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Minimizing expected time to reach a given capital level before ruin
Minimization of the coefficient of variation for patient waiting system governed by a generic maximum waiting policy
1.  Department of Health Services and Outcomes Research, National Healthcare Group, 138543, Singapore 
2.  Department of Emergency Medicine, Tan Tock Seng Hospital, 308433, Singapore 
Timely access of care has been widely recognized as an important dimension of health care quality. Waiting times can affect patient satisfaction and quality of care in the emergency department (ED). This study analyzes a general patient waiting policy such that ED patients who wait beyond a threshold have their wait shortened. Assuming that the policy is implemented to accelerate the longwaiting cases within a short time interval, we transform the original waiting distribution to a piecewise distribution. The objective of this paper is to examine the reliability of the induced waiting system by minimizing the coefficient of variation (CV) of waiting time. We convert the CV minimization problem to an approximation counterpart using the sampling technique. With patient waiting time data from an emergency department in Singapore, we derive the optimal values of parameters, such as the threshold and the length of the underlying time interval, needed in the policy. Numerical results show that CV and variance of new waiting time will be reduced remarkably by 38% and 58% respectively, in comparison with the original ones.
References:
[1] 
Australian Institute of Health and Welfare (AIHW), Australian Hospital Statistics 2003. 04. (Canberra: AIHW, 2005) (AIHW cat. no. HSE 37; Health Services series no. 23). 
[2] 
B. Bursch, J. Beezy, R. Shaw, Emergency department satisfaction: What matters most?, Ann. Emerg. Med., 22 (1993), 586591. doi: 10.1016/S01960644(05)81947X. 
[3] 
L. G. Connelly, A. E. Bair, Discrete event simulation of emergency department activity: A platform for systemlevel operations research, Acad. Emerg. Med., 11 (2004), 11771185. 
[4] 
B. L. ConnerSpady, C. Sanmartin, G. H. Johnston, J. J. McGurran, M. Kehler, T. W. Noseworthy, The importance of patient expectations as a determinant of satisfaction with waiting times for hip and knee replacement surgery, Health Policy, 101 (2011), 245252. doi: 10.1016/j.healthpol.2011.05.011. 
[5] 
P. De, J. B. Ghosh, C. E. Wells, Scheduling to minimize the coefficient of variation, Int. J. Prod. Econ., 44 (1996), 249253. doi: 10.1016/09255273(96)000564. 
[6] 
R. L. Gardner, U. Sarkar, J. H. Maselli, Factors associated with longer ED lengths of stay, Am. J. Emerg. Med., 25 (2007), 643650. doi: 10.1016/j.ajem.2006.11.037. 
[7] 
L. V. Green, S. Savin, Reducing delays for medical appointments: A queueing approach, Oper. Res., 56 (2008), 15261538. doi: 10.1287/opre.1080.0575. 
[8] 
A. Guttmann, M. J. Schull, M. J. Vermeulen and T. A. Stukel, Association between waiting times and short term mortality and hospital admission after departure from emergency department: Population based cohort study from Ontario, Canada, BMJ, 342 (2011). 
[9] 
P. R. Harper, H. M. Gamlin, Reduced outpatient waiting times with improved appointment scheduling: A simulation modelling approach, OR Spectrum, 25 (2003), 207222. doi: 10.1007/s002910030122x. 
[10] 
S. A. Hemaya, T. E. Locke, How accurate are predicted waiting times, determined upon a patient's arrival in the Emergency Department?, Emerg. Med. J., 29 (2012), 316318. doi: 10.1136/emj.2010.106534. 
[11] 
E. T. Ho, Improving waiting time and operational clinic flow in a tertiary diabetes center BMJ Qual. Improv. Rep. , 2 (2014). 
[12] 
M. A. Kallen, J. A. Terrell, P. LewisPatterson, J. P. Hwang, Improving wait time for chemotherapy in an outpatient clinic at a comprehensive cancer center, J. Oncol. Practice, 8 (2012), e1e7. doi: 10.1200/JOP.2011.000281. 
[13] 
K. Kelly, Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach, Behav. Res. Methods, 39 (2007), 755766. doi: 10.3758/BF03192966. 
