According to D'Ambrósio (1986, p.11) "Modeling is a very rich process of facing situations and culminates with the effective solution of the real problem and not with the simple formal resolution of an artificial problem." Already for Biembengut and Hein (2003, p. 16) "Mathematical Modeling is the art of transforming problems of reality into mathematical problems and solving them by interpreting their solutions in real-world language." Given these concepts, it is clear that mathematics and reality are disjoint sets that can be brought into interaction through modeling. And also, the need to promote the teaching of mathematics so that real models are worked.

          The authors Biembengut and Hein (2003) also mention that the modeling process can be divided into three steps, which are interconnected and can be resumed if necessary, as follows:

• First step: The interaction with the problem, which consists in making the first contact with the problem situation and also, if necessary, the search for more information for a better understanding and familiarization of the theme in question.

• Second stage: Mathematization, which consists in identifying and formulating the problem through a set of formulas, graphs or arithmetic expressions that can solve the problem.

• Third Stage: Mathematical Model, which is validated in this stage, the model obtained in the second stage, besides analyzing its reliability in the use of the model situation. And if unreliable it is necessary to return to the second step and improve suitability.

       According to DEMO (1991) apud SILVA et al. (2015), modeling enables an investigative action as a way to know, understand and act in reality, also through field research, being possible to collect data and information to perform qualitative analyzes, generating the need for investigative actions. , assumed as a scientific and educational principle. Accordingly, with him, Bassanezi (2002, p. 33-34) some positive points to work with modeling, they are:

• It can stimulate new ideas and experimental techniques;

• It may give information on different aspects from those originally intended;

• It can be a method for making tweens, extrapolations, and predictions;

• May suggest priorities for resource and research applications and eventual decision making;

• It can fill in gaps where experimental data is lacking;

• It can serve as a resource for a better understanding of reality;

       And yet, SILVA, et al. (2015) apud Blum (1989), in the beneficial perspective to work with modeling in mathematics teaching, there are six favorable arguments:

1. Formative argument: Students become explorative, creative, and problem-solving through mathematical applications through modeling.

2. Critical competence argument: students become active in society;

3.  Utility Argument: Using Math as a Tool for Solving Different Problems

4. Intrinsic Argument: Provides the opportunity to understand and interpret mathematics from all angles;

5. Learning argument: enables a better understanding of mathematical arguments, concepts and results.

6. Epistemological alternative argument: modeling acquires an ethnomathematic characteristic, in which a part of reality naturally comes to the rigor and mathematical formalism.

        Therefore, according to SILVA et al. (2015) Given this, the Mathematical Modeling is essential for the investigative environment, allowing a greater understanding of the mathematical concepts viewed daily.

       

        REFERENCES: 

​

BIEMBENGUT, M. S. & Hein.N. Mathematical Modeling in teaching. São Paulo: Context, 2003.

​

D'AMBRÓSIO, U. From reality to action: reflections on education and mathematics. Sao Paulo: Summus, 1986.

​

SILVA, et al. Math Modeling: Theoretical  Reflections and Applications. Juiz de Fora, 2015.

​

​

​

​

​

​

​

​

 

• 1st Stage: Definition of the subject, production of the blog and direcionamento of the work to be done.​

• 2nd Stage: Definition of the problem looking for what wants to be modelled.

• 3rd Stage: General observation of the model, defining component, hypotheses, variables and deciding on the objective of the considered model.

• 4th Stage: Data collection, analysing the situation of the hospital service in one day, from literatures, articles, sites.

• 5th Stage: Analysis of the data collected, producing equations for the modeling of the process.

• 6th Stage: Modeling of the project in function of the Theorem of lines.

• 7th Stage: Creation of the article, discussions of results and conclusion of the project Modeling Hospital Emergence.

IMPORTANT: During the project the group found difficulties for the development of the project. However, the monitorias and classrooms helped, so that the project was developed.

