OPTIMIZING THE PRE-HOSPITAL MOBILE CARE SYSTEM : A CASE STUDY

Freidenfields (1980) introduced for the modeling of several transport systems demands the concepts of queueing theories and studied the problem of the capacity expansion of the transport system as a random process of life and death, showing that it is possible to adapt the stochastic model of demand growth into a deterministic model. Souza (1996) applied this theory to predict the expansion of the emergency care systems. The modeling of the supply of Us integrated into the hospitals – emergency care and inter-hospital removals – despite being considered a restricted market service, as new solutions are developed new knowledge is aggregated into an increasingly lower cost (GOLDBERG, 2004). The dimensioning, allocation and distribution of the supply of Us developed for the pre-hospital mobile care system, utilizing data based on the Brazilian situation, is a field that deserves extreme attention. That will allow the assessment of the present situation and can lead to new routes in terms of public policies. Thus, the distribution of service stations of the regulation centers represents the ordering and orienting element of the State Systems of Urgency and Emergency. These centers must be structured in all levels, organizing the relation between several services, qualifying the flux of patients in the system and generating an integrative gateway for the hospitals, by which distress signals are received, evaluated and ranked. These rules must be followed by all services, both public or private. It can be mentioned, as an example, that for the emergency services a widely used measure is the maximization use of the Us or the minimization of response time (TR), between any user of the transport system and the nearest hospital. 1Universidade Federal de Pernambuco UFPE, Departamento de Engenharia Civil, CEP: 50670901, Recife, PE, Brasil. E-mail: dmc@poli.br 2Universidade Federal de Pernambuco UFPE, Departamento de Estat́ıstica, CEP: 50670-901, Recife, PE, Brasil. E-mail: gausscordeiro@gmail.com 3Instituto Materno Infantil de Pernambuco IMIP, Recife, PE, Brasil. E-mail: danieldiscunha@hotmail.com Rev. Bras. Biom., Lavras, v.36, n.1, p.157-187, 2018 doi: 10.28951/rbb.v36i1.161 157


Introduction
The first models developed for care emergency services were deterministic according to (CRONK et al., 1986;GOODMAN et al., 1986) and were important for planning and investigation analysis, ignoring the stochastic considerations about the problem.A revision of the specific models of localization of service unit (U s ) for medical emergency is addressed by (BROTCORNE et al., 2003).
A great disadvantage of these models is that they come from the hypothesis that U s are available when requested, which is not always reasonable in practical applications.The congestion in emergency care services, that can induce the unavailability of U s , motivated the development of probabilistic models.In probabilistic modeling of emergency services, some simplifying hypotheses allow the use of mathematical programming.Nonetheless, in more general situations some hypotheses are not applicable; conducting the treatment of these problems through the use of stochastic process.
Various probabilistic models have been developed, considering the stochastic nature of the events (accidents), such as the fact that U s operate as servers in a queue system and sometimes are not available for operation, as occurred in the case study developed by GOLDBERG et al. (1990) and CHING (1997).The hypercube model, proposed by Larson (1982), and analyzed by many authors (SWERSEY, 1994) is an important mathematical tool for the emergency systems planning, especially the urban systems in which the U s moves to assist in some type of accident.The model is adequate for analyzing coordinated or centralized systems, where the user who needs assistance is demanded through a care center system.The system manager then dispatches the U s of a facility that is close to the occurrence.In the case of no U s being available, the request is put into a waiting queue so that it can be fulfilled as soon as the U s is available.
The applications of the hypercube model are innumerable.In Brazil, some examples are: the ambulance localization in a tract of the BR-111 -SC road (GONSALVES et al., 1994(GONSALVES et al., , 1995)), the balancing of the usage factor of ambulances in the "Asphalt Angels" system of the Presidente Dutra road (MENDONC ¸A andMORABITO, 2000, 2001), and the performance analysis of the SAMU-192 system of Campinas -São Paulo (TAKEDA et al., 2004).
Initially the intention is of calculating the minimum number of U s necessary from each region R, from estimated parameters of this region, in a way that there is enough confidence to not have any missing U s for assisting the patient of a specific region.
To maximize the efficiency of the assistance, it is made necessary to adequately spread the U s of a city, minimizing the response time T R, defined as the time between the request of U s and their arrival at the accident location.The most severe problem is that, in general, between the reception of the request in the screening center or service station and the sending of the service unit -U s , some time is unnecessarily lost as a result of a lack of a perfect operational system in the screening center.

Dimensioning of stations/Integration centers -Hospital beds
Consider a given region R with initial population h.Each user who belongs to R has constant long time coordinates (λ, µ), defined by: λ: user accident average rate; µ: off-care user average rate.
(λ) and (µ) are denominated average rates because it is necessary to estimate them from a known population.The accident event is defined as any entry in the hospital through a service unit -U s .The following hypotheses are premises for the system demand dimensioning: 1. User of region R can only suffer accidents by being out of hospitalization; 2. All injured and not injured users (operational) are considered as random independent variables; 3. Individuals who arrive at hospital using a U s as means of transport are considered users of the system; 4. Population h is considered a function that does not vary during the time of the modeling.
By the first hypothesis we do not compute as accidents the removals (user patients that are transferred from a hospital to another), because they are interactions that do not alter the number of hospitalized individuals (system users).The definitions hereafter are to be considered: T M H = λ −1 , average time of system user's hospitalization and T M F = µ −1 , average duration time of a not injured patient -user's out of system (healthy user).
Considering T V as the travel time of a U s , it can be then written that λ −1 = T V + T M H, because at the moment the individual suffers an accident, he/she becomes the system user and at the moment he/she gets out of the hospital, the user is cured, exiting the system.An approximation of this equation to reality can be made, because as quoted in (CORDEIRO, 2012) the T M H in Brazil varies from 6, 10 days to 8, 65 days; in other words, it is possible to consider that T M H >> T V .Thus, this variable can be initially overlooked by doing, T M H = λ −1 .The presumption that T V = 0 is coherent with the third hypothesis, due to the fact that possible improvements in the U s s services of type GP S are not being considered because these only alter the T V , that is being relaxed in the initial modeling.
