3.1. Methods

SD is a computer-facilitated approach to policy analysis and design with a focus on modeling stocks (accumulations) and flows (rates) of systems. Typically, SD is applied to dynamic problems that are characterized by interdependence and mutual interaction of elements, information feedback, and circular causality [29, 30]. Virtual worlds, i.e., simulation models, created with SD can act as learning laboratories with the purpose of developing and testing strategies before they are implemented in practice. This is highly relevant for organizations nowadays considering the fact that many of them operate within increasingly dynamic environments and, therefore, strategies have to be evaluated and adjusted constantly [31, 32].

We chose SD as our modeling approach for the following reasons. First, it seemed important to us that ED operations are not only analyzed from one methodological viewpoint, that is, discrete event modeling, but tackled by a diversity of simulation methods in order to generate new insights. Second, agreeing with [21], we think that considering aggregated variables (e.g., aggregated flows of patients) which is the focus of SD encourages both a systemic view of the interactions of patient flows and information, and a more strategic perspective of the management of the system. Third, due to its accessible graphical iconography, SD is particularly useful to engage stakeholders both in the model building and in the model analysis phase [33]. In SD, the model structure can be explained and presented in simple mathematical terms which facilitates communication with a non-technical audience. Additionally, SD models take high-level policies as inputs making them accessible for interpretation and fostering dialogue between hospital stakeholders and the modeling team. This was a key aspect to us because we intended to involve EPs, nurses, and ED managers throughout the entire modeling process. SD is still our method of choice when it comes to stakeholder involvement, despite recent efforts in facilitated discrete-event simulation modeling [34].

3.2. ED under study

Singapore is a city state with a population of 5.61 million people [16]. The study institution is the largest hospital in the country with 1’600 inpatient beds and provides tertiary care to a significant share of the population. The hospital is part of the Singhealth Regional Health System (RHS) which covers a population of more than 1.1 million people and handles more than 4 million patient visits yearly. The hospital-based ED cares for more than 140’000 patient visits annually, with about 350 visits per day in 2019 [35, 36]. The ED is equipped with 25 specialist EPs who work an average of 180 clinical shift hours per 28 days, along with roughly 40 non-specialists who clock an average of 216 clinical shift hours in the same period [35].

3.3. Overall structure of the ED

Patients come to the ED by ambulance or other forms of transportation (walk-in) from the community to seek care. Upon arrival, ED patients go through a brief registration before the triage processes commence. Triage refers to the categorization of ED patients for treatment in situations of scarce resources according to the patients’ medical conditions and established sorting plan. In Singapore, the Patient Acuity Category Scale (PACS) which prioritizes patients into four main priorities is used to triage patients at the ED [37]. The priorities are: (1) Priority 1 are patients in a state of cardiovascular or imminent collapse. They are the most serious, time-critical patients who require immediate attention or resuscitation—examples of conditions are heart attack, severe injuries, severe bleeding, shock and severe asthma attack; (2) Priority 2 patients are non-ambulant patients with acute medical conditions who appear to be in a stable state with no immediate danger of collapse—examples of conditions are major limb fracture/dislocation, moderate injuries, severe abdominal pain and other severe medical illnesses; (3) Priority 3 refers to ambulant patients with acute symptoms who are in a stable condition. These patients could be treated by general practitioners, family physicians with acute care resources—example of conditions are sprains, minor injuries, minor abdominal pain, vomiting, fever, rashes and mild headaches; Finally, (4) Priority 4 are non-emergency patients with old injuries or conditions that have been present for a long time—examples include chronic joint pain, chronic skin rash, long-term nasal discharge, old scars, cataracts, removal of tattoos and sore throats.

A trained nurse evaluates the patient’s condition, takes his or her medical history, initiates diagnostic measurements, and determines the priority for treatment, i.e., P1, P2, P3 or P4. Patients with fever, irrespective of treatment priority—P1, P2, P3, or P4—are sent to the isolation area to be seen by a physician to reduce the risk of infecting other patients in the ED. Non-ambulant or trolley-based patients in priority 1 and 2 are treated at the critical care area, while ambulatory patients, irrespective of their treatment priority are sent for treatment at the ambulatory care area.

Each treatment area—critical care, ambulatory care, and isolation care—has a dedicated waiting area and allocated ED nurses and physicians. The average waiting time to consult an ED physician depends on the number of ED patients waiting for consultation and the number of ED physicians available. The higher the ED patient’s acuity or priority, the greater the average physician consultation time. For P1 and P2 patients receiving treatment in the critical care area, after initial consultation, almost all patients are admitted to the observation ward for observation. During observation, patients who require laboratory services undergo the investigation there and wait for the results. If there are no beds in the observation ward, patients are observed in the waiting area. For ambulatory patients, after initial consultation, laboratory services are provided. Those who require observation are admitted into the observation ward, while others wait for laboratory investigation results in the waiting area. Lastly for isolation patients, after initial consultation, patients are observed before a decision to discharge is made.

For patients in the observation ward, a decision to send them home or admit them into the hospital is made after a further review by ED physicians. To that end, laboratory results (if available) are reviewed. Discharged ED patients proceed to the pharmacy for medication and payment. For those who require hospital admission, arrangement is made with the appropriate hospital ward for patient transfer. An overview of the principal processes in a hospital-based ED in Singapore is shown in Fig 3.

