Analysis of Arrival Data from an Endocrinology Clinic

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Transcription:

Analysis of Arrival Data from an Endocrinology Clinic Song-Hee Kim, Ponni Vel, Ward Whitt, and Won Chul Cha January 7, 2 Abstract Queueing models of service systems typically assume that the arrival process is a non-homogeneous Poisson process (NHPP). Kim and Whitt [24] previously tested that hypothesis for arrival data from a call center, and concluded that the data were indeed consistent with an NHPP within each day, but found significant over-dispersion over multiple days. Kim et al. [24] provides data analysis of arrival data from an endrocrinology clinic where arrivals are by appointment, and apply the same statistical methods of Kim and Whitt (24). Despite the intended deterministic pattern, the appointment-generated arrival data are also consistent with an NHPP within a single day, but now there is significant under-dispersion over multiple days. This appendix provides additional materials for the main paper. Keywords: fitting queueing models to data, queues with scheduled arrivals, appointments, statistical tests, Poisson processes, dispersion Yale School of Management, Yale University, Email: hailey.kim@yale.edu Industrial Engineering and Operations Research, Columbia University, Email: pvv2@columbia.edu Industrial Engineering and Operations Research, Columbia University, Email: ww24@columbia.edu Department of Emergency Medicine, Samsung Medical Center, Seoul, Korea, Email: docchaster@gmail.com

Contents Overview 3 2 The Study Data 3 3 Appointment Slots 8 4 Arrivals within Each Shift 2 4. Estimated Arrival Rate Functions................................ 2 4.2 No-Shows and Lateness..................................... 28 4.3 Testing An NHPP Within Shifts................................ 37 Under-Dispersion Over Multiple s 39 2

Overview We present supporting material in this document to the main paper [Kim et al. 24]; the main paper has data analysis for only one doctor, and this document provides the results for all of the 6 doctors. Here is how this document is organized: In 2 we introduce our study data from an endocrinology outpatient clinic and discuss discuss different schedules of 6 doctors. In 4, we compare each doctor s scheduled arrivals and actual arrivals, show the presence of no-shows and early and late arrivals, and conduct statistical tests that show the arrivals are consistent with a PP within shifts. We substantiate under-dispersion over multiple days in. 2 The Study Data The appointment arrival data are from an endocrinology outpatient clinic of a major teaching hospital in South Korea, collected over a 3-wesek period from July 23 to September 23. Sixteen doctors work in this clinic and patients arrive to the clinic knowing which doctor they will meet; hence, each doctor operates as a single-server system. Each doctor works in a subset of available days and shifts. There are three shifts: morning (am) shifts, roughly from 8:3 am to 2:3 pm, afternoon (pm) shifts, roughly from 2:3 pm to 4:3 pm, and full-day shifts. During the weekdays of the 3-week study period, the 6 doctors worked for a total of 228 am shifts, 22 pm shifts, 2 full-day shifts. The shifts are not evenly distributed among the doctors; the numbers ranged from to 46. Given the small number of full-day shifts, this documents will focus on am and pm shifts from now on. 3

Table : Shifts by Doctor over 3 Weeks (from July to September 23) Mon Tues Wed Thurs Fri All Weekdays Doc AM PM Both AM PM Both AM PM Both AM PM Both AM PM Both AM PM Both 2 7 4 9 7 7 9 2 2 6 6 9 2 8 6 3 2 9 2 2 4 2 2 34 2 2 9 2 23 9 2 6 6 4 4 4 2 7 4 7 8 7 3 8 8 9 9 9 9 9 2 2 9 22 22 9 2 9 2 2 2 2 2 3 3 2 3 4 3 23 6 2 2 All 4 74 2 49 38 2 47 4 44 36 48 8 9 228 22 2 4

Figure : Daily totals on am (top) and pm (bottom) shifts for doctors -6. Same Schedule Scheduled Actual Aug Sep Oct Aug Sep Oct

Figure 2: Daily totals on am (top) and pm (bottom) shifts for doctors 7-2. Jul Aug Sep Jul Aug Sep 6

Figure 3: Daily totals on am (top) and pm (bottom) shifts for doctors 3-6. Sep Oct 7

3 Appointment Slots Table 2: B s,j for Doctor 9 - am shift. Slot Different s Avg Var c 2 8..32.23 2.24 8.7.. 22. 8.33. 8.. 8.67. 8.83 3 4 4 4 4 4 4 4 3 2 4 2 4 4 2 4 4 3 3.4.3. 9. 3 4 2 3 3 2 3 3 3 3 3 2 2 2 2 3 4 3 2 3 4 2 2.77.47.6 9.7 3 3 3 2 2 2 4 2 2 3 2 3 2 3 3 3 2 2 3 2 3 3 2.9.3. 9.33 2 2 4 2 3 2 3 2 2 3 3 3 2 3 2 3 3 3 3 2 3 2 2.9.3. 9. 3 2 3 4 3 3 4 3 3 3 3 3 3 2 2 2 2 3 3 3 3 2.77.47.6 9.67 3 3 3 2 2 2 2 3 3 2 2 3 2 3 2 2 2 2 3 2 2 2 2.36.24.4 9.83 3 3 3 3 2 3 3 3 3 3 3 2 2 3 3 3 3 3 2 2 3 3 2.77.8.2. 3 2 3 3 2 3 2 3 2 3 3 3 3 3 3 3 4 4 3 3 3 3 2.9.28.3.7 3 3 3 3 3 3 3 3 2 2 3 3 3 3 3 3 3 3 3 3 3 3 2.9.9..33 2 3 3 3 3 3 3 2 3 3 2 3 3 3 3 2 3 2 3 4 3 3 2.82.2.3. 3 2 3 3 3 2 4 2 3 2 3 3 3 3 3 2 3 3 2 4 3 3 2.82.3.4.67 3 3 3 3 3 2 3 2 3 3 2 3 2 3 2 3 3 3 2 2.4..9.83 2 3 3 3 2 3 2 3 3 3 2 3 3 3 3 3 3 2 3 3 3 2.68.32.4. 3 2 3 2 3 2 3 2 2 4 4 4 2 3 3 3 3 3 3 4 3 4 2.9.2.6.7 3 3 3 3 3 3 3 2 3 3 2 3 2 3 2 3 3 3 3 3 2.64.43.6.33 2 3 3 3 3 3 3 3 3 3 3 3 3 2 2 3 3 3 3 3 3 4 2.9.8.2. 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 3 2 2.77.8.2.67 3 2 3 3 2 3 3 3 3 2 3 3 2 3 3 3 3 3 3 2 3 2.68.32.4.83 3 3 3 3 3 2 2 3 3 2 3 2 4 3 3 3 2 2 3 3 3 2.68.42.6 2. 2 3 3 2 3 3 4 3 3 2 3 3 3 3 3 3 3 3 2 2 3 4 2.86.3.4 2.7 3 3 3 2 3 3 2 3 2 3 3 2 3 3 4 3 2 3 2 3 3 2.68.42.6 2.33 2 4 3 2 3 3 3 3 4 3 3 3 3 2 2 3 3 4 3 3 2.77.66.9 2. 2 3 3 3 3 2 2 2 2 3 3 3 2 4 3 2 3 2.4.27.28 2.67 2 2 4 3 3 2 2 3 3 4 2 3 3.68 2.3.7 2.83 4 3 3 2 4 4.9 2.43 2.66 3..9.9.48 Daily Total 63 62 67 9 6 62 73 7 72 9 69 64 9 7 67 67 68 64 69 66 67 7 66. 2.9. [9, 2) Total 47 3 49 47 44 4 47 48 48 46 46 48 48 49 2 49.9 6.9.3 All slot avg 2. 2. 2.2.9 2. 2. 2.4 2.3 2.3.9 2.2 2..9 2.3 2.2 2.2 2.2 2. 2.2 2. 2.2 2.4 2.3.64.36 All slot var..9 2.2.9.8..8.3..7...6..3.7.8.6.6 2.2.8.6 (across all days) All slot c 2.4.....4.3.3.3..3.4.4.3.3.4.4.4.3..4.3 [9, 2) avg 2.8 2.6 2.9 2.7 2.6 2.4 3. 2.6 2.7 2.7 2.8 2.8 2.6 2.8 2.6 2.7 2.8 2.7 2.7 2.9 2.8 2.8 2.73.34. [9, 2) var.2..2.4.4.4.4.3.2..3.3..2.4.4.4.4.2..4.4 (across all days) [9, 2) c 2.2.7.2.6..6.4.4.3.7.3.4.8.2.6....3... 8

