Discrete-Event Simulation Modeling of Patient Flow in Healthcare Systems Ugbebor O. Olabisi Department of Mathematics, University of Ibadan, Nigeria
[email protected], +2348026437109 Nwonye Chukwunoso Department of Mathematics, University of Ibadan, Nigeria
[email protected], +2348102648514
Abstract Healthcare plays a critical role in determining both the quality and longevity of our lives. The delivery of health care services has been under pressure due to limited funding and increasing demand. This has highlighted the need to increase not only the effectiveness but also the efficiency of health care delivery. Discrete-event simulation has been suggested as an analysis tool in healthcare management to support the planning of health care resources. This paper focuses on the analysis and discrete-event simulation modeling of the flow of patients via the Antenatal department of a university teaching hospital in Nigeria. In this paper, we model and analyze the queue dynamics at the Antenatal department of a university teaching hospital in Nigeria. The goal of this research work is to study and model the stochastic interactions between the Antenatal department and other bottleneck care units in the hospital with the intention of providing tools to support and evaluate management decisions on reengineering the healthcare processes, resource planning and capacity allocation. AMS MSC (2010): 60K25; 68M20; 90B22; 90B15; 37M05; 93C65 Keywords: Discrete-Event Simulation Modeling, Healthcare Systems, Queueing Models, Delays, Stochastic Processes, Patient Flow.
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1. Introduction At some point in our lives, we have all visited some healthcare systems. Most of the time, we rely on both public and private hospitals to provide preventive care (such as vaccinations) and to treat our injuries and illnesses. Healthcare is riddled with delays. Almost all of us have waited for days or weeks to get an appointment with a physician or schedule a procedure, and upon arrival we wait some more until we have been seen. In hospitals, it is not unusual to find patients waiting in hallways for beds and delays for surgery or diagnostic tests are common. Delays are the result of a disparity between demand for a service and the capacity available to meet that demand. Usually this mismatch is temporary and due to natural variability in the timing of demands and in the duration of time needed to provide service. A simple example would be a healthcare clinic where patients walk in without appointments, in an unpredictable fashion, and require anything from a flu shot to the setting of a broken limb. This variability and the interaction between the arrival and service processes make the dynamics of service systems very complex. Consequently, it is impossible to predict levels of congestion or to determine how much capacity is needed to achieve some desired level of performance without the help of a queueing model and discrete event simulation modeling (Hall et al., 2006; Sheldon et al., 2006; Green, 2006; Creemers & Lambrecht, 2007, 2011; Jackson, 1963; Alexander, 2010; Hall, 2006; Steins, 2010). Patient flow represents the ability of the healthcare system to serve patients quickly and efficiently as they move through stages of medical care. For an efficient system, patients’ have a smooth sail via the different stages of care with minimal delay. When bottlenecks exist in the system, patients accumulate at the chokepoints, and experience protracted delays. Put differently, good patient flow means that patient queueing is minimized; poor patient flow means that patients suffer considerable queueing delays. The problem of prolonged waiting in medical facilities in Nigeria has become a common phenomenon. Much attention has been given to procuring medical equipment but no attention has been directed towards easing the flow of patients through the different stages of treatment in the hospitals. Queueing models have been applied in numerous industrial settings and service industries. The number of applications in healthcare, however, is relatively small. This is probably due to a number of unique healthcare related features that make queueing problems particularly difficult to solve (Green, 2006; Creemers & Lambrecht, 2007, 2008, 2011). This paper aims at proffering qualitative solutions to the existing problems. The prompt delivery of healthcare services to ailing individuals has become a global concern. Currently, patients in public hospitals in Nigeria are experiencing protracted waiting times. The number of patients waiting increases on a daily basis due to: (1) the randomness inherent in the arrival and service processes, (2) the limited availability of human resources and (3) the finite capacity of medical facilities (Hall et al., 2006; Hall, 1989, 2006; Green, 2006). It is universally acknowledged that a healthcare system should treat its patients and especially those in need of critical care in a 2
timely manner. However, this is often not achieved in practice, particularly in state-run public healthcare systems that suffer from high patient demand and limited resources (Au-Yeung et al., 2006, 2007, 2008). Traditionally, research focus has been on optimizing the performance of different departments in the treatment chain, thus minimizing waiting time for a specific unit. Hence, the utilization of every care unit is maximized. The current trend is to observe the whole process and optimize on the time it takes an arbitrary patient to transit from admission to discharge. The remainder of this paper is organized as follows. Section 2 presents the research design. Section 3 describes in details the model assumptions, input parameters, and the simulation model of the flow of patients through our case study department using SIMUL8. In section 4, we present the results of the input data analysis (illustrated in figures and tables). Section 5 presents the output data analysis in the form of simulated results and scenario analysis. In section 6 we validate the model by comparing the simulated results with the actual data. Section 7 concludes the paper with recommendations on improving the performance of the system and directions for further work.
