National Tsing-Hua University 2020.07.09 Equilibrium and Data-analytics Laboratory A prehospital-stroke-scale parameterized bypass-protocol model for suspected stroke patient 建立繞道協議模型結合疑似中風病人到院前 中風程度量表 國立清華大學 工業工程與工程管理學系 Presenter: 張祐禎 指導教授: 李雨青 博士 

National Tsing-Hua University 2020.07.09 2 Equilibrium and Data-analytics Laboratory Outline 1 Current situation 2 3 Our Model About Previous models 4 Results Discussions Limitations Conclusions 

National Tsing-Hua University 2020.07.09 3 Equilibrium and Data-analytics Laboratory Part 1 Current situation 

National Tsing-Hua University 2020.07.09 4 Equilibrium and Data-analytics Laboratory About suspected stroke patients A suspected stroke patient he/she can get the definitive-treatment only in comprehensive stroke centers (CSCs) who are capable for proving EVT treatment. (CSCs→大醫院) If he/she is AIS without LVO,. (小中風) If he/she is AIS with LVO, (大中風) he/she can get the definitive-treatment in both CSCs and rt-PA hospitals who are capable for proving rt-PA treatment.. (rt-PA hospitals→小醫院) 

National Tsing-Hua University 2020.07.09 5 Equilibrium and Data-analytics Laboratory Current decision-making process add your words here，according to your need to draw the text box size Door: the patient arrives at the hospital (CSC or rt-PA hospital) The call is received by the dispatcher EMTs leave the scene Choosing the receiving hospital based on the number of the CPSS symptoms a patient had Three symptoms of the CPSS: arm drift, facial palsy, and speaking 

National Tsing-Hua University 2020.07.09 6 Equilibrium and Data-analytics Laboratory Part 2 Previous models 

National Tsing-Hua University 2020.07.09 7 Equilibrium and Data-analytics Laboratory Literature Review • RACE (Ossa et al., 2013) [1, 2, 3] • FAST (Rozycki et al., 1993) • NIHSS (Lyden et al., 1999) • CP-SSS (Katz et al., 2015) 01 Stroke scales: There are several kinds of the stroke scales used in the world. [1] Ali et al., 2018, Stroke [2] Jumaa et al., 2019, Journal of NeuroInterventional Surgery. [3] Schlemm et al., 2017, Stroke 

National Tsing-Hua University 2020.07.09 8 Equilibrium and Data-analytics Laboratory Literature Review 02 How do they set the value of the parameters, transport time and process time in hospitals, in the model? [1] Schlemm et al., 2017, Stroke [2] Xu et al., 2019, Stroke [3] Holodinsky et al., 2017, Stroke [4] Jumaa et al., 2019, Journal of NeuroInterventional Surgery. [5] Ali et al., 2018, Stroke • Ideal transport time [1, 2] • Actual or median transport time [3-5] • Ideal process time recommended from the guideline (Powers et al., 2018) [1] 

National Tsing-Hua University 2020.07.09 9 Equilibrium and Data-analytics Laboratory Literature Review Cost [5] 03 Present the results of the optimization models in different forms Time [1-5] Distribution (% CSC, % rt-PA hospitals) [5] [1] Schlemm et al., 2017, Stroke [2] Xu et al., 2019, Stroke [3] Holodinsky et al., 2017, Stroke [4] Jumaa et al., 2019, Journal of NeuroInterventional Surgery. [5] Ali et al., 2018, Stroke 

National Tsing-Hua University 2020.07.09 10 Equilibrium and Data-analytics Laboratory Part 3 Model 

National Tsing-Hua University 2020.07.09 11 Equilibrium and Data-analytics Laboratory Model 

National Tsing-Hua University 2020.07.09 12 Equilibrium and Data-analytics Laboratory Notations H set of hospitals that provide both the rt-PA treatment and the EVT C set of hospitals that provide only the rt-PA treatment Ω set of patients Sets: 𝑋𝑎,𝑖 𝑖 ∈ Ω, 𝑎 ∈ 𝑯 ∪ 𝑪 1 if patient 𝑖 is sent to hospital 𝑎 from the scene; 0 otherwise. Variables: 

National Tsing-Hua University 2020.07.09 13 Equilibrium and Data-analytics Laboratory Parameters: 𝑝𝑖 𝑖 ∈ Ω the probability that patient 𝑖 with AIS without LVO 1 − 𝑝𝑖 𝑖 ∈ Ω the probability that patient 𝑖 with LVO 𝑇𝑎,𝑖 𝑖 ∈ Ω, 𝑎 ∈ 𝑯 ∪ 𝑪 first transport time from getting patient 𝑖 on the scene to hospital 𝑎 𝑇𝑎,𝑏 𝑎 ∈ 𝑪, 𝑏 ∈ 𝑯 the second transport time from rt-PA hospital 𝑎 to a CSC 𝑏 