Consider now X t defined as a random variable that represents the number of cured citizens or non-injured in the region R in the period of time t.The maximum user population of the system (maximum demand of the system -h max ) represents the whole community (population h) of region R, while the user population of the system (effective demand) refers to the injured individuals (users) of the region R that were transferred to a hospital using U s as means of transport.Being j the state it can be said that the system is in state j -E j in time t if, and only if, X t = j.In this way, it can be defined that the probability of an individual in state j is cured is represented by: P j (t) = P (X t = j).
First and second hypotheses conditioned to the system user rates -injured individuals (λ j ) and operational -cured individuals (µ j ), represent a process of life and death, respectively.A simplified equation, for the calculation P j (t) = P (X t = j), is detailed in (CORDEIRO, 2002), in other words: (1) Defining p as being the probability of an individual in R of being cured in the long run, in other words, p is the proportion of the time he remains out of the system.This equation represents the quotient between the T M F and (T M F + T M H).Therefore, Taking the complement of the Equation (2), it turns out that (1 − p) = µ λ + µ , which represents the probability of an individual in region R of becoming a user of the system.It is important to highlight that these probabilities are stabilized in the long run.This way, after the replacement of the term λ µ in Equation (1), we come from the probability function of P j , to a very known distribution, which is the binomial distribution with success probability p and failure probability (1 − p).
After the definition of the X distribution, it is important to calculate the statistics distribution.This way, the number of cured individuals (X) after the system reaches the stable equilibrium follows a binomial distribution of average E(X) = hp and variance Being Y another random variable, which represents the number of users relocated from the system to a hospital, using U s as means of transport.Then, the population variable can be written as H = X + Y ; and as Y is a binomial, H also follows the binomial law (HOEL, 1996).Therefore, the probability of j injured individuals in region R being users of the system can be calculated by, Consider now Y = h i=0 Y i and Y i as independent.The "dummy" variable concept is adopted, in which Y i = 1 if the ith individual is a user of the system (relocated to the hospital, using U s as means of transport) or Y i = 0, if not.In virtue of h being big in comparison with Y , we understand by the central limit theorem (HOEL, 1996) that the random variable Y will be a normal one with the following representation: In order to gain a more accurate picture, suppose that the population of region R is h = 15.000-order of magnitude 10 4 ; it can be understood that the probability of an individual to be cured (not injured) in region R is of p = 0, 999.So it is important to know what is the probability of an individual never being a user of the system -there never is an accident with this individual inside the region R, in a way that, by suffering an accident, he/she will be relocated to a hospital using U s as a means of transport.Therefore we have, where Z ∼ N (0, 1).Making the appropriate substitutions, we can find that P (Y ≤ 0) = P (Z ≤ −3, 8749) = 0, 0001 = 0. Thus, using a Z ∼ N (0, 1) -reduced normal , does not bring significant results, because it imputes P (Y ≤ 0), a 0, 0001 probability -only 10 times smaller than the order of magnitude of (1 − p) = 0, 001, to an event that surely can never happen.Consequently, for bigger improvements in the pre-hospital care services (MCALEER and NAQVI, 1994) and (TAVAKOLI and LIGHTNER, 2003) suggest that the regions corresponding to a city division must be smaller in a way that: h ↓ implies a P (Y ≤ 0) ↑.In order to correct the distortion of P (Y ≤ 0), it is necessary to adopt the following procedure: instead of Y being treated as a normal Z ∼ N (0, 1), treating Y as a truncated normal, which probability density function (p.d.f.) is given by: with Y > 0 and c representing the correction constant for the p.d.f. of Y , calculated by: To dimension h it is necessary to know the number of hospital beds (n L ) being supplied by the hospital beforehand, which are situated in the region R. Knowing n L , the problem consists now in calculating the maximum user population of the transport by U s , in a way that the probability of not having missing hospital beds is known (α).
The solution for this problem will be important in the division of a city in assistance zones (ZA) which will correspond to the hospitals in a one-to-one relation.That way, knowing n L , we can arrive to the calculation of its user population and the corresponding zone.Consider that Y ∼ N (h(1 − p), h(1 − p)p).Then, we can calculate the probability P (Y ≤ n L ) = α, by means of: Resolving the Equation ( 7) we can obtain the estimation of the n L equation.
The Equation ( 8) is very general, because it allows to calculate n L from h and vice-versa under any circumstances.However, if the condition h ≥ 16 p 1 − p is satisfied, the problem inverts, and the calculation is made by Z ∼ N (0, 1).That way, knowing h and p, we can now calculate n L by: Considering that there must not be any beds missing in the hospitals, then α = 1.Therefore, any of the equations can be used for the definition of n L .And based on the normal distribution table On this case, the search for the minimum value of n L is obtained from the equality: The city of Recife, currently, provides 8.089 hospital beds; in practice when dividing the city in ZAs (zones), The n L s being provided are certainly greater than 40.This value was calculated considering the extreme case, in which each Zone represents a city neighborhood (there are in Recife at least 200 neighborhoods).Setting the average variability of (1 − p) between 4, 2 × 10 −4 (national average) and 1, 0 × 10 −2 , the condition h ≥ 16 p 1 − p will always be verified.Thus, the search for h always comes from the Equation (8).Now, doing the proper isolation of h, we have: In case of interest -no missing beds, α = 1.Replacing Φ −1 (1) ≥ 4 in Equation ( 11), the value of h is determined according to n L , and vice-versa and, therefore The maximum user population value (maximum demand) h max is calculated by considering the strict equality in the above equation.
