Medical testing services’ quality enhancement with discrete-event simulation

doi:10.21203/rs.3.rs-2735245/v1

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Medical testing services’ quality enhancement with discrete-event simulation

https://doi.org/10.21203/rs.3.rs-2735245/v1

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In recent years, the demand for health care has increased owing to the ongoing introduction of new health problems and the intensification of existing illnesses. As research in this area progresses, knowledge becomes the primary motivation for patients to adopt a better quality of life. Consequently, many health units have struggled to develop solutions to manage excessive demand while maintaining a high quality of service to enhance patient satisfaction and boost revenues. One of these tactics is using simulation models as a method of operational research to analyze the flow of patients and predict potential solutions to enhance resource efficiency while decreasing waiting time. This work aims to develop and demonstrate a simulation model that replicates the functioning of a hospital unit. For this purpose, data derived from the health services of the Imaging Department of a private hospital, located in Lisbon, Portugal, were used. This model will replicate the daily flow that this department is subjected to by its patients and personnel and depict how the system evolves due to the management of the department's number of resources and examination rooms. At the end of the analyses, several solutions are presented and discussed, from level to level, concerning improvements in the performance of services, resources and patients. However, only the increase in one unit of examination rooms translates into a huge positive impact on waiting times in the respective queues.

Imaging

Process improvement

Private hospital

Discrete-event simulation models

Patients are increasingly curious and apprehensive about receiving healthcare due to global competitiveness in a rising industry (Li et al., 2015). Manzoor et al. (2019) concluded that increasing health consciousness and affluence in the contemporary world have dramatically increased the demand for healthcare and moved demographic trends toward a healthier way of life. As a consequence, it has been producing a demanding environment that has an impact on local businesses, including (and especially) medical services. Due to the growing rivalry between hospitals, the healthcare relationship has evolved to emphasize the provision of outstanding healthcare services (Bleustein et al., 2014) leading patients to pick the most suitable, efficient, and timely healthcare unit.

To deal with the competition, hospitals and other healthcare units must provide patients with better healthcare services to meet their demands due to heightened consumer expectations and more demand for standard services (Zarei et al., 2015). As a result, hospitals confront significant obstacles. As the need for medical treatment develops, hospitals may dissatisfy patients with prolonged length of stay (LOS) caused by excessive waiting times (Manzoor et al., 2019). Consequently, Wang et al. (2012) agree that a number of these health care entities have made substantial efforts to improve efficiency, LOS, and other issues such as patient waiting times, patient satisfaction, spending levels, utilized capacity, resource consumption and occupancy rate, working conditions, staff morale, and accessibility to error-free and evidence-based treatment (Barjis, 2011).

Healthcare institutions often base their continuous improvement activities on human experiences and qualitative notions. However, such strategies may not deliver the big improvement one would have hoped for, and it is impossible to make a quantitative forecast of the outcome of such an effort (Wang et al., 2012). A simulation model may mimic the process and its dynamics under certain random distributions, show patient flow and care delivery techniques, and provide forecasts for performance evaluation (Gaba, 2004). With the support of such a tool, healthcare management can assess the effectiveness of existing procedures, enable the studying of potential modifications, experience circumstances that would not otherwise be possible without spending a significant amount of money on system development, training, and equipment acquisition, or look into the relationships or trade-offs among system variables (Manzoor et al., 2019). Furthermore, the typical functions of comparing situations or visualizing processes may be expanded upon by using healthcare simulation. A simulation model may be used as a continuous initiative to assess and enhance performance and boost efficiency.

Motivated by this context, the primary objective of this research is to develop a simulation model capable of portraying the functioning of an Imaging Department in a given hospital. This model will be used to test various combinations of internal factor changes, such as the number of personnel resources and facilities of a particular service, to examine the development of system performance under different scenarios. Thus, for each tested combination, it will be possible to predict how the unit shall behave, determining the number of resources sufficient to reach a state that already reflects a good performance for the decision-maker.

In addition, this study complements the evaluation of the effectiveness of human resources in each scenario, as assessed by the occupancy rate, indicating the proportion of execution time that a particular resource was dedicated to an activity. It should be observed that the occupancy rate increases when the amount of a given resource allocated to a given number of activities decreases, but this does not equate to more excellent performance in terms of waiting times, thus indicating a possible trade-off. In conclusion, the constructed model should offer the hospital department helpful information to enhance its resource management. This model's precision matches demand and supply, allowing for the optimal allocation of resources in time and space while reducing queueing times.

This paper is organized as follows. Section 2 details the construction of a Discrete-Event Simulation (DES) model. Section 3 describes and formulates the problem, along with the research questions, hypotheses, and assumptions to be followed and presents the framework that will guide the whole process until the analysis of the results. Section 4 focuses on data treatment in which examine the data given through papers and files to find trends and comprehend patient behavior creating the model. Section 5 details the model construction, demonstrating how the model was configured in SIMUL8 software. Section 6 provides and discusses the results acquired from the preceding step to determine the influence of each testing scenario on the system's performance and the advantage of utilizing such a combination of activities and resources to support the operations of the healthcare unit. Finally, Section 7 concludes the study with the key insights and examples of complementing future work.

Simulation models can be used to evaluate and improve performance and productivity of many (if not all) systems. A simulation model is created to test and integrate with the organization's operational information systems (Darema, 2004). This is done to examine a system's longitudinal behavior and recommend changes while it works and generates dynamic data. Simulation's true value can be realized when simulation models are fully integrated into the current information system applications that support healthcare providers' daily operations (Gaba, 2004).

2.1. Discrete-Event Simulation

Simulation models can be distinguished between static or dynamic, deterministic or stochastic and continuous or discrete. The most used simulation technique is DES, which combines dynamic, stochastic and discrete properties (Mourtzis, 2019), which is the base of this research. DES is a cheap, secure, and fast tool for analyzing complex systems and assessing performance indicators. It is an easy and adaptable computer-based modelling technique for decision-making that simulates the dynamic behavior of complex systems and the interactions between people, communities, and their surroundings (Huerta-Torruco et al., 2022; Karnon et al., 2012), reflecting the system's actual behavior (Urbani et al., 2020). DES is better for modelling complex systems at the individual level than at the cohort level than aggregate models without interaction, such as decision trees or Markov models (Karnon et al., 2012; Zhang, 2018). It shows the flow of individual entities through discrete events (activities) in a system that evolves over time (Law, 2015). Due to resource scarcity, entities must line up between these events, in the so-called queueing networks (Brailsford & Hilton, 2000).

In hospital simulation, (temporary) entities are often patients. This simulation allows traits to affect system progression and resource limits (Huerta-Torruco et al., 2022). Thus, it accurately depicts how patient characteristics affect the health system. Since DES results are samples from a distribution, multiple simulation runs may be needed to accurately measure output parameters. This is especially problematic in unstable systems when the arrival rate is near the service rate or activity time fluctuations are large (Renard et al., 2013). DES models allow patients to be unique and interact with resource supply (Manzoor et al., 2019). Although testing and running these models takes longer, they help model healthcare delivery systems, especially when resources are scarce (Forbus & Berleant, 2022).

2.2. Framework for Developing GE-DES in Healthcare Systems

The framework for Generalizable Discrete Event Simulation (GE-DES) for Healthcare Systems consists of four sections: (I) understanding the issue scenario; (II) establishing the modelling objectives; (III) choosing the model content; and (IV) developing the model. The first section, (I) understanding the issue scenario, is also divided into four steps: (I.1) study the population; (I.2) evaluating the system performance; (I.3) status quo process map; and (I.4) experimental decision factors. The information gathered for each of the four domains is intended to guide modelling operations. The comprehensive process mapping lists relevant components to help selecting model content. The modeler must understand the topic to build an accurate model that addresses the concerns (Robinson, 2008).

To determine if DES is the best option, consumers and subject matter experts must understand and express the problem (Boyle et al., 2022). Section (II), establishing the modelling objectives, describes how simulation research analyses alternative system configurations based on a performance metric to aid client decision- making. Typically, configuration options are constrained by hospital available financial resources, physical space, and laws (Takakuwa & Shiozaki, 2004). Modelling objectives, level of detail, and generality may often be interrelated (Günal et al., 2010), but more details may make a less generic model. However, the model must fit the decision-maker goals, which are classified either as general (run-time and visualization needs, development effort, and re-use flexibility) or modelling (answering "what are the most important issues to be addressed by experimenting with the model?") (Furian et al., 2018). The generic and reusable content of the model is related to the general objectives. The primary function of selected models determines their modelling objectives.