National Tsing-Hua University 2020.07.09 14 Equilibrium and Data-analytics Laboratory 𝑄𝑎 𝑎 ∈ 𝑯 ∪ 𝑪 door-to-test time in hospital 𝑎 𝐷𝑎 𝑎 ∈ 𝑯 ∪ 𝑪 test-to-treatment time in hospital 𝑎 for a patient with AIS without LVO ഥ 𝐷𝑎 𝑎 ∈ 𝑯 test-to-treatment time in hospital 𝑎 for patient with LVO 𝐸𝑎 𝑎 ∈ 𝑪 the shortest possible time for a patient transferred from hospital a to a CSC until receiving definitive treatment, i.e. min 𝑏∈𝑯 ( 𝑇𝑎,𝑏 + 𝑄𝑏 + ഥ 𝐷𝑏 ) 𝐴 administration time of hospital transfer 

National Tsing-Hua University 2020.07.09 15 Equilibrium and Data-analytics Laboratory 𝑆𝑖 𝑖 ∈ Ω response time for the ambulance to reach the site of patient 𝑖 plus on-scene time 𝑈 threshold (If the transport time difference between the scene to the nearest rt-PA hospital and the scene to the nearest CSC is not more than 𝑈 seconds, then bypass the nearest rt-PA hospital to the nearest CSC.) 𝑀 a large number 

National Tsing-Hua University 2020.07.09 16 Equilibrium and Data-analytics Laboratory Model 𝐌𝐢𝐧𝐢𝐦𝐢𝐳𝐞 ෍ 𝑎∈𝑯 (𝑆𝑖 +𝑇𝑎,𝑖 + 𝑄𝑎 +𝑝𝑖 𝐷𝑎 + 1 − 𝑝𝑖 ഥ 𝐷𝑎 )𝑋𝑎,𝑖 + ෍ 𝑎∈𝑪 (𝑆𝑖 + 𝑇𝑎,𝑖 + 𝑄𝑎 + 𝑝𝑖 𝐷𝑎 + 1 − 𝑝𝑖 𝐴 + 𝐸𝑎 )𝑋𝑎,𝑖 (1) Subject to ෍ 𝑎∈𝑯∪𝑪 𝑋𝑎,𝑖 = 1 ∀ 𝑖 ∈ Ω (2) min𝑎∈𝑯 𝑇𝑎,𝑖 − 𝑇𝑐,𝑖 −𝑈 ≥ (−𝑀)(1 − 𝑋𝑐,𝑖 ) ∀ 𝑐 ∈ 𝑪 , 𝑖 ∈ Ω (3) 

National Tsing-Hua University 2020.07.09 17 Equilibrium and Data-analytics Laboratory Source of the value of the parameters 1 − 𝑝𝑖 When a patient has 3, 2, or 1 symptoms of CPSS, the probability that a patient with LVO is 0.310, 0.265 or 0.239 ( 0.727, 0.343 or 0.343) from Scheitz et al.(Richards et al.) Scheitz et al., 2017, Stroke Richards et al., 2018, Prehospital Emergency Care. 𝑇𝑎,𝑖 𝑇𝑎,𝑏 The off-peak driving time from Google Maps. 𝑄𝑎 𝐷𝑎 ഥ 𝐷𝑎 The processing time in each hospital is the four-year median data during the periods from 2016 to 2019. 𝐴 46.5 minutes. (Ng et al.) Ng et al., 2017, Stroke 𝑈 𝑈 was initially set at 15 minutes (900 seconds) because AHA guideline proposed that the good outcome might decrease for every 15 minutes (900 seconds) delay. 

National Tsing-Hua University 2020.07.09 18 Equilibrium and Data-analytics Laboratory About the patient data and the software Patient data used in this research 6-year 7,678 historical patients’ data set which contains patients who were suspected stroke in Taipei City Among the 7,678 patients, 4,037 patients have three symptoms of the CPSS indicators, 1,319 patients have two symptoms of the CPSS indicators and 2,322 patients have one symptom of the CPSS indicators. Software The model is implemented with AMPL, which is an intuitive algebraic modeling system. Solver CPLEX (V12.8) is the solver used in this research to solve the underlying mathematical programming model. 

