**Postprandial Hypotension as a Risk Factor for the Development of New Cardiovascular Disease: A Prospective Cohort Study with 36 Month Follow-Up in Community-Dwelling Elderly People**

#### **Aelee Jang**

Department of Nursing, University of Ulsan, 93, Daehak-ro, Nam-gu, Ulsan 44610, Korea; aeleejang@ulsan.ac.kr; Tel.: +82-52-259-1252

Received: 2 January 2020; Accepted: 23 January 2020; Published: 27 January 2020

**Abstract:** Postprandial hypotension (PPH) is common among the elderly. However, it is unknown whether the presence of PPH can predict the development of new cardiovascular disease (CVD) in the elderly during the long-term period. This study aimed to prospectively evaluate the presence of PPH and the development of new CVD within a 36 month period in 94 community-dwelling elderly people without a history of CVD. PPH was diagnosed in 47 (50.0%) participants at baseline and in 7 (7.4%) during the follow-up period. Thirty participants (31.9%) developed new CVD within 36 months. We performed a time-dependent Cox regression analysis with PPH, hypertension, diabetes, and body mass index (BMI) as time-varying covariates. In the univariate analyses, the presence of PPH, higher BMI, hypertension, diabetes mellitus, and higher systolic and diastolic blood pressure were associated with the development of new CVD. The multivariate analysis indicated that the relationship between PPH and the development of new CVD remained (adjusted hazard ratio 11.18, 95% confidence interval 2.43–51.38, *p* = 0.002) even after controlling for other variables as covariates. In conclusion, the presence of PPH can predict the development of new CVD. Elderly people with PPH may require close surveillance to prevent CVD.

**Keywords:** cardiovascular disease; postprandial; hypotension; blood pressure; elderly

#### **1. Introduction**

Postprandial hypotension (PPH) is a common but often unrecognized disorder in the elderly [1]. PPH is defined as a ≥20 mmHg decrease in systolic blood pressure (SBP) within 2 h after a meal [2]. The prevalence of PPH is 20–91% in hospitalized geriatric patients [3–6]. The pathogenesis of PPH is unclear, but an inadequate cardiovascular response to postprandial splanchnic blood pooling is regarded as a primary mechanism [2]. In the elderly, sympathetic activation to decrease the effective circulating volume is often suppressed; hence, a continuous pooling of blood in the splanchnic bed may result in a significant decrease in blood pressure (BP) after a meal [7,8]. Changes in cardiovascular function and the neurohormonal response with aging can also contribute to diverse comorbidities associated with the dysregulation of BP homeostasis [9].

Cardiovascular diseases (CVDs), such as stroke, transient ischemic attack, angina, and myocardial infarction, are the main cause of mortality and work disability in the elderly [10,11]. Early recognition of subclinical risk factors is essential for the proper surveillance and prevention of new CVD. PPH is also recognized as an important clinical issue, particularly as a significant risk factor for subsequent CVD, but this has not yet been accurately confirmed [2,12]. Some studies have suggested an association between PPH and CVD and mortality [9,13]. However, it is difficult to determine the casual relationship between PPH and CVD development owing to differences in patient selection, study design, and diagnosis criteria among previous studies [9,14–16]. Furthermore, the diagnosis of PPH was heterogeneous in previous studies, such as with regard to the time interval between meal consumption and BP measurement [5]. Therefore, to investigate whether the presence of PPH can predict the development of new CVD in the elderly, this study aimed to prospectively evaluate the existence of PPH and the development of new CVD among community-dwelling elderly people in a period of 36 months.

#### **2. Materials and Methods**

#### *2.1. Study Design and Participants*

This prospective cohort study enrolled 94 participants from three senior community centers in South Korea between 2011 and 2015. The inclusion criteria were (1) age ≥65 years, (2) ability to eat independently and maintain a sitting position for 2 h after a meal, and (3) ability to perform activities of daily living independently. The exclusion criteria included (1) impaired cognitive function, (2) psychiatric illness, (3) recent hospitalization owing to acute illness within 1 month prior to the study, (4) medical history of CVD (congestive heart failure, myocardial infarction, angina, or cerebrovascular accidents, including transient ischemic attack), or (5) use of medications affecting gastrointestinal motility. After evaluating the presence of PPH at baseline, the new onset of PPH was evaluated during the follow-up period of 36 months. Body mass index (BMI) and the presence of hypertension and diabetes were also assessed regularly. This study was conducted following the guidelines of the Declaration of Helsinki of 1975, revised in 2013, and was approved by the institutional review board of Pusan National University Hospital (E-2014007), located in B city, South Korea. All participants provided written informed consent.

#### *2.2. Evaluation of PPH and Acquisition of Demographic Data*

BP was measured using an automated sphygmomanometer (A&D UA-851; A&D Company, Japan) following the European Society of Hypertension guidelines [17]. BP was measured by a well-trained nurse with the participant in a sitting position. Baseline BP was measured before a meal. Participants were asked to refrain from eating food, drinking coffee or alcohol, and smoking 4 h before measurement. All participants took any prescription medications after the evaluation of PPH, including antihypertensive drugs and hypoglycemic agents. BP was measured twice with a 5 min interval before a meal, and the average of the two measurements was used as the baseline BP. To measure postprandial BP, participants were provided the same standardized meal comprising 210 g rice, 100 g soup, and 70 g side dishes, which was approximately 500 kcal. After the participant consumed the meal, postprandial BP was measured every 15 min for 2 h. PPH was defined as a decrease in SBP of ≥20 mmHg from the baseline within 2 h after a meal [2]. We analyzed the reliability of measurements and found that the intraclass correlation coefficients (ICCs) for intra-rater reliability for SBP and diastolic blood pressure (DBP) were 0.995 (95% confidence interval (CI) 0.975–0.998, *p* < 0.001) and 0.989 (95% CI 0.942–0.996, *p* < 0.001), respectively.

#### *J. Clin. Med.* **2020**, *9*, 345

Demographic and clinical information was also obtained by interviewing the participants using a questionnaire. Information on age, living conditions, alcohol intake, and smoking habits was collected, and information on medical history was obtained from hospital records after consent was obtained from the participants and their families. Bodyweight and height were measured using an automated measuring machine (G-Tech International Co., Ltd., Uijeongbu, Korea) to determine the BMI.

#### *2.3. Surveillance of CVD Development*

The primary endpoint was the occurrence of new CVD within 36 months. CVD in this study included coronary heart disease (such as congestive heart failure, angina, and myocardial infarction) and cerebrovascular disease (such as stroke and transient ischemic attack) according to the World Health Organization definition of CVD [18]. Participants were regularly followed every 3 months, and CVD-related information was obtained from them or their families during their visits to the centers or via telephone calls. If the participant was diagnosed with a new disease, additional information was also obtained from the medical record or prescription from the hospital at which the participant was diagnosed with new conditions, including the new onset of hypertension, diabetes, and CVDs.

#### *2.4. Statistical Analysis*

The reliability of BP measurements was calculated by determining the ICC (two-way random effects model). Variables are expressed as frequencies and percentages for categorical data, and means ± standard deviation for numerical data. Group differences were assessed using the chi-squared test or Fisher's exact test for categorical data and the independent t test or Mann–Whitney U test for numerical data as appropriate. To determine whether the distribution was normal, we used the Shapiro–Wilk test. Time to CVD was estimated using Kaplan–Meier curves. Survival curves were compared between groups using the log-rank test. Cox regression models with time-varying covariates were used to identify prognostic factors that were independently related to CVD. The main predictors were baseline PPH and PPH as a time-varying covariate (incorporating new onset PPH during the follow-up and duration of PPH as a time-varying covariate). In addition, hypertension, diabetes mellitus, and BMI were monitored during the follow-up period. These predicting variables were considered as time-varying covariates in time-dependent Cox regression models. All statistical analyses were performed using SPSS version 24.0 (IBM Corp., Armonk, NY, USA) and R version 3.5.1, and *p*-values <0.05 were considered statistically significant.

#### **3. Results**

#### *3.1. Baseline Characteristics*

The participants comprised 79 (84.0%) females and 15 (16.0%) males. The mean age was 73.1 <sup>±</sup> 4.8 years, and the mean BMI was 23.7 <sup>±</sup> 2.5 kg/m2. Among the participants, 67 (71.3%) had lower than middle school education, and 62 (66.0%) were currently living with their family or spouse. The most common comorbidity was hypertension (50.0%), followed by diabetes mellitus (19.1%). At baseline, 47 (50%) participants were diagnosed with PPH, but there were no significant differences in baseline characteristics, with the exception of BP, between participants with and without PPH (Table 1).


**Table 1.** Baseline characteristics of participants with and without PPH.

Values are either frequency with percentage in parentheses or mean ± standard deviation. PPH, postprandial hypotension; SBP, systolic blood pressure; DBP, diastolic blood pressure. <sup>1</sup> P values were derived from independent t tests. <sup>2</sup> P values were derived from Mann–Whitney's U test. <sup>3</sup> P values were derived using chi-square tests. <sup>4</sup> P values were derived using Fisher's exact test. Shapiro–Wilk's test was employed for test of normality assumption.

#### *3.2. Incidence of CVD*

During the follow-up, 30 (31.9%) patients developed new CVD, of whom 4 passed away owing to acute myocardial infarction (*n* = 2) and ischemic stroke (*n* = 2). The CVD incidence was significantly higher in the group with PPH than the group without PPH at baseline (55.3% vs. 8.5%, *p* < 0.001, Figure 1).

**Figure 1.** Kaplan–Meier curve of the incidence of new CVD between participants with and without PPH at baseline. The 3 year incidence of CVD was significantly higher in the group with PPH than in the group without PPH at baseline (55.3% vs. 8.5%, log-rank test, *p* < 0.001). PPH, postprandial hypotension; CVD, cardiovascular disease.