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Fig 3. Overall structure of patient processing in a hospital-based ED in Singapore.

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3.4. Model structure

SD models consist of an interconnecting set of differential and algebraic equations developed from a broad range of empirical data. SD models comprise of stocks, interconnected flows and auxiliary variables. A general mathematical representation of stocks and flows are: (1) (2) (3) where N and M are the system parameters. The flows are the derivatives or rates of change of the associated stocks. Stocks create disequilibrium dynamics as they decouple flows. As a consequence, typically, inflows and outflows differ and are governed by different decision rules. The overall model structure of an ED in Singapore is presented in Fig 4. For a list with all variables and their respective abbreviations see S2 Table.

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Fig 4. Overall model structure of an ED in Singapore.

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3.4.1. Registration and triage.

For ED patients, the journey begins when they arrive at the ED to seek care (see Fig 5). New patient arrivals a(t) at any time (t) proceed for registration and quickly transition from registration to triage. The equation for patients waiting for registration P(t) at time (t) is:

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Fig 5. Registration and triage sub-model.

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(4) where P(t₀) is patients waiting for registration at time (t₀) and (5) (6)

New patient arrivals a(t) is an exogenous input and is fed into the simulation model as historical time-series data; g(t) is patients moving from registration to triage and is represented herein as a pipeline delay function of new patient arrivals a(t) and average registration time RT.

After registration at the ED, patients wait to be triaged. Patients are normally triaged into four treatment priorities (j)—P1, P2, P3, and P4; and three care areas—critical care, ambulatory care, and isolation care. B(t), that is, patients waiting for triage, increases as patients move from registration to triage g(t) and decreases as patient are triaged to critical care cca_j(t), ambulatory care ab_j(t), and isolation care is_j(t). The equation for patients waiting for triage B(t) is: (7) where B(t₀) is patients waiting for triage at time (t₀) and (8) (9) (10)

Patients triaged to critical care cca_j(t), ambulatory care ab_j(t), and isolation care is_j(t) are modeled herein as pipeline delay functions of patients moving from registration to triage g(t) and average triage time TT, adjusted by the fraction of patients sent to each care area; fcca_j(t) is the fraction of patients triaged to critical care, fab_j(t) is the fraction of patients triaged to ambulatory care, and fis_j(t) is the fraction of patients triaged to isolation care. All three fractions sum up to one.

3.4.2. Critical care pathways.

Patients triaged to the critical care area wait in queue for consultation. In total, there are four main patient pathways within the critical care area (see Fig 6):

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Fig 6. Critical care area sub-model.

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1. Waiting for consultation → consultation → discharge
2. Waiting for consultation → consultation → laboratory investigation → discharge
3. Waiting for consultation → consultation → observation → discharge
4. Waiting for consultation → consultation → laboratory investigation → observation → discharge

The number of patients waiting for consultation C_j(t) increases by patients triaged to critical care cca_j(t) and new ambulance arrivals nab_j(t), and decreases as patients start consultation cs_j(t). Patient consultation cs_j(t) is initiated when an ED doctor becomes available and initiates consultation nc_j(t). The equation for patients waiting for consultation C_j(t) is: (11) where C_j(t₀) is patients waiting for consultation at time (t₀) and (12) (13) nab_j(t) is the exogenous historical ambulance arrival data; ppd is the patient per doctor ratio in the critical care area.

Initiation of consultation requires an ED doctor. The number of ED doctors consulting PCC(t) increases as ED doctors initiate consultation nc_j(t) and decreases as consultation is completed cc_j(t). ED doctors available to initiate consultation pa(t) is the difference between the number of ED doctors allocated to critical care NP(t) and ED doctors consulting PCC(t). An average consultation time is assumed for each patient by treatment priority. P1 patients are assumed to require longer consultation time followed by P2, P3, and P4 patients. However, only P1 and P2 patients are triaged to the critical care area. The equation for the ED doctors consulting PCC(t) is: (14) where PCC(t₀) is ED doctors consulting at the critical care area at time (t₀) and (15) (16) (17) (18) C_p1(t) and C_p2(t) are P1 and P2 patients waiting for consultation in the critical care area; AT is adjustment time—a model artifact to ensure unit consistency. The value of AT is 1. CT is consultation time; dpp is the doctor per patient ratio in the critical care area.

A co-flow structure was used to model patients in consultation. As an ED doctor initiates consultation nc_j(t), a patient moves from the stock of patients waiting for consultation C_j(t) to the stock of patients in consultation EP_j(t). Hence, completion of consultation cc_j(t) decreases the number of patients in consultation EP_j(t)via to observation co_j(t), to laboratory and investigation cl_j(t) or to home ch_j(t). The equation for patients in consultation EP_j(t) is: (19) where EP_j(t₀) is ED patients in consultation at time (t₀) and (20) (21) (22) (23) (24) avb(t) is available beds in the observation ward; cpo_p1(t) and cpo_p2(t) are P1 and P2 patients requiring referral to the observation ward, co_j(t) is the patients from consultation to observation, fb is the fraction of patients who require observation, fh is the fraction of patients discharged home after consultation.