National Tsing-Hua University 2020.07.09 19 Equilibrium and Data-analytics Laboratory Part 4 Results、Discussions Limitations、Conclusions 

National Tsing-Hua University 2020.07.09 20 Equilibrium and Data-analytics Laboratory Results of the model with initial parameter settings The simulation of the proposed protocol sends all the patients who are suspected stroke directly to one of the CSCs and 2,643 (34.42%) of them actually bypass the nearest rt-PA hospital. (For both probability measures based on Scheitz et al. and Richards et al..) What’s the implication? The difference of transport times between from the scene to the nearest rt-PA hospital and from the scene to any one of the CSCs is rarely more than 15 minutes (900 seconds). Administration time 𝐴 for hospital transfer will affect the results as well. 

National Tsing-Hua University 2020.07.09 21 Equilibrium and Data-analytics Laboratory Sensitivity of the two critical parameters (The probability is according to the study of Scheitz et al.) Method- decreasing threshold 𝑈 and administration time 𝐴 by 1 minute (60 seconds) gradually. When threshold 𝑈 and administration time 𝐴 are reduced to 14 minutes (840 seconds) and 41.50 minutes (2,490 seconds), respectively, a few patients will be taken to the rt-PA hospitals. When threshold 𝑈 is 9 minutes (540 seconds) and administration time 𝐴 is 30.50 minutes (1,830 seconds), there are 228 patients who will be decided to be sent to the rt-PA hospitals at first When threshold 𝑈 is 6 minutes (360 seconds) and administration time 𝐴 is 30.50 minutes (1,830 seconds), there are 378 patients who will be sent to rt-PA hospitals according to the results of the proposed model. 

National Tsing-Hua University 2020.07.09 22 Equilibrium and Data-analytics Laboratory Sensitivity of the two critical parameters (The probability is according to the study of Scheitz et al.) 7678 7678 7655 7628 7619 7595 7574 7563 7553 7543 7542 7542 7542 7542 7542 7678 7677 7645 7602 7574 7497 7450 7402 7357 7300 7291 7285 7276 7270 7262 7000 7100 7200 7300 7400 7500 7600 7700 7800 900 840 780 720 660 600 540 480 420 360 300 240 180 120 60 Number of patients are sent to CSCs directly. 𝑈 A=2790 A=1830 

National Tsing-Hua University 2020.07.09 23 Equilibrium and Data-analytics Laboratory Threshold 𝑈 (seconds) Administration time 𝐴 (seconds) Numbers of patients who were sent to rt-PA hospitals at first Numbers of patients who were sent to CSCs at first Expected time that patients get the definitive treatment (seconds) estimated by the model 900 2,790 0 7,678 6,107 540 1,830 228 7,450 6,101 360 1,830 378 7,300 6,098 Sensitivity of the two critical parameters (The probability is according to the study of Scheitz et al.) For the purpose of decreasing the time needed for a patient to get the definitive treatment →the 228 patients who will be sent to rt-PA hospitals can reduce 202 seconds per patient to receive the definitive treatment For the purpose of balancing the medical resources provision →although these 378 patients can only reduce 183 seconds before receiving the definitive treatment, there are 150 more patients being sent to the rt-PA hospitals at first to mitigate congestions in the CSCs. 

National Tsing-Hua University 2020.07.09 24 Equilibrium and Data-analytics Laboratory Sensitivity of the two critical parameters (The probability is according to the study of Richards et al.) Method- decreasing threshold 𝑈 and administration time 𝐴 by 1 minute (60 seconds) gradually. When threshold 𝑈 is 13 minutes (780 seconds) and administration time 𝐴 is 34.50 minutes (2,070 seconds), a few patients (12 people) start to be sent to rt-PA hospitals at first and they can accept transfer. When threshold 𝑈 is 7 minutes (420 seconds) and administration time 𝐴 is 23.50 minutes (1,410 seconds), there are 132 patients being sent to rt-PA hospitals with same threshold 𝑈. 

National Tsing-Hua University 2020.07.09 25 Equilibrium and Data-analytics Laboratory Sensitivity of the two critical parameters (The probability is according to the study of Richards et al.) 7678 7678 7678 7678 7678 7678 7678 7678 7678 7678 7678 7678 7678 7678 7678 7678 7678 7655 7625 7615 7585 7562 7549 7546 7543 7542 7542 7542 7542 7542 7450 7500 7550 7600 7650 7700 900 840 780 720 660 600 540 480 420 360 300 240 180 120 60 Number of patients are sent to CSCs directly. 𝑈 A=2790 A=1410 

