Volume 13 • Number 4 • 2010 VA L U E I N H E A LT H
Tutorial in Medical Decision Modeling Incorporating Waiting Lines and Queues Using Discrete Event Simulation vhe_707
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Beate Jahn, PhD,1,2 Engelbert Theurl, PhD,3 Uwe Siebert, MPH, MSc,1,4 Karl-Peter Pfeiffer, PhD2,5 1
Institute of Public Health, Medical Decision Making and Health Technology Assessment, Department of Public Health, Information Systems and Health Technology Assessment, UMIT—University for Health Sciences, Medical Informatics and Technology, Hall i.T., Austria; 2Department of Medical Statistics, Informatics and Health Economics, Innsbruck Medical University, Innsbruck, Austria; 3Department of Public Finance, Leopold-Franzens-University of Innsbruck, Innsbruck, Austria; 4Cardiovascular Research Program, Institute for Technology Assessment and Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Boston, MA, USA; Center for Health Decision Science, Department of Health Policy and Management, Harvard School of Public Health, Boston, MA, USA; 5FH Johanneum University of Applied Sciences, Graz, Austria
A B S T R AC T In most decision-analytic models in health care, it is assumed that there is treatment without delay and availability of all required resources. Therefore, waiting times caused by limited resources and their impact on treatment effects and costs often remain unconsidered. Queuing theory enables mathematical analysis and the derivation of several performance measures of queuing systems. Nevertheless, an analytical approach with closed formulas is not always possible. Therefore, simulation techniques are used to evaluate systems that include queuing or waiting, for example, discrete event simulation. To include queuing in decisionanalytic models requires a basic knowledge of queuing theory and of the
underlying interrelationships. This tutorial introduces queuing theory. Analysts and decision-makers get an understanding of queue characteristics, modeling features, and its strength. Conceptual issues are covered, but the emphasis is on practical issues like modeling the arrival of patients. The treatment of coronary artery disease with percutaneous coronary intervention including stent placement serves as an illustrative queuing example. Discrete event simulation is applied to explicitly model resource capacities, to incorporate waiting lines and queues in the decision-analytic modeling example.. Keywords: discrete event simulation, modeling, queue, waiting time.
Introduction
in the model. DES allows modeling of waiting lists or queues and resource use, waiting time and costs are provided [4–8]. Appropriate queue modeling requires knowledge of characteristics and mechanisms of queuing systems. To support the conceptualization of economic evaluation that includes limited capacities and waiting lines, this tutorial article provides an introduction to queuing. The focus is on modeling aspects and performance measures using DES. First, the queue modeling example (treatment of coronary artery disease (CAD)) is introduced. Thereafter, an introduction to queuing theory and attributes of queues are given. Data sources, modeling of arrival, and performance measures are discussed. Finally, methods and results of the CAD modeling example are presented.