2. Research Design In reality, healthcare systems are complex and are usually characterized by stochastic arrival processes, interruptible service processes, probabilistic patient routing, finite waiting areas, finite number of staff members, and complicated shift patterns of resource personnel, among others (Green, 2006; Hall et al., 2006; Au-Yeung et al., 2006, 2007, 2008; Creemers & Lambrecht, 2007, 2009, 2011). For simplicity, table 1 below shows the departments visited by different patient types. Total treatment cost and total time spent in the system depend on the care units on a patient’s trajectory. Suppose we have 6 healthcare units and 3 specialists. Specialist 1 attends to patient types a, b, and c, specialist 2 attends to patient types d, e, and f and specialist 3 handles patient types w, y, and z. Table 1 shows the treatment stages of different patient types. For example, patient type a visits units A, C, D and F while patient type z visits units B, C, E and F.
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Unit A Specialist 1 Patient Type a Patient Type b Patient Type c Specialist 2 Patient Type d Patient Type e Patient Type f Specialist 3 Patient Type w Patient Type y Patient Type z
Unit B
x
Healthcare Units Unit C Unit D x
x x
x
x
x x
x x
Unit F x
x x
x
x
x
x
x
x x
x
x x
x
x
x x x
Unit E
x x x
Table 2.1: Example of treatment process
The Diagnostic Related Group (DRG) of a patient type is defined by its trajectory. The total treatment cost is obtained using a patient’s DRG. The model incorporates the interactions between the Antenatal department and other bottleneck units such as the Medical Records department, the Pharmacy, the Operation Theater, and the Bank Pay Point. These units all inhibit the efficient flow of patients through the system. At each of these units, patients spend an appreciable amount of time waiting on the queue before receiving service. In the simulation model, patient types are defined by their sequential visits to the aforementioned care units. The performance of the system is measured in terms of waiting time, sojourn time, throughput rate, utilization (and percentage of idle times) of care units and medical personnel. Considering the degree of complexity of the system being studied and the inflexibility of queue analytic methods, we have resorted to the use of discrete-event simulation modeling. The simulation study of healthcare systems can be broken down into a series of steps. These steps are grouped as Initial, Model construction and analysis as follows: Initial Steps
Identify and study the process to be simulated Define the objectives of the research project Formulate and define the simulation (or queueing) model
Model Development
Conduct data collection exercise and perform data analysis Develop the model using the input parameters obtained from empirical data Verify and debug the model Validate the model (comparing simulated results with actual data) 4
Analysis
Set up evaluation alternatives (scenarios) Run multiple simulations on each scenario and evaluate results Present the best alternative
These steps are sequential in nature and skipping a step often results in unnecessary additional work. Failure to plan the model structure prior to data collection often results in both inadequate and excessive data collection efforts.
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3. The Simulation Model In this section, we describe the details of the simulation model, the input parameters and the assumptions made in the process of model development.
3.1.
Model Description
The figure below depicts the directed flow of patients through the Antenatal department, Pharmaceutical unit, Medical Lab, and the Operation Theater.