National Tsing-Hua University 2020.07.09 26 Equilibrium and Data-analytics Laboratory Sensitivity of the two critical parameters (The probability is according to the study of Scheitz et al.) The total expected time reduced for 7,678 patients are 255.93 minutes (15,356 seconds) than the results of the model with its initial parameters. →That is, the 132 patients who are sent to rt-PA hospitals can save 116 seconds per person to receive the definitive treatment. Threshold 𝑈(seconds) Administration time 𝐴(seconds) Numbers of patients who are sent to rt- PA hospitals at first Numbers of patients who are sent to CSCs at first Expected time for a patient to get the definitive treatment (seconds) from the model 900 2,790 0 7,678 7,027 420 1,410 132 7,546 7,025 

National Tsing-Hua University 2020.07.09 27 Equilibrium and Data-analytics Laboratory Comparisons with other strategies for deciding the receiving hospitals a. A patient who was suspected stroke with at least one symptoms of the CPSS was sent to the nearest hospital including CSCs and rt-PA hospitals on the first occasion. If a patient with LVO was sent to an rt-PA hospital, the patient should be transferred to the nearest CSC. b. A patient who was suspected stroke with at least one symptom of the CPSS was sent to the nearest CSC directly. c. A patient who was suspected stroke with at least one symptom of the CPSS was sent to a hospital according to the result of the proposed hospital selection model. d. A patient who was suspected stroke was sent to a hospital based on number of symptoms of the CPSS indicators he/she had. If a patient had three symptoms of the CPSS indicators, he/she would be sent to the nearest CSC directly. If a patient had one or two symptoms of the CPSS indicators, he/she would be sent to the nearest hospital including CSCs and rt-PA hospitals. e. A patient who was suspected stroke was sent to a hospital based on how many symptoms of the CPSS indicators he/she had. If a patient had three or two symptoms of the CPSS, he/she would be sent to a CSC directly. If a patient had one symptom of the CPSS indicator, he/she would be sent to the nearest hospitals including CSCs and rt-PA hospitals. 

National Tsing-Hua University 2020.07.09 28 Equilibrium and Data-analytics Laboratory Random sampling method (the probability measure is from Scheitz et al.) Number of patients with 3 symptoms of CPSS : 4,037 Number of patients with 2 symptoms of CPSS : 1,319 Number of patients with 1 symptom of CPSS : 2,322 The probability a patient with LVO : 0.310 That is, there would be 31% of 4,037 patients (1,251 patients) having AIS with LVO. The probability a patient with LVO : 0.265 The probability a patient with LVO : 0.239 That is, there would be 23.9% of 2,322 patients (555 people) having AIS with LVO. There are totally 2,156 patients who are determined as with LVO, and 5,522 patients are without LVO. That is, there would be 26.5% of 1,319 patients (350 people) having AIS with LVO. Randomly extract 350 patients from 1,319 patients and determine they are with LVO Randomly extract 555 patients from 2,322 patients and determine they are with LVO Randomly extract 1,251 patients from 4,037 patients and determine they are with LVO 

National Tsing-Hua University 2020.07.09 29 Equilibrium and Data-analytics Laboratory Random sampling method (the probability measure is from Richards et al.) Number of patients with 3 symptoms of CPSS : 4,037 Number of patients with 2 symptoms of CPSS : 1,319 Number of patients with 1 symptom of CPSS : 2,322 The probability a patient with LVO : 0.727 That is, there would be 72.7% of 4,037 patients (2,935 patients) having AIS with LVO. The probability a patient with LVO : 0.343 The probability a patient with LVO : 0.343 That is, there would be 34.3% of 2,322 patients (452 people) having AIS with LVO. There are totally 3,816 patients who are determined as with LVO, and 3,862 patients are without LVO. That is, there would be 34.3% of 1,319 patients (429 people) having AIS with LVO. Randomly extract 429 patients from 1,319 patients and determine they are with LVO Randomly extract 452 patients from 2,322 patients and determine they are with LVO Randomly extract 2,935 patients from 4,037 patients and determine they are with LVO 

National Tsing-Hua University 2020.07.09 30 • We used the sampling method five times to randomly generate 5 different patients profiles for each probability measure. • Then, we simulate the EMS process under the use of 5 different strategies to determine patients’ first receiving hospitals and compute the time for each patient to receive treatment over these 5 patients profiles. • Running 5 strategies on one patients profile is called a trial. We run 5 trials for each probability measure. Equilibrium and Data-analytics Laboratory Random sampling method 

