Using a Discrete-event Simulation to Balance Ambulance Availability and Demand in Static Deployment Systems Ching-Han Wu, EMT-P, and Kevin P. Hwang, PhD
Abstract Objectives: To improve ambulance response time, matching ambulance availability with the emergency demand is crucial. To maintain the standard of 90% of response times within 9 minutes, the authors introduce a discrete-event simulation method to estimate the threshold for expanding the ambulance fleet when demand increases and to find the optimal dispatching strategies when provisional events create temporary decreases in ambulance availability. Methods: The simulation model was developed with information from the literature. Although the development was theoretical, the model was validated on the emergency medical services (EMS) system of Tainan City. The data are divided: one part is for model development, and the other for validation. For increasing demand, the effect was modeled on response time when call arrival rates increased. For temporary availability decreases, the authors simulated all possible alternatives of ambulance deployment in accordance with the number of out-of-routine-duty ambulances and the durations of three types of mass gatherings: marathon races (06:00–10:00 hr), rock concerts (18:00–22:00 hr), and New Year’s Eve parties (20:00–01:00 hr). Results: Statistical analysis confirmed that the model reasonably represented the actual Tainan EMS system. The response-time standard could not be reached when the incremental ratio of call arrivals exceeded 56%, which is the threshold for the Tainan EMS system to expand its ambulance fleet. When provisional events created temporary availability decreases, the Tainan EMS system could spare at most two ambulances from the standard configuration, except between 20:00 and 01:00, when it could spare three. The model also demonstrated that the current Tainan EMS has two excess ambulances that could be dropped. The authors suggest dispatching strategies to minimize the response times in routine daily emergencies. Conclusions: Strategies of capacity management based on this model improved response times. The more ambulances that are out of routine duty, the better the performance of the optimal strategies that are based on this model. ACADEMIC EMERGENCY MEDICINE 2009; 16:1359–1366 ª 2009 by the Society for Academic Emergency Medicine Keywords: ambulance availability, emergency medical services, response time, static ambulance deployment models, discrete-event simulation.
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mergency medical services (EMS) administrators continually endeavor to provide the best service they can, despite limited resources. To decrease the morbidity and mortality of casualties, response time—the interval between when the dispatcher receives a call and the ambulance arrives at the scene—is one of the most important factors.1–4 From the Department of Transportation & Communication Management Science, National Cheng Kung University, Tainan, Taiwan. Received April 18, 2009; revisions received June 9 and June 30, 2009; accepted July 1, 2009. Address for correspondence and reprints: Ching-Han Wu, EMT-P; e-mail:
[email protected].
ª 2009 by the Society for Academic Emergency Medicine doi: 10.1111/j.1553-2712.2009.00583.x
EMS systems are service-oriented, and services cannot be inventoried.5,6 A request for ambulance service when the system is full will be rejected or set in a queue, which increases response time. This may occur even if, on average, demand does not exceed ambulance availability. Thus, to improve response time, the ability to maintain equilibrium between capacity and demand is the key to success.5,6 EMS systems have traditionally been designed with redundancy, so-called buffer capacity, to cope with fluctuations in call volume and changes in ambulance availability and to provide some degree of preparedness for unexpected demand.7,8 In static ambulance deployment models, ambulances are on standby at fixed-post stations between calls.
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Deployment is inflexible and difficult to modify because it takes time to set up new stations. As demand increases year by year,9–12 the ambulance fleet becomes incapable of meeting it. EMS authorities must therefore estimate in advance how many increments beyond their buffer capacity the new demand will be. This may permit the ambulance fleet to be expanded to meet basic requirements. Moreover, EMS must account not only for routine daily emergencies, but also for some provisional events (e.g., vehicle breakdown, mass gatherings, mutual aid assistance) that cause temporary decreases in ambulance availability,7,13,14 even though everyday management is not focused on these nonroutine events. Although EMS systems typically have extra ambulances (back-up vehicles) available, if the number of ambulances needed for provisional events is greater than the number of back-up ambulances, managers must spare some ambulances from the standard configuration, which interferes with the designed layout and increases the response time in routine daily emergencies. Thus, administrators need to proactively plan which ambulances to mobilize for events so that response times in routine daily emergencies can be minimized. Furthermore, managers must anticipate how many ambulances the system can afford and then determine the critical threshold that prompts the system to look for outsourced support to minimize the disruption of routine operations.13 However, studies on the capacity management of ambulance fleet have focused mainly on where to locate EMS resources or how large they should be to fit the spatial pattern and the volume of calls, or both,15–25 as opposed to shaping strategies to proactively plan for demand increases and temporary availability decreases in static models. To maintain the required level of response time, the objectives of the present study were: 1) to develop a discrete-event simulation model to evaluate the effect of various scenarios on response time, 2) to estimate the threshold for expanding the ambulance fleet when demand increases, and 3) to find the optimal dispatching strategies when provisional events create temporary availability decreases so that response times in routine daily emergencies are minimized. METHODS Study Design This study was a computer-based simulation study, not a clinical trial, and it did not involve human subjects; it did not require institutional review board approval under our regulations in Taiwan. Study Setting and Population The Tainan EMS system was the subject of the study. Tainan City (population 764,000; area 175.65 km2), in southern Taiwan, has a fire department–based EMS system, which is a free social service12 and uses static ambulance deployment. There are six designated hospitals and 14 EMS stations (Figure 1), with 12 basic life support (BLS)-defibrillator teams and two advanced life support (ALS) teams. Each station has only one ambulance around the clock. Emergency calls are collectively
Figure 1. Distribution of Tainan City EMS stations.