[14] 
D. Kozlowski, D. Worthington, Use of queue modelling in the analysis of elective patient treatment governed by a maximum waiting time policy, European Journal of Operational Research, 244 (2015), 331338. doi: 10.1016/j.ejor.2015.01.024. 
[15] 
J. Librero, M. Marin, S. Peiro, A. V. Munujos, Exploring the impact of complications on length of stay in major surgery diagnosisrelated groups, Int. J. Qual. Health Care, 16 (2004), 5157. doi: 10.1093/intqhc/mzh008. 
[16] 
F. Meng, K. L. Teow, C. K. Ooi, B. H. Heng, S. Y. Tay, Analysis of patient waiting time governed by a generic maximum waiting time policy with general phasetype approximations, Health Care Manag. Sci., 18 (2015), 267278. doi: 10.1007/s1072901493089. 
[17] 
F. Meng, J. Qi, M. Zhang, J. Ang, S. Chu, M. Sim, A robust optimization model for managing elective admision in a public hospital, Oper. Res., 63 (2015), 14521467. doi: 10.1287/opre.2015.1423. 
[18] 
MOH (Ministry of Health) Singapore, 2012. Available from: http://www.moh.gov.sg/content/moh_web/home/statistics/healthcare_institutionstatistics/Waiting_Time_for_Admission_to_Ward.html. 
[19] 
Ministry of Health, MOH Statistics Bulletin, Singapore, 2009. 
[20] 
J. C. Mowen, J. W. Licata, J. McPhail, Waiting in the emergency room: how to improve patient satisfaction, J. Health Care Mark., 13 (1993), 2633. 
[21] 
H. C. Ndukwe, S. Omale, O. O. Opanuga, Reducing queues in a Nigerian hospital pharmacy, Afr. J. Pharm. Pharmacol., 5 (2011), 10201026. 
[22] 
H. P. Phua, Waiting Times at Public Sector Emergency Departments, Working paper, Health Information Management Branch, Ministry of Health, Singapore, 2005. 
[23] 
P. K. Plunkett, D. G. Byrne, T. Breslin, K. Bennett, B. Silke, Increasing wait times predict increasing mortality for emergency medical admissions, Eur. J. Emerg. Med., 18 (2011), 192196. doi: 10.1097/MEJ.0b013e328344917e. 
[24] 
W. Rauf, J. J. Blitz, M. M. Geyser, A. Rauf, Quality improvement cycles that reduced waiting times at Tshwane district hospital emergency department, S. Afr. Fam. Pract., 50 (2008), 4343e. doi: 10.1080/20786204.2008.10873781. 
[25] 
G. F. Reed, F. Lynn, B. D. Meade, Use of coefficient of variation in assessing variability of of quantitative assays, Clin. Diagn. Lab. Immunol., 9 (2002), 12351239. 
[26] 
D. Ru, M. L. McCarthy, J. S. Desmond, Characterizing waiting room time, treatment time, and boarding time in the emergency department using quantile regression, Acad. Emerg. Med., 17 (2010), 813823. 
[27] 
L. Siciliani, J. Hurst, Tackling excessive waiting times for elective surgery: A comparative analysis of policies in 12 OECD countries, Health Policy, 72 (2005), 201215. doi: 10.1016/j.healthpol.2004.07.003. 
[28] 
Y. Sun, K. L. Teow, B. H. Heng, C. K. Ooi, S. Y. Tay, Real time prediction of waiting time in the emergency department using quantile regression, Ann. Emerg. Med., 60 (2012), 299308. doi: 10.1016/j.annemergmed.2012.03.011. 
[29] 
H. Tekiner, D. Coit, System reliability optimization considering uncertainty: minimization of a coefficient of variation measure, Proceedings of the 2008 Industrial Engineering Research Conference, (2008), 9951000. 
[30] 
D. A. Thompson, P. R. Yarnold, D. R. Williams, S. L. Adams, Effects of actual waiting time, perceived waiting time, information delivery, and expressive quality on patient satisfaction in the emergency department, Ann. Emerg. Med., 28 (1996), 657665. doi: 10.1016/S01960644(96)700902. 
[31] 
D. A. Thompson, P. R. Yarnold, Relating patient satisfaction to waiting time perceptions and expectations: the disconfirmation paradigm, Acad. Emerg. Med., 2 (1995), 10571062. doi: 10.1111/j.15532712.1995.tb03150.x. 
