Multiple Table Models Based on Queuing Theory for Firefighting Traffic Problems

With the development of global urbanization, urban planning has been a hot spot of most concern. Since the traditional urban fire station setting method has been gradually unable to meet the emergency needs of a city. In this paper, a series of methods are developed to achieve the best balance between economic expenditure and reliability. According to the heat maps about emergency calls in a city in recent years, different types of alarm phones have obvious regional distribution characteristics. Inspired by the Queuing Theory algorithm, we abstract this complex problem into a parallel multi-server waiting queuing model to find each average queue length of three kinds of alarm calls and success in converting the multiplicity problem into solving the vehicle number and location problems respectively. Based on this planning problem, new combinations with different numbers of vehicles can be gotten. Then based on the assignment model, the optimal deployment plan can also be different with different numbers and geographical locations of the three kinds of stations. Our method significantly improves the firefighting traffic problems and can be applied in the actual situation.


Introduction
Responding to fire alarms [1][2][3], medical alarms [4][5][6] and false alarms [7][8][9] in a timely manner is a problem with many complexities.According to statistics, around 10000 fires occur every day around the world, resulting in over 2000 deaths and 3000-4000 injuries, resulting in incalculable property losses.So once there is a fire, it is necessary to immediately report it to the police.The earlier the alarm is triggered, the smaller the loss.False alarms refer to information that is falsely declared to be a real threat or danger without actual threat or danger.The consequences of false alarms can be very serious, not only wasting public resources and time, but also causing unnecessary panic and chaos.It is crucial to establish an effective emergency drill plan for handling false alarms in response to such incidents.
Firstly, determining whether it is a false alarm should focus on identifying and evaluating false alarms.By simulating various false alarm scenarios, relevant personnel are trained to judge and evaluate false information, in order to reduce misjudgment and misleading.We must establish a sound command system and emergency plans.When conducting alarm simulation, attention should be paid to joint collaboration and resource integration among various departments.At the same time, we establish communication simulations between different departments to strengthen information sharing, in order to timely assess the situation and work together to solve the problem of false alarms.
Through in-depth analysis and research on the background of the problem, combined with the specific constraints given, the restate of the problem can be expressed as follows: (1) Build a mathematical model to determine the optional fire department number and combination of the ladder tracks, paramedic vehicles and engine tracks.The model should balance several factors.
(2) Based on the model, explain how it adapts to the higher travel cost through change.
(3) Based on the model, explain how it adapts to the higher private provider cost through change.
(4) Considering whether all vehicles should be used to serve the public.
(5) Considering whether all vehicles should be used to serve the public.

Notations
Some important mathematical notations used in this paper are listed in Table 1.
Table 1.Notations used in this paper.Note: There are some variables that are not listed here and will be discussed in detail in each section.

Rasterization of Regions
The work of the predecessors has given us some inspiration [10][11][12][13].For this problem, we should first determine its spatial mode.The purpose is to use a mathematical model to express the actual physical environment, which is convenient for computer processing.So, this article uses grid method to tackle this problem.Therefore, the grid site ) , ( y x of any point can be expressed as: where ) , ( y x is the coordinate of a potential place, S G represents the grid size, here S G can be identified as 1 mile.
The grid is divided into three types, Existing Grid, Possible Grid and Common Grid, the specific introduction is as follows: (1) Common Grid: The grid of common place is used to represent the places other than the location of the fire station.
(2) These 10 zones are marked by the four-color method of the map.Fire alarms, medical alarms and false alarms will be generated in these areas, and these alarms be sent to the nearest emergency center or private supplier.
(3) Existing Grid: The Existing Grid is a location for the existing emergency station settings.Each of them can hold two vehicles and be dispatched when emergency calls are generated.They cost $20,000 per year.
(4) Possible Grid: As the name suggests, these locations can set up stations as needed.Similarly, they can each hold two vehicles and be dispatched when necessary.They spend a certain amount of construction cost every year.
These Grid can be easily displayed by Figure 1:

