A QUEUING-ASE STATISTICAL APPROXIMATION OF HOSPITAL EMERGENCY DEPARTMENT OARDING James R. royles a Jeffery K. Cochran b a RAND Corporation, Santa Monica, CA 90401, james_broyles@rand.org b Department of Operational Sciences, Graduate School of Engineering and Management, Air Force Institute of Technology, Wright-Patterson AF, OH 45433, jeffery.cochran@afit.edu The views expressed in this article are those of the authors and do not reflect the official policy of the United States Air Force, Department of Defense, or the United States Government. Abstract: A hospital Emergency Department s ( s) wait times can be driven by high occupancy in its downstream InPatient hospital (IP). patients not admitted to the IP due to high IP occupancy or transfer delays are termed to be 'boarding' in the. boarding causes a decrease in service capacity, longer wait times, increased fractions of patients that leave without treatment, and increased ambulance diversion. oarding is a 'whole hospital' effect subject to dynamic staffing levels, demand, and capacity in a hospital. In this paper, a statistical approximation of boarding that has a queuing theory foundation is created and used to analyze data from a large hospital. The model produced is generic in queuing structure but specific to a hospital s data. Such approximations enable a hospital to estimate the effects of IP length of stay reduction on boarding and to quantify the reduction of waiting times where boarding is reduced or removed. Keywords: Emergency department, boarding, queuing theory, statistical estimation 1. Introduction and Motivation Hospital Emergency Departments (s) are a critical component to patient flow within a hospital. s typically have large patient volumes, operate under high utilizations, service many types of patients, and pass discharged patients to most other areas of the hospital. Although the majority of patients are discharged home, approximately 2 of the patients are admitted to the InPatient hospital (IP). The IP is quite different from the in terms of patient flow. In the IP, patients typically have Length Of Say durations (LOSs) of days whereas, in the, patients typically stay for hours. The IP comprises of many different locations (e.g., surgery, intensive care units, telemetry units, obstetric units) which patients flow between the locations. [1] IP patients typically consume more resources and physical hospital space than the patients. boarding is a problem for many s. A patient is boarded in the if the patient is admitted to an IP unit and the IP does not have a bed available in which to place the patient. In other words, s are finished treating the patient yet the patient waits in an bed for an IP bed to become available. boarding decreases the service capacity by consuming the beds with boarded patients. boarding causes a variety of undesired patient flow and safety effects including admitted patients not being placed in the appropriate IP units in a timely manner, increase patient waiting times, and an increase in patients that leave without treatment due to long waiting times. This could cause an admitted critically ill patient to not be treated by the intensive care IP unit in a timely manner, [2] patients to wait too long to receive treatment, [2] and patients to leave the without receiving treatment. [3] 122
A 2003 GAO report found that boarding is a major contributing factor to excessive crowding. [4] The GAO s survey reported that more that more than 75% of boarding patients were boarded for longer than two hours. They also found that one in every five s had an average boarding time of eight hours or more per admitted patient. boarding is the results of a whole hospital effect and cannot be reduced by process improvements in a localized area of the hospital. As a whole hospital effect, the causes of boarding are the following: Insufficiently low IP patient throughput rates Low IP throughput rates are likely caused by not having enough IP beds, not having reasonable IP patient LOSs, or both. Due to the large capital expensive nature of IP units, hospitals strive to keep their IP full of patients thereby increasing IP utilization and increasing boarding. Daily seasonality mismatch between the and IP peak arrival rates are generally in the morning through the evening. IP peak patient discharge activity is in the late afternoon and evening. Therefore, most of the boarding occurs in the middle of the day when arrival rates are large and IP discharge rates are low. Differing sense of urgencies IP units managers and clinicians