When selecting the model content, Section (III), the framework distinguishes between the model scope – identifying the model boundaries by including or omitting a representation of fragments of the system under investigation as model components, – and the model detail/depth – which focuses on characteristics (Monks et al., 2017), including model assumptions and potential simplifications (Boyle et al., 2022). Before evaluating the extent and amount of complexity of the suggested simulation model, its use should be questioned (Robinson, 2008). Variability, interconnectedness, and complexity of the modelled system determine simulation selection. DES is a widely applicable technique because many operational systems are queuing systems. In addition, the issue scenario, modelling objectives, experimental factors, and responses will determine whether simulation is the right method.

Finally, in section (IV), healthcare decision-making involves complex social interactions, with healthcare service delivery and patient flow management issues being hard to define. Understanding the healthcare process is essential for making sound, defensible decisions and achieving success (Abo-Hamad & Arisha, 2013). Consequently, the issue must be developed from the service delivery perspective. After defining the problem, identifying inputs, outputs, assumptions, and entities, the model can finally be developed. Quantitative data (observations) are saved in databases or recorded on any form of storage media (records), while qualitative data can be collected by direct observation of the system and expert interviews (Steuben et al., 2019). Doctors, nurses, consultants, administrators, and managers are hospital experts (Baesler et al., 2003). The hospital information system records patient treatment route, arrival method, referral type, and discharge or admission time (Abo-Hamad & Arisha, 2013). Staff enter patient data (e.g., administrators, doctors, and nurses through the stages of patient care). Hospital records lack precision and uniformity due to healthcare system restrictions. Before extracting data, data mining must extract the most dedicated group of documents. Expert and clinician observations and interviews are used to gather model inputs.

This chapter explains all of the information provided by the Imaging Department management of a private hospital located in Lisbon, Portugal, regarding the unit under study for developing the plan and subsequent conceptual model.

3.1. Service Functioning Description

The Imaging service offers about ten types of examinations. The decision-maker suggested Mammography, X-Ray, Magnetic Resonance Imaging (MRI), Computerized Tomography (CAT), and Ultrasound to reduce waiting times.

The department has two Mammography equipment, four X-Ray appliances, six MRI machines, three CAT devices, and eight Ultrasound scanners. Each equipment serves a single patient. Each office has its own exam equipment. Regarding the health personnel assigned to supervise and conduct each examination, it is necessary to have: one technician in Mammography, X-Ray, MRI, and CAT; one assistant in X-Ray, MRI, CAT, and Ultrasound; and a physician in Ultrasound. In addition, only a secretary is required to perform the check-in of the patient.

Once entered into the system (Imaging service), the patient gets a ticket at a check-in location to start the journey. The unit has two entrances with check- in counters. Examinations are inserted in zones A (Mammography, XRay, MRI, and CAT examinations) and B (Ultrasound). Patients must enter the entrance room concerning the respective zone. A patient with zone A and zone B examinations will enter any zone and be directed to a waiting room after check-in. In case of the patient enters zone A but has an examination scheduled in zone B, the check-in service is usually done there, but after that, the patient will go to the other waiting room. The patient must wait until the healthcare personnel responsible for the examination summoned them. The duration of each exam is predetermined and corresponds to the time slot provided for each patient. This period will include completing the exam and all associated activities (including the preparation of equipment, instructing the patient, and filling out the medical report). By the end of this period, the patient will leave the office to make room for the next one. The patient's number of examinations does not affect the prescribed duration for all examinations except MRI, i.e., if the patient has two MRI examinations, the specified period will be doubled. After leaving the office, the patient can wait for another exam or leave the unit. In the first scenario, they will go to the waiting area for the other exam and repeat. The patient does not return to the queue because all scheduled examinations of the same type are done simultaneously.

3.2. Objectives

As stated before, the decision-maker's primary objective is to decrease patients' general average waiting time in the examination queue in order to improve their satisfaction. However, sub-goals can be set to help achieve the main goal. Sub-objectives may include reducing the average queueing and maximum time for check-in and examination activities separately, while improving the efficiency of resources. Increasing the number of facilities inside the unit to have a more significant patient flow (fewer patients in queues) so they can be treated more quickly may also boost patient satisfaction and increase the occupancy rate of some resources to increase profitability.

3.3. Key Performance Indicators

Performance metrics must be established after problem definition and decision-maker’s objectives. These will determine the system's performance throughout the project by assessing the degree to which the predetermined objectives have been met. Like the pre-objectives, the KPIs must measure service quality to reduce waiting times in service queues. The average and maximum queueing time for each check-in service or examination and resources’ occupancy rate are used for this purpose. These metrics will be reviewed after several system runs to determine if the system’s performance is improving or deteriorating.

3.4. Activities

Activities represent a period with a given duration. They refer to a group of operations that modify the state of an object. Although activities and events are often used interchangeably, they have distinct meanings in the context of simulating. An event is a change in the system's state caused by an activity, with the event serving as the consequence and the activity as the cause. Each activity that will be analyzed is associated with each service under study. In this case, there will be seven activities, corresponding to: Mammography, X-Ray, MRI, CAT, and Ultrasound examinations, plus the Imaging and Ultrasound Check-ins.

3.5. Assumptions

To design the model, some basic assumptions were needed to guide its construction, analysis, and solution-finding. The issue was formulated without mentioning these assumptions, which helped create the model and assume the right conditions for modelling.

A patient goes through the check-in procedure only once each day for an unlimited number of examinations.

A patient with records on several days will be considered as one new patient each day they attended the unit.

Every patient exits the department after completing at least one examination.

The patient waits in the queue for an examination for as long as required.

The patient entering the unit on a particular day is required to be seen on the same day and, therefore, must exit the unit on the same day, never being allowed to remain there from day to day.

A patient cannot perform simultaneously more than one examination.

The duration of each examination activity corresponds to the slot provided for the patient (and its associated resources).

A patient leaving an examination may proceed to the exit or to another waiting queue for another examination, without returning to the check-in area.

If a patient has several examinations of the same sort, they will be done inside the same time slot with no change in duration. In the case of MRI, the activity duration is multiplied by the number of times the patient has more than one examination in it.

3.6. Sketch of the Conceptual Model

From a macro perspective, the model to be developed may be divided into two sections. The first section corresponds to the time between the patients' entry (the period corresponding to the withdrawal of the ticket from the kiosk) and their departure from the check-in service to join one of the queues heading to an examination. Consequently, the second section of the model addresses the remainder of the journey, from the patients' departure from the check-in until they depart from the system, with the transit of patients through various examinations as its primary focus.

In the first section of the model, when patients arrive at the service, they are directed to either the IS (Imaging Service) or the US (Ultrasound check-in). In any case, there is a check-in activity service linked to each zone and a queue before each of these activities. After abandoning the check-in activities, patients are sent to one of the queues for one of the five facilities. There are four facilities within IS, corresponding to Mammography, X-Ray, MRI, and CAT examinations, although only one facility for US, which is the ultrasound itself. After the patient has left the examination activity, they will have the option of proceeding to another queue of an examination or exiting the system. Figure 1 portrays the life cycle diagram of a patient entering the service.

This phase focuses on collecting data and how they were treated to implement in the conceptual model.

Before beginning the data treatment procedure, it is essential to be aware of the data privacy, and security law created and approved by the European Union, which became effective on May 25, 2018, known as General Data Protection Regulation (GDPR). The legislation imposes requirements on companies worldwide as long as they target or collect data on European Union citizens. If this law's privacy and security criteria are broken, penalties of up to tens of millions of euros may be applied. Considering the mentioned regulation and the topic of this work, it is essential to note that every clinical study must adhere to the regulation. Since the provided data does not include any meaningful information about the users, their anonymity is likewise protected. One could not directly tell them that the information about their movements inside the unit is being utilized. Therefore, it is only necessary to maintain the confidentiality and anonymity of data concerning the provider, which is the hospital board of directors in this situation.

4.1. Patient Entries

The entries of the patients concern the patient's arrival at the unit depending on the time of the day. The given data included the moment each patient arrives at the unit, more specifically, the timestamp (composed of date and time) the patient receives the ticket for check-in. Since obtaining the check-in ticket is a zero-duration action, this seemed to be a viable method for defining the instant a patient enters the system. A few more data procedures enabled the arrival date to be matched with the day of the week, revealing a greater influx of patients on weekdays than on weekends.

