable markings from M in N is denoted as R (N , M ).
To clearly indicate the location of the components in a marking or a Parikh vector, the marking or the Parikh vector can be described using a special multiset. Let N = (P, T , F, W , C) be a finite capacity Petri net with P = {p 1 , p 2 , p 3 , · · · , p m } and T = {t 1 , t 2 , t 3 , · · · , t n }. We use (M (p 1 )p 1 , M (p 2 )p 2 , M (p 3 )p 3 , · · · , M (p m )p m ) to denote marking M . Similarly, we use (σ (t 1 )t 1 , σ (t 2 )t 2 , σ (t 3 )t 3 , · · · , σ (t n )t n ) to denote Parikh vector σ . Let p ∈ P be a place in N . All transitions in (1) and (2), if all transitions in • p ∪ p • are enabled, it can obtain Let t 1 and t 2 be two transitions, M be a marking, and σ be a transition sequence. If t 1 and t 2 fire at M , it is denoted as if t 2 fires before t 1 , or if t 1 and t 2 fire at the same time. To facilitate the description of synchronous discrete events in this paper, we have a following assumption.   The initial marking is M 0 = (9, 5, 7) T . Therefore, transitions t 1 and t 2 can both fire simultaneously since M 0 (p 1 ) If transitions t 1 and t 2 simultaneously fire at marking M 0 , a new marking M 1 can be obtained by In the same way, only the transition t 2 can fire at marking M 1 . Let σ 2 = t 2 . We have σ 2 = (0, 1) T . A new marking M 2 can be obtained by 2 ) = (2, 2, 17) T if transition t 2 fires at marking M 1 . Finally, neither t 1 nor t 2 can fire at marking M 2 . The whole process can be denoted as

III. THE SATURATED CONGESTION PROBLEMS
In a flight process guidance system, as shown in Fig. 2, its all processes can be classified into two major steps as follows.
Step 1: During the passenger boarding process, passengers first need to purchase tickets in terminal T 3 -C. They should take the transition buses to terminal T 3 -D to wait for the boarding announcements. Next, they begin to check-in and wait in line for boarding after they hear the boarding announcements.
Step 2: During the passenger landing process, passengers begin to disembark from the airplane and enter the terminal T 3 -D in an orderly manner to take transition buses to the terminal T 3 -C after the airplane arrived at the destination. Finally, they leave terminal T 3 -D by transition buses in an orderly manner. To facilitate a formal description, we assume that passengers who enter terminal T 3 -D are formally divided into several groups with equal numbers. Each group should queue up to enter the terminal T 3 -D to avoid unordered congestion.  shows the Petri net model for the flight process guidance system that is shown in Fig. 2. It contains 12 places and 16 transitions. Each place has a capacity that represents the capacity of the terminal T 3 -D, terminal T 3 -C, and airplanes. Table 1 shows their descriptions.
In order to make the Petri net model of a flight process guidance system more clearer in Fig. 4, we simplify the Petri net model, as shown in Fig. 5, where path p t3c → t edi → p t3d → t exi → p ai is corresponding to step 1 of Fig. 2, denoted as −−−→ p t3c p ai , and the path p ai → t cdi → p t3d → t lei → p t3c is similar with step 2, denoted as −−−→ p ai p t3c . In path −−−→ p ai p t3c , the capacity of p t3d is limited because its building costs are proportional to its capacity. Therefore, a limited flow structure is defined based on finite capacity Petri nets.
Definition 1: A limited flow structure is defined as a finite capacity Petri net N s = (P, T , F, W , C), where is a mapping that assigns a number of passengers to a passenger flow arc. 6) C : F → N is a mapping that assigns a capacity to a limited capacity place p l . In a limited flow structure N s = (P, T , F, W , C) with P = {p i , p l , p o } and T = {t i , t o }, p l has an input flow that represents the number of passengers who are entering the p l , denoted as p I l , and an output flow that represents the number of passengers who are leaving p l by transition buses, denoted as p O l , such that Generally, p I l < p O l , it is a normal state. However, we have p I l p O l during the peak periods. In a limited flow structure, it may not only suffer from the saturated congestion problems but also exist the waste of transition bus resources if many passengers are waiting in terminals and the capacities of the terminals are unsuitable. For example, Fig. 6(a) Fig. 6(b). This means that there exists a saturated congestion problem in the limited flow structure N s .