From Equation (11) it is concluded: if α ↑ then h ↓; when α → 0, h → +∞.In Table 1, the results of the simulation for obtaining the values of h are found from (1 − p), α and n L .The quantile function of the normal Φ −1 (α) = z, being Z ∼ N (0, 1) was calculated using the routines "Cumulative probability" and "Inverse cumulative probability" (MINITAB, 2011).Values of Φ(z) were generated between 0 and 3.9999.Afterwards Φ −1 (α) was selected for α = 0, 8, α = 0, 95 and α = 0, 99999 (h max -maximum user population or maximum demand).The values of (1−p) -the probabilities of an individual being a system user were fixated in 0, 001, 0, 002, 0, 004, 0, 008 and 0, 01.The World Health Organization (WHO) recommends for the hospital conjuncture that the number of hospital beds of a city can be determined by the ratio of 3 beds for each 1.000 inhabitants.In Brazil, the adopted ratio is usually 2,53 beds for each 1.000 inhabitants (IBGE, 2005).However, each country adopts a different ratio.It is a fact that the conditions for an inhabitant to suffer an accident and be taken to a hospital vary according to the region.
In major cities the probability is way higher than in the respective states.Under the Equation (12), if n L is a lot bigger (n L → +∞) and α is any positive value greater than zero, we have: and, this way, (1 − p) can be defined as being the bed per inhabitant unit factor, and that way, the number of beds is equal to the expected number of injured individuals E(Y ) (system users).
Therefore, it can be concluded that the greater n L is, the smaller the average percentage of idleness will be (tendency of system user population being equal to the bed supply); this average idleness time tends towards zero when n L → +∞.That way, the usage of hospital beds percentage is given by h (1 − p) n L .Thus, for α and (1 − p) being constants, the expected percentage of idle beds is 1 which increases when n L increases.
3 Estimates of Pp Consider Y t be the random variable that represents total the number of injured individuals (system users) and hospitalized in the period of time t and being I n the number of hospitalized users of a given region R's hospital in the period of time ∆t = (t 2 − t 1 ), which in regards to the survey made by IBGE is annual (365 days).
Consider, now, I n1 , the number of hospitalized users remaining in the period of time t 1 .The area of the curve Y t between t 2 and t 1 , according to Figure 1, represents the total of injured users and hospitalization in period of time ∆t (user x day).So, the average time of hospitalization -T M H for all individuals (cured individuals + system users) of the region R, after the stable equilibrium, will be calculated by: Solving it, we come to: and, then, the estimation for p is given by the following equation: This would be the correct estimation procedure for T M H -Equation ( 13), that is, knowing the curve Y in function of t, the value of A would be calculated by the Y integral between t 1 and t 2 .Therefore, with the result of A (users x day) we can calculate the value of E(Y ) -average number of injured individuals in region R, who are taken to the hospital using U s as a means of transport.In practice, the variation of the Y coefficient -relative dispersion of Y -is very small compared with h.This coefficient by definition is . Thus, c Y in function of A and ∆t is given by: To clarify the developed methodology -estimations of T M H, p, number of hospital beds n L , n Lmin , T M F , T M H the three main capitals of states of the Northeast region of Brazil (region that presented the highest T M H) were chosen along with Belo Horizonte and Curitiba cities.For the estimations of T M H ( T M H) it was considered that the relocation of the injured individual will be made to any hospital institution.In all calculations made for the minimum number of hospital beds n Lmin , the Equation ( 10) is utilized and for the considerations with reliability 100% (α = 1).This result is compared with the effective beds with consideration to the missing beds estimation (deficit).All calculations above are represented in Table 2.
In relation to the entry data for making the proper estimations, they were obtained from the Medical Sanitary Assistance -AM S (IBGE, 2005).The data of Table 2 was calculated from the estimated values of p(p) and (1 − p) ((1 − p)) of the cities of Recife, Fortaleza, Salvador, Belo Horizonte and Curitiba.This table also presents the total number of hospital beds represented by the private and official establishments (Municipal, State, Federal and partnered with SUS -Health Unic System).
For instance, the city of Salvador has a bigger population than Recife; but the probability of an individual being injured and relocated to hospitalization is 0, 00292 in Salvador city, while in Recife it is 0, 00544.That is why the minimum number of hospital beds required is greater than Recife (8.448) in comparison with Salvador (8.026).In regards to the estimated values of T M H, n Lmin and T M F there are no abnormalities.
Observe that for the set of elements constituted by the estimated probabilities (p), the value of the variance is of order of magnitude 10 −7 .As the probabilities are situated in the same reliability range, it can be stated that the probabilities of an injured user being relocated to hospitalization is, in the statistical point of view, practically the same for all analyzed cities, no matter if it is a public or private institution.
Therefore, hospital beds can not be guaranteed to any individual from the five analyzed cities.Maybe these arduous deficits are caused by predominant factors, that hinder the increase of n L .However, if the assistance policy is to guarantee hospitalization to the insured people of SUS, or to any individual in general -as predicted in the Federal Constitution -the hospitalization can be made in any institution -therefore, any variant for the estimation of T M H and p must be pondered with all hospitals.
For example, the hospital of the University of Pernambuco -"Hospital da Restauração" -HR -situated in one of the most heavily-trafficked avenues of Recife, has currently 535 beds.How many patients can utilize the hospital with 100% reliability?Applying the Equation ( 12), we have: h max = 83.397users.This way, this hospital can serve to an area with 84 thousand inhabitants.If the former is destined only to the SUS insured, this area will be bigger and, in any case, this area can be calculated by the estimation of the user density correspondent to the assistance policy adopted by the hospital.
To estimate p (p) we must define the equations before and after the time for the stable equilibrium (t e ).Being h the population of region R with J non-injured individuals, then: λ j = (h − j)λ and µ j = jµ, for j = 0, 1, . . ., h.The average value of λ (λ) can be calculated using the concept of weighted average.That is, λ = h j=0 λ j P j , in which P j represents the probability of an individual in state j not being a user of the system.Then, Using Equation (1), for the calculation of P j and making the proper substitutions, we arrive to the following expression for the average rate of injured individuals (λ): With regards to the average rate of non-injured individuals (µ), it is calculated using a procedure similar to the calculation of (λ).Making the proper substitutions, we find the reduced form for µ.
