Outpatient Appointment Scheduling with
Different Arrival Rates of Walk-ins in Taiwan
Fenghueih Huarng1 & Ming-Te Chen2
1. Dept. of Business Adm, Southern Taiwan Univ. of Technology
2. Dept. of Business Adm, National Chung Cheng Univ.
Why Walk-in?

Patient’s habit (Taiwan starts pre-register since 1980.)
 Lack of good appointment system—
‧pre-register given only sequence number ( no appointed time)
‧late penalty for pre-register ( every 10th, more 3,etc.)
 Different clinic nature
‧Fetter & Thompson (1996)
— two air force hospitals.
average 37% walk-in, pediatric 55~58% walk-in and call-in,
urology 7~11%, dermatology 37.5%
—clinic TV
pediatric 15.2% walk-in, 42.7% call-in
‧Babes & Sarma (1991) — Algeria
‧Liu & Liu (1998) — Hong Kong
Motivation

Lack of good pre-registration system

High percentage of walk-in

Time lag between registration & consultation
‧accumulation of walk-in patients before consultation
‧schedule late arrival of first pre-registered

Understand the impacts of walk-in arrival rate & preregister ratio
Simulation Setting






Register 8:00 Am ~ 11:30 (210min)
Consult 8:30 Am ~ 12:00 Noon (210min)
# of patient per session (N):
20(  s =10.5 min), 40 (  s =5.25 min), 60(  s =3.5 min)
Service-time distribution:
exponentially, cv =1  s
uniformally , cv = 0.2  s
No-show ratio: ρ= 0.1, 0.2
Pre-registration ratio: α = 0.3, 0.5, 0.7
Walk-in arrival rate: λ=1.5, 2.0
mean inter-arrival time depends on (N,α)
λ =1.5
λ =2.0
λ =1.0
N\α
0.3
0.5
0.7
0.3
0.5
0.7
0.3
0.5
0.7
20
10
14
23.33
7.5
10.5
17.5
15
21
35
40
6.67
9.33
15.55
5
7
11.67
10
14
23.33
60
3.33
4.67
7.78
2.5
3.5
5.84
5
7
11.67
Benchmark ASR (given even # to pre-register)



A2   s
If   0.5
A2 k  A2 k 2  2s ........... k : 2 ~  N
If   0.5
A2 k  A2 k 2  2  s ............ k : 2 ~ (1   ) N
Ai  Ai 1   s .............

i:
 2 1    N  1 ~ N
Note:
(1) the patient with least sequence # has highest priority
(2) no penalty for pre-register ( punctuality assumption)
(3) the best rule has been used in practice in Taiwan
Table Ⅰ . Experimental results using benchmark ASR.
UNIT=MINUTE
N
α ρ λ
20 0.3 0.1 1.5
2
μs
0.2 1.5
=
2
10.5
0.5 0.1 1.5
2
0.2 1.5
2
0.7 0.1 1.5
2
0.2 1.5
2
40 0.3 0.1 1.5
2
μs
0.2 1.5
=
2
5.25
0.5 0.1 1.5
2
0.2 1.5
2
0.7 0.1 1.5
2
0.2 1.5
2
60 0.3 0.1 1.5
2
μs
0.2 1.5
=
2
3.5
0.5 0.1 1.5
2
0.2 1.5
2
0.7 0.1 1.5
2
0.2 1.5
2
TIQ
55.89
66.01
54.04
64
34.27
40.14
32.86
38.63
20.62
22.34
18.91
20.38
55.33
66.84
52.39
64.86
31.82
39.15
29.37
37.34
15.88
18
13.98
16.4
54.82
67.45
52.41
64.79
30.31
38.81
28.25
36.75
13.41
16.18
11.89
14.63
E(F)
3.03
0.91
3.6
1.11
25.86
24.46
30.2
28.72
37.2
34.72
45.05
43.18
0.57
0.06
0.89
0.08
19.46
19.11
25.55
24.76
30.17
28.26
39.3
37.56
0.17
0
0.21
0.02
17.31
16.65
23.47
23.44
26.77
25.09
36.48
35.68
CV=1.0
E(L)
236.62
234.47
231.07
228.1
254.05
253.33
248.77
247.31
261.58
260.15
254.98
253.64
235.04
233.71
227.61
227.63
248.83
248.72
244.32
244.11
254.83
253.44
249.42
248.41
233.58
233.79
227.33
226.88
246.29
246.03
242.77
242.61
251.31
250.34
246.97
245.91
TIQa
14.78
15.18
14.25
14.33
15.07
14.93
13.71
13.42
16.58
16.11
14.16
13.49
9.19
9.08
8.19
8.31
9.51
9.47
7.89
7.88
11.51
10.47
9.1
8.24
6.65
6.72
5.84
5.83
7.01
6.98
5.62
5.56
8.82
7.96
6.77
5.89
TIQw
71.8
85.6
67.72
81.05
51.67
62.85
48.27
58.83
29.21
35.45
27.88
33.27
73.12
89.1
67.55
84.25
51.93
65.87
46.54
60.9
25.1
33.8
23.15
31.66
73.4
90.83
68.38
85.02
51.3
67.45
46.38
61.71
23.07
33.43
21.47
30.92
TIQ
54.29
65.92
52.18
63.65
29.83
37.07
28.11
35.16
12.49
14.15
11.78
13.51
54.71
66.83
52.37
64.86
29.16
37.49
26.98
35.52
10
12.66
9.12
11.94
54.97
67.38
52.36
65.17
28.95
37.91
26.46
35.65
8.69
11.95
8
11.36
E(F)
0.46
0.04
0.73
0.06
11.61
10.43
18.44
17.4
20.58
17.99
31.4
29.57
0.03
0
0.06
0
10.33
9.98
18.85
18.67
18.13
16.05
30.36
29.35
0
0
0.01
0
10.03
9.99
19.58
19.35
16.95
15.66
29.75
29.26
CV=0.2
E(L)
233.7
233.82
227.74
227.48
240.3
239.84
236.82
236.39
245
243.13
241.46
240.17
233.54
233.69
227.4
227.3
239.69
239.64
237.72
237.59
243.15
241.49
240.65
239.85
233.62
233.81
227.48
227.46
239.59
239.6
238.34
238.25
242.08
240.91
240.35
239.87
TIQa
5.51
5.52
5.46
5.47
4.98
4.9
4.49
4.43
5.27
4.32
4.29
3.43
2.87
2.88
2.78
2.78
2.62
2.62
2.32
2.29
3.43
2.28
2.56
1.7
1.95
1.95
1.86
1.86
1.78
1.78
1.52
1.52
2.42
1.45
1.76
1.09
TIQw
73.19
89.26
68.27
83.62
52.33
65.98
47.16
59.75
27.87
34.84
25.97
32.36
74.71
91.5
69.39
86.13
53.08
68.9
46.76
62.09
23.89
34.47
21.42
31.05
75.44
92.63
69.72
86.85
53.44
70.45
46.38
62.94
21.89
33.99
19.67
30.51
Benchmark rule results


