# Decision Support for the Optimization of Provider Staffing for Hospital Emergency Departments with a Queue-Based Approach

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## Abstract

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## 1. Introduction

#### 1.1. Background

#### 1.2. Contribution Profile

## 2. Related Work

## 3. The Proposed Model of Medical Emergency Services

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

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

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

#### 4.1. Theoretical Analysis

_{n}= λ if 0 ≤ n ≤ N and λ

_{n}= 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

_{n}= µ. The service volume can be classified into two parts as follows:

_{S}= 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$$\mathrm{P}\left(\mathrm{n}\right)=\frac{{\mathsf{\lambda}}_{0}\xb7{\mathsf{\lambda}}_{1}\xb7{\mathsf{\lambda}}_{2}\text{}\cdots \text{}{\mathsf{\lambda}}_{\mathrm{n}-1}}{{\mathsf{\mu}}_{1}\xb7{\mathsf{\mu}}_{2}\xb7{\mathsf{\mu}}_{3}\text{}\cdots \text{}{\mathsf{\mu}}_{\mathrm{n}}}\mathrm{P}\left(0\right)=\frac{{\mathsf{\lambda}}^{\mathrm{n}}}{\mathsf{\mu}\text{}\left(2\mathsf{\mu}\right)\left(3\mathsf{\mu}\right)\text{}\cdots \text{}\left(\mathrm{n}\mathsf{\mu}\right)}\mathrm{P}\left(0\right)=\frac{{\mathsf{\lambda}}^{\mathrm{n}}}{{\mathsf{\mu}}^{\mathrm{n}\text{}}\mathrm{n}!}\mathrm{P}\left(0\right)=\frac{{\mathsf{\rho}}^{\mathrm{n}}}{\mathrm{n}!}\mathrm{P}\left(0\right)$$

- (B)
- Segment (2): (R+1) ≤ n ≤ N,$$\begin{array}{c}\mathrm{P}\left(\mathrm{n}\right)=\frac{{\mathsf{\lambda}}_{0}\xb7{\mathsf{\lambda}}_{1}\xb7{\mathsf{\lambda}}_{2}\text{}\cdots \text{}{\mathsf{\lambda}}_{\mathrm{n}-1}}{({\mathsf{\mu}}_{1}\xb7{\mathsf{\mu}}_{2}\text{}\cdots \text{}{\mathsf{\mu}}_{\mathrm{R}})({\mathsf{\mu}}_{\mathrm{R}+1}\text{}\cdots \text{}{\mathsf{\mu}}_{\mathrm{n}})}\mathrm{P}\left(0\right)=\frac{{\mathsf{\lambda}}^{\mathrm{n}}}{[\mathsf{\mu}\xb7\left(2\mathsf{\mu}\right)\cdots \left(\mathrm{R}\mathsf{\mu}\right)](\mathrm{R}\mathsf{\mu}\cdots \mathrm{R}\mathsf{\mu})}\mathrm{P}\left(0\right)=\\ =\frac{{\mathsf{\lambda}}^{\mathrm{n}}}{\mathrm{R}!\text{}{\mathsf{\mu}}^{\mathrm{R}}\text{}{(\mathrm{R}\mathsf{\mu})}^{\mathrm{n}-\mathrm{R}}}\mathrm{P}\left(0\right)=\frac{{\mathsf{\rho}}^{\mathrm{n}}}{\mathrm{R}!\text{}{(\mathrm{R})}^{\mathrm{n}-\mathrm{R}}}\mathrm{P}\left(0\right)\end{array}$$

#### 4.2. System Performance Measures

- 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,
- P
_{B}= Probability that all servers are busy, - Ws = average waiting times in the system,
- Wq = average waiting times in the queue buffer.

_{S}, P

_{0}), the system performance measures can be derived as follows:

**j**=

**n**—R so that n = R is changed to j = 0, and n = N is changed to j = N—R,

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

_{S}= ρ/R= 0.5, which is less than unity for the stable system.