After consultation, patients are either referred to the observation ward co_j(t), to laboratory and investigation cl_j(t) or discharged home ch_j(t) depending on their care needs The number of patients waiting for laboratory and investigation PLI_j(t), i.e., patients who have to go through the laboratory investigation process and wait for their results, increases as patients are referred to laboratory and investigation cl_j(t) and decreases as patients are either discharged after laboratory and investigation ld_j(t), transferred to the observation ward lo_j(t), or observed at the waiting area due to lack of beds in the observation ward lw_j(t). Patients under observation in the waiting area POW_j(t) are discharged dw_j(t) as their conditions improve or admitted to the hospital cad(t). The equations for patients waiting for laboratory and investigation PLI_j(t) and patients under observation in the waiting area POW_j(t) are: (25) (26) where PLI_j(t₀) is patients waiting for laboratory and investigation at time (t₀), POW_j(t₀) is patients under observation in the waiting area at time (t₀), and (27) (28) (29) (30) (31) (32) (33) (34) al_j(t) is patients who have completed laboratory and investigation; fob_j is the fraction of patients who need to go to the observation ward after laboratory and investigation; LIT is the average waiting time for laboratory and investigation, fdd is the fraction of patients waiting in the waiting area admitted; and wt is the average observation time.

3.4.3. Ambulatory care pathways.

Patients triaged to ambulatory care, like other care pathways, wait in queue for consultation. In total, there are two main pathways for patients triaged to the ambulatory care area (see Fig 7):

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Fig 7. Ambulatory care area sub-model.

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1. Waiting for consultation → consultation → laboratory investigation → discharge
2. Waiting for consultation → consultation → laboratory investigation → observation → discharge

The number of patients waiting for consultation CAB_j(t) increases by patients triaged to ambulatory care ab_j(t) and decreases as consultation starts csAB_j(t). Patient consultation is initiated when an ED doctor becomes available and starts consultation ncAB_j(t). The number of ED doctors consulting PCAB(t) increases as an ED doctor initiates consultation ncAB_j(t) and decreases as consultation is completed ccAB_j(t). Available ED doctors to initiate consultation paAB(t) is the difference between ED physicians allocated to ambulatory care NPAB(t) and the number of ED physician consulting PCAB_j(t). The equations for ED patients waiting for consultation CAB_j(t) and ED physicians consulting PCAB(t) are: (35) (36) where CAB_j(t₀) is the number of patients in ambulatory care waiting for consultation at time (t₀), PCAB(t₀) is the number of ED physicians consulting at time (t₀), and (37) (38) (39) (40) (41) (42) (43) ppd is the patient per doctor ratio in the ambulatory care area; dpp is the doctor per patient ratio; CT is consultation time, and AT is adjustment time.

Similar to the critical care area, a co-flow structure was developed to track patients in consultation in ambulatory care. As an ED physician initiates consultation ncAB_j(t), a patient moves from the stock of patients waiting for consultation CAB_j(t) to the stock of patients in consultation EPAB_j(t). Consequently, completion of consultation ccAB_j(t) decreases the stock of patients in consultation EPAB_j(t) via to laboratory and investigation clAB_j(t). The equation illustrating this dynamic is: (44) where EPAB_j(t₀) is ED patients in consultation at the ambulatory care area at time (t₀) and (45)

After consultation, patients proceed to laboratory and investigation. Patients waiting for laboratory and investigation PHIAB_j(t)–a procedure which includes various tests and examinations as well as waiting for test results to be discussed with the ED physician–increases as patients are referred to laboratory and investigation clAB_j(t) and decreases as patients are referred to the observation ward ao_j(t), discharged home ldAB_j(t) or transferred to be observed in the waiting area due to limited beds in the observation ward lwAB_j(t). Patients under observation in the waiting area POWAB_j(t) due to capacity constraints in the observation ward decreases via discharge dwAB(t) or hospital admission aAB_j(t). The equations for patients in laboratory and investigation PHIAB_j(t) and patients under observation in the waiting area POWAB_j(t) are: (46) (47) where PHIAB_j(t₀) is the initial number of patients in the ambulatory care area waiting for laboratory and investigation at time (t₀), POWAB_j(t₀) is the number of patients under observation at time (t₀), and (48) (49) (50) (51) (52) (53) (54) ala_j(t) is the number of patients who have finished laboratory and investigation, and foba_j is the fraction of patients who have completed laboratory and investigation and are admitted to the observation ward, while famb is the fraction of patients observed in the waiting area and are admitted to the hospital.

3.4.4. Observation ward and discharge.

The observation ward receives patients from critical care and ambulatory care areas—patients triaged to isolation care have a separate observation ward (see Fig 8). The number of patients in the observation ward OW_j(t) increases as critical care patients are referred to it immediately after consultation co_j(t) or after laboratory and investigation lo_j(t), as well as referral of ambulatory patients after laboratory and investigation ao_j(t), and decreases as patients are admitted into the hospital ah_j(t) or discharged home via pharmacy and payment do_j(t). Admission into the observation ward depends on the available beds avb(t). Available beds avb(t) is the difference between observation bed capacity bc(t) and the number of patients in the observation ward OW_j(t). After observation, discharged patients go through pharmacy and payment PHAB_j(t) for payment and collection of prescribed medication. The number of patients in pharmacy and payment PHAB_j(t) increases as patients are discharged from the observation ward do_j(t), as patients are released from laboratory and investigations both in critical care ld_j(t) and ambulatory care ldAB_j(t), as well as patients are discharged from observation in waiting areas, both in critical care dw_j(t) and ambulatory care dwAB_j(t), as well as patients discharged after consultation from critical care ch_j(t) and decreases as patients leave for home hab_j(t). The equations for observation ward admission and discharge are:

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Fig 8. Observation ward and discharge area sub-model.