Thus, to define the average of p (p), that is, the average probability of an individual being out of the system, the same formalized development for Equation (2) is utilized: This way, it is determined that p = p -Equation ( 2).Thus, it is important for eliminating any doubts about the values of p t to establish a function that explains the behavior of p t with t.Being q t = (1 − p t ) the probabilistic distribution of the number of injured individuals in a period of time.Consider that g(t) represents the probability density function (p.d.f.) of the distribution q t , and g(t) is the p.d.f. of the exponential distribution, represented below: To calculate (1 − p t ), it is necessary to use the concept of cumulative distribution G(t) for g(t).For this, we must calculate the g(t) integral in its domain.
Solving the integral in the interval of (0, t), the expression that defines the probability of an individual suffering an accident before the stable equilibrium is found.
As the probability of an individual suffering an accident after t ≥ t e (stable equilibrium) is (1 , the probability of an individual suffering an accident in the period of time t is: Consequently, the probability of an individual not suffering an accident in the period of time t is: If X t has a binomial distribution, then E(X t ) = hp t and V (X t ) = hp t (1 − p t ) = hp t q t .The probability of existing j cured individuals -out of the system in time t is calculated through the Equation (3): After the stable equilibrium we get the limit of the Equation ( 22), that is, lim The Equation ( 24) is invariant with t, because lim This way, it is shown that the Equations ( 24) and ( 4) are equal; and that p represents the proportion of time an individual stays non-injured.Now, as Y t is the number of system user individuals in time t, that are relocated to hospitals using the U s , then, the probability of j injured individuals in time t also follows a binomial distribution.Substituting now the expression (T M F −1 + T M H −1 ) by (µ + λ) and later calculating for each city the function G(t) = 1 − e −(λ+µ)t , with t varying between 0 and +∞, we arrive to the equilibrium when the distribution function G(t) = 1, that is, when the result of the expression (1 − e −(λ+µ)t ) is equal to 1.In Table 3, the results of t e s calculated for each city are presented, considering the value of the distribution function G(t) = 1 − e −(λ+µ)t = 1.The values of the t e s vary from 70 days to 130 days.Curitiba has the shortest time and Fortaleza the longest, which is tied with Salvador.Recife is equal to Belo Horizonte, with a t e = 110 days.Calculating the percentages of participation of the t e with the respective T M F s, we come to a variation of 3,8% (Salvador) to 6,70% (Recife).The average of these percentages of participation is 4,86% with a standard deviation of 0, 96% and the dispersion coefficient 0, 19 from the average.
In the case of regions with shorter T M H, the stable equilibrium t e is reached faster.This way, it is viable to establish conditions, in the sense of defining a strategy for the values of p before and after the stable equilibrium.In Cordeiro (2012), a detailed behavior analysis of the curve E(Y t ) is made with the intention of making a more robust estimation of p in the period of time.The obtained results suggest a difference in the average percentage of the relative error varying from 0, 06743% to 0, 07106%.However, when crossing the averages of the values of ps there is a difference, which order of magnitude is 10 −6 .

Random variable time between successive entries in hospitalization -T EESH
The time between successive entries in hospitalization (T EESH) is a concept that is independent of how the patient got to the hospital, by using U s or not.But the time between successive entries in emergency (T EESE) for patients that need emergency treatment must have the transport service guaranteed -pre-hospital assistance via a U s .This provided service satisfying serves to all emergency patients, who are its potential users.
Thus, the other patients enter hospitalization via queues, with elaborated guidelines, being booked for hospitalization, without emergency.To dimension the number of U s capable of serving all injured individuals in region R, that are hospitalized with emergency, it is important to define a variable that explains the time between successive entries for these potential users; patients that arrive via pre-hospital mobile care denominated U s .Initially, this variable -time between successive entries for patients that arrive via U s (T EESU s ) is equal to T EESE.
Considering that the region reached the stable equilibrium and being X the number of non-injured individuals, then the average value of this time conditioned to j cured individuals is: E(T EESH)| j cured) = E(T EESH(j)).Therefore, the average value E(T EESH) is calculated weighting the E(T EESH)| j cured) with the P j : Observe that in a given region R of population h in time t has j cured individuals equal to E(X) = hp e (h − j) and injured equal to E(Y ) = h(1 − p).T N A is the variable that measured the time of a non-injured individual out of the system.This way, for hp cured individuals: which is the Equation ( 14) for the calculation of p.
For the ZAs of the U s it is of good measure to affix to h an order of magnitude that varies between 10 4 and 10 5 inhabitants.This way, it is possible to decentralize the U s or as a single region to concentrate all U s in a single spot.Distributing them in assistance zones is a more viable solution both economically and efficiently in relation to the response time.This way, the travel times computed by the U s are shorter with the decentralization, when compared to the travel times of the U s concentrated in a single spot (GOLDBERG et al., 1990).
A structure equation, similar to Equation ( 27) -P (T EESH(hp) ≥ t) = e −µhpt , can be obtained considering the proportion of the T EESH in relation to ∆t, that means, 1 − t ∆t .This way, the P (T EESH(hp) ≥ t) can be calculated also through the equation below: Differentiating in relation to t, the distribution function accumulated from the variable T EESH, we obtain the p.d.f.f T EESH (t) (remembering that I n is big), The statistics E(T EESH) and V (T EESH) from this distribution f T EESH (t) are similar to the Equation ( 27).As I n is big, then the relation When replacing I n ∆t for µhp, we obtain the expected value E(TESEH) = (µhp) −1 , which is the same found from E(T EESH) of Equation ( 27).In the long run, ∆t = 1 year, the distribution processes of T EESH are coincidental.Actually, the expected value E(T EESH) is greater for the exponential -most favorable case -; on the good side, the standard deviation is greater.
Based on the average of the I n s related to the five cities, the imputed values for the increase percentages are: %∆(E(T EESH)) = 0, 0002973% and %∆σ T EESH = 0, 0004450%.This way, working for the distribution of the variable T EESH as exponential -Equation ( 27) is closer to reality and, therefore, its sudden variations are predicted, resulting from a higher standard deviation.
