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

Abstract

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.

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References

  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):324–327

    Article  Google Scholar 

  2. Brandeau ML, Sainfort F, Pierskalla WP (2004) Operations research and health care: a handbook of methods and applications. Springer, Berlin

    Google Scholar 

  3. Brindley PG (2010) Patient safety and acute care medicine: lessons for the future, insights from the past. Crit Care 14(2):217–221

    Article  Google 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):298–304

    Article  Google 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):18–26

    Article  Google 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):2076–2082

    Article  Google 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):2463–2478

    Article  Google Scholar 

  8. DeVita MA, Hillman K, Bellomo R (2011) Textbook of rapid response systems: concept and implementation. Springer, Berlin

    Google 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):477–481

    Article  Google 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, MA

  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):155–169

    Article  Google Scholar 

  12. Green L (2006) Queueing analysis in healthcare. In: Hall RW (ed) Patient flow: reducing delays in healthcare delivery. Springer, Berlin, pp 281–307

    Google 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):42–51

    Article  Google Scholar 

  14. Hall RW (2006) Patient flow: reducing delays in healthcare delivery. Springer, Berlin

    Google 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):343–348

    Article  Google 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):2091–2097

    Article  Google Scholar 

  17. Jacobson SH, Hall SN, Swisher JR (2006) Discrete-event simulation of health care systems. Patient Flow Reducing Delay Healthc Deliv 91:211–252

    Article  Google Scholar 

  18. Kohn LT, Corrigan JM, Donaldson MS (2000) To err is human: building a safer health system. Institute of Medicine, National Academy Press, Washington

    Google Scholar 

  19. Lakshmi C, Iyer SA (2013) Application of queueing theory in health care: a literature review. Oper Res Health Care 2(1):25–39

    Google Scholar 

  20. Leape LL, Berwick DM (2005) Five years after to err is human: What have we learned? J Am Med Assoc 293:2384–2390

    Article  Google Scholar 

  21. Li J, Meerkov SM (2005) On the coefficients of variation of up- and downtime of manufacturing equipment. Math Probl Eng 2005:1–6

    Article  Google 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):3260–3273

    Article  Google 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):11–21

    Article  Google 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):905–906

    Article  Google 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):696–704

    Article  Google 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):255–259

    Google 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):1398–1404

    Article  Google 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):422–432

    Article  Google 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 593–612

    Google 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):108–120

    Article  Google 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):341–351

    Article  Google 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):379–393

    Article  Google Scholar 

  33. Watcher RM (2004) The end of the beginning: patient safety five years after “To err is human”. Health Aff W4:534–545

    Google Scholar 

  34. Whitlock J (2017) Doctors, residents, interns, and attendings: What’s the difference? The doctors on your healthcare team. https://www.verywell.com/types-of-doctors-residents-interns-and-fellows-3157293 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):1371–1379

    Article  Google 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):1238–1243

    Article  Google 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):215–225

    Article  Google 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):1240–1249

    Article  Google 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):514–528

    Article  Google 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 399–426

    Google Scholar 

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Acknowledgements

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.

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Correspondence to Xiaolei Xie.

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Appendices

Appendix 1: Iteration procedures

Three-patient example: Level-1 iteration procedure

Denote \(\tau _{k,r}^{(j)}\), \(k=1,2,3\), \(r \in X\), as the mean decision time that includes patient k’s waiting time for provider \(r\) during the j-th iteration, \(j=1,2,\ldots\). Let \(p_{k,r}^{(j)}\) be the probability that provider r is treating patient k and there is another request for provider r during the j-th iteration. At the beginning of iteration, assume

$$\begin{aligned} \tau _{k,r}^{(0)}=T_d \quad \text{ and }\quad p_{k,r}^{(0)}=0, \qquad k=1,2,3, \quad r \in X. \end{aligned}$$

First, consider patient 1. During the first iteration, \(\tau _{1,int}^{(1)}\) can be updated as:

$$\begin{aligned} \tau _{1,int}^{(1)} = T_d + (p_{2,int}^{(0)}+p_{3,int}^{(0)})(p_{int} \tau _{int}+ p_{rrt \& int} \tau _{rrt \& int}). \end{aligned}$$

The \(p_{1,int}^{(1)}\) can be updated as:

$$\begin{aligned} p_{1,int}^{(1)} = \frac{p_{int}^2 \tau _{int}+ p_{rrt \& int}^2 \tau _{rrt \& int}}{\tau _{1,int}^{(1)}}. \end{aligned}$$

Next, Consider patient 2. Decision time \(\tau _{2,int}^{(1)}\) and probability \(p_{2,int}^{(1)}\) can be calculated.

$$\begin{aligned} \tau _{2,int}^{(1)}&= T_d + (p_{1,int}^{(1)}+p_{3,int}^{(0)})(p_{int} \tau _{int}+ p_{rrt \& int} \tau _{rrt \& int}),\\ p_{2,int}^{(1)}&= \frac{p_{int}^2 \tau _{int}+ p_{rrt \& int}^2 \tau _{rrt \& int}}{\tau _{2,int}^{(1)}}. \end{aligned}$$

Lastly, consider patient 3, we have

$$\begin{aligned} \tau _{3,int}^{(1)}&= T_d + (p_{1,int}^{(1)}+p_{2,int}^{(1)})(p_{int} \tau _{int}+ p_{rrt \& int} \tau _{rrt \& int}),\\ p_{3,int}^{(1)}&= \frac{p_{int}^2 \tau _{int}+ p_{rrt \& int}^2 \tau _{rrt \& int}}{\tau _{3,int}^{(1)}}. \end{aligned}$$

This completes the update of the intern.