forwarded to the dispatch center. Without a standard ALS-prioritized dispatch protocol, Tainan uses a onetiered dispatch rule: the nearest available ambulance is dispatched without considering the severity of the patients’ condition or the level of care that can be provided by the ambulance. Study Protocol Simulation Model. The state of evolution of an EMS system depends on the occurrence of asynchronous discrete events (e.g., call arrivals, ambulance transportation, and treatment for patients) over time. Accordingly, we developed a discrete-event simulation model of ambulance operations to forecast response times in possible scenarios. The EMS simulation model (Figure 2) consists of three modules: a call generator, a dispatcher, and a time interval calculator. A set of theoretical distributions governs event flow (EMS operation process); parameters were calculated from empirical data. The components of the model were calibrated based on information from the literature, as explained below. Call Generator. This module generates incoming calls along with their attributes: locations and types. 1) Call arrival time: Because the call arrival rate may vary by the time of day, day of the week, and month of the year, we have presented it as a nonhomogeneous Poisson process17,26,27 and assumed that the arrival rate was constant after each time period was divided into several shorter durations.23 Month of the year and day of the week were divided using ANOVA,24 and a 24-hour day was broken into shorter durations by measuring the coefficient of variation.23,24 2) Spatial distribution of calls: For computational purposes, the calls in a small area were aggregated to the centroid28,29 using Nearest Neighbor Hierarchical clustering (NNH).30 NNH clusters points together based on a threshold distance for all pairs of points; points that are closer than the threshold distance (set
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Time Interval Calculator. This module calculates the time that an ambulance spends on each stage.
Figure 2. Flowcharts of the simulation process and EMS operations process. Random processes, marked by circled numbers, drive all aspects of ambulance flow. 1 = call arrival as a nonhomogeneous Poisson process, spatial distributions of calls as multinomial distributions, call types as multinomial distributions; 2 = closest free ambulance dispatched; 3 = transport interval from station to scene, speeds as lognormal distributions; 4 = on-scene intervals as lognormal distributions; 5 = probabilities of destination hospitals as a multinomial distribution; 6 = the transport interval from scene to hospital, speeds as lognormal distributions; 7 = in-hospital intervals as lognormal distributions; 8 = transport intervals from hospital to station, speeds as lognormal distributions.