Departures RW
SG
TP
Stochastic Arrivals
PP PH SN
TP
PP
Departures ML OU Stochastic Arrivals MR
Stochastic Arrivals
Stochastic Arrivals GY
AN
Departures
Figure 3.1: A Schematic representation of patient flow via the Antenatal department
Index: MR – Medical Records, AN – Antenatal Nurses, GY - Gynaecologists, OU – Other Units, SN – Surgery Nurses, SG – Surgeons, PH – Pharmacy, TP – Teller Point, PP – Pay Point, ML – Medical Laboratory, RW – Recovery Wards; Flow Direction;
Waiting Rooms;
Discharge/Departures;
Medical Personnel (Offices and Consultation Rooms)
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As mentioned earlier, there are several patient types depending on their DRG. In this model, patients are grouped according to their directed routing. We will describe the simulation model depicted in Figure 3.1 from the standpoint of an arbitrary arriving patient. When a patient arrives at the hospital, the first port of call is the medical records department to obtain his/her medical documents. External arrivals also occur at the medical lab and pharmacy. Upon arrival at the medical records department, the patient either gets service immediately or joins the queue if staff members are busy. The medical records department is characterized by long queues and prolonged waiting due to the time-dependent arrival surge at the department, especially in the morning hours. Also, the medical records department is responsible for the medical records of all patient types visiting all the departments in the hospital. Upon exit from the medical records, the patient proceeds to the unit of his/her choice. This project is focused on the flow of patients via the Antenatal department and its interactions with the associated departments. If the patient proceeds to the Antenatal department, she will first be attended to by the nurses to ascertain the patient’s health condition before proceeding to the doctor’s office for consultation and treatment. On exit from the doctor’s consultation room, the patient might be directed to the pharmacy or medical lab or to the theater for surgical operations, or may leave the system. The flow of patients at the pharmacy department is also not straightforward. Patients visiting the pharmacy proceed to the drug point for drug costing, then to a teller point to obtain payment documents, proceed to the pay point for payment and back to the drug point for drug collection before exiting from the system. Patients from other units visit the pharmacy, resulting in increased congestion at the pharmacy. The routing of patients at the medical lab is similar to what happens at the pharmacy. Upon service completion at the operation theater, patients are moved to the medical wards for recovery.
3.2.
Model Assumptions and System Constraints
Processes in hospitals are characterized by uncertainties; hence require stochastic modeling and analysis. Patients’ arrivals at the hospital are assumed to be unscheduled, hence random. The inter-arrival times of the different patient types were found to be exponentially distributed; hence, the arrival process follows a Poisson distribution (see Figure 4.1). External arrivals occur at the medical records, medical lab and pharmacy. In reality, patients can become impatient and leave (e.g., balk or renege) the system without receiving service. In this research work, we assume that patients remain in queue until they are served. The queue discipline at all treatment stages is modeled using first-in first-out (FIFO) queueing discipline. In a real hospital setting, the waiting area is finite but in this model, we assume that all waiting areas have infinite capacities. The service time of patients at different stages of treatment is modeled using the fitted probability distributions (see section 4). The service process at a medical care unit can be interrupted by a number of factors, for example, the arrival an emergency patient, phone calls, and so on. In this model, we utilize the service time of a patient at a care unit i.e., the pure service time plus interruptions. 7
The Antenatal department of the hospital is an outpatient department, hence it starts empty and closes empty. The sequential flow of a patient type via the treatment stages is assumed to be predefined. Direct reentry of patients is assumed not to occur at the care units and patients who bypass the Antenatal department are not considered. The resource persons at the Medical Records, Medical Lab, Antenatal Clinic, Bank Pay Point, and Pharmacy department work for 9 hours daily (8am – 5pm) while the recovery wards and the operating theater are open 24 hours. The main concern of this research work is to model and evaluate management schemes aimed at reducing the protracted waiting experienced by ailing patients in our case study department.
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4. Input Data Analysis The credibility of a simulation model largely depends on the quality of the input parameters used in developing the model. Below we itemize some of the input parameters used in the model development:
Percentages of different patient types flow through the various departments under consideration The external arrival rates of different patient types at different care units Capacity of each unit and the associated number of resident medical personnel Shift patterns of the care units and the resource persons Service times distributions for different patient types at different departments
4.1.