​

​

​

​

​

​

​

• REPORT:

There is common in the services of emergences of hospitals the use of techniques of business and industrial managements to organize the patients flow. Recently, the Ministry of Health announced the adoption of the system Lean in six hospitals of the Country. The Japanese method created in the factory of cars of the Toyota, aims to identify and to reduce the wastes. In the health area, the mark of the SUS is to administer the overcrowding.
Teacher of the Department of Politics, Management and Health of the University of São Paulo, Marilia Louvison one tells “preoccupied with idioms” and it emphasizes that “these initiatives exist already in the health system”.
“It is basic that the health services, in individual the hospital ones, organize his entrance doors to minimize the onus of to be a service of door confided in a certainty an imprevisibilidade of the demand”, it recognizes.
However, the most important thing, according to the specialist, is to integrate effectively several service levels to the population to ease the lines. Over there, it is clear, of providing the deficiencies. “If there is no answer flow, I can organize a line of the best possible way. If she does not walk, it does not decide.”

Link of report: https://jornal.usp.br/atualidades/filas-em-hospitais-publicos-tem-causas-complexas/

​

​

​

​

​

​

​

​

The lines are present in day by day as with example: in supermarkets, banks, traffic, or still, in any situation in which it is necessary to hope for a servço or opportunity.
 "A lines system can be described like clients who arrive for one determined service in which they are attended immediately or wait, going out after the service. (Unicamp, 2012, p.3)

"The main reason of Theory of Lines is studied is to optimize the system, that if
it characterizes for:
- better use of the available services,
- less wait time,
- bigger speed in the service." (Unicamp, 2012, p.3)

​

It imagines 1: Structure of a system of waiting line

​

​

​

​

​

​

​

​

​

​

​

GARAY, A.W.M. An introduction to the Theory of the Lines. University State of Campinas.2012. Available in: <http://www.ime.unicamp.br/~hlachos/ME323-Teoria%20Filas >. Acesso em :22 Nov 2019.

​

Tax of use of the system (), calculated for: ρ = λ ÷ μ. She shows if the system produces or does not seize. For ρ> 1, it has that there is a number bigger of persons who arrive of which attended persons, in other words, the use of the service system is bigger than his capacity. 
 Middle number of clients in the line (Lq), calculated for: Lq = λ ² / (μ (μ-λ)). It does an average of the size of the produced line. In cases where is limit for the size of the line, this calculation returns necessarily so that it knows in which point it has to in order that of clients were received or which point becomes critical the clients' reception. 
 Middle wait time in the line (Wq), calculated for: Wq = λ / (μ (μ-λ)). It does an average of the time in which a client / patient has to wait to receive the given service. Besides having an average of the number of clients in the line, the clients time also is necessary for some taking decision or even to revise the service process so that it has an every time bigger profit. It is always good to have the reduced wait time, but in some cases this wait time can become the most important factor for the one who waits, be a patient needing treatment or someone in the line of a restaurant for the lunch. So, the observation of this parameter is necessary for the good functioning of the service.
 The middle number of persons in the system (Ls), calculated for: Ls = λ / (μ-λ). He indicates the number of present persons thinking those that are in state of wait, in the line, up to the persons who are being attended.
 And for last, the middle persons time in the system (Ws), calculated for: Ws=1 / (μ-λ). He calculates the average of time that a person takes from the moment in which this reprimand, up to the end of his service.

 

SANTOS, C.M.S.; LYRA, L.M.L. Avaliação do Atendimento em Unidades de pronto atendimento: estudo de caso aplicando teoria das filas. XXVIII Encontro Nacional de Engenharia de Produção. Joinville, SC. 2017. 

​

​

 

 

​

​

 

The factor Ws is tied the Wq of analogous way to inductors in parallel since both "divide" the persons flow in the system (Ls). And this parallel is tied to ρ (tax of use). For end, who gives rise to the persons flow in the system (service) is the line (middle number of patients in the reception).

​

It imagines 2 – Analogy of the Hospital System with Electric Circuit

​

​

​

​

​

​

​

​

The equacionamento of the electric circuit is given for:  VR + VL = VF (1)​

In what: VR = tension in the resistor;
              VL = tension in the inductor;
              VF = tension of the fountain.

 VL1 = VL2 (2)
Essa equação pode ser melhorada da seguinte maneira:
R.i + L.di/dt =VF (3)
Adaptando para esse modelo fica:
ρ.Ls + Ws.dLs/dt = Lq (4)
Ws.dLs/dt = Wq.dLs/dt (5)

The function of transfer that models the system for the adopted method is given for:
Ls (s) = 0,0039/s (s 0,04) (6)
Then the entry on her the exit of the system is:
(Ls (s)) / Lq (s) =1 / (s (s 0,04)) (7)

​

References: final article of developed by the group

​

​

​

​

​

​

​

​

​

​

​

​

​