at 500 meters) to one or more other points are selected for clustering. We then used multinomial distributions to model the spatial distribution.26 3) Call types: Call types (ALS case ⁄ BLS case and trauma ⁄ nontrauma) may affect on-scene and in-hospital intervals17 and were treated as multinomial distributions.26 Dispatcher. This module, which handles the incoming calls, decides which ambulance responds to a call and to which hospital the patient is transported. 1) Ambulance dispatching: This module assigns the closest free ambulance to the call. If there are no available ambulances, the call will be put in a queue. To estimate the distances, we used linear regression to analyze the relationship between the shortest network distances and straight-line distances. 2) Destination hospital: After a victim has been initially treated at the emergency scene, the ambulance transports the victim to a hospital. Although the distances from scenes to hospitals are important determinants for choosing the destination hospital, other factors are also important.17 We used multinomial distributions to model the destination probabilities for each responding ambulance.26
1) Travel intervals: Because traffic patterns change during the day, we classified travel speeds (stations to scenes, scenes to hospitals, and hospitals back to stations) into daily shifts based on call-arrival times. An ambulance responds primarily to emergencies within the neighborhood of its station; therefore, we regarded the neighborhood as the designated district of the station. Based on station locations, the whole city was conceptually divided into several smaller districts, and the traffic patterns of these smaller districts may be more homogeneous than those of the whole city. The categorization may therefore more accurately reflect the traffic patterns of each smaller district than taking the average from the whole city. Thus, to accurately estimate response times, speeds from stations to scenes were further divided by activating stations to differentiate the traffic patterns of every district. We then calibrated the distribution to fit the speeds. 2) On-scene intervals and in-hospital intervals: These intervals were categorized into eight groups based on the two levels of emergency medical technicians (EMTs) that responded (EMT-Basic and EMT-Intermediate ⁄ EMT-Paramedic) and the four call types. We assumed these intervals were lognormally distributed.31,32 We implemented the simulation using C++ language and used common random numbers to reduce variances from random variates.31 Random numbers were generated using the Mersenne Twister algorithm.33 To verify its accuracy, the model was presented to experts familiar with EMS, statistical analysis, or simulation programming. We replicated the simulation five times, each covering 1 year. To validate the model, we split the empirical data into two parts: one part was for model development, and then the simulated response times were compared with those in the other part for validation.31,34–36 Data Collection. We collected data from 16,984 cases from January through December 2004. The records included such information as 1) the times for the following rescue segments (in minutes): notification of calls, arrival at the scene, departure from the scene, arrival at the hospital, departure from the hospital, and return to the base station; and 2) call locations, station notified, destination hospitals, chief complaints, vital signs, and so on. Although the times of each segment were manually recorded, EMTs were requested to communicate with the center by radio in real time to ensure accuracy. The radio communications were recorded for reconfirmation. Measures. To estimate the effect of increasing demand on response time, we gradually increased callarrival rates and assumed that spatial distributions remained the same. For temporary decreases in ambulance availability, we simulated alternatives of ambulance deployment in accordance with the durations and the number of out-of-routine-duty ambulances to find the optimal dispatching strategies that minimized
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mated 1.2 times their straight-line distances (R2 = 0.98), and most travel speeds were lognormally distributed. Chi-square tests showed that the observed distributions of the spatial distributions of calls, call types, on-scene intervals, in-hospital intervals, and destination hospitals closely matched theoretical distributions. The simulation model reasonably represented the operations of the actual Tainan EMS system, according to a t-test comparing the simulated response times with empirical data from November through December, in which there were no significant differences between simulated results and their empirical counterparts (Table 1).
response times in routine daily emergencies. Although the model was able to simulate any specific durations, we tested for time periods during which three types of three mass gatherings often occur, marathon races (06:00–10:00 hr), rock concerts (18:00–22:00 hr), and New Year’s Eve parties (20:00–01:00 hr), and assumed that all events occurred on Saturdays. While lacking a universal standard in Taiwan, we used the criterion of at least 90% of response times within 9 minutes18 to evaluate performance. Data Analysis. Statistical analyses were done using commercial software (Minitab 14.1, Minitab Inc., State College, PA), and the significance level (a) was set at 0.05 throughout this study.
Increasing Demand for Emergency Services The simulated results showed that the response times increased with the increase in demand (Figure 3). With
RESULTS Empirical Data Analysis and Model Validation The empirical data were divided into two parts: one part (January through October: 13,871 run-reports, of which 2,871 records had no concrete response time) was for model development, and the other (November through December: 3,113 run-reports, of which 183 records had no concrete response time) was for model validation. There were no significant differences in call volumes between months (p = 0.248), i.e., seasonal variability was insignificant, but there were significant differences between weekdays and between times of the day.24 When we divided the data into 12 durations (hours of the day: 03–07, 07–11,…, 23–03; days of the week: Tuesday through Thursday and Friday through Monday), chi-square tests showed that call arrivals during each duration followed the homogeneous Poisson process.24 The shortest network distances approxi-
Figure 3. The effect of increasing demand on the mean response time (solid line) and on the percentage of response time within 9 minutes (dashed line).
Table 1 Comparison of Simulated Response Time With Empirical Data From November Through December 2004 Week Tuesday through Thursday
Duration*
Sample
90th Percentile
03–07
Empirical Simulated Empirical Simulated Empirical Simulated Empirical Simulated Empirical Simulated Empirical Simulated Empirical Simulated Empirical Simulated Empirical Simulated Empirical Simulated Empirical Simulated Empirical Simulated
9.00 8.39 8.00 8.53 8.00 8.22 8.00 8.58 8.00 8.01 8.00 8.27 9.00 8.84 8.00 8.32 8.00 8.32 9.00 8.77 8.00 8.22 8.00 8.73
07–11 11–15 15–19 19–23 23–03 Friday through Monday
03–07 07–11 11–15 15–19 19–23 23–03
*Duration is shown in 24-hour units.
