1. Study Design
The first step in the process is that patients who come to the ED have to register with the health informatics employee, except for some urgent cases where they are received by ambulance through a special gate. At the emergency department, there is the health informatics desk. The employee is the frontline and does lots of work at the same time, such as registering for coming patients, doing admission procedures, coordinating with nurse staff, checking the insurance or payments before the registration, entering invoices, closing the final visits, giving reports, and answering inquiries. The second spot is the triage room. Patients after registration go to the triage room to be estimated and prioritized according to the illness acuity (low, medium, and high). Thirdly, patients sit for a while in the waiting area to get into a vacant bed in the emergency bed area. At times, the bottleneck might be seen at the health informatics employee desk or the waiting area for emergency beds. Generally, the ED's physicians are 2, but from 12 pm to 7 pm during the week, and except Friday, they become 3. The available beds are 20, 1 CPR, and 1 isolation bed. There is 1 health informatics employee per shift, and 11 nurse staff and one of them covers the triage room. The average number of visits per 24 hours is 100, and about 60% are during peak hours.
2. Data Collection
Depending on the given statistics at the KHCC, the peak hours during the previous 3 years, 2019 to 2021, were from 10 am to 6 pm, except Friday, there was no overload. Thus, the researchers have collected data by counting patients who came to the ED during peak hours and days, from 3 areas (health informatics desk, triage room, and emergency bed area). According to [5,16], the authors of this study used equations of queuing theory, as shown in Table 1.
Table 1. Queuing theory equations
|
Code
|
Definition
|
Equation
|
Ρ
|
Operation rate
|
ρ = λ /μ
|
Wq
|
The average waiting time in a queue
|
Wq = ρ /μ – λ
|
Ws
|
The average whole time in the ED
|
Ws = 1 /μ – λ
|
Lq
|
The average number of patients in a queue
|
Lq = ρλ/ μ – λ
|
Ls
|
The average number of patients at a specific point
|
Ls = λ /μ – λ
|
Source: [5,16]
|
3. Data Analysis
1) Description of the sample
As shown in Table 2 and Figure 1, the authors have collected data from the records in the past three years, 2019, 2020, and 2021. Indeed, the number of patients has been growing dramatically, from 21947 in 2019 to 32841 in 2021. Again, the average visits during these years were 1661, 2065, and 2568 respectively. In 2019, May and June were the most crowded months, with 2045 and 2164 visits in a row. In 2020, the minimum number of visits was in February with 1582 visits, whereas the maximum was in August with 2498 visits. Unlike in 2021, visit numbers have surged to reach their maximum in August and September, with 2936 and 2911 visits in a sequence. Importantly, by calculating the average visits in the aforementioned years together per month, the peak influx of patients was in August with 2403 visits, and the least was in February with 1667 visits.
Table 2. Patient visits from 2019 until 2021
|
Month
|
2019
|
2020
|
2021
|
Average
|
Jan
|
1620
|
1633
|
2345
|
1866
|
Feb
|
1388
|
1582
|
2031
|
1667
|
Mar
|
1718
|
1700
|
2244
|
1887
|
Apr
|
1599
|
1705
|
2299
|
1868
|
May
|
2045
|
2036
|
2629
|
2237
|
Jun
|
2164
|
2337
|
2539
|
2347
|
Jul
|
1686
|
2424
|
2812
|
2307
|
Aug
|
1775
|
2498
|
2936
|
2403
|
Sep
|
1307
|
2255
|
2911
|
2158
|
Oct
|
1495
|
2256
|
2721
|
2157
|
Nov
|
1479
|
2121
|
2645
|
2082
|
Dec
|
1652
|
2232
|
2708
|
2197
|
Total
|
21947
|
26799
|
32841
|
81587
|
Average
|
1660.667
|
2064.917
|
2568.333
|
2097.97222
|
Source: (The KHCC statistics)
|
However, as shown in Figure 2., calculating the aggregation visits per hour during the past three years gave the authors clear information about the peak hours. Visits’ rising began gradually from 9 am, where the most pressure on the facility was from 10 am until 6 pm. Also, the highest point was at 11 a.m. with 6052 visits, and after that, it dropped 2 times, at noon with 5656 visits and at 1 pm with 4648, to start leveling out until 6 pm with 3949 visits.