5 Studies of the random variables that represent the time between the entries in hospitalization (T EESHU s ) and in emergency (T EESE) Consider the random variable that represents the time between successive entries for injured individuals that need emergency treatment -T EESE.Supposing I n entries in hospitalization during the period ∆t.Being K entries in emergencies then (I n − K) represents the entries in the hospitals without emergency.For the hypothesis based on a service policy for the pre-hospital assistance, the equation T EESE = T EESHU s , is feasible, in which the T EESHU s represents the time between successive entries in hospitalization for the injured that need a mobile pre-hospital assistance -U s .It is very hard to determine T EESHU s beforehand, from statistics made for the entries with the service units (U s ), because there is no availability to quantify the data relative to the number of emergency calls that were not answered and effected to SAMU-192.The (I n -K) entries in the hospitals are statistics of easy determination and more reliability, because they can be made through the hospital inpatient admission order.Therefore, the hypothesis of T BESE = T EESHU s is the most unfavorable case of assistance through U s , that is, assisting every emergency accidents.
Two types of failure can occur in the assistance given by the service units.The first comes from the lack of assistance by the U s to the injured, that is, the injured dies before receiving medical assistance by the U s .On the second type, the injured is assisted by the U s , but dies on the way to the hospital -dies before being hospitalized.It is aggravating that there is no statistics from SAMU that makes it possible to estimate the failure occurrence percentages.
The assistance requests through a U s in which the injured dies before arriving for hospitalization are inserted into the K emergencies.By exclusion, Z 1 is the variable that represents the number of entries with emergency that are not realized, via U s , although they are done more efficiently via other means of transport.This can occur sometimes, because the patient's transportation is done more quickly by a vehicle that is closer to him/her -private vehicle.
On the other hand, Z 2 also exists, which represents the number of entries without emergency in the hospitals, that due to the patient's health condition, the transportation must be done by a U s , although the time of the transportation can be booked beforehand.
In general, the values of Z 1 and Z 2 are negligible when compared to I n and K; this way, we can suppose that Z 1 = Z 2 , making the equation T EESE = T EESHU s valid, and consequently, the expected values as well: E(T EESE) = E(T EESHU s ).
Considering that the K entries in emergency are independent and distributed in a random way at any moment, in a period ∆t and, being both probable, that one entry can be in any point of ∆t, then we can establish for the expected value E(T EESE) = 1 × ∆t K + 1 analogous to the Equation ( 29), that is to say, . Therefore, crossing the equation E(T EESE) with Equation ( 29) comes Equation ( 30).

E(T EESE)
As a result of the similarity of the equations mentioned, we can state that the process of distribution of K emergencies in the I n entries in hospitals -injured individuals that arrive to the hospitals -spaced out from E(T EESH) (discreet process) is equivalent to considering that the emergency entries are independent and randomly distributed in a period of time ∆t (continuous process).Therefore, it is important to state that the random variable T EESE has a distribution analogous to the random variable T EESH.
This way, we can make for the equations P (T EESE), f T EESE and V (T EESE) from distribution T EESE, expressions analogous to the expressions (28) and ( 29) for the distribution T EESH.Therefore, to establish these equations we only need to change the variable I n for the variable K: Thefore, Consider that ψ 100 represents the proportion of entries, which purpose is the emergency assistance.As ∆t = I n µhp and K = ψ 100 I n , it occurs that: As verified before, the exponential represents an excellent approximation for the variable T EESH and also for the variable T EESE, whose average is 100 ψµhp .This approximation is favorable, because it imputes in the long run ∆t = 1 year a more expected value for E(T EESE): In a very long ∆t period, K K + 1 ≈ 1.This way, the Equation ( 34) is in function of the statistic ψ and the entries in hospitals µhp.
For the exposed reasons, the adoption of the exponential distribution, also, for the variable T EESE is more appropriate, because it provokes less risks derived from a higher variation for the expected value, that is %∆(E(T EESE)) = 1 K .However for the variable T EESE, the number of emergency entries in a long period of time ∆t > 0 also follows a Poisson distribution ψ 100 µhp∆t .It is known that K ∼ P ψ 100 µhp , then the probability of the hp cured individuals having k hospitalized individuals in emergency in ∆t is: and k = 0, 1, 2, . . .This way, we have the mathematical formalization of the variable distributions of T M F , T EHS, T EESH and T EESE, as well as their respective statistics, all inherent to the hospitals.These hospitals will be treated as integration stations, whose means of transport are the service units U s s, for the pre-hospital assistance.All in all, to compute all statistics, it is fundamental to safely estimate the variable ψ, that represents the proportion of the entries that are destined strictly to hospital emergencies.

Study of the variable ψ -Proportion of entries destined to hospital emergencies
There are some difficulties on the construction of the variable ψ.The first one refers to the data collection relative to strict emergency; since the last statistical yearbook made by (IBGE, 2005) does not contain any information for this purpose.The second is that in hospitals, the hospital inpatient admission order reflects in its majority normal entries.The emergency entries established in the medical record receive a mark of adequate or inadequate characterization.Although the variable ψ can easily be superiorly limited -security factor favorable for the dimensioning of the U s -its big imprecision can generate big idleness and majority encumber the costs associated to transportation.That is because an improvement in the U s service can significantly increase the number of emergency entries.
To define the estimation for ψ in favor of security, we use the law of large numbers.The estimation is based on the study developed about "The profile of emergency in the "Hospital da Restauração" -HR: an analysis of the possible impacts after the municipalization of health services", (CAVALCANTI and ARA ÚJO, 2004).The authors emphasize that between 1993 and 2001 the number of emergency entries increased almost 50%.The traumatology procedure participates with 35%, probably in its vast majority caused by traffic accidents.
Almost all patients of HR (90%) come from Recife or from the Metropolitan Region, being Monday and daytime the periods with highest demand.The HR is characterized as an emergency hospital, with a monthly average of 25.800 entries being 12.000 (46, 51%) strictly emergencies.Not including patients diagnosed with cholelithiasis submitted to operation on emergences (CUNHA et al., 2016).What is most surprising in the assistance point of view is that in the computed entries destined to HR's emergency, 74, 5% are characterized as inadequate for this emergency -patients whose diagnoses could be cured by basic treatment.(SALLA et al., 2002) found in a broad hospital research a 60, 6% percentage of entries in emergencies that were characterized as inadequate.
