# Physician-Customized Strategies for Reducing Outpatient Waiting Time in South Korea Using Queueing Theory and Probabilistic Metamodels

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## Abstract

**:**

## 1. Introduction

- To analyze the actual outpatient waiting time data of a tertiary hospital;
- To apply queueing theory to the data of each physician to analyze the effect of influencing parameters on outpatient waiting time;
- To suggest ways to improve the waiting time in the OPD without the need for additional resources or reducing the number of patients.

## 2. Theory and Data

#### 2.1. Queueing Theory

#### 2.2. Outpatient Record Data

## 3. Data Processing and Problem Formulation

#### 3.1. Data Pre-Processing and Parameter Extraction

_{est}, which is calculated as follows, was used.

_{est}has the same value as R unless R is 0, in which case, the value estimated using a 95% confidence interval is used. Parameter R

_{est}is calculated as follows.

_{est}in this manner because when its value is 0, μ

_{est}becomes infinite, which is not realistic. Finally, P, the probability of over-waiting when x equals 0.5, is calculated using μ

_{est}instead of μ in Equation (1). The calculated probability of over-waiting can be changed when adjusting the number of patients in a session, which helps calculate the expected number of patients who will wait for more than 30 min.

#### 3.2. Statistical Analysis of Effective Parameters

#### 3.3. Regression Modeling of Waiting Time Reduction

^{2}of 0.99 or more.

#### 3.4. Formulation of the Waiting Time Reduction Problem

_{ij}(i = 1 to 7, j = 1, 2, …)

_{i}(x

_{ij}) = P

_{1}(x

_{11}, x

_{12}), P

_{2}(x

_{21}, x

_{22}, x

_{23}), P

_{3}(x

_{31}, x

_{32}), …

_{ij}≤ 1.1

_{ij}) = constant;

_{ij}were used to represent the weighting of the patients according to each category of the effective parameters, 7 objective functions P

_{i}(x

_{ij}) were used to represent the probability of over-waiting according to the 7 parameters, and 1 constraint was used to keep the total number of patients constant. Here, i indicates the parameter, and j indicates the category of each parameter. For example, the session time of day parameter had morning and afternoon categories, whereas the day of the week parameter had Monday, Tuesday, Wednesday, Thursday, Friday, and Saturday as categories.

## 4. Results

#### 4.1. Validation of the Effective Parameters

^{2}) of the statistical model was not found to be sufficiently high. Specifically, the result of multiple regression analysis of the entire OPD shows that the significance probability of the number of walk-in patients parameter is 0.001, which is significant at 0.01, but the R

^{2}of the statistical model is 0.026, which is extremely low. The ANOVA results of department A reveal that the significance probability of the day of the week parameter is 0.029, which is significant at 0.05, but none of the days have a significance probability of 0.1 or less in the post hoc test. The results of multiple regression analysis of department A show that the significance probability of the number of walk-in patients parameter is 0.043, which is significant at 0.05, but the R

^{2}of the statistical model is 0.026, which again is extremely low. The multiple regression analysis results of department B show that the significance probability of the number of no-shows parameter is 0.005, which is significant at 0.01, but the R

^{2}of the statistical model is 0.059, which is also extremely low. The results of multiple regression analysis of department C show that the significance probabilities of the number of no-shows and proportion of first-time patients parameters are 0.009 and 0.016, respectively, which are significant at 0.01 and 0.05, but the R

^{2}of the statistical model is low at 0.108.

#### 4.2. Results of Waiting Time Reduction

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Effective Parameters | Probability Statistics Verification Methods | All Three Medical Dept. | A Dept. | B Dept. | C Dept. | |
---|---|---|---|---|---|---|

1. ○ | Session running time | t test | X | X | X | X |

2. ● | Day of the week | ANOVA | X | O | X | X |

3. ◇ | Session running month | ANOVA | X | X | X | X |

4. ◆ | Lateness of start time | Multiple Regression | X | X | X | X |

5. □ | Number of receipts on the day | Multiple Regression | O | O | X | X |

6. ■ | Number of no-shows | Multiple Regression | X | X | O | O |

7. ☆ | Proportion of first-timers | Multiple Regression | X | X | X | O |

Physician ID Code | Number of Sessions | Total Number of Patients | Normalized Probability of Over-Waiting Patients |
---|---|---|---|

