A two-level iteration approach for modeling and analysis of rapid response process with multiple deteriorating patients

  • Zexian Zeng
  • Zhenghao Fan
  • Xiaolei XieEmail author
  • Colleen H. Swartz
  • Paul DePriest
  • Jingshan Li


In acute care, a patient’s clinical deterioration is often a precursor to serious and often fatal outcomes. To reduce the severity and frequency of negative outcomes, care providers need to response rapidly by providing quick evaluation, triage, and treatment to patients with declining conditions. However, a provider’s availability to respond can be constrained when multiple patients are deteriorating at the same time. To study the multiple patients rapid response process, we introduce a network model with complex structures, such as split, merge, and parallel. Iterative methods are presented to evaluate the mean decision time (i.e., the average time from the detection of a patient’s declining to a physician’s treatment decision being made). It is shown that such methods lead to convergent results and high accuracy in performance evaluation. Such a model provides a quantitative tool for healthcare professionals to design and improve rapid response systems.


Rapid response Decision time Mean waiting time Multiple patients Patient deterioration Iterations 



This work is supported in part by National Science Foundation Grant No. CMMI-1536987 and by National Natural Science Foundation of China Grant No. 71501109.


  1. Berwick DM, Calkins DR, McCannon CJ, Hackbarth AD (2006) The 100,000 lives campaign: setting a goal and a deadline for improving health care quality. J Am Med Assoc 295(3):324–327CrossRefGoogle Scholar
  2. Brandeau ML, Sainfort F, Pierskalla WP (2004) Operations research and health care: a handbook of methods and applications. Springer, BerlinzbMATHGoogle Scholar
  3. Brindley PG (2010) Patient safety and acute care medicine: lessons for the future, insights from the past. Crit Care 14(2):217–221CrossRefGoogle Scholar
  4. Buchman TG, Coopersmith CM, Meissen HW, Grabenkort WR, Bakshi V, Hiddleson CA, Gregg SR (2017) Innovative interdisciplinary strategies to address the intensivist shortage. Crit Care Med 45(2):298–304CrossRefGoogle Scholar
  5. Chan PS, Jain R, Nallmothu BK, Berg RA, Sasson C (2010) Rapid response teams: a systematic review and meta-analysis. Arch Intern Med 170(1):18–26CrossRefGoogle Scholar
  6. Dacey MJ, Mirza ER, Wilcox V, Doherty M, Mello J, Boyer A, Gates J, Brothers T, Baute R (2007) The effect of a rapid response team on major clinical outcome measures in a community hospital. Crit Care Med 35(9):2076–2082CrossRefGoogle Scholar
  7. DeVita MA, Bellomo R, Hillman K, Kellum J, Rotondi A, Teres D, Auerbach A, Chen W-J, Duncan K, Kenward G (2006) Findings of the first consensus conference on medical emergency teams. Crit Care Med 34(9):2463–2478CrossRefGoogle Scholar
  8. DeVita MA, Hillman K, Bellomo R (2011) Textbook of rapid response systems: concept and implementation. Springer, BerlinCrossRefGoogle Scholar
  9. Downey A, Quach J, Haase M, Haase-Fielitz A, Jones D, Bellomo R (2008) Characteristics and outcomes of patients receiving a medical emergency team review for acute change in conscious state or arrhythmias. Crit Care Med 36(2):477–481CrossRefGoogle Scholar
  10. Fomundam S, Herrmann J (2007) A survey of queuing theory applications in health care. Technicial report no. 2007-24, the Institute for Systems Research, University of Maryland, College Park, MAGoogle Scholar
  11. Garg L, McClean S, Meenan B, Millard P (2010) A non-homogeneous discrete time Markov model for admission scheduling and resource planning in a cost or capacity constrained healthcare system. Health Care Manag Sci 13(2):155–169CrossRefGoogle Scholar
  12. Green L (2006) Queueing analysis in healthcare. In: Hall RW (ed) Patient flow: reducing delays in healthcare delivery. Springer, Berlin, pp 281–307CrossRefGoogle Scholar
  13. Gunal MM, Pidd M (2010) Discrete event simulation for performance modelling in health care: a review of the literature. J Simul 4(1):42–51CrossRefGoogle Scholar
  14. Hall RW (2006) Patient flow: reducing delays in healthcare delivery. Springer, BerlinCrossRefGoogle Scholar
  15. Hillman K, Bristow P, Chey T, Daffurn K, Jacques T, Norman S, Bishop GF, Simmons G (2001) Antecedents to hospital deaths. Inter Med J 31(6):343–348CrossRefGoogle Scholar
  16. Hillman K, Chen J, Cretikos M, Bellomo R, Brown D, Doig G, Finfer S, Flabouris A (2005) Introduction of the medical emergency team (MET) system: a cluster-randomised controlled trial. Lancet 365(9477):2091–2097CrossRefGoogle Scholar
  17. Jacobson SH, Hall SN, Swisher JR (2006) Discrete-event simulation of health care systems. Patient Flow Reducing Delay Healthc Deliv 91:211–252CrossRefGoogle Scholar
  18. Kohn LT, Corrigan JM, Donaldson MS (2000) To err is human: building a safer health system. Institute of Medicine, National Academy Press, WashingtonGoogle Scholar