[32] 
E. U. Weber, S. Shafir, A.R. Blais, Predicting risk sensitivity in humans and lower annimals: Risk as variance or coefficient of variation, Psychol. Rev., 111 (2004), 430445. 
[33] 
L. Zhao and B. Lie, Modeling and simulation of patient flow in hospitals for resource utilization, 2008. Available from: http://www.scansims.org/sims2008/02.pdf. 
[34] 
Z. C. Zhu, B. H. Heng and K. L. Teow, Reducing consultation waiting time and clinic overtime in outpatient clinic: challenges and solution, in Management Engineering for Effective Healthcare Delivery: Principles and Applications, Medical Information Science Reference (eds. A. Kolker and P. Story), Hershey, Pennsylvania, (2011), 229245. 
show all references
References:
[1] 
Australian Institute of Health and Welfare (AIHW), Australian Hospital Statistics 2003. 04. (Canberra: AIHW, 2005) (AIHW cat. no. HSE 37; Health Services series no. 23). 
[2] 
B. Bursch, J. Beezy, R. Shaw, Emergency department satisfaction: What matters most?, Ann. Emerg. Med., 22 (1993), 586591. doi: 10.1016/S01960644(05)81947X. 
[3] 
L. G. Connelly, A. E. Bair, Discrete event simulation of emergency department activity: A platform for systemlevel operations research, Acad. Emerg. Med., 11 (2004), 11771185. 
[4] 
B. L. ConnerSpady, C. Sanmartin, G. H. Johnston, J. J. McGurran, M. Kehler, T. W. Noseworthy, The importance of patient expectations as a determinant of satisfaction with waiting times for hip and knee replacement surgery, Health Policy, 101 (2011), 245252. doi: 10.1016/j.healthpol.2011.05.011. 
[5] 
P. De, J. B. Ghosh, C. E. Wells, Scheduling to minimize the coefficient of variation, Int. J. Prod. Econ., 44 (1996), 249253. doi: 10.1016/09255273(96)000564. 
[6] 
R. L. Gardner, U. Sarkar, J. H. Maselli, Factors associated with longer ED lengths of stay, Am. J. Emerg. Med., 25 (2007), 643650. doi: 10.1016/j.ajem.2006.11.037. 
[7] 
L. V. Green, S. Savin, Reducing delays for medical appointments: A queueing approach, Oper. Res., 56 (2008), 15261538. doi: 10.1287/opre.1080.0575. 
[8] 
A. Guttmann, M. J. Schull, M. J. Vermeulen and T. A. Stukel, Association between waiting times and short term mortality and hospital admission after departure from emergency department: Population based cohort study from Ontario, Canada, BMJ, 342 (2011). 
[9] 
P. R. Harper, H. M. Gamlin, Reduced outpatient waiting times with improved appointment scheduling: A simulation modelling approach, OR Spectrum, 25 (2003), 207222. doi: 10.1007/s002910030122x. 
[10] 
S. A. Hemaya, T. E. Locke, How accurate are predicted waiting times, determined upon a patient's arrival in the Emergency Department?, Emerg. Med. J., 29 (2012), 316318. doi: 10.1136/emj.2010.106534. 
[11] 
E. T. Ho, Improving waiting time and operational clinic flow in a tertiary diabetes center BMJ Qual. Improv. Rep. , 2 (2014). 
[12] 
M. A. Kallen, J. A. Terrell, P. LewisPatterson, J. P. Hwang, Improving wait time for chemotherapy in an outpatient clinic at a comprehensive cancer center, J. Oncol. Practice, 8 (2012), e1e7. doi: 10.1200/JOP.2011.000281. 
[13] 
K. Kelly, Sample size planning for the coefficient of variation from the accuracy in parameter estimation approach, Behav. Res. Methods, 39 (2007), 755766. doi: 10.3758/BF03192966. 
[14] 
D. Kozlowski, D. Worthington, Use of queue modelling in the analysis of elective patient treatment governed by a maximum waiting time policy, European Journal of Operational Research, 244 (2015), 331338. doi: 10.1016/j.ejor.2015.01.024. 