Waiting times in health care have become a major political topic. Waiting times can lead to poor health status and a reduced treatment success. Strategies to reduce waiting times for surgery have been implemented successfully, for example, in Australia, in New Zealand, and in England [1]. Nevertheless, before a strategy is implemented, the impact should be analyzed. Queuing theory could support this analysis [2]. Providers of pharmaceuticals and medical devices are forced to demonstrate health benefits, safety, and effectiveness of new developments. In economic evaluations that compare treatment alternatives (e.g., cost-effectiveness analysis), waiting or queuing is often not considered in detail. Therefore, results of such evaluations can be biased and might not provide proper information for health-care providers. In health care, computer simulation supports analyzing treatment scenarios and patient flows. Discrete event simulation (DES) offers a wide range of useful characteristics. Disease progression or remission can be captured, and long-term illnesses or complex chronic diseases can be modeled. Patients can be modeled as individuals with specific characteristics (e.g., age, sex, severity of disease) [3]. These characteristics influence the treatment process and the flow of the patient through the model. A main advantage of DES compared to Markov state-transition models is that resource constraints can be naturally incorporated Address correspondence to: Beate Jahn, Institute of Public Health, Medical Decision Making and Health, Technology Assessment, Department of Public Health, Information Systems and Health Technology Assessment, UMIT—University for Health Sciences, Medical Informatics and Technology, Eduard Wallnoefer Center I, A-6060 HALL i.T., Austria. E-mail:
[email protected] 10.1111/j.1524-4733.2010.00707.x
Example of Queue Model for Treatment of CAD “CAD is a condition caused by a narrowing or occlusion of coronary arteries” [9]. Main treatment interventions are percutaneous coronary intervention (PCI) with stent placement and coronary artery bypass graft (CABG). PCI involves the inflation of a balloon catheter to widen the narrowed artery. To decrease the risk of restenosis (reocclusion), a mesh tube (stent) is used. With bare-metal stents (B), the risk of restenosis may still be more than 30% [10]. Newly developed drug-eluting stents (D) have shown promising results leading to a decreasing number of repeated interventions. In a real-world health-care setting, the existing facilities, staff, etc., determine the possible number of percutaneous interventions. Interventions are delayed if the demand exceeds the treatment capacity. Subsequently, waiting lists are built up. While patients are waiting, they receive additional treatment. Waiting
© 2010, International Society for Pharmacoeconomics and Outcomes Research (ISPOR)
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502 patients are assumed to suffer from angina symptoms and distressing uncertainty. An increasing number of restenosis leads to an increased number of repeated interventions. Consequently, if the new device reduces the number of repeated interventions, it could show a positive effect on waiting lists, costs, and health outcomes (e.g., quality of life). In the model, new patients (A1) that are referred to PCI and patients requiring a repeated intervention (A2) are scheduled for stent implantation [11]. Treatment is provided on the day of arrival, if treatment capacity is available. Otherwise, patients may wait and move to a queue (or waiting list). If treatment is not successful, patients are either sent to be scheduled for stenting again (second-line treatment, A2) or to the surgery department for CABG (third-line treatment). CABG is assumed to be the final intervention.
Queue Modeling and Queuing Theory Queuing models are widely used in service facilities, production, and material-handling systems, and in general situations where congestion or competition of scarce resources can occur [12]. Applications in health care ranges from appointment systems to pharmacokinetics [13].
Attributes of a Queuing System Generally, a queuing system is described by defining its population, the nature of arrival, the service time and mechanism, queuing behavior, and the queuing discipline. The population in the stent example (Fig. 1) comprises all people that are referred to PCI. Therefore, patients for first-line treatment (stent placement) and patients for repeated intervention are considered.
Patients arrive at random for first-line stent treatment depending on the distribution of the risk factors (e.g., diabetes) in the population. The arrival of patients for second-line treatment depends on first-line treatment. In other settings, arrival might be deterministic (e.g., scheduled patients at a dentist). Queue behavior refers to the actions of patients while they are queuing (or on a waiting list). They might leave the queue because of a sudden adverse event (myocardial infarction). Long waiting lists might encourage patients to search for treatment alternatives. Therefore, they do not enter the queue. In the stent example, patients enter the queue and wait until treatment capacities are available and stent placement is done. Queuing discipline determines the order of treated patients (prioritization). The stent placement is assumed to be a “firstcome, first-served” order synonymous to a waiting list that is processed from the top to the bottom. Newly arriving patients are included at the bottom of the list. This simplification does not always reflect health-care practice pattern. Therefore, queuing discipline could in addition reflect urgent needs (e.g., unstable patients waiting) or a severity score could determine the order of patients. The server refers to any resource or set of resources which are necessary for the treatment (e.g., staff, devices, medical units). Service time may be constant or random, depending on patient characteristic or learning curves. The service mechanism describes how the service is provided. It is assumed that stent treatment is provided to a predefined daily number of patients at the cardiology unit. Kendall’s notation is a standard to describe and classify queuing models by arrival process/service process/number of servers (see Appendix).
Patient Population and the Arrival Process repeated interventions (A2)
new CAD patients referred to PCI (A1)
Cardio labs busy ?