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(55) (56) where OW_j(t₀) is the initial number of patients in the observation ward at time (t₀), PHAB_j(t₀) is the initial number of patients in the stock pharmacy and payment at time (t₀), and (57) (58) (59) (60) (61) (62) (63) (64) where avb(t) is the total available beds in the observation ward; ala_j(t) is the number of patients who have finished laboratory and investigation; foba_j is the fraction of patients who have completed laboratory and investigation and are admitted to the observation ward; df is the fraction of discharged patients from the observation ward; ot is the average observation time for patients in the observation ward; ppb is the patients per bed ratio; ttba is the time to make a bed available for patients to use.

3.4.5. Isolation care pathways.

Patients triaged to isolation care, like other care pathways, wait in queue for consultation. In total, there are two main pathways for patients triaged to isolation care (see Fig 9):

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Fig 9. Isolation area sub-model.

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1. Waiting for consultation → consultation → discharge
2. Waiting for consultation → consultation → observation → discharge

The number of patients waiting for consultation CIS_j(t) increases by patients triaged to isolation care is_j(t) and decreases as consultation starts csIS_j(t). Patient consultation is initiated when an ED physician becomes available and initiates consultation ncIS_j(t). The number of ED physicians consulting PCIS(t) at the isolation care area increases as an ED physician starts consultation ncIS_j(t) and decreases as consultation is completed ccIS_j(t). Available ED physicians to initiate consultation paIS(t) is the difference between ED physicians allocated to isolation care NPIS(t) and ED physicians currently consulting with patients PCIS_j(t). The equations for patients waiting for consultation CIS_j(t) and ED doctors consulting in the isolated care area PCIS(t) are: (65) (66) where CIS_j(t₀) is patients in the isolation care area waiting for consultation at time (t₀), PCIS(t₀) is the number of ED physicians consulting at time (t₀), and (67) (68) (69) (70) (71) (72) (73) where ppd is the patient per doctor ratio in the isolation care area; dpp is the doctor per patient ratio; AT is adjustment time; and CIS_P1, CIS_p2, CIS_P3, and CIS_P4 are the stocks of patients waiting for consultation.

Similar to other care areas, a co-flow structure was developed to model patients in consultation in the isolation care area. As an ED physician initiates consultation ncIS_j(t) a patient moves from the stock of patients waiting for consultation csIS_j(t) to the stock of patients in consultation EPIS_j(t). Hence, completion of consultation ccIS_j(t) decreases the stock of patients in consultation via to observation coIS_j(t) and for discharge cpIS_j(t). After consultation, patients referred to the observation ward coIS_j(t) are observed in the observation ward OWIS_j(t). After a period of observation time otis, patients in the observation ward are either discharged oph_j(t) or admitted into the hospital ahis_j(t). Likewise, patients discharged from the isolation care area, i.e., via observation ward oph_j(t) and or after consultation cpIS_j(t), proceed to the pharmacy and payment PHIS_j(t) for prescribed medicine and payment and then leave his_j(t). The equations illustrating the stock of patients in consultation EPIS_j(t), the stock of patients in observation OWIS_j(t), and the stock of patients in pharmacy and payment PHIS_j(t) are: (74) (75) (76) where EPIS_j(t₀) is the initial number of patients in consultation at the isolation care area at time (t₀), OWIS_j(t₀) is the number of patients under observation at time (t₀), PHIS_j(t₀) is the number of patients in pharmacy and payment at time (t₀), and (77) (78) (79) (80) (81) (82) where fis is the fraction of patients who need to go to the observation ward; owd_j(t) is the number of patients who were observed and proceed to discharge or hospital admission; fah is the fraction of patients who were observed and are admitted to the hospital; otis is the average time patients were observed; ppt is the average time at the pharmacy and payment.

3.5. Data sources

We parameterized a simulation model that runs for 24 hours. To that end, we had access to ED data for the period of June–August 2017. The key parameter values essential for estimating ALOS are average registration time, average triage time, consult time (for critical care, ambulatory care and isolation care), average waiting time for observation ward, lab and investigation waiting time, pharmacy and payment waiting time, time to make bed available, and average time to admit patients. The distribution of the key parameter values is assumed to follow a triangular distribution [38]. The estimated values used for the lower limit a, upper limit b, and mode c are as follows: For registration and triage, the values are—average registration time [Min = 3; Median = 5; Max = 7], and average triage time [Min = 4; Median = 5; Max = 7]. For critical care the input values are—consult time [Min = 10; Median = 15; Max = 20], average observation waiting time [Min = 30; Median = 60; Max = 90], and laboratory and investigation [Min = 35; Median = 45; Max = 60]. For ambulatory care, the input values are—consult time [Min = 10; Median = 15.5; Max = 20], and average observation waiting time [Min = 30; Median = 60; Max = 90]. For isolation care, the values are—consult time [Min = 20; Median = 30; Max = 45], average observation waiting time [Min = 30; Median = 60; Max = 90], and pharmacy and payment waiting time [Min = 10; Median = 15; Max = 30]. For observation ward and discharge, the input values are—time to make beds available [Min = 10; Median = 15; Max = 25], and average time to admit patients [Min = 100; Median = 120; Max = 150].

