Optimal Control Of An Emergency Room Triage And Treatment .

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Optimal control of an emergency roomtriage and treatment processGabriel Zayas-Cabán1Mark E. Lewis1Jungui Xie2Linda V. Green31 CornellUniversityIthaca, NY2 Universityof Science and Technology of ChinaBeijing, China3 ColumbiaUniversityNew York, NYTriage and Treatment()1 / 41

O UTLINEOptimal Control of Triage and TreatmentBackgroundModeling ApproachNumerical StudyConcluding RemarksOngoing and Future WorkTriage and Treatment()2 / 41

Optimal Control of Triage and TreatmentBackgroundOptimal Control of Triage and TreatmentBackgroundModeling ApproachNumerical StudyConcluding RemarksOngoing and Future WorkTriage and Treatment()3 / 41

Optimal Control of Triage and TreatmentTriage and Treatment()Background4 / 41

Optimal Control of Triage and TreatmentTriage and Treatment()Background4 / 41

Optimal Control of Triage and TreatmentTriage and Treatment()Background4 / 41

Optimal Control of Triage and TreatmentBackgroundN EW C ARE M ODELS IN THE EDEmergency Department (ED):IIn 2010, number of visits in the U.S. around 129.8 million and increasing2–3% per year.INumber of ED beds decreasing.IOvercrowded departments, long waiting times, overworked staff, patientdissatisfaction, and abandonments (LWOT).[NAMCS]Triage and Treatment()5 / 41

Optimal Control of Triage and TreatmentBackgroundN EW C ARE M ODELS IN THE EDIMany ED patients present with low-acuity conditions and do not requirehospitalization.ILow-acuity ED patients have to be treated, diverting resources frommore critical patients.IEDs developing new models of care to handle these lower-acuitypatients to facilitate patient flow. [Helm et al. 2011, Saghafian et al. 2012, Saghafian et al. 2014 ]Triage and Treatment()6 / 41

Optimal Control of Triage and TreatmentBackgroundT HE L UTHERAN M EDICAL C ENTEROVERVIEWImage available at http://www.lutheranmedicalcenter.com; downloaded June 2013.Lutheran Medical Center (LMC) Triage-Treat-and-Release (TTR) program:I Developed in 2010.I Multiple providers (physicians or physician assistants) who handle both phasesof service.Triage and Treatment()7 / 41

Optimal Control of Triage and TreatmentBackgroundT HE L UTHERAN M EDICAL C ENTERTTRPROGRAM1. Patients arrive to ED and are registered.2. Patients proceed to triage (phase-one service) on afirst-come-first-served (FCFS) basis.3. After triage, high severity patients are assigned to another part of the EDfor testing and/or treatment.4. Low severity and low complexity patients await treatment (phase-twoservice) in triage area.Triage and Treatment()8 / 41

Optimal Control of Triage and TreatmentBackgroundT HE L UTHERAN M EDICAL C ENTERTTRIPROGRAMMay help reduce long waiting times in the ER.IEarlier patient contact with a physician and, hence, earlier decision-making.IPhysicians and physician assistants are more reliable in assessingpatients during triage.[Soremkun et al. 2012, Burströ et al. 2012]IDecoupled (“Fast-track system”) vs. coupled (TTR Program) triage andtreatment.IOther examples: Health clinics, other ER operations.Triage and Treatment()9 / 41

Optimal Control of Triage and TreatmentBackgroundInterested inItwo-phase stochastic service systems,Ihaving single medical service provider, andIwhere patients may renege or abandon before completing service.Triage and Treatment()10 / 41

Optimal Control of Triage and TreatmentBackgroundInterested inItwo-phase stochastic service systems,Ihaving single medical service provider, andIwhere patients may renege or abandon before completing service.Broad issueHow should we prioritize the work by medical service providers to balanceinitial delay for care with the need to discharge patients in a timely fashion.Triage and Treatment()10 / 41

Optimal Control of Triage and TreatmentModeling ApproachA P RIMER ON Q UEUEING S YSTEMTriage and Treatment()11 / 41

Optimal Control of Triage and TreatmentModeling ApproachA P RIMER ON Q UEUEING S YSTEMQueueing system,IOne or more servers (physicians, physician assistants) providing serviceto arriving customers (patients).IIf all servers busy, customer (patient) join one or more queues (or lines)in front of servers, hence the name.IThree components: arrival process, service mechanism, and queuediscipline.Triage and Treatment()12 / 41

Optimal Control of Triage and TreatmentModeling ApproachA P RIMER ON Q UEUEING S YSTEMQueueing System,IArrival process: how customers arrive to the system.IIAi — interarrival time between customer i 1 and i.1λ E(A: the arrival rate.i)Triage and Treatment()13 / 41