NO
YES
Waiting List (Queue)
Failure
(reocclusion)
NO
YES
YES
treatment
CABG NO
End of treatment
Cardio Lab PCI + Stent
Figure 1 Example: patient flow of coronary artery disease (CAD) treatment including waiting line. CABG, coronary artery bypass graft; PCI, percutaneous coronary intervention.
Data source. How can the population, that is, patients for stent treatment, be described? Where can we get information about the size and arriving patterns of the patients in the model? The patient population can be described from different perspectives: need, demand, and provision of treatment. Need is defined by a body of professionals. Demand describes what prospective patients want, depending on personal (income) and institutional (price, access) characteristics. Provision describes the way resources are provided or organized [14]. These perspectives are not independent. They provide a classification of sources from which estimates for model inputs can be derived. Need can be derived from epidemiology-based statistics (incidence and prevalence). Nevertheless, incidence and prevalence are not always directly related to the number of patients who need and seek treatment. The ability to detect diseases, for example, influences these parameters. The evaluation of demand is relatively complex because of the strong influence of the provision of health services and the need. Waiting lists are a way of rationing that may discourage demand. High demand might lead to increased provision. Databases of hospital appointments or accounting systems of health insurance agencies provide retrospective information about provision and utilization of the provided resources, which is an indication of demand and need. The numbers of patients for first-line treatment in the stent model are derived from the Austrian Cardiac Care registry [15]. Revascularization rates and number of repeated interventions are based on the CARDIACCESS database (Cardiac Care Network of Ontario [CNN], Toronto, Canada) [16]. Arrival process. Frequently, a Poisson process is used to model the arrival of a patient at a discrete point in time [17–19]. The
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Tutorial in Medical Decision Modeling Poisson process assumes that 1) the number of patients who seek treatment on 1 day is independent from the number the following day; and 2) the number of arrivals in any time interval only depends on the length of the interval and not on its starting point. This means that there is no rush or slack period. Further background of this important stochastic process is presented in the Appendix and [29,30].
Performance Measures Performance measures of queue models are for example utilization, waiting time, or number of waiting people (averaged over a specific period or current). Utilization denotes the percentage of time the server or the clinical unit is busy. Queuing theory provides an analytic approach to estimate long-term performance. Estimates of further performance measures require assumptions on arrival and service time distributions. Examples that can be solved by an analytic approach (i.e., closed formulas) are given in the Appendix. Nevertheless, as systems become more complex, simulation is used instead of this analytic approach. For computational background, see Appendix.
Simulation of Queue Model for CAD and Stent Treatment
Table 1 Summary of model input
Revascularization sate in subgroup [16] (S1) (S2) (S3) (S4) Resource utilization* Cost of intervention (EUR) Cost of hospital stay (EUR) Time of hospital stay (days) Cost while waiting (per day EUR)† No. of stents per intervention Quality of life utility values (EQ-5D utilities) [28] Baseline 1 month 6 months 12 months
Coronary artery bypass graft
Drug-eluting stent
Bare-metal stent
0.054 0.054 0.069 0.051
0.095 0.051 0.143 0.055
2,648 2,808 4.9 15
1,953 2,808 4.9 15
6,560 16,180 15.2 n.a.
1.24
1.24
n.a.
0.69 0.84 0.86 0.86
0.69 0.84 0.86 0.86
0.68 0.78 0.86 0.87
*Average per patient, Accounting System, Innsbruck Medical University Hospital, unpublished report. †Expert opinion Division of Cardiology, Medical University Innsbruck, Austria. Note: During waiting time utility values assumed to be equal to baseline. n.a., not applicable.