plan for IP discharges on a daily basis, whereas, s plan for patient discharges in terms of hours and minutes. Reducing the IP LOS is recognized as a feasible method for reducing blocking. The literature highlights widespread efforts to reduce IP LOS by streamlining the IP patient discharge process, reducing patient waiting for tests, waiting for pharmacy, and increasing staff. Modeling boarding can be quite difficult due to the complex network of IP units and the input estimates required in queuing theory models. oarding is called blocking in queuing theory nomenclature. The queuing theory literature has developed blocking models in a variety of different queuing structures. Published queuing blocking models include two tandem queues, [5][6][7] Markovian queuing networks, [8][9] and a Kanban controlled production system. [10] The queuing theory literature assumes that inputs such as mean service time and service capacity are known when the queuing models are applied. In hospitals however, it is difficult to accurately estimate the mean service time and service capacity. In an, the average time a patient stays in an cannot be used as an estimate for the mean service time, because, beds are not available for service during the custodial cleaning and room reset that occurs in between each patient visit. This effect is also present in IP beds. Also, sometimes and IP beds are not fully staffed and operational thereby under utilizing their bed capacity. The true service time and service capacity are difficult to estimate for these reasons. Recent research has used queuing theory based regressions to estimate these unknown input parameters in hospitals under a queuing theory foundation. [1][11] This paper presents a queuing theory based statistical estimation to quantify the effects of IP LOS on boarding and boarding on wait times. The statistical point estimations fits the IP service rates and service rates to match existing data of boarding time and waiting times using the method of moments. This methodology is applied to data collected from a large hospital in Arizona. This paper demonstrates how the estimated queuing theory model can be used to quantity the reduction in boarding and waiting time resulting from a decrease in IP LOS. 2. A Queuing Theory ased Statistical Estimation of Emergency Department oarding This section presents a queuing theory model and an estimation methodology to quantify boarding. This methodology views the hospital as two multi-server Markovian queues in series. Reference [12] 123
introduces the two node view to model patient flow as a whole hospital effect. Figure 1 displays the and IP as two queues in series. λ DA λ M/M/c μ, W q f A λ IP M/M/c IP μ IP, W q λ DA + f A λ (1 f A )λ Figure 1: The and IP displayed as two queues in series Table 1 lists the variables used in this formulation and a description of each. Symbol Description Units Support Type λ Observed patients arrival rate to the patients/time period [0, c μ ) Hospital Data μ Service rate per bed including boarding time patients/time period (λ /c, ) Model Fit μ Service rate per bed excluding boarding time patients/time period (λ /c, ) Model Fit c Number of beds beds [1, 2, ) Hospital Data f A Observed fraction of patients admitted proportion [0, 1] Hospital Data W q Expected value of waiting time time [0, ) Model Estimate w Observed average wait time time [0, ) Hospital Data λ DA Observed direct admit IP arrival rate not through the patients/time period [0, c IP μ IP f A λ ) Hospital Data μ IP Service rate per IP bed patients/time period ((λ DA +f A λ )/c IP, ) Model Fit c IP Number of IP beds beds [1, 2, ) Hospital Data W q Expected value of boarding time time [0, ) Model Estimate b Observed average boarding time for admitted patients time [0, ) Hospital Data α Fraction of boarding time reduction proportion [0, 1] Model Estimate Table 1: List of variables in the queuing base statistical estimation model Variables with the type Hospital Data are variables that are estimated directly from the hospital data collected. Variables with type Model Fit are variables fit using the queuing based statistical estimations described below. Variables with type Model Estimate are variables that are estimated by the queuing post statistical fitting. This model statistically fits the and IP service rates (i.e., µ, µ IP ) to match the queuing model estimates of boarding W q and wait time W q to that of observed average boarding time b and the observed average wait time w. Under the queuing formulation in Figure 1, the expected wait time W q is Equation 1.!! In Equation 1, r = λ /µ, ρ = r /c, and p 0 is Equation 2.! Equation 2 is the probability that the is empty. In practice, the service time is the addition of the time that the patient is treated in the and the time the patient is boarded in the. Therefore, the expected service rate µ can be written as a function of the expected treatment time without boarding 1/µ and the expected boarding time W q as Equation 3. (1) (2) (3) 124