Similar updating process for the resident can be carried out. First, we study patient 1:

$$\begin{aligned} \tau _{1,res}^{(1)}&= T_d + (p_{2,res}^{(0)}+p_{3,res}^{(0)})(p_{res} \tau _{res} +p_{rrt \& res} \tau _{rrt \& res}),\\ p_{1,res}^{(1)}&= \frac{p_{res}^2 \tau _{res}+ p_{rrt \& res}^2 \tau _{rrt \& res}}{\tau _{1,res}^{(1)}}. \end{aligned}$$

Next, consider patient 2:

$$\begin{aligned} \tau _{2,res}^{(1)}&= T_d + (p_{1,res}^{(1)}+p_{3,res}^{(0)})(p_{res} \tau _{res} +p_{rrt \& res} \tau _{rrt \& res}),\\ p_{2,res}^{(1)}&= \frac{p_{res}^2 \tau _{res}+ p_{rrt \& res}^2 \tau _{rrt \& res}}{\tau _{2,res}^{(1)}}. \end{aligned}$$

Then, patient 3 is included:

$$\begin{aligned} \tau _{3,res}^{(1)}&= T_d + (p_{1,res}^{(1)}+p_{2,res}^{(1)})(p_{res} \tau _{res} +p_{rrt \& res} \tau _{rrt \& res}),\\ p_{3,res}^{(1)}&= \frac{p_{res}^2 \tau _{res}+ p_{rrt \& res}^2 \tau _{rrt \& res}}{\tau _{3,res}^{(1)}}. \end{aligned}$$

Similarly, all the rest of providers are updated. Particularly, for the RRT, considering patient 1, we obtain:

$$\begin{aligned} \tau _{1,rrt}^{(1)}&= T_d + (p_{2,rrt}^{(0)}+p_{3,rrt}^{(0)})(p_{rrt} \tau _{rrt} + p_{rrt \& int} \tau _{rrt \& int} + p_{rrt \& res} \tau _{rrt \& res} \\&\quad +\,p_{rrt \& fel}\tau _{rrt \& fel}+ p_{rrt \& atn} \tau _{rrt \& atn}),\\ p_{1,rrt}^{(1)}&= (p_{rrt}^2 \tau _{rrt} + p_{rrt \& int}^2 \tau _{rrt \& int}^2 + p_{rrt \& res}^2 \tau _{rrt \& res} + p_{rrt \& fel}^2 \tau _{rrt \& fel} \\&\quad +\, p_{rrt \& atn}^2 \tau _{rrt \& atn}) / \tau _{1,rrt}^{(1)}. \end{aligned}$$

Regarding patient 2, we have

$$\begin{aligned} \tau _{2,rrt}^{(1)}&= T_d + (p_{1,rrt}^{(1)}+p_{3,rrt}^{(0)})(p_{rrt} \tau _{rrt} + p_{rrt \& int} \tau _{rrt \& int} + p_{rrt \& res} \tau _{rrt \& res} \\&\quad +\, p_{rrt \& fel} \tau _{rrt \& fel} + p_{rrt \& atn} \tau _{rrt \& atn}),\\ p_{2,rrt}^{(1)}&= (p_{rrt}^2 \tau _{rrt} + p_{rrt \& int}^2 \tau _{rrt \& int}^2 + p_{rrt \& res}^2 \tau _{rrt \& res} + p_{rrt \& fel}^2 \tau _{rrt \& fel} \\&\quad +\, p_{rrt \& atn}^2 \tau _{rrt \& atn}) / \tau _{2,rrt}^{(1)}. \end{aligned}$$

Furthermore, parameters of patient 3 are updated:

$$\begin{aligned} \tau _{3,rrt}^{(1)}&= T_d + (p_{1,rrt}^{(1)}+p_{2,rrt}^{(1)})(p_{rrt} \tau _{rrt} + p_{rrt \& int} \tau _{rrt \& int} + p_{rrt \& res} \tau _{rrt \& res} \\&\quad +\, p_{rrt \& fel} \tau _{rrt \& fel} + p_{rrt \& atn} \tau _{rrt \& atn}),\\ p_{3,rrt}^{(1)}&= (p_{rrt}^2 \tau _{rrt} + p_{rrt \& int}^2 \tau _{rrt \& int}^2 + p_{rrt \& res}^2 \tau _{rrt \& res} + p_{rrt \& fel}^2 \tau _{rrt \& fel} \\&\quad +\, p_{rrt \& atn}^2 \tau _{rrt \& atn}) /\tau _{3,rrt}^{(1)}. \end{aligned}$$

Then for the fellow, patients 1 to 3 are considered:

$$\begin{aligned} \tau _{1,fel}^{(1)}&= T_d + (p_{2,fel}^{(0)}+p_{3,fel}^{(0)})(p_{fel} \tau _{fel}+ p_{rrt \& fel} \tau _{rrt \& fel}),\\ p_{1,fel}^{(1)}&= \frac{p_{fel}^2 \tau _{fel}+ p_{rrt \& fel}^2 \tau _{rrt \& fel}}{\tau _{1,fel}^{(1)}},\\ \tau _{2,fel}^{(1)}&= T_d + (p_{1,fel}^{(1)}+p_{3,fel}^{(0)})(p_{fel} \tau _{fel}+ p_{rrt \& fel} \tau _{rrt \& fel}),\\ p_{2,fel}^{(1)}&= \frac{p_{fel}^2 \tau _{fel}+ p_{rrt \& fel}^2 \tau _{rrt \& fel}}{\tau _{2,fel}^{(1)}},\\ \tau _{2,fel}^{(1)}&= T_d + (p_{1,fel}^{(1)}+p_{2,fel}^{(1)})(p_{fel} \tau _{fel}+ p_{rrt \& fel} \tau _{rrt \& fel}),\\ p_{2,fel}^{(1)}&= \frac{p_{fel}^2 \tau _{fel}+ p_{rrt \& fel}^2 \tau _{rrt \& fel}}{\tau _{3,fel}^{(1)}}. \end{aligned}$$