Table 2 indicates the differences between groups with respect to CVD development. The presence of PPH at baseline (86.7% in the CVD group vs. 32.8% in the non-CVD group, *p* < 0.001), higher BMI (24.5 ± 2.1 in the CVD group vs. 23.3 ± 2.7 in the non-CVD group), hypertension (66.7% in the CVD group vs. 42.2% in the non-CVD group), higher baseline SBP (139.3 ± 20.9 mmHg in the CVD group vs. 123.6 ± 17.9 mmHg in the non-CVD group), and higher baseline DBP (79.0 ± 8.6 mmHg in the CVD group vs. 73.4 ± 10.1 mmHg in the non-CVD group) were significantly related to the development of new CVD. The percentage of participants with an SBP change of ≥20 mmHg (86.7% vs. 32.8%, respectively, *p* < 0.001) and a DBP change of ≥10 mmHg (73.3% vs. 48.4%, respectively, *p* = 0.023) were significantly higher among participants who developed new CVD than among those who did not.




**Table 2.** *Cont.*

Values are either frequency with percentage in parentheses or mean ± standard deviation. PPH, postprandial hypotension; CVD, cardiovascular disease; SBP, systolic blood pressure; DBP, diastolic blood pressure. <sup>1</sup> P values were derived from independent t tests. <sup>2</sup> P values were derived from Mann–Whitney's U test. <sup>3</sup> P values were derived using chi-square tests. <sup>4</sup> P values were derived using Fisher's exact test. Shapiro–Wilk's test was employed for test of normality assumption.

#### *3.3. Predictive Factors for the Development of New CVD*

Ninety-four patients were available for the Cox regression models to assess the relationship between PPH and CVD using PPH as the time-varying covariate. There were 30 cases of CVD. We considered time-varying covariates, including PPH, hypertension, diabetes, and BMI, in the time-dependent Cox regression models. Among the 47 patients without PPH at baseline, 7 were diagnosed with PPH during the follow-up period. The main predictive factors were baseline PPH, with PPH as a time-varying covariate incorporating new onset PPH during follow-up, and the duration of PPH as a time-varying covariate. New onset hypertension and diabetes mellitus during the follow-up period were observed in 12 and 11 patients, respectively. In addition, BMI was measured at baseline, 1 year, and 2 years. All predictive variables were considered as time-varying covariates in time-dependent Cox regression models. In the univariate analyses, the presence of PPH, higher BMI, hypertension, diabetes mellitus, and higher SBP and DBP were found to be associated with the development of new CVD, while age, sex, education level, living status, alcohol consumption, and smoking were not. Patients with PPH were more likely to develop new CVD (crude hazard ratio (HR) 15.97, 95% CI 3.80–67.08, *p* < 0.001). The multivariate analysis that included significant factors in the univariate analyses as covariates revealed that the relationship between PPH and the development of new CVD remained (adjusted HR 11.18, 95% CI 2.43–51.38, *p* = 0.002) even after controlling for other variables as covariates in the multivariate analysis. Hypertension was also found to be a significant factor affecting CVD (adjusted HR 3.26, 95% CI 1.22–8.76, *p* = 0.019) (Table 3).


**Table 3.** Summary of time-dependent Cox regression analyses using time-varying covariates (a total of 94 subjects, 30 CVDs).

HR, hazard ratio; CVD, cardiovascular disease; BMI, body mass index; SBP, systolic blood pressure; DBP, diastolic blood pressure; PPH, postprandial hypotension.

#### **4. Discussion**

This study investigated the time-varying effect of PPH associated with new CVD in community-dwelling elderly people. The study indicated that approximately half (55.3%) of the participants with PPH developed new CVD within the 36 month follow-up period, whereas only 8.5% of participants without PPH developed CVD. After adjustment for other covariates, the time-varying effect of PPH on the development of new CVD was found to be significant (HR 11.18, 95% CI 2.43–51.38). These results suggest that elderly people with PPH were more likely to develop new CVD and, thus, required close surveillance.

The reported prevalence of PPH varies widely in previous studies depending on the patient population and diagnostic methods [5]. Studies of institutionalized elderly patients reported a PPH prevalence of 24–38% [3,4]. Meanwhile, the incidence of PPH among hospitalized geriatric patients was considerably high (up to 91%) [5,6]. However, studies investigating the incidence of PPH among healthy elderly people are relatively rare [19]. Unlike previous studies, the present study investigated PPH in relatively healthy elderly people who were not institutionalized or hospitalized. Although the study population comprised elderly patients, the PPH incidence was 50%, which seems to be slightly higher than that reported in previous studies. The development of PPH is highly dependent on meal composition. Carbohydrate-rich meals predispose patients to a more immediate decrease in BP than do meals containing primarily protein or fat [20]. Therefore, it is expected that PPH might be more prevalent in people from Asian regions owing to their carbohydrate-dominant diets. However, few studies have investigated PPH in Asian patients [21]. To date, this is the first prospective cohort study to investigate PPH and its association with CVD in relatively healthy Asian elderly people. In this study, a stringent diagnostic approach was used for PPH, such as provision of a standardized meal and more frequent BP measurements. Furthermore, in this study, variance in BP changes related to food composition was reduced by administering the same standardized meal to all participants. Therefore, this study may provide relevant insights into the epidemiology of PPH in Asian regions. Large cohort studies are recommended to further determine the prevalence of PPH in different patient populations using a standardized diagnostic method.

In this study, 30 (31.9%) participants developed new CVD during the 36 month follow-up. In one study on PPH as a predictor of new CVD [14], 40.8% of older nursing home residents developed CVD during a 29 month follow-up. The patients with new CVD exhibited a significantly greater decline in postprandial SBP than did those without (*p* < 0.001). The other previous study found a 59.2% occurrence of new CVD during a 4.7 year follow-up period in older low-level-care residents in long-term health facilities [9]. The occurrence of new CVD cases was lower in this study (31.9%) than in previous studies (40.1–59.2%) because this study was conducted in a relatively healthy cohort from a community setting. Furthermore, the previous studies did not fully exclude populations with a history of CVD. In contrast, this prospective study excluded people with a history of CVD to clearly demonstrate the relationship between PPH and CVD.

In this study, the presence of PPH, higher BMI, hypertension, diabetes mellitus, and higher SBP and DBP were predictive factors for the development of CVD in the univariate analysis using the time-dependent Cox proportional hazard model. After adjustment for these factors, the time-varying effect of PPH and hypertension on the occurrence of new CVD still remained in the elderly people. According to Aronow et al. [14], the mean maximal decrease in postprandial SBP was a significant risk factor for developing CVD. Similarly, in previous cross-sectional studies, PPH was considered an independent predictor of asymptomatic cerebrovascular damage in healthy participants and hospitalized patients with essential hypertension [15,16]. Despite the small sample size, this study is meaningful as a prospective cohort study for the risk factors of new CVD in the presence or absence of PPH among community-dwelling elderly people without a history of CVD.

The total mortality (7/94, 7.4%) and CVD-related mortality (4/94, 4.3%) rates in this study were relatively lower than those reported in previous studies (16.9–54.2%) [9,13,14]. Again, this is likely because this study included relatively healthy older adults (i.e., from a community setting) in contrast to previous studies, which generally included hospitalized geriatric patients or patients in nursing homes. The mean age of the participants in this study (73.1 years) was also relatively lower than that in previous studies (77.8–83.2 years) [9,13,14]. In a previous study of hypertensive elderly patients who were followed up at the cardiology clinic for 4 years, the total mortality rate was 16.9%, and a greater decrease in postprandial SBP was associated with a greater increase in CVD mortality. In that prospective study, CVD mortality constituted 50% of the total mortality rate, while PPH accounted for 52.9% of deaths owing to CVD [13], which was similar to the findings of the present study. In a study by Fisher et al. [9], PPH was the only risk factor for all-cause mortality, and 50% of deaths were attributed to CVD. Participants with a postprandial SBP difference of ≥20 mmHg had the lowest survival during the 4.7 year follow-up period. They also reported that PPH accounted for 47% of deaths in long-term healthcare facilities, which was lower than the 57.1% (4 of 7 deaths) found in the present study. Populations in long-term healthcare facilities typically require nursing care for monitoring and preventing serious diseases, including CVD. These perspectives may explain the difference in attributing the total mortality rate to CVD mortality between the present study and the previous study mentioned above [9]. Therefore, prevention and close monitoring are needed to reduce deaths due to new CVD by classifying patients diagnosed with PPH as high risk for CVD.

This study had some limitations. First, the sample size was relatively small. However, compared to previous studies, this study was conducted over a long-term follow-up with the time-varying effect of PPH. The post hoc power for investigating the primary objective of this study was approximately 99%. Second, despite efforts to recruit a similar ratio of males and females, the participants were predominantly female because of the higher proportion of elderly women in the Korean elderly population. Future studies should confirm CVD risk factors in sex-balanced populations with postprandial SBP reduction during long-term follow-up. Third, the level of physical activity was not considered in this study, which could affect the development of CVDs. Nevertheless, the participants had similar lifestyle patterns because they resided in the same town and enjoyed the same activity programs provided by the community welfare center. Therefore, their physical activity levels may be similar. Fourth, the pathophysiological relationship between PPH and CVD remains vague despite PPH being identified as an independent time-varying variable for the development of new CVD. Autonomic dysfunction may play a major role in the development of PPH-related new CVD because PPH has been reported to be pathologically associated with autonomic dysfunction. This needs further

evaluation using animal models. Fifth, the positive association between PPH and CVD may only reflect the high baseline SBP on the risk of CVD because the baseline SBP level in the PPH group was much higher than that in the non-PPH group. Although hypertension was adjusted in the multivariate model, based on the small sample size, the effect of high baseline SBP may persist. As it is impossible to stratify the analysis by hypertension status at baseline, owing to the limited sample size, further research in the form of a well-designed, large-sample study in a normotensive elderly population is necessary to definitively establish the effect of PPH on the risk of CVD.

This study investigated the relationship between PPH and the development of new CVD in elderly people during a long-term follow-up period. The results also suggest that the underlying reason of CVD development could be the greater decline in postprandial SBP (such as that occurring in PPH). This study demonstrated that increases in SBP variations after a meal may be a manifestation of CVD development and may reflect the presence of subclinical cardiovascular damage.

#### **5. Conclusions**

This study revealed that PPH was an independent predictor of new CVD among community-dwelling elderly people during a 36 month follow-up. Thus, CVD development may be prevented or monitored based on the presence of PPH. However, longitudinal studies in larger samples are needed to clarify the prognostic significance of PPH in the development of new CVD.