Mean (±SD) 5.09 4.65 4.64 4.70 4.30 4.49 4.47 4.63 4.37 4.26 4.29 4.49 5.26 4.89 4.38 4.57 4.38 4.47 4.49 4.73 4.30 4.51 4.46 4.68
(2.16) (3.06) (2.17) (3.25) (1.82) (3.16) (2.21) (3.31) (1.97) (3.21) (1.63) (3.09) (2.18) (3.09) (1.73) (3.11) (1.84) (3.28) (2.15) (3.62) (1.96) (3.19) (2.01) (3.24)
p-value 0.099 0.727 0.198 0.315 0.442 0.286 0.070 0.112 0.500 0.093 0.107 0.191
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a 17% increase in demand, the response times were not significantly longer than those of the empirical data, despite their statistical significance when the incremental ratio exceeded 18% (p = 0.04). The percentage of response times within 9 minutes was less than 90% when the incremental ratio exceeded 56%. Therefore, a 56% increase in demand appears to be the threshold value for the current Tainan EMS before having to implement a major expansion. Furthermore, under the call volume of the empirical data, when the system dropped any one or two ambulances in one of the three sets of ambulance stations—(1) 3, 8, 9; (2) 7, 8, 9; and (3) 8, 9, 14—the 90%-within-9-minutes standard was maintained. Temporary Decreases in Ambulance Availability When one, two, or three of 14 ambulances must be set aside for provisional events, there are 14 (C114 ), 91 (C214 ), and 364 (C314 ) combinations of possible alternatives:
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Crn ¼
n! r!ðn rÞ!
denotes the number of ways that a subset of r elements can be chosen out of a set of n elements). We simulated all possible alternatives to search for the optimal one: the one in which the percentage of response times within 9 minutes was highest (Table 2). For the optimal alternatives, the percentage-within9-minutes was more than 90% when the number of out-of-routine-duty ambulances was one or two for all durations and three for 20:00–01:00. When more than three ambulances were out of routine duty, the 90%within-9-minutes standard could not be maintained. When randomly choosing one of the possible alternatives, the percentage-within-9-minutes and the expected response time are listed. For the percentage-within-9minutes, the optimal alternatives yielded better results than those of randomly chosen alternatives. The
Table 2 The Response Time of the Optimal Alternatives of Ambulance Deployment for Saturdays with One, Two, and Three Ambulances Out of Service for Routine Daily Emergencies 06:00–10:00 The normal state Mean (±SD) 90th percentile Percentage* Temporary availability decreases One ambulance out of service Random choice Expected response time 90th percentile Percentage* Optimal alternative Ambulance sentà Mean (±SD) 90th percentile Percentage* Two ambulances out of service Random choice Expected response time 90th percentile Percentage* Optimal alternative Ambulances sentà Mean (±SD) 90th percentile Percentage* Three ambulances out of service Random choice Expected response time 90th percentile Percentage* Optimal alternative Ambulances sentà Mean (±SD) 90th percentile Percentage*
18:00–22:00
20:00–01:00
4.57 (3.07) 8.80 90.7%
4.38 (2.96) 7.98 93.1%
4.49 (3.20) 8.39 91.9%
4.85 (3.38) 9.69 88.8%
4.72 (3.38) 8.84 91.0%
4.76 (3.50) 8.88 90.3%
8 4.62 (3.05) 8.80 90.7%
9 4.48 (3.01) 8.39 92.6%
9 4.56 (3.21) 8.39 91.7%
5.16 (3.72) 10.22 86.6%
5.09 (3.80) 9.60 88.6%
5.07 (3.82) 9.46 88.5%
8, 9 4.77 (3.17) 8.93 90.2% 5.51 (4.08) 10.83 84.3% 6, 8, 9 4.88 (3.29)§ 9.59 89.1%
3, 9 4.68 (3.15) 8.60 91.7% 5.51 (4.26) 10.53 85.9% 3, 8, 9 4.98 (3.40)§– 9.02 89.7%
9, 10 4.79 (3.25) 8.68 91.0% 5.42 (4.18) 10.31 86.2% 9, 10, 14 5.06 (3.36)– 8.88 90.0%
*Percentage of response time within 9 minutes. Randomly choosing one from the possible alternatives. àThe optimal ambulances sent for provisional events. For example, should the Tainan EMS system provide three ambulances for provisional events at 06:00–10:00 on Saturdays, the optimal dispatching strategy that minimizes response times in routine daily emergencies is to send ambulance Nos. 6, 8, and 9. §p < 0.05, one-tailed t-test between the randomly chosen alternative and the optimal one. –p < 0.05, one-tailed t-test between the normal state and the optimal alternative.