Finally, for the attending, we again address all three patients:

$$\begin{aligned} \tau _{1,atn}^{(1)}&= T_d + (p_{2,atn}^{(0)}+p_{3,atn}^{(0)})(p_{atn} \tau _{atn}+ p_{rrt \& atn} \tau _{rrt \& atn}),\\ p_{1,atn}^{(1)}&= \frac{p_{atn}^2 \tau _{atn}+ p_{rrt \& atn}^2 \tau _{rrt \& atn}}{\tau _{1,atn}^{(1)}},\\ \tau _{2,atn}^{(1)}&= T_d + (p_{1,atn}^{(1)}+p_{3,atn}^{(0)})(p_{atn} \tau _{atn}+ p_{rrt \& atn} \tau _{rrt \& atn}),\\ p_{2,atn}^{(1)}&= \frac{p_{atn}^2 \tau _{atn}+ p_{rrt \& atn}^2 \tau _{rrt \& atn}}{\tau _{2,atn}^{(1)}},\\ \tau _{3,atn}^{(1)}&= T_d + (p_{1,atn}^{(1)}+p_{2,atn}^{(1)})(p_{atn} \tau _{atn}+ p_{rrt \& atn} \tau _{rrt \& atn}),\\ p_{3,atn}^{(1)}&= \frac{p_{atn}^2 \tau _{atn}+ p_{rrt \& atn}^2 \tau _{rrt \& atn}}{\tau _{3,atn}^{(1)}}. \end{aligned}$$

When the first iteration is finished, all the updated parameters will be used for the second iteration to calculate \(\tau _{k,r}^{(2)}\), \(k=1,2,3\), \(r \in X\), and \(p_{k,r}^{(2)}\). The process is repeated until procedure converges. Let \(\delta = 10^{-5}\). When

$$\begin{aligned} | \tau _{i,r}^{(j+1)} - \tau _{i,r}^{(j)} | \le \delta , \qquad | p_{i,r}^{(j+1)} - p_{i,r}^{(j)} | \le \delta , \qquad i=1,2,3, \quad r\in X, \end{aligned}$$

the procedure is convergent, i.e.,

$$\begin{aligned} \lim _{j \rightarrow \infty } \tau _{i,r}^{(j)} = \tau _{i,r}, \qquad \lim _{j \rightarrow \infty } p_{i,r}^{(j)} = p_{i,r}, \qquad i=1,2,3. \end{aligned}$$

In particular, all \(\tau _{i,r}, i=1,2,3,\) are identical and all \(p_{i,r}, i=1,2,3,\) are the same. Then the mean decision time (including waiting time) \({\mathrm {T}}_r\) and provider utilization \(P_r\) can be obtained:

$$\begin{aligned} \tau _{1,r}=\tau _{2,r}=\tau _{3,r}:={\mathrm {T}}_{r},\qquad p_{1,r}=p_{2,r}=p(3,r):=P_{r}. \end{aligned}$$

The mean decision time \(T_{in}\) includes the additional waiting time.

$$\begin{aligned} T_{in} = T_d + \Sigma _{r, r \in X} P_{r} {\mathrm {T}}_{r}. \end{aligned}$$

Three-patient example: Level-2 iteration procedure

Denote \(\rho _k^{(l)}\), \(k=1,2,3\), \(l=1,2,\ldots\), as the percentage of time the patient is in a deteriorating status in iteration j, and \(\lambda _k^{(l)}\), \(k=1,2,3\), \(l=1,2,\ldots\), as the updated mean decision time in iteration j by including the time percentage patient k is declining. When the iteration starts, assume all

$$\begin{aligned} \rho _k^{(0)}=0 \quad \text{ and } \quad \lambda _k^{(0)}=T_{in}, \qquad k=1,2,3. \end{aligned}$$

Considering patient 1, \(\lambda _1^{(1)}\) can be updated as:

$$\begin{aligned} \lambda _1^{(1)} = T_{in}\left[ 1+\rho _1^{(0)}\left( \rho _2^{(0)}+\rho _3^{(0)}\right) \right] . \end{aligned}$$

The time percentage that the first patient is in deteriorating status can be calculated as

$$\begin{aligned} \rho _1^{(1)} = \frac{\lambda _1^{(1)}}{\lambda _1^{(1)}+T_{normal}}. \end{aligned}$$

Next consider patients 2 and 3, where \(\lambda _i^{(1)}\) and \(\rho _i^{(1)}\), \(i=2,3\), can be obtained:

$$\begin{aligned} \lambda _2^{(1)}&= T_{in}\left[ 1+\rho _2^{(0)}\left( \rho _1^{(1)}+\rho _3^{(0)}\right) \right] ,\\ \rho _2^{(1)}&= \frac{\lambda _2^{(1)}}{\lambda _2^{(1)}+T_{normal}},\\ \lambda _3^{(1)}&= T_{in}\left[ 1+\rho _3^{(0)}\left( \rho _1^{(1)}+\rho _2^{(1)}\right) \right] ,\\ \rho _3^{(1)}&= \frac{\lambda _3^{(1)}}{\lambda _3^{(1)}+T_{normal}}. \end{aligned}$$