Optimal Control of Triage and TreatmentModeling ApproachA P RIMER ON Q UEUEING S YSTEMQueueing System,IArrival process: how customers arrive to the system.IIIAi — interarrival time between customer i 1 and i.1λ E(A: the arrival rate.i)Service mechanism: how many servers, how are they organized.IISi — service time of the ith arriving customer.µ E(S1 i ) : the service rate.Triage and Treatment()13 / 41

Optimal Control of Triage and TreatmentModeling ApproachA P RIMER ON Q UEUEING S YSTEMQueueing System,IArrival process: how customers arrive to the system.IIIService mechanism: how many servers, how are they organized.IIIAi — interarrival time between customer i 1 and i.1λ E(A: the arrival rate.i)Si — service time of the ith arriving customer.µ E(S1 i ) : the service rate.Queue discipline: rule used to choose next customer from queue whenserver completes service of current customer (e.g. FCFS).Triage and Treatment()13 / 41

Optimal Control of Triage and TreatmentModeling ApproachTypically,IFix queueing system/model configuration.IUse model to help evaluate and predict performance of existing andproposed system (e.g. waiting times, queue length, utilization).Triage and Treatment()14 / 41

Optimal Control of Triage and TreatmentModeling ApproachTypically,IFix queueing system/model configuration.IUse model to help evaluate and predict performance of existing andproposed system (e.g. waiting times, queue length, utilization).ITheory and/or simulation experimentation.IGoal: Improve the design of a system.Triage and Treatment()14 / 41

Optimal Control of Triage and TreatmentModeling ApproachHowever,IThe parameters of the system (e.g. the arrival and service rates, queuedisciplines) can be varied dynamically over time.ICan significantly improve performance (e.g. reduced congestion, timespent waiting to be served).Triage and Treatment()15 / 41

Optimal Control of Triage and TreatmentModeling ApproachHowever,IThe parameters of the system (e.g. the arrival and service rates, queuedisciplines) can be varied dynamically over time.ICan significantly improve performance (e.g. reduced congestion, timespent waiting to be served).IMarkov decision processes.Triage and Treatment()[M. Puterman 2005]15 / 41

Optimal Control of Triage and TreatmentModeling ApproachM ARKOV D ECISION P ROCESS P RIMER[Bäurle and Rieder, Markov Decision Processes with Applications to Finance]Triage and Treatment()16 / 41

Optimal Control of Triage and TreatmentModeling ApproachOptimal Control of Triage and TreatmentBackgroundModeling ApproachNumerical StudyConcluding RemarksOngoing and Future WorkTriage and Treatment()17 / 41

Optimal Control of Triage and TreatmentModeling ApproachSingle-server tandem queue:?Triage and Treatment()18 / 41

Optimal Control of Triage and TreatmentModeling ApproachSingle-server two-phase stochastic service system model:I Rate λ Poisson arrival process.I FCFS phase-one service (triage).I After phase-one:IIIpatients leave the system (w/ probability 1 p), orpatients wait for FCFS phase-two service (w/ probability p).0 p 1.Triage and Treatment()19 / 41

Optimal Control of Triage and TreatmentModeling ApproachSingle-server two-phase stochastic service system model:I Patients wait for phase-two service (treatment) according to an exponentiallydistributed random variable with rate β before abandoning.I Services in both phases are exponential with rates µ1 and µ2 .I After phase-two service, patient leaves the system.Triage and Treatment()20 / 41

Optimal Control of Triage and TreatmentModeling ApproachDecision-making scenario:1. Decision-maker (medical service provider) views number of patients ateach station.Triage and Treatment()21 / 41

Optimal Control of Triage and TreatmentModeling ApproachDecision-making scenario:1. Decision-maker (medical service provider) views number of patients ateach station.2. Decides where to serve next, assuming preemptive service disciplinesand rewards R1 and R2 .Triage and Treatment()21 / 41

Optimal Control of Triage and TreatmentModeling ApproachDecision-making scenario:1. Decision-maker (medical service provider) views number of patients ateach station.2. Decides where to serve next, assuming preemptive service disciplinesand rewards R1 and R2 .Specific objectiveWant service disciplines that maximize total discounted expected reward orlong-run average reward of the system.Triage and Treatment()21 / 41

Optimal Control of Triage and TreatmentModeling ApproachState Space:X : {(i, j) i, j Z },where i (j) represents number of patients at station 1 (2).Decision epochs:T : {tn , n 1},sequence of times of events.Triage and Treatment()22 / 41