In a survey made in the 62.500 entries in HR -about 73 days, in each 2 emergencies of 100 observed entries, in almost 99% of them, the relative frequency observed K I n would be at less than 2% from 46, 51%.It can be claimed that ψ = 48, 51% is a safe estimation for a hospital with a strict emergency characteristic.This way, we observe for ψ the variation interval (44, 51%, 48, 51%).
Based on the calculations already properly computed, it is verified that in Recife, in the year of 2005, the expected value E(T EESE) = 9, 32 minutes, that is to say, averagely in each 9,32 minutes an injured individual is entered in emergency.Furthermore, the probability that the T EESE in 2005 is superior to 10 minutes is of 0, 007821.It can be observed that the tendency of the expected value for the variable T EESE is to reduce substantially with the increase of h; particularly for constant ψ.

Study of the service time variable T S of Any U s
The care policy for the service units (U s ) is to assist only emergency patients.Therefore, considering Z 1 -the number of entries in emergency not carried out by U s , and Z 2 -the number of non-emergency entries using the U s , to maximize the emergency care via U s we must look for Z 1 = Z 2 = 0.This way this situation converges to the most unfavorable case for the dimensioning of the U s aimed at pre-hospital care.This way, it can be considered that the variable Time Between Successive Entries in Hospitalization Using Service Unit U s (T EESHU s ) represents the same variable T EESE.So the equations for the variable T EESHU s are the same defined for T EESE.It is worth to emphasize that the values of Z 1 and Z 2 participate with very small representativeness, when compared to the volume of I n -entries in hospitals and K -entries in emergency.Therefore it is appropriate to set the U s only to care for emergencies, and this way we can consider Z 1 Z 2 0.
Being T S a random variable that establishes the service time of any U s and W , an also random variable that represents the wait time for a U s to be available and be assigned to the call (wait time in queue).Initially, the variable T S is defined as the sum of a set of random independent variables defined below: • T P : Time spent for the U s to leave; that is, the time between the occupation of the U s and its exit from the storage location; • T V : Travel time of departure or arrival; • T A: Time spent for picking up the injured patient and helping him on spot, that is, time spent between the arrival to the location of the accident and the departure of the U s from the location; • T D: Time spent to deliver the patient to the hospital until the clearance of the U s -service unit ready for return.
Besides those, the variable T S is defined by the sum below: he variables T P , T V , T P and T D depend a lot on the work dynamic established for the collection and confirmation of the results.As a consequence of that, the estimations produced can provoke big distortions, especially because of the strong sensitivity of these variables.This way, the estimations to be obtained are originated from a probabilistic distribution properly adjusted to the observed statistics.A feasible distribution to be adjusted to a specific interval of time ∆t is the beta distribution.Suppose, the following time measurements, all obtained from a sample of ∆t: ∆t (average), ∆t min , ∆t max and σ ∆t (standard deviation).Consider the following distribution for Ω defined by the equation below: Exemplifying, if ∆t corresponds to the sequence (t 1 , t 2 , t 3 , t 4 , t 5 ), the assumed values for Ω are in between 0 ≤ Ω ≤ 1.This is valid for any sequence.Therefore, it is possible to consider that Ω ∼ β(a, b), where the a and b are real positive parameters of the distribution denominated scale and declivity, respectively.These parameters are determined from the equations below: With relation to the variable T V , it depends on the geometry of the assistance service zone (ZA) made by the service unit (U s ) and also of the location where the hospital is.As the clearance of the U s obey a queueing system, it is possible to calculate through the queueing model the average and variance of the variable T S.
The response time T R, as defined previously, represents the time between the U s request and its arrival at the patient, This variable is represented by the following equation: After the definitions of the equations for the variables T S and T R and, based on the distribution that represents the variable T EESE, we seek for a representation for the variable distribution that represents the time between the successive requests of U s in the screening center for assistance to the injured -T ECSU s .
7 Study of the variable time between successive request of U s -T ECSU s For the purpose of studying the distributions for the variables T ECSU s and T EESHU s we must consider a hospital that is assisting a small user population situated in a determined ZA, whose dimension reproduces a smaller T R. Thus, the number of service units n s , allocated to the hospital tends to be small.For example, n s = 1 to n s = 3.It is important to emphasize that to maximize the care system's efficiency, it is necessary to increase the idleness factor (1-ρ) or to decrease the usage ρ; or to change the wait time in queue to W = 0 and consequently the following inequations must be obeyed, that is T S < T ECSU s and T S < T EESE.The equation that measures this idleness is expressed by the usage factor ρ, being determined in function of the variables T S, T ECSU s and n s -number of service units, according to the expression below: As seen before, the distribution of the variable T EESE is recognized and represented by the Poisson distribution.This way, the behavior of the system will be represented by the variable T ECSU s -system entry, by the variable T S -service assistance, and by the variable T EESE -system exit.Being the system exit a Poisson distribution and the variable T S ruled by the exponential distribution, then the variable T ECSU s follows a Poisson with the same exit rate.Therefore, we have an stigmatic system and, this way, we can state that T ECSU s = T EESE.A detailed demonstration about this theme is found in (HILLIER et al., 1986).This way we stumble over a problem that is based on the strict queueing theory.This queueing system is characterized by being a Markov process.Still in relation to the queueing system, the moments (expected value and variance) of the chosen distributions T ECSU s , T EESE and T S are extremely necessary as an optimization tool.In case of finding difficulties in the expression of the moments of these distributions, it is necessary to find approximated methods for the acquisition of the moments, as for example, the Laplace Transform.Making the response time exponentially distributed to a queueing system of type M/M/n s , then the probability distribution function of the variable T R is given by: Observe that the minimization of the response time is caused due to the increase of the idleness factor (1 − ρ) or usage minimization.In the same way, it can be demonstrated that for the system M/M/n s the probability distribution of the random variable W -wait time is represented by the equation: This way it is shown that the idleness increase or the usage decrease of the U s responds sensitively to the wait time reduction W and also to the behavior change of the response time function (T R); this time rapidly decreases with the increase of the idleness.To demonstrate that these are exponential times, it is necessary to apply the Laplace Transform to f W (t) e f T R (t).It is shown that having as exponential the f W (t). The same procedure for the response time, that is This way, it can be generalized for queuing system with assistance through U s , the representation M/G/n s , where the M represents the Poisson's entry distribution, G a general distribution for the service time T S and n s the number of units U s , that represent the service servers.Therefore, it is possible to admit for this general system, that the entry distribution is equal to the exit distribution; therefore, T ECSU s = T EESE.It can be observed that for the variable distribution T ECSU s the Equations ( 31) and ( 35) explain this distribution.