**Funding:** This work was supported by the 2019 Research Fund of University of Ulsan.

**Conflicts of Interest:** The author declares no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Relationship between Morbidity and Health Behavior in Chronic Diseases**

**Munjae Lee 1,2, Sewon Park 1,2 and Kyu-Sung Lee 2,3,\***


Received: 15 November 2019; Accepted: 26 December 2019; Published: 2 January 2020

**Abstract:** This study aimed to analyze the demographic characteristics and health behaviors related to chronic diseases and to identify factors that may affect chronic diseases. Data from the Seventh Korea National Health and Nutrition Examination Survey were used, and 3795 adults aged above 40 years were included. The following demographic variables were obtained: sex, age, education, income, type of health insurance, and private insurance. The following health behavior factors were also analyzed: medical checkup, drinking, smoking, exercise, obesity, and hypercholesterolemia. Participants with lower socioeconomic status had a higher risk of developing chronic diseases. Meanwhile, those with private health insurance had a lower risk of developing chronic diseases. In addition, participants who underwent medical checkups and performed exercises had a lower risk, while those with obesity and hypercholesterolemia had a higher risk of developing chronic diseases. It is necessary to manage chronic diseases through comprehensive programs, rather than managing these diseases individually, and through community primary care institutions to improve health behaviors.

**Keywords:** chronic disease; health behavior; socioeconomic status; primary care; Korea

#### **1. Introduction**

An individual's behavior related to health may have an effect on their physical health or ability to recover from illness. In particular, health-related behavior, such as a lack of exercise, smoking, and drinking, are some of the main factors that can contribute to morbidity and mortality [1–3]. Health behavior affects 40% of premature deaths; in order to reduce premature mortality, improving health behaviors is more cost-effective than improving the social and physical environments or health-care systems [4]. These health behaviors are important in maintaining good health, which is influenced by biological and socioeconomic factors, among others [5,6]. Rapid economic growth, high health-care costs, lifestyle changes, and population aging have been associated with an increased prevalence of chronic diseases worldwide. Chronic diseases may cause complications, and thus, require continuous care and are among the types of diseases with high health-care costs due to their long disease duration [7–9].

Chronic diseases, one of the leading causes of death worldwide, especially cardiocerebrovascular diseases, diabetes, and hypertension, have a high mortality rate. However, the mortality rate of chronic diseases can be reduced through prevention [10–12]. Chronic disease is closely related to changes in health behaviors; the main health behaviors affecting the development of chronic diseases include health risk behavioral factors, such as smoking, drinking, and physical activities, and clinicopathologic factors, including obesity, hypertension, and hypercholesterolemia [13,14]. In particular, since health-related lifestyles have increased the risk of mortality, the significance of managing health risk behavioral factors

has also been increasing. Thus, it is necessary to prevent chronic diseases and delay the aggravation of symptoms by improving individual lifestyles [15–17]. In addition, individual health behaviors may differ according to sociodemographic characteristics including age and sex [18]. In the identification of the individual physical condition, sociodemographic and socioeconomic factors are known to act as important factors, and prevalence rates vary in accordance with the individual's income level, education level, and socioeconomic factors [19].

Previous studies have analyzed the relationship between chronic diseases and health promotion behaviors, but were only conducted in predetermined age groups, such as in older patients, or examined the relationship between chronic diseases and health behaviors while only targeting certain chronic diseases [20–24]. As the number of polychronic patients has increased, a comprehensive analysis of chronic diseases is required. To date, the number of studies evaluating patients with chronic diseases is limited. Accordingly, in this study, we aimed to analyze the sociodemographic characteristics and health behaviors related to the development of chronic diseases and to identify factors that may have an effect on the morbidity of chronic diseases. Through this and by suggesting measures to contribute to the effective management and prevention of chronic diseases, we intend to promote the health of the people.

#### **2. Experimental Section**

#### *2.1. Data Source and Research Participants*

In this study, we utilized the source data from the 2nd year (2017) of the 7th period of the Korea National Health and Nutrition Examination Survey, performed by the Korea Centers for Disease Control and Prevention. The Korea National Health and Nutrition Examination Survey (KNHANES) is a nationwide national survey, conducted to determine health-related parameters including the prevalence of chronic diseases and health behaviors based on Article 16 of the National Health Promotion Act. A total of 10,430 individuals from 3580 households were surveyed, but only 8127 participated in the study. Of them, 5159 aged above 40 years were extracted. In addition, 658 individuals whose answers were not related to chronic diseases were excluded; hence, only 3795 participants were analyzed, after further excluding 706 who did not respond to the questions related to health behaviors. The KNHANES is approved by the ethical committee of the Korea Centers for Disease Control and Prevention. The requirement for informed consent was waived because data in the KNHANE database are anonymized in adherence to strict confidentiality guidelines. The flowchart is shown in Figure 1.

**Figure 1.** Flowchart of the study.

#### *2.2. Description of Variables*

In this study, questions related to sociodemographic characteristics, morbidity, and presence of chronic diseases, and health behaviors were utilized. Sex, age, education level, level of income, type of health insurance, and private insurance policy were used as sociodemographic variables. In terms of age, participants aged 19 years or older were divided into two groups: adults and older adults (aged 65 years and above). The education level was stratified into middle school graduates or less and high school graduates or more. Income status was determined by the monthly mean household gross income and was classified based on 3 million won as the cut-off point. The patients with health insurance were classified as health insurance subscribers and medical care beneficiaries [23,25–28].

Hypertension and diabetes are the main causes of cardiovascular disease, and the number of patients continues to rise due to the increase in obesity rate. In addition, the cost of medical care is proliferating more rapidly than the number of patients. Therefore, it is significant to prevent it by analyzing the factors influencing chronic diseases. Hitherto, chronic disease was defined as hypertension, dyslipidemia, stroke, myocardial infarction, and diabetes. The presence of chronic disease was determined based on a response of "Yes" to the question related to a doctor's diagnosis. Health behaviors included health checkups, drinking, smoking, exercise, obesity, and hypercholesterolemia [28–31]. Health checkup status was classified as patients who underwent health checkups and those who did not undergo health checkups. Drinking status was classified as non-drinkers and drinkers based on the monthly drinking rate; smoking status was classified as non-smokers and smokers using the current smoking rate. Exercise history was stratified as those who performed exercises and those who did not perform exercises based on the aerobic physical activity practice rate [32,33]. Furthermore, the prevalence of obesity was determined and obesity was stratified based on the following indices: a body mass index of 18.5 kg/m2 or higher or a body mass index of 23 kg/m2 or lower indicates normal weight, while a body mass index of 25 kg/m2 or higher indicates obesity [34–36]. Hypercholesterolemia was stratified based on its prevalence (Table 1).


**Table 1.** Classification and definition of variables.


**Table 1.** *Cont.*

#### *2.3. Statistical Analysis*

In order to analyze the relationship among sociodemographic characteristics, health behaviors, and the presence of chronic diseases, statistical analyses were conducted using the SPSS (version 25.0, https://www.ibm.com/kr-ko/analytics/spss-statistics-software).

First, cross-analysis was performed to analyze the relationship between chronic diseases and sociodemographic characteristics and between health behaviors and chronic diseases. In order to determine the relationship between sociodemographic characteristics and health behaviors and the risk for developing chronic diseases, a logistic regression analysis was performed.

#### **3. Results**

#### *3.1. Participants' Demographic Characteristics*

Of the total participants, 56% were women and the proportion of women was higher than that of men; older adults aged 65 years or higher accounted for 33% of the total study population. Most of the participants were middle school graduates or had obtained higher education (2269 persons, 59.8%) and had an income of more than 3 million won (2107 persons, 55.5%). With regard to the type of health insurance, national health insurance subscribers accounted for 95.7% of the total participants according to the characteristics of health insurance in Korea. Meanwhile, private insurance subscribers accounted for 74.2%, even though the proportion of health insurance subscribers corresponded to a majority; this finding indicates that most of the patients took a private medical insurance policy due to the lack of coverage by the national health insurance. A total of 857 patients (22.6%) underwent medical checkups, which suggests that only a few patients were able to undergo medical checkups. A total of 1762 patients (46.4%) developed chronic diseases, of whom 21.4% had two or more chronic diseases (Table 2).


**Table 2.** Demographic characteristics (*n* = 3795).


**Table 2.** *Cont.*

#### *3.2. Relationship between Demographic Characteristics and Chronic Diseases*

In this study, we intended to analyze the relationship between sociodemographic characteristics and chronic diseases, and the results are shown in Table 3. Among chronic disease patients, 955 (44.9%) were women, this proportion being higher than that of men. Meanwhile, 807 (48.4%) of 862 male participants had chronic diseases, which indicates that men had a higher rate of chronic disease morbidity. Approximately 71.3% of the participants aged 65 years or higher had chronic diseases. In addition, most of the patients with a lower educational level and lower-income level had chronic diseases (981 patients (64.3%)and 1008 patients (59.7%), respectively). A total of 1648 (45.4%) health insurance subscribers were chronic disease patients, while 114 (69.9%) medical care recipients were chronic disease patients. Furthermore, 1128 (40.1%) chronic disease patients were private insurance subscribers.


**Table 3.** Relationship between demographic characteristics and chronic diseases.

\*\* *p* < 0.05, \*\*\* *p* < 0.001.

#### *3.3. Relationship between Health Behavior and Chronic Diseases*

We analyzed the relationship between health behaviors and chronic diseases; the results are shown in Table 4. Of the total chronic disease patients, 944 (50.6%) were alcohol drinkers, 1503 (47%) were smokers, and 605 (40.1%) performed exercises, which is less than the number of patients who did not perform exercises (1157, 50.6%). Moreover, 1263 (52.7%) and 823 (73.9%) patients with obesity and hypercholesterolemia, respectively, had chronic diseases.


**Table 4.** Relationship between health behavior and chronic diseases.

\* *p* < 0.1, \*\*\* *p* < 0.001.

#### *3.4. Factors A*ff*ecting Chronic Diseases*

In order to determine the factors that may affect the development of chronic diseases, logistic regression analysis was performed, and the results are shown in Table 5. The factors with statistically significant effects in patients with chronic disease included sex, age, education, income, types of health insurance, decision to take a private insurance policy, health checkups, exercise, obesity, and hypercholesterolemia.