National Tsing-Hua University 2020.07.09 31 Equilibrium and Data-analytics Laboratory Comparisons with other strategies for deciding the receiving hospitals strategy a. strategy b. strategy c. strategy d. strategy e. trial 1 6,715 6,803 6,106 6,746 6,767 trial 2 6,700 6,793 6,105 6,732 6,750 trial 3 6,714 6,803 6,106 6,744 6,764 trial 4 6,741 6,823 6,114 6,772 6,784 trial 5 6,708 6,803 6,106 6,745 6,759 B1 B2 B3 B4 B5 B6 strategy b. 983 2,104 836 1,234 1,397 1,124 strategy c. for all trials 80 5,277 2,321 0 0 0 The mean time (in sec.) for a patient to receive definitive treatment under the 5 different strategies for deciding receiving hospital. ( 𝑈 = 900, 𝐴 = 2,790 and the probability measure is from Scheitz et al.) Number of patients sent to each receiving CSCs under different strategies and trials. ( 𝑈 = 900, 𝐴 = 2,790 and the probability measure is based on Scheitz et al.) 

National Tsing-Hua University 2020.07.09 32 Equilibrium and Data-analytics Laboratory Comparisons with other strategies for deciding the receiving hospitals strategy a. strategy b. strategy c. strategy d. strategy e. trial 1 8,412 8,375 7,022 8,338 8,352 trial 2 8,418 8,371 7,028 8,339 8,348 trial 3 8,421 8,393 7,025 8,352 8,368 trial 4 8,436 8,384 7,026 8,350 8,362 trial 5 8,410 8,368 7,027 8,338 8,347 The mean time (in sec.) for a patient to receive definitive treatment under the 5 different strategies for deciding receiving hospital. ( 𝑈 = 900, 𝐴 = 2,790 and the probability measure is based on Richards et al.) B1 B2 B3 B4 B5 B6 strategy b. 983 2,104 836 1,234 1,397 1,124 strategy c. for all trials 0 3,802 3,876 0 0 0 Number of patients sent to each receiving CSCs under different strategies and trials. ( 𝑈 = 900 , 𝐴 = 2,790 and the probability measure is based on Richards et al.) 

National Tsing-Hua University 2020.07.09 33 Equilibrium and Data-analytics Laboratory Threshold 𝑈 (seconds) Administratio n time 𝐴 (seconds) Expected time for a patient to get the definitive treatment (seconds) from the model trial 1 trial 2 trial 3 trial 4 trial 5 900 2,790 7,027 7,022 7,028 7,025 7,026 7,027 420 1,410 7,025 7,022 7,028 7,024 7,026 7,026 The mean time (in sec.) for a patient to receive definitive treatment under the 5 different trials. (The probability measure is from Richards et al.) Threshold 𝑈 (seconds) Administratio n time 𝐴 (seconds) Expected time for a patient to get the definitive treatment (seconds) from the model trial 1 trial 2 trial 3 trial 4 trial 5 900 2,790 6,107 6,106 6,105 6,106 6,114 6,106 540 1,830 6,101 6,099 6,103 6,101 6,113 6,097 360 1,830 6,098 6,101 6,103 6,104 6,120 6,098 The mean time (in sec.) for a patient to receive definitive treatment under the 5 different trials. (The probability measure is from Scheitz et al.) Comparisons between actual time and expected time 

National Tsing-Hua University 2020.07.09 34 Equilibrium and Data-analytics Laboratory Discussions • EMTs choose the receiving hospitals based on how many symptoms of the CPSS a patient has and the time or distance from the scene to the hospitals. → not only the transport time and the number of the symptoms of the CPSS need to be considered, but also the time after entering the hospitals and the transfer time between hospitals should be cared • The results of our model showed that door-to-treatment time can affect the optimal receiving hospital significantly because the door- to-treatment time in the 6 CSCs in Taipei City have gap with each other. The difference between the shortest door-to-treatment time and the longest door-to-treatment time of these 6 CSCs is about 1 hour. 

National Tsing-Hua University 2020.07.09 35 • Although no patients are sent to rt-PA hospitals when using the model (strategy c.) with the initial parameters sets, the time to get the definitive treatment is lower than the results of using strategy b. • Whether patients can be assigned to rt-PA hospitals to balance the usage of medial resources and mitigate the potential crowdedness in the CSCs → shortening the administration time of hospital transfer is a resolution and the administration time for transfer is improved to the designed level according to our sensitivity analysis • A web-based system accessible by EMTs’ mobile devices has been developed for Taipei City. Equilibrium and Data-analytics Laboratory Discussions 

National Tsing-Hua University 2020.07.09 36 Equilibrium and Data-analytics Laboratory 

National Tsing-Hua University 2020.07.09 37 Equilibrium and Data-analytics Laboratory 

National Tsing-Hua University 2020.07.09 38 Equilibrium and Data-analytics Laboratory 