If the capacity of p 4 is expanded to C(p 4 ) = 1500, as shown in Fig. 7(a), where M 0 (p 4 ) = C(p 4 ), the similar scenario begins as follows.  Fig. 7(b). This means that there exists the waste of transition bus resources in the limited flow structure N s . For a limited flow structure N s = (P, T , F, W , C) with the initial marking M 0 (p l ) = C(p l ). We can conclude that N s may exist a marking M ∈ R(N s , M 0 ) such that p I l + M (p l ) > C(p l ) and M (p l ) < p O l . It may suffer a saturated congestion problem, as shown in Fig. 6(b). In addition, N s may also exist another marking M ∈ R(N s , M 0 ) such that M (p l ) < p O l . It may exist the waste of transition bus resources, as shown in Fig. 7(b). This means that the inconsistent among p I l , p O l and C(p l ) will cause the above two urgent problems during peak periods.

IV. CAPACITY OPTIMIZATION FOR TERMINALS
In order to resolve the two urgent problems during peak periods, it is necessary to calculate appropriate capacities for terminals in limited flow structures. Therefore, a calculation scheme is proposed to calculate the minimum capacities for terminals by also considering their building costs. The proposed scheme not only can avoid the saturated congestion problems but also reduce the waste of transition bus resources during peak periods. The minimized capacities can also economize the costs for building or expanding the terminals.
It means that the number of passengers in p l is reduced gradually. Definition 3: Let N s = (P, T , F, W , C) be a limited flow structure and p l ∈ P be a terminal. Interval [p O l , C(p l ) − p I l ] represents the passenger increasing region of p l , denote as p ↑ l . Interval [C(p l ) − p I l + 1, C(p l )] represents the passenger decreasing region of p l , denote as p ↓ l . Fig. 8 shows the partition of error marking region, legal marking region, passenger increasing region, and passenger decreasing region for p l .
Property 1: Let N s = (P, T , F, W , C) be a limited flow structure, p l ∈ P be a terminal, M 0 be an initial marking such Proof: M 0 (p l ) ∈ p ↓ l since M 0 (p l ) = C(p l ). This means that the passenger numbers in p l will be reduced gradually. Let M 1 ∈ R(N s , M 0 ) be a reachable marking such that M 1 (p l ) = C(p l ) − p I l + 1 is the minimum number in the passenger decreasing region p ↓ l . At marking M 1 , a new marking M can be obtained after transition t ∈ p • l fires by It means that the number of passengers in p l will be increased gradually. Therefore, is the minimum number in p l . For example, we scale down the data to quickly visualize the results, as shown in Fig. 9. According to Property 1, we can obtain that C(p l ) − p I l − p O l + 1 is the minimum number in p l if M 0 (p l ) = C(p l ). Therefore, we still need to prove that ∃M ∈ R(N s , M 0 ) such that Lemma 1: Let x and y be two prime numbers and a 1 , a 2 , · · · , a y be a complete system of residues for a module y. Therefore, xa 1 , xa 2 , · · · , xa y are also a complete system of residues for a module y. This means that xa 1 , xa 2 , · · · , xa y can traverse a 1 , a 2 , · · · , a y [45]. Property 2: Let N s = (P, T , F, W , C) be a limited flow structure, p l ∈ P be a terminal, M 0 be an initial marking such that M 0 (p l ) = C(p l ), p I l and p O l be two prime numbers. Then, ∃M ∈ R(N s , M 0 ) such that M (p l ) = C(p l ) − p I l − p O l + 1. Proof: It is known that p I l and p O l are two prime numbers, 0, 1, 2, · · · , p I l − 1 are a typical complete system of residues for a module p I l and M 0 (p l ) = C(p l ). It is true that the number of passengers is changed between the interval [C(p l ) − p I l − p O l + 1, C(p l )] according to Property 1. Therefore, it is true that ∀M ∈ R(N s , M 0 ) such that where m and n are two positive integers and the increment of n is one. It can obtain that It means that C(p l )−M (p l ) will traverse the complete system of residues 0, 1, 2, · · · , p I l − 1 for a module p I l according to Lemma 1. We can obtain that ∃M ∈ R(N s , M 0 ) such that C where p I l = kx, p O l = ky, k is the greatest common divisor of p I l and p O l , x and y are two prime numbers. Moreover, k, x, and y are positive integers and greater than or equal to one.