Methods Figure 1 displays the assumed path of the patients. Two arrival streams are pooled (A1-new cases, A2-repeated intervention/ second-line treatment). A2 depends on A1. The arrival of new patients is assumed to be a Poisson process and daily arrival is modeled with rate l = 14,600/365 [15]. To account for higher risk of revascularization, the patient cohort is assumed to be split into four mutually exclusive subgroups: (S1) nondiabetics with long lesions or narrow vessels; (S2) nondiabetics with short lesions and wide vessels; (S3) diabetics with long lesions or narrow vessels; and (S4) diabetics with short lesions and wide vessels. In the simulation, patients are “created” and randomly assigned to these subgroups Si,i = 1,2,3,4(P1 = 0.38, P2 = 0.4, P3 = 0.12, P4 = 0.10[16]. It is assumed that restenosis occurs at any time within 1 year at constant rates (Table 1). The rate of repeated interventions/ number of second-line treatments depends on the provided type of the stent and on the patient-specific risk factors (according to subgroup). Treatment (D or B) can differ for subgroups. Treatment allocation scenarios describe the allocation of all patient subgroups to the two treatment alternatives. Each subgroup is assumed to receive either D or B for first- and second-line treatment. CABG is the third line and final intervention if patients still face renarrowing. The allocation scenario is defined as = {Stent type for subgroup (S1), (S2), (S3), (S4)}. In the scenario {DBDD}, for example, subgroup (S2) receives B and (S1), (S3), (S4) receive D (Table 2). Patients from all subgroups line up for stenting in the cardiology unit. In the analysis of treatment outcomes, different scenarios are evaluated and the impact of stent alternatives (different arrival A2) is analyzed. The analysis consists of two parts. The base case analysis assumes immediate treatment. This implies unlimited capacities for stent interventions. Utilization of the cardiology labs is measured in four scenarios. Treatment cost and quality of life utility values are calculated. In the second part of the analysis, limited capacities for stenting are assumed (38, 40, 42 patients per day). If patients arrive and the system capacity is reached, they are put on the waiting list. While they are waiting, they receive drug treatment which
leads to additional costs. Pain and uncertainty lead to a reduced quality of life. The improvements of drug-eluting stents are derived from a broader perspective. Besides the direct effect on patients, also the benefit for patients who subsequently could be treated earlier (because of less patients waiting for treatment in total) are estimated. The DES model is built by using ARENA (Rockwell Software, Inc., Wexford, PA).
Results Utilization estimates (Fig. 2) indicate that drug-eluting stents reduce long-term utilization of the cardiology labs. Long-term average costs (Table 3) indicate a trade-off between the reduction in utilization and increased average costs per patients for the new stent device. Average patient costs under the assumption of limited capacities are also presented in Table 3. Quality of life utility values are presented in Table 4 and the incremental cost-effectiveness ratio is given in Table 5. Figure 3 displays the average waiting time per patient. With decreasing resource capacities, the waiting time and the number of waiting patients increase. Costs increase with decreasing capacities because of additional drugs. Under specific circumstances, this leads to substantial differences in the outcomes of the treatment alternatives. For example, a resource capacity of 38 stented patients per day is assumed. Hence, the treatment alternative DBDB and DBDD not only do imply reduced waiting lists compared with BBBB, but also do they lead to less or equal average patient costs.
Discussion This article provides a tutorial on the application of queue modeling for decision analytic models in health care. Queue modeling and optimizing are discussed to a broad extent for industrial applications. The challenging task is to transfer concepts and ideas to the health-care sector. Patients are not widgets. Factory workflow differs from patient pathways with respect to the com-
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504 Table 2 Example treatment allocation scenario DBDD Treatment Patient cohort consists of subgroups (defined by baseline risk factors) (S1) (S2) (S3) (S4)
nondiabetes nondiabetes diabetes diabetes
long lesion or narrow vessel short lesion and wide vessel long lesion or narrow vessel Short lesion and wide vessel
38% 40% 12% 10%
First
Second*
Third
D B D D
D B D D
CABG CABG CABG CABG
*Type of stent for first line assumed to be type of stent for second-line treatment. B, bare-metal stents; CABG, coronary artery bypass graft; D, drug-eluting stents; DBDD, subgroup (S2) receives B and (S1), (S3), (S4) receive D.