Table 3: B u,j for Doctor 9 - am shift. Slot Different s Avg Var c 2 8... 22. 8.7. 8.33. 8.. 8.67. 8.83.4.2 6.63 9..9.9.48 9.7.4.2 6.63 9.33.. 22. 9... 22. 9.67.9.9.48 9.83.9.9.48... 22..7.. 22..33.4.2 6.63..9.9.48.67.4.2 6.63.83.9.9.48..9.9.48.7.9.9.48.33.. 22....67..83.9.9.48 2.. 2.7.4.2 6.63 2.33.. 22. 2..4.2 6.63 2.67 2.4.22.76 2.83.9.9.48 3..9.9.48 Daily Total 2 3 3 2 3 8 3 2 2 2 3 4 2 3 2.8 2.82.9 [9, 2) Total 3 2 2 3 2 2 2 2 2.36.72.39 9

Table 4: B s+u,j for Doctor 9 - am shift. Slot Different s Avg Var c 2 8. 2.36.34 2. 8.7.. 22. 8.33. 8.. 8.67. 8.83 3 4 4 4 4 4 4 4 3 3 4 3 4 4 3 4 4 3 3..2.8 9. 3 4 2 3 3 2 3 3 3 3 3 3 2 2 2 3 4 3 2 3 4 3 2.86.4. 9.7 3 3 3 2 2 2 4 2 3 3 3 3 2 3 3 3 3 2 3 2 3 3 2.73.3.4 9.33 3 2 4 2 3 2 3 2 2 3 3 3 2 3 2 3 3 3 3 2 3 2 2.64.34. 9. 3 3 3 4 3 3 4 3 3 3 3 3 3 2 2 2 2 3 3 3 3 2.82.44.6 9.67 3 3 3 2 2 2 2 3 3 2 2 3 2 3 2 3 3 2 3 2 2 2 2.4.26.4 9.83 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 3 3 2.86.2.2. 3 2 3 3 2 3 2 3 3 3 3 3 3 3 3 3 4 4 3 3 3 3 2.9.24.3.7 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 2.9...33 2 3 3 3 3 3 3 2 3 3 3 3 3 3 3 2 3 3 3 4 4 3 2.9.24.3. 3 2 3 3 3 2 4 2 3 2 3 4 3 3 3 2 3 4 2 4 3 3 2.9.47.6.67 3 2 3 3 3 2 3 2 3 2 3 3 2 3 2 2 3 2 3 3 3 2 2.9.2.4.83 2 3 3 3 2 2 3 3 3 3 3 2 3 3 3 3 3 3 2 3 3 3 2.77.8.2. 3 3 3 2 3 2 3 2 2 4 4 4 3 3 3 3 3 3 3 4 3 4 3..43..7 3 3 3 2 3 3 3 3 2 3 3 2 3 3 3 2 3 3 3 3 3 2.73.3.4.33 2 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 4 2.9.4.2. 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 3 2 2.77.8.2.67 3 2 3 3 2 3 3 3 3 2 3 3 2 3 3 3 3 3 3 2 3 2.68.32.4.83 3 3 3 3 3 2 3 3 3 2 3 2 4 3 3 3 2 2 3 3 2 3 2.77.28.4 2. 2 3 3 2 3 3 4 3 3 2 3 3 3 3 3 3 3 3 2 2 3 4 2.86.3.4 2.7 3 3 3 2 3 3 2 3 2 3 3 3 3 3 4 3 2 2 3 3 3 3 2.82.2.3 2.33 3 4 3 2 3 3 3 3 4 3 3 3 3 2 2 3 3 4 3 3 2.82.63.8 2. 2 3 3 4 3 2 2 3 2 3 3 3 2 4 3 2 2 3 2.27.3.26 2.67 2 2 4 3 4 4 2 3 3 4 2 3 3.82 2.4.77 2.83 4 3 3 2 4 4. 2.33 2.3 3..8.6 4.7 Daily Total 6 6 67 6 62 63 74 73 74 6 72 72 62 72 69 69 7 68 69 68 7 76 68.23 22.28. [9, 2) Total 3 48 4 48 49 3 3 48 47 3 49 2 3 2.4.88. All slot avg 2. 2. 2.2.9 2. 2. 2.4 2.4 2.4.9 2.3 2.3 2. 2.3 2.2 2.2 2.3 2.2 2.2 2.2 2.3 2. 2.2.66.34 All slot var.6 2. 2.2.9.7..8.3..7.6.8..4.4.6.7.6.6 2.2.8.6 (across all days) All slot c 2.4.4...4.4.3.2.3..3.3.4.3.3.3.3.3.3..4.3 [9, 2) avg 2.8 2.8 2.9 2.8 2.7 2. 3. 2.7 2.8 2.7 2.9 2.9 2.7 2.8 2.6 2.8 2.9 2.8 2.7 2.9 2.9 2.9 2.8.29.4 [9, 2) var..3.2.3.2.3.3.2.2.4.2.3...4.2.3.4.2..3.3 (across all days) [9, 2) c 2.2.4.2.4.3.4.3.3.2.6.2.3.7.2..2.3..3..3.4

Table : B a s,j for Doctor 9 - am shift. Slot Different s Avg Var c 2 8.. 8.7. 8.33. 8.. 8.67. 8.83 2 4 3 4 4 2 4 4 3 2 4 2 4 4 2 4 4 2 3.8.49. 9. 3 3 2 2 3 2 3 3 3 2 3 2 2 2 2 3 4 3 2 2 4 2 2.9.44.7 9.7 3 3 3 2 2 2 2 2 3 3 3 3 3 2 3 2 3 3 2.32.6. 9.33 2 2 4 2 3 2 3 2 2 3 3 2 2 3 3 3 2 3 2 3 2 2.4.63. 9. 3 2 3 2 3 3 3 2 3 3 3 3 3 2 2 2 2 3 3 3 3 2.9.3. 9.67 3 3 2 2 2 2 3 3 2 2 2 2 3 2 2 2 2 3 2 2 2.8.3.7 9.83 3 3 3 3 2 3 3 3 3 3 3 2 2 3 3 3 3 3 2 2 3 3 2.77.8.2. 3 2 3 3 2 3 2 3 2 2 3 2 3 3 3 3 4 2 3 3 3 2.64.43.6.7 3 3 3 2 3 3 3 2 2 2 2 2 3 3 3 3 2 3 3 3 3 2.9.3..33 2 3 2 2 3 3 3 2 2 3 2 2 3 3 3 2 3 2 2 4 3 3 2.9.3.. 3 2 3 3 3 2 4 2 2 3 3 3 3 2 2 3 3 2 4 2 3 2.64.3.8.67 3 3 3 3 3 2 3 2 2 3 2 3 2 3 3 2 3 2.23.8.7.83 2 3 2 3 2 2 3 3 3 2 3 2 3 3 3 3 2 3 3 2.4.4.9. 3 2 3 2 2 3 2 2 3 3 4 3 3 2 3 3 3 3 3 4 2.64.62.9.7 3 3 3 3 3 3 3 2 3 3 2 3 2 2 2 3 2 2 3 2 2.4.4.7.33 2 3 3 3 3 2 3 3 3 3 3 2 3 2 3 2 3 3 2 3 4 2.68.42.6. 3 3 2 3 3 3 3 3 3 3 3 2 2 3 3 2 2 2 2 3 2 2..3..67 3 2 2 3 2 3 2 2 2 2 3 3 2 3 3 3 3 3 2 2 3 2.4.3.6.83 3 3 3 3 3 2 2 3 3 2 3 2 4 3 3 2 2 2 3 3 2...8 2. 2 3 2 2 3 3 3 3 2 3 2 3 3 2 3 2 2 2 3 2.32.. 2.7 3 3 3 2 3 3 2 3 2 3 3 2 3 3 4 3 2 3 3 3 2.64.3.8 2.33 4 3 2 3 3 3 3 4 3 3 2 3 2 2 3 3 4 3 3 2.68.8. 2. 2 3 2 2 3 2 2 2 3 3 3 2 4 3 2 3 2..24.3 2.67 2 2 4 3 3 2 3 3 4 2 3 2.9 2.6.82 2.83 4 3 3 2 4 3.9 2.8 2.64 3.. Daily Total 6 6 6 9 6 63 63 67 63 4 3 68 62 62 62 9 6 9 62 6 6.9 8.73. [9, 2) Total 4 48 42 46 42 49 43 4 4 47 42 42 49 42 44 47 43 4 46 48 46 4.27 6.9. All slot avg.9.9 2..6.9.9 2. 2. 2.2.8 2..7.7 2.2 2. 2. 2..9 2..9 2. 2..9.9.42 All slot var.6.9 2...8......6.2...6..6.7. 2..9.6 (across all days) All slot c 2.4...6..4.4.4.3..4.4..3.4.4.4..3.6..4 [9, 2) avg 2.8 2. 2.7 2.3 2.6 2.3 2.7 2.4 2. 2. 2.6 2.3 2.3 2.7 2.3 2.4 2.6 2.4 2. 2.6 2.7 2.6 2.4.47.7 [9, 2) var.2.6..6.4..3.4.3.4.4..7.2.8.4.4.8.3..6.8 (across all days) [9, 2) c 2.2..7..6.9.4.6.4.6..9.3.3..6...4.8.8.3