- (1)
- 0 ≤ n ≤ (R–1) = 3, $\mathrm{P}\left(\mathrm{n}\right)=\frac{{\mathsf{\rho}}^{\mathrm{n}}}{\mathrm{n}!}\mathrm{P}\left(0\right)=\frac{{2}^{\mathrm{n}}}{\mathrm{n}!}\mathrm{P}\left(0\right)$$$\mathrm{P}\left(1\right)=\frac{{2}^{1}}{1!}\mathrm{P}\left(0\right)=2\mathrm{P}\left(0\right);\text{}\mathrm{P}\left(2\right)=\frac{{2}^{2}}{2!}\mathrm{P}\left(0\right)=2\mathrm{P}\left(0\right);\text{}\mathrm{P}\left(3\right)=\frac{{2}^{3}}{3!}\mathrm{P}\left(0\right)=\left(1.33\right)\mathrm{P}\left(0\right)$$

- (2)
- R ≤ n ≤ N, i.e., For 4 ≤ n ≤ 5, $\mathrm{P}\left(\mathrm{n}\right)=\frac{{\mathsf{\rho}}^{\mathrm{n}}}{\mathrm{R}!{\text{}\mathrm{R}}^{\mathrm{n}-\mathrm{R}}}\text{}\mathrm{P}\left(0\right)$$$\mathrm{P}\left(4\right)=\frac{{2}^{4}}{4!\text{}{4}^{4-4}}\text{}\mathrm{P}\left(0\right)=0.667\text{}\mathrm{P}\left(0\right);\text{}\mathrm{P}\left(5\right)=\frac{{2}^{5}}{4!\text{}{4}^{5-4}}\text{}\mathrm{P}\left(0\right)=0.333\text{}\mathrm{P}\left(0\right)$$⇒[P(1), P(2), P(3), P(4), P(5)] = = [2, 2, 1.333, 0.667, 0.333] P(0).

## 5. Issue on Decision Support for HED Management

#### 5.1. Evaluation Formulation on Cost

_{W}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,
- (C
_{B}, C_{I}) = cost per unit time when one server is (busy, idle).

_{B}, 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).

_{S}) =(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

- (a)
- Average arrival rate of patients (λ) = 2.5, 3.0, and 3.5,
- (b)
- Average service rate of a server (µ) = 1,
- (c)
- Cost rate: (Cq, Cs, C
_{B}, C_{I},) = (200, 150, 120, 100), - (d)
- N = 15 for emergency departments of small and medium size.

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

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

_{B}(average arrival rate) and µ

_{B}(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

## Author Contributions

## Funding

## Conflicts of Interest

## References

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

**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.

**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%.

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

R | Cost Values | AWT | R | Cost Values | AWT |
---|---|---|---|---|---|

1 | 2990.0 | 388.57 | 7 | 1309.9 | 2.13 |

2 | 2873.9 | 333.42 | 8 | 1399.5 | 0.65 |

3 | 2345.8 | 220.77 | 9 | 1496.3 | 0.19 |

4 | 1530.2 | 78.57 | 10 | 1595.4 | 0.05 |

5 | 1250.7 | 22.51 | 11 | 1695.1 | 0.01 |

6 | 1242.5 | 6.84 | 12 | 1795.0 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Jiang, F.-C.; Shih, C.-M.; Wang, Y.-M.; Yang, C.-T.; Chiang, Y.-J.; Lee, C.-H.
Decision Support for the Optimization of Provider Staffing for Hospital Emergency Departments with a Queue-Based Approach. *J. Clin. Med.* **2019**, *8*, 2154.
https://doi.org/10.3390/jcm8122154

**AMA Style**

Jiang F-C, Shih C-M, Wang Y-M, Yang C-T, Chiang Y-J, Lee C-H.
Decision Support for the Optimization of Provider Staffing for Hospital Emergency Departments with a Queue-Based Approach. *Journal of Clinical Medicine*. 2019; 8(12):2154.
https://doi.org/10.3390/jcm8122154

**Chicago/Turabian Style**

Jiang, Fuu-Cheng, Cheng-Min Shih, Yun-Ming Wang, Chao-Tung Yang, Yi-Ju Chiang, and Cheng-Hung Lee.
2019. "Decision Support for the Optimization of Provider Staffing for Hospital Emergency Departments with a Queue-Based Approach" *Journal of Clinical Medicine* 8, no. 12: 2154.
https://doi.org/10.3390/jcm8122154