This finishes the first iteration. Then \(\rho _k^{(1)}\), \(k=1,2,3\), and \(\lambda _k^{(1)}\) are used for the second iteration to evaluate \(\rho _k^{(2)}\) and \(\lambda _k^{(2)}\). The process is repeated until the procedure converges. When the following criteria is met:

$$\begin{aligned} | \lambda _{i}^{(j+1)} - \lambda _{i}^{(l)} | \le \delta ,\qquad | \rho _{i}^{(j+1)} - \rho _{i}^{(l)} | \le \delta , \qquad i=1,2,3, \end{aligned}$$

the procedure is convergent. Again \(\delta = 10^{-5}\). Upon converges, we have

$$\begin{aligned} \lim _{l \rightarrow \infty } \lambda _{i}^{(l)} = \lambda _{i}, \qquad \lim _{l \rightarrow \infty } \rho _{i}^{(l)} = \rho _{i}, \qquad i=1,2,3. \end{aligned}$$

The final mean decision time can be obtained:

$$\begin{aligned} \lambda _{1}=\lambda _{2}=\lambda _3=T_{final}. \end{aligned}$$

General iteration procedure

Procedure 1

(1) Level-1 iteration

Step 1.1 Initialization: Calculate \(p_i\), \(i \in X\), and \(T_d\) using the results in Xie et al. (2012). Set \(j=0\) and

$$\begin{aligned} \tau _{k,i}^{(j)}=p_{k,i}^{(j)}=0. \end{aligned}$$

Step 1.2 Update \(\tau _{k,i}^{(j)}\) and \(p_{k,i}^{(j)}\): For patient 1,

$$\begin{aligned} \tau _{1,int}^{(j+1)}&= T_d + \Sigma _{i=2}^n p_{i,int}^{(j)} (p_{int} \tau _{int}+ p_{rrt \& int} \tau _{rrt \& int}), \\ p_{1,int}^{(j+1)}&= (p_{int}^2 \tau _{int}+ p_{rrt \& int}^2 \tau _{rrt \& int})\big /\tau _{1,int}^{(j+1)}, \\ \tau _{1,res}^{(j+1)}&= T_d + \Sigma _{i=2}^n p_{i,res}^{(j)} (p_{res} \tau _{res}+ p_{rrt \& res} \tau _{rrt \& res}), \\ p_{1,res}^{(j+1)}&= (p_{res}^2 \tau _{res}+ p_{rrt \& res}^2 \tau _{rrt \& res})\big /\tau _{1,res}^{(j+1)}, \\ \tau _{1,rrt}^{(j+1)}&= T_d + \Sigma _{i=2}^n p_{i,rrt}^{(j)} (p_{rrt} \tau _{rrt} + p_{rrt \& int} \tau _{rrt \& int} +p_{rrt \& res} \tau _{rrt \& res} \\&\quad +\, p_{rrt \& fel} \tau _{rrt \& fel}+ p_{rrt \& atn} \tau _{rrt \& atn}), \\ p_{1,rrt}^{(j+1)}&= (p_{rrt}^2 \tau _{rrt} + p_{rrt \& int}^2 \tau _{rrt \& int}^2 + p_{rrt \& res}^2 \tau _{rrt \& res} \\&\quad +\, p_{rrt \& fel}^2 \tau _{rrt \& fel} + p_{rrt \& atn}^2 \tau _{rrt \& atn}) \big / {\tau _{1,rrt}^{(j+1)}}, \\ \tau _{1,fel}^{(j+1)}&= T_d + \Sigma _{i=2}^n p_{i,fel}^{(j)} (p_{fel} \tau _{fel}+ p_{rrt \& fel} \tau _{rrt \& fel}), \\ p_{1,fel}^{(j+1)}&= (p_{fel}^2 \tau _{fel}+ p_{rrt \& fel}^2 \tau _{rrt \& fel})\big /\tau _{1,fel}^{(j+1)}, \\ \tau _{1,atn}^{(j+1)}&= T_d + \Sigma _{i=2}^n p_{i,atn}^{(j)} (p_{atn} \tau _{atn}+ p_{rrt \& atn} \tau _{rrt \& atn}), \\ p_{1,atn}^{(j+1)}&= (p_{atn}^2 \tau _{atn}+ p_{rrt \& atn}^2 \tau _{rrt \& atn})\big /\tau _{1,atn}^{(j+1)}. \end{aligned}$$
(4)

