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

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 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

- Zengin, S.; Guzel, R.; Al, B.; Kartal, S.; Sarcan, E.; Yildirim, C. Cost analysis of a university hospital’s adult emergency service. J. Acad. Emerg. Med.
**2013**, 12, 71–75. [Google Scholar] [CrossRef] - Simonet, D. Cost reduction strategies for emergency services: Insurance role, practice changes and patients accountability. Health Care Anal.
**2009**, 17, 1–19. [Google Scholar] [CrossRef] [PubMed] - Cremonesi, P.; Di Bella, E.; Montefiori, M. Cost analysis of emergency department. J. Prev. Med. Hyg.
**2010**, 51, 157–163. [Google Scholar] [PubMed] - Cooper, R.B. Introduction to Queuing Theory, 2nd ed.; Elsevier Science Publishing Co., Inc.: New York, NY, USA, 1990. [Google Scholar]
- Gross, D.; Shortle, J.F.; Thompson, J.M.; Harris, C. Fundamentals of Queuing Theory, 4th ed.; John Wiley &Sons, Inc.: New York, NY, USA, 2008. [Google Scholar]
- Finamore, S.R.; Sheila, S.A. Shorting the wait: A strategy to reduce waiting times in the emergency department. J. Emerg. Nurs.
**2009**, 35, 509–514. [Google Scholar] [CrossRef] [PubMed] - Green, L.V.; Soares, J.; Giglio, J.F.; Green, R.A. Using queuing theory to increase the effectiveness of emergency department provider staffing. Acad. Emerg. Med.
**2006**, 13, 61–68. [Google Scholar] [CrossRef] [PubMed][Green Version] - Derlet, R.W.; Richards, J.R.; Kravitz, R.L. Frequent overcrowding in U.S. emergency departments. Acad. Emerg. Med.
**2001**, 8, 151–155. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kennedy, J.; Rhodes, K.; Walls, C.A.; Asplin, B.R. Access to emergency care: Restricted by long waiting times and cost and coverage concerns. Ann. Emerg. Med.
**2004**, 43, 567–573. [Google Scholar] [CrossRef] [PubMed] - Hoot, N.R.; Aronsky, D. Systematic review of emergency department crowding: Causes, effects, and solutions. Ann. Emerg. Med.
**2008**, 52, 126–136. [Google Scholar] [CrossRef] [PubMed] - Ding, R.; Zeger, S.L.; McCarthy, M.L.; Pines, J.M. Comparison of methods for measuring crowding and its effects on length of stay in the emergency department. Acad. Emerg. Med.
**2011**, 18, 1269–1277. [Google Scholar] - Kaushal, A.; Zhao, Y.; Peng, Q.; Strome, T.; Weldon, E.; Zhang, M.; Chochinov, A. Evaluation of fast track strategies using agent-based simulation modeling to reduce waiting time in a hospital emergency department. Socio Econ. Plan. Sci.
**2015**, 50, 18–31. [Google Scholar] [CrossRef] - Vass, H.; Szabo, Z.K. Application of queuing model to patient flow in emergency department. Case study. Procedia Econ. Financ.
**2015**, 32, 479–487. [Google Scholar] [CrossRef][Green Version] - Taichung Veterans General Hospital (TVGH): Departments of Medical Services. Available online: http://www.vghtc.org.tw/ (accessed on 5 December 2019).
- Fletcher, A.; Worthington, D. What is a ‘generic’ hospital model?—A comparison of ‘generic’ and ‘specific’ hospital models of emergency patient flows. Health Care Manag. Sci.
**2009**, 12, 374–391. [Google Scholar] [CrossRef][Green Version] - Hoot, N.R.; Leblanc, L.J.; Jones, I.; Levin, S.R.; Zhou, C.; Gadd, C.S.; Aronsky, D. Forecasting emergency department crowding: A discrete event simulation. Ann. Emerg. Med.
**2008**, 52, 116–125. [Google Scholar] [CrossRef] - Cochran, J.K.; Roche, K.T. A multi-class queuing network analysis methodology for improving hospital emergency department performance. Comput. Oper. Res.
**2009**, 36, 1497–1512. [Google Scholar] [CrossRef] - McManus, M.L.; Long, M.C.; Cooper, A.; Litvak, E. Queuing theory accurately models the need for critical care resources. Anesthesiology
**2004**, 100, 1271–1276. [Google Scholar] [CrossRef] - Marmor, Y.N.; Wasserkrug, S.; Zeltyn, S.; Mesika, Y.; Greenshpan, O.; Carmeli, B.; Shtub, A.; Mandelbaum, A. Toward simulation-based real-time decision-support systems for emergency departments. In Proceedings of the 2009 Winter Simulation Conference (WSC), Austin, TX, USA, 13–16 December 2009. [Google Scholar]
- Oh, C.; Novotny, A.M.; Carter, P.L.; Ready, R.K.; Campbell, D.D.; Leckie, M.C. Use of a simulation-based decision support tool to improve emergency department throughput. Oper. Res. Health Care
**2016**, 9, 29–39. [Google Scholar] [CrossRef]

**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 |

© 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/).

## 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