Probability Distributions of Interarrival Times and Service Times
Given the data sets for the external interarrival times and service times at the workstations; we performed statistical analysis on these samples. The outcomes of the data analysis reveal that the probability distributions (shown in the tables and figures below) accurately model the interarrival and service time random variables. Below we present the tables and figures depicting the summary statistics and probability distributions obtained for and subsequently used to model the interarrival and service times at the various workstations. These distributions were used to generate service times at the respective workstations during simulation runs. Probability Density Function 0.52
Statistic
Value
0.48
Sample Size
5279
0.44
Range
9.07
0.36
Mean
1.05
Variance
1.13
0.24
Standard Deviation
1.06
0.16
Coefficient of Variation
1.01
Inverse Scale Parameter
=0.95
0.4
f(x)
0.32 0.28
0.2
0.12 0.08 0.04 0 0
0.8
1.6
2.4
3.2
4
4.8
5.6
6.4
7.2
8
8.8
x Histogram
Table 4.1: Summary Statistics of MR Interarrival Times
Exponential
Figure 4.1: Probability Density Function of the MR Interarrival Times
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Statistic
Value
Statistic
Value
Sample Size
4783
Sample Size
4381
Range
8.82
Range
57.28
Mean
20.33
Mean
29.97
Variance
3.37
Variance
60.50
Standard Deviation
1.84
Standard Deviation
7.78
Coefficient of Variation
0.09
Coefficient of Variation
0.26
Model Parameter
m=20.09
Scale Parameter
=2.02
Lower Boundary Parameter
a=15.98
Shape Parameter
=14.85
Upper Boundary Parameter
b=24.92
Table 4.3: Summary Statistics of Pharmacy Service Times
Table 4.2: Summary Statistics of Medical Records Service Times
Probability Density Function
0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0
0.24 0.22 0.2 0.18 0.16 f(x)
f(x)
Probability Density Function
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
16
16.8
17.6
18.4
19.2
20
20.8
21.6
22.4
23.2
24
24.8
x Histogram Triangular
Figure 4.2: Probability Density Function of the Medical Records Service Times
10
15
20
25
30
35
40
45
50
55
60
65
x Histogram Gamma
Figure 4.3: Probability Density Function of the Pharmacy Service Times
10
Statistic
Value
Statistic
Value
Sample Size
3471
Sample Size
3374
Range
14.78
Range
38.05
Mean
21.42
Mean
30.03
Variance
11.95
Variance
30.04
Standard Deviation
3.46
Standard Deviation
5.48
Coefficient of Variation
0.16
Coefficient of Variation
0.183
Shape Parameter 1
1=1.51
Scale Parameter
=1.00
Shape Parameter 2
2=2.01
Shape Parameter
=30.01
Lower Boundary Parameter
a=15.01
Table 4.5: Summary Statistics of Doctors Service Times
Upper Boundary Parameter
b=29.91
Table 4.4: Summary Statistics of Medical Lab Service Times
Probability Density Function
Probability Density Function 0.24
0.13 0.22
0.11
0.2
0.1
0.18
0.09
0.16
0.08
0.14 f(x)
f(x)
0.12
0.07 0.06
0.12 0.1
0.05
0.08
0.04
0.06
0.03 0.02
0.04
0.01
0.02
0
0
16
18
20
22
24
26
28
x Histogram Beta
Figure 4.4: Probability Density Function of the Medical Lab Service Times
16
20
24
28
32
36
40
44
48
52
x Histogram Gamma
Figure 4.5: Probability Density Function of the Doctors Service Times
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In addition, we performed goodness of fit test (using Kolmogorov-Smirnov, Anderson-Darling, and Chi-Squared) to ascertain the probability distribution that best models the random (arrival or service) process in question.
5. Research Results and Discussions In this section of the paper, we present the simulated results portraying the current situation of the system under study and the results obtained from the various “what-if” scenario analyses conducted. As mentioned earlier, the performance of the system is measured in terms of waiting time, sojourn time, and throughput rate. Considering the complexity of the system being studied and the intractability of the application of queueing analytic techniques to model healthcare systems, we have resorted to discrete-event simulation modeling. As a first step towards developing the simulation model of the Antenatal department, we developed submodels of the pharmacy and medical lab units since these units play a critical role in a patient’s journey through the system. Put differently, the development of these sub-models was crucial in order to investigate in details, the intricacies inherent in system’s operations. Thereafter, we merged these sub-models to form the complete simulation model of the Antenatal clinic. Below we present plots of an arbitrary simulation run depicting the queue dynamics at various bottleneck units in the system.