It was revealed that the department started accepting patients at 07:30 am each day (Monday through Friday) and continued until 8:30 pm. Therefore, since the system operates 13 hours/day, it was necessary to divide it into several time intervals to better analyze the events that act in each of them and a way to maintain tracking control for when validation occurs later on. Figure 2 presents the histogram regarding the number of arrivals as a function of the extracted timestamps divided into 112 bins (classes), i.e., the 13 hours of labor split into 112 time periods, which gives approximately seven minutes for each time slot. Data used to construct this histogram refer to all entries in the system from January to July 2022, and the size of each bar regard the average of all observations for each time interval (bin).

Upon the first examination of the plot, it is apparent that the number of arrivals begins to rise gradually over time, i.e., the rate of patients per time interval rises until it stabilizes for some time and then drops slowly until half day of labor. In other words, the time gap between two successive arrivals decreases progressively until it stabilizes, at which point it begins to climb. The rate of patients at each time interval is inversely proportional to the period between successive arrivals, as the longer this delay, the fewer patients come per interval (Ranjkesh et al., 2019). During the second half of the laboring day, the procedure is similar to that of the first half. However, the stabilization is less pronounced since it starts to drop very slowly and then gradually intensifies, although always to a lower extent than in the first half.

One way to study arrival periods in isolation is to cluster the areas of the graphic that have typical behavior and studying them separately. In this study, it was easy to do this separation by eye, as shown in Fig. 2. Therefore, there are seven areas, two smaller than the remaining. By further examining the data, it was possible to set proper intervals for the samples limited by each area in red. In a simplified way of doing those computations, one can check that the first region of the graphic contains twelve bars. As indicated earlier, each bar is equivalent to seven minutes. Then, for the first region, the interval has a length of 84 minutes which might be extended to 90 minutes (one hour and a half). The procedure is repeated for the last region of the graphic. This way, three of the 13 hours were allotted to the intervals, leaving ten hours for five intervals or an average of two hours for each interval (or 120 minutes).

For each workday, the number of arrivals within each time slot was counted and stored in a vector comprising these numbers. Each vector element represents the number of arrivals from that time slot on a given day. For example, the vector corresponding to the first time slot (07:30 AM – 09:00 AM) could be represented as 𝐴𝑉1 = {39, 67, 4, 49, 43, 53, 60, …, 55, 54}. This means that on the first day, there were precisely 39 patients that arrived within that time slot, 67 on the second day, and from then on. Along with the vector containing the arrivals for each time slot, it is essential to keep the effective hours that the department had worked for each day. With this, a second vector, relative to the first time slot, containing this information (converted to decimals) could be represented as 𝑇𝑉1 = {1.47, 1.43, 0.09, 1.43, 1.33, 1.42, 1.43, …, 1.38, 1.40}. With the two vectors, it was possible to find the rate at which the patients came to the unit (expressed in patients/hour) by simply dividing the number of arrivals by the hours the department had been working during the specific time slot. A third vector was then emerged storing those values, 𝑅𝑉1 = {27, 47, 44, 34, 32, 37, 42, …, 40, 39}. However, it was necessary to define a probability density function for the inter-arrival time. It is well known that, should the number of arrivals in a given moment follow a Poisson distribution with average λ (e.g., customers per hour), then the time between two consecutive arrivals follows an exponential with average 1/ λ (e.g., hour) – the reciprocal is also true. Therefore, each vector was subjected to an examination to determine whether or not its data were consistent with a Poisson distribution. Table 1 presents the empirical average arrival rate and inter-arrival time per time slot. A simple chi-square test was applied to the data of each time slot and six (out of seven) times the null hypothesis was not rejected at the significance level of 5%, with p-values larger than or equal to 0.07. In one case (third time slot), the p-value associated with the null hypothesis was 0.02, thus it may not be rejected it by lowering the significance level to 1%. After defining the time slots and calculating the average inter-arrival time for each, the proportion of patients who proceeded to IS or US was determined within each time slot. The results are presented in Table 2.

Table 1

– Rate of patient per hour and inter-arrival time, in minutes.
# Time Slot	Interval	Arrival Rate (Patients/h)	Inter-arrival Time (min)
1	07:30 AM – 09:00 AM	35.37	1.70
2	09:00 AM – 11:00 AM	57.22	1.05
3	11:00 AM – 01:00 PM	47.49	1.26
4	01:00 PM – 03:00 PM	41.45	1.45
5	03:00 PM – 05:00 PM	51.53	1.16
6	05:00 PM – 07:00 PM	39.00	1.54
7	07:00 PM – 08:30 PM	13.45	4.46

Table 2

– Proportion of patients that proceeded to one of the services per time slot in the moment of arrival.
	Percentage of patients
Time Slot	Imaging (IS)	Ultrasound (US)
07:30 AM – 09:00 PM	81.19	18.81
09:00 AM – 11:00 PM	77.79	22.21
11:00 AM – 01:00 PM	77.86	22.14
01:00 AM – 03:00 PM	83.57	16.43
03:00 AM – 05:00 PM	77.01	22.99
05:00 AM – 07:00 PM	77.96	22.04
07:00 AM – 08:30 PM	92.71	7.29

4.2. Check-In Service

In addition to the patients' entrance timestamps in the unit, each patient's admission duration at check-in was also accessible. As indicated in the preceding section, all check-in admission durations of less than two minutes were deemed outliers and were eliminated with the agreement of department managers once they most likely referred to a patient who had previously been registered into the system. When making the first analysis of these durations, one might verify that the bulk of these durations falls between five and 10 minutes. Moreover, when looking into more detail, some durations above 20 minutes were observed. These values are a little out of place. They might be related to the fact that the system has considered the period during which an employee was absent from the workplace after finishing their shift with the previous patient and before returning to the workplace to resume their job. Other possibilities could be the overlapping durations of different patients or the existence of dead time between one patient leaving the check-in activity and the next one, who comes considerably later. This second scenario is possible during periods with a lower overall number of patients. Consequently, it was necessary to take precautions to ensure that these values were not taken into consideration to avoid producing disruptive outcomes. Subsequently, it was necessary to detect and eliminate outliers from the dataset. In this research, statistical principles such as the computation of quartiles and the upper limit calculation were used (the lower boundary was not calculated since there are only acceptable values above two minutes). Consequently, the time set followed a log-normal distribution with a mean of 7.53 minutes and a standard deviation of 3.65 minutes. The goodness of fit was also assessed through a chi-square test (p-value = 0.74), after the Akaike information criterion has concluded that the log-normal distribution was the best fitting function over a complete set of families of probability density functions.

4.3. Imaging and Ultrasound Services

This step aims to determine the probability that a specific patient will use each service. These calculations are crucial because they will enable the simulation to determine whether the patient will choose one route or another depending on the actual use of each service.

Table 3

– Number of patients who used two different services according to the combined pairs
# Combination	Services	Number of Patients	Total Patients	Intersection Probability
1	X-Ray & MRI	1,366	56,045	0.024
2	X-Ray & CAT	728		0.012
3	X-Ray & Mammography	531		0.009
4	X-Ray & Ultrasound	3,078		0.055
5	MRI & CAT	332		0.006
6	MRI & Mammography	63		0.001
7	MRI & Ultrasound	311		0.006
8	CAT & Mammography	185		0.003
9	CAT & Ultrasound	1,029		0.018
10	Mammography & Ultrasound	5,917		0.106
11	X-Ray & Exit	12,985		0.232
12	MRI & Exit	7,765		0.139
13	CAT & Exit	9,238		0.165
14	Mammography & Exit	161		0.003
15	Ultrasound & Exit	14,432		0.258

The total number of patients that entered the hospital for any examination was 56,045. In terms of the overall number of days examined, a total of 32,254 patients entered the IS, which is about 58% of the general patients. The Imaging Service (IS) is divided into four facilities, which received 17,885 X-Ray patients, 9,627 MRI patients, 11,119 CAT patients, and 6,102 Mammography patients. Consequently, the percentage of patients who underwent each of these facilities might be estimated by dividing the 49 number of patients that entered each facility by the total patients that entered the IS, being approximate 55%, 34%, 30%, and 19% for X-Ray, CAT, MRI and Mammography, respectively. The same thing might be done for the Ultrasound Service (US). During the period under review, 23,991 patients used this service, which is about 42% of the overall patients. Since the percentage of usage of each Imaging facility differs from the one in the US, it was required to adjust them to the same denominator, i.e., the overall patients. This resulted in 11%, 32%, 17%, 20% and 42% of usage for Mammography, X-Ray, MRI, CAT and Ultrasound, respectively.