Table 6: B a u,j for Doctor 9 - am shift. Slot Different s Avg Var c 2 8.. 8.7. 8.33. 8.. 8.67. 8.83.4.2 6.63 9..9.9.48 9.7.4.2 6.63 9.33.. 22. 9... 22. 9.67.9.9.48 9.83.9.9.48... 22..7..33.4.2 6.63... 22..67.4.2 6.63.83.9.9.48..9.9.48.7.. 22..33.. 22....67..83.9.9.48 2.. 2.7.4.2 6.63 2.33.. 22. 2..9.9.48 2.67 2.4.22.76 2.83.9.9.48 3..9.9.48 Daily Total 2 3 2 2 3 7 3 2 2 3 4 2 2.9 2.43.64 [9, 2) Total 3 2 2 2 2 2 2 2 2.23.76. 2

Table 7: B a,j for Doctor 9 - am shift. Slot Different s Avg Var c 2 8.. 8.7. 8.33. 8.. 8.67. 8.83 2 4 3 4 4 2 4 4 3 3 4 3 4 4 3 4 4 2 3.32.27.2 9. 3 3 2 2 3 2 3 3 3 2 3 3 2 2 2 3 4 3 2 2 4 3 2.68.42.6 9.7 3 3 3 2 2 2 2 3 3 2 3 3 3 3 3 3 2 3 3 2.4..9 9.33 3 2 4 2 3 2 3 2 2 3 3 2 2 3 3 3 2 3 2 3 2 2.4.64. 9. 3 3 3 2 3 3 3 2 3 3 3 3 3 2 2 2 2 3 3 3 3 2.64.34. 9.67 3 3 2 2 2 2 3 3 2 2 2 2 3 2 3 3 2 3 2 2 2.27.4.8 9.83 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 3 3 2.86.2.2. 3 2 3 3 2 3 2 3 3 2 3 2 3 3 3 3 4 2 3 3 3 2.68.42.6.7 3 3 3 2 3 3 3 2 2 2 2 2 3 3 3 3 2 3 3 3 3 2.9.3..33 2 3 2 2 3 3 3 2 2 3 3 2 3 3 3 2 3 3 2 4 4 3 2.73.4.. 3 2 3 3 3 2 4 2 2 3 3 3 3 2 2 3 4 2 4 2 3 2.68.6.8.67 3 3 3 3 2 3 2 3 2 2 3 2 3 2 2 3 3 2 3 2.36.3.9.83 2 3 2 3 2 2 3 3 3 3 2 3 2 3 3 3 3 2 3 3 2..4.7. 3 3 3 2 2 3 2 2 3 3 4 2 3 3 2 3 3 3 3 3 4 2.73.49.7.7 3 3 3 2 3 3 3 3 2 3 3 2 3 2 2 2 3 2 2 3 2 2..36.6.33 2 3 3 3 3 2 3 3 3 3 3 2 3 2 2 3 2 3 3 2 3 4 2.73.3.4. 3 3 2 3 3 3 3 3 3 3 3 2 2 3 3 2 2 2 2 3 2 2..3..67 3 2 2 3 2 3 2 2 2 2 3 3 2 3 3 3 3 3 2 2 3 2.4.3.6.83 3 3 3 3 3 2 3 3 3 2 3 2 4 3 3 2 2 2 3 2 3 2.64.43.6 2. 2 3 2 2 3 3 3 3 2 3 2 3 3 2 3 2 2 2 3 2.32.. 2.7 3 3 3 2 3 3 2 3 2 3 3 3 3 3 4 3 2 2 3 2 3 3 2.77.28.4 2.33 2 4 3 2 3 3 3 3 4 3 3 2 3 2 2 3 3 4 3 3 2.73.68.9 2. 2 3 2 2 3 2 3 2 3 3 3 2 4 3 2 2 3 2.9.23.28 2.67 2 2 4 3 4 4 3 3 4 2 3 2.73 2.49.84 2.83 4 3 3 2 4 3. 2. 2. 3..9.9.48 Daily Total 62 63 6 2 6 6 64 6 69 66 6 6 69 64 64 6 63 6 6 64 66 62. 7.2. [9, 2) Total 48 48 43 47 43 44 47 4 49 44 44 49 43 46 49 4 4 46 47 46. 6.26. All slot avg 2. 2. 2..7.9 2. 2. 2. 2.2.8 2. 2..8 2.2 2. 2. 2. 2. 2. 2. 2. 2. 2.2.6.39 All slot var.6.9 2...8...4.6..6.6.4.4.6..6.8. 2..9.6 (across all days) All slot c 2.4.....4.4.3.3..4.4.4.3.4.4.4.4.3...4 [9, 2) avg 2.8 2.7 2.7 2.4 2.6 2.4 2.8 2.4 2.6 2. 2.7 2.4 2.4 2.7 2.4 2.6 2.7 2. 2. 2.6 2.8 2.6 2.8.42.6 [9, 2) var..4...3.4.3.4.3.4.2..6.2.7.3.3..3...8 (across all days) [9, 2) c 2.2..7.9.4.6.4.6.4.6.3.8..3.3.4.4.6.4.8.7.2 3

Table 8: B s,j for Doctor 9 - pm shift. Slot Different s Avg Var c 2... 22..7.. 22..33....67..83. 2.. 2.7. 2.33. 2.. 2.67. 2.83 6 4 3 4 4 4 4 3 4 4 4 3 4 3 3 4 2 3.9 2.8.3 3. 3 4 2 3 3 3 3 4 2 2 3 3 3 3 3 3 3 3 2.68.37.9 3.7 3 2 3 2 2 3 2 3 3 2 2 3 3 3 3 3 2 3 3 3 2.4.82.4 3.33 2 3 3 3 3 3 3 2 3 2 2 3 3 2 2 2 3 4 2 2 2.36.9.6 3. 3 3 4 3 3 3 3 3 2 3 2 2 2 3 3 3 3 2 3 3 3 3 2.82.2.3 3.67 3 3 2 3 3 2 3 3 2 2 3 3 3 2 3 2 3 2 3 3 3 3 2.68.23.3 3.83 3 3 3 2 3 3 3 3 4 2 2 3 3 3 3 2 2 3 3 2 2..74. 4. 2 2 2 2 3 2 3 3 3 3 3 3 3 2 3 2 2 2 2 3 3 2.4.3.6 4.7 3 2 2 2 2 3 3 2 3 3 2 4 3 3 2 3 3 3 2 3 3 3 2.68.32.4 4.33 2 3 3 3 3 3 2 3 3 2 3 3 3 2 3 3 3 2 3 2 2..4.7 4. 2 2 3 3 3 3 3 3 3 3 2 4 3 3 2 2 3 2 3 3 2.9.4.8 4.67 3 3 2 2 2 3 3 3 3 3 2 2 2 2 3 2 3 2 3 3 2.4.44.8 4.83 2 3 3 2 3 3 2 2 2 2 2 3 3 3 2 3 3 3 2 3 2 2.4.3.6. 3 2 2 3 3 2 3 3 3 3 3 3 2 3 3 3 3 2 3 2...9.7 3 3 2 2 3 2 2 3 3 3 3 4 3 2 3 3 2 2 2 2.36.8..33 3 2 3 2 3 3 3 2 3 3 2 3 3 3 3 3 2 3 3 3 2 2.64.34.. 2 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 2.73.4..67 3 2 3 2 3 3 3 3 3 3 3 2 2 3 3 3 3 3 2 2 3 4 2.77.28.4.83 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 3 3 3 3 2 2 2.77.28.4 6. 3 3 2 2 3 3 3 3 2 3 2 3 2 3 2 3 2 3 3 3 3 2.9.3. 6.7 3 3 2 3 3 3 2 3 3 2 3 2 3 3 2 2 3 3 2.32.8. 6.33 3 2 3 3 3 3 2 2 3 2 3 3 2 2 3.9.32.36 6. 3 2 2 3 2 3 2 2 3 2 3 2.4.49.7 6.67 3 3 2 3 2.68.27 2.74 6.83.9.9.48 7. 2.9.8 22. 7.7 3.4.4 22. Daily Total 7 63 2 4 7 6 3 66 6 68 8 4 43 6 67 6 6 62 63 8 9 8.73 3.4. [3, 6) Total 4 49 4 44 47 47 42 49 48 2 4 4 4 47 2 44 48 42 48 44 46.4.4. All slot avg.6.8...6.7..9.7.9.7..2.7.9.7.9.8.8.7.7...96.82 All slot var 2.4 2..9.8 2..9 2. 2..9 2.3.7.8 2. 2. 2..9.8.8 2. 2.3 2..9 (across all days) All slot c 2.9.7.9.7.8.7.9.6.7.6.6.7.3.7.6.6..6.6.8.7.9 [3, 6) avg 2. 2.7 2. 2.4 2.6 2.6 2.3 2.7 2.7 2.9 2. 2.3 2.3 2.6 2.9 2.4 2.7 2.3 2.7 2.8 2.8 2.4 2.8.2.8 [3, 6) var..3.7.3...3.2.4.3.3.6...2.6.2.8.2.6.2.6 (across all days) [3, 6) c 2.8.4.2.4.7.7.24.3..4.4..29.7.3..3..3.8.2. 4