The Multi-Server Desk Queuing Model
Queuing theory [14-15] (stochastic service system theory), is based on the statistical research on the arrival of service objects and service time.This paper simulates the multi-server waiting queuing model M/M/s/∞/∞/FCFS.The system space is infinite and the customer source is infinite.Let the system have s service windows arranged in parallel, each service window works independently, and the service time of each window obeys the negative exponential distribution.The calculation is brought into the average service time in the setting conditions.It is assumed that customers arrive according to the Poisson distribution with parameter λ , which is that the interval of customers' arrival follows an exponential distribution.If the s service windows are busy when customers arrive at the system, customers queue and wait.It is assumed that each service window works independently according to the first-come-first-serve (FCFS) principle, allowing for permanent queuing.In the case of global dispatching, FCFS is used to solve the problem, while use PS in local dispatching.Since the nearest rescue vehicle response is required for each alarm call, we take all the alarms in the whole city as customer sources, and all the corresponding response vehicles as service counters to establish a multi-server waiting queuing model.
The important process in queuing theory is deduced below: Suppose that the customer arrives individually, and the successive arrival time interval obeys the negative exponential distribution with the parameter λ .There are s service desks in the system, and the service time of each service desk is independent of each other and obeys the negative exponential distribution with the parameter μ.When the customer arrives, if there is a free service desk, they will immediately accept the service, otherwise they will wait in a queue, and the waiting time is infinite.
So, the following formula as: where n represents the number of customers in the system under equilibrium ρ indicates the average number of customers being served, q L is the average queue length.
Based on the formula (2), we build three vehicles respond model [16] as follows: (1) Ladder Tracks Response Model For the 10 regions of the whole city, the number of fire alarms i a per 12 hours in a year is obtained by the sum of the second column of the table.Therefore, the average arrival rate of customers per hour in this model and the service rate of the service counter is: By constructing a multi-service counter waiting queuing model, we obtain the average queue length q L based on formula (2), ( 3), ( 4).
(2) Paramedic Vehicles Response Model For the 10 regions of the whole city, the number of medical alarms i b per 12 hours in a year is obtained by summing up the fourth column of the table.Therefore, the average arrival rate of customers per hour in this model and the service rate of the service counter is: By constructing the queuing model of multiple service counters, we obtain the average queue length q L based on formula (2), ( 5), ( 6). (

3) Engine Tracks Response Model
The engine trucks will be used to respond to all alarm calls, including fire alarms, medical alarms and false alarms.Therefore, for the 10 regions of the whole city, the total number of alarms per 12 hours in a year c is obtained by summing the second, third and fourth columns of the table.Therefore, c is used to obtain the average hourly arrival rate of customers in this model: The average time of engine trucks alarm processing can be obtained by weighted sum of three alarm processing times [17]: so, the service rate of the server counters is: By constructing a multi-server waiting queue model, we get the average queue length 3 q L based on formula ( 8), (9).
We abstract this complex problem into a parallel multiple servers waiting queue model to find the average queue length for each of the three alarm calls.So, we need to use different service desks to set up three curves.For the three multi-server models, considering the different number of servers, the number-average queuing length curve can be drawn as follows in Figure 2:

Results and Analysis
We first use the multi-server waiting system queuing model proposed to separate the problem conditions, abstract the queuing process, analyze the vehicle scheduling under three alarm situations, and obtain the average queuing length under the existing conditions.The results are shown in Figure 3.
That's great.We've made sure how many cars we need!Next, we can get the final solution by simply assigning the location of the existing vehicle using the assignment problem.Although we first consider the number of vehicles and then consider the location, it does not affect the rationality of our proposal, because we comprehensively consider the response alarm scheduling, private provider cost, and scheduling travel cost factors in turn.
As can be seen, the more expensive the private supplier's prices, the more vehicles need to be taken into use to make sure there are adequate vehicles at all times.Also, as the number of vehicles increases, more station is put into use.
For the traffic cost cases, we chose the changing factor to vary in the interval from 0.1 to 15 with a 0.1 increasing gap.The increase in the traffic cost mainly changes the balance between distance and the number of stations.When it is cheap for the vehicles to take long-term travel, the base cost for an extra station will appear more expensive which means fewer stations in feasible range may bring greater benefit.On the contrary, when it is more expensive travel, short distance travel will be less expensive and there will be more stations be taken into use correspondingly.
As we abstract the deployment problem of vehicles into an integer programming one, the changing of the deployment plan is discrete.Two of the typical deployment plans will be shown in follows which can effectively embody features.By cleverly establishing constraints and objective functions, we transform the actual problem into the sequential solution process of multiple mathematical models, analyze and solve the problem layer by layer, and finally obtain the optimal solution.Based on the existing setup conditions, our solution for the Fire Department is shown below.

Conclusion
In this paper, by combining the queuing theory, the complex problem is abstracted into a parallel multiserver waiting queuing model to find each average queue length of three kinds of alarm calls and success in converting the multiplicity problem into solving the vehicle number and location problems respectively.By solving the planning problem, new combinations with different numbers of vehicles can be calculated.Then by the assignment model, the optimal deployment plan will also be different with different numbers and geographical locations of the three kinds of stations.Finally, the actual situation is analysed by the model.Due to the problem limitation, when vehicles are dispatched on a call, the closest idle vehicle is dispatched first.If no vehicles are idle, then the call must be sent to a private provider.So, we take the average queue length as the service request severed by a private provider and bring it into the Multi-objective Planning Model.Considering that both the station cost and the number of calls sent to a private provider are optimization objectives and the two are linked by the queue model, we use the idea of assigning weights to optimize the objectives and determine the optimal number of vehicles.
Future work will use the new artificial intelligence methodology and framework such as convolutional neural network and generative adversarial network to solve the firefighting traffic problems, and compare with the conclusion of this paper.
ij D the distance between the i-th grid and the j-th grid i BC the i-th station's base cost per year jk DC the demand call from the j-th grid to the k-th station

Figure 3 .
Figure 3. Result of the Question.