The objective is also to determine the proportion of patients who move from one service to another without leaving the hospital unit. Following that, all combinations of two-by-two services were included with the exact number of patients who underwent these two services and the probability of intersection was computed, as demonstrated in the Table 3.

Through the individual probabilities of the usage of each service as well as the probabilities of the intersection of the usage in two different services, one was able to arrive at the calculation of all conditioned probabilities that will serve as input data for the simulator. What is important to know is the likelihood that a certain patient may switch services. Mathematically speaking, one wants to determine the chance of a patient, for instance, going to a Mammography after having had an X-Ray. Since these probabilities cannot be determined just by their intersection, because the likelihood of one event may vary depending on the occurrence of another event first, it is reasonable to evaluate the chance of occurrence of event 𝐴 given the occurrence of event 𝐵. This indicates that the chance of a patient leaving a Mammography for an X-Ray may vary from the probability of a patient leaving an X-Ray for a Mammography, for example. The conditional probabilities are presented in Table 4.

Table 4

– Conditional probabilities of a patient leaving one service and entering another/leave the unit.
1st Examination	2nd Examination	Intersection Probability	Conditional Probability	% of Patients
X-Ray	MRI	0.024	0.075	8
	CAT	0.012	0.038	4
	Mammography	0.009	0.028	3
	Ultrasound	0.055	0.172	17
	Exit	0.232	0.719	72
MRI	X-Ray	0.024	0.142	14
	CAT	0.006	0.035	4
	Mammography	0.001	0.006	1
	Ultrasound	0.006	0.035	4
	Exit	0.139	0.808	81
CAT	X-Ray	0.012	0.061	6
	MRI	0.006	0.030	3
	Mammography	0.003	0.015	2
	Ultrasound	0.018	0.091	9
	Exit	0.165	0.833	83
Mammography	X-Ray	0.009	0.082	8
	MRI	0.001	0.009	1
	CAT	0.003	0.028	3
	Ultrasound	0.106	0.972	97
	Exit	0.003	0.028	3
Ultrasound	X-Ray	0.055	0.130	13
	MRI	0.006	0.014	1
	CAT	0.018	0.042	4
	Mammography	0.106	0.250	25
	Exit	0.258	0.608	61

In terms of durations of each activity, the decision-makers and experts decided that each activity time would be well modelled by a fixed value that, of course, depends on the service: 10min for an X-Ray, 15min for a mammography or an ultrasound, 20min for a CAT scan, and 40min for a MRI.

Once gathered, data (from observations and interviews with experts) were combined to conceive the conceptual model using the SIMUL8 software, where processes and activities could be defined. The conceptual model guides the actual simulation model, which contains more system detail. Besides that, it serves as a communication mechanism for validating the model.

5.1. Simulation Clock

Before beginning the simulation, it was important to establish the system's unit of measurement. Given that most of the information provided is presented in minutes and that the pre-established performance metrics also use this time unit as a reference, it was determined that the minute would be the most appropriate unit of measurement for evaluating the acquired outcomes. Concerning the simulation's running hours on working days, according to the information provided, it was specified that the simulation period begins each day at 07:30 AM and ends at 09:00 PM. The duration of each working day is 13 hours and 30 minutes.

5.2. Arrival Shifts

A total of seven shifts were created with the weekly pattern option enabled. The shifts created follow the same logic as the time intervals of the arrival time slots, so a specific shift covers each time slot.

5.3. Resources

Throughout the simulation, the resources serve as permanent entities that the system retains and employs for numerous actions. An activity that relies on them demands using personnel (or equipment not exclusive) to this activity alone. In general, a resource is shared by several activities, with the others on active standby until it becomes available for use by another activity. Technicians, assistants, physicians and secretaries will be able to comprehend the four distinct and potential resources described in this research for application in examination-related tasks. Therefore, four resources were created. However, knowing their roles in the system is irrelevant to the problem. The only essential information is to understand how many resources of each type are required for each activity that depends on them and when they are no longer needed.

5.4. Labels

Labels allow for adding more control to the system. Labels can be attached to work items when they enter the simulation or at any point. The label's value can be tested and/or changed at any workstation. In this context, the labels will serve as a trail for each patient, indicating the latest examination they passed as well as the other tests where they have gone, with the latter incrementing the value of the exam-specific label each time the patient enters this activity. Since the patient should not re-enter the queue for an examination in which they have previously participated, this will also function as a debug to determine whether the patient has repeated an examination. Using labels, conditions will be set that prevent these incidents from occurring.

5.5. Activities

Below is a list of model activities and how they were modelled.

1. Arrivals: This activity receives a distribution that specifies the inter-arrival time at each time slot, so all patient arrival information, including temporal distributions, must be stored here. This activity is, therefore, time-dependent, with seven smaller exponential distributions comprising each time slot. The parameter of each smaller distribution is expressed in the last column of Table 1. Here, all the created labels were set to zero along, to indicate that the patient has not been into any kind of examination. When designing the system, only one entry for all patients was considered. Next activity will cover dividing patients by entry zone.

2. Shift and Sep: Patients will be transferred to one of seven fictitious activities based on their arrival time. Thus, each shift created had to be assigned with each of these shift activities, meaning each activity will only take patients during its assigned shift. Since percentages vary by slot, this was necessary to accommodate patients who choose Imaging or Ultrasound. Thus, each Shift activity is linked to two fictional activities, Sep_I or Sep_US, with percent mode enabled in the routing out of the first. Table 2 shows the Shift activity input values according to the time slot.

3. Wait_Check_In: After getting the ticket at the checkpoint, the patient waits to be called to the service for check-in. Two queues of this type, Wait_Check_In_I and Wait_Check_In_US, one for each zone, are connected to one of the immediately preceding activities, Sep_I and Sep_US, respectively, and only pass patients designated for the respective services.

4. Check_In: Here, the log-normal distribution specified in the section 4.2. of this paper is employed for the duration of both activities. The resource responsible for this activity are the secretaries, with the standard option to pick it up and release it when the patient exits the activity.

5. Go_Wait_Exam: This fictitious activity follows the patient from the check-in desk to the examination waiting room. In reality, a patient may be served at a service check-in and wait in the opposite waiting room of the other service with a 15% probability defined in the routing out dialogue. Patients who leave the Go_Wait_Exam_I activity have four pathways that match the four exam waiting rooms. Each examination's percentages had to be adjusted to the main service's patient count. Thus, 19%, 55%, 30%, and 34% of Imaging patients go to Mammography, X-Ray, MRI, and CAT, respectively, while 42% go to Ultrasound. This activity's routing dialogue includes the percentage values provided.

6. Wait_Exam: There are five queues of this kind, each preceding the appropriate examination-related activity.

7. Exam: There are five activities of this kind, each corresponding to one examination with the respective duration as a fixed distribution and with connected to the corresponding resources. The resources are required and released here, apart for the MRI, which is released ahead. The labels relatively to this examination is set to one. The labels relatively to the last examinations of the other activities are set to zero to indicate that this was the last activity the patient has been.

8. Dispatch_MRI: This activity is responsible for moving the patient back to this examination, with a probability of 32%, which corresponds to the probability of a patient repeat that examination.

9. Other_Exam: The patient that finishes its examination, will have the option to moving to another queue or leave the unit. The conditional probabilities previously mentioned in section 4.3. are applied in the routing out of this activity. The MRI resources are released here.

10. Mammo, XRay, MRI, CAT and US: These fictitious queues are imposed only to apply visual code to prevent patients from entering the waiting queues where they have already been. In case a patient is sent to one of these queues, they will go back to the previous activity to re-sent them back again.

11. Exit_Mammo, Exit_XRay, Exit_MRI, Exit_CAT and Exit_US: This exit point receives the patients who abandon the unit.

12. Go_Wait: This fictitious activity directs the patient towards the specific queue to perform other examination.

The final model implemented in Simul8 software is exhibited in Fig. 2.

5.6. Model Validation

To validate the model, it was necessary to run the simulation for a sufficiently large time and pause to confirm that the system's behavior adheres to the previously specified requirements during various times of the day on different days. Thus, a simulation monitoring was conducted with a running period of a week, beginning at 07:30 AM on Monday and finishing at 08:30 PM on Friday, with the primary purpose of detecting any implementation faults for different operational conditions.