**Table 5.** Factors affecting the development of chronic diseases.


\* *p* < 0.1, \*\* *p* < 0.05, \*\*\* *p*< 0.001.

In men, the risk of developing chronic diseases was higher by 1.498 times. Further, as age increased, the risk of developing chronic diseases also increased by 3.145 times. In participants with a higher education level, the risk of developing chronic diseases increased by 0.535 times. In participants with higher income, the risk of developing chronic disease reduced by 0.773 times. With regard to the type of health insurance, the risk of developing chronic diseases increased by 1.727 times among medical care beneficiaries. In addition, for those who took a private insurance policy, the risk of developing chronic diseases increased by 0.782 times. Meanwhile, the risk of developing chronic diseases decreased by 0.782 times and 0.861 times among those who underwent medical checkups and who performed exercises, respectively. In normal-weight people, the risk of developing chronic diseases reduced by 0.544 times. In patients with hypercholesterolemia, the risk increased by 5.444 times.

#### **4. Discussion**

In this study, we analyzed the factors affecting the development of chronic diseases through logistic regression analysis using the data from the Korea National Health and Nutrition Examination Survey (2017). Of the sociodemographic characteristics, sex, age, education and income level, types of health insurance, and private insurance were found to have an effect on chronic diseases. In terms of sex, the proportion of women with chronic diseases was higher than that of men. Compared with women, men had a higher rate of chronic disease morbidity and the risk of developing chronic diseases. These results are inconsistent with those of previous studies, which reported that the prevalence of chronic diseases is higher among women than in men because men can maintain their economic level for longer than women. Women who have a lower income level than men have relatively low medical accessibility and find it difficult to manage their chronic diseases [37]. The number of chronic disease patients is increasing due to the lack of physical activity and the increasing prevalence of hypercholesterolemia and obesity, and considering that previous studies have shown that the prevalence of chronic disease was lower among men who received management, managing chronic diseases according to sex seems to be of utmost importance [38]. In addition, the number of patients aged 65 years or older who had chronic diseases was higher; therefore, the higher the age, the higher the risk of developing chronic diseases. This finding is consistent with those of a previous study, which reported that as age increases, the prevalence of chronic diseases also increases due to the decreased amount of physical activities and habit-based health risk behaviors [8,39].

It was also found that the higher the income and education levels, the lower the risk of chronic diseases. This finding is consistent with those of previous studies reporting that socioeconomic status, including income, education, and occupation levels, affects the health-related lifestyles and risk of chronic diseases [40]. Because of the low rates of physical activity and exercise practice and as the provision of medical services for managing chronic diseases has still not been ensured owing to lower educational levels or living standards, the prevalence of chronic diseases is increasing. Among medical care beneficiaries, the risk of developing chronic diseases was high, which was similar to the results of a previous study reporting that the incidence of chronic disease increased among individuals who belonged to the lower social class, like those in the low-income bracket. Social determinants, such as income, education, and social class, may cause health-related inequality but create an environment in which quality medical care can be provided for the treatment of chronic diseases. In addition, non-medical factors, such as social determinants, play a more substantial role in the management of chronic diseases than medical factors. It seems that medical care beneficiaries with low income may have more difficulty in managing chronic diseases [41–43]. There were many chronic disease patients who obtained a private medical insurance policy; the results showed that patients with private medical insurance had a lower risk of developing chronic diseases. These findings are similar to those of a previous study, which indicated that those who have private medical insurance policies tend to receive outpatient and inpatient treatments. In line with these findings, among patients with chronic diseases who require continuous health care, those with private medical insurance have a reduced burden in terms of medical expenses, leading to better health-care outcomes [44–46]. Considering these results, there are limitations in managing chronic diseases with national health insurance only. Furthermore, it is estimated that people obtain commercial medical insurance policies due to the burden of medical expenses caused by the recent increase in polychronic diseases. Therefore, since health-related inequalities in the low-income group patients, who find it difficult to pay the private medical insurance premiums, will become a serious problem if we only rely on private medical insurance for the management of chronic diseases, the coverage of the national health insurance should be reinforced for the management of chronic diseases.

Among health behaviors, the factors affecting the risk of developing chronic diseases included health checkups, exercise, obesity, and hypercholesterolemia. Those who underwent periodic health checkups had a risk of developing chronic diseases, which is similar to previous findings showing that periodic health checkups promote health and help prevent chronic diseases [8,47]. In addition, considering the results of previous studies reporting that those who benefit from health insurance are more likely to receive health checkups depending on the nature of the health insurance system in Korea, chronic diseases could be effectively managed through modifying the nature of the insurance provided. Previous studies have shown that health behavior factors related to chronic diseases include smoking, drinking, exercise, body mass index, and regular life and eating habits [7,29,36,48,49]. However, in this study, drinking and smoking did not have a statistically significant effect on the prevalence of chronic diseases, and these results are different from those of existing research. Furthermore, exercise, obesity, and hypercholesterolemia were associated with the risk of developing chronic diseases, consistent with existing research. Among those who performed exercises, the risk of developing chronic diseases was lower, while among those with obesity and hypercholesterolemia, the risk of developing chronic diseases was higher. Weight loss via exercise programs reduces the risk of developing chronic diseases. Maintaining a standard body weight can prevent chronic diseases by alleviating hypercholesterolemia. Management of chronic diseases should be comprehensively performed with weight management through exercise; however, there seems to be a limitation in this regard according to patients' behavioral changes [50,51]. In order to overcome this limitation, wearable medical devices, which use ICT (Information & Communication Technology), have recently been developed for chronic disease management. Prevention and management of chronic diseases can be ensured through exercise [52–55]. The use of medical devices to promote physical activity leads to obesity and hypercholesterolemia management, and through the linkage between these medical devices and local clinic-centered, effective management of chronic diseases can be achieved through periodic monitoring. The results of this study also suggest that gender, age, education, and income levels have impacts on chronic disease, and it is significant to add these as risk factors and to continue monitoring in local clinic-centered facilities. Through this, a personalized chronic disease management system could be established.

This study has some limitations. First, chronic disease patients aged 40 years or below were not included. Recently, the number of younger chronic disease patients has increased owing to changes in lifestyle, therefore, further studies to analyze the factors influencing the risk of developing chronic diseases in this age group will be required, with the patients stratified as follows: youth, middle-aged, and older adults. Second, analyses according to the number of chronic diseases were not performed. In this study, only the presence or absence of chronic diseases in patients was assessed. Further studies to determine the influencing factors according to the number of chronic diseases are required. Third, there was no analysis of factors affecting chronic disease according to the residential area. Accessibility to medical services varies depending on where you live; therefore, chronic disease management may be different. Hence, it is necessary to analyze the factors affecting chronic diseases according to urban and rural areas. Despite these limitations, we comprehensively analyzed the factors influencing the prevalence of chronic diseases. Our study is significant as we were able to determine the risk factors for chronic diseases, which can be used as a basis for developing policies for the comprehensive management of chronic diseases, based on sex, age, and social factors.

#### **5. Conclusions**

In order to manage chronic diseases, the management approach should be based on patients' socioeconomic characteristics to address the differences related to sex, education, income, and medical care. The management should also include approaches to improve health behaviors, including the use of wearable medical devices and digital healthcare products. Based on our findings, we presume that chronic diseases develop due to a combination of factors. Age, socioeconomic factors, obesity, and hypercholesterolemia are factors that can be controlled to prevent and manage chronic diseases through comprehensive programs rather than through individual management. Moreover, those who belong to the lower social class, are more likely to require chronic disease management via primary healthcare institutions in the community. In order to improve health behaviors, continuous observation is required, and local clinic-centered chronic disease management can help improve health

behaviors. It is significant to establish a comprehensive management system and promote efficient medical delivery systems for chronic diseases focused on local clinic-centered facilities. However, Korea's medical delivery system urgently needs reorganization due to the concentration of university hospitals and the weakening of a local clinic-centered structure. Therefore, in order to expand the role of local clinic-centered facilities and to efficiently manage chronic diseases, the integrated local clinic-centered care chronic disease management project is being implemented. Through this, medical treatment for chronic disease management and education for improving lifestyle, are applied to lower the patient's copayment. If the burden reduction of chronic disease management is expanded, the dependency on private health insurance will be reduced, which will prevent excessive medical expenses for chronic patients. In addition, strengthening the role of local clinic-centered facilities will lead to strengthening medical access for low-income people, thereby relieving health inequalities. For older adults, when included in the community care project in line with community-based primary healthcare service, comprehensive management of chronic diseases, including health improvement and lifestyle modification, could be implemented. In particular, in Europe, where public health policies are in place, chronic diseases are effectively managed by strengthening the local clinic-centered services, such as the attending physician, to manage chronic diseases. For common goals such as chronic disease management, community care is implemented to ensure continuous health care. In view of this, chronic disease management through public health policy should be implemented prior to private medical insurance. Patients with private medical insurance have a lower risk of developing chronic diseases, but this can be seen as a problem of low insurance coverage for chronic diseases. This can be resolved through community care projects such as in Europe. Because of this, patient-centered chronic disease management will ultimately improve the health of chronic disease patients.

**Author Contributions:** Conceptualization, M.L.; methodology, S.P.; software, S.P.; validation, M.L. and K.-S.L.; formal analysis, M.L.; writing—original draft preparation, S.P.; writing—review and editing, M.L.; supervision, K.-S.L.; project administration, M.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea (NRF-2019S1A5A2A03040304).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Decision Support for the Optimization of Provider Sta**ffi**ng for Hospital Emergency Departments with a Queue-Based Approach**

**Fuu-Cheng Jiang 1,**†**, Cheng-Min Shih 2,4,**†**, Yun-Ming Wang 2, Chao-Tung Yang 1,\*, Yi-Ju Chiang <sup>3</sup> and Cheng-Hung Lee 4,5**


Received: 6 October 2019; Accepted: 24 November 2019; Published: 5 December 2019

**Abstract:** Deployment or distribution of valuable medical resources has emerged as an increasing challenge to hospital administrators and health policy makers. The hospital emergency department (HED) census and workload can be highly variable. Improvement of emergency services is an important stage in the development of the healthcare system and research on the optimal deployment of medical resources appears to be an important issue for HED long-term management. HED performance, in terms of patient flow and available resources, can be studied using the queue-based approach. The kernel point of this research is to approach the optimal cost on logistics using queuing theory. To model the proposed approach for a qualitative profile, a generic HED system is mapped into the M/M/R/N queue-based model, which assumes an R-server queuing system with Poisson arrivals, exponentially distributed service times and a system capacity of N. A comprehensive quantitative mathematical analysis on the cost pattern was done, while relevant simulations were also conducted to validate the proposed optimization model. The design illustration is presented in this paper to demonstrate the application scenario in a HED platform. Hence, the proposed approach provides a feasibly cost-oriented decision support framework to adapt a HED management requirement.