In addition, if p I l and p O l are not prime numbers, it obtains that k = 1, p I l = kx, and p O l = ky. It can be considered as that x and y are the new inputs and outputs. Similarly, it can obtain that C(p l ) k(x + 2y − 1).
Let N s = (P, T , F, W , C) be a limited flow structure, p l ∈ P be a terminal and M 0 be an initial marking such that M 0 (p l ) = C(p l ). ∀M ∈ R(N s , M 0 ), it is impossible to suffer a saturated congestion problem at marking M in the terminal p l if C(p l ) satisfies the conditions of Property 4. Moreover, it is also impossible to exist the waste of transition bus resources if C(p l ) k(x + 2y − 1). Therefore, the proposed scheme not only can calculate the minimum capacities for terminals but also can reduce the waste of transition bus resources.

V. EXAMPLE ANALYSIS
As one of the three complex airport hubs in China, Beijing Capital International Airport includes three airlines, i.e., China Southern Airlines, Air China Airlines, and China Eastern Airlines. Among the three airlines, the mainstream model of China Eastern Airlines is the Air bus A320 that is a typical seating 2-class with a capacity range of 150-180. The mainstream model of Air China Airlines is the Boeing B737-800 that is also a typical seating 2-class with a capacity range of 162-178 and the maximum capacity is 210. In this example, we assume that the average capacity of the mainstream models in Beijing Capital International Airport is 163 and the average capacity of the transition buses is 102 by analyzing their aviation resources.
The statistical results show that approximately five airplanes were arrived at the airport during fifteen minutes and passengers should enter terminal T 3 -D as quickly as possible during the peak periods. At the same time, passengers should leave the terminal T 3 -D by five transition buses. The Petri net model for the above scenario is shown in Fig. 9 and Table 2 shows its descriptions. This Petri net model contains seven places {p a1 -p a5 , p 4 , p 5 } and ten transitions {t a1 -t a5 , t t1 -t t5 }. The inputs of p 4 is 163 × 5 = 815, i.e., p I 4 = 815, and the outputs of p 4 is 102 × 5 = 510, i.e., p I 4 = 510.  Fig. 10. In order to make the Petri net model more clearer in Fig. 10, it can be simplified to a flight process guidance system that is a limited flow structure N s = ({p a , p 4 , p 5 }, {t a , t t }, F, W , C), as shown in Fig. 11, where place p a is associated with places p a1 − p a5 , transition t a is associated with transitions t a1 − t a5 , and transition t t is associated with transitions t t1 − t t5 . Let M 0 be the initial marking of N s . ∀M ∈ R(N s , M 0 ), it is necessary to ensure M [t t . The main purpose is to reduce the waste of transition bus resources. Meanwhile, it is also necessary to ensure M [t t or M [{t a , t t } . The purpose is to avoid the saturated congestion problem. Both the necessary requirements require an appropriate capacity for p 4 Assume M 0 (p 4 ) = C(p 4 ) = 1830, as shown in Fig. 11. after a series of transitions among t a , t t , or {t a , t t } fire, where σ = t t t t {t a t t }t t · · · {t a t t }t t {t a t t }{t a t t } = σ 1 + σ 2 + σ 3 + σ 4 + · · · + σ 162 + σ 163 + σ 164 = σ 5 + 100t a + 159t t .
The firing processes are modeled by a tool that is developed by programming language C + + [46]. It forms a loop from marking M 1 to marking M 164 . Therefore, it can effectively avoid the saturated congestion problem. By carefully verifying the results, we know that transition t t can fire at marking M 0 . Moreover, ∀M ∈ {M 1 , M 2 , M 3 , M 4 , M 5 · · · , M 164 }, transition t t can always fire at marking M . This means that passengers in p 4 are continuously leaving terminal p 4 by transition buses and the transition buses are fully loaded. It can reduce the waste of transition bus resources. It knows from the example above that can avoid the saturated congestion problem if C(p l ) satisfies the conditions of C(p 4 ) k(x +2y−1) in Property 4. If C(p 4 ) < k(x +2y−1), that exists the waste of transition bus resources, the examples are as follows.