During the monitoring procedure, the pauses made at various times and days helped to validate some of the requirements needed by the decision-maker; therefore, the implemented model was subjected to four verifications. The first verification had to do with the average hourly arrival of patients at the unit. The second verification was related to the portion of patients who went to the check-in of each zone (IS or US), depending on the shifts in which these arrivals occur. The third with the proportion of patients who go to each of the queues for the respective examinations after leaving the Imaging check-in service. The fourth and final verification tried to ensure that patients do not return to the queue where they were before to complete a test if they have taken other routes than departing immediately after completing the examinations.

The rate of patient arrival to the service between 07:30 AM and 09:00 AM is 35.37 per hour, as stated in Table 1. Around 1.5 x 35.37 should come within this time slot, resulting in an average of 53.05 patients. The expected arrival value for the hour between 3:00 PM and 5:00 PM is 2 x 51.53, which equals an average of 103.06 patients. From 7:00 PM to 8:30 PM, there is an arrival rate of 13.45 patients per hour, averaging 1.5 x 13.45 = 20.18 patients. Using the average predicted values, the system was executed throughout the above-described execution time. The values on the number of patients who arrived within each shift were all close to the expected values.

In the scenario described, a patient's likelihood of visiting one of the check-in stations depends on the arrival shift. For instance, in the 1:00 PM – 3:00 PM shift, the probability that a patient would visit the Imaging check-in service or the ultrasound check-in service is 83.6% and 16.4%, respectively. Given that the arrival rate for that shift is 41.45 patients/h, a total of 2 x 41.45 = 82.89 patients is expected to arrive at that time. The system was executed for five days, and the number of patients who moved to IS or US within that time period was tallied to compare with the theoretical values. It should be emphasized, however, that we are not dealing with fixed values. Thus, the percentages may change but will remain near the predicted values.

Similar to when patients are distributed to one of the check-in stations after entering the system, the percentage of patients going to each of the queues after leaving the check-in follows the same logic. For this reason, the premise for the third verification is similar to what was done with the second one, along with the explanation provided regarding the allowed values being approximately in the predicted range. In this particular scenario, the number of patients in queues was tallied at the end of each day. To eliminate possible patients going to the queue for each examination by other paths, not directly from the check-in service, it was necessary to perform a simulation with these paths cut, thus leaving only the values of interest.

With the apparent possibility of the patient moving to a queue where they have been previously, through other examinations, it must be verified whether or not this situation occurs. As previously described in section 5.5., code snippets were written in the software's Visual Logic so that the passage to one of the queues where the patient has previously been denied, making their return to the last activity to choose a different path. Thus, the system was run within the allocated time, with periodic execution pauses to consult the labels of patients either in the queue or engaged in an activity connected to the examination shortly after. In all the pauses to which the system was subjected, it was checked that the labels assigned to the activities where the patients were all contained the zero value, indicating that the patient had not yet completed these activities while inside the system having the system been validated.

5.7. Minimum Number of Resources

When examining the system's long-term behavior, the results produced can be affected by the starting circumstances specified, hence modifying the system's behavior. Define a strategy to prevent the scenario and assure the quality and dependability of the outcomes. In order to tackle this issue, a warm-up period will be considered, during which no data will be taken, guaranteeing that the results will only pertain to the model's stable states. To define this period, it was necessary to deduct the minimum number of required resources.

This number reflects the minimum resources the system can minimally work (without meaningful congestion cases) and serves as the basis for future resource alterations. The minimum number of resources needed to conduct each of them in each activity will be measured. When determining this number, one of the variables must be considered is the amount of time devoted by each resource to a particular task. As soon as a patient begins an activity, resources start to be used and are released when the patient departs after the activity's length has passed. Thus, for each activity, each resource spends the same amount of time as the activity's duration.

Table 5

– Parameters to determine the minimum number of resources for examination activity
Activity	Resources	Average Arrival Rate (patients/h) – Simulated	% of Patients – Simulated	Duration (min)
Exam_Mammo	1 Technician	38.39	19	15
Exam_XRay	1 Technician 1 Assistant		42	10
Exam_MRI	1 Technician 1 Assistant		22	40
Exam_CAT	1 Technician 1 Assistant		22	20
Exam_Ultrasound	1 Physician 1 Assistant		45	15

Taking the number of patients who entered the system in the first day and dividing it by the working hours it yields an average rate of 38.39 patients/h. Although the acquired average arrival rate value is universal, the number of minimal resources for each activity cannot be computed using this value since there are patients who are split among the current activities and should not be considered for this reason. Therefore, the proportion of total patients who travel through each activity must be determined through the simulation results and multiplied by the average arrival rate. This final value is multiplied by the time each resource spent on the relevant activity and then divided by 60 to get the minimal number of resources required per activity each hour. The parameters to compute the minimal number of resources required for each examination activity are summarized in Table 5.

For Mammography, the calculated minimum number of resources (technicians) is 1.823. This value should thus be rounded up to the closest integer. Therefore, the minimum for this activity to work decently would be to have two technicians (equivalent to two activities of this type taking place simultaneously). The same reasoning used in this activity is also applied to the other services. For the X-Ray, the minimum number of resources is three, i.e., three technicians and three assistants. For the MRI are six of each, i.e., six technicians and six assistants. For the CAT, three of each, i.e., three technicians and three assistants. Finally, for Ultrasound, there should exist five of each, i.e., five physicians and five assistants.

The minimum number of resources of a certain kind in the system is equal to the sum of the minimum resources present in each activity. In total, there will need to be at least 14 technicians, 17 assistants and six physicians, which translates into having simultaneously two activities for Mammography, three for X-Ray, six for MRI, three for CAT, and five for Ultrasound.

The same procedure needed to be done for the check-in activities since these activities also required resources to be carried out. Unlike the other resources previously listed for the examinations, these resources are exclusive for the check-in activities, with no resource sharing across activities of different types.

Table 6

– Parameters to determine the minimum number of resources for each Check-In activity.
Activity	Average Arrival Rate (patients/h) – Simulated	% of Patients – Simulated	Duration (min)
Check_In_I	38.385	80	7.527
Check_In_US	38.385	20	7.527

For the Imaging check-in activity, the minimum number of human resources required for the system to work appropriately is four. On the other hand, for the check-in activity in the ultrasound facility, the minimum number of activities required for the system to work correctly is one. In conclusion, the system will require five employees to meet the minimum conditions for working on these two activities.

5.8. Warm-Up Period

After determining the minimal number of resources required for each activity, the next step was to set these number of resources and replicas of activities within the system and determine when the occupancy rate of each resource along with the average queueing time in the corresponding activity began to stabilize (KPIs). Therefore, multiple runs were conducted weekly to determine that period of stabilization. The results of the simulation after 13 weeks of execution seem to be entirely stable after a period of small oscillations relating to the average waiting times. On the other hand, the occupancy rates of the resources have stabilized since the first week. Due to the need of accounting for an extra safety margin to guarantee those beginning circumstances do not impact the results as much as possible, it was determined that the warm-up period would be 26 weeks (twice the 13 weeks verified), which corresponds to 105,300 minutes.

5.9. Termination Conditions Definition

The results collection period must be determined after the warm-up period. Thus, it is supposed to examine, across different time intervals, the variation related to all the KPIs considering the resources and activities at their minimum. To ensure confidence interval precision around the simulation results' estimated mean, each experiment's number of trials was first determined. Thus, the 5% mean value accuracy is applied. This implies that a genuine KPI result will fall within its minimum and maximum values 95% of the time. The Simul8 software demonstrated that three trials were enough, but five were added for safety. Therefore, each data collection period will include five runs from the same experiment, each with some variability due to random numbers regulating its behavior. The mean and values within each KPI's confidence interval will be displayed in SIMUL8's results manager. Then, the collection periods for results are arbitrarily set at 5, 10, and 20 weeks. This duration was altered following each trial's five executions. The objective is to compare KPI confidence intervals across periods of data collection.

After running the system for each period, the variations between each KPI's upper and lower bound were computed. Observations indicate that the gap of each confidence interval decrease as the period of data collection lengthens, indicating that the system becomes more stable over time. Therefore, it makes sense to analyze the system over longer collection periods. Long-term system operation necessitates consideration of the CPU's computationally intensive tasks. Thus, it was sufficient to establish a collection period of 20 weeks.