Along with that, in Figure 3., looking at the aggregation of visits per day during the past three years, it seems that the highest ratio of visits to the hospital was on Sundays, with 12054 visits, and the lowest on Friday, with 8943, while on the other days of the week, with an insignificant difference, the ratio was between 10684 and 11071 visits. Drawing on the given numbers, the authors have collected data during a chosen week of July, except for Friday, since it is not significant, and from 10 am until 6 pm during peak hours.
2) Analysis of queuing theory and other relativistic equations
(1) Health Informatics Desk: Table 3. shows the average number of patients who arrived at the health informatics desk during the peak hours, from 10 am to 6 pm, was a minimum of 51 patients on Thursday and 73 patients on Wednesday, where the employee at this point served 111 patients on Sunday compared to the number of patients arriving, which was 63. However, the average number of patients on the line was about 1 patient with an operation of 57% on Sunday, and on Thursday it was roughly 3 patients with an operation of 80%. The residence time, between the arrival and giving the service, reached the highest value of 5 minutes.
Table 3. Health informatics desk
|
Day and date
|
Arrival time ʎ
|
service time µ
|
Average operation
|
The average number of patients in the line(average queue length)
|
The average number of patients in the system
|
The average wait time in the line
|
Average time in the system(average residence time in the system)
|
Saturday
16-7-2022
|
67
|
94
|
0.712766
|
1.768715524
|
2.48148148
|
1.58392435
|
2.222222222
|
Sunday
17-7-2022
|
63
|
111
|
0.5675676
|
0.744932432
|
1.3125
|
0.70945946
|
1.25
|
Monday
18-7-2022
|
60
|
83
|
0.7228916
|
1.885804086
|
2.60869565
|
1.88580409
|
2.608695652
|
Tuesday
19-7-2022
|
62
|
85
|
0.7294118
|
1.966240409
|
2.69565217
|
1.9028133
|
2.608695652
|
Wednesday
20-7-2022
|
73
|
93
|
0.7849462
|
2.865053763
|
3.65
|
2.35483871
|
3
|
Thursday
21-7-2022
|
51
|
63
|
0.8095238
|
3.44047619
|
4.25
|
4.04761905
|
5
|
2) Triage room: Importantly, Table 4. shows the number of patients who come to the triage after registering at the health informatics desk during peak hours was the maximum on Wednesday, a total of 68 patients, and the lowest on Thursday, 50 patients. The operation during the week of the study was 100% with no numbers or times of waiting. The patients with low acuity were about 75%; the medium, with 21%; and 4% with high acuity.
Table 4. Triage room
|
Day and Date
|
Arrival time ʎ
|
service time µ real
|
Average operation
|
The average number of patients in the line(average queue length)
|
The average number of patients in the system
|
The average waiting time in the line
|
Average time in the system(average residence time in the system)
|
Low acuity, according to nurse evaluation
|
Medium acuity, according to nurse evaluation
|
High acuity, according to nurse evaluation
|
Total patients
|
Saturday
16-7-2022
|
54
|
54
|
1
|
0
|
0
|
0
|
0
|
22
|
30
|
2
|
54
|
Sunday
17-7-2022
|
57
|
57
|
1
|
0
|
0
|
0
|
0
|
45
|
11
|
1
|
57
|
Monday
18-7-2022
|
58
|
58
|
1
|
0
|
0
|
0
|
0
|
40
|
15
|
3
|
58
|
Tuesday
19-7-2022
|
58
|
58
|
1
|
0
|
0
|
0
|
0
|
56
|
1
|
1
|
58
|
Wednesday
20-7-2022
|
68
|
68
|
1
|
0
|
0
|
0
|
0
|
61
|
4
|
3
|
68
|
Thursday
21-7-2022
|
50
|
50
|
1
|
0
|
0
|
0
|
0
|
36
|
11
|
3
|
50
|
3) Emergency bed area: as it is shown in Table 5., the average operation was 56% on Saturday, with a minute wait time in line, and about 90% on Monday, Tuesday, and Wednesday. From Sunday to Thursday, the average time in the system was between 4 and 10 minutes, and about 3 to 9 patients per hour waited for service in line.