**Keywords:** hospital emergency department; queuing theory; decision support; cost optimization

#### **1. Introduction**

#### *1.1. Background*

Hospitals play an important role in the healthcare system of society. They have changed rapidly in parallel with improvements in medical instruments and medicine. Administrators of hospital institutions should have spared no effort in developing strategies that enable the provision of high-quality services, while operating with increased costs and pressure from competition [1]. One of the most demanding departments in terms of economic resource consumption and programming is the

hospital emergency department (HED). To this extent its operational profile should be monitored and optimized in order to provide the optimal quality of medical service subject to the budget constraint.

HEDs must be operational 24 h, and, moreover, should respond to multiple demands requiring sophisticated technical equipment and the manpower to operate them, all of which imply higher costs. Large HEDs have even higher costs because they offer a wide range of services that would be unavailable in a small rural HED [2]. Emergency Departments have traditionally been a crucial issue in the hospitals' cost containment and management. The optimization of patient flow and bottleneck elimination in key departments could be a viable way for policy makers to decrease operational costs and boost quality of care [3]. In the interest of improving patient throughput and resource utilization, appropriate key performance measures are selected, like the size of staffing providers, HED patient arrival patterns, service rate of staffing providers, waiting time, etc. To explore the tradeoff among them, the proposed queue-based optimization technique on cost may provide hospital management with a decision support for deploying staffing providers under the constraints of the kernel performance parameters.

An M/M/R/N queuing model was adopted to explore the cost profile from analyzing the relationships among relevant performance parameters in a HED, such as the number of staffing providers (i.e., servers) needed during each staffing interval. This model assumes a single queue with a limited system capacity of N that feeds into R identical medical servers (i.e., staffing providers). The fourth symbol "N" of the notation M/M/R/N indicates the restriction on the system capacity of the HED. The value of (N−R) gives the number of waiting rooms for incoming patients when all staffing providers in the HED are busy. Arrivals occur according to a time-homogeneous Poisson process with a constant rate, and the service duration (e.g., provider time associated with a patient) has an exponential distribution. In the language of queuing theory [4,5], these two assumptions are often called Markovian, hence the use of the two "M's" in the notation used for the model. One advantage of adopting the M/M/R/N model is that given an arrival rate, an average service duration, and the number of servers, formulae for performance measures such as the cost profile, the average number of patients, or the average waiting time can be easily obtained.

#### *1.2. Contribution Profile*

This novel idea in this work originated from the theory of an M/M/R/N queuing system (QS), which is used to estimate the optimal number of providers needed during each staffing interval [4,5]. At some pre-configured period (say a shift, or a day), a finite quantity of staffing providers exists to provide medical services for patients under limited waiting rooms in HEDs. On application modeling, such a finite quantity of staffing providers (i.e., R) can be regarded as the term "server" in the M/M/R/N model of queuing theory. The quantity of (N−R) can be considered to be the rather limited waiting rooms in HEDs regulated by each hospital.

The research goal for this work is to explore whether, with cost-based deployment, how many sets of staffing providers in the HED schedule would be optimal if a certain level of the server availability is kept? To explore the tradeoff between them, the proposed optimization technique may provide the HED management with decision support on the number of staffing providers. The key contributions of this paper are threefold: (1) This work provides HED administrators with an efficient deployment of staffing providers for the HED platform to optimize the cost improvement. In regards to management, the proposed system can be adopted as a decision-making methodology approaching predictive management, rather than reactive or chaotic management. (2) For quantitative analysis, the M/M/R/N queue model was applied and derived, and then the relevant system metrics were established in a brand-new manner. The mathematical expression of the cost function was established for the evaluation requirement. (3) In regards to verification, relevant experimental results were obtained in terms of configurations on cost optimization and average waiting time. The simulated results indicate that the proposed approach may provide a feasible decision support for deployment on quantities of standby servers.

The rest of the paper is organized as follows: Section 2 describes related work and the motivation behind this research. To demonstrate the framework qualitatively, an M/M/R/N model of queuing theory is adopted and the mapping profile is demonstrated in Section 3. Quantitative work is presented in Section 4, where the mathematical analysis is conducted in detail and the relevant system performance measures, such as the expected number of online servers, the expected number of spares, etc., are derived. Following this, in Section 5, the queue-based model is further addressed in terms of cost function, and the simulations of the feasibility of the proposed scheme are conducted. Finally, some concluding remarks are made in Section 6.

#### **2. Related Work**

HED crowding represents an important issue that may affect the quality and access of health care. Accordingly, the optimization of average waiting times has become a focus across many mainstream hospitals. As defined by the Canadian Association of Emergency Physicians [6], HED overcrowding is a situation in which the demand for services exceeds the ability of health care professionals to provide care within a reasonable length of time. As stated in [7], significant variation in HED patient arrival rates necessitates the adjustment of staffing patterns to optimize the timely care of patients. Green et al. [7] collected detailed HED arrival data from an urban hospital and used a queue-based analysis to gain insights on how to change provider staffing to decrease the proportion of patients who leave without being seen. However, no optimization materials in terms of mathematical theory were addressed at all in these studies [8–10].

Finamore et al. [6] described an innovative use of a satellite clinic to prevent patients returning to HED for care on a scheduled basis. Their strategy allows patients returning for follow-up diagnostics or treatment to bypass the main HED. The proposed HED satellite clinic may shorten the waiting times in multiple ways, such as increasing the capacity to remove returning patients from the pool of patients requiring care in the HED, and creating a separate registration area and a separated staffed treatment area. The visit data in the HED were used to measure crowding and completion of waiting room time, treatment time, and boarding time for all patients treated and released or admitted to a single HED during 2010. In [11], the authors conducted a relevant statistical analysis and concluded that a HED census at arrival demonstrated variation in crowding exposure over time. In the work of Wiler et al. [12], the authors developed an agent-based simulation model for the evaluation of the FTS (fast track strategies) scheme applied in the HED to reduce patient waiting time. By and large, the issues regarding cost optimization on the HED management cost are not a concern for these open studies [5,11,12].

Vass and Szabo [13] evaluated 2195 questionnaires in the HED situated in Mures County, Romania, for a period three years (2010–2013). Their research reported that long waiting times were the most important complaint in patient's satisfaction surveys. To perceive the waiting times, only a specific M/M/3 queuing model was considered in their work to demonstrate the computation details. The work of [13] has motivated us to consider whether it is possible to provide an effective and feasible approach to decision support for the optimization of provider staffing under cost constraints for the HEDs with more elaborative queue-based frameworks. This research generalizes the queuing model of [13] into the M/M/R/N queuing framework in terms of three practical aspects: (1) Numbers of medical servers (provider staffing) can be configured to one of the system parameters instead of a fixed quantity. Such a dynamic staffing level enables a hospital to quantify the cost patterns and the alleviation of HED waiting times. (2) The space available in the HED would be limited for every hospital management. The fourth factor (N) in the notation of the M/M/R/N model symbolizes the fact that only N patients can be allowed to enter the waiting rooms of the HED in order not to exacerbate the issue of overcrowding. (3) The exact mathematical expressions would be derived in an elaborative manner and the relevant cost formulation would be used to provide the generic decision support for the hospital administrators.

#### **3. The Proposed Model of Medical Emergency Services**

#### *3.1. The Generic Platform of Medical Emergency Services*

This research explores feasible decision support that is proposed to optimize running costs under the constraint of the waiting time at a HED using queue-based models. The exemplified HED is in a metropolitan hospital (Taichung Veterans General Hospital or TVGH) located in central Taiwan. It began offering medical services on 16 September 1982. Since 1991, it has been accredited as a "Medical Center and First-Class Teaching Hospital" by the Department of Health, Taiwan. Taichung Veterans General Hospital is a 1500-bed hospital with up to 3900 employees. According to the latest statistical average data of registration accessed in TVGH, it offers a capacity of about 7000 outpatients and 190 patients in the emergency room daily [14]. As a public medical center, it provides safe, high-quality medical services with advanced facilities and training programs, as well as outstanding research and development programs.

The HED building, with eight floors at the TVGH (TVGH-ED), provides comprehensive emergency services 24 h a day. The functional deployment on the ground floor of the TVGH-ED building, as shown in Figure 1, is composed of different zones, including the Registration and Triage Area, Resuscitation Areas, Internal Medicine Areas, the Pediatric Treatment Area, Waiting Areas, Clinic Areas, Monitor Rooms and the Fever Screen Center, and relevant auxiliary service units such as the X-ray service and nursing stations. As this HED is a rather complicated service system due to random arrivals, various disease chains, uncertain service times of care, and randomness in human decision-making, it is difficult to model the whole HED with a single operational model. From the perspective of the model attribute [15], a generic operational model is defined as a formal description of operations performed to deliver a health service that is applicable over a wide range of health service delivery settings. For the sake of simplicity, this research concentrates the optimization issue on a specific platform of medical service, which is used hereafter to model staffing providers for a single disease chain.

**Figure 1.** The functional deployment on the ground floor of the TVGH-ED building. TVGH-ED, Taichung Veterans General Hospital - Emergency Department.