Table 9: B u,j for Doctor 9 - pm shift. Slot Different s Avg Var c 2... 22..7..33....67..83. 2.. 2.7. 2.33. 2.. 2.67.. 22. 2.83.4.2 6.63 3... 22. 3.7.9.9.48 3.33.. 22. 3.. 3.67.8.6 4.7 3.83.. 22. 4..9.9.48 4.7.. 22. 4.33.9.9.48 4..9.9.48 4.67.9.9.48 4.83.. 22.. 2.4.22.76.7..33.... 22..67.. 22..83.. 22. 6..4.2 6.63 6.7 2.4.22.76 6.33.8.6 4.7 6..23.8 3.6 6.67.. 22. 6.83. 7.. 7.7. Daily Total 2 3 2 2 3 3 2 3 4 4 7 2 2 2 2. 2.7.6 [3, 6) Total 2 3 3 4 3.4.36.

Table : B s+u,j for Doctor 9 - pm shift. Slot Different s Avg Var c 2..9.9.48.7.. 22..33....67..83. 2.. 2.7. 2.33. 2.. 2.67.. 22. 2.83 6 3 4 4 4 4 3 4 4 4 4 4 3 3 4 3 3.23 2.9.28 3. 3 4 2 3 3 3 3 4 3 2 3 3 3 3 3 3 3 3 2.73.3.8 3.7 3 2 3 2 2 3 3 3 3 2 2 3 3 3 3 3 3 3 3 3 2..83.3 3.33 2 3 3 3 3 3 3 2 3 3 2 3 3 2 2 2 3 4 2 2 2.4.92.6 3. 3 3 4 3 3 3 3 3 2 3 2 2 2 3 3 3 3 2 3 3 3 3 2.82.2.3 3.67 3 3 2 3 3 3 3 3 2 3 3 3 3 2 3 2 4 3 3 3 3 3 2.86.22.3 3.83 3 3 3 2 3 3 3 3 4 2 2 3 3 3 3 2 2 3 3 2 2.9.4.8 4. 2 3 2 3 3 2 3 3 3 3 3 3 3 2 3 2 2 2 2 3 3 2..3. 4.7 3 2 2 2 2 3 3 2 3 3 2 4 3 3 3 3 3 3 2 3 3 3 2.73.3.4 4.33 2 3 3 3 3 3 2 3 3 3 3 3 3 2 2 3 3 3 2 3 2 2.64.34. 4. 2 2 2 3 3 3 3 3 3 3 3 2 4 3 3 2 2 2 3 2 3 3 2.68.32.4 4.67 3 3 2 2 2 3 3 3 3 3 3 2 2 2 3 2 2 3 2 3 3 2..36.6 4.83 2 3 3 2 3 3 2 2 3 2 2 3 3 3 2 3 3 3 2 3 2 2..36.6. 3 2 2 3 3 2 3 3 3 3 3 2 3 2 3 3 3 3 3 2 3 2.64.34..7 3 3 2 2 3 2 2 3 3 3 3 4 3 2 3 3 2 2 2 2.36.8..33 3 2 3 2 3 3 3 2 3 3 2 3 3 3 3 3 2 3 3 3 2 2.64.34.. 3 3 3 3 3 3 2 3 3 3 3 3 3 3 3 3 3 3 3 3 2.77.37..67 3 2 3 2 3 3 3 3 3 3 3 2 2 3 3 3 3 3 2 3 3 4 2.82.2.3.83 3 3 3 3 3 3 3 3 3 3 3 2 3 3 4 3 3 3 3 2 2 2.82.3.4 6. 3 3 2 2 3 3 3 3 2 3 2 3 3 3 3 3 2 3 3 3 2 3 2.73.2.3 6.7 3 3 2 3 3 3 2 3 3 2 3 3 3 3 3 2 2 3 3 2.4.74.2 6.33 3 2 2 3 3 3 3 2 2 3 3 3 3 2 2 2 3 2.9.3.26 6. 2 3 2 3 3 2 3 3 2 3 2 3 3.64.67.62 6.67 3 3 2 3 2.73.26 2.37 6.83.9.9.48 7. 2.9.8 22. 7.7 3.4.4 22. Daily Total 9 66 2 6 8 6 3 68 6 7 6 4 64 7 6 66 69 6 6 6 6.77 46.66. [3, 6) Total 47 4 4 47 48 42 48 48 4 42 48 3 48 49 4 49 2 44 47. 2.64. All slot avg.7.9..6.7.7..9.7 2..7.6.3.8 2..9.9 2..9.7.7..6 2..78 All slot var 2. 2.3.9.8 2. 2. 2. 2..9 2.4.8.7 2. 2. 2.2 2..9.7 2. 2.3 2..9 (across all days) All slot c 2.9.6.9.7.8.6.9..7.6.6.7.3.6..6..4.6.8.7.9 [3, 6) avg 2.6 2.8 2. 2. 2.6 2.7 2.3 2.8 2.7 3. 2.7 2.3 2.3 2.7 2.9 2.7 2.7 2. 2.7 2.9 2.8 2.4 2.64.48.7 [3, 6) var.4.3.7.3...3.2.4.2.2.6.4.4.2..3..2.6.2.6 (across all days) [3, 6) c 2..4.2.4.7.7.24.2..2.3..26..2.7.4.8.3.7.2. 6

Table : B a s,j for Doctor 9 - pm shift. Slot Different s Avg Var c 2...7.. 22..33....67..83. 2.. 2.7. 2.33. 2.. 2.67. 2.83 6 4 3 3 4 4 3 3 4 4 3 2 4 3 3 4 2 2.9 2.7.33 3. 3 3 2 3 2 3 2 4 2 3 3 3 3 3 2 4 3 3 2.4.2.2 3.7 3 2 3 2 2 3 2 3 3 2 2 3 3 3 3 3 2 3 3 2 2.36.8. 3.33 2 2 3 3 3 3 2 2 3 2 2 3 3 2 2 2 4 2 2 2.8.92.9 3. 3 3 3 3 3 3 3 3 3 2 2 2 3 3 3 3 2 2 2 3 3 2.64.34. 3.67 3 3 2 2 3 2 3 3 2 2 3 3 3 2 3 2 2 2 3 3 3 2..36.6 3.83 2 3 2 3 3 3 3 4 2 3 3 3 3 2 2 3 3 2 2.36.9.6 4. 2 2 2 2 3 3 3 2 3 3 3 2 3 2 2 2 2 3 3 2.27.49. 4.7 3 2 2 2 3 3 2 2 3 2 3 2 3 2 3 2 3 2 3 3 3 2.4.3.6 4.33 3 3 2 3 3 2 3 3 2 3 3 2 2 3 3 2 2 2 2.27.9. 4. 2 2 3 3 3 3 3 3 3 3 2 4 3 2 2 2 3 3 3 2..64. 4.67 3 3 2 2 2 3 2 3 3 2 2 2 2 2 3 2 3 3 3 2.23.66.3 4.83 2 3 3 2 3 3 2 2 2 2 2 3 2 3 2 3 3 3 2 2 2.32.42.8. 3 2 2 2 2 2 3 3 3 3 2 3 3 3 2 2 2 2.9.7.7.7 3 3 2 3 2 2 2 3 3 3 4 2 2 3 3 2 2.4.98.2.33 3 2 3 2 3 3 3 2 3 3 2 2 3 2 3 3 2 3 3 2 2.4.4.7. 2 3 2 3 3 3 3 2 3 3 3 3 3 3 2 3 2.8..23.67 3 2 3 2 3 3 3 3 3 2 2 2 2 2 3 2 3 2 3 4 2.4..9.83 3 3 3 3 3 2 3 3 3 2 3 3 2 3 3 3 3 2 2 2...9 6. 3 2 2 2 3 3 3 3 2 2 2 3 3 2 3 2 3 2 3 2 2.32.6. 6.7 3 3 2 3 3 2 2 3 3 2 3 2 3 3 2 2 2 3 2.23.76. 6.33 3 2 3 2 3 3 2 3 3 3 2 3.73.3.4 6. 3 2 2 2 2 3 2 2 3 2 3 2.36.39.74 6.67 3 3 2 3 2.68.27 2.74 6.83.9.9.48 7. 2.9.8 22. 7.7 3.4.4 22. Daily Total 2 8 47 49 3 8 6 7 62 4 48 39 6 6 7 6 62 6 48 2 4 3.86 37.93. [3, 6) Total 4 4 4 4 43 4 4 46 4 48 4 37 38 43 47 4 43 42 43 4 4 39 42.32 8.89. All slot avg..7.3.4..7.4.7.6.8..4..6.7.6.7.8.6.4..3.42.78.88 All slot var 2.4.9.6..9.9 2..8.8 2...6.8.8.9.8.6.8.8.8.8.7 (across all days) All slot c 2..7.9.8.8.7..6.7.7.6.8.4.7.6.7..6.7..8. [3, 6) avg 2.2 2. 2.2 2.2 2.4 2. 2.2 2.6 2. 2.7 2.3 2. 2. 2.4 2.6 2.2 2.4 2.3 2.4 2.3 2. 2.2 2.3.66.2 [3, 6) var..4.8.4...4.4...3.8.4.6..8.4.8.4... (across all days) [3, 6) c 2.2.6.6.8.9.8.28.6.8.7.6.8.3..7.6.6..6.2.8.23 7