  19. Lakshmi C, Iyer SA (2013) Application of queueing theory in health care: a literature review. Oper Res Health Care 2(1):25–39Google Scholar
  20. Leape LL, Berwick DM (2005) Five years after to err is human: What have we learned? J Am Med Assoc 293:2384–2390CrossRefGoogle Scholar
  21. Li J, Meerkov SM (2005) On the coefficients of variation of up- and downtime of manufacturing equipment. Math Probl Eng 2005:1–6CrossRefzbMATHGoogle Scholar
  22. Massey D, Aitken LM, Chaboyer W (2010) Literature review: Do rapid response systems reduce the incidence of major adverse events in the deteriorating ward patient? J Clin Nurs 19(23–24):3260–3273CrossRefGoogle Scholar
  23. Mayhew L, Smith D (2008) Using queuing theory to analyse the government’s 4-h completion time target in accident and emergency departments. Health Care Manag Sci 11(1):11–21CrossRefGoogle Scholar
  24. Meyers MO, Sarosi GA, Brasel KJ (2017) Perspective of residency program directors on accreditation council for graduate medical education changes in resident work environment and duty hours. JAMA Surg 152(10):905–906CrossRefGoogle Scholar
  25. McArthur-Rouse F (2001) Critical care outreach services and early warning scoring systems: a review of the literature. J Adv Nurs 36(5):696–704CrossRefGoogle Scholar
  26. McGloin H, Adam SK, Singer M (1999) Unexpected deaths and referrals to intensive care of patients on general wards. Are some cases potentially avoidable? J R Coll Phys Lond 33(3):255–259Google Scholar
  27. Priestley G, Watson W, Rashidian A, Mozley C, Russell D, Wilson J, Cope J, Hart D, Kay D, Cowley K, Pateraki J (2004) Introducing critical care outreach: a ward randmized trial of phased introduction in a general hospital. Intensive Care Med 30(7):1398–1404CrossRefGoogle Scholar
  28. Ranji S, Auerbach A, Hurd C, O’Rourke K, Shohania K (2007) Effects of rapid response systems on clinical outcomes: systematic review and meta analysis. J Hosp Med 2(6):422–432CrossRefGoogle Scholar
  29. Schaefer AJ, Bailey MD, Shechter SM, Roberts MS (2005) Modeling medical treatment using Markov decision processes. In: Brandeau ML et al (eds) Operations research and health care. Springer, Berlin, pp 593–612CrossRefGoogle Scholar
  30. Wang J, Quan S, Li J, Hollis A (2012) Modeling and analysis of work flow and staffing level in a computed tomography division of University of Wisconsin Medical Foundation. Health Care Manag Sci 15(2):108–120CrossRefGoogle Scholar
  31. Wang J, Li J, Howard PK (2013) A system model of work flow in the patient room of hospital emergency department. Health Care Manag Sci 16(4):341–351CrossRefGoogle Scholar
  32. Wang J, Zhong X, Li J, Howard PK (2014) Modeling and analysis of care delivery services within patient rooms: a system-theoretic approach. IEEE Trans AutomSci Eng 11(2):379–393CrossRefGoogle Scholar
  33. Watcher RM (2004) The end of the beginning: patient safety five years after “To err is human”. Health Aff W4:534–545Google Scholar
  34. Whitlock J (2017) Doctors, residents, interns, and attendings: What’s the difference? The doctors on your healthcare team. Accessed Jan 2018
  35. Wiler JL, Griffey RT, Olsen T (2011) Review of modeling approaches for emergency department patient flow and crowding research. Acad Emerg Med 18(12):1371–1379CrossRefGoogle Scholar
  36. Winters BD, Pham JC, Hunt E, Guallar EA, Berenholtz S, Pronovost PJ (2007) Rapid response systems: a systematic review. Crit Care Med 35(5):1238–1243CrossRefGoogle Scholar
  37. Xie X, Li J, Swartz CH, Depriest P (2012) Modeling and analysis of rapid response process to improve patient safety. IEEE Trans Autom Sci Eng 9(2):215–225CrossRefGoogle Scholar
  38. Xie X, Li J, Swartz CH, Depriest P (2014) Improving response-time performance in acute care delivery: a systems approach. IEEE Trans Autom Sci Eng 11(4):1240–1249CrossRefGoogle Scholar
  39. Xie X, Li J, Swartz C, Dong Y, DePriest P (2016) Modeling and analysis of ward patient rescue process on the hospital floor. IEEE Trans Autom Sci Eng 13(2):514–528CrossRefGoogle Scholar
  40. Zhong X, Williams M, Li J, Kraft S, Sleeth J (2015) Primary care redesign: review and a simulation study at a pediatric clinic. In: Yang H, Lee E (eds) Healthcare data analytics, Wiley series on operations research and management science (WORMS). Wiley, Hoboken, pp 399–426Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Feinberg School of MedicineNorthwestern UniversityEvanstonUSA
  2. 2.Department of Industrial EngineeringTsinghua UniversityBeijingPeople’s Republic of China
  3. 3.University of Kentucky Chandler Medical CenterLexingtonUSA
  4. 4.Baptist Memorial Health Care CorporationMemphisUSA
  5. 5.Department of Industrial and Systems EngineeringUniversity of WisconsinMadisonUSA

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