It is worth to emphasize that there are particular difficulties in adopting the same distribution T ECSU s throughout the entire day.These difficulties are inherent to the abrupt variations that occur in the demand -higher search of the U s at rush hours -periods with larger traffic volumes.This way, it is possible to partition the day in periods of time.For example, the T ECSU s1 of smaller value and constant for the period ∆t 1 = n d P 1 ; the T ECSU s2 of greater value and constant for period ∆t 2 = n d P 2 .Being n d the number of days in ∆t = ∆t 1 + ∆t 2 and P 1 two periods of the day defined as the first shift from 6:00H to 14:00H and the second shift from 14:00H to 22:00H.While the second period P 2 refers to the third shift from 22:00H to 6:00H.Therefore, ∆t 1 = 2∆t 2 e P 1 = 2P 2 .In this sense, there is the necessity to guarantee a homoscedasticity of the request rates during the study period, in such a way that guarantees the Poisson process.later in the second period ∆t 2 , all the consecutive P 2 s are also formed.The request rates λ 1 and λ 2 of the U s in the independent process and identically distributed over ∆t 1 and ∆t 2 , respectively and, admitting that T ECSU S , T ECSU s1 and T ECSU s2 are exponentials.As the variable K represents the number of emergency entries using as means of transport the U s , and making the request probability of this U s being in ∆t 1 , the expected values for the variables T ECSU s , T ECSU s1 and T ECSU s2 are: Having considered ∆t 1 = 2 3 ∆t and ∆t 2 = 1 3 ∆t, that is, in K hospitalizations with service units, it is seen that K hospitalizations are at day and (1 − )K at night.Thus, making the proper substitutions in the previous equations: Observe that E(T ECSU s ) = 100 ψµhp and V (T ECSU s ) = 10000 (ψµhp) 2 .Therefore, it is possible to put E(T ECSU s1 ) and E(T ECSU s2 ) in function of 100 ψµhp .What happens is: After the expressions of the expected values of the variables T ECSU s1 and T ECSU s2 we come to the result of the variances and with that the distributions' characteristics: The variables T ECSU s1 and T ECSU s2 have exponential distributions of parameters λ 1 and λ 2 .In all explaining equations of the variables T ECSU s1 and T ECSU s2 it is necessary to estimate the value of the proportion .The methodology for this estimation is similar to the one described for the variable ψ.Therefore the estimated values of the average rates for the attendances in ∆t 1 and ∆t 2 are: 9 Dimensioning of the U s to efficiently help the users of an assistance zone ZA In Cordeiro (2012), it is demonstrated that for each station of an service unit (U s ) that belongs to an assistance zone we must use the queuing system structured in the form of M/G/n s (∞, First in First out -F IF O) with entries (λs) given by the Equations ( 47 W D , W M and W are the average wait time in the queue of an assistance request of an injured individual using U s , considering the queuing models M/D/n s , M/M/n s and M/G/n s , respectively.Owen (1971) while studying the average wait time in queue, in the M/E m /n s model through theoretical results obtained with simulation, concluded that it is possible to acquire an adequate approximation for this W , from the interpolation between W M and W D , as long as the average wait time in queue is small (W ≈ 0).This way, it is possible to consider for the queuing system the model M/E m /n s adjusted to the general model M/G/n s and, as a consequence of that, obtain the W interpolating between W M , W D by the means of: Remembering that T S ∼ β(a, b), then c = 1 a ab a + b + 1 . The probability density function of the Erlang of order m with request rate (call rate) of U s constant λ is: The expected value of X is given by:E(X) = m λ and the variance, by: V (X) = m λ 2 .For m = 1, we have the exponential distribution and a Poisson arrival process.As m increases, the distribution's relative dispersion decreases, reaching the deterministic situation (constant intervals between requests) when m → ∞.As m = 1, 2, . . .n s , then what shall be the probability of all n s being occupied -a U s request to enter the queue or the probability of the number of injured users being greater or equal to n s ?It is used, in this case, the Erlang's delay formula, where P (n s , a) is equal to this probability with a = λ µ , that represents the number of U s requests per service time.Based on the usage factor (utilization factor) ρ = λ µn s and on the idleness factor (1 − ρ), we have Erlang's delay formula given by the equation: Considering the constant population density d = 4.832 hab Km 2 , it happens that the coverage area of this hospital is estimated to Â = 91.5934.832 Km 2 = 18, 85 Km 2 .So we have to define a rule to make the knowledge of these variables tangible via their estimations, because they are important for the acquisition of the service time (T S) and of the response time (T R).Consider that these variables are evenly distributed over the 24 hour period (BURNS et al., 1985), inside the following variation intervals in minutes: T P ∼ U (0, 4), T A ∼ U (0, 8) and T D ∼ U (0, 4).So, E(T P ) = 2 min, E(T A) = 4 min and E(T D) = 2 min.The variances are: V (T P ) = 1, 33 min 2 , V (T A) = 5, 33 min 2 and V (T D) = 1, 33 min 2 .For the characterization of the assistance zone, consider that ZA has geometrical shape different from the square with spin of ∠45 degrees in relation to the horizontal axis.Thus, to analyze the compatibility of ZA, it is necessary to verify if the equations below demonstrated in (CORDEIRO, 2012) are satisfied, that is: As the coverage area is estimated for 18, 85 Km 2 , the equation is satisfied, because its result is of 21, 72 Km 2 ; superior to the 18, 85 Km 2 -which completely satisfies the zone ZA.The following stages refer to the sensitivity analysis for the quantification or dimensioning of the number of U s (n s ), corresponding to the equations ( 50), ( 51), ( 52) and ( 53).