Table 2: B a u,j for Doctor 9 - pm shift. Slot Different s Avg Var c 2...7..33....67..83. 2.. 2.7. 2.33. 2.. 2.67.. 22. 2.83.4.2 6.63 3... 22. 3.7.. 22. 3.33.. 22. 3.. 3.67.8.6 4.7 3.83.. 22. 4..9.9.48 4.7.. 22. 4.33.9.9.48 4..9.9.48 4.67.. 22. 4.83.. 22.. 2.4.22.76.7..33....67.. 22..83.. 22. 6..4.2 6.63 6.7.9.9.48 6.33.4.2 6.63 6..23.8 3.6 6.67.. 22. 6.83. 7.. 7.7. Daily Total 3 2 3 2 2 3 4 4 6 2 2 2.82 2.44.74 [3, 6) Total 3 2 4 3..24.24 8

Table 3: B a,j for Doctor 9 - pm shift. Slot Different s Avg Var c 2...7.. 22..33....67..83. 2.. 2.7. 2.33. 2.. 2.67.. 22. 2.83 6 3 3 4 4 3 3 4 4 3 3 4 3 3 4 3 3. 2.8.3 3. 3 3 2 3 2 3 2 4 3 3 3 3 3 3 2 4 3 3 2.4.2.2 3.7 3 2 3 2 2 3 2 3 3 2 2 3 3 3 3 3 3 3 3 2 2.4.82.4 3.33 2 2 3 3 3 3 2 2 3 3 2 3 3 2 2 2 4 2 2 2.23.9.9 3. 3 3 3 3 3 3 3 3 3 2 2 2 3 3 3 3 2 2 2 3 3 2.64.34. 3.67 3 3 2 2 3 3 3 3 2 3 3 3 3 2 3 2 3 3 3 3 3 2.68.32.4 3.83 2 3 2 3 3 3 3 4 2 3 3 3 3 2 2 3 3 2 2.4.73.3 4. 3 3 2 2 3 3 3 2 3 3 3 2 3 2 2 2 2 3 3 2.36.3.9 4.7 3 2 2 2 3 3 2 2 3 2 3 2 3 3 3 2 3 2 3 3 3 2..36.6 4.33 3 3 2 3 3 2 3 3 3 3 3 2 2 2 3 3 2 2 2 2.36.3.9 4. 2 2 2 3 3 3 3 3 3 3 3 2 4 3 2 2 2 2 3 3 3 2.9.44.7 4.67 3 3 2 2 2 3 2 3 3 2 2 2 2 2 3 2 2 3 3 3 2.27.9. 4.83 2 3 3 2 3 3 2 2 3 2 2 3 2 3 2 3 3 3 2 2 2.36.43.8. 3 2 2 2 2 2 3 3 3 3 2 2 3 3 3 3 2 2 2 2.23.66.3.7 3 3 2 3 2 2 2 3 3 3 4 2 2 3 3 2 2.4.98.2.33 3 2 3 2 3 3 3 2 3 3 2 2 3 2 3 3 2 3 3 2 2.4.4.7. 2 3 2 3 3 3 3 2 3 3 3 3 3 3 2 3 2.8..23.67 3 2 3 2 3 3 3 3 3 2 2 2 2 2 3 2 3 2 2 3 4 2..4.7.83 3 3 3 3 3 2 3 3 3 2 3 3 2 4 3 3 3 2 2 2..64. 6. 3 2 2 2 3 3 3 3 2 2 2 3 2 3 3 3 2 3 2 3 2 2.4.3.6 6.7 3 3 2 3 3 2 2 3 3 2 3 3 3 3 2 2 2 2 3 2.32.7.3 6.33 3 2 2 3 2 3 3 2 3 2 3 3 2 2 3.86.27.36 6. 2 3 2 3 2 2 3 3 2 3 2 3 3.9.9.63 6.67 3 3 2 3 2.73.26 2.37 6.83.9.9.48 7. 2.9.8 22. 7.7 3.4.4 22. Daily Total 3 6 47 3 9 62 7 6 6 49 4 9 6 6 6 68 8 4 4.68 49.8.2 [3, 6) Total 4 46 4 4 43 46 4 46 4 43 37 39 44 48 44 44 4 44 42 4 39 43.32.7. All slot avg..7.3...7.4.8.6.9.6.4.2.7.9.7.7.9.7.4..3.47.83.8 All slot var 2.4 2..6.6.9.9 2..8.8 2.2.6..8.9 2..9.7.6.9.8.8.7 (across all days) All slot c 2..7.9.7.8.7..6.7.6.6.8.3.7.6.6..4.7.9.8. [3, 6) avg 2.3 2.6 2.2 2.3 2.4 2.6 2.2 2.6 2. 2.8 2.4 2. 2.2 2.4 2.7 2.4 2.4 2. 2.4 2.3 2. 2.2 2.4.64. [3, 6) var.9.4.8.4...4.4..4.4.8.3...7.4..4.9.. (across all days) [3, 6) c 2.8.6.6.9.9.8.28.6.8..6.8.28.8.7.2.6.8.6.7.8.23 9

4 Arrivals within Each Shift 4. Estimated Arrival Rate Functions Figure 4: Scheduled arrivals () and actual arrivals () during the morning shifts of doctors -4. Avg or Avg Avg 2 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2

Figure : Scheduled arrivals () and actual arrivals () during the morning shifts of doctors -8. 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2

Figure 6: Scheduled arrivals () and actual arrivals () during the morning shifts of doctors 9-2. 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 22

Figure 7: Scheduled arrivals () and actual arrivals () during the morning shifts of doctors 3-6. 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 2 Avg or 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7. 8. 9... 2. 3. 7 8 9 2 3 4 23

Figure 8: Scheduled arrivals () and actual arrivals () during the afternoon shifts of doctors -4. 2 8 6 4 2 2 8 6 4 2 Avg or 2 8 6 4 2 Avg Avg 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 24

Figure 9: Scheduled arrivals () and actual arrivals () during the afternoon shifts of doctors -8. 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2

Figure : Scheduled arrivals () and actual arrivals () during the afternoon shifts of doctors 9-2. 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 26

Figure : Scheduled arrivals () and actual arrivals () during the afternoon shifts of doctors 3-6. 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 2 2 2 2 8 6 4 2 8 6 4 2 Avg or 8 6 4 2 9 2 3 4 6 7 9 2 3 4 6 7 9 2 3 4 6 7 7 8 9 2 3 4 27