The symbol for the expected boarding time is W q because it, in essence, is the expected waiting time to be admitted into the IP hospital. Viewing this as such and the IP as a Markovian queue as in Figure 1, the expected boarding time is written as Equation 4.! (4) In Equation 4, r IP = (λ DA+f A λ )/µ IP, ρ IP = r IP/c IP, and p IP 0 is Equation 5.!! Equation 5 is the probability that the IP is empty. (5) As mentioned in the introduction, the and IP service rates (i.e., µ, µ IP ) and service capacities (i.e., c, c IP ) are difficult to estimate from hospital patient LOS data. This estimate uncertainty is caused by beds being unavailable beyond the patients visits (e.g., for cleaning purposes) and beds not being staffed to full service capacity. Therefore, we take the hospital s bed estimates (regardless of if they are staffed) as a given and statistically fit the and IP services rates to compensate for this uncertainty and match model estimates to observed average boarding times and average wait times. To fit the service rates, we use the method moments statistical estimation. The method of moments is a statistical estimation approached that estimates distribution parameters based on sample moments observed in the data. [13] In our methodology, we estimate the and IP services rates such that the first moments of the waiting time and boarding time (i.e., W q and W q ) are approximately equal the sample average wait time w and sample average boarding time b. ecause Wq and Wq are nonlinear functions of the service rates, we define a squared objective function and use a numerical analysis to computational find the method of moment estimates. Given the sample average boarding time b, Equation 6 is the method of moments estimate of the IP service rate µ IP. argmin (6) Given the sample average waiting time w and the µ IP * in Equation 6, Equation 7 is the method of moments estimate of the service rate µ. argmin (7) As mentioned in the introduction, the IP hospital may attempt reduce IP LOS to decrease boarding and wait times. Equations 8 and 9 estimate the percent reduction in boarding α and waiting δ respectively given a 100κ percent reduction in IP LOS. 100 1 / 100 1 / (8) (9) 3. A Hospital s Estimation of oarding This section applies the queuing theory based statistical estimation detailed above to a large hospital in Arizona. The data used and available to the authors for this research was from select time periods over the course of a year. Using the available hospital data, Table 2 displays the input estimates needed for this formulation. 125
Symbol λ c f A w λ DA c IP b Hospital Data 8.5 53 0.22 1.6 0.3 174 2.6 Units patients/hour beds proportion hours patients/hour IP beds hours Table 2: Estimates from hospital data Estimates in Table 2 indicate that, on average, the has 117 bed-boarding hours per day which about 9.2% of the total bed-hours offered per day. Using the data in Table 2 and the Newton-Raphson method to solve Equations 6 and 7, Table 3 displays the estimates for µ and µ IP. Symbol μ μ IP Model Estimate 0.17 0.01 Units patients/hour patients/hour Table 3: Queuing theory base estimates of and IP service rates If this hospital were to reduce their inpatient LOS by 100κ percent, Figures 2a and 2b display the percent reduction of boarding and the nominal and percent reduction of the expected waiting time. (a) (b) % IP LOS Reduction (100κ) 13% 12% 11% 1 9% 8% 7% 6% 5% 4% 3% 2% 1% 1 % IP LOS Reduction % Wq Reduction 2 3 4 5 6 7 8 9 % Reduction in oarding Time (α) 10 10 9 8 7 6 5 4 3 2 1 % Reduction of Wq (δ) Wq (minutes) Figure 2: W q and W q as a function of a 100κ percent reduction in IP LOS Figure 2 shows that a relatively small decrease in this hospital s IP expected LOS can cause significant reductions in boarding and waiting. For this hospital under this formulation, a 1% reduction in IP LOS (i.e., κ = 0.01) will yield a 3 reduction in boarding time (Figure 2a), a 54% reduction in waiting times (Figure 2a), and a decrease of the average wait time from 98.5 minutes to 44.2 minutes. Similarly, a 2% reduction in IP LOS (i.e., κ = 0.02) will yield a 5 reduction in boarding time, a 73% reduction in waiting times, and a decrease of the average wait time from 98.5 minutes to 27.9 minutes. 