Referring to patient arrivals, we picked the highest daily patient arrival pattern in the 3-months period from June to August 2017 because we intentionally wanted to stress-test the system. Operational problems in the ED become only visible when the workload is high, and the system is stretched to its limits.

To supplement the data requirements, an observational study on other process timings that were not captured in the patient records was conducted over a 2 weeks period in November 2017 to estimate some of the model parameters. Finally, for all parameters that could not be observed nor estimated from the data, we had to rely on expert judgment. More information about model inputs, their values, units, and sources can be found in S1 Table. The simulation model is provided in the supplementary file for review.

3.6. Model validation

To ensure that the ED model developed herein is fit for purpose and robust and that the results could be used to inform ED policies, structure and behavior validation tests were conducted [39–41]. Structure validation tests focused on engaging ED doctors with significant experience in the operations of the ED in Singapore to verify the model structure and its assumptions regarding causal relationships, feedbacks, time delays, and patient flows. This validation process was conducted in four different meetings—where the model structure was thoroughly reviewed—to ensure that the model structure is as close to reality as possible. Hence, we believe that the current model structure is firmly grounded in current operations of a hospital-based ED in Singapore.

On behavior validation tests, the simulation results were compared to available data of selected outcomes (average length of stay across all venues—CCA, ambulatory, and isolation). In addition, a mean absolute percentage error (MAPE) and a Theil statistic [42, 43] analysis were conducted to check the behavioral validity of the model. The MAPE—which is a measure of prediction accuracy—for the selected outcomes were; 9.78% for ALOS in the critical care area, 12.5% for ALOS in the ambulatory care area, and 8.85% for ALOS in the isolation care area. Given that MAPE of 30% is considered to be good, our results—which have a maximum MAPE of 12.5%—indicate that the simulated model compares well with available data considering that available ALOS data used was an average across venues over an hour. For the Theil statistic, the error due to bias (U^M) for CCA, ambulatory, and isolation care areas were 11%, 4.5%, and 17.7% respectively; while that for unequal variance (U^S) for CCA, ambulatory, and isolation care areas was 2.4%, 7.1%, and 9.4% respectively; and the error for covariation component (U^C) for CCA, ambulatory, and isolation care areas were 87.1%, 88.4%, and 72.9% respectively. Thus, critical care, ambulatory care, and isolation care areas have most of the error within the covariation component (U^C) as compared to bias (U^M) and unequal variance (U^S). For Theil statistics, if majority of the errors comes from covariation components, it indicates that the simulated variables track the underlying trend well, but diverge when comparing point-by-point, indicating that majority of the errors are unsystematic with respect to the purpose of the model.

5.1. Business-as-usual (BAU)

As shown in Tables 1–4, in the BAU case, a critical care patient that goes through critical care pathway 1 for emergency care is projected to experience an ALOS of 25 minutes from 00:00 am to 08:00 am; 39 minutes from 08:00 am to 04:00 pm; and 27 minutes from 04:00 pm to 00:00 am. The estimated ALOS for critical care pathway 2 patients is 72 minutes from 00:00 am to 08:00 am; 85 minutes from 08:00 am to 04:00 pm; and 73 minutes from 04:00 pm to 00:00 am. The ALOS for critical care pathway 3 patients is 149 minutes from 00:00 am to 08:00 am, 162 minutes from 08:00 am to 04:00 pm and 150 minutes from 04:00 pm to 00:00 am. Lastly, the ALOS for critical care pathway 4 patients is 195 minutes from 00:00 am to 08:00 am, 209 minutes from 08:00 am to 04:00 pm and 197 minutes from 04:00 pm to 00:00 am.

For patients seeking emergency care at the ambulatory care venue, and who go through ambulatory care pathway 1 are estimated to experience an ALOS of 72 minutes from 00:00 am to 08:00 am; 166 minutes from 08:00 am to 04:00 pm; and 85 minutes from 04:00 pm to 00:00 am. For ambulatory pathway 2 patients, under the base-case, a projected ALOS of 196 minutes from 00:00 am to 08:00 am; 290 minutes from 08:00 am to 04:00 pm; and 208 minutes from 04:00 pm to 00:00 am is expected.

Lastly, patients at the isolation care area following the isolation care pathway 1 are projected to experience an ALOS of 42 minutes at all phases. For isolation care pathway 2 patients, ALOS is projected to be 101 minutes from 00:00 am to 04:00 pm and 102 minutes from 04:00 pm to 00:00 am.