[15] 
J. Librero, M. Marin, S. Peiro, A. V. Munujos, Exploring the impact of complications on length of stay in major surgery diagnosisrelated groups, Int. J. Qual. Health Care, 16 (2004), 5157. doi: 10.1093/intqhc/mzh008. 
[16] 
F. Meng, K. L. Teow, C. K. Ooi, B. H. Heng, S. Y. Tay, Analysis of patient waiting time governed by a generic maximum waiting time policy with general phasetype approximations, Health Care Manag. Sci., 18 (2015), 267278. doi: 10.1007/s1072901493089. 
[17] 
F. Meng, J. Qi, M. Zhang, J. Ang, S. Chu, M. Sim, A robust optimization model for managing elective admision in a public hospital, Oper. Res., 63 (2015), 14521467. doi: 10.1287/opre.2015.1423. 
[18] 
MOH (Ministry of Health) Singapore, 2012. Available from: http://www.moh.gov.sg/content/moh_web/home/statistics/healthcare_institutionstatistics/Waiting_Time_for_Admission_to_Ward.html. 
[19] 
Ministry of Health, MOH Statistics Bulletin, Singapore, 2009. 
[20] 
J. C. Mowen, J. W. Licata, J. McPhail, Waiting in the emergency room: how to improve patient satisfaction, J. Health Care Mark., 13 (1993), 2633. 
[21] 
H. C. Ndukwe, S. Omale, O. O. Opanuga, Reducing queues in a Nigerian hospital pharmacy, Afr. J. Pharm. Pharmacol., 5 (2011), 10201026. 
[22] 
H. P. Phua, Waiting Times at Public Sector Emergency Departments, Working paper, Health Information Management Branch, Ministry of Health, Singapore, 2005. 
[23] 
P. K. Plunkett, D. G. Byrne, T. Breslin, K. Bennett, B. Silke, Increasing wait times predict increasing mortality for emergency medical admissions, Eur. J. Emerg. Med., 18 (2011), 192196. doi: 10.1097/MEJ.0b013e328344917e. 
[24] 
W. Rauf, J. J. Blitz, M. M. Geyser, A. Rauf, Quality improvement cycles that reduced waiting times at Tshwane district hospital emergency department, S. Afr. Fam. Pract., 50 (2008), 4343e. doi: 10.1080/20786204.2008.10873781. 
[25] 
G. F. Reed, F. Lynn, B. D. Meade, Use of coefficient of variation in assessing variability of of quantitative assays, Clin. Diagn. Lab. Immunol., 9 (2002), 12351239. 
[26] 
D. Ru, M. L. McCarthy, J. S. Desmond, Characterizing waiting room time, treatment time, and boarding time in the emergency department using quantile regression, Acad. Emerg. Med., 17 (2010), 813823. 
[27] 
L. Siciliani, J. Hurst, Tackling excessive waiting times for elective surgery: A comparative analysis of policies in 12 OECD countries, Health Policy, 72 (2005), 201215. doi: 10.1016/j.healthpol.2004.07.003. 
[28] 
Y. Sun, K. L. Teow, B. H. Heng, C. K. Ooi, S. Y. Tay, Real time prediction of waiting time in the emergency department using quantile regression, Ann. Emerg. Med., 60 (2012), 299308. doi: 10.1016/j.annemergmed.2012.03.011. 
[29] 
H. Tekiner, D. Coit, System reliability optimization considering uncertainty: minimization of a coefficient of variation measure, Proceedings of the 2008 Industrial Engineering Research Conference, (2008), 9951000. 
[30] 
D. A. Thompson, P. R. Yarnold, D. R. Williams, S. L. Adams, Effects of actual waiting time, perceived waiting time, information delivery, and expressive quality on patient satisfaction in the emergency department, Ann. Emerg. Med., 28 (1996), 657665. doi: 10.1016/S01960644(96)700902. 
[31] 
D. A. Thompson, P. R. Yarnold, Relating patient satisfaction to waiting time perceptions and expectations: the disconfirmation paradigm, Acad. Emerg. Med., 2 (1995), 10571062. doi: 10.1111/j.15532712.1995.tb03150.x. 
[32] 
E. U. Weber, S. Shafir, A.R. Blais, Predicting risk sensitivity in humans and lower annimals: Risk as variance or coefficient of variation, Psychol. Rev., 111 (2004), 430445. 