Table 3 Cost per patient (EUR) (CAD example) Cost per patient (EUR) No scarcity Mean 95% Cl upper bound 95% Cl lower bound 42 patients daily stented Mean 95% Cl upper bound 95% Cl lower bound 40 patients daily stented Mean 95% Cl upper bound 95% Cl lower bound 38 patients daily stented Mean 95% Cl upper bound 95% Cl lower bound
BBBB
BBDB
Table 4 Average quality of life utility value (CAD example) DBDB
DBDD
5269 5271 5268
5285 5287 5283
5446 5447 5444
5516 5518 5515
5457 5466 5448
5420 5429 5411
5491 5496 5486
5561 5566 5556
5950 5963 5938
5894 5905 5882
5922 5932 5911
5988 5999 5978
6585 6599 6571
6519 6533 6506
6532 6545 6519
6600 6613 6587
Quality of life utility values No scarcity Mean 95% Cl upper bound 95% Cl lower bound 42 patients daily stented Mean 95% Cl upper bound 95% Cl lower bound 40 patients daily stented Mean 95% Cl upper bound 95% Cl lower bound 38 patients daily stented Mean 95% Cl upper bound 95% Cl lower bound
BBBB
BBDB
DBDB
DBDD
0.86296 0.86296 0.86296
0.86302 0.86301 0.86302
0.86311 0.86311 0.86311
0.86312 0.86311 0.86312
0.86089 0.86080 0.86099
0.86153 0.86143 0.86162
0.86262 0.86257 0.86267
0.86262 0.86257 0.86267
0.85551 0.85538 0.85564
0.85635 0.85622 0.85647
0.85790 0.85779 0.85802
0.85795 0.85784 0.85807
0.84855 0.84841 0.84869
0.84948 0.84934 0.84963
0.85120 0.85106 0.85135
0.85124 0.85110 0.85138
B, bare-metal stents; D, drug-eluting stents; BBBB, subgroup (S1), (S2), (S3), (S4) receive B; BBDB subgroup (S3) receives D and (S1), (S3), (S4) receive B; CI, confidence interval; DBDB subgroup (S1), (S3) receives D and (S2), (S4) receive B; DBDD, subgroup (S2) receives bare-metal stents and (S1), (S3), (S4) receive drug-eluting stents.
plexities of treatment success and clinicians’ decisions. Quality of life is a patient-relevant outcome used in economic evaluations. Data of quality of life in situations where medical procedures are delayed are insufficient. The assumption of capacities (supplies) that are unequal to the current need is very sensitive from an ethical and political
perspective. Nevertheless, a simulation model provides an ideal environment to render (hidden) costs and health consequences and to test alternatives and their impact on resource utilization. They can identify bottlenecks and help optimize patient flows or test prioritization rules for specific patient groups. DES models have been applied to optimize the performance of clinics with
Average number of daily stented patients (95% CI)
B, bare-metal stents; D, drug-eluting stents; BBBB, subgroup (S1), (S2), (S3), (S4) receive B; BBDB subgroup (S3) receives D and (S1), (S3), (S4) receive B; CI, confidence interval; DBDB subgroup (S1), (S3) receives D and (S2), (S4) receive B; DBDD, subgroup (S2) receives bare-metal stents and (S1), (S3), (S4) receive drug-eluting stents.
43.0 42.8 42.6 42.4 42.2 42.0 41.8 BBBB
BBDB
DBDB
Treatment allocation scenario
DBDD
Figure 2 Utilization of the cardiology department in the coronary artery disease example.
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Tutorial in Medical Decision Modeling Table 5 Incremental cost-effectiveness ratio (CAD example) Incremental cost-effectiveness Cost [EUR]/quality of life utility values
Max. no. of daily stented patients
BBBB-BBDB
BBBB-DBDB
BBBB-DBDD
No limit 42 40 38 36
275,939 Dominated Dominated Dominated Dominated
1,143,679 19,749 Dominated Dominated Dominated
1,580,492 60,527 15,557 5,587 Dominated
B, bare-metal stents; D, drug-eluting stents; BBBB, subgroup (S1), (S2), (S3), (S4) receive B; BBDB subgroup (S3) receives D and (S1), (S3), (S4) receive B; DBDB subgroup (S1), (S3) receives D and (S2), (S4) receive B; DBDD, subgroup (S2) receives bare-metal stents and (S1), (S3), (S4) receive drug-eluting stents.