5.2. Co-location (policy 1)

As indicated in Table 1, under policy 1, where 10% to 30% of P4 and P3 patients from each venue of care are decanted from the ED to a GP clinic co-located in the ED to provide needed care, the ALOS for critical care patients going through critical care pathways 1 to 4 are projected to be the same as BAU. However, for patients going through ambulatory care pathway 1 and 2, ALOS was projected to reduce under policy 1. In the scenario where 10% of P4 and P3 patients were decanted, ALOS for ambulatory care pathway 1 is projected to decrease by 24.6% from 08:00 am to 04:00 pm, and by 4.7% from 04:00 pm to 00:00 am, compared to the BAU. The ALOS of patients in ambulatory care pathway 1 arriving in the time period between 00:00 am and 08:00 am remains unchanged for the 10%, 20%, and 30% scenario; for ambulatory care pathway 2 ALOS is projected to decrease by 14.1% from 08:00 am to 04:00 pm, while that for 04:00 pm to 00:00 am is 1.44%, compared to the BAU. The ALOS of patients in ambulatory care pathway 2 presenting themselves between 00:00 am and 08:00 am remains unchanged for the 10%, 20%, and 30% scenario. In the scenario where 20% of P4 and P3 patients were removed from the ED, ALOS for ambulatory care pathway 1 is projected to fall by 36.1% from 08:00 am to 04:00 pm, while that for 04:00 pm to 00:00 am is 8.2%, compared to the BAU; for ambulatory care pathway 2 ALOS is forecasted to diminish by 20.7% from 08:00 am to 04:00 pm, while that for 04:00 pm to 00:00 am is 3.4%, compared to the BAU. Finally, in the scenario where 30% of P4 and P3 patients are redirected to an inhouse GP clinic, ALOS for ambulatory care pathway 1 is projected to fall by 42.2% from 08:00 am to 04:00 pm and by 11.8% from 04:00 pm to 00:00 am, compared to the BAU; for ambulatory care pathway 2 ALOS is forecasted to reduce by 24.5% from 08:00 am to 04:00 pm and by 4.3% from 04:00 pm to 00:00 am, compared to the base case. Lastly, policy 1 does not show an impact on isolation care pathways in our experimental set-up. On average, only 6% of all presenting patients are triaged to isolation care (i.e., fever patients) and so patient flow is smooth through this care venue. This means that fever patients typically do not need to wait until they see a doctor. Consequently, reducing the inflow of patients into isolation care (which is the effect of policy 1) does not change the ALOS of patients going through this venue of care. ALOS is driven by the waiting time for lab and investigation and if necessary, by the waiting time for a bed in the hospital. Both waiting times are not influenced by policy 1.

5.3. Capacity of doctors (policy 2)

As shown in Table 2, under this policy where the number of ED doctors is gradually increased by 10%, 20% and 30%, a critical care patient going through critical care pathways 1 to 4 is projected to experience ALOS similar to that of the BAU from 00:00 am to 08:00 am and 04:00 pm to 00:00 am. However, during the time period of 08:00 am to 04:00 pm, with a scenario where the doctors allocated to each venue is increased by 10%, ALOS is projected to decrease by 20.5%, 8.2%, 4.3% and 3.8% respectively for care pathways 1 to 4. In the scenarios where a 30% increase of doctor’s allocation was experimented, ALOS is projected to decrease by 28.2%, 12.9%, 6.7% and 5.2% respectively for care pathways 1 to 4.

Similarly, under the ambulatory care pathways 1 and 2, ALOS is projected to be similar to that of the BAU from 00:00 am to 08:00 am. However, ALOS is projected to decrease under the 10% increase in doctor’s capacity scenario by 34.3%, and 20% for ambulatory care pathways 1 and 2 respectively from 08:00 am to 04:00 pm; while that for 04:00 pm to 00:00 am was 7.05% and 2.4%. Under the 30% increase in doctor’s capacity scenario, ALOS is expected to reduce by 49.3% from 08:00 am to 04:00 pm and 12.9% from 04:00 pm to 00:00 am for ambulatory care pathway 1; and 28.2% from 08:00 am to 04:00 pm and 5.2% from 04:00 pm to 00:00 am for ambulatory care pathway 2. Lastly, ALOS for patients going through isolation care pathways are expected to experience ALOS like that of the BAU. The drivers of ALOS for patients in isolation care are the waiting times for lab and investigation and admission to the hospital. Both waiting times are not changed (reduced) by increasing the capacity of doctors in the ED.

5.4. Observation ward and laboratory (policy 3)

As shown in Table 3, under the observation ward and laboratory policy, a critical care patient going through critical care pathways 1 is projected to experience ALOS comparable to that of the BAU across all time phases. But, for critical care pathway 2, 3 and 4, ALOS is projected to decrease as observation ward and laboratory and investigation waiting times are reduced. Under the scenario where observation ward and laboratory waiting times were reduced by 10%, compared to the BAU, ALOS is expected to decrease by 6.9%, 4.7% and 5.4% for critical care pathway 2 across the three-time phases; 8%, 7.4% and 8% for critical care pathway 3 across the three-time phases; while that for critical care pathway 4 was 8.7%, 8.1% and 8.6% respectively across the three phases. Under the scenario where observation ward and laboratory waiting times were reduced by 30%, ALOS for critical care pathway 2 is projected to reduce by 19.4% from 00:00 am to 08:00 am, 16.4% from 08:00 am to 04:00 pm and 19.1% from 04:00 pm to 00:00 am; while that for critical care pathway 3 was 24.8% from 00:00 am to 08:00 am, 22.8% from 08:00 am to 04:00 pm and 24.6% from 04:00 pm to 00:00 am. Likewise, the ALOS for critical care pathway 4 is projected to reduce by 26.15% from 00:00 am to 08:00 am, 24.4% from 08:00 am to 04:00 pm and 25.8% from 04:00 pm to 00:00 am.