[33] 
L. Zhao and B. Lie, Modeling and simulation of patient flow in hospitals for resource utilization, 2008. Available from: http://www.scansims.org/sims2008/02.pdf. 
[34] 
Z. C. Zhu, B. H. Heng and K. L. Teow, Reducing consultation waiting time and clinic overtime in outpatient clinic: challenges and solution, in Management Engineering for Effective Healthcare Delivery: Principles and Applications, Medical Information Science Reference (eds. A. Kolker and P. Story), Hershey, Pennsylvania, (2011), 229245. 
Sample Size  27,689 
Min  0 (minute) 
25th Percentile  21.1 (minutes) 
Median  40.1 (minutes) 
Mean  50.6 (minutes) 
75th Percentile  69.6 (minutes) 
95th Percentile  128.3 (minutes) 
Max  321.8 (minutes) 
Standard Deviation  39.1 (minutes) 
Variance  1528.7 
Coefficient of Variation  0.772 
Skewness  1.454 
Sample Size  27,689 
Min  0 (minute) 
25th Percentile  21.1 (minutes) 
Median  40.1 (minutes) 
Mean  50.6 (minutes) 
75th Percentile  69.6 (minutes) 
95th Percentile  128.3 (minutes) 
Max  321.8 (minutes) 
Standard Deviation  39.1 (minutes) 
Variance  1528.7 
Coefficient of Variation  0.772 
Skewness  1.454 
the length  
10  12  14  15  16  17  18  19  21  23  25  
128  0.624  0.618  0.613  0.610  0.606  0.603  0.600  0.596  0.591  0.582  0.576 
127  0.620  0.614  0.608  0.605  0.601  0.598  0.595  0.591  0.586  0.579  0.571 
126  0.615  0.610  0.604  0.600  0.597  0.594  0.590  0.588  0.579  0.573  0.567 
125  0.611  0.606  0.599  0.596  0.592  0.589  0.586  0.583  0.576  0.568  0.562 
124  0.607  0.601  0.594  0.591  0.587  0.585  0.582  0.577  0.570  0.563  0.556 
123  0.603  0.596  0.589  0.586  0.583  0.580  0.575  0.573  0.565  0.558  0.551 
122  0.598  0.591  0.584  0.581  0.579  0.574  0.571  0.567  0.560  0.553  0.546 
121  0.593  0.586  0.580  0.577  0.572  0.570  0.565  0.562  0.555  0.548  0.541 
120  0.588  0.581  0.575  0.571  0.568  0.564  0.560  0.557  0.549  0.543  0.535 
119  0.584  0.577  0.569  0.566  0.562  0.559  0.555  0.551  0.544  0.538  0.528 
118  0.578  0.572  0.565  0.561  0.557  0.553  0.550  0.546  0.539  0.531  0.523 
117  0.573  0.566  0.559  0.555  0.552  0.548  0.544  0.541  0.534  0.525  0.515 
116  0.569  0.561  0.554  0.550  0.546  0.542  0.539  0.535  0.528  0.519  0.508 
115  0.563  0.556  0.548  0.544  0.541  0.537  0.533  0.530  0.521  0.512  0.503 
114  0.558  0.550  0.543  0.539  0.535  0.531  0.528  0.524  0.515  0.504  0.496 
113  0.552  0.545  0.537  0.533  0.529  0.526  0.522  0.517  0.508  0.499  0.489 
112  0.547  0.539  0.531  0.528  0.524  0.520  0.515  0.511  0.500  0.492  0.483 
111  0.541  0.533  0.526  0.522  0.518  0.513  0.509  0.504  0.495  0.485  0.473 
110  0.535  0.527  0.520  0.516  0.511  0.507  0.502  0.497  0.488  0.478  0.466 
109  0.529  0.522  0.514  0.510  0.505  0.500  0.495  0.490  0.481  0.469  0.461 
108  0.524  0.516  0.508  0.503  0.498  0.493  0.488  0.484  0.474  0.462  0.452 
107  0.518  0.510  0.501  0.496  0.491  0.486  0.481  0.476  0.465  0.456  0.444 