Optimal Control of Triage and TreatmentModeling ApproachState Space:X : {(i, j) i, j Z },where i (j) represents number of patients at station 1 (2).Decision epochs:T : {tn , n 1},sequence of times of events.Available actions in state x (i, j): {0, 1, 2} {0, 1}A(x) {0, 2} {0}if i, j 1,if i 1, j 0,if j 1, i 0,if i j 0,where 0, 1, and 2 denote idling, serving at station 1, and serving at station 2.Triage and Treatment()22 / 41

Optimal Control of Triage and TreatmentModeling ApproachReward: Ri received after completing phase i service, i 1, 2.Expected reward function:r((i, j), a) µ1 R1 λ µ1 jβµ2 R2 λ µ2 jβ 0Triage and Treatment()if i 0, a 1,if j 0, a 2,if a 0.23 / 41

Optimal Control of Triage and TreatmentModeling ApproachOptimal Control of Triage and TreatmentBackgroundModeling ApproachNumerical StudyConcluding RemarksOngoing and Future WorkTriage and Treatment()24 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 2 (P2)Triage and Treatment()25 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachP RIORITIZE S TATION 1 (P1)Triage and Treatment()26 / 41

Optimal Control of Triage and TreatmentModeling ApproachS OME RESULTSPropositionThere is an optimal policy which does not idle the server whenever there are patientswaiting.Triage and Treatment()27 / 41

Optimal Control of Triage and TreatmentModeling ApproachS OME RESULTSPropositionThere is an optimal policy which does not idle the server whenever there are patientswaiting.TheoremThe following hold:1. If µ2 R2 µ1 R1 implies it is optimal to prioritize station 2. 2. If λ µ11 µ2 1 β 1 and there is no discounting, then it is optimal to prioritizestation 2.Triage and Treatment()27 / 41

Optimal Control of Triage and TreatmentModeling ApproachS OME RESULTSPropositionThere is an optimal policy which does not idle the server whenever there are patientswaiting.TheoremThe following hold:1. If µ2 R2 µ1 R1 implies it is optimal to prioritize station 2. 2. If λ µ11 µ2 1 β 1 and there is no discounting, then it is optimal to prioritizestation 2.PropositionIf patients do not abandon, then µ1 R1 µ2 R2 implies that it is optimal to prioritizestation 1.Triage and Treatment()27 / 41

Optimal Control of Triage and TreatmentModeling ApproachF INAL REMARKSIDenote prioritizing station 1 by P1 and prioritizing station 2 by P2.IBenefits of P2:IIIEasy to implement.Follows patient throughout her/his service “cycle”.Drawbacks of P2:IIRestrictive condition.P2 spends highest proportion of time at station 2.Triage and Treatment()28 / 41

Optimal Control of Triage and TreatmentModeling ApproachN UMERICAL S TUDY: P RELUDET HRESHOLD P OLICIESIThreshold policy with level T: medical service provider works at station 2untilIIStation 2 is empty orNumber of patients at station 1 reaches T.Triage and Treatment()29 / 41

Optimal Control of Triage and TreatmentModeling ApproachN UMERICAL S TUDY: P RELUDET HRESHOLD P OLICIESIThreshold policy with level T: medical service provider works at station 2untilIIStation 2 is empty orNumber of patients at station 1 reaches T.IExhaustive Policy (E)IP2 (T ), P1 (T 1), spend, respectively, highest and leastproportion of effort at station 2.IBetween these two extremes are threshold policies with higherthresholds spending more time at station 2.Triage and Treatment()29 / 41

Optimal Control of Triage and TreatmentNumerical StudyOptimal Control of Triage and TreatmentBackgroundModeling ApproachNumerical StudyConcluding RemarksOngoing and Future WorkTriage and Treatment()30 / 41

Optimal Control of Triage and TreatmentParameter Symbolµ1µ2βpR1R2λNumerical StudyValue(s)8.574.620.15, 0.3, 0.5, 0.8110, 15200.5, 1.5, 3,4.5, 6.5, 8.5Table: List of Parameters and their valuesTriage and Treatment()31 / 41

Optimal Control of Triage and TreatmentParameter Symbolµ1µ2βpR1R2λNumerical StudyValue(s)8.574.620.15, 0.3, 0.5, 0.8110, 15200.5, 1.5, 3,4.5, 6.5, 8.5From LMC’s TTR.Table: List of Parameters and their valuesTriage and Treatment()31 / 41

Optimal Control of Triage and TreatmentParameter Symbolµ1µ2βpR1R2λNumerical StudyValue(s)8.574.620.15, 0.3, 0.5, 0.8110, 15200.5, 1.5, 3,4.5, 6.5, 8.5From LMC’s TTR.Mandelbaum and Zeltyn (2007);Batt and Terwiesch (2013).Table: List of Parameters and their valuesTriage and Treatment()31 / 41

