# MA0261 – Coursework Assignment

## Problem description

A hospital wants to analyze the performance of its emergency department (ED) and is interested in the following questions, among others:

• How are the waiting times distributed across different patient groups?
• To what extent would varying staff capacity reduce waiting times across different patient groups?

You are asked to help the hospital to answer those questions by building a simulation model of the ED. The following information is available:

## Patients, categories (specialties) and priority levels

Each patient arriving in the ED belongs to exactly one of the following types: (a) surgical inpatient, (b) surgical outpatient, (c) internal medicine inpatient and (d) internal medicine outpatient. The patient types (a)–(d) are further divided into priority levels 1, 2, 3 and 4. Higher priority levels correspond to smaller numbers.

### Patient arrivals

Arrivals occur at random, and the interval between successive arrivals follows the Negative Exponential distribution with parameter λ across priorities and specialties as given by the following table:

 Priority number: outpatient surgical inpatient surgical outpatient internal medicine inpatient internal medicine 1 0.3 per hour 0.1 per hour 0.2 per hour 0.1 per hour 2 0.4 per hour 0.2 per hour 0.3 per hour 0.2 per hour 3 0.5 per hour 0.3 per hour 0.4 per hour 0.3 per hour 4 0.6 per hour 0.4 per hour 0.5 per hour 0.4 per hour

## Resources and their availabilities

### Physicians

There are two types of physicians relevant for the ED: (a) surgeons and (b) internists. Surgeons treat surgical patients and internists treat internal medicine patients. Nurses treat both types of patients.

Shift schedule There is always one physician of each specialty available which means that from 0.00am – 12pm (midnight), one surgical and one internal medicine physician is present. From 8:30am – 4:30pm there is one additional physician available for each specialty. This means that, for example, at 1pm, there are two surgical physicians and two internal medicine physicians available.

### Nurses

There is always at least one nurse at the emergency department. The availability of nurses is shown in the following:

• nurse: 0:00 am - 8:30 am and 4:30 pm - 24:00 pm
• nurses: 8:30 am - 1:30 pm and 3:00 pm - 4:30 pm
• nurses: 1:30 pm - 3:00 pm

### Rooms

Rooms can be divided into: (a) examination/treatment rooms, (b) X-ray rooms, (c) CT-rooms, (d) laboratories (e) and break rooms.

Examination/treatment rooms There are seven rooms in which at most one patient can be treated at a time. Every room has the equipment needed to treat all patients (besides X-ray, CT and laboratory equipment).

X-ray room One room is equipped to diagnose patients by X-ray. Only one patient can be treated at a time. The transport of the patient to the X-ray room and x-raying the patient is done by external staff that does not have to be considered in the model. The process of x-raying requires ERLA(0.0391, 3) hours.

CT room One room where CT scans can be performed. Only one patient can be treated at a time. The transport of the patient to the CT room and the CT scan is done by external staff that does not have to be considered in the model. The process of CT-scanning requires 0.1 + EXPO(0.2) hours.

Laboratories One room for different laboratory activities, e.g. test of blood samples. Several laboratory activities can be performed simultaneously. Only a nurse is needed to perform the laboratory activities. During this time the patients stays in the examination room. The laboratory activities requires LOGN(1.1, 0.813) hours.

Break rooms When physicians and nurses are idle they stay at the break room. Assume that the break rooms are sufficiently large enough to accommodate all physicians and nurses, respectively.

## Patient Flow in the ED

### Patient pathways (Examination → Additional tests → Treatment)

Patients who arrive are examined in one of the seven examination/treatment rooms. Then it is decided whether they require additional tests (e.g. X-ray, lab test). After the examination and any additional tests have been carried out, patients are treated in an examination/treatment room. Finally, they leave the ED. Patients may have to wait in the waiting area until e.g. the X-ray is free or the lab test results arrive. Every patient has an examination and then a treatment. Whether additional tests are carried out between examination and treatment is chosen randomly, see below.