For patient \(k=2,\ldots ,m-1\),

$$\begin{aligned} \tau _{k,int}^{(j+1)}&= T_d + \left( \Sigma _{i=1}^{k-1} p_{i,int}^{(j+1)}+\Sigma _{i=k+1}^m p_{i,int}^{(j)}\right) \cdot (p_{int} \tau _{int}+ p_{rrt \& int} \tau _{rrt \& int}), \\ p_{k,int}^{(j+1)}&= (p_{int}^2 \tau _{int}+ p_{rrt \& int}^2 \tau _{rrt \& int})\big /\tau _{k,int}^{(j+1)}, \\ \tau _{k,res}^{(j+1)}&= T_d + \left( \Sigma _{i=1}^{k-1} p_{i,res}^{(j+1)}+\Sigma _{i=k+1}^m p_{i,res}^{(j)}\right) \cdot (p_{res} \tau _{res}+ p_{rrt \& res} \tau _{rrt \& res}), \\ p_{k,res}^{(j+1)}&= (p_{res}^2 \tau _{res}+ p_{rrt \& res}^2 \tau _{rrt \& res})\big /\tau _{k,res}^{(j+1)}, \\ \tau _{k,rrt}^{(j+1)}&= T_d + \left( \Sigma _{i=1}^{k-1} p_{i,rrt}^{(j+1)} +\Sigma _{i=k+1}^m p_{i,rrt}^{(j)}\right) \cdot (p_{rrt} \tau _{rrt} + p_{rrt \& int} \tau _{rrt \& int} \\&\quad +\, p_{rrt \& res} \tau _{rrt \& res}+ p_{rrt \& fel} \tau _{rrt \& fel} + p_{rrt \& atn}\tau _{rrt \& atn}), \\ p_{k,rrt}^{(j+1)}&= (p_{rrt}^2 \tau _{rrt} + p_{rrt \& int}^2 \tau _{rrt \& int}^2 + p_{rrt \& res}^2 \tau _{rrt \& res} + p_{rrt \& fel}^2 \tau _{rrt \& fel} \\&\quad +\, p_{rrt \& atn}^2 \tau _{rrt \& atn}) / \tau _{k,rrt}^{(j+1)}, \\ \tau _{k,fel}^{(j+1)}&= T_d + (\Sigma _{i=1}^{k-1} p_{i,fel}^{(j+1)}+\Sigma _{i=k+1}^m p_{i,fel}^{(j)}) \cdot (p_{fel} \tau _{fel}+ p_{rrt \& fel} \tau _{rrt \& fel}), \\ p_{k,fel}^{(j+1)}&= (p_{fel}^2 \tau _{fel}+ p_{rrt \& fel}^2 \tau _{rrt \& fel})\big /\tau _{k,fel}^{(j+1)}, \\ \tau _{k,atn}^{(j+1)}&= T_d + \left( \Sigma _{i=1}^{k-1} p_{i,atn}^{(j+1)}+\Sigma _{i=k+1}^m p_{i,atn}^{(j)}\right) \cdot (p_{atn} \tau _{atn}+ p_{rrt \& atn} \tau _{rrt \& atn}), \\ p_{k,atn}^{(j+1)}&= (p_{atn}^2 \tau _{atn}+ p_{rrt \& atn}^2 \tau _{rrt \& atn})\big /\tau _{k,atn}^{(j+1)}. \end{aligned}$$
(5)

For patient m,

$$\begin{aligned} \tau _{m,int}^{(j+1)}&= T_d + \Sigma _{i=1}^{m-1} p_{i,int}^{(j+1)} (p_{int} \tau _{int} + p_{rrt \& int} \tau _{rrt \& int}), \\ p_{m,int}^{(j+1)}&= \frac{p_{int}^2 \tau _{int}+ p_{rrt \& int}^2 \tau _{rrt \& int}}{\tau _{k,r}^{(j+1)}}, \\ \tau _{m,res}^{(j+1)}&=T_d + \Sigma _{i=1}^{m-1} p_{i,res}^{(j+1)} (p_{res} \tau _{res} + p_{rrt \& res} \tau _{rrt \& res}), \\ p_{m,res}^{(j+1)}&= (p_{res}^2 \tau _{res}+ p_{rrt \& res}^2 \tau _{rrt \& res})\big /\tau _{k,res}^{(j+1)}, \\ \tau _{n,rrt}^{(j+1)}&=T_d + \Sigma _{i=1}^{m-1} p_{i,rrt}^{(j+1)} (p_{rrt} \tau _{rrt} + p_{rrt \& int} \tau _{rrt \& int} + p_{rrt \& res} \tau _{rrt \& res} \\&\quad +\, p_{rrt \& fel} \tau _{rrt \& fel} + p_{rrt \& atn} \tau _{rrt \& atn}), \\ p_{m,r}^{(j+1)}&= (p_{rrt}^2 \tau _{rrt} + p_{rrt \& int}^2 \tau _{rrt \& int}^2 + p_{rrt \& res}^2 \tau _{rrt \& res} + p_{rrt \& fel}^2 \tau _{rrt \& fel} \\&\quad +\, p_{rrt \& atn}^2 \tau _{rrt \& atn}) \big / \tau _{m,rrt}^{(j+1)}, \\ \tau _{m,fel}^{(j+1)}&= T_d + \Sigma _{i=1}^{m-1} p_{i,fel}^{(j+1)} (p_{fel} \tau _{fel}+ p_{rrt \& fel} \tau _{rrt \& fel}), \\ p_{m,fel}^{(j+1)}&= (p_{fel}^2 \tau _{f}+ p_{rrt \& fel}^2 \tau _{rrt \& fel})\big /\tau _{m,fel}^{(j+1)}, \\ \tau _{m,atn}^{(j+1)}&= T_d + \Sigma _{i=1}^{m-1} p_{i,atn}^{(j+1)} (p_{atn} \tau _{atn}+ p_{rrt \& atn} \tau _{rrt \& atn}), \\ p_{m,atn}^{(j+1)}&= (p_{atn}^2 \tau _{atn}+ p_{rrt \& atn}^2 \tau _{rrt \& atn})\big /\tau _{m,atn}^{(j+1)}. \end{aligned}$$
(6)