Optimal Control of Triage and TreatmentParameter Symbolµ1µ2βpR1R2λNumerical StudyValue(s)8.574.620.15, 0.3, 0.5, 0.8110, 15200.5, 1.5, 3,4.5, 6.5, 8.5From LMC’s TTR.Mandelbaum and Zeltyn (2007);Batt and Terwiesch (2013).µ1 R1 µ2 R2 ; µ1 R1 µ2 R2 .Table: List of Parameters and their valuesTriage and Treatment()31 / 41

Optimal Control of Triage and TreatmentParameter Symbolµ1µ2βpR1R2λNumerical StudyValue(s)8.574.620.15, 0.3, 0.5, 0.8110, 15200.5, 1.5, 3,4.5, 6.5, 8.51µ11 µ 1 βFrom LMC’s TTR.Mandelbaum and Zeltyn (2007);Batt and Terwiesch (2013).µ1 R1 µ2 R2 ; µ1 R1 µ2 R2 . λ µ1 .2Table: List of Parameters and their valuesTriage and Treatment()31 / 41

Optimal Control of Triage and TreatmentNumerical StudyPercent of the baseline reward100%95%90%85%70%0.511.52Arrival rate λ2.53Percent of the baseline reward (β 0.8, R1 10)Triage and Treatment()32 / 41

Optimal Control of Triage and TreatmentNumerical StudyAverage reward13012011010090803.54.55.56.5Arrival rate λ7.58.5Average reward (β 0.8, R1 15)Triage and Treatment()33 / 41

Optimal Control of Triage and TreatmentNumerical StudyR EMARKSWhen P2 is stable:I Decreasing the threshold makes the average reward worse.IIn all instances, P1 (T 1) performed the worst.I If λ {0.5, 1}, all policies comparable to P2 – within 6% of the optimalI Similar observations hold for R1 15.Triage and Treatment()34 / 41

Optimal Control of Triage and TreatmentNumerical StudyR EMARKSWhen P2 is stable:I Decreasing the threshold makes the average reward worse.IIn all instances, P1 (T 1) performed the worst.I If λ {0.5, 1}, all policies comparable to P2 – within 6% of the optimalI Similar observations hold for R1 15.When P2 is not stable, and P1 is used:I Gains in average reward can be obtained if we are close to stability by usingthreshold policies but at the cost of larger queue lengths.Triage and Treatment()34 / 41

Optimal Control of Triage and TreatmentConcluding RemarksOptimal Control of Triage and TreatmentBackgroundModeling ApproachNumerical StudyConcluding RemarksOngoing and Future WorkTriage and Treatment()35 / 41

Optimal Control of Triage and TreatmentConcluding RemarksR ECOMMENDATIONS FOR A TTR SYSTEMThreshold policies with parameter T - reasonable alternatives to P1 (T 1) and P2(T )I P2 is stableIIIf system is lightly loaded, no significant loss of optimality.If system is highly loaded, there is significant loss of optimality.Triage and Treatment()36 / 41

Optimal Control of Triage and TreatmentConcluding RemarksR ECOMMENDATIONS FOR A TTR SYSTEMThreshold policies with parameter T - reasonable alternatives to P1 (T 1) and P2(T )I P2 is stableIIIf system is lightly loaded, no significant loss of optimality.If system is highly loaded, there is significant loss of optimality.I P2 is unstable – impracticalIAverage reward of alternative policies are not too different – a providermight consider policies with the lowest average total number in the system,say.Triage and Treatment()36 / 41

Ongoing and Future WorkA DDITIONAL C HALLENGES FROM THE ERI Arrival processes are non-stationary (time-dependent) and often periodicIReplace homogeneous Poisson process with a non-homogeneous Poissonprocess or Markov modulated processI Patients/customers are impatientIModels should include abandonments at both stagesI Health can be deterioratingIService times are usually not exponential.Triage and Treatment()37 / 41

Ongoing and Future WorkA DDITIONAL C HALLENGES FROM THE ERI Arrival processes are non-stationary (time-dependent) and often periodicIReplace homogeneous Poisson process with a non-homogeneous Poissonprocess or Markov modulated processI Patients/customers are impatientIModels should include abandonments at both stagesI Health can be deterioratingIService times

Optimal control of an emergency room triage and treatment process Gabriel Zayas-Cabán 1Mark E. Lewis Jungui Xie2 Linda V. Green3 1Cornell University Ithaca, NY 2University of Science and Technology of China Beijing, China 3Columbia University New York, NY Triage and Treatment 1 / 41

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