106  0.512  0.504  0.494  0.489  0.484  0.479  0.474  0.469  0.457  0.447  0.434 
the length  
10  12  14  15  16  17  18  19  21  23  25  
128  0.624  0.618  0.613  0.610  0.606  0.603  0.600  0.596  0.591  0.582  0.576 
127  0.620  0.614  0.608  0.605  0.601  0.598  0.595  0.591  0.586  0.579  0.571 
126  0.615  0.610  0.604  0.600  0.597  0.594  0.590  0.588  0.579  0.573  0.567 
125  0.611  0.606  0.599  0.596  0.592  0.589  0.586  0.583  0.576  0.568  0.562 
124  0.607  0.601  0.594  0.591  0.587  0.585  0.582  0.577  0.570  0.563  0.556 
123  0.603  0.596  0.589  0.586  0.583  0.580  0.575  0.573  0.565  0.558  0.551 
122  0.598  0.591  0.584  0.581  0.579  0.574  0.571  0.567  0.560  0.553  0.546 
121  0.593  0.586  0.580  0.577  0.572  0.570  0.565  0.562  0.555  0.548  0.541 
120  0.588  0.581  0.575  0.571  0.568  0.564  0.560  0.557  0.549  0.543  0.535 
119  0.584  0.577  0.569  0.566  0.562  0.559  0.555  0.551  0.544  0.538  0.528 
118  0.578  0.572  0.565  0.561  0.557  0.553  0.550  0.546  0.539  0.531  0.523 
117  0.573  0.566  0.559  0.555  0.552  0.548  0.544  0.541  0.534  0.525  0.515 
116  0.569  0.561  0.554  0.550  0.546  0.542  0.539  0.535  0.528  0.519  0.508 
115  0.563  0.556  0.548  0.544  0.541  0.537  0.533  0.530  0.521  0.512  0.503 
114  0.558  0.550  0.543  0.539  0.535  0.531  0.528  0.524  0.515  0.504  0.496 
113  0.552  0.545  0.537  0.533  0.529  0.526  0.522  0.517  0.508  0.499  0.489 
112  0.547  0.539  0.531  0.528  0.524  0.520  0.515  0.511  0.500  0.492  0.483 
111  0.541  0.533  0.526  0.522  0.518  0.513  0.509  0.504  0.495  0.485  0.473 
110  0.535  0.527  0.520  0.516  0.511  0.507  0.502  0.497  0.488  0.478  0.466 
109  0.529  0.522  0.514  0.510  0.505  0.500  0.495  0.490  0.481  0.469  0.461 
108  0.524  0.516  0.508  0.503  0.498  0.493  0.488  0.484  0.474  0.462  0.452 
107  0.518  0.510  0.501  0.496  0.491  0.486  0.481  0.476  0.465  0.456  0.444 
106  0.512  0.504  0.494  0.489  0.484  0.479  0.474  0.469  0.457  0.447  0.434 
length  12  14  15  16  17  18  19  21  23 
CV  0.50  0.49  0.49  0.48  0.48  0.47  0.47  0.46  0.45 
New mean  52.22  52.36  52.43  52.56  52.64  52.71  52.79  52.95  53.10 
New variance  691.37  669.14  656.33  647.02  636.20  624.37  613.49  586.38  563.09 
CV reduction  34.8%  36.1%  36.8%  37.4%  38.0%  38.7%  39.3%  40.8%  42.2% 
Mean increase  3.1%  3.4%  3.5%  3.6%  3.9%  4.0%  4.2%  4.4%  5.0% 
Var reduction  54.1%  55.4%  56.2%  57.1%  58.0 %  58.4%  59.2%  61.0%  62.0% 
length  12  14  15  16  17  18  19  21  23 
CV  0.50  0.49  0.49  0.48  0.48  0.47  0.47  0.46  0.45 
New mean  52.22  52.36  52.43  52.56  52.64  52.71  52.79  52.95  53.10 
New variance  691.37  669.14  656.33  647.02  636.20  624.37  613.49  586.38  563.09 
CV reduction  34.8%  36.1%  36.8%  37.4%  38.0%  38.7%  39.3%  40.8%  42.2% 
Mean increase  3.1%  3.4%  3.5%  3.6%  3.9%  4.0%  4.2%  4.4%  5.0% 
Var reduction  54.1%  55.4%  56.2%  57.1%  58.0 %  58.4%  59.2%  61.0%  62.0% 
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