Figure 5.1: Queue Dynamics at the Medical Records Waiting Room
Figure 5.3: Queue Dynamics at the Gynaecologists Waiting Room
Figure 5.2: Queue Dynamics at the Antenatal Nurses Waiting Room
Figure 5.4: Queue Dynamics at the Pharmacy waiting line
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Figure 5.5: Queue Dynamics at the Medical Lab waiting line
Figure 5.6: Queue Dynamics at the Pay Point waiting line
The queue dynamics shown in the figures above depict an increasing trend in the queue length and waiting times of patients at the various stages of treatment. The figures also show that a substantial number of patients are unattended to at the end of a 9-hour day. In an attempt to proffer a solution to this undesirable situation, we have performed a number of “what-if” scenario analyses.
5.1.
SCENARIO A
In scenario A, we progressively increase the number of healthcare givers across all the care units under consideration. Table 5.1 displays the current and suggested capacities of the units. Medical Care Units Medical Records Antenatal Nurses Gynaecologists Pharmacists Teller Points 1&2 Pay Points Medical Lab 1&2 Surgeons Surgery Nurses Recovery Beds
Current Settings 4 2 2 2 1&2 2 2&2 3 3 Ample
Capacities of Care Units Suggested (1) Suggested (2) 5 6 3 4 3 4 3 4 2&2 3&3 3 4 3&2 4&2 3 3 3 3 Ample Ample
Suggested (3) 7 5 5 5 4&4 5 5&2 3 3 Ample
Table 5.1: Capacities of care units in the current setting and proposed settings
The results of scenario A analyses (including the current situation) are presented in the figures below.
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Antenatal Clinic Departures
Antenatal Clinic Departures
240
70
235
65
230
60
225
55
220
50
215
45
210
Average Time in the System
205
35
200
30
195
25
190
20
Figure 5.7: Average Sojourn time of patients that successfully completed all the stages of treatment and exited via the pharmacy
Average Number Successfully Completed
40
Figure 5.8: Average number of patients that successfully completed all the stages of treatment and exited via the pharmacy
Figures 5.7 and 5.8 portray an improvement in the system performance since there is a consistent decrease in the sojourn time of patients and increase in the number of patients that successfully go through all stages of treatment, as we vary the unit capacities from the current situation to the Suggested 3 settings.
Medical Records Queue
Medical Records Queue
150
300
130
250
110
200
90
150
70
Average Queuing Time
100
50
50
30
0
Figure 5.9: Average Waiting Time of patients at the medical records waiting area
Average Number of Unattended Patients Average Queue Length
Figure 5.10: Average Queue Size at the medical records waiting area and the average number of patients unattended to at the end of a 9-hour day
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The plots in figures 5.9 and 5.10 express an improvement in the system performance indicators since there is a reproducible decrease in average waiting time, average queue length, and average number of unattended patients as the different unit capacities is increased progressively. Antenatal Nurses Queue
Antenatal Nurses Queue
70
35
60
30
50
25
40
20
30
Average Number of Unattended Patients
15 Average Queuing Time
20
10
10
5
0
0
Figure 5.11: Average Waiting Time of patients at the Antenatal Nurses waiting room
Average Queue Length
Figure 5.12: Average Queue Size at the Antenatal Nurses waiting room and the average number of patients unattended to at the end of a 9-hour day
Gynaecologists Queue
Gynaecologists Queue 40
85 35 75
30
65 55
25 Average Queuing Time
20
45
15
35
10
Figure 5.13: Average Waiting Time of patients at the Gynaecologists’ waiting room
Average Number of Unattended Patients Average Queue Length
Figure 5.14: Average Queue Size at the Gynaecologists’ waiting room and the average number of patients unattended to at the end of a 9-hour day
15
Pharmacy Queue
Pharmacy Queue
90
110 100
85
90 80
80
Average Number of Unattended Patients
70 75
Average Queuing Time
70
60
Average Queue Length
50 40
65
30
Figure 5.15: Average Waiting Time of patients at the Pharmacy waiting room
Figure 5.16: Average Queue Size at the Pharmacy waiting room and the average number of patients unattended to at the end of a 9-hour day
Medical Lab Queue 1
Medical Lab Queue 1
100
40
90
35
80
30
70 60
25
50
20
40 30
Average Queuing Time
20
15 10
10
5
0
0
Figure 5.17: Average Waiting Time of patients at the Medical Lab waiting room