Considering the ambulatory care pathways, under the 10% reduction in observation ward and laboratory waiting times ALOS is projected to decrease 6.9%, 3% and 5.8% respectively across the time phases for care pathway 1; while that for care pathway 2 is projected to decrease by 8.6%, 5.8% and 8.1% respectively across the time phases. Under the 30% reduction in waiting times, ALOS is projected to decrease by 19.4%, 8.4% and 16.4% respectively for care pathway 1, while projections for care pathway 2 are reductions of 26%, 18.6% and 24.5% respectively.

For isolation care pathways, while isolation care pathway 1 is projected to remain unchanged relative to the BAU, isolation care pathway 2, under the 10% reduction in observation ward and laboratory waiting times is projected to decrease ALOS by 5.9% across all time phases. However, under the 30% reduction in observation ward and laboratory waiting times, ALOS is projected to decrease by 16.8%, 17.8% and 17.6% respectively across the time phases.

5.5. Combined interventions (policy 4)

As indicated in Table 4, under the combined interventions where policies 1 to 3 are implemented simultaneously, under the 10% assumptions—where 10% of P4 and P3 patients are decanted to a GP clinic in the ED, 10% increase in doctors allocated to each care venue and 10% reduction in observation ward and laboratory waiting times—a critical care patient that goes through critical care pathway 1 for emergency care is projected to experience 20.5% reduction in ALOS from 08:00 am to 04:00 pm; while that for 04:00 pm to 00:00 am is 3.7%. Interestingly, ALOS rises by 4% for patients arriving between 00:00 am and 08:00 am. For critical care pathway 2 a reduction in ALOS of 6.9%, 14.1%, and 6.8% are projected across the three-time phases. Patients going through the critical care pathway 3 are projected to experience 8%, 12.3% and 8.6% reduction in ALOS across the time phases. Lastly, patients going through critical care pathway 4 are projected to experience 8.7%, 11.9% and 9.1% reduction in ALOS across the time phases. Under the 30% assumptions, patients going through the critical care pathways 1 to 4 are projected to experience greater reductions in ALOS as indicated in Table 4. For critical care pathway 1 a reduction in ALOS of 28.2% is projected from 08:00 am to 04:00 pm and of 3.7% for 04:00 pm to 00:00 am. The ALOS of patients arriving between 00:00 am and 08:00 am remains unchanged compared to the BAU. For critical care pathway 2 a reduction of 19.4%, 29.4%, and 20.5% is projected across the three-time phases; while that for critical care pathway 3 are 24.8%, 29.6% and 25.3% respectively across the time phases. Lastly, patients going through critical care pathway 4 are projected to experience 26.1%, 29.6% and 26.9% reduction in ALOS across time phases under policy 4.

For patients going through the ambulatory care venue, ambulatory care pathway 1 patients, under the 10% assumptions, are projected to experience a reduction in ALOS by 6.9% from 00:00 am to 08:00 am, 43.9% from 08:00 am to 04:00 pm, and 15.2% from 04:00 pm to 00:00 am; while the reduction in ALOS for ambulatory care pathway 2 patients is 8.6% from 00:00 am to 08:00 am, 29.6% from 08:00 am to 04:00 pm and 12% from 04:00 pm to 00:00 am. As is the case with all the policies, under the 30% assumptions, ALOS is projected to reduce much more compared to the 10% assumptions as indicated in Table 4.

Finally, for patients going through the isolation care venue, ALOS for isolation care pathway 1 patients is projected to remain unchanged relative to the BAU across all time phases. For isolation care pathway 2, under the 10% assumptions, ALOS is projected to decline by 5.9%, 5.9% and 5.8% respectively across time phases, whereas under the 30% assumptions, ALOS is projected to decline by 16.8%, 17.8% and 17.6% respectively across time phases.

6.1. General remarks

ED crowding is a multifaceted issue and so far, many solutions have failed because they ignored or underestimated the dynamic complexity of the problem [5]. After thoroughly reviewing the literature on ED crowding [47], recommends to ‘research systems-wide solutions on the basis of existing evidence and operations theory, with the aim of mitigating the risk/problem of crowding.’ For this reason, in the present study, we analyzed ED crowding from a systems thinking perspective to explicitly account for the problem’s dynamic complexity caused by a web of interrelated influencing factors [48]. More specifically, we used SD to map and simulate the interrelations among variables affecting and affected by ED crowding. The resulting simulation model helps to better understand the dynamic nature of this phenomenon and it can serve as an effective decision support system because of its capability to test policy proposals in silico. This has the pivotal advantage that ED mangers can experiment with policy proposals and study their consequences in a risk-free environment.

Given the current arrival pattern of patients and the configuration of the hospital-based ED in Singapore, a patient seeking emergency care, with the exception of a few periods of the day, is highly probable to receive care within the target time of 4 hours if triaged to the critical care or ambulatory care units or within 4.5 hours if triaged to the isolation care unit. As expected, the longer the care pathway of the patient (which includes consultation, laboratory, and observation), irrespective of the care venue, the larger the ALOS. Patients triaged to the critical care unit have the shortest ALOS compared to ambulatory and isolation care patients. Policies that focus on decanting P4 and P3 patients to a GP clinic co-located within the ED are more likely to reduce the ALOS of ambulatory patients, since all the P4 and P3 patients are triaged to the ambulatory care unit. Likewise, a policy that increases the efficiency of patient transfer from the observatory ward in the ED to the hospital ward, as well as decreasing the waiting time of laboratory investigation is more likely to reduce the ALOS of patients whose care pathway includes the observation ward and laboratory investigations.