Many software-based experiments were conducted, based on the data collection period previously set, to examine all possibilities regarding the quantity of human resources to be compared and assessed. In the initial phase of the analysis, the number of replicas and resources will be kept at minimum values, except for the studied activity and its associated resources.

As the number of minimum resources is based on the minimum number of activities working in parallel (so the system can minimally work without considerable congestion), each type of activity has defined a collection of values corresponding to the exact number of replicas of that activity that will be used for each experiment run. Along with replicas, resources are defined. For each execution, a pair is determined by the number of replicas and resources. These values are the resource's "quantity."

Because the number of resources is defined by the number of replicas, it can be proven, using check-in-related activities as an example, that when the number of resources increases with the same number of replicas, the resource's utilization rate will decrease (there are more resources for the same number of replicas, reducing the load). When there are more replicas than resources, resource utilization increases (the load increases between them). To ensure sufficient resources for all activities, it is necessary to proportionally adjust the number of resources and replicas (although the aspect of decreasing resources and maintaining the same number of replicas will be evaluated later). This helps determining how the resource occupancy rate changes as the number of resources and replicas increases proportionally. In each of the next three runs, the number of replicas and resources will increase by one unit while resources are reduced to a minimum. Each iteration retrieves all three KPIs. An analysis will determine the system's optimal levels.

Due to the model's two parts, the above procedure must be performed separately, as one may congest the other or the first's congestion may affect the second. Due to the activities' varying durations, this can happen in a system with a single entry point and a queue. If the first activity lasts longer than the second, its departure rate is lower than its entrance rate, leaving the first activity to condition the rest of the system (bottleneck). The wait between these two activities will be shorter than before the first. The number of replicas (and, ultimately, resources) is also a key parameter controlling the system's flow. To accurately interpret KPI values after adding replicas and executing resources, the preceding activities must have a natural flow. If check-in activities are thought to be "bottlenecks" in the system, exam waiting lines would not behave similarly to what really happens. It would be a bad practice to believe the KPIs results are the real ones and then make system adjustments based on them. To avoid this issue, it was suggested that modifications to exam activities be evaluated and assessed while keeping an optimal number of replicas and resources for check-in activities so they are not conditioned.

6.1. Changes in the Imaging Check-In Service

Initially, four activities of this type were designated as minimum resources and four secretary resources, one for each replica. Under the starting conditions of limited resources and a 20-week execution period, the average and maximum queueing time for this activity were observed to be 63.05 minutes and 219.72 minutes, respectively.

With the addition of one more replica, and, consequently, one more secretary, the average queueing time dropped to 10.60 minutes and the maximum queueing time to 82.03 minutes, demonstrating 83% and 63% reductions, respectively. With the increase from five to six replicas, the average waiting time decreased by 83% to 1.77 minutes, while the maximum waiting time decreased by 64% to 29.60 minutes. If a seventh replica is added, the average queueing time was reduced by 69% to 0.55 minutes, while the maximum queueing time was reduced by 41% to 17.33 minutes.

For this activity, increasing the number of replicas from four to five had the same impact as increasing it from five to six, i.e., the benefit rose correspondingly. The effect is diminished when the number of replicas increases from six to seven. The initial occupancy rate of the resources present in this activity was 96.10% and, depending on the increase in the number of replicas and resources, decreased to 82.53%, 70.74% and 61.90%, respectively. Table 7 summarizes these results.

For these reasons, the decision-makers concluded to be preferable to parameterize the system with six replicas (experiment #2). In fact, although the waiting time considerably decreases when the quantity of secretariats increase, the same does happen with the resource’s occupancy rate (which should be neither very high to avoid congestion nor very low to avoid small productivity rates). Therefore, we are beholding a trade-off between customer satisfaction and asset profitability. Nonetheless, the decision-makers concluded that the best compromise solution would be six secretariats for the imaging check-in service.

Table 7

– Evolution of KPIs with the different experiments for the imaging check-in service.
Imaging check-in service	Initial conditions (min. resources = 4 secretariats)	Experiment #1 (5 secretariats)	Experiment #2 (6 secretariats)	Experiment #3 (7 secretariats)
Average waiting time	63.05 min	10.60 min (-83%)	1.77 min (-83%)	0.55 min (-69%)
Maximum waiting time	219.72 min	82.03 min (-63%)	29.60 min (-64%)	17.33 min (-41%)
Resource’s occupancy rate	96.10%	82.53% (-14%)	70.74% (-12%)	61.90% (-9%)

6.2. Changes in the Ultrasound Check-In Service

We conducted a similar exercise as in subsection 6.1, this time concerning the ultrasound check-in service with a single secretariat as the minimum quantity of human resources for this activity. Results of the three experiments are expressed in Table 8. Each experiment corresponds to an increase of a resource unit, from one (the minimum) to four secretariats. It can be seen massive waiting times associated with a single secretariat, which drastically drop when the quantity of resources increases. For this case study, raising the number of replicas and associated resources from one to two proved to have the highest impact on the KPIs.

Table 8

– Evolution of KPIs with the different experiments for the ultrasound check-in service.
Ultrasound check-in service	Initial conditions (min. resources = 1 secretariat)	Experiment #1 (2 secretariats)	Experiment #2 (3 secretariats)	Experiment #3 (4 secretariats)
Average waiting time	101.65 min	3.90 min (-96%)	0.58 min (-85%)	0.10 min (-83%)
Maximum waiting time	343.35 min	57.58 min (-83%)	23.64 min (-59%)	11.85 min (-50%)
Resource’s occupancy rate	96.10%	81.87% (-14%)	70.74% (-11%)	61.40% (-9%)

6.3. Changes in the Mammography Facility

Initially, two mammography facilities were designated to operate concurrently, which makes a total of two technicians working simultaneously, one in each room. Under starting conditions of minimal resources and a period of 20 weeks, after execution, the average and maximum queueing times for this activity were determined to be 61.49 minutes and 244.22 minutes, respectively. With the increase of one more facility there is a massive improvement in the KPIs results, as shown in Table 7, with a decrease of the waiting time in queues by 93%. Naturally, the increase of resources (alongside the number of replicas or offices) is accompanied by a reduction of both waiting time and occupancy rate. The average waiting time is vastly decreased every time a new unit of resource is added to the system, but the decrease of occupancy rate is about 4–5% per new unit. With this, it was verified that the percentage of improvement had the highest impact from the transition from two to three replicas. As this number increased, the benefit to the decision-makers becomes only marginal.

Table 9

– Evolution of KPIs with the different experiments for the mammography facility.
	Initial conditions (min. resources = 2 technicians)	Experiment #1 (3 technicians)	Experiment #2 (4 technicians)	Experiment #3 (5 technicians)
Average waiting time	61.49 min	4.51 min (-93%)	0.83 min (-82%)	0.18 min (-78%)
Maximum waiting time	244.22 min	52.60 min (-78%)	24.90 min (-53%)	13.30 min (-47%)
Resource’s occupancy rate	91.92%	87.42% (-4.5%)	82.03% (-5.39%)	77.17% (-4.86%)

6.4. Changes in the X-Ray Facility

Initially, three X-Ray facilities were accounted for to function concurrently, requiring three technicians and three assistants in each room. After the first execution under initial conditions of minimum resources, the average and maximum queueing times for this activity were found to be 33.72 minutes and 174.97 minutes, respectively. As one can conclude from the results in Table 10, there was a seemingly more significant percentage difference when increasing from three to four replicas (experiment #1), which appears to translate into a better gain for the same increase of one resource of each kind and one replica in each of the executions.

Table 10

– Evolution of KPIs with the different experiments for the X-Ray facility.
	Initial conditions (min. resources = 3 technicians + 3 assistants)	Experiment #1 (4 technicians + 4 assistants)	Experiment #2 (5 technicians + 5 assistants)	Experiment #3 (6 technicians + 6 assistants)
Average waiting time	33.72 min	3.05 min (-91%)	0.66 min (-78%)	0.22 min (-67%)
Maximum waiting time	174.97 min	43.31 min (-75%)	18.80 min (-57%)	15.60 min (-17%)
Resource’s occupancy rate	91.92% (techn.), 91.50% (assist.)	86.88% (techn.), 87.40% (assist.)	81.62% (techn.), 82.92% (assist.)	77.08% (techn.), 79.01% (assist.)