#### *3.2. Mapping Profile between the HED Service Platform and the M*/*M*/*R*/*N Queuing System*

The proposed generic framework on the HED service platform is considered to be modeled as an M/M/R/N queuing system (QS), which is used to estimate the optimal number of providers needed during each staffing interval. An input-throughput-output framework of HED operations is used as the prototype shown in Figure 2 for a generic profile [16]. The ambulance icon symbolizes the arrivals of HED patients. Practically, patient arrivals are hard to schedule, or even control significantly. Arrivals may surge in some unpredictable time windows due to a short-term disaster, car accidents, and seasonal influences [17]. In modeling language, the busy and regular time windows can be associated with high and normal arrival rates, respectively. Patient arrivals in the proposed model are assumed to be Poisson processes [18], with average hourly rates that are forecasted for each future hour in question (say a shift, or a day) [19].

The itinerary for HED patients from arrival to exit can conceptually be divided into three phases [12]. The first phase, named "Waiting for treatment phase (*waiting phase*)", is symbolized by the icon HED Waiting Rooms in Figure 2. In the waiting phase, the patient goes through some standard processes that assist the HED to grasp the record of patients and their current medical condition. These are termed the Registration and Triage process, respectively. Registration guarantees administratively that patient demographics are captured accurately for billing and maintaining the record. Triage is the first assessment conducted by a healthcare professional after the patient arrives in the HED. The second phase (*treatment phase*) begins when the patient is placed in bed. For simplicity, the treatment phase is represented by the icons of provider staffing (medical servers) in Figure 3 for a generic profile. The whole medical service largely depends on patient acuity and physician activities. In modeling language, the duration of treatment can be regarded mathematically as the service rate of (medical) servers. The treatment phase is followed by the post-treatment phase, which is represented by the expression HED patient departures in Figure 3. Exiting from the treatment area of the HED

is reasonably assumed to mean that the patients are discharged, either as an outpatient or into the hospital [16].

+('3ODWIRUPPRGHOHGDVDQ00514XHXH

**Figure 3.** An M/M/R/N queue system mapped by the HED service platform.

The mapping scenarios for the theoretical approach are illustrated in Figures 2 and 3. An M/M/R/N queuing model was used to estimate the number of providers needed during each staffing time window. In Figure 3, the proposed model assumes a single queue with regulated and finite waiting rooms that feed into R identical servers with blue highlights, which is mapped to providers in the HED. The walking-man (customers) icons symbolize HED patient arrivals. Based on the proposed queuing model, relevant system metrics, such as average waiting times, expected number of customers in the queue buffer, and the probability that all servers are busy, can be analyzed and derived mathematically [7]. For instance, a patient's total length-of-stay from arrival to departure from the HED platform is termed as the patient throughput time, which is equivalent to the waiting time in the QS. Patient throughput time has a significant impact on operational and economic efficiency as well as overall patient satisfaction, which is a measure of medical service quality [20].

Generally, the performance metric on average waiting times may provide the HED administrator with decision support on how to alleviate patients' complaints. To avoid the deterioration of average patient throughput time (i.e., the average wait times in the QS), the optimization approach on the average waiting times, under some constraints such as a limited number of servers in the QS (i.e., mapped counterpart: level of staffing in the HED platform), is explored further in this article. The metric is the probability that all servers can be used to reveal the possibility and scenario in which notorious HED crowding may occur. The question is how to reduce this HED crowding phenomenon in some specific time windows. Such a metric can provide decision support for the administrator in order to properly configure or deploy hospital resources.

#### **4. Quantitative Modeling and System Measures for the HED Platform**

#### *4.1. Theoretical Analysis*

The service-oriented model on the HED platform in Figure 3 is considered to have R servers with an adequate level of staffing and a finite size (N) of waiting rooms for HED arrivals. The birth-and-death process is adopted to derive analytic steady-state solutions to the M/M/R/N queuing system (QS). Let the states n (*n* = 0, 1, 2, ... , N) represent the number of customers in the QS. McManus et al. [18] studied all admissions to the medical–surgical intensive care unit (ICU) of a large, urban children's hospital during a 2-year period. Their statistical analysis confirmed that the arrival rate of patients to ICUs follows a Poisson distribution, and the durations of stay (service times) were found to follow an exponential distribution. Hence, it is reasonably assumed that the customers arrive according to a Poisson process with mean arrival rate λ<sup>n</sup> = λ if 0 ≤ n ≤ N and λ<sup>n</sup> = 0 if n > N due to a finite system capacity. The QS has R servers, each having an exponential distribution of service times with an identical service rate μ<sup>n</sup> = μ. The service volume can be classified into two parts as follows:

Mean Service Rate:

$$\mu\_{\mathbf{n}} = \begin{cases} \begin{array}{c} \text{n } \mu, \text{ if } 1 \le \mathbf{n} \le \mathbf{R} \\\ R\mu, \text{ if } (\mathbf{R}+1) \le \mathbf{n} \le \mathbf{N} \end{array} \tag{1} $$

To approach analytic steady-state results for the proposed model, we first construct the state-transition-rate diagram depicted in Figure 4. The number inside the circle represents the number of customers (patients) in the system. Each circle in Figure 4 shows the steady-state probability scenario that may occur during the service period in the system. For each circle except the first one (n = 0) and the last one (n = N), there are four arrows marked with the corresponding values of the state-transition rate. The quantity marked along each arrow gives either the flow-in probability into that state or the flow-out probability off that state.

**Figure 4.** State-transition-rate diagram for the proposed model.

Let the notation P(n) = the probability that there are n customers in the system, where *n* = 0, 1, 2, ... , N. Hence, for a steady-state case, the state probability functions P(n) can be obtained from the birth-and-death formula [5] in association with the state-transition-rate diagram shown in Figure 4. We define notation ρ = λ/μ for the server utilization and ρ<sup>S</sup> = ρ/R = λ/(Rμ) for the system utilization. According to the value n (number of customers in the QS that may be present), two segments are defined by the vector: (Segment 1, Segment 2) = (1 ≤ *n* ≤ R, (R+1) ≤ *n* ≤ N). The state probability functions P(n) can then be derived in terms of two segments as follows:

(A) Segment (1): 1 ≤ n ≤ R

$$\mathbf{P}(\mathbf{n}) = \frac{\lambda\_0 \cdot \lambda\_1 \cdot \lambda\_2 \cdot \cdots \cdot \lambda\_{n-1}}{\mu\_1 \cdot \mu\_2 \cdot \mu\_3 \cdot \cdots \cdot \mu\_n} \mathbf{P}(0) = \frac{\lambda^n}{\mu \left(2\mu\right) \left(3\mu\right) \cdots \left(n\mu\right)} \mathbf{P}(0) = \frac{\lambda^n}{\mu^n \cdot \mathbf{n}!} \mathbf{P}(0) = \frac{\mathbf{p}^n}{\mathbf{n}!} \mathbf{P}(0) \tag{2}$$

(B) Segment (2): (R+1) ≤ n ≤ N,

$$\begin{split} \mathbf{P}(\mathbf{n}) &= \frac{\frac{\lambda \cdot \mu\_1 \cdot \lambda\_2 \cdots \lambda\_{n-1}}{(\mu\_1 \cdot \mu\_2 \cdots \mu\_R)(\mu\_{R+1} \cdots \mu\_n)} \mathbf{P}(\mathbf{0})}{\mathbf{A}^n} \mathbf{P}(\mathbf{0}) = \frac{\lambda^n}{(\mu \cdot (2\mu) \cdots (\mathbf{R}\mu))(\mathbf{R}\mu \cdots \mathbf{R}\mu)} \mathbf{P}(\mathbf{0}) = \\ &= \frac{\lambda^n}{\mathbf{R}! \cdot \mu^R \ (\mathbf{R}\mu)^{n-R}} \mathbf{P}(\mathbf{0}) = \frac{\rho^n}{\mathbf{R}! \cdot (\mathbf{R})^{n-R}} \mathbf{P}(\mathbf{0}) \end{split} \tag{3}$$

There are (n–R) terms of Rμ in the parenthesis (Rμ. Rμ) of the above denominator. Equations (2) and (3) are the closed-forms for the state probability functions P(n) in terms of two segments in which the number of customers may be present. To obtain P(0), we substitute expressions (2) and (3) in the normalizing equation <sup>N</sup> <sup>n</sup>=<sup>0</sup> P(n) = 1, which yields:

$$\sum\_{n=0}^{\mathcal{R}} \frac{\rho^{\mathcal{R}}}{n!} \mathcal{P}(0) + \sum\_{n=\mathcal{R}+1}^{\mathcal{N}} \left(\frac{\rho^{\mathcal{R}}}{\mathcal{R}! \ R^{\mathcal{R}-\mathcal{R}}}\right) \mathcal{P}(0) = 1$$

$$\mathcal{P}(0) = \left[\sum\_{n=0}^{\mathcal{R}} \frac{\rho^{\mathcal{R}}}{n!} + \sum\_{n=\mathcal{R}+1}^{\mathcal{N}} \left(\frac{\rho^{\mathcal{R}}}{\mathcal{R}! \ R^{\mathcal{R}-\mathcal{R}}}\right)\right]^{-1} = \left[\sum\_{n=0}^{\mathcal{R}} \frac{\rho^{\mathcal{R}}}{n!} + \frac{\rho^{\mathcal{R}} \Big(1 - \rho\_{\mathcal{S}}^{\mathcal{N}-\mathcal{R}+1}\Big)}{\mathcal{R}! \ (1-\rho\_{\mathcal{S}})}\right]^{-1} \tag{4}$$

*4.2. System Performance Measures*

Mathematical expectations are crucially important for the long-run theoretical average values of relevant parameters in the system. To formulate the expressions of the system performance metrics, it is necessary to construct average-based functions, such as the expected number of customers in the queue, expected number of busy servers in the system, etc. The following mathematical analyses are all necessary for the system performance measures of an M/M/R/N QS.

Let

Ls = expected number of customers in the system,

Lq = expected number of customers in the queue buffer,

E[I] = expected number of idle servers,

E[B] = expected number of busy servers,

PB = Probability that all servers are busy,

Ws = average waiting times in the system,

Wq = average waiting times in the queue buffer.