The observed results can be explained by the interaction between the implemented policies (i.e., co-location policy, optimal allocation of doctor’s policy, and observation ward and laboratory policy) and available ED capacity. For example, as patients are decanted to GP clinics on arrival at the ED, the share of patients triaged for emergency care decreases; therefore, waiting time for consultation will decrease resulting in a drop in ALOS. The decline in consultation waiting time is due to the fall in the number of patients waiting for consultation. Thus, available resources (ED doctors) are able to care for fewer patients demanding ED services. In addition, raising the number of ED doctors (via the optimal allocation policy) increases the resources (ED doctors) available to provide care, hence a reduction in ALOS as consultation waiting time declines. Lastly, an efficient transfer of patients from the observation ward to the acute hospital wards, as well as reduction of waiting time for laboratory and investigation is expected to decrease the ALOS of patients whose pathway includes the observation ward and laboratory and investigation. These policies were implemented based on the identification of ED bottlenecks—waiting time for consultation, laboratory and investigation waiting time and observation ward waiting time. These were identified as bottlenecks due to the significant time patients spend in these venues, thus increasing the ALOS of ED patients.

6.2. Main findings of this study

Ambulatory care patients with priority 3 and 4 make up the lion’s share of all attending patients in our ED under study. On average, 55% of all patients are triaged to the ambulatory care area. Consequently, even a small reduction in the number of incoming patients into the ambulatory care area has a significant impact on waiting times and ALOS. This key finding has policy implications. Overall, the finding suggests that a comprehensive ED system that anticipates inappropriate self-referral of patients and makes provisions for such patients—by transferring such patients to a GP clinic in the ED—is more likely to triage only appropriate ED patients for emergency care, thus reducing the pressure on available ED resources. The lesson from this finding is that, there is a significant proportion of ED patients who inappropriately self-refer to the ED. Policymakers should anticipate that behavior and either provide GP care co-located at the ED or incentive non-emergency patients—by reducing the out-of-pocket-cost—to seek GP care first before coming to the ED.

In Singapore, a pilot intervention called GP first, which incentivizes patients with less serious conditions to see their GP first before going to the ED was shown to reduce the number of non-emergency patients seeking care at the ED. In addition, EDs should be incentivized (by sharing savings from non-emergency patients triaged to co-located GPs) to triage patients accurately—to prevent up-coding where patients are assigned to higher severity than the actual condition to justify their use of ED services—and to ensure that patients are right-sited for care, that is, patients receive care at the appropriate venue with the lowest cost. The implication of this finding is that if EDs are inappropriately incentivized referring to the triage function, e.g., punished for providing ED care to non-emergency patients (P4), the EDs are likely to up-code non-emergency patients preventing the opportunity to improve care efficiency and reduce cost. Lastly, it is important to emphasis that, a sustainable approach to reduce ED crowding will require a well-functioning enhanced primary care system that improves health outcomes of the population and significantly lessens ED care demand among non-emergency patients. This is vital for countries with an aging population where demand for healthcare services is expected to increase. If the primary care system is not strengthened to provide appropriate care for the elderly population with multiple chronic diseases, inappropriate demand for ED care is expected to increase with its consequences of ED crowding.

Given that the main bottlenecks in most EDs are significant waiting times for consultation, laboratory investigation, and for hospital admission (mostly through the observation ward), it is important to ensure that proactive and innovative interventions are explored to reduce waiting times in these locations of ED care. Interventions that focus on the optimal allocation of ED doctors (typically increasing their numbers) should be explored to reduce consultation waiting time. In addition, efficient operating systems that ensure speedy transfer of patients from the observation ward in the ED to hospital wards should be implemented. For instance, interventions that focus on (i) categorization of wards to medical specialty; (ii) instituting a no reject policy; and (iii) performing ward level audits have been shown to improve waiting time for hospital admission [49].

6.3. Limitations of this study

The model presented here has some limitations. First, the use of an SD modeling approach for modeling ED patient flows introduces patient mixing that makes it difficult to track each patient individually; however, there are other modeling forms—such as agent-based modeling (ABM)—that focus on simulating the actions and interactions of autonomous agents that address this limitation. Patient mixing and the assumption that patients triaged into the same category have similar characteristics is an oversimplification that may affect the results. Second, transfer of patients to other hospitals—which was not relevant in our case—was not included in the model; whereas the likely impact of nurses and other allied health workers on waiting time was not included. Third, modeling results are reported as single values (ALOS) without an indication of the statistical uncertainty for the different venues of care and time phases. This could be improved by deriving confidence intervals for the modeling results through Monte Carlo simulation. Despite these limitations, the model presented herein remains useful for policymakers to test and evaluate innovative policies. For instance, the model could be used as an exploratory tool to search for high-leverage policies and to evaluate the likely impact of alternative policies on specific outcomes of interest. In addition, the model could help policymakers to design and communicate policy insights to stakeholders to build consensus and inform policy implementation.