4.2 No-Shows and Lateness Table 4: Details of am shifts for doctors -8. Averages over shifts with 9% confidence intervals are shown. (X measured in minutes.) Doc Interval Avg # Scheduled % No-show % Late Avg(X + ) Avg(X ) Avg(X) [8, 9].2±.4 6.7±2.8 7.7 -.4±7.7-8.±34. [9, ] 9.2±.7.4±3.4 9.±3.7 22.3±3. -27.8±3.9-7.6±4.2 [, ] 8.±.4 6.2±3. 2.6±.3 6.2±4.7-32.7±4. -22.3±4. [, 2] 9.2±.2 6.4±2.4 2.6±4.2.6±4. -4.6±6.3-33.7±.3 [2, 3] 8.9±2..±2.3.2±. 9.4±3. -4.7±8.9-47.6±8. [3, 4].7±.6-76.3±87.9-76.3±87.9 T otal 76.9±4..8±.8.6±2. 6.±4.8-39.±2.9-3.8±2.9 2 [8, 9] 2.6±.4.8±6.8 8.7±3. 37.4±39. -2.±6.4-2.8±.9 [9, ] 4.4±. 7.±4. 4.±6. 24.9±3.9-33.4±3.4-2.±4. [, ] 6.3±.8 8.±3.6.8±. 4.9±8. -43.2±4. -3.8±3.8 [, 2].±.9 6.6±2.8.8±4.4 2.7±4. -48.6±8.2-42.±9. [2, 3].±.7.6±6..2±9. 6.2±4.6-7.±2.8 -.3±3. T otal 3.9±2.6 7.8±2. 2.6±3.3 8.±4. -42.±2.9-34.9±3.8 3 [8, 9].6±. -4.6±9.8-4.6±9.8 [9, ] 3.±.9 4.7±3.6.3±4.6 2.±7.4-36.±3. -3.4±4.7 [, ] 3.2±.2 9.6±.4 6.6±3. 24.2±2. -48.7±6.7-43.6±6. [, 2] 2.6±.9 9.±4.4.±4.8 7.7±.8 -.±. -48.±2.3 [2, 3].±..7±.8 7.4±6.2 6.4±4.9-6.2±3.9-6.3±6.6 [3, 4].6±.4 33.±. 6.9±3. -3.±6.9-9.3±23.2 T otal.4±3.3 6.9±2.4 9.2±2.3 7.2±.7 -.±3.9-43.9±4.8 4 [9, ] 2.9±. 7.±.9 9.9±2. 9.4±4.3-29.±2. -9.8±2.3 [, ] 2.2±.7 8.9±2. 8.6±3.9 4.2±2.4-3.4±2.4-23.±3.3 [, 2] 22.2±.9 6.2±.4.7±2.4 9.6±.8-4.9±3.3-34.7±2.8 [2, 3].±.9 27.2±8.2 6.7±.8-49.8±3. -3.±8.6 T otal 6.4±.8 7.2±. 6.9±.9.6±.9-34.7±.7-26.2±.8 [8, 9] 3.±.9.3±2.4 22.3±7.4 34.8±2.7-2.±.6-8.±4.3 [9, ] 24.8±.7.8±2. 22.2±3.7 9.3±6.8-24.±2. -4.3±3.3 [, ] 23.3±. 4.3±.4 22.±3.8 3.8±4. -22.9±.7-4.9±2.6 [, 2] 22.3±. 4.7±.7 9.4±4.6 3.±4.3-28.±3.2-2.6±3. [2, 3] 2.2±2. 4.8±2..9±2.9.6±3.8-33.±3.4-28.±3.4 [3, 4].4±.4.±27.8. -44.8±67.7-43.4±69.2 T otal.±..±.9 9.±2.7 6.7±3. -26.4±.4-7.9±2.2 6 [8, 9] 4.±2.7 7.4±2.9 2.3±3.6 49.3±22.2-2.±.8-6.±4.8 [9, ] 26.8±. 8.3±2.7.8±2. 27.6±3.2-3.±3. -28.±3.6 [, ] 2.4±2.6 6.6±2.4.2±3.8.4±3.4-47.7±3.8-4.9±4. [, 2] 23.±4.4 7.9±2.4 9.±6.7 2.8±6.6-66.±7.6-6.±9.2 [2, 3] 2.9±.2.4±7.3 9.2±6.3 8.±.4-76.6±3.8-68.9±3.9 T otal 2.8±2. 7.9±.4.2±.8 24.±7.9-48.2±2.9-4.±2.4 7 [8, 9] 2.8±.4 8.8±4.3 3.3±26.8-23.8±7.6-4.3±.7 [9, ] 22.±. 8.6±3.4 3.2±. 37.7±2.6-36.±4. -27.7±4.8 [, ] 22.8±.8 3.6±2.7 9.7±3.7 29.2±6. -48.7±3.4-4.±4.8 [, 2] 24.±.4.±2.9 7.±4.2 29.6±9.2-6.4±6. -.±7.3 [2, 3] 7.9±4. 6.7±4.4.8±3.7.4±6. -6.4±.8-7.±.2 [3, 4].7±. -82.±46.2-82.±46.2 T otal 9.3±7..9±.6 9.6±2.2 27.7±8.8 -.8±4. -44.±4.4 8 [8, 9].7±. 6.7±42.8-4.±. -4.±. [9, ].4±2. 3.±2.8 6.6±9. 8.±4.9-36.±4. -27.3±6.7 [, ] 4.9±2.7 7.4±4.6 3.7±8.3 8.6±9.2-49.±. -4.2±2. [, 2].2±2.8.7±3.8 8.7±3.7 6.4±8.7-2.3±. -46.2±3.7 [2, 3].6±3.4 7.±4. 8.3±7.3 2.4±4. -62.8±8.8 -.±8.3 T otal 6.8±..9±2.2 2.4±. 6.2±.7-48.2±4. -4.4±.9 28

Table : Details of am shifts for doctors 9,, 2, 3,, and 6. Averages over shifts with 9% confidence intervals are shown. (X measured in minutes.) Doc Interval Avg # Scheduled % No-show % Late Avg(X + ) Avg(X ) Avg(X) 9 [8, 9] 3.8±.6 3.6±8.3 2.3±.9 3.8±22. -23.3±4. -.±6.8 [9, ].9±.8 6.3±3. 6.4±2.8 3.4±6.7-32.6±3.2-22.±3. [, ] 6.6±.8 8.9±2.6 6.4±3. 3.9±.2-43.6±4.6-34.±4.7 [, 2] 6.6±. 7.8±2.7 2.±3.7 6.±7.9-7.8±6.2-48.3±6. [2, 3] 3.±.7 7.3±2.7 7.7±2.9.±.9-6.2±8.3 -.4±8.8 [3, 4].±.. T otal 66.±2. 8.2±.6 4.±.6 2.8±. -46.2±2.9-36.7±3. [9, ] 4.±..±3.6 9.3±7.9 2.±9.3-3.3±4.2-2.±4.9 [, ].7±.4 7.±3.9.7±4. 8.6±9. -4.2±8.2-34.2±6.8 [, 2].2±. 4.6±3.8 7.8±3.6 8.8±22.3-6.±8.7-3.6±7.4 [2, 3].2±. 6.7±42.8 2. -63.±6.8-49.±8.2 T otal 46.6±3..8±2. 2.±3.4 2.±7. -4.7±4.3-37.4±4.4 2 [8, 9].±.2-2.9±. -2.9±. [9, ].±..8±. 2.9±7. 32.3±37.7-29.9±3. -8.6±.8 [, ] 3.4±.4.9±7.3 4.7±6.4 24.3±9.4-44.7±8.8-34.9±7.7 [, 2] 3.±.8 9.±.4 3.3±7.4 6.9±7.3 -.6±. -4.4±.2 [2, 3] 4.4±.4.8±9.9.2±9.4.±.4-6.4±2.4-47.±22.7 T otal 42.±2.7.3±4.9.7±3.4 8.7±6.8-44.2±. -34.±3.6 3 [9, ].8±. 8.9±3.7 2.4±. 3.8±2.9-33.2±4.4-2.3±7.3 [, ] 2.±. 8.3±6. 6.4±7. 2.7±.3-46.3±8.4-3.9±.6 [, 2] 2.3±.2.4±7.6.9±6.2 9.3±4. -8.8±.6 -.4±2. [2, 3] 4.3±.3.±.7 6.±7.4 2.8±34.2-2.4±23.2-49.7±24. T otal 4.8±2.8 9.4±2.7.±. 24.9±7. -47.6±6.3-37.6±7.2 [9, ].±.8.±4.9 23.2±9.6 27.7±. -3.9±. -6.7±9. [, ].7±. 4.±4. 6.6±7.7.6±9.4-47.9±4. -37.4±2. [, 2].8±..3±6.3 4.8±9. 6.6±6.4-62.3±. -.3±.9 [2, 3] 3.±.8 6.3±4.8.9±.3 6.9±.4-46.8±29.7-38.±23.9 T otal 38.±2.8 8.±2.9 8.2±4.6 2.2±9.2-48.±8.8-3.6±7. 6 [9, ].3±.9 8.±..±7.7 8.±4.9-28.7±.7-2.±7. [, ].6±.8.±.3 3.8±8.9 7.8±3. -43.3±9.2-3.±9.9 [, 2].±.2.±4.9 4.9±6.6 7.4±2.8-49.7±4.2-4.2±3.4 [2, 3] 3.4±.2 4.2±4.9 29.9±23.4 7.9±9. -42.8±8.2-28.2±7. T otal 37.8±3..±2.2.2±3.8 6.6±6.2-4.9±4. -32.3±.4 29