C-007 | 12 | 401 | 1.00 |

A-009 | 13 | 815 | 0.91 |

C-004 | 8 | 348 | 0.84 |

C-003 | 12 | 495 | 0.76 |

C-001 | 7 | 198 | 0.57 |

C-002 | 19 | 715 | 0.49 |

C-018 | 10 | 290 | 0.43 |

A-008 | 12 | 274 | 0.41 |

A-014 | 10 | 532 | 0.38 |

B-006 | 12 | 572 | 0.37 |

C-013 | 9 | 234 | 0.38 |

B-019 | 11 | 484 | 0.34 |

B-017 | 13 | 532 | 0.30 |

B-008 | 11 | 431 | 0.29 |

B-003 | 10 | 328 | 0.27 |

A-006 | 9 | 181 | 0.26 |

B-010 | 12 | 569 | 0.24 |

C-014 | 12 | 443 | 0.23 |

A-015 | 18 | 1033 | 0.17 |

A-001 | 14 | 807 | 0.15 |

C-012 | 12 | 390 | 0.14 |

Effective Parameters | Number of Patients | $\mathit{a}$ | $\mathit{b}$ | |
---|---|---|---|---|

○ | Morning | 134 | 0.002121 | 6.212 |

Afternoon | 267 | 0.001059 | 6.818 | |

● | Monday | 0 | - | - |

Tuesday | 136 | 0.000941 | 7.112 | |

Wednesday | 134 | 0.002121 | 6.212 | |

Thursday | 131 | 0.001335 | 6.364 | |

Friday | 0 | - | - | |

Saturday | 0 | - | - | |

◇ | June | 104 | 0.001854 | 6.526 |

August | 107 | 0.00091 | 6.843 | |

September | 102 | 0.001416 | 6.777 | |

December | 88 | 0.001515 | 5.943 | |

◆ | Lateness of start time 0 or less | 327 | 0.000986 | 6.688 |

Lateness of start time bigger than 0 | 74 | 0.003026 | 6.449 | |

□ | Number of receipts on the day 0 or less | 401 | 0.001353 | 6.604 |

Number of receipts on the day 1 or more | 0 | - | - | |

■ | Number of no-shows 4 or less | 175 | 0.001632 | 6.766 |

Number of no-shows 5 or more | 226 | 0.001191 | 6.339 | |

☆ | Proportion of first-timers less than 0.1 | 167 | 0.001706 | 6.212 |

Proportion of first-timers more than 0.1 | 234 | 0.001206 | 6.819 |

Effective Parameters | Number of Patients | Weighting of Number of Patients | Improved Probability of Over-Waiting | |
---|---|---|---|---|

○ | Morning | 134 | 0.9979 | 0.9985 |

Afternoon | 267 | 1.0007 | ||

● | Monday | 0 | 1.0000 | 0.9792 |

Tuesday | 136 | 0.9638 | ||

Wednesday | 134 | 1.0035 | ||

Thursday | 131 | 1.0333 | ||

Friday | 0 | 1.0000 | ||

Saturday | 0 | 1.0000 | ||

◇ | June | 104 | 0.9489 | 0.9554 (3rd priority) |

August | 107 | 1.0401 | ||

September | 102 | 0.9628 | ||

December | 88 | 1.0537 | ||

◆ | Lateness of start time 0 or less | 327 | 1.0223 | 0.9347 (1st priority) |

Lateness of start time bigger than 0 | 74 | 0.9000 | ||

□ | Number of receipts on the day 0 or less | 401 | - | - |

Number of receipts on the day 1 or more | 0 | - | ||

■ | Number of no-shows 4 or less | 175 | 0.9028 | 0.9443 (2nd priority) |

Number of no-shows 5 or more | 226 | 1.0748 | ||

☆ | Proportion of first-timers less than 0.1 | 167 | 1.0338 | 0.9848 |

Proportion of first-timers more than 0.1 | 234 | 0.9755 |

Effective Parameters | Number of Patients | Weighting of Number of Patients | Improved Probability of Over-Waiting | |
---|---|---|---|---|