4. Conclusion and Future Research This paper presents a queuing theory statistical estimation that quantifies the effect of boarding on patient delays. The queuing theory formulation makes this methodology a generic construct and the statistical estimation customizes the formulation to a specific hospital. The queuing formulation considers the and IP as two Markovian queues in series. The statistical estimation used is the method of moments where and IP service rates are estimated such that the resulting queuing estimates of wait time and boarding time are approximately equal to hospital data estimates. This formulation was motivated by the research that finds boarding is problematic in many hospitals. The hospital case study found that small percent reductions in the average IP LOS will yield significant reductions in boarding and waiting. 100 90 80 70 60 50 40 30 20 10 0 1 2 Wq % Wq Reduction 3 4 5 6 7 8 % Reduction in oarding Time (α) 9 10 10 9 8 7 6 5 4 3 2 1 % Reduction of Wq (δ) 126
Future modeling research of boarding is needed in a several directions. As mentioned in the introduction and explicitly ignored in the formulation, hospitals experience daily seasonality of patient arrival rates and discharge rates. Research is needed to determine if seasonally and its effects on boarding is necessary need to be modeled explicitly. There is some indication in the literature that Markovian properties in patient arrival and service processes exist. However, future research is needed to assess the appropriateness of Markovian assumptions when modeling boarding. Research of patient arrival and services processes that are phase type or general distributions may be necessary. 5. Acknowledgements We would like to thank the leadership of anner Health for their support and expertise. In particular, we would like to thank Twila urdick of anner Health s Care Management Division and Dick Andrews, Mary Ellen ucco, and Steve Kisiel of anner Health s Management Engineering. Without them, this research and our related research would not have been completed. We also would like to thank Aseem harti and Kevin Roche for their related research contributions and support. Last, we would like to thank the Agency for Healthcare Research and Quality (AHRQ) for the Partnerships in Patient Safety Grant HS015921-01 which yielded the foundational work for this research. 6. References 1. royles JR, Cochran JK, Montgomery DC (2010). A statistical Markov chain approximation of transient hospital inpatient inventory. European Journal of Operational Research. 207(3):1645-1657. 2. Chalfin D, Trzeciak S, Likourezos A, aumann M, Dellinger RP (2007). Impact of delayed transfer of critically ill patients form the emergency department to the intensive care unit. Critical Care Medicine. 35(6):1477-1483. 3. Cochran JK, urdick TL (2011). The impact of the Door-To-Doc emergency department patient flow model. Proceedings of the 2011 Industrial Engineering Research Conference. T. Doolen and E. Van Aken, eds. 4. United States General Accounting Office (2003). Hospital emergency departments: crowded conditions vary among hospitals and communities. GAO-03-460. 5. Gomez-Corral A (2002). A Tandem Queuing with locking and Markovian Arrival Process. Queuing Systems. 41:343-370. 6. Gomez-Corral A (2004). Sojourn Times in a Two-Stage Queuing Network with locking. Wiley Inter-Science. 7. Gomez-Corral A, Martos ME (2006). Performance of Two-Stage Tandem Qeueues with locking: The Impact of Several Flows of Signals. Performance Evaluation. 63:910-938. 8. Perros HG (1994). Queueing Networks with locking. Oxford University Press. New York. 9. Ramesh S, Perros HG (2000). A Two-level Queueing Network Model with locking and Non- for Performance Evaluation of blocking Messages. Annals of Operations Research. 93:357-372. 10. Mascolo MD, Frein Y, Dallery Y (1996). An Analytical Method Kanban Controlled Production Systems. Operations Research. 44(1):50-64. 11. Cochran JK, royles JR. (2010). Developing nonlinear queuing regressions to increase emergency department patient safety: approximating reneging with balking. Computers and Industrial Engineering. 59(3):378-386. 12. Cochran JK, Roche KT (2009). Determining bed capacities using queuing theory: a whole hospital view. IIE Industrial Engineering Research Conference. 856-861. 13. ain LJ, Engelhardt M (1992). Introduction to probability and mathem atical statistics. Duxbury. 127