National Tsing-Hua University 2020.07.09 39 • The historical patients data and the median time in hospitals will change gradually. To ensure effectiveness of the method, the model parameters need to be adjusted periodically according to the latest situation. • The optimal hospital of the model under different probability measures about level of the stroke would not be the same → The more accurate the estimate of the probability measure for the target region, the better the savings of patients’ time to receive the definitive treatment by the proposed hospital selection model. Equilibrium and Data-analytics Laboratory Limitations 

National Tsing-Hua University 2020.07.09 40 • A mathematical optimization model is proposed in this research and the model considers not only the probability of patient with LVO and the real-time transport time but also the door-to-treatment time in hospitals and the transfer time (second-transport time + administration time). • After adjusting the parameters, some patients will be sent to the rt-PA hospital instead and accept transfer. Also, the results of our model are better than the results of other typical 4 strategies. • This model can help EMTs to decide the receiving hospitals and make patients with suspected stroke receive the definitive treatment in the shortest time. In addition, the model has generality to be used in other regions. Equilibrium and Data-analytics Laboratory Conclusions 

National Tsing-Hua University 2020.07.09 41 Equilibrium and Data-analytics Laboratory Q & A 

National Tsing-Hua University 2020.07.09 42 Equilibrium and Data-analytics Laboratory Thank you for listening! 

National Tsing-Hua University 2020.07.09 43 Equilibrium and Data-analytics Laboratory Appendix 1 (The probability measure is from Scheitz et al.) A\U (sec.) 900 840 780 720 660 600 540 480 2790 7678 7678 7655 7628 7619 7595 7574 7563 2730 7678 7678 7655 7626 7616 7590 7569 7551 2670 7678 7678 7655 7626 7616 7590 7569 7549 2610 7678 7678 7655 7625 7615 7585 7564 7544 2550 7678 7678 7655 7625 7615 7585 7562 7539 2490 7678 7677 7654 7622 7611 7580 7556 7533 2430 7678 7677 7645 7608 7596 7565 7541 7516 2370 7678 7677 7645 7608 7596 7561 7537 7511 2310 7678 7677 7645 7607 7595 7555 7530 7502 2250 7678 7677 7645 7607 7591 7550 7525 7494 2190 7678 7677 7645 7607 7585 7540 7515 7483 2130 7678 7677 7645 7605 7583 7535 7507 7475 2070 7678 7677 7645 7604 7580 7528 7498 7466 2010 7678 7677 7645 7603 7577 7520 7489 7456 1950 7678 7677 7645 7603 7577 7516 7484 7447 1890 7678 7677 7645 7602 7575 7509 7473 7431 1830 7678 7677 7645 7602 7574 7497 7450 7402 1770 7678 7677 7645 7602 7573 7486 7436 7380 1710 7678 7677 7645 7602 7573 7480 7429 7366 1650 7678 7677 7645 7602 7573 7472 7421 7354 1590 7678 7677 7645 7602 7573 7464 7413 7341 1530 7678 7677 7645 7602 7573 7462 7410 7336 1470 7678 7677 7645 7602 7573 7457 7402 7327 1410 7678 7677 7645 7602 7573 7455 7399 