○ | Morning | 558 | 0.9973 | 0.3646 |

Afternoon | 14 | 1.0998 | ||

● | Monday | 213 | 1.0158 | 0.3615 (3rd priority) |

Tuesday | 0 | 1.0000 | ||

Wednesday | 236 | 0.9623 | ||

Thursday | 0 | 1.0000 | ||

Friday | 0 | 1.0000 | ||

Saturday | 123 | 1.0441 | ||

◇ | June | 127 | 1.0850 | 0.3372 (1st priority) |

August | 121 | 0.9502 | ||

September | 162 | 0.9856 | ||

December | 162 | 0.9851 | ||

◆ | Lateness of start time 0 or less | 254 | 1.0000 | - |

Lateness of start time bigger than 0 | 318 | 1.0000 | ||

□ | Number of receipts on the day 0 or less | 310 | 1.0000 | - |

Number of receipts on the day 1 or more | 262 | 1.0000 | ||

■ | Number of no-shows 9 or less | 263 | 0.9542 | 0.3549 (2nd priority) |

Number of no-shows 10 or more | 309 | 1.0386 | ||

☆ | Proportion of first-timers less than 0.3 | 319 | 1.0000 | - |

Proportion of first-timers more than 0.3 | 253 | 1.0000 |

Physician ID Code | Normalized Probability of Over-Waiting | Three Most Effective Parameters Obtained from the Optimization | ||||
---|---|---|---|---|---|---|

Initial | Optimal | Improvement | #1 | #2 | #3 | |

C-007 | 1.00 | 0.82 | 0.18 | ◆ | ■ | ◇ |

A-009 | 0.91 | 0.77 | 0.14 | ■ | ● | □ |

C-004 | 0.84 | 0.77 | 0.07 | ○ | ○ | ◇ |

C-003 | 0.76 | 0.60 | 0.16 | ● | ◇ | ○ |

C-001 | 0.57 | 0.26 | 0.31 | ◆ | ● | ○ |

C-002 | 0.49 | 0.44 | 0.05 | ◇ | ● | ○ |

C-018 | 0.43 | 0.13 | 0.30 | ● | ◇ | ■ |

A-008 | 0.41 | 0.15 | 0.26 | ◇ | ● | ■ |

A-014 | 0.38 | 0.18 | 0.20 | ● | ■ | ◇ |

B-006 | 0.37 | 0.31 | 0.06 | ◇ | ■ | ◆ |

C-013 | 0.38 | 0.06 | 0.32 | ☆ | ● | ◇ |

B-019 | 0.34 | 0.19 | 0.15 | ◇ | □ | ● |

B-017 | 0.30 | 0.12 | 0.18 | ◇ | ■ | ○ |

B-008 | 0.29 | 0.08 | 0.21 | ◇ | ☆ | ● |

B-003 | 0.27 | 0.13 | 0.14 | ● | ■ | ◇ |

A-006 | 0.26 | 0.06 | 0.20 | ● | ◇ | □ |

B-010 | 0.24 | 0.13 | 0.11 | ● | ○ | □ |

C-014 | 0.23 | 0.08 | 0.15 | ◇ | ● | ○ |

A-015 | 0.17 | 0.12 | 0.05 | ● | ◇ | ☆ |

A-001 | 0.15 | 0.05 | 0.10 | ● | □ | ◇ |

C-012 | 0.14 | 0.05 | 0.09 | ○ | ■ | ● |

Department | Normalized Probability of Over-Waiting | ||
---|---|---|---|

Prior | Improvement | Difference | |

All of A, B, and C | 0.232 | 0.162 | 0.070 |

A | 0.154 | 0.107 | 0.047 |

B | 0.170 | 0.099 | 0.071 |

C | 0.513 | 0.387 | 0.126 |

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## Share and Cite

**MDPI and ACS Style**

Lee, H.; Choi, E.K.; Min, K.A.; Bae, E.; Lee, H.; Lee, J.
Physician-Customized Strategies for Reducing Outpatient Waiting Time in South Korea Using Queueing Theory and Probabilistic Metamodels. *Int. J. Environ. Res. Public Health* **2022**, *19*, 2073.
https://doi.org/10.3390/ijerph19042073

**AMA Style**

Lee H, Choi EK, Min KA, Bae E, Lee H, Lee J.
Physician-Customized Strategies for Reducing Outpatient Waiting Time in South Korea Using Queueing Theory and Probabilistic Metamodels. *International Journal of Environmental Research and Public Health*. 2022; 19(4):2073.
https://doi.org/10.3390/ijerph19042073

**Chicago/Turabian Style**

Lee, Hanbit, Eun Kyoung Choi, Kyung A. Min, Eunjeong Bae, Hooyun Lee, and Jongsoo Lee.
2022. "Physician-Customized Strategies for Reducing Outpatient Waiting Time in South Korea Using Queueing Theory and Probabilistic Metamodels" *International Journal of Environmental Research and Public Health* 19, no. 4: 2073.
https://doi.org/10.3390/ijerph19042073