 23 min ,
TIQ  52 min ,
TIQ
for α=0.7
for α=0.3
Leave 3~13 min before noon when α=0.3
Leave even 21 min after noon when α=0.7
TIQ_N=20
TIQ_N=40
TIQ_N=60
70
TIQ ( minutes )
60
50
40
30
20
10
0
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
1
1
1
1
1
1
1
1
1
1
1
1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
Figure1. the average waiting time in queue per patient using Benchmark ASR
cv
TIQ_N=20
TIQ_N=40
TIQ_N=60
TIQ/mean service time
20
18
16
14
12
10
8
6
4
2
0
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
1
1
1
1
1
1
1
1
1
1
1
1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
Figure2. TIQ in terms of mean service time using Benchmark ASR
cv
E(F)_N=20
E(F)_N=40
E(F)_N=60
50
E(F) ( minutes )
40
30
20
10
0
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
ρ
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
λ
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
α
1
1
1
1
1
1
1
1
1
1
1
1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
cv
Figure3. physician average idle time per session using Benchmark ASR
E(F)_N=20
E(F)_N=40
E(F)_N=60
E(F)/mean service time
12
10
8
6
4
2
0
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
ρ
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
λ
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
α
1
1
1
1
1
1
1
1
1
1
1
1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
cv
Figure4. physician idle time in terms of mean service time using Benchmark ASR
E(L)_N=20
E(L)_N=40
E(L)_N=60
270
E(L) ( minutes )
260
250
240
230
220
210
200
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
0.1
0.2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
1.5
1.5
2
2
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
0.3
0.3
0.3
0.3
0.5
0.5
0.5
0.5
0.7
0.7
0.7
0.7
1
1
1
1
1
1
1
1
1
1
1
1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
Figure5. the closing time using Benchmark ASR
cv
Improve ASR(1)

kfirst =expected # of walk-in before consultation

na = # of pre-register   N

tlag=time lag between register and consult
A2  tlag  kfirst  p  s


int v  ( Noon  A2 ) / na
A2 k  A2 k 2  int v..............k : 2 ~  N
cv
α
p
1
0.5
1
0.7
1
0.3
3
0.5
2.4
0.7
1.8
0.2
Improve ASR(4)
— when
  0.3, cv  1.0,  ,  , N

A2  tlag  kfirst  p  s

A2 k  A2 k 2  2 s ..............k : 2 ~  N
cv
α
p
1.0
0.3
1.5
Benchmark vs. ASR(1) & (4)
TIQ_Benchmark ASR
TIQ (minutes)
TIQ_Improved ASR
70
60
50
40
30
20
10
0
0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2
1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2
0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7
20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 N
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 cv
Figure1. Comparisons of TIQ between Benchmark ASR and Improved ASR
TIQ/mean service time
TIQ_Benchmark ASR
TIQ_Improved ASR
20
18
16
14
12
10
8
6
4
2
0
0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2
1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2
0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7
20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 N
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 cv
Figure2. Comparisons of TIQ/mean service time between Benchmark ASR and Improved ASR
E(L) (minutes)
E(L)_Benchmark ASR
E(L)_Improved ASR
270
260
250
240
230
220
210
0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2
1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2 1.5 1.5 2
2
0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7 0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.5 0.7 0.7 0.7 0.7
20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 20 20 20 20 20 20 20 20 20 20 20 20 40 40 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 60 60 N
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 cv
Figure3. Comparisons of the closing time between Benchmark ASR and Improved ASR
Conclusions

Walk-in practice need academic research

Improving Benchmark ASR
→ ”closing time” move toward noon.

α is the most influential factor,
α↑,TIQ↓ , IDLE↓ , Leave↓

walk-in arrival rate λ↑, TIQ↑ , IDLE↓ , Leave↓
Future Research

Time lag ↑, more accumulation of walk-ins before
consultation, the impact of delaying A2↑.

“time lag=0” fits for other country Appointment
problem with walk-in(Fetter & Thompson,1996) and
for Taiwan with electronical records.

Need to develop different ASR for different
appointment ratio(α)

May consider different arrival distribution for walk-ins.

More ASRs should be tested and created.