Table 5. Applying queuing theory to the emergency bed area
|
Day and Date
|
Arrival time [ʎ] to the emergency bed
|
Service time [µ] real
|
Average operation
|
The average number of patients in the line(average queue length)
|
The average number of patients in the system
|
The average wait time in the line
|
The average time in the system (average residence time in the system)
|
Saturday
16-7-2022
|
40
|
72
|
0.555555556
|
0.694444444
|
1.25
|
1.041666667
|
1.875
|
Sunday
17-7-2022
|
48
|
64
|
0.75
|
2.25
|
3
|
2.8125
|
3.75
|
Monday
18-7-2022
|
57
|
63
|
0.904762
|
8.595238
|
9.5
|
9.047619
|
10
|
Tuesday
19-7-2022
|
53
|
59
|
0.898305
|
7.935028
|
8.833333
|
8.983051
|
10
|
Wednesday
20-7-2022
|
61
|
69
|
0.884058
|
6.740942
|
7.625
|
6.630435
|
7.5
|
Thursday
21-7-2022
|
46
|
60
|
0.766667
|
2.519048
|
3.285714
|
3.285714
|
4.285714286
|
4) Relativistic analysis of the emergency bed area: In Table 6. below, the average patient’s residence was 21 to 36 minutes. On top of that, the average number of patients receiving the service was about 2 to 3 patients per hour. The low acuity ratio, as per physicians’ assessment, was 61%, the medium 25%, and the high 14% of the total. The patient-physician ratio was between 2 and 3 patients per physician, and the patient-nurse ratio was less than 1 patient per hour.
Table 6. Applying relativistic equations to the emergency bed area
|
Day and date
|
Low acuity, according to physician evaluation
|
Medium acuity, according to physician evaluation
|
High acuity, according to physician evaluation
|
Total of patients who received the service
|
Patients departure
|
The average patients’ residency per hour (aggregation of the number of patients' arrivals to the beds per hour divided by the departure per hour)
|
The average time for patient residency per hour ((60 min/(aggregation of the number of patients' arrivals to the beds per hour divided by the departure per hour))
|
The physicians treat patients per hour. ((2 physicians available, except (from noon and until 19 o’clock, 3 physicians))
|
The nurse treats patients per hour. (10 nurses) during the day shift
|
Saturday
16-7-2022
|
43
|
17
|
12
|
72
|
38
|
1.894736842
|
31.66666667
|
3.375
|
0.9
|
Sunday
17-7-2022
|
34
|
22
|
8
|
64
|
28
|
2.285714
|
26.25
|
1.93125
|
0.8
|
Monday
18-7-2022
|
34
|
16
|
13
|
63
|
38
|
1.657895
|
36.19048
|
3.020833
|
0.7875
|
Tuesday
19-7-2022
|
33
|
17
|
9
|
59
|
33
|
1.787879
|
33.55932
|
2.75
|
0.7375
|
Wednesday
20-7-2022
|
51
|
11
|
7
|
69
|
28
|
2.464286
|
24.34783
|
3.25
|
0.8625
|
Thursday
21-7-2022
|
41
|
14
|
5
|
60
|
21
|
2.857143
|
21
|
2.854167
|
0.75
|
Source: (authors’ elaboration)
|