With steady-state probability functions (2) and (3), it yields

$$\mathcal{L}\_{\sf s} = \sum\_{n=0}^{N} \mathbf{n} \, \mathbf{P}(\mathbf{n}) \tag{5}$$

$$\mathbf{L\_{q}} = \sum\_{\mathbf{n}=\mathbf{R}}^{\mathbf{N}} (\mathbf{n} - \mathbf{R}) \, \mathbf{P}(\mathbf{n}) \tag{6}$$

$$\operatorname{E}[\mathbf{I}] = \sum\_{\mathbf{n}=0}^{\mathbf{R}-1} (\mathbf{R} - \mathbf{n}) \operatorname{P}(\mathbf{n}) \tag{7}$$

$$\text{E[B]} = \text{R} - \text{E[I]} \tag{8}$$

$$P\_{\mathcal{B}} = \sum\_{\mathbf{n}=\mathcal{R}}^{\mathcal{N}} \mathcal{P}(\mathbf{n})\tag{9}$$

To express the above parameters in terms of (R, N, ρ, ρS, P0), the system performance measures can be derived as follows:

$$\mathbf{L}\_{\mathbf{s}} = \sum\_{\mathbf{n}=0}^{\mathbf{N}} \mathbf{n} \ P(\mathbf{n}) = \sum\_{\mathbf{n}=0}^{\mathbf{R}-1} \mathbf{n} \ P(\mathbf{n}) + \sum\_{\mathbf{n}=\mathbf{R}}^{\mathbf{N}} \mathbf{n} \ P(\mathbf{n}) = \sum\_{\mathbf{n}=0}^{\mathbf{R}-1} \mathbf{n} \cdot \frac{\rho^{\mathbf{n}}}{\mathbf{n}!} \mathbf{P}(\mathbf{0}) + \sum\_{\mathbf{n}=\mathbf{R}}^{\mathbf{N}} (\mathbf{n} - \mathbf{R} + \mathbf{R}) \mathbf{P}(\mathbf{n}) = \sum\_{\mathbf{n}=0}^{\mathbf{R}-1} \mathbf{n} \cdot \frac{\rho^{\mathbf{n}}}{\mathbf{n}!} \mathbf{P}(\mathbf{n}) + \frac{\mathbf{N}}{\mathbf{n} - \mathbf{R}} \mathbf{N} \mathbf{P}(\mathbf{n}) + \mathbf{R} \sum\_{\mathbf{n}=\mathbf{R}}^{\mathbf{N}} \mathbf{P}(\mathbf{n}) = \sum\_{\mathbf{n}=0}^{\mathbf{R}-1} \mathbf{n} \cdot \frac{\rho^{\mathbf{n}}}{\mathbf{n}!} \mathbf{P}(\mathbf{0}) + \mathbf{L}\_{\mathbf{q}} + \mathbf{R} \mathbf{P}\_{\mathbf{B}} \tag{10}$$

$$\mathbf{P\_B = \sum\_{n=R}^{N} \mathbf{P(n)} = \sum\_{n=R}^{N} \frac{\rho^n}{\mathbf{R!} \, \mathbf{R^{n-R}}} \mathbf{P(0)} = \frac{\rho^{\mathbf{R}}}{\mathbf{R!}} \frac{[1 - (\rho\_s)^{\mathbf{N} - \mathbf{R} + 1}]}{(1 - \rho\_s)} \mathbf{P(0)}\tag{11}$$

By changing the indices of **j** = **n**—R so that n = R is changed to j = 0, and n = N is changed to j = N—R,

$$\mathbf{L}\_{\mathbf{q}} = \sum\_{\mathbf{n}=\mathbf{R}}^{\mathbf{N}} (\mathbf{n} - \mathbf{R}) \, \mathbf{P}(\mathbf{n}) = \sum\_{\mathbf{n}=\mathbf{R}}^{\mathbf{N}} (\mathbf{n} - \mathbf{R}) \frac{\rho^{\mathbf{n}}}{\mathbf{R}! \, \mathbf{R}^{\mathbf{n}-\mathbf{R}}} \mathbf{P}(0) = \frac{\rho^{\mathbf{R}} \mathbf{P}(0)}{\mathbf{R}!} \sum\_{j=0}^{\mathbf{N}-\mathbf{R}} \left[ \mathbf{j} \, (\rho\_{\mathbf{s}})^{j} \right] \tag{12}$$

The average waiting times in the system and in the queue buffer (Ws, Wq) can be derived by applying Little's formula, which gives Ws = Ls <sup>λ</sup> and Wq <sup>=</sup> Lq <sup>λ</sup> , respectively.

#### *4.3. An Illustrative Example with Computation Details*

To gain prompt perception on the theoretical implication of the quantitative modeling, an example is given by a detailed calculation. Let (R, N) = (4, 5) and (λ, μ) = (2, 1), then the server utilization ρ = λ/μ= 2 and the system utilization ρ<sup>S</sup> = ρ/R= 0.5, which is less than unity for the stable system.

$$\begin{array}{ll} \text{(1)} \quad 0 \le \mathbf{n} \le (\mathbf{R} - \mathbf{1}) = \mathbf{3}, \mathbf{P}(\mathbf{n}) = \frac{\mathbf{p}^{\mathbf{n}}}{\mathbf{n}!} \mathbf{P}(0) = \frac{\mathbf{2}^{\mathbf{n}}}{\mathbf{n}!} \mathbf{P}(0) \\\\ \mathbf{P}(1) = \frac{\mathbf{2}^{1}}{\mathbf{1}!} \mathbf{P}(0) = 2\mathbf{P}(0); \ \mathbf{P}(2) = \frac{\mathbf{2}^{2}}{\mathbf{2}!} \mathbf{P}(0) = 2\mathbf{P}(0); \ \mathbf{P}(3) = \frac{\mathbf{2}^{3}}{\mathbf{3}!} \mathbf{P}(0) = (1.33)\mathbf{P}(0) \end{array} \tag{13}$$

$$\text{(2)}\quad \mathbb{R} \le \mathbf{n} \le \text{N}, \text{ i.e., } \text{For 4} \le \mathbf{n} \le \text{5}, \text{P}(\mathbf{n}) = \frac{\rho^{\mathbf{n}}}{\mathbb{R}! \ R^{\mathbf{n}-\overline{\mathbf{R}}}} \text{ P}(\mathbf{0}).$$

$$\text{P(4)} = \frac{2^4}{4! \cdot 4^{4-4}} \text{ P(0)} = 0.667 \text{ P(0)}; \text{ P(5)} = \frac{2^5}{4! \cdot 4^{5-4}} \text{ P(0)} = 0.333 \text{ P(0)}\tag{14}$$

$$\implies \text{[P(1), P(2), P(3), P(4), P(5)]} = \text{[2, 2, 1.333, 0.667, 0.333] P(0)}.\tag{15}$$

The complete five state probabilities are assembled and expressed in terms of P(0) as follows: Using the normalization condition, <sup>N</sup> <sup>n</sup>=<sup>0</sup> <sup>P</sup>(n) <sup>=</sup> <sup>1</sup> <sup>⇒</sup> <sup>5</sup> <sup>n</sup>=<sup>0</sup> P(n) = (7.33)P(0)

$$\implies \mathbf{P}(0) = 0.136 \,\text{\#} \tag{16}$$

And from Equations (5)–(9), the system metrics can be computed sequentially as follows:

$$\text{L}\_{\bullet} = \sum\_{\mathbf{n}=0}^{N} \text{n P(n)} = \text{P(1)} + 2\text{P(2)} + 3\text{P(3)} + 4\text{P(4)} + 5\text{P(5)} = (14.333) \cdot (0.136) = 1.949 \tag{17}$$

$$\text{L}\_{\text{q}} = \sum\_{\mathbf{n}=\mathbf{R}}^{\text{N}} (\mathbf{n} - \mathbf{R}) \text{ P}(\mathbf{n}) = \sum\_{\mathbf{n}=\mathbf{4}}^{\text{5}} (\mathbf{n} - \mathbf{4}) \text{P}(\mathbf{n}) = \text{P}(\mathbf{5}) = 0.0453 \tag{18}$$

$$\mathrm{E}[\mathrm{I}] = \sum\_{\mathrm{n}=0}^{\mathrm{R}-1} (\mathrm{R}-\mathrm{n})\mathrm{P}(\mathrm{n}) = \sum\_{\mathrm{n}=0}^{3} (4-\mathrm{n})\mathrm{P}(\mathrm{n}) = 4\mathrm{P}(0) + 3\mathrm{P}(1) + 2\mathrm{P}(2) + \mathrm{P}(3) = 2.085 \tag{19}$$

$$\text{E[B]} = \text{R} - \text{E[I]} = 4 - 2.085 = 1.915\tag{20}$$

$$P\_{\rm B} = \sum\_{\rm n=R}^{N} P(\rm n) = \sum\_{\rm tn=4}^{5} P(\rm n) = P(\rm 4) + P(\rm 5) = 0.136\tag{21}$$

$$\text{Ws} = \text{L}. \text{s/} \lambda = 1.949/2 = 0.9745 \text{ and } \text{Wq} = \text{L} \text{q/} \lambda = 0.0453/2 = 0.0223 \tag{22}$$

The distribution of steady-state probabilities is depicted in Figure 5. The relevant system performance measures, such as Ls, Lq, E[B] and Ws, are shown in the left-lower part of Figure 5. The average number of customers in the QS and the queue buffer are 1.949 and 0.0457, respectively. The average waiting times in the system and the queue buffer are 0.9745 and 0.0223, respectively.

**Figure 5.** Steady-state probabilities with parameters (R, N, λ, μ) = (4, 5, 2, 1).

#### **5. Issue on Decision Support for HED Management**

#### *5.1. Evaluation Formulation on Cost*

The strategy to minimize the total cost of the operating horizon is referred to as the optimal policy. Like all medical institutions, the cost pattern is important for gaining long-term and stable hospital management. To optimize the cost, we developed a steady-state expected cost function per unit time for an M/M/R/N queuing system, in which the parameter vector of (R, N, λ, μ) and the average waiting times (Lq) are considered as decision variables. The cost element CW is defined as the waiting cost per unit time (or cost rate) per customer (HED patient) present in the system. Our goal is to provide decision support for determining the optimal number of servers R, say R\*, to optimize the cost function. To formulate the cost function, some cost parameters are defined in the following vector form as follows:

Cq = cost per unit time when one customer is waiting for service, Cs = cost per unit time when one customer joins the system and is served, (CB, CI) = cost per unit time when one server is (busy, idle).

