Queueing Theory Applications in HospitalsGarett Robertson

Introduction and Outline 2

1. Hospital Challenges• Congestion• Process Changes• Queuing

Applications

2. Queueing Theory• What is it• Model Variables• Model Equations

3. Hospital Model• Example

Simulation• Parameters• Explanation

4. Example Scenarios• Demand Shifts• Process Changes• Service Pathway

Engineering

5. Benefits and Challenges• Summary of Benefits• Data Requirements• System

Requirements

6. Uses in Other Industries• Hospitality• Other Industries

Hospital Operations Challenges 3

Predicting congestion in hospitals or clinics is very difficult. Patients arrive for treatment at random intervals seeking different levels of treatment. Factors such as the time of day, or the season are related to changes in admission patterns as well. Because the services required by each patient are different, there is a level of variability and randomness in service delivery. Mismatches between service times and patient arrival times cause congestion.Resultant delays in care can cause patient conditions to worsen making care delivery more complicated and more expensive.Additionally, anticipating demand that does not arrive increases fixed costs per patient also making care delivery more expensive and inefficient in an industry where margins are already very low.

Congestion

Service delivery in healthcare is highly complicated with each patient’s care requiring coordination from many departments such as the clinic, surgery, labs, or recovery. Each of these departments acts independent of the others, and often sees patients with vary different needs. Each department delivers highly customized services with constrained resources. Hospitals, in an effort to decrease the time patients are in their system, may evaluate many different care pathways including but not limited to creating dedicated hospital services to fast track care for certain patients or partnering with specialty clinics and services such as home health agencies.In these or other cases, the goal is the same. The hospital must strive to get the patients through the system more quickly without compromising care.

Process Changes Capacity Planning

Queuing Models can help solve these critical issues.

Demand ForecastingWhile these models can predict the average demand in a given department, they also can show the expected random noise around the mean. By understanding these fluctuations, a hospital could make more efficient decisions about what level of congestion is acceptable and then allocate hospital resources such as staff, beds, wheelchairs or other equipment.

Process ImprovementThese models can also be used to understand how changes in one departmental operations can accelerate service times allowing the hospital to handle greater patient loads more efficiently despite resource constraints such as hospital bed licenses or staff availability.

Queuing models are mathematical models of random processes. In the case of a hospital, the model shows the current volume of patients being served given an average rate at which patients arrive, and an average rate at which they are discharged. The arrival and discharge times are modeled as exponential random variables. Exponential random variables have the key property of not being dependent on any previous events making them ideal for simulations like hospitals where one arrival comes because of the flu and another because of a broken arm. Additionally, each patient’s care is independent of any other patients care allowing their discharge times to modeled this way as well. The difference in the two rates creates a service queue that can be modeled and studied.When the arrival rate on average is greater than the discharge rate, the queue grows. When it is smaller, the queue shrinks. Because the rates are random variables, the process will eventually oscillate around a steady state known as the equilibrium.

What is it?

Model Inputs:

Model Inputs

Queueing Theory 4

Model Equations:With these inputs a model can be built where arrivals arrive at intervals determined by the first equation and are discharged at intervals set by the second. In the second equation, it is important to note that the average departure time is divided by the current queue length. This is because if there are more patients receiving service at a given moment, it is more likely that one will complete service and discharge. Thus, the average rate of discharge accelerates as the overall patient volume increases. The final model constraint is that the queue cannot grow beyond the number of patients the hospital is licensed to serve at a given moment. Excess capacity would be turned away to receive service elsewhere.

Example Hospital Queueing Model 5

https://www.hcup-us.ahrq.gov/reports/statbriefs/sb180-Hospitalizations-United-States-2012.pdf

0 30 60 90 120 150 180 2100

15

30

45

60# of Patients in Hospital over

Time

Average Simulation Output

Time (Days)

In this model, arrivals to a hypothetical 50 bed hospital are modeled over 210 days.Patients arrive for treatment on average 10 times per day while the average patient then stays for 4.5 days. According to the Agency for Healthcare Research and Quality, 4.5 days was the average length of stay in the US in 2012. Additionally, when the hospital reaches 50 patients, it has to refer patients elsewhere.Using these parameters, I created this simulation. From this, the time to reach equilibrium as well as the degree of noise around the mean can be evaluated. If the hospital chooses to staff to accommodate the average then they will have congestion problems when the actual level is above the mean.By analyzing this data, a hospital could determine an acceptable level of congestion and staff accordingly.If the patient level drops below the average, the hospital would be able to more properly assess whether staffing should be adjusted and if the adjustment should be temporary or longer term.