7323 1350 7678 7677 7645 7602 7573 7450 7394 7316 1290 7678 7677 7645 7602 7573 7448 7392 7313 1230 7678 7677 7645 7602 7573 7446 7389 7310 1170 7678 7677 7645 7602 7573 7446 7388 7309 1110 7678 7677 7645 7602 7573 7444 7386 7306 1050 7678 7677 7645 7602 7573 7439 7381 7301 990 7678 7677 7645 7602 7573 7434 7372 7290 930 7678 7677 7645 7602 7573 7434 7370 7287 870 7678 7677 7645 7602 7573 7434 7370 7286 810 7678 7677 7645 7602 7573 7432 7367 7282 750 7678 7677 7645 7602 7573 7431 7366 7281 690 7678 7677 7645 7602 7573 7430 7365 7280 630 7678 7677 7645 7602 7573 7430 7364 7278 570 7678 7677 7645 7602 7573 7430 7362 7276 510 7678 7677 7645 7602 7573 7430 7362 7276 450 7678 7677 7645 7602 7573 7430 7362 7276 390 7678 7677 7645 7602 7573 7430 7362 7274 330 7678 7677 7645 7602 7573 7430 7362 7274 270 7678 7677 7645 7602 7573 7430 7362 7274 210 7678 7677 7645 7602 7573 7430 7362 7274 150 7678 7677 7645 7602 7573 7430 7362 7274 90 7678 7677 7645 7602 7573 7430 7362 7274 30 7678 7677 7645 7602 7573 7430 7362 7274 A\U (sec.) 420 360 300 240 180 120 60 2790 7553 7543 7542 7542 7542 7542 7542 2730 7540 7529 7527 7527 7527 7527 7527 2670 7537 7524 7522 7521 7521 7521 7521 2610 7525 7512 7510 7509 7509 7509 7509 2550 7516 7501 7497 7496 7495 7495 7495 2490 7509 7493 7487 7486 7484 7484 7484 2430 7488 7468 7462 7461 7459 7458 7458 2370 7480 7459 7453 7450 7448 7447 7447 2310 7467 7440 7434 7430 7427 7426 7426 2250 7459 7422 7416 7411 7408 7407 7407 2190 7446 7408 7402 7397 7393 7392 7391 2130 7436 7395 7389 7384 7378 7377 7376 2070 7425 7384 7376 7371 7364 7362 7360 2010 7414 7371 7363 7357 7349 7346 7344 1950 7404 7360 7352 7346 7338 7333 7329 1890 7387 7337 7328 7322 7313 7308 7300 1830 7357 7300 7291 7285 7276 7270 7262 1770 7335 7272 7260 7251 7241 7231 7221 1710 7318 7252 7239 7229 7217 7205 7195 1650 7305 7234 7219 7204 7190 7176 7154 1590 7288 7213 7194 7177 7163 7149 7119 1530 7278 7202 7179 7162 7147 7133 7098 1470 7262 7184 7160 7142 7120 7103 7063 1410 7251 7171 7145 7126 7103 7085 7045 1350 7242 7130 7102 7082 7059 7038 6994 1290 7238 7112 7083 7063 7039 7013 6960 1230 7234 7100 7068 7048 7024 6995 6931 1170 7231 7094 7060 7040 7016 6986 6920 1110 7227 7089 7050 7026 6998 6965 6890 1050 7222 7084 7043 7018 6990 6957 6878 990 7211 7070 7018 6980 6952 6917 6829 930 7208 7066 7013 6973 6943 6905 6813 870 7206 7059 7000 6958 6927 6889 6785 810 7199 7052 6986 6941 6903 6863 6747 750 7195 7048 6980 6931 6892 6845 6700 690 7193 7044 6975 6923 6876 6822 6664 630 7191 7037 6964 6907 6860 6805 6630 570 7188 7032 6959 6898 6848 6788 6597 510 7184 7028 6954 6892 6839 6774 6569 450 7184 7027 6950 6886 6828 6756 6531 390 7182 7025 6945 6879 6818 6742 6511 330 7182 7024 6943 6875 6808 6727 6489 270 7182 7024 6940 6867 6796 6711 6464 210 7179 7021 6936 6863 6792 6705 6444 150 7179 7021 6936 6858 6780 6685 6406 90 7179 7021 6935 6851 6771 6675 6382 30 7179 7021 6935 6845 6762 6661 6358 