Step 1.3 Iteration: Set \(j=j+1\). If the terminating criteria is not met, go back to Step 1.2. Let \(\delta = 10^{-5}\), the Level-1 iteration is finished if

$$\begin{aligned} | \tau _{i,r}^{(j+1)} - \tau _{i,r}^{(j)} | \le \delta ,\qquad | p_{i,r}^{(j+1)} - p_{i,r}^{(j)} | \le \delta ,\qquad i=1,2,\ldots ,m. \end{aligned}$$

Step 1.4 Termination: If the stopping conditions are met, set

$$\begin{aligned} \tau _{i,r}^{(j+1)}&={\mathrm {T}}_{r}, \qquad p_{i,r}^{(j+1)}=P_{r}, \qquad i=1,\ldots ,m, \\ T_{in}&= T_d + \Sigma _{r, r \in X} P_{r} {\mathrm {T}}_{r}. \end{aligned}$$
(7)

(2) Level-2 iteration

Step 2.1 Initialization: Set \(l=0\) and

$$\begin{aligned} \rho _1^{(l)}=0,\qquad \lambda _1^{(l)} = T_{in}. \end{aligned}$$

Step 2.2 Update \(\rho _k^{(l)}\) and \(\lambda _k^{(l)}\): For patient 1,

$$\begin{aligned} \lambda _1^{(l+1)}&= T_{in}(1+\rho _1^{(l)} \Sigma _{i=2}^m \rho _i^{(l)}), \\ \rho _1^{(l+1)}&= \frac{\lambda _1^{(l+1)}}{\lambda _1^{(l+1)}+T_{normal}}. \end{aligned}$$
(8)

For patient \(k=2,\ldots ,m-1\),

$$\begin{aligned} \lambda _k^{(l+1)}&= T_{in}(1+\rho _k^{(l)} (\Sigma _{i=1}^{k-1} \rho _i^{(l+1)}+\Sigma _{i=k+1}^m \rho _{i}^{(l)})), \\ \rho _k^{(l+1)}&= \frac{\lambda _k^{(l+1)}}{\lambda _k^{(l+1)}+T_{normal}}. \end{aligned}$$
(9)

For patient m,

$$\begin{aligned} \lambda _m^{(l+1)}&= T_{in}(1+\rho _k^{(l)} \Sigma _{i=1}^{m-1} \rho _i^{(l+1)}), \\ \rho _m^{(l+1)}&= \frac{\lambda _k^{(l+1)}}{\lambda _m^{(l+1)}+T_{normal}}. \end{aligned}$$
(10)

Step 2.3 Iteration: Set \(l=l+1\). If the terminating criteria is not met, go back to Step 2.2.

$$\begin{aligned} | \lambda _{i}^{(l+1)} - \lambda _{i}^{(l)} | \le \delta , \qquad | \rho _{i}^{(l+1)} - \rho _{i}^{(l)} | \le \delta , \quad i=1,\ldots ,m. \end{aligned}$$

Step 2.4 Termination: If the terminating condition is met, set

$$\begin{aligned} \lambda _{i}^{(l+1)}=\lambda _{i}, \quad \rho _{i}^{(l+1)}=\rho _{i}, \quad \lambda _{i}=T_{final},\quad i=1,\ldots ,m. \end{aligned}$$

Appendix 2: Proofs

To prove Proposition 1, Lemmas 1 and 2 are needed.

Lemma 1

Under assumptions (1)–(5), when\(m=2\), if\(p_{2,r}^{(j)} > p_{2,r}^{(j-1)}\), \(r \in X\), \(j=1,2,\ldots\), then\(\tau _{1,r}^{(j+1)} > \tau _{1,r}^{(j)}\), \(p_{1,r}^{(j+1)} < p_{1,r}^{(j)}\), \(\tau _{2,r}^{(j+1)} < \tau _{2,r}^{(j)}\), \(p_{2,r}^{(j+1)} > p_{2,r}^{(j)}\).

Lemma 2

Under assumptions (1)–(5), when\(m=2\), the sequences\(p_{1,r}^{(j)}\) and\(\tau _{2,r}^{(j)}\) are monotonically decreasing, while the sequences\(p_{2,r}^{(j)}\) and\(\tau _{1,r}^{(j)}\) are monotonically increasing.

Proof of Lemma 1

From all the equations related to the update of \(\tau _{i,r}^{(j)}\) and \(p_{i,r}^{(j)}\), which are from (4) to (6), define \(C_{1,r}\) and \(C_{2,r}\) as constants related to resource r, \(r \in X\). We have