Model Variables

10 Arrivals per Day 50 Max Hospital CapacityAvg stay per patient: 4.5 days 210 Day Simulation Length

0 Starting Queue

Example Hospital Demand Shift 6

In this model, shifts in demand can be modeled by adjusting the arrival rates.Modeling demand shifts can be useful to assess the impact of new competition, seasonalities, demand during different times of the day or days of the week, access to an additional health insurance network, or changes in perception that could accompany a renovation project, or poor health quality survey.Additionally, healthcare often faces labor shortages. Nursing shortages are often filled with expensive agency staff or overtime which drives up costs and can decrease quality of care. Anticipating the affects of demand shifts can help avoid these inefficiencies.With a queuing model, the effect of the shift in demand on overall patient levels can be modeled by simply changing the arrival rates. In this case, the arrival rates were decreased in month 2 and then increased in month 3 to simulate the effects on overall patient load.

0 15 30 45 60 75 9020

30

40

50

60# of Patients in Hospital over

Time

30 Day Average Simulation Output

Time (Days)Model VariablesArrivals per day: 10,8.3,12.5 50 Max Hospital CapacityAvg stay per patient: 4.5 days 90 Day Simulation Length43 Starting Queue

Example Hospital Service Shift 7

In this model, shifts in service time can be modeled by changing the average days to discharge.This graph shows how small changes in length of stay of a patient can lead to significant changes in service levels and overall congestion.

As an example, administrative paperwork can take many hours per patient and cause delays in final discharge times. Queueing models can be used to model how changes in these processes can alter overall patient loads and levels of congestion.

Changes in regulations have over the years caused procedural changes that add administration time. The increased congestion as a result of these changes can also be modeled and used to help evaluate the overall benefits or problems caused by these regulatory changes.

0 15 30 45 60 75 9020

30

40

50

60# of Patients in Hospital over

Time

30 Day Average Simulation Output

Time (Days)Model VariablesArrivals per day: 10 50 Max Hospital CapacityAvg Stay per patient: 4.5, 4, 5 days 90 Day Simulation Length

43 Starting Queue

Process Improvement: Fast Track Care Pathways

Queuing models can be used to model the affects of process improvement initiatives on hospital capacity and costs.Care delivery in hospitals consists of complex chains of interdependent processes that vary significantly from one patient to the next. Congestion and capacity problems can lead to problems with correctly triaging critical patients.By using queueing models to analyze admissions and patient data, it is possible to determine arrival rates and lengths of stay for patients with specific conditions and needs and prioritize their care with dedicated hospital resources.Using this information, it is possible to redesign critical processes to facilitate delivering care to specific groups more efficiently without creating service bottlenecks that slow down and worsen care delivery.The model to the right is an example of a hospital dedicating a unit to certain high risk patients. These patients could be shunted off from the main service area to a dedicated unit that could be staffed and equipped to suit these particular patients. Queueing models could be used to analyze these demographics and plan efficiently how to allocate resources to this dedicated critical care pathway. While some patients may experience increased congestion, critical patients would experience much shorter waits.

Surgery

Admit from Cardiology,

Oncology, etc

InPatient Recovery

Discharge to Home

Health for recovery

Eval

Process Improvement: Outsourcing Care Pathways

Discharge

Queuing models can be used to model the affects of outsourcing process improvement initiatives on hospital capacity and costs.An example flowchart here shows the flow from admissions to discharge for a patient needing surgery. The total time they spend in the hospital consists of:• Admission and Pre-surgery Prep• Surgery• Post Surgery Evaluation• In Patient Recovery• DischargePatients from multiple hospital units such as cardiology, oncology or neurology could compete for hospital resources. Capacity limitations could cause congestion in shared units such as in-patient recovery.Because recovery times vary from one patient to the next. It can be difficult anticipating utilization or congestion in these shared units. Queuing models can be used to accurately assess alternatives to hospital resources including home health agencies. In some cases, patients could be discharged more quickly to these alternative settings preventing congestion and aiding in resource planning.