6.5. Changes in the MRI Facility

At first, six MRI facilities were proposed to operate simultaneously, necessitating the presence of six technicians and six assistants, one of each in each room. The average and maximum queueing times for this activity were found to be 34.95 minutes and 209.58 minutes, respectively. The improvements resulting from increasing one unit of each type of resource are detailed in Table 11. According with the decision-makers, only experiments #2 and #3 produced acceptable outcomes, although only the latter resulted into a good value for the maximum waiting time in the queue. However, a closer look at the distribution of queueing times revealed that the tails are not heavy, which means that there is just a small chance of a patient waiting over half an hour in the queue for eight or nine technicians and assistants. As such, and given financial constraints, the decision-makers concluded that experiment #2 was the one guaranteeing simultaneously low levels of customer dissatisfaction and resources’ efficient utilization (profitability).

Table 11

– Evolution of KPIs with the different experiments for the MRI facility.
	Initial conditions (min. resources = 6 technicians + 6 assistants)	Experiment #1 (7 technicians + 7 assistants)	Experiment #2 (8 technicians + 8 assistants)	Experiment #3 (9 technicians + 9 assistants)
Average waiting time	34.95 min	12.84 min (-63%)	5.25 min (-59%)	2.42 min (-54%)
Maximum waiting time	209.58 min	97.49 min (-53%)	66.04 min (-32%)	34.50 min (-48%)
Resource’s occupancy rate	91.92% (techn.), 91.50% (assist.)	89.96% (techn.), 89.87% (assist.)	87.73% (techn.), 88.08% (assist.)	85.68% (techn.), 86.26% (assist.)

6.6. Changes in the CAT Facility

Initially, three CAT facilities were assigned to run concurrently, resulting in three technicians and three assistants operating simultaneously in each room. Table 12 shows the evolution of the three KPIs with the different experiments for the CAT facility. Although a massive improvement occurred when there are four technicians and four assistants (compared with the baseline conditions), for the decision-makers only experiments #2 and #3 produce interesting outcomes, as happened to the MRI facility. As before, the tails of the queueing time’s distribution are not heavy, meaning that waiting almost an hour for this exam is unlikely in practice. So, the decision-makers considered that the service must have five technicians and five assistants.

Table 12

– Evolution of KPIs with the different experiments for the CAT facility.
	Initial conditions (min. resources = 3 technicians + 3 assistants)	Experiment #1 (4 technicians + 4 assistants)	Experiment #2 (5 technicians + 5 assistants)	Experiment #3 (6 technicians + 6 assistants)
Average waiting time	78.71 min	15.32 min (-81%)	2.90 min (-81%)	0.91 min (-69%)
Maximum waiting time	291.57 min	127.88 min (-56%)	50.95 min (-60%)	26.25 min (-48%)
Resource’s occupancy rate	91.92% (techn.), 91.50% (assist.)	86.13% (techn.), 89.24% (assist.)	83.70% (techn.), 84.74% (assist.)	79.19% (techn.), 80.83% (assist.)

6.7. Changes in the Ultrasound Facility

At first, five ultrasound facilities were assigned to operate simultaneously, resulting in five assistants and five physicians working simultaneously, one in each room. According with the data in Table 13, adding a sixth replica, with the respective physician and assistant, has a more substantial influence on the system than increasing from six to seven or even from seven to eight replicas. However, the second greatest benefit regarding the maximum queueing time is raising the number of facilities from seven to eight (55% against 48%), which makes this transition considered the second-highest improvement.

6.8. Results Interpretation and Analysis

As was have noted, the most significant improvement in the KPIs occurred almost immediately after the first run, where the values corresponding to the minimum number of replicas and resources were raised by one unit each. Nevertheless, these improvements were still significant, so it was impossible to entirely rule out the potential of climbing to a higher level. The person in charge of making a choice should determine how much of an increase in costs they are ready to accept to make this change in the system.

Table 13

– Evolution of KPIs with the different experiments for the Ultrasound facility.
	Initial conditions (min. resources = 5 physicians + 5 assistants)	Experiment #1 (6 physicians + 6 assistants)	Experiment #2 (7 physicians + 7 assistants)	Experiment #3 (8 physicians + 8 assistants)
Average waiting time	31.24 min	5.01 min (-84%)	1.29 min (-74%)	0.37 min (-71%)
Maximum waiting time	142.62 min	52.74 min (-63%)	27.41 min (-48%)	12.25 min (-55%)
Resource’s occupancy rate	91.50% (assist.), 90.79% (physic.)	86.78% (assist.), 76.06% (physic.)	90.79% (assist.), 65.16% (physic.)	78.12% (assist.), 57.06% (physic.)

In general, the percentage of the extra benefit tends to stabilize as the number of replicas and resources increases. However, after a few runs, it is possible to see that for certain KPIs, the value between the last two runs changes by just a tiny amount. However, as the proportion of benefit decreases until it stabilizes, there will come a time when increasing the number of replicas and resources is no longer advantageous, either in this context or in any other where a comparable simulation experience is conducted. From this point on, raising the number of replicas or resources is not advised since the utilization (occupancy) rate would fall considerably. In other words, the proportion of replicas and resources to work items would be 1:1, queueing times would go toward zero, and the system would be under-crowded, which would be a benefit for patients.

Considering the number of replicas and resources where the benefit is more significant in each activity, these values were included in the model to examine the effect of all modifications in concert. Table A.1 in Appendix compares the results of the KPIs when each activity was analyzed individually and in combination with the other activities, always employing the number of replicas and resources for which there was a more significant benefit.

Looking at the first greatest benefit column of the table, the performance of the KPIs has deteriorated, as predicted, when the number of replicas and resources are optimized for all activities concurrently. When reviewing the KPI results for a particular activity while limiting the others to the bare minimum of resources, this conditioning in the waiting queues of these activities obstructed the regular flow. As a result of fewer patients circulating from these activities, fewer patients were added to the waiting queue for the "optimal" activity (via other examinations). On the other hand, when all activities are optimized simultaneously, the number of patients circulating between examinations increases. More patients are sent for examinations, causing more patients to move to other waiting rooms.

Consequently, it may be required to investigate, for each activity, the number of replicas and resources for which the second-greatest benefit will be realized. According to Tables A.2-A.8, the second highest benefit occurs most frequently immediately after increasing the number of existing replicas from the first benefit by one. For MRI and Ultrasound examinations, however, Table A.6 and A.8 reveal that the total benefit (relative to the two KPIs in the queues) is more significant when the number of replicas is increased by two rather than one. Recall that this extra added benefit per unit stabilizes at a specific value. Increasing the number of replicas is advantageous if the added benefit is equal to or greater than the immediately preceding value.

Then, the number of replicas will grow for each activity to reach the second-highest benefit, and the resource allocation will be readjusted. Consequently, the system was rerun for the same period (20 weeks), with the parameters presented in the second column of the Table A.1. The results of the KPIs are shown in the last column of that table. As expected, the results revealed decreased queueing times for all queues; thus, the KPIs performed better with increased replicas and their resources in a single unit for the second-highest benefit. This provides a far more convenient solution to the system resulting from shorter times in queues. Notably, the difference between the individual and collective analyses for the second-highest benefit is lower than those for the first more substantial benefit, anticipating that the difference would decrease as replicas and resources rise.

When analyzing resource efficiency, one may directly notice a decline when the queues' KPIs for a single activity improve. Lower occupancy rates are a direct result of lower efficiency and vice-versa. This indicates that more resources can be wasted in a specific period for operations to be carried out efficiently. The fact that resources are less efficient throughout the tests, as a consequence of their increase based upon the number of replicas, is because there are more of them for the same number of patients on the waiting queue. The workload of each resource is decreased because, as predicted, the same number of patients is more evenly distributed among them. With the assumption that the activity time is constant, there is, thus, more free time for the resource. More free time denotes increased resource waste or resources awaiting allocation to an activity.

Even though the number of resources is proportionate to the number of activities, the workload is reduced even further when all activities are examined simultaneously because more resources are available. The fact that the activities frequently share a resource is the only reason for this to occur. There is a propensity for stabilization between individual and collective analyses when a physician is involved in the Ultrasound activity because it is the only exclusive resource of this activity. Since more patients have this examination simultaneously due to the increased flow, there is still a chance to detect a slight, nearly imperceptible rise, which was expected given the increased demand for the resource.