$$\begin{aligned} C_{1,r} &= {} \left\{ \begin{array}{ll} p_{int} \tau _{int}+ p_{rrt \& int} \tau _{rrt \& int}, &{}\text{ if } r=int, \\ \\ p_{res} \tau _{res}+ p_{rrt \& res} \tau _{rrt \& res} &{}\text{ if } r=res, \\ \\ p_{rrt} \tau _{rrt} + p_{rrt \& int} \tau _{rrt \& int} &{}\\ + p_{rrt \& res} \tau _{rrt \& res}\\ + p_{rrt \& fel} \tau _{rrt \& fel}+ p_{rrt \& atn} \tau _{rrt \& atn} &{}\text{ if } r=rrt, \\ \\ p_{fel} \tau _{fel}+ p_{rrt \& fel} \tau _{rrt \& fel} &{}\text{ if } r=fel, \\ \\ p_{atn} \tau _{atn}+ p_{rrt \& atn} \tau _{rrt \& atn} &{}\text{ if } r=atn. \end{array}\right. \\ C_{2,r} &= {} \left\{ \begin{array}{ll} p_{int}^2 \tau _{int}+ p_{rrt \& int}^2 \tau _{rrt \& int}, &{}\text{ if } r=int, \\ \\ p_{res}^2 \tau _{res}+ p_{rrt \& res}^2 \tau _{rrt \& res} &{}\text{ if } r=res, \\ \\ p_{rrt}^2 \tau _{rrt} + p_{rrt \& int}^2 \tau _{rrt \& int}^2 \\ + p_{rrt \& res}^2 \tau _{rrt \& res}\\ + p_{rrt \& fel}^2 \tau _{rrt \& fel}+ p_{rrt \& atn}^2 \tau _{rrt \& atn} &{}\text{ if } r=rrt, \\ \\ p_{fel}^2 \tau _{fel}+ p_{rrt \& fel}^2 \tau _{rrt \& fel} &{}\text{ if } r=fel, \\ \\ p_{atn}^2 \tau _{atn}+ p_{rrt \& atn}^2 \tau _{rrt \& atn} &{}\text{ if } r=atn. \end{array}\right. \end{aligned}$$

For iteration j, if \(p_{2,r}^{(j)} > p_{2,r}^{(j-1)}\), then for patient 1:

$$\begin{aligned} \tau _{1,r}^{(j)} &= {} T_d + p_{2,r}^{(j-1)} C_{1,r} < T_d + p_{2,r}^{(j)} C_{1,r} = \tau _{1,r}^{(j+1)}, \end{aligned}$$
(11)
$$\begin{aligned} p_{1,r}^{(j)} &= {} \frac{C_{1,r}}{\tau _{1,r}^{(j)}} > \frac{C_{1,r}}{\tau _{1,r}^{(j+1)}} = p_{1,r}^{(j+1)}. \end{aligned}$$
(12)

This leads to, for patient 2,

$$\begin{aligned} \tau _{2,r}^{(j)} &= T_d + p_{1,r}^{(j)} C_{1,r} > T_d + p_{1,r}^{(j+1)} C_{1,r} = \tau _{2,r}^{(j+1)}, \end{aligned}$$
(13)
$$\begin{aligned} p_{2,r}^{(j)} &= {} \frac{C_{2,r}}{\tau _{2,r}^{(j)}} < \frac{C_{2,r}}{\tau _{2,r}^{(j+1)}} = p_{2,r}^{(j+1)}. \end{aligned}$$
(14)

The obtained results in the above four inequations complete the proof. \(\square\)

Proof of Lemma 2

Induction is used for the proof of the lemma.

Initial Step: When \(j=1\), since \(p_{2,r}^{(0)}=0\), from Eq. (14), we have

$$\begin{aligned} p_{2,r}^{(1)} > p_{2,r}^{(0)}=0. \end{aligned}$$

Then, from Lemma 1, we obtain

$$\begin{aligned} \tau _{1,r}^{(2)}> \tau _{1,r}^{(1)},\qquad p_{1,r}^{(2)}< p_{1,r}^{(1)},\qquad \tau _{2,r}^{2)} < \tau _{2,r}^{(1)},\qquad p_{2,r}^{(2)} > p_{2,r}^{(1)}. \end{aligned}$$

The base case is proved.

Inductive Step: Assume when \(j=k\), we have

$$\begin{aligned} \tau _{1,r}^{(k+1)}> \tau _{1,r}^{(k)},\qquad p_{1,r}^{(k+1)}< p_{1,r}^{(k)},\qquad \tau _{2,r}^{(k+1)} < \tau _{2,r}^{(k)},\qquad p_{2,r}^{(k+1)} > p_{2,r}^{(k)}. \end{aligned}$$

From Lemma 1, this leads to

$$\begin{aligned} \tau _{1,r}^{(k+2)}> \tau _{1,r}^{(k+1)},\qquad p_{1,r}^{(k+2)}< p_{1,r}^{(k+1)},\qquad \tau _{2,r}^{(k+2)} < \tau _{2,r}^{(k+1)},\qquad p_{2,r}^{(k+2)} > p_{2,r}^{(k+1)}. \end{aligned}$$

Thus, the case of \(j=k+1\) also holds.

By induction, we obtain that, when \(m=2\), the sequences \(p_{1,r}^{(j)}\) and \(\tau _{2,r}^{(j)}\) are monotonically decreasing, while the sequences \(p_{2,int}^{(j)}\) and \(\tau _{1,int}^{(j)}\) are monotonically increasing, \(r\in X\), \(j=1,2,\ldots\). \(\square\)

Proof of Proposition 1

From Lemma 2, we obtain the monotonicity of decreasing sequences \(p_{1,r}^{(j)}\) and \(\tau _{2,r}^{(j)}\) and increasing sequences \(p_{2,int}^{(j)}\) and \(\tau _{1,int}^{(j)}\), \(r\in X\), \(j=1,2,\ldots\). Next we show that the sequences \(\tau _{i,r}^{(j)}\) and \(p_{i,r}^{(j)}\), are bounded from above and below. For \(p_{i,r}^{(j)}\)s, from Eqs. (12) and (14), we have

$$\begin{aligned} 0< p_{i,r}^{(j)} < 1. \end{aligned}$$

For \(\tau _{i,r}^{(j)}\)s, from Eqs. (11) and (13), since \(0<p_{i,r}^{(j)}<1\), we obtain

$$\begin{aligned} T_d< \tau _{i,r}^{(j)} < T_d + C_{i,r}. \end{aligned}$$

Since the sequences \(\tau _{i,r}^{(j)}\) and \(p_{i,r}^{(j)}\), \(r \in X\); \(j=1,2,\ldots\), are monotonic and bounded from above and below, they are convergent. Thus, Level-1 iteration is convergent. \(\square\)

To prove Proposition 2, Lemma 3 is needed.