Summary of Benefits to Hospitals 10

Adjustment Times

Regulation Evaluation

Demand Forecasting

Daily, weekly, or seasonal demand can be forecasted using queueing models allowing hospitals to plan revenues and resources more accurately. Ultimately this results in:

decreasing waste from over or under anticipating patient load

preventing congestion delivering better care saving money increasing margins

01

When any factor causes changes to the arrival or discharge rates, there will be a period of adjustment. Patient levels will gradually adjust as the existing patients discharge and new patients arrive. It is possible to see and model this adjustment period. This would be particularly valuable in the following scenarios: New Hospital Opening Opening a new hospital

unit Altering a key process Adjusting for

seasonalities

Hospital regulation can create significant delays in providing services to patients, or prevent timely discharges. Paper work can add hours of work to care delivery. Changes in these process can have significant impacts a significant impact on patient length of stay in a hospital.

Hospitals and regulators can use these models to assess the impact of regulation changes and use them to craft better policies. 0

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Process Improvement

As hospitals use data to evaluate care delivery pathways, queueing models can be valuable in determining the benefit of altering these pathways.

Altering these pathways through dedicating resources for patient fast tracking or outsourcing certain portions of care illustrate how a hospital can save money and/or serve additional patients thus increasing profitability.

02

Implementation Challenges 11

In order to most effectively analyze and use the simulation data, a hospital must determine what level of congestion and waste are acceptable. As an example, the hospital must decide whether to staff at the mean from the output, 10% above the mean, or something else. This decision will have a significant impact on not only hospital expenditures, but also the quality of care delivered to patients.

Congestion StandardThe only data needed for the model are the rates of arrival and discharge. The model can be improved by drilling down to arrival and discharge time by patient demographic or condition. This data is ultimately limited though because the data for arrivals that are turned away from service due to capacity constraints is not routinely collected.

Model Inputs and Accuracy

As with all statistical models, sample size is important. The more data is available, the more accurate these models will be. For smaller hospitals and clinics that do not have large quantities of data, the model loses accuracy and may lead to miscalculations of demand or resource utilization.

Data Sample Sizes Although the algorithm is relatively simple, software systems will still need significant enhancement in order to collect the data needed. Additionally, systems may need significant redesign in order to fully realize the benefits. Modules linking staffing, equipment management and other areas must be linked to the algorithm through an ERP to realize these benefits.

Software Development

Comparison to Other Cases and Industries 12

The classic queueing model forecasts queue length at a bank with one or multiple tellers. This is because arrival times for new customers are random as are the service times for each customer. In theory, any industry that relies on a combination of random arrivals for customers or random service times could be modeled with a queueing model.Some example uses could be:• determining staffing in retail

based on changes in arrival rates in customers based on hour, day or season

• determining inventory and product mix in retail stores based on shifts in anticipated arrival times due to factors such as the weather.

• Optimizing food service at a restaurant where customers arrive randomly and order meals randomly with different preparation times.

• Managing wait times for rides and service levels at theme parks.

Industry CasesHospitality

More than room and board, the hospitality industry sells service. Unpredicted or unusual demand can cause problems in service delivery. These events cause service spikes that, while good for revenue, can cause congestion resulting in room service errors, uncomfortably long waits for concierge, or overall weak service. Queuing models can be used to help plan appropriate resources for these moments.Additionally, if service fails causing arrival rates to slow down, it is possible to estimate the ultimate impact to the bottom line with these models.Finally, opening a new hotel comes with its own unique challenges from understanding how big the hotel should be to anticipating the time before the property is considered at functional capacity. Queueing models can be used to more accurately assess the market opportunity ensuring that projects are executed successfully.

Thank You

References:Green, Linda. Queueing Theory and Modeling. Graduate School of Business, Columbia University, New York, New York 10027,https://www0.gsb.columbia.edu/mygsb/faculty/research/pubfiles/5474/queueing%20theory%20and%20modeling.pdfhttps://www.isixsigma.com/industries/retail/queuing-theory-and-practice-source-competitive-advantage/