Table 6: Details of pm shifts for doctors -9. Averages over shifts with 9% confidence intervals are shown. (X measured in minutes.) Doc Interval Avg # Scheduled % No-show % Late Avg(X + ) Avg(X ) Avg(X) [2, 3] 2.8±2..4±3. 2.2±34. 9.±64.3-4.±23.6-2.8±32.8 [3, 4] 2.±.7 4.9±2.7 2.8±4. 36.4±2.6-4.±4.6-36.2±. [4, ] 8.9±.9 8.2±3..6±3.3 26.2±. -43.7±6.8-36.9±7.3 [, 6] 7.8±. 7.9±3.9.8±4.3.±. -47.9±6. -4.±6. [6, 7] 2.±2. 2.9±2.2 7.2±4. 2.±.6-43.±8.2-4.2±8. [7, 8].6±.8-47.±28. -47.±28. T otal 72.9±3.4 6.±.4.4±.9 23.2±7.2-4.7±3. -38.6±3.3 2 [2, 3].9±.6.±22.6.±2.6 23.6±NaN -3.±33.8-44.±3. [3, 4] 4.7±.3 9.3±4.2 9.±3.9 7.6±9. -6.6±8.9-3.6±8.6 [4, ] 3.8±.3 2.3±.7 2.3±.3 6.9±.3 -.8±8.2-47.2±8.3 [, 6] 4.4±.9 9.4±3.9 3.±4.8.9±.9-9.±2.9-49.4±.7 [6, 7] 3.9±.4 4.4±4.8 2.±. 8.2±9.3-6.±3.6-2.9±3.7 T otal 47.8±3.8.2±2..6±2.2 4.8±4.4-7.9±.8-49.±.2 3 [2, 3].4±.. 3.±29.2-6.6±698.2-23.8±334. [3, 4] 4.8±.8.2±6..9±6.3.±2.2-66.4±.6-7.6±3.2 [4, ].3±.3.6±7.2 9.±6.4 3.9±6.6-63.8±6.4-3.7±2.8 [, 6] 7.7±2.3.7±6.8.4±9.3 23.2±43. -76.±26. -69.±22. [6, 7] 6.±. 9.7±6.6 4.2±4.8 7.±28.4-8.9±6.8-77.9±4.7 T otal 4.3±4.6.4±4. 9.±4. 9.2±7. -69.2±. -6.±9.3 4 [3, 4] 2.9±.4 7.6±3.8 7.±.3 24.±8.6-44.±7.7-33.±8.8 [4, ] 2.3±. 2.±4. 3.3±4. 2.2±9. -37.4±6.3-29.6±7. [, 6] 2.4±.8 4.7±2.2 3.2±7.7 8.±3.2-4.±7.6-33.±6. T otal 64.6±2.6 8.±2.2 4.4±4.3 8.±.7-4.6±. -32.±6.3 [2, 3] 2.9±..2±3.3 4.±4.9.±6. -38.3±.8-3.3±6.9 [3, 4] 24.±.8 4.±2. 3.2±4. 24.2±2.7-37.8±4.9-29.±.3 [4, ] 23.3±..8±2. 6.9±3.2 8.3±.2-34.±4.2-2.7±4. [, 6] 23.±.3 4.9±2. 8.9±3.9 4.±3.3-34.6±4.6-2.7±4.7 [6, 7] 7.2±.8 7.6±2.9 2.±3.9 9.±3. -4.8±7.7-39.±6.8 T otal.9±4..4±..4±.7 6.2±3. -38.±2.3-29.7±2.4 6 [3, 4] 26.8±3. 3.4±2. 7.3±.3.±.8-6.±3. -9.3±24.8 [4, ] 27.±2..9±9.2 8.±3.6 2.3±9.9-6.±26. -.3±27.4 [, 6] 26.±3.4 6.4±3.7 4.±.2 7.±9. -49.4±22.8-4.±2.4 [6, 7] 6.±4. 6.3±.8 4.2±3.3 2. -68.4±82. -67.±84.3 T otal 86.3±.4 3.±6. 9.±9.2 4.±.9-8.7±2.3 -.9±22. 7 [2, 3] 4.3±.7.3±.2 8.3±3.3 29.6±29.9-4.±22.8-49.±27.8 [3, 4] 2.4±3.6.6±4..±4.9 7.±7.8-6.9±. -62.3±2.3 [4, ] 23.±.2 7.8±6.2 9.8±4.8 2.2±6. -64.4±4.4-6.7±4.2 [, 6] 23.±.2 7.±4.7.9±3.9 4.8±7.8-82.±2. -77.±22. [6, 7].9±4.8.7±7.2 9.2±8.4 2.8±3.7-78.6±24.6-7.±22. T otal 83.6±7.2 8.±2.4 7.3±2..±7.4-72.4±9. -66.±8.8 8 [2, 3].±.2-29.6-29.6 [3, 4].4±2. 8.±4. 6.4±2.3 3.6±3. -6.3±.4 -.2±. [4, ] 4.2±.9 8.3±3.3 2.8±.2 2.7±9.2-9.8±2.3 -.±.2 [, 6] 4.6±.9 9.2±4. 4.3±4.9.±4. -7.±4.2-7.8±.4 [6, 7] 6.6±. 3.8±4.6 3.±6.7 9.±9. -8.±. -48.±.9 T otal.8±6. 8.±2..±2. 22.7±. -62.±6. -3.±.4 9 [, 2].±.2. -2.2-2.2 [2, 3] 3.±.7 6.±.8 8.8±9.3 6.7±2. -67.4±8.8-7.8±7.9 [3, 4].±.4 6.3±2.8.4±4. 2.4±. -6.4±7. -3.2±6.4 [4, ].±.7 7.2±2.4 8.4±3.2 2.7±9.6-6.3±9. -8.±9.6 [, 6].8±.6 2.4±3..4±4. 3.6±.8-6.7±. -2.4±9.3 [6, 7] 9.±.4 8.±. 3.3±..4±6.9-9.8±3.8 -.6±3.4 [7, 8].2±. -34.6-34.6 T otal 8.8±2.7 8.4±..9±2.2 7.8±4.9-6.9±4.6-3.3±4. 3