National Tsing-Hua University 2020.07.09 44 Equilibrium and Data-analytics Laboratory Appendix 2 (The probability measure is from Richards et al.) A\U (sec.) 900 840 780 720 660 600 540 480 2790 7678 7678 7678 7678 7678 7678 7678 7678 2730 7678 7678 7678 7678 7678 7678 7678 7678 2670 7678 7678 7678 7678 7678 7678 7678 7678 2610 7678 7678 7678 7678 7678 7678 7678 7678 2550 7678 7678 7678 7678 7678 7678 7678 7678 2490 7678 7678 7678 7678 7678 7678 7678 7678 2430 7678 7678 7678 7678 7678 7678 7678 7678 2370 7678 7678 7678 7678 7678 7678 7678 7678 2310 7678 7678 7678 7678 7678 7678 7678 7678 2250 7678 7678 7678 7678 7678 7678 7678 7678 2190 7678 7678 7678 7678 7677 7677 7677 7677 2130 7678 7678 7678 7678 7677 7677 7677 7677 2070 7678 7678 7666 7656 7654 7654 7654 7654 2010 7678 7678 7657 7639 7637 7637 7637 7637 1950 7678 7678 7656 7638 7636 7636 7636 7636 1890 7678 7678 7656 7636 7633 7633 7633 7633 1830 7678 7678 7656 7633 7626 7625 7625 7625 1770 7678 7678 7656 7632 7625 7620 7618 7618 1710 7678 7678 7655 7629 7621 7613 7610 7610 1650 7678 7678 7655 7629 7621 7612 7604 7604 1590 7678 7678 7655 7625 7617 7607 7595 7595 1530 7678 7678 7655 7625 7616 7604 7587 7583 1470 7678 7678 7655 7625 7615 7598 7580 7576 1410 7678 7678 7655 7625 7615 7585 7562 7549 1350 7678 7678 7655 7625 7615 7584 7561 7541 1290 7678 7678 7655 7625 7615 7578 7554 7529 1230 7678 7678 7655 7625 7615 7578 7554 7528 1170 7678 7678 7655 7625 7615 7578 7554 7528 1110 7678 7678 7655 7625 7615 7575 7551 7521 1050 7678 7678 7655 7625 7615 7568 7544 7512 990 7678 7678 7655 7625 7615 7567 7543 7511 930 7678 7678 7655 7625 7615 7563 7539 7507 870 7678 7678 7655 7625 7615 7560 7536 7503 810 7678 7678 7655 7625 7615 7556 7532 7499 750 7678 7678 7655 7625 7615 7551 7524 7490 690 7678 7678 7655 7625 7615 7550 7521 7487 630 7678 7678 7655 7625 7615 7549 7519 7482 570 7678 7678 7655 7625 7615 7549 7519 7480 510 7678 7678 7655 7625 7615 7549 7518 7478 450 7678 7678 7655 7625 7615 7548 7515 7475 390 7678 7678 7655 7625 7615 7548 7515 7475 330 7678 7678 7655 7625 7615 7548 7513 7473 270 7678 7678 7655 7625 7615 7548 7512 7471 210 7678 7678 7655 7625 7615 7548 7512 7470 150 7678 7678 7655 7625 7615 7548 7512 7468 90 7678 7678 7655 7625 7615 7548 7512 7468 30 7678 7678 7655 7625 7615 7548 7512 7468 A\U (sec.) 420 360 300 240 180 120 60 2790 7678 7678 7678 7678 7678 7678 7678 2730 7678 7678 7678 7678 7678 7678 7678 2670 7678 7678 7678 7678 7678 7678 7678 2610 7678 7678 7678 7678 7678 7678 7678 2550 7678 7678 7678 7678 7678 7678 7678 2490 7678 7678 7678 7678 7678 7678 7678 2430 7678 7678 7678 7678 7678 7678 7678 2370 7678 7678 7678 7678 7678 7678 7678 2310 7678 7678 7678 7678 7678 7678 7678 2250 7678 7678 7678 7678 7678 7678 7678 2190 7677 7677 7677 7677 7677 7677 7677 2130 7677 7677 7677 7677 7677 7677 7677 2070 7654 7654 7654 7654 7654 7654 7654 2010 7637 7637 7637 7637 7637 7637 7637 1950 7636 7636 7636 7636 7636 7636 7636 1890 7633 7633 7633 7633 7633 7633 7633 1830 7625 7625 7625 7625 7625 7625 7625 1770 7618 7618 7618 7618 7618 7618 7618 1710 7610 7610 7610 7610 7610 7610 7610 1650 7604 7604 7604 7604 7604 7604 7604 1590 7593 7593 7593 7593 7593 7593 7593 1530 7581 7581 7581 7581 7581 7581 7581 1470 7573 7573 7573 7573 7573 7573 7573 1410 7546 7543 7542 7542 7542 7542 7542 1350 7538 7534 7533 7533 7533 7533 7533 1290 7520 7512 7511 7511 7511 7511 7511 1230 7513 7501 7496 7495 7495 7495 7495 1170 7506 7492 7487 7485 7485 7485 7485 1110 7494 7478 7472 7469 7469 7469 7469 1050 7481 7439 7433 7430 7428 7428 7428 990 7470 7427 7421 7417 7413 7413 7413 930 7465 7418 7412 7408 7404 7403 7403 870 7461 7411 7405 7401 7396 7392 7392 810 7456 7401 7395 7391 7386 7382 7379 750 7447 7389 7383 7378 7372 7368 7365 690 7444 7383 7375 7370 7364 7359 7356 630 7439 7376 7365 7358 7351 7346 7342 570 7437 7374 7358 7350 7343 7338 7332 510 7433 7370 7353 7345 7337 7328 7317 450 7427 7362 7343 7333 7320 7308 7273 390 7427 7362 7342 7329 7312 7297 7256 330 7425 7357 7337 7322 7304 7287 7240 270 7423 7354 7331 7316 7293 7274 7221 210 7422 7353 7330 7314 7289 7268 7210 150 7420 7351 7325 7309 7283 7257 7192 90 7419 7349 7319 7301 7274 7243 7169 30 7412 7341 7309 7290 7263 7229 7145