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differences in mean response times between optimal and randomly chosen alternatives became larger when the number of out-of-routine-duty ambulances increased, and the response time of the optimal alternatives was significantly shorter for 18:00–22:00 with two or three ambulances (p = 0.04, p = 0.02) and 06:00–10:00 with three ambulances (p = 0.01). Compared with the normal state, when more than two ambulances were out of routine duty, the response time of every optimal alternative was significantly longer, except for 06:00–10:00 (p = 0.09). An increase in the number of out-of-routine-duty ambulances led to the deterioration in mean response time and percentage-within-9-minutes. In addition, if the number of temporary availability decreases exceeded three ambulances, there would probably be no ambulances available for other emergency calls.
ambulance No. 6 dealt with the fewest incidents (14th in descending order: 2.3%), but it was assigned in only one alternative. Ambulance No. 9 was involved in most optimal alternatives, but the number of calls it responded to was 12th (2.9%). If EMS response times are to be optimized, appropriate ambulance deployment must be determined by using scientific methods and by quantitatively analyzing empirical data. Computerbased simulation modeling is a capable tool for EMS ambulance management.17,39 Furthermore, the optimal alternatives performed better than those randomly chosen from the possible alternatives, and the differences became greater when the number of out-of-routine-duty ambulances increased. These findings revealed that the simulated strategies improved response times. The more ambulances that are out of routine duty, the stricter the planning needed.
DISCUSSION Managers should plan, rather than react, to efficiently balance ambulance availability with demand.5,37,38 It seems difficult to use valid mathematical models to efficiently solve real EMS system management problems, because the system is relatively complex and demands on its services are always uncertain. Therefore, we designed and tested a discrete-event simulation for proactive capacity management of the ambulance fleet and used the Tainan EMS system as a subject. The present ambulance fleet may not be sufficient to efficiently handle the constantly growing demand. Managers can roughly perceive the changing trends in response times by studying historical statistics, but the exact threshold of excessive demand is still unknown. The simulation method developed in the present study enables managers to precisely estimate the timing for expanding the ambulance fleet. In the case of Tainan, consistent with our intuition, the response times increased along with the increase in demand. When the incremental ratio of demand exceeded 56%, the percentage of response time within 9 minutes was less than 90%. Therefore, a 56% increase in demand is the threshold value for the current Tainan EMS before having to implement a major expansion. In addition, the Tainan EMS presently has two more ambulances than it needs and can meet the 90%-within-9-minutes standard even if one or two ambulances in one of the three sets of ambulance stations is unavailable for any reason. Because of the distribution of the Tainan EMS stations (Figure 1), we conjecture that the service areas of No. 7, 8, and 9 as well as those of No. 3, 7, and 14 overlap. To meet the 90%-within-9-minutes standard, the Tainan EMS can spare at most two ambulances for provisional events, except for 20:00–01:00, when it can spare at most three ambulances. If the number of out-ofroutine-duty ambulances exceeds the system’s capacity, EMS administrators must look for outsourced support to maintain an acceptable level of service quality. We also suggest dispatching strategies for provisional events to minimize the response times in routine daily emergencies. It is difficult to judge optimal deployment by experience or by the number of incidents to which the station responds. For example,
LIMITATIONS Our simulation model was developed without ad hoc parameters, and we found in the validation phase no evidence that explicitly indicated how well the model would generalize to other EMS systems. However, the model design, which includes only operations processes, would more than likely apply to other communities, as long as the probability distributions governing the simulation were reestimated with site-specific data to adjust for differences between communities. Besides, the simulated results are based on empirical data and may not be applicable after a long period of time because of changes in the system. Thus, the parameters of the model must be constantly updated and simulations should be repeated at regular intervals to account for changes in the system, such as the spatial distribution of calls, traffic patterns, call types, etc. We estimated the threshold for expanding the ambulance fleet, but when to begin expanding was not examined. However, managers can, using methods that forecast the changing trend of call volume,20,25 estimate when to expand by coordinating the time required to build a new station with the estimated increase in demand. Furthermore, we cannot model the effects on response time of increasing the number of ambulances, because the new locations are not determined. Although how to locate ambulances has been well studied,15–19 it is beyond the scope of the current study. CONCLUSIONS This study developed a discrete-event simulation model to evaluate the effect on response times for demand increases and for temporary decreases in ambulance availability. This model reasonably represented the operations of one EMS system. Our model is capable of estimating the exact threshold for expanding the ambulance fleet. When provisional events created temporary availability decreases, the accommodating strategies minimized the response times in routine daily emergencies. The more ambulances that are out of routine duty, the better the performance of the optimal strategies that are based on our model.
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