Average Number of Unattended Patients Average Queue Length
Figure 5.18: Average Queue Size at the Medical Lab waiting room and the average number of patients unattended to at the end of a 9-hour day
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Medical Lab Departures
Medical Lab Departures 70
240
60
220
50
200
40 30
180
Average Time in the System
160 140
Figure 5.19: Average Sojourn time of patients that successfully completed all the stages of treatment and exited via the medical lab
20
Average Number Successfully Completed
10 0
Figure 5.20: Average number of patients that successfully completed all the stages of treatment and exited via the medical lab
Obviously, the nurses’ subsection of the Antenatal department responded positively to the intervention since there is consistent decrease in the average waiting time, average queue length, and average number of unattended patients (see figures 5.11 and 5.12). The intervention has also shown a positive impact on the medical lab unit (see figures 5.17, 5.18, 5.19, and 5.20). There is no clear trend in the average waiting time of patients at the pharmacy unit as we consistently increase the capacities of the care units. We also notice an increasing trend in the average queue length and average number of unattended patients. This increase can be attributed to fact that each patient visits the pharmacy twice, thereby doubly the arrival rate at the pharmacy. In addition, patients from other departments visit the pharmacy, hence, increasing the arrival rate at the pharmacy. In an attempt to remedy the situation at the pharmacy and medical lab, we propose a modification of patients’ routing at the said units.
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5.2.
SCENARIO B
In this section of the paper, we consider some modification of patients’ flow via the pharmacy and medical lab. In scenario B, the capacities of the care units remain unchanged. It is crucial to investigate, modify and model the intricacies in the routing of patients at the pharmacy and medical lab because these units are chokepoints in the system being studied. For the purpose of illustration, we present a schematic representation of the pharmacy sub-model. The routing of patients via the medical lab is modified in a similar fashion.
Departures
AD
PH
Stochastic Arrivals from other sources
TP
PP
Figure 5.21: The Pharmacy sub-model (Current Situation)
Stochastic Arrivals from other sources
AD
TP
PP
PH
Departures Figure 5.22: The Pharmacy sub-model (Proposed Setting)
The figures below are results of the intervention (and modifications) at the pharmacy and medical lab units.
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Pharmacy Queue (Scenario B) 90 80 70 60
Average Number of Unattended Patients
50
Average Queue Length
40
Average Queuing Time
30 20 10 0 Current Situation
Modified PH Routing
Modified PH & ML Routing
Figure 5.23: Average Waiting Time of patients and Average Queue Size at the Pharmacy waiting room, and the average number of patients unattended to at the end of a 9-hour day
Medical Lab Queue (Scenario B) 100 90 80 Average Number of Unattended Patients
70 60
Average Queue Length
50
Average Queuing Time
40 30 20 10 0 Current Situation
Modified PH Modified PH & Routing ML Routing
Figure 5.24: Average Waiting Time of patients and Average Queue Size at the Medical lab waiting room, and the average number of patients unattended to at the end of a 9-hour day
19
Considering the graphs in figures 29 and 30, there is a tremendous decrease in the average waiting time, average queue lengths, and the average number of unattended patients at the pharmacy and medical lab care units. It shows that these bottleneck units have strong influence on a patient’s flow via the system. However the striking improvements shown in figures 29 and 30 are local and do not positively impact on the performance of the upstream units namely the medical records, Antenatal nurses, and doctors.
5.3.
SCENARIO C
In scenario C, we further modify the settings of the care units under consideration. Scenario C is a further improvement on scenario B. In addition to the improved routing at the pharmacy and medical lab, we progressively increase the capacities of the bottleneck units; medical records, Antenatal nurses, and doctors. The table below depicts a summary of the care unit capacities in the current situation and in the proposed layout.