Average waiting time per patient (days) 95% CI
regard to access to care and financial viability [20], to analyze surgical sequencing rules [21], to optimize organ allocation processes [22], and to evaluate treatment of cardiac disease [23,24]. The modeling example of CAD treatment points out the importance of queue modeling for economic evaluation of treatment alternatives. The treatment scenarios differ in waiting times and consequently show differences in the absolute and relative cost outcomes. In addition, limited resources influence quality of life utility values. The example yields for improvements. Revascularization rates could account for time dependencies (e.g., for bare-metal stents higher in the first 3–6 months). Furthermore, long-term trends and seasonal effects can lead to time dependent arrival
rates. Therefore, generalized nonstationary Poisson Processes could be applied [22,25,26]. The arrival process of patients is a substantial queuing characteristic. Underlying assumptions, such as incidence or prevalence, can be regionally specific. Therefore, the transferability of results is limited. Models should be built with the necessary flexibility to adapt input values and to incorporate expected long-term trends which influence arrival (e.g., aging society). The teaching example follows a simplified queuing discipline (first come–first serve). It does not reflect urgent need of unstable PCI patients. Serious adverse events resulting from delays remain unconsidered. The analysis does not consider drug-eluting stents to be the second-line treatment if bare-metal stents are not successful. Decision-analytic modeling supports decision-makers in many fields of health care [27]. DES is very suitable for simulation of queue models, which is shown by the wide use for optimizing service systems, production lines, or hospital units. Nevertheless, such detailed decision-analytic models in health care require information that might not be collected yet in clinical trials or databases (e.g., prioritization rules and queue discipline). DES allows for modeling of individuals and interaction of individuals. An alternative individual-based modeling technique with the focus on behavioral factors is agent-based modeling. This approach could be considered for the evaluation of infectious diseases. Decisions on simple queuing problems might be evaluated using an analytic approach to estimate longterm results. A Markov Model cohort simulation does not
125
100
75
50
25
0
D BD D 8 _3
38
Figure 3 Waiting time of patients (coronary artery disease example).
38 B_ BD
D
B_
38
0 _4 D
B_
D BB
B BB
BD
D
40
40 B_
BD D
B_
40
2 _4
B_
D BB
B BB
42
42 B_
BD
D BD
D
D
42
B_
B_
D BB
B BB
Scenario capacity (no. of daily steted patients)
506 allow for dynamic queues. In addition, a state transition microsimulation with fixed time cycles appears to be not suitable (e.g., with respect to flexible arrival and service time distribution).
Conclusions The validity of the results of economic evaluation can be improved by explicitly taking queues into account. Queuing models should be applied when a substantial difference in resource utilization is expected between the compared healthcare interventions. Source of financial support: Financial support for this study was provided in part by the Medical University of Innsbruck, Leopold-FranzensUniversity of Innsbruck and the Canadian Study Centre Innsbruck, and ONCOTYROL—Center for Personalized Cancer Medicine (Austria). ONCOTYROL is a K1-COMET Center and funded by the Federal Ministry for Transport Innovation and Technology (BMVIT) and the Federal Ministry of Economics and Labour/the Federal Ministry of Economy, Family and Youth (BMWA/BMWFJ), the Tyrolean Future Foundation (TZS), and the State of Styria, represented by the Styrian Business Promotion Agency (SFG), and supported by UMIT—University for Health Sciences, Medical Informatics and Technology and TILAK-Hospital Holding Company. The funding agreement ensured the authors’ independence in designing the study, interpreting the data, writing, and publishing the report. Supporting information for this article can be found at: http:// www.ispor.org/Publications/value/ViHsupplementary/ViH13i4_Jahn.asp
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