Using the definitions of each cost element with its corresponding feature, the cost function F(R, N) can be developed in association with the system metrics Ls, PB, Lq, E[I], and E[B], which are given in Equations (10)–(12), (7) and (8), respectively. It is noted that the steady-state probabilities for two segments are given in Equations (2) and (3). The probability that there is no customer in the system, P(0), is given by Equation (4).

$$\begin{array}{l} \mathrm{F}(\mathrm{R},\mathrm{N}) = \mathrm{C}\mathrm{q}\,\mathrm{L}\mathrm{q} + \mathrm{C}\mathrm{s}\,\mathrm{(Ls-L\,\mathrm{q}\,)} + \mathrm{C}\_{\mathrm{B}}\mathrm{E}[\mathrm{B}] + \mathrm{C}\_{\mathrm{I}}\mathrm{E}[\mathrm{I}] = \left(\mathrm{C}\mathrm{q} - \mathrm{C}\mathrm{s}\right)\mathrm{L}\mathrm{q} + \mathrm{C}\mathrm{s}\,\mathrm{Ls} + \mathrm{C}\_{\mathrm{B}}\mathrm{E}[\mathrm{B}] + \mathrm{C}\_{\mathrm{I}}\mathrm{E}[\mathrm{I}] \\ = \left(\mathrm{C}\mathrm{q} - \mathrm{C}\mathrm{s}\right)\frac{\mathrm{p}^{\mathrm{R}}\mathrm{P}(0)}{\mathrm{R}!}\sum\_{j=0}^{\mathrm{N}-\mathrm{R}}\left[\mathrm{j}\cdot(\rho\_{s})^{\dagger}\right] + \mathrm{C}\mathrm{s}\left(\mathrm{L}\_{\mathrm{n}=0}^{\mathrm{R}-1}\mathrm{n}\cdot\frac{\mathrm{p}^{\mathrm{R}}}{\mathrm{R}!}\mathrm{P}(0) + \frac{\rho^{\mathrm{R}}\mathrm{P}(0)}{\mathrm{R}!}\sum\_{j=0}^{\mathrm{N}-\mathrm{R}}\left[\mathrm{j}\cdot(\rho\_{s})^{\dagger}\right] + \mathrm{R} \cdot \mathrm{C} \end{array} \tag{23}$$

The cost function F(R, N) in Equation (23) is expressed in terms of basic parameters, such as (R, N, λ, μ), and cost elements. It is noted that the utilization parameters of the unit server and system is given by (ρ, ρS) =(λ/μ, λ/(Rμ)), respectively. The state probability functions P(n) for two segments are given in Equations (2) and (3), which are quite complex for the control parameter R. To find the optimal profile on the cost function, it is necessary to show the existence of convexity or unimodality of F(R, N). However, this mathematical task is difficult to implement. The cost function F(R, N) is unimodal; that is, it has a single relative minimum.

#### *5.2. Evaluation of Cost Optimization*

Equation (23) shows that the parameter R occurs not only at the location of in-line items, but also at the upper limit of the summation symbol Σ, which makes F(R, N) a highly nonlinear and complex function. Instead, practical numerical examples are presented and intensively studied by applying the proposed models. The optimization evaluation is firstly probed in terms of cost patterns in this subsection. For illustrative purposes, we first study the effect of varying R while keeping N constant, and then varying N while keeping R constant. All simulations are performed with the MATLAB platform with custom MATLAB scripts. The exemplified system parameters are listed as vector forms as follows:


Contour plots may provide the best graphical representation of the optimization problem, and also possess a powerful visualization that permits the solutions of the optimization problem by inspection. To validate the analytical solution, the graphical results were obtained and are shown in Figure 6A, where three cost contours with the black box, red circle, and blue triangle icons are depicted along the Y-axis in terms of λ = 2.5, 3.0, and 3.5, respectively. Generally, a higher patient arrival rate implies that the medical service cost is higher, so the blue line marked with the triangle icon (λ = 3.5) is situated over the red line marked with a circle (λ = 3.0). To clearly show the crucial region surrounded by the dashed-line rectangle in Figure 6A, enlarged detail is depicted in Figure 6B. In Figure 6B, the critical region is between R = 3 and R = 9 on the X-axis. The optimal cost value with the corresponding optimal R\* is shown for each contour.

**Figure 6.** (**A**). Optimal cost patterns shown in terms of three average arrival rates. (**B**) An enlarged diagram showing the optimal cost data from Figure 6A.

#### *5.3. Issues on Cost Profile under the Constraint of Average Waiting Time*

In view of performance evaluation, the average waiting time (AWT) can also be regarded as a measure of performance committed to the HED patients, and of a yardstick for comparing the effectiveness of the deployment of the staffing providers in a quantitative manner. Practically, it is reasonable for management experts to guarantee an AWT target level when they want to alleviate the sense of worry for potential HED patients. Logically, the higher the number of staffing providers deployed, the higher the cost. Hence, the proposed approach explores the issue of decision support for optimal cost under the constraint of the AWT at some target level.

In Figure 7 with the double-Y axis, the left Y-axis and the right Y-axis are set to be the cost values and the average waiting time (AWT), respectively. Observing the solid-line contour marked with black rectangles (i.e., the left Y-axis), the optimal cost value F(R, N) = 1242.5, which occurs at R\* = 6 based on the similar parameters in Figure 6A with the average arrival rate λ = 3.5. However, the corresponding AWT approaches 6.84 units, which is a reference metric for decision-making. The proposed generic model could be used for general insights into the issues faced in deploying multiple staffing providers for a disease chain or a single department like the Department of Pediatrics shown in the middle right-handed location of the ground floor in Figure 1. On the right Y-axis, the dash-curve marked with a red star shows the variation profile of the performance metric for AWT.

During the busy time-window for a specific disease chain in HED, patients may spend hours in crowded waiting rooms before seeing a doctor. Those who choose to tolerate longer waiting times expose themselves to others who may have a contagious illness. To alleviate such an occasional impact, one straight approach for reducing the waiting time is to deploy more staffing providers for that specific disease chain. The simulation results in Figure 7 provide an exemplified decision reference on re-deploying the amount of staffing providers to alleviate the waiting time.

**Figure 7.** Decision support on optimal cost at R\* = 7 under the constraint of reduction of AWT (average waiting time) by 68.9%, which is calculated from ((6.84–2.13)/6.84) × 100%.

Then an issue emerges from the judgment: how many extra staffing providers are needed to gain a reduction in the AWT by some level (for example, 50%) without over-provisioning? Observing the red-star contour with the right Y-axis in Figure 7, it is found that the AWT can be reduced by 68.9% at R\* = 7 (shown by the red dash-line) at the expense of only adding one staffing provider and cost values F(R\* = 7, N) = 1309.8, compared to the minimum cost F(R = 6, N) = 1242.5. The detailed numerical data are listed in Table 1 with a range of R from unity to 12. The value of AWT for R = 12 is less than 0.01 and then marked to be 0 for clarity. In other words, the proposed approach can provide a quantitative decision support on the trade-off study between the cost profile and the amount of staffing providers in HED deployment.

**Table 1.** Numerical data on AWT and the corresponding cost values for the range of R from unity to 12.


AWT, average waiting time.

#### *5.4. Application Profile in a Window-by-Window Way*

This work addressed the issue of the mathematical modelling to evaluate scenarios for deployment of medical resources to the HED, and also aimed to provide feasible applications iteratively to approaching an effective decision support in terms of deploying appropriate staffing providers to alleviate the impact on HED crowding. Patients who want to receive medical services always arrive randomly, and they require immediate services available at that time. If the service facility is operating at peak capacity when they arrive, they are obliged to wait in line (queue) with patience in the case of a shortage in staffing level. The surges and changes in HED activity may occur from time to time in terms of various time-widows with associated system parameters, as shown in Figure 8.

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**Figure 8.** Iterative applications exemplified by various time-windows.

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In Figure 8, each unique time-window may represent a specific surge time-period, wherein larger numbers of patients nearby the hospital are delivered to HED after some disaster or traffic event has occurred. In modeling language, the proposed modeling approaches can be applied in a window-by-window way that each specific time window can be approximated by its λ (average arrival rate) and μ (average service rate) in association with various practical historical data. For example, supposing that the surge in time window B of Figure 8 represents some middle-level traffic event, then the system parameters λ<sup>B</sup> (average arrival rate) and μ<sup>B</sup> (average service rate) may be approximated by some existing past and experienced parameters for the baseline, and then the cost function F(R, N) (23) may be applied iteratively to approach the cost optimization in a window-by-window way.

#### **6. Conclusions**

In terms of patient flow and available resources, an efficient generic methodology to optimize the performance of the HED platform has been addressed in this research. The proposed queue-based approach provides HED administrators with an efficient deployment of staffing providers to optimize the cost profile. Conceptually, the HED service platform was mapped into an M/M/R/N queuing system, and illustrated using appropriate figures and materials in the work. To gain insight into the queuing model, the mathematical derivation was detailed for the application need as well.

Based on the quantitative analysis, the M/M/R/N queue model was applied and derived, and then the relevant system metrics were established in a brand-new manner. The mathematical expression for cost function was established for evaluation requirements. In regards to verification, the relevant experimental results were obtained in terms of integration configurations on cost optimization and average waiting time. Instead of chaotic management, the proposed generic methodology may provide feasible applications for approaching an effective decision support in terms of deploying appropriate staffing providers to alleviate the impact on HED crowding.

**Author Contributions:** Conceptualization, F.-C.J.; Data curation, Y.-J.C. and C.-H.L.; Formal analysis, C.-M.S.; Investigation, C.-H.L.; Methodology, C.-M.S. and Y.-M.W.; Project administration, Y.-J.C.; Resources, C.-T.Y.; Software, C.-T.Y.; Supervision, F.-C.J.; Validation, C.-M.S.; Writing—original draft, F.-C.J. and Y.-M.W.; Writing—review & editing, F.-C.J. and C.-T.Y.

**Funding:** This research was supported in part by the Ministry of Science and Technology research grants MOST 108-2119-M-029-001-A and 108-2221-E-029-010. This research was also supported in part from Taichung Veterans General Hospital (TCVGH-1075103B to Cheng-Min Shih).

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*