Lemma 3

Under assumptions (1)–(5), if\(\rho _{i}^{(l)} > \rho _{i}^{(l-1)}\), \(i=1,\ldots ,m\), \(l=1,2,\ldots\), then\(\rho _{i}^{(l+1)} > \rho _{i}^{(l)}\).

Proof of Lemma 3

From Eq. (8), we obtain

$$\begin{aligned} \lambda _1^{(l+1)} = T_{in}(1+\rho _1^{(l)} \Sigma _{i=2}^m \rho _i^{(l)}) > T_{in}(1+\rho _1^{(l-1)} \Sigma _{i=2}^m \rho _i^{(l-1)}) = \lambda _1^{(l)}. \end{aligned}$$

This implies that

$$\begin{aligned} \rho _1^{(l+1)} &= {} \frac{\lambda _1^{(l+1)}}{\lambda _1^{(l+1)}+T_{normal}}=\frac{1}{1+\frac{T_{normal}}{\lambda _1^{(l+1)}}}>\frac{1}{1+\frac{T_{normal}}{\lambda _1^{(l)}}}=\rho _1^{(l)}. \end{aligned}$$

When \(2 \le k \le m-1\), from (9), we have

$$\begin{aligned} \lambda _k^{(l+1)} &= T_{in}(1+\rho _k^{(l)} \left( \Sigma _{i=1}^{k-1} \rho _i^{(l+1)}+\Sigma _{i=k+1}^{m} \rho _i^{(l)}\right) \\ &> T_{in}(1+\rho _1^{(l-1)} \left( \Sigma _{i=1}^{k-1} \rho _i^{(l)}+\Sigma _{i=k+1}^{m} \rho _i^{(l-1)}\right) \\ &= {} \lambda _k^{(l)},\\ \rho _k^{(l+1)}&= \frac{\lambda _k^{(l+1)}}{\lambda _k^{(l+1)}+ T_{normal}}>\frac{1}{1+\frac{T_{normal}}{\lambda _k^{(l)}}}=\rho _k^{(l)}. \end{aligned}$$

Finally, for \(k=m\), from (10), it follows that

$$\begin{aligned} \lambda _m^{(l+1)}&= T_{in}\left( 1+\rho _m^{(l)} \Sigma _{i=1}^{m-1} \rho _i^{(l+1)} \right)> T_{in}\left( 1+\rho _m^{(l-1)} \Sigma _{i=1}^{m-1} \rho _i^{(l)}\right) =\lambda _m^{(l)},\\ \rho _m^{(l+1)} &= \frac{\lambda _m^{(l+1)}}{\lambda _m^{(l+1)}+T_{normal}}> \frac{1}{1+\frac{T_{normal}}{\lambda _m^{(l-1)}}}=\rho _m^{(l)}. \end{aligned}$$

The arguments follow directly. \(\square\)

Proof of Proposition 2

First we prove that the sequences \(\lambda _{i}^{(l)}\) and \(\rho _{i}^{(l)}\), \(i=1,2,\ldots ,m\); \(l=1,2,\ldots\), are monotonically increasing using mathematical induction.

Initial Step: When \(l=1\), since \(\rho _{i}^{(0)}=0\), from Lemma 3,

$$\begin{aligned} \rho _{i}^{(1)}>\rho _{i}^{(0)}=0. \end{aligned}$$

This leads to

$$\begin{aligned} \rho _{i}^{(2)}> \rho _{i}^{(1)}, \qquad \lambda _{i}^{(2)} > \lambda _{i}^{(1)}. \end{aligned}$$

The base case is proved.

Inductive Step: Assume when \(l=k\), we have

$$\begin{aligned} \lambda _{i}^{(k)}> \lambda _{i}^{(k-1)},\qquad \rho _{i}^{(k)}>\rho _{i}^{(k-1)}, \qquad i=1,2,\ldots ,m. \end{aligned}$$

Then from Lemma 3, we have

$$\begin{aligned} \lambda _{i}^{(k+1)}> \lambda _{i}^{(k)}, \qquad \rho _{i}^{(k+1)}>\rho _{i}^{(k)}. \end{aligned}$$

Therefore, the case where \(l=k+1\) also holds. Then, the sequences \(\lambda _{i}^{(l)}\) and \(\rho _{i}^{(l)}\), \(i=1,2,\ldots ,m\); \(l=1,2,\ldots\), are monotonically increasing.

For boundedness, it is clear that \(\rho _{i}^{(l)}\)s are bounded between 0 and 1 from Eqs. (8), (9), and (10), while \(\lambda _{i}^{(l)}\)s are also bounded according to equations (8) and (9).

Since the sequences \(\lambda _{i}^{(l)}\) and \(\rho _{i}^{(l)}\), \(i=1,2,\ldots ,m\), are both monotonic and bounded from above and below, they are convergent. \(\square\)

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Zeng, Z., Fan, Z., Xie, X. et al. A two-level iteration approach for modeling and analysis of rapid response process with multiple deteriorating patients. Flex Serv Manuf J 32, 35–71 (2020). https://doi.org/10.1007/s10696-019-09347-6

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Keywords

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