Table 7: Details of pm shifts for doctors -6. Averages over shifts with 9% confidence intervals are shown. (X measured in minutes.) Doc Interval Avg # Scheduled % No-show % Late Avg(X + ) Avg(X ) Avg(X) [3, 4].±.7 6.9±.3 6.9±8. 36.±86.9-66.9±32.2-9.3±3.3 [4, ].7±.9.±.6 2.4±.8 6.7±.6-74.7±23.9-6.3±24.8 [, 6] 9.9±.4.3±. 4.±. 2.±4.4-6.9±4.8-4.9±. [6, 7] 3.6±2. 6.±.7 9.2±2. 4.9±2.2-4.6±2. -33.8±.4 T otal 29.2±3.2 2.9±3..8±6.2 8.3±8.6-64.±6. -4.6±. [3, 4] 7.±.8.±7.6 7.3±9.3 2.7±. -78.±24.6-9.9±8.9 [4, ] 4.2±.4.9±3.9 7.3±7.2 24.±4.8-69.±. -2.2±.2 [, 6] 3.3±.6 7.±. 2.±4.7 7.4±. -7.2±3.4-6.8±3. [6, 7] 6.9±.8 8.±6. 3.6±.3 4.9±2.6-6.7±26. -2.±9.7 T otal 4.±3. 7.3±.7 4.9±2.6 9.±6. -7.2±6.8-7.7±6.4 2 [3, 4] 6.3±.7 2.±8.4 3.6±9.7.9±9.4-64.7±22. -4.±9.4 [4, ].4±2.6 2.3±6.8 2.6±7..±4.9-64.±29.2 -.6±26.9 [, 6].8±.8.7±9.7 3.7±.6 7.±7. -7.±8.7-7.2±3. [6, 7] 6.4±.9.9±9.8 7.9±..2±3.4-46.3±7.2-3.2±6.6 T otal 34.9±..2±.6 4.6±4. 2.8±. -67.±2.9 -.±.8 3 [3, 4].8±.9.8±.9 2.2±8.3 9.±3. -64.9±9.4-7.±2.4 [4, ].9±.4.±. 6.7±9. 22.3±7.4-6.4±7. -4.9±7.8 [, 6] 9.3±. 8.3±6. 9.2±.9.3±.7 -.±4.8-43.7±3.6 [6, 7].±.4 8.±7.8.7±.6 2.3±2. -38.±.9-3.9±.8 T otal 3.±2.9 9.±4.6.6±3. 4.6±6.2-4.4±.6-46.6±.4 4 [3, 4] 6.3±.6 3.±6.6.±.8 2.±6. -7.7±.8 -.8±.3 [4, ].±..8±3.3.±.4.±7. -7.3±.7-6.4±. [, 6].6±..±..±3.7 2.6±6.7-2.4±9.9-4.±9. [6, 7] 6.±.9 4.3±3.2 8.9±.2.±4. -42.±7.3-38.2±6.7 T otal 34.4±2.7 9.7±2.4.3±2.9.9±4. -7.8±4.7 -.8±4.8 [3, 4].±.3 7.±7.7.4±. 9.±.8-67.9±33.4-2.±2.6 [4, ].8±.7.±6.2.4±6.6 9.8±27.2-6.4±.7-7.3±2.8 [, 6].±. 7.±6.6.4±6.7 3.2±7.6-47.±9.4-4.±8.2 [6, 7] 6.±. 9.9±9.4 9.±. 9.8±22. -.9±7.8-4.9±8. T otal 34.±3.4.7±2.7 2.±.2 3.6±7. -7.9±7.8-49.±7.8 6 [3, 4] 6.±. 2.±6.2.±.9 4.3±9.3-7.9±24.3-7.3±23.4 [4, ] 2.±. 8.±..6±7.3 4.8±7. -6.±7.4-49.8±6. [, 6].3±. 7.3±.3 2.±7..9±4.6-4.±3.4-46.±2. [6, 7] 6.2±.9 7.6±7. 9.±8. 9.±9.6-4.3±29. -49.9±29.6 T otal 34.6±3.6 8.6±3.6 2.±4.3 2.9±6. -9.4±9. -.9±8.7 3

Figure 2: Lateness CDFs for All Doctors (Left: AM, Right: PM).8.6.4 8 9 2.8.6.4 2 3 4 6.2.2 24 8 2 6 6 24 8 2 6 6 32

Figure 3: Lateness CDFs (From top to bottom: doctors, 2, 3, 4. From left to right: AM, PM).8.6.4 9 2.8.6.4 2 3 4 6.2.2 24 8 2 6 6 24 8 2 6 6.8.6.4 8 9 2.8.6.4 3 4 6.2.2 24 8 2 6 6 24 8 2 6 6.8 9 2.8 3 4 6.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6.8 9.8 3 4.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6 33

Figure 4: Lateness CDFs (From top to bottom: doctors, 6, 7, 8. From left to right: AM, PM).8.6.4 8 9 2.8.6.4 2 3 4 6.2.2 24 8 2 6 6 24 8 2 6 6.8.6.4 8 9 2.8.6.4 3 4.2.2 24 8 2 6 6 24 8 2 6 6.8 9 2.8 3 4 6.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6.8 9 2.8 3 4 6.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6 34

Figure : Lateness CDFs (From top to bottom: doctors 9,, 2, 3. From left to right: AM, PM).8.6.4 8 9 2.8.6.4 2 3 4 6.2.2 24 8 2 6 6 24 8 2 6 6.8 9.8 3 4 6.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6.8 9.8 3 4 6.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6.8 9 2.8 3 4 6.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6 3

Figure 6: Lateness CDFs (From top to bottom: doctors, 6. From left to right: AM, PM).8 9.8 3 4 6.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6.8 9.8 3 4 6.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6 Figure 7: Lateness CDFs (From left to right: doctors, 4. Both PM shifts.).8 4.8 3 4 6.6.4.6.4.2.2 24 8 2 6 6 24 8 2 6 6 36

4.3 Testing An NHPP Within Shifts Table 8: P-values of the Lewis KS Test of NHPP Scheduled Arrivals Actual Arrivals Shift Doctor n L = L = 3 L = T n L = L = 3 L = T AM 93... 94.7.3.3 2 93... 89.2.9.7 3 46... 48.39.26.94 4 27... 822... 4... 632...22 6 226... 2.6.. 7 863... 823.8.92.29 8 436... 43.2.2.48 9 4... 997.2.6.4 37... 4.3.8.27 442... 366.3.. 2 44... 377.6.39.99 3 423... 363.27.9.83 4 2... 27.72.7.7 34... 288.43.46.4 6 4... 33.74.22.42 PM 3... 922.3.6.46 2 744... 64..6.9 3 33... 262.2.3.7 4 742... 76.2.3.27 294... 242.44.36.2 6 36... 22.27..4 7 9... 4.2.2. 8 8... 64.62..2 9 984... 776..49.8 24... 27.6.7.49 427... 366.22.2.32 2 29... 2..8.3 3 3... 32.24.39.7 4 66... 67...3 324... 29.47.78.46 6 294... 266.38.42.4 37

Table 9: P-values of the CU KS Test of NHPP Scheduled Arrivals Actual Arrivals Shift Doctor n L = L = 3 L = T n L = L = 3 L = T AM 93...2 94.9..7 2 93...9 89...2 3 46... 48.23..98 4 27...3 822...92 4...84 632.3..6 6 226...3 2... 7 863... 823.2..73 8 436..8.2 43...37 9 4... 997...93 37.3..96 4.48.32.87 442... 366.3..84 2 44..2.9 377...69 3 423..3.73 363.2..69 4 2..72.72 27.67.92.92 34... 288...92 6 4..6.9 33.23..93 PM 3...6 922.3..3 2 744...24 64...89 3 33...9 262.6..66 4 742..2.98 76...8 294...46 242.4.26.26 6 36..29.8 22.7..4 7 9..7. 4.4..7 8 8...83 64...6 9 984... 776...8 24...9 27.9.3.99 427...74 366.37.3.8 2 29...49 2.33.8.72 3 3...87 32..6.99 4 66...22 67.29..68 324... 29.73.38.3 6 294...6 266..22.76 38

Under-Dispersion Over Multiple s Table 2: Under-dispersion in the number of scheduled arrivals (appointments made before the appointment day) and actual arrivals over different days AM Shift PM Shift Scheduled Actual Scheduled Actual Doc n µ σ2 D µ σ2 D n µ σ2 D µ σ2 D 7 76.9 6..8 73. 48..7 9 72.9.8.7 7.6 4.7.8 2 2 3.9 33.6.6.8 38.6.8 8 47.8 8.3.2 4.4 23.2. 3 2.4 26.6.. 29.7.6 4.3 46.4.2 37.7 3.2.8 4 34 6.4 27.9.4 63.4 9.6.3 2 64.6 6.4.3 6. 4.6.2 23. 33.8.3.8 34.2.3 9.9 7.4.7 97.2 7..8 6 7 2.8 7.4.4 96. 493.4. 4 86.3 42.9. 76. 73.. 7 3 9.3.6.7 86.7 46..7 8 83.6 7..9 79.6 82.6. 8 6.8 98.2 3. 7.4 9.4.7 9.8 8.4 3.6 48.8 3.6 2. 9 22 66. 2.3.3 62.6 7.4.3 22 8.8 3.9.6.7 49..9 2 9. 8..4 24. 24.. 9 29.2 6.9.6 28.4..4 46.6 7.2.4 44.4 24.7.6 2 4. 22.3. 4.3 7.8.4 2 2 42. 8.6.4 39.3 33.8.9 34.9.2. 33..9.3 3 2 4.8 9.. 38.3 8.6.2 3 3. 23.7.8 32. 9.7.6 4 2. 28. 23 34.4 38.2. 34. 22.8.7 38..8.4 36.9 3.7.4 34. 26..8 33.6 24.9.7 6 2 37.8 23.8.6 3. 7.. 34.6 24.9.7 33.3 26..8 39

References Kim, S.-H., P. Vel, W. Whitt, W. C. Cha. 24. Poisson and non-poisson properties in appointmentgenerated arrival processes: The case of an endocrinology clinic. Columbia University, http://www.columbia.edu/ ww24/allpapers.html. Kim, S.-H., W. Whitt. 24. Choosing arrival process models for service systems: Tests of a nonhomogeneous Poisson process. Naval Research Logistics 6() 66 9. 4

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