Medical Care Units Medical Records Antenatal Nurses Gynaecologists Pharmacists Teller Points 1&2 Pay Points Medical Lab 1&2 Surgeons Surgery Nurses Recovery Beds
Current Settings 4 2 2 2 1&2 2 2&2 3 3 Ample
Capacities of Care Units Modified (1) Modified (2) 2 2 3 4 4 5 3 5 2&2 3&3 3 4 2&2 2&2 3 3 3 3 Ample Ample
Modified (3) 2 5 6 5 4&4 4 3&2 3 3 Ample
Table 5.31: Capacities of care units in the current setting and suggested settings
Notice that the staff strength in the medical records unit has decreased from 4 to 2. This is because the operations at the medical records unit are computerized and the service time is reduced to 5 minutes deterministic.
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The figures below show plots of the average total time spent by patients exiting via the Antenatal clinic and medical lab. The figures also depict the total number of patients that successfully passed through required treatment stages.
Antenatal Clinic Departures 300 250 200 Average Time in the System
150
Average Number Successfully Completed
100 50 0 Current Modified 1 Modified 2 Modified 3 Situation
Figure 5.31: Average Sojourn time and average number of patients that successfully completed all the stages of treatment and exited via the pharmacy
Medical Lab Departures 250
200
150
Average Time in the System
100
Average Number Successfully Completed
50
0 Current Modified 1 Modified 2 Modified 3 Situation
Figure 5.32: Average Sojourn time and average number of patients that successfully completed all the stages of treatment and exited via the medical lab
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6. Model Validation The simulation model of our case study department (the Antenatal department) is validated by comparing simulated results (for example, sojourn times) with actual data obtained during data collection. To further ascertain the credibility of the simulation model, we conducted multiple comparative simulation runs (experiments) using the empirical data (interarrival times and service times) versus the fitted probability distributions. The outcomes (i.e. system performance measures) of these two scenarios were close; within ±0.2 error margin. It also shows that the probability distributions used to model the interarrival and service times were the right options.
Departing Patients via the Pharmacy Departing Patients via the Medical Lab
Average Sojourn Time (Actual Data)
Average Sojourn Time (Simulated Results)
238.98
239.21
228.04
227.72
Table 6.1: Total time spent in the system (Simulated results compared with actual data)
7. Recommendations Healthcare systems are intrinsically complex and are thus analytically intractable. In the medical setting, Markov models and decision trees have been extensively used despite their limitations in reproducing healthcare problems accurately. We conclude first, by stressing the strength and flexibility of the application of discrete event simulation modeling in queueing problems. Although discrete event simulation modeling is taxing, expensive, and time consuming, its gains are evident in its ability to accurately model the intricacies and complexities in queueing systems such as healthcare systems. The model (and the scenario analysis) of our case study department has shown that there is room for improvement in the performance of the system being modeled. In scenario A, we noticed a tremendous improvement (in terms of average waiting time, average queue length, average sojourn time, average number of patients unattended to) on the overall performance of the system as we gradually increase the capacities of the units in the system. Scenario B highlighted a local intervention in the performance of the pharmacy and medical lab units. Conducting scenario B was crucial considering the routing complications at the pharmacy and medical lab units. More importantly, the health condition of patients can deteriorate while waiting on queue. Hence a more detailed investigation and improvement of the medical lab and pharmacy is paramount. In scenario C, we modified the service process at the medical records (service time is 5 minutes fixed) and consistently increased the capacities of the care units as shown in table 8. Scenario C shows an improved performance of the system (in terms of
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average waiting time, average queue length, average sojourn time, average number of patients unattended to). We recommend a further investigation on the arrival process at the care units. We also recommend an application of discrete event simulation in studying other healthcare units, for example, emergency department, paediatrics, orthopedics, surgery among others.