Examination Arriving patients who cannot be treated immediately wait in the waiting area which has an infinite capacity. Patients are first seen by human resources in one of the examination/treatment rooms according to the priority. If two patients have the same priority, the patients are served in a first come first served fashion. If a patients waits for more than two hours, the priority is increased by one priority level.

Priority 1 and 2 patients must be examined by 1 nurse and 1 physician (sequentially, first by the nurse and then by the physician corresponding to whether the patient is a surgical or internal medicine patient). Patients having priority 3 are examined by the corresponding type of physician, only. Patients belonging to priority 4 are examined by a nurse, only.

Additional tests The following probabilities apply for the patients and we assume (for simplification) that these probabilities are independent of the priority level:

 Surgical patients: inpatients outpatients X-ray (yes?) 47% 60% CT-scan (yes?) 1% 17% Laboratory 10% 67% Internal medicine patients: inpatients outpatients X-ray (yes?) 13% 63% CT-scan (yes?) 1% 20% Laboratory 51% 80%

### Treatment

After any lab, X-ray and other test has been finished, the treatment starts. Similar to the examinations, priority 1 and 2 patients must now be treated by 1 nurse and 1 physician (sequentially, first by the nurse and then by the physician corresponding to whether the patient is a surgical or internal medicine patient). Patients having priority 3 are (now) treated by the corresponding type of physician, only. Patients belonging to priority 4 are (now) treated by a nurse, only.

For example, once a surgical inpatient has arrived, there is a 47% chance that he gets an X-ray scan. Alternatively, if an internal medicine outpatient arrives, there is an 80% chance that a laboratory test is required.

Treatment durations by nurses The following durations (in hours) apply for examinations and treatments by nurses.

Surgery: Treatment duration: LOGN(0.254, 0.315)

Examination duration: WEIB(0.494, 0965)

Internal: Treatment duration: BETA(0.749, 1.17451)

Examination duration: 1.97*BETA(0.999, 2.09)

Treatment durations by physicians The following durations (in hours) apply for examinations and treatments by physicians.

Surgery: Treatment duration: LOGN(0.157, 0.173)

Examination duration: BETA(0.754, 1.62207)

Internal: Treatment duration: BETA(1.02, 1.63319)

Examination duration: BETA(0.869, 1.29613)

## Some assumptions and simplifications

• A nurse at the reception who prioritizes patients does no have to be modeled.
• We assume that if a patient is examined by one human resource (nurse, physician or both) a different human resource of the same type can carry out the treatment. For example, if patient 1 is examined by nurse 1, a different nurse (e.g. nurse 2) can treat patient 1. • We do not distinguish between examination and treatment rooms. One room type can be used to carry out examinations and treatments. • For simplification, we assume that patients exceeding five hours waiting time leave the ED (e.g. to another hospital). This shall be independent of the patients’ priority level.
• We simulate 24 hours and patients who are in the system at midnight leave the system. However, if they have experienced waiting time, this should be taken into account when calculating average waiting times.
• For simplification, we can assume that the flow between X-ray, lab test and CT is random. Patients may wait in the waiting area before having e.g. access to the CT.

### Tasks to be Completed for the Coursework

1. Model the process described above in Simul8. Justify any assumption and simplification that you made during the modelling process. (5 credits)
2. Explain how you determine the warm up time and the number of replications (2 credits)
3. Give the following statistical outputs for every hour of the day: (3 credits)
• A diagram displaying the average total waiting time, the average waiting for physicians, the average waiting for nurses and the average waiting for rooms.
• A diagram displaying the average overall waiting for physicians, the average waiting for surgeons, the average waiting for internists and the average waiting for nurses.
• A diagram displaying the average utilization of rooms and average availability of rooms.
1. A hospital manager has two ideas how the waiting time can be reduced. The first idea is to assign an additional nurse from 4 pm - 10 pm. The second idea is to assign an additional internist from 12 (noon) – 4 pm. Do those suggestions make sense? To what extend is overall waiting and waiting for physicians, nurses and rooms reduced by this? (1 credit)
2. Considering the setting described above, what extensions have to be made for a more realistic simulation? (1 credit)
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