It is anticipated that the difference between the two collective analyses of the first and second- highest benefits will reflect the same logic given what has already been stated about efficiency significantly declining as the number of resources increases. Table A.1 shows a noticeable decrease in the utilization rate between the two strategies, proving what was predicted.

To promote resource efficiency, the system was put into place to study the impact of reducing resources by resource type each while keeping the number of activities. The resource utilization rate is expected to grow. Because it is not possible to simultaneously improve resource efficiency and service delivery, there is a trade-off between the two. In order to do this, it is important to strike a balance between the two and determine a level of comfort where there are neither benefits nor losses concerning one another. Thus, the value of the performance of the KPIs of the second-highest benefit was used as a starting point, and three experiments were conducted, each with one less resource of each type compared to the previous experience, to determine how the efficiency of resources improved concerning the decrease in performance of the remaining KPIs. The results are presented in Table A.9.

The analysis of the results reveals that there are still some improvements in the average queueing times for the Mammography and MRI queues. However, the values of the KPIs associated with other examination queues (except for the Ultrasound) indicate weak variances compared to the base experience. One of the reasons this occurs is because the check-in activities, which condition the system, have fewer resources. As a result, the system is more crowded in these queues, which lowers the flow in the second part of the system. It can be seen from that table for these queues that performance, namely the maximum queueing time, actually improves dramatically. Although the number of patients who wait longer than the average value is relatively small, it is still a case that merits consideration because the average waiting time is still low (Figures A.1 and A.2). Although the performance of the KPIs associated with the Ultrasound queue deteriorated significantly due to the reduction in the number of physicians, it became evident that this resource is very dependent on the system's functioning as a whole. To circumvent this issue, a second execution was carried out, taking as the basis the experiment's values with less than three resources (specified in Table A.10), increasing only the number of physicians by two.

According with Table A.10, a reduction from 24 to 21 technicians increased their efficiency from 68.22–75.90% following a final general analysis based on the values of the second-highest benefit. With the reduction from 28 to 25 assistants, this resource's efficiency increased from 67.97–74.33%. The change in efficiency caused by the drop from 8 to 7 physicians was from 58.89–67.30%. A reduction from 10 to 7 secretaries increased productivity from 50.59–72.22%. In addition, it was demonstrated that the increase in resource efficiency had a minor effect on the performance of the queues, although some have varied more than others.

This study addressed the need to enhance the quality of service provided to patients by the Imaging Department of a private hospital. The objective was to reduce the amount of time customers must wait for services in order to satisfy patients' needs for a higher quality of service and increase demand. In order to provide a set of solutions, the metrics for minimizing these wait times were explored in a variety of methods. These solutions will strengthen the decision-making ability of the individual responsible for making the call following the presentation of these results, as only the decision-maker will be able to determine which option is ideal for keeping the system operational.

Sometimes the decision-making process can be more difficult than expected. In situations where a clear trade-off is apparent, where a gain in performance in one factor leads to a decrease in another, it is crucial for the decision-maker to reflect and consider, for instance, which of the two factors has the greater impact for them. During the analysis, it was always observed the effect of the trade-off between keeping patients pleased and making better use of resources, given that increasing the quantity of one requires decreasing the amount of the other. If more resources are available, there will be a better flow of patients through the waiting queues, lowering the queueing times and, therefore, their length of stay in the system, so boosting patient satisfaction in this regard. Satisfied patients are more likely to return, leading to an increase in demand and higher revenue. On the other hand, the quantity of money supplied to employees in the form of wages, for example, will grow. This will lead to an increase in variable costs for each extra resource or activity. In addition, the fixed costs may be adjusted, but only to a small degree than variable costs. The fixed costs include equipment maintenance and electricity bills (more equipment running).

During the analysis, it was possible to observe that the bottleneck effect on check-in activities is not necessarily detrimental to the system, as it frequently prevents examination waiting rooms from being overcrowded, which may be preferable given that patients may remain in more than one queue for an examination during their stay. As witnessed, the results of the KPIs were only able to fluctuate slightly between resource removal trials due to the conditioning of the check-in activities with the flow. A limitation on check-in activities might also be a successful strategy for reducing the number of health personnel (technicians, assistants and physicians). After taking this into consideration, as observed in the last experiment, reducing the number of resources did not result in a substantial difference in the performance of the examination queues other than enhancing the efficiency between the resources, so it may be advantageous to go this way.

Furthermore, when the aim of the modeler is to improve the performance of the queues, as was done in the initial analyses of the preceding chapter, it is crucial to underline the importance of reducing the bottleneck effect. It is important to reiterate the point that system enhancements will need to be made in phases. To put it another way, increasing the performance of examination-related tasks would be pointless if check-in procedures delayed patients. Because of this, it is crucial to evaluate and test the other activities from a position where patients are not backed up in the check-in lines, i.e., where the flow of these activities is already judged acceptable, and the exit rate is, in the best scenarios, close to the entrance rate. Therefore, it is argued that congestion will always occur in a system where an activity's exit rate is larger than the entry rate of the activity that precedes it.

Table A.9 reveals that the resource utilization is approximately 70%, for the maximum reduction the model was subjected, indicating that 30% of the time is spent waiting to be allocated to activities owing to a lack of patients or an excess of resources. Studying the periods when these resources are most in demand and the contrary, i.e., when they are less needed, is one strategy to improve these resources’ efficiency even more. Since this research deals with a limited number of patients whose admission follows particular temporal distributions, the number of patients fluctuates with the time of day, and the flow of patients through the system will not be constant throughout the day. However, the number of patients is similar from time to time. In other words, there is a propensity for patterns to repeat themselves based on the time of day. When keeping the same activities and resources throughout a changeable patient flow, it is only reasonable that the utilization rate would oscillate. There may be certain times when the utilization rate reaches close to 100% as well as other periods when it reaches a value below 50%, indicating that an adjustment could be made in the number of resources according to the time of day when this occurs so that there is a more efficient use of resources. In light of this, one could divide the system into many shifts and propose the necessary number of resources for each shift to battle the patient flow. Thus, it was guaranteed that less resources would be wasted, thereby boosting their efficiency without impacting the performance of the queues.

a. Competing Interests

Not applicable.

b. Authors contribution

Conceptualization, D.C.F. and J.M.; methodology, J.M.; software, J.M.; validation, D.C.F., P.C. and M.V.; formal analysis, J.M. and D.C.F.; investigation, J.M.; resources, D.C.F. and P.C.; data curation, J.M.; writing—original draft preparation, J.M.; writing—review and editing, D.C.F., P.C. and M.V.; visualization, M.V.; supervision, D.C.F.; project administration, D.C.F.; funding acquisition, D.C.F, P.C., and M.V. All authors have read and agreed to the published version of the manuscript.

c. Funding

This work is part of the research activity carried out by D.C. Ferreira at Civil Engineering Research and Innovation for Sustainability (CERIS) and has been funded by Fundação para a Ciência e a Tecnologia (FCT) in the framework of project UIDB/04625/2020.

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No competing interests reported.

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Medical testing services’ quality enhancement with discrete-event simulation

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Abstract

Figures

1. Introduction

2. On Constructing A Discrete-event Simulation

2.1. Discrete-Event Simulation

2.2. Framework for Developing GE-DES in Healthcare Systems

3. Problem Formulation, System Analysis And Conceptual Model Presentation

3.1. Service Functioning Description

3.2. Objectives

3.3. Key Performance Indicators

3.4. Activities

3.5. Assumptions

3.6. Sketch of the Conceptual Model

4. Data Treatment And Model Development

4.1. Patient Entries

4.2. Check-In Service

4.3. Imaging and Ultrasound Services

5. Simulation Model Development

5.1. Simulation Clock

5.2. Arrival Shifts

5.3. Resources

5.4. Labels

5.5. Activities

5.6. Model Validation

5.7. Minimum Number of Resources

5.8. Warm-Up Period

5.9. Termination Conditions Definition

6. Results, Discussion And Presentation

6.1. Changes in the Imaging Check-In Service

6.2. Changes in the Ultrasound Check-In Service

6.3. Changes in the Mammography Facility

6.4. Changes in the X-Ray Facility

6.5. Changes in the MRI Facility

6.6. Changes in the CAT Facility

6.7. Changes in the Ultrasound Facility

6.8. Results Interpretation and Analysis

7. Concluding Remarks And Future Work

Declarations

References

Additional Declarations

Supplementary Files

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