References Alexander Kolker (2010). Queuing Theory and Discrete Events Simulation for Health Care: From Basic Processes to Complex Systems with Interdependencies; Handbook of Research on Discrete Event Simulation Environments: Technologies and Applications, Information Science Reference; pp 443-483 Au-Yeung Susanna Wau Men (2007). Response Times in Healthcare Systems. PhD Dissertation, Department of Computing, Imperial College London Au-Yeung S.W.M., U. Harder, E.J. McCoy, and W.J. Knottenbelt (2008). Predicting patient arrivals to an Accident and Emergency department. Emergency Medicine Journal. Au-Yeung S.W.M., P.G. Harrison, and W.J. Knottenbelt (2006). A Queueing Network Model of Patient Flow in an Accident and Emergency Department. In Proceedings of the 20th Annual European and Simulation Modeling Conference, pages 60–67,Toulouse, France Au-Yeung S.W.M., Harrison P.G., and Knottenbelt W.J. (2007). Approximate queueing network analysis of patient treatment times. In Proceedings of the Second International Conference on Performance Evaluation Methodologies and Tools (VALUETOOLS 2007), Nantes, France Hall W. Randolph (1989). Queueing Methods for Services and Manufacturing, Prentice Hall, Englewood Cliffs, New Jersey Hall W. Randolph (2006). Patient Flow: The New Queueing Theory for healthcare. OR/MS Today 23:36–40 Hall Randolph, David Belson, Pavan Murali and Maged Dessouky (2006). Modeling patient flows through the healthcare system; Reducing delay in healthcare delivery: International Series in Operations Research & Management Science, Springer; pp 1-44 Jackson R. James (1957). Network of waiting lines. Operations Research 5:518–521
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Jackson R. James (1963). Jobshop-like queueing systems. Management Science 10:131– 142 Johnston M. J., P. Samaranayake, A. Dadich1 and J.A. Fitzgerald (2009). Modeling radiology department operation using discrete event simulation. 18th World IMACS / MODSIM Congress, Cairns, Australia Krisjanis Steins (2010). Discrete-Event Simulation for Hospital Resource Planning – Possibilities and Requirements. PhD Dissertation, Department of Science and Technology, Linköping University, SE-601 74 Norrköping, Sweden Linda Green (2006). Queueing analysis in healthcare; Reducing delay in healthcare delivery: International Series in Operations Research & Management Science, Springer; pp 281-307 Lloyd G. Connelly (2004). Discrete Event Simulation of Emergency Department Activity: A Platform for System level Operations Research. ACAD EMERG MED Vol. 11, No. 11, pp 1177-1185 Marek Laskowski, Robert D. McLeod, Marcia R. Friesen, Blake W. Podaima, Attahiru S. Alfa (2009). Models of Emergency Departments for Reducing Patient Waiting Times. PLoS ONE 4(7): e6127.doi:10.1371/journal.pone.0006127 Reetu Mehandiratta (2011). Applications of queueing theory in healthcare. International Journal of Computing and Business Research Sheldon H. Jacobson, Shane N. Hall and James R. Swisher (2006). Discrete-event simulation of healthcare Systems; Reducing delay in healthcare delivery: International Series in Operations Research & Management Science, Springer; pp 211-252 Stefan Creemers and Marc Lambrecht (2007) Modeling a healthcare system as a queueing network: the case of a Belgian hospital. In FBE publications: Research Reports and Discussion papers. Department of Decision Sciences & Information Management, Research Center for Operations Management, Catholic University Leuven Stefan Creemers and Marc Lambrecht (2009). Healthcare queueing models. In: FBE publications: Research Reports and Discussion papers. Department of Decision Sciences & Information Management, Research Center for Operations Management, Catholic University Leuven
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Stefan Creemers and Marc Lambrecht (2011). Modeling a Hospital Queueing Network; Queueing Networks A Fundamental Approach: International Series in Operations Research & Management Science, Springer; pp 767-798 Stefan Creemers and Marc Lambrecht (2011). The Modeling of Interrupts and Unplanned Absences in HealthCare Operations. Supply Chain Forum; An international Journal. pp 32-40
About the Authors Ugbebor O. Olabisi is an Associate Professor of Probability Theory and Stochastic Processes at the University of Ibadan, Ibadan, Nigeria. She obtained PhD in Mathematics from the University of London in 1976. Ugbebor teaches courses on probability theory and stochastic analysis and she has published numerous articles in leading international journals.
Nwonye Chukwunoso is a researcher at the department of Mathematics, University of Ibadan, Ibadan, Nigeria. He holds a Professional Doctorate Degree in Engineering (PDEng, Mathematics for Industry) obtained from the University of Technology, Eindhoven, The Netherlands. Nwonye’s field of research interest is Queueing Modeling, Stochastic Processes, and Discrete-Event Simulation Modeling. He has done a number of academic, industrial, and simulation based research projects in the areas of applications of queueing theory and stochastic processes to solving real world problems.
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