1) The values of the average request rates for assistance in ∆t 1 (day) and ∆t 2 (night) are: 2) Consider that the RP A1 is divided in a single assistance zone ZA, that is N = 1.Therefore, the values referring to the metrical D are given by the equations below: 3) The travel time T V considered constant in every way is calculated in function of 4) After determining the travel time and the estimation for the times T P , T A, and T D, we find the expected value of the service time E(T S) = 17, 30 mins and V (T S) = 31, 91 mins 2 .
5) The calculation of the dispersion coefficient c = 0, 326 is used for the determination of the interpolation of W . 6) After the calculation of the service time E(T S) it is necessary to calculate the average assistance rate µ.Thus with the probability P (n s , a), n s , λ 1 , λ 2 and µ we determine the values of W M , W D and, afterwards W : µ = 1 E(T S) = 0, 0578 calls min .

Conclusions
a) The effective need of service units for the day period is of n s = 3.But, with n s = 2 -reduction of 33, 33% it is necessary to neglect the travel time (E(T V ) = 4, 65 min) in a percentage of 29, 08 sec 60 × 4, 65 sec = 10, 42%.This negligence causes a risk to the injured in a way, for not assisting him/her in the critical time -time to save the life; b) In the night period the need is also of n s = 3.In case of choosing 2 service units, the travel time must be neglected at about 2,27%, but with more ease because of a possible increase in the commercial speed of the U s .Thus, adopting the variance interval as the average wait time in queue W = 29, 08 seconds would augment the E(T V ) to about 10,42%.So for a user population of the hospital HR with 91.047 users, 3 service units are satisfactory for the day period and 3 service units are satisfactory for the night period considering W ≈ 0; c) Observe that in the day period, considering two (n s = 2) or three (n s = 3) service units, the probability of a request in the service station of the ZA, that contemplates the HR, of not finding a U s is of 0, 0627 and of 0, 000716, respectively.This way, the increase of 50% in the service units provokes a reduction of 99% in this probability, since it is little representative.It is emphasized that the usage factor, that measures the effectiveness of the fixed costs of the U s , represents an efficiency indicator in relation to the costs.With the increase of this n s this usage factor decreases at 33,33%.As the fixed costs represents about 75% of the total cost, this consideration (n s = 3) imputes in an increase of the total cost for the U s at about 25%; d) For the night period there are 3 (n s = 3) service units.The probability of a request in the service station of the ZA, that contemplates the HR, of not finding a U s is of 0, 0158 (n s = 2) and 0, 00095 (n s = 3).All the probabilities are very small; configuring as excellent results.It is highlighted that for the efficiency indicator of the system, the choice of n s = 3 is much more favorable; e) The RP A1 has a population of 77.607 inhabitants.Thus, the ZA that contemplates the user population of HR has 91.047 users.This way, the 13.440 remaining users can be considered in an extra area of 13.440 hab 4832 hab Km 2 = 2, 89 Km 2 , where a satellite station with one or two service units would be allocated, probably in an area with high chances of accidents, as for example the "Conde da Boa Vista" avenue where the occurrence of people getting run over by cars has increased in a significant way.Another option is the "Complexo Salgadinho" in the division between the cities of Recife and Olinda.All in all, this user demand of 13.440 much likely come from other assistance zones; f) In case of the user population being smaller than the population of the ZA it is necessary to relocate this assistance deficit to another close zone and recalculate its coverage area, as well as the increase of beds directed at the hospital where the assistance units station is located.Then, it is necessary to recalculate the increase of U s units that must supply these new attendances; g) When estimating the speed v = 20Km/h, the results of the U s 's dimensioning differ very little.For the day period, the effective necessity of service units remains in n s = 3.But, for n s = 2 and due to the substantial increase in the E(T V ), the risk in not assisting the injured individual in critical time due to traffic jams is 35, 88 seg 488, 4 seg = 7, 36%.This number in relation to the previous result (10, 42%) is about 30% smaller.The impedance of adopting n s = 2 is based on the high probability (0, 0685) of lacking U s , when the requests are made.
, then: P (T EESH(hp) ≥ t) = e −µhpt .(27) Thus, there are always hp cured individuals susceptible to accidents.Therefore, the p.d.f. of the variable T EESH is exponential, because the T EESH(hp) is exponential.Considering that I n represents the number of hospitalizations in the period of ∆t = 1 year, then I n = µhp∆t; as µp = λ(1 − p), then I n ∆t = λh(1 − p) and being A the area below the curve Y(t) -Figure 1, then h(1 − p) = A ∆t , n = µhp∆t.As I n -hospitalizations or entries of injured individuals in the hospitals -is very big because the period ∆t is long (1 year), then P (T EESH(hp) ≥ t) is approximated to exponential.Remember that lim x→+∞ 1 T EESH(hp) > t) = e −µhpt .
The service time -Equation (36) given by T S = T P + 2T V + T A + T D. All those times are estimated from a function adjusted to the observed values and E(T V ) and V (T V ) are determined by the equation -E(D) = 0, 50 √ A and V (D) = A 36 .Depending on the layout of the ZA, V (D) = 0, 108 A N (N, number of sub-zones of the ZA).More details consult (CORDEIRO, 2015).

Table 1 -
The estimated values for h in function of alpha, n L and (1 − p)

Table 2 -
System User Population × Minimum Number of Beds with 100% Reliability Cities P op − h Hospitalizations-I n Beds-n L Beds-n

Table 3 -
Values of Time for the Stable Equilibrium -t e