Proceedings of the 2021 Winter Simulation Conference
S. Kim, B. Feng, K. Smith, S. Masoud, Z. Zheng, C. Szabo, and M. Loper, eds.

SIMULATING NEW YORK CITY HOSPITAL LOAD BALANCING DURING COVID-19
Enrique Lelo de Larrea
Henry Lam
Elioth Sanabria
Jay Sethuraman

Sevin Mohammadi
Audrey Olivier
Andrew W. Smyth

Department of IE&OR
Columbia University
500 West 120th Street
New York, NY 10027, USA

Department of Civil Eng. & Eng. Mechanics
Columbia University
500 West 120th Street
New York, NY 10027, USA

Edward M. Dolan
Nicholas E. Johnson
Timothy R. Kepler
Afsan Quayyum
Kathleen S. Thomson
Bureau of Management Analysis and PlanningFire Department, City of New York
9 MetroTech Center
Brooklyn, NY 11201, USA

ABSTRACTIn most emergency medical services (EMS) systems, patients are transported by ambulance to the closest
most appropriate hospital. However, in extreme cases, such as the COVID-19 pandemic, this policy may
lead to hospital overloading, which can have detrimental effects on patients. To address this concern, we
propose an optimization-based, data-driven hospital load balancing approach. The approach finds a trade-off
between short transport times for patients that are not high acuity while avoiding hospital overloading. In
order to test the new rule, we build a simulation model, tailored for New York City’s EMS system. We
use historical EMS incident data from the worst weeks of the pandemic as a model input. Our simulation
indicates that 911 patient load balancing is beneficial to hospital occupancy rates and is a reasonable rulefor non-critical 911 patient transports. The load balancing rule has been recently implemented in New
York City’s EMS system.
1

INTRODUCTIONIn New York City (NYC), the emergency medical services (EMS) system is operated by the city’s Fire
Department (FDNY). Each year, the system receives approximately 1.5 million medical incident calls,
1.1 million of which result in a patient transport to a hospital. Given the scale of this operation, it isbeneficial for the FDNY to rely on quantitative tools, especially when considering the implementation ofpolicy changes. These tools currently include the tracking of several performance metrics via computational
dashboards, which get the necessary inputs from the ambulance EMS (computer-aided) dispatch system.

Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
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Bronx
Brooklyn
Manhattan
QueensStaten Island

Figure 1: (Left) The 5 boroughs of NYC. (Right) The ratio of number of patient transports to number ofavailable beds for NYC hospitals during the first wave of the COVID-19 pandemic in spring 2020. Larger
circles indicate more transports per bed.
This system logs historical details of EMS dispatch operations, and its current configuration is not integrated
with real-time data analytics or optimization. Therefore, for testing new policies, simulation and statisticalmethods need to be developed. In this paper, we describe an in-house EMS simulation model to help the
FDNY assess a change in the hospital assignment rule.
The desire for a change in the hospital assignment rule was driven by the COVID-19 pandemic. In
normal times, the system recommends to transport patients to the closest most appropriate hospital (closest
in terms of time, not distance). In this paper, we refer to this rule as the closest hospital rule. However,during the first wave of the pandemic, in March and April 2020, the EMS system experienced a spikein incident calls, and certain hospitals suffered a considerable overload due to both EMS transports andwalk-in patients. Two dynamics during this time contributed to an inefficiency for hospital capacity: (1) The
downtown core of Manhattan and Brooklyn was vacated of its normal working population and therefore
medical incident density reduced. (2) Outer-borough hospitals in residential areas became COVID-19
hotspots and quickly overwhelmed. To illustrate this overload, we plot in Figure 1 the number of EMS
patient transports per available hospital bed during one day in April 2020. Larger values are indicators ofpossible hospital overloading. This is the case for some hospitals in Queens and the Bronx. To address
this concern, the FDNY partnered with Columbia University in the late spring to explore an alternative
hospital assignment rule which takes into account both the closeness of the hospital and its capacity levelin preparation for a second pandemic surge. We refer to this approach as the load balancing rule and we
assess its impact using the simulation model. Similar results to the ones presented here assisted in the
FDNY’s decision to actually implement the load balancing rule in response to lessons learned during thespring 2020 pandemic surge.

Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
Smyth, and Thomson
The use of simulation in the EMS context has a long and rich history. See, for instance, the survey
article by Aboueljinane et al. (2013) for a detailed comparison of EMS simulation models in terms of
input data, assumptions, and performance metrics. Here, we mention only a few relevant examples. In
NYC, for example, Savas (1969) was one of the first to use simulation and a cost analysis to assess thebenefits of creating a satellite ambulance station in Brooklyn and dispersing ambulance standby locations
across the city (a practice that is still in use today). Ingolfsson et al. (2003) analyze, via simulation,
the impact of consolidating all of Edmonton’s ambulance stations into a “single start” station, especiallyin regards to ambulance utilization and ambulance coverage (defined as the proportion of incidents towhich an ambulance can respond in less than a specific target time). Henderson and Mason (2005) build asimulation model for the Auckland area (which was later expanded for Melbourne). This model features
a complex network-based travel time model, a visualization interface, and the use of trace simulation(historical data) for the incident call process. More recently, for the French Val-de-Marne department,
Aboueljinane et al. (2014) evaluate several strategies to improve ambulance coverage. These strategies
include adding ambulances in certain shifts, relocating ambulances to new stations, and reducing dispatch
times. They also consider a simulation optimization technique to find better ambulance standby locations
according to the time of day. Olave-Rojas and Nickel (2021) implement a hybrid simulation model for the
EMS system in an area of northern Germany. They use machine learning techniques to predict ambulance
average travel speeds.
The work previously mentioned focuses on improving the dispatch side of the EMS operations. This
relates to setting ambulance standby locations, selecting the staffing levels, and determining dispatch(ambulance selection) policies. Less attention has been given to hospital selection rules. As mentioned
by Aboueljinane et al. (2013), many authors assume the closest hospital rule in their models. That being
said, some simulation studies have been done on hospital selection. Wears and Winton (1993) study, inthe northeast Florida and southeast Georgia area, the effect of modifying (a) the necessary severity level
for a trauma patient to be transported to a specialized hospital (maybe bypassing closer hospitals), and(b) the helicopter dispatch policy. Unlike us, their approach does not take into account the capacity of
the hospitals. Wang et al. (2012) analyze patient mortality by comparing twelve hospital selection rulesin the aftermath of a single mass casualty incident in Pittsburgh. Some of these rules take into accountthe capacity of hospitals and the length of waiting queues. This analysis, however, focuses on a single
event, not on normal day-to-day operations. Finally, Aringhieri et al. (2018) test several dispatching,routing, and redeployment policies in a simulation scheme. Some of these modify the closest hospital ruleby considering waiting times at the hospital. These strategies resemble the current practice of hospital
redirection and hospital diversion policies present in NYC’s EMS system. Our load balancing rule canwork together with both of these existing practices; while redirection and diversion are reactive measures
to stabilize hospital capacities, load balancing can be seen as proactive, since it anticipates the behavior
during the next day, thereby creating an opportunity to mitigate real-time overload.
The rest of the paper is organized as follows. In Section 2, we describe the main elements of the
EMS simulation model. Section 3 presents the simulation-based comparison between the closest hospitaland load balancing rules. We conclude in Section 4. For more technical details, Appendix A includes theprocedure to calibrate the hospital discharge process and Appendix B has the optimization formulation of
the load balancing problem.
2

THE SIMULATION MODEL

The simulation model is a computational program which represents the main objects, agents, and dynamics
of an EMS system. Although this model is tailored to the structure, operations, and needs of NYC’s EMS
system, it can be modified accordingly to other cities or geographies. The code is written in Python, whichallows for user-friendly visualizations.

Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
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2.1 New York City Geography
For EMS operations, the FDNY divides NYC into nearly 2,400 geographical areas (polygons), also known
as atoms. Certain decisions, such as ambulance dispatch to an incident or hospital selection for a transport,
are made at the atom level. For instance, two incidents in the same atom, that require a hospital transportwith the same (critical) care category, will usually be assigned to the same hospital. Given the relevance ofatoms in the EMS system, the simulation model reads the atom geographical information (boundaries, anda user-defined centroid) from an external file. All the virtual objects of the model will be located within theboundaries established by this file (with the exception of some EMS hospitals which are located outside
NYC proper).
2.2 EMS Objects
The simulation model is built using the discrete event simulation methodology and has an object-oriented
structure. We define several classes representing real-life EMS objects. The objects’ attributes can be readby the model from external files. The following is a brief description of the main objects of the model.
Ambulances: also called units (we use either term indistinguishably from now on). There are two
types of ambulances: basic life support (BLS) and advanced life support (ALS). BLS units treat low
acuity incidents, while ALS units are designated to high acuity ones. For the most critical incidents, twoambulances (one BLS and one ALS) will be dispatched to the scene. Ambulances work in schedules (alsoknown as tours) which are either 8 or 12 hours long. Each unit has an assigned standby position, known
as the cross-street location (CSL). Between incident responses, the ambulance remains at its CSL, waiting
for the next incident. Once the tour ends, the ambulance goes back to its assigned station, where a crew
shift occurs and a new tour begins. Finally, the EMS system includes both ambulances managed either
directly by the FDNY (municipal units) or by the hospitals (voluntary units). Both municipal and voluntary
units are coordinated for dispatch and hospital transport through NYC’s 911 system via the EMS dispatch
system. The model contains the information and CSL of around 500 units distributed across the city.
Hospitals: In general, hospitals receive patients via walk-in, inter-facility transport, and 911 transport.
In this model however, we only keep track of 911 transports, which are the ones directly related to EMS
operations. Each hospital has a different capacity level which evolves through time. On one hand, newpatient transports arrive to the hospital, a percentage of which (currently set to 40%) will occupy a bed,
thus reducing the hospital capacity. On the other hand, we assume a random hospital discharge process
which simulates patient discharges, increasing the hospital capacity. See Section 2.4 for more details on
the discharge process. Even when the hospital is at zero capacity, the model allows for new patients to betransported to it. In that case, the hospital becomes overloaded and the hospital’s bed occupancy surpasses
100%. This behavior is reflective of real-life situations (see Figure 1). Additionally, not all hospitals havethe resources to accept all patient types. For instance, not all hospitals are equipped to treat severe burns
or stroke patients. All 911 receiving hospitals in NYC do accept General Emergency Department patients.
The model contains the information of more than 60 hospitals that are part of the EMS system (not all ofthem in NYC proper).
Ambulance Stations: These buildings house FDNY ambulances when they are out of service or between
tours. Municipal units are assigned to an ambulance station, whereas voluntary units use their hospital asbase. The model contains the information of 40 ambulance stations.
Incidents: After a medical incident is reported to the 911 system, unit(s) will be dispatched basedon the location and severity of the incident. Upon arrival, the crew spends time on-scene tending to andevaluating the patient. If the patient needs to go to a hospital, the unit will proceed with the transport. The
incident’s acuity or severity is assessed in two stages. An initial severity level (coded as 1–8), which inpractice is selected by a medically trained call taker utilizing computerized triage, determines the resource
type (BLS or ALS) and number of units that are dispatched. Once at the scene, the EMS crew will reassess
the situation and, if a hospital transort is necessary, assign a care category to the patient. The atom location

Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
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and care category enable the EMS dispatch system to provide a list of recommended hospitals in order ofclosest to furthest from the incident location. The model allows for the use of either historical or random
incidents. Feeding historical incidents into the model is particularly useful when the user wishes to “replay”
certain periods of time, while changing certain policies or model inputs. See Section 2.4 for more details
on how to simulate random incidents.
Dispatcher: This object represents a fictional “main” agent who acts as the system decision maker. The
dispatcher decides which unit(s) will be dispatched to an incoming incident and, if a transport is required,to which hospital the patient will be sent. In real life, the role of the dispatcher is a joint effort between
human dispatchers, the EMS dispatch system, and a set of policies.
EMS System: a parent object that contains all of the other EMS objects. It defines a simulation window,
sets up an initial configuration for the objects, and starts the simulation clock. During a simulation run, it
also gathers several metrics and statistics for posterior analysis.
2.3 Model Dynamics
The main dynamics of the model are standard and similar across different EMS systems; see Figure 2. We
summarize them as follows:
1. Incident arrival: A new incident is generated and its location and severity level are informed to thedispatcher.
2. Unit assignment (dispatch): According to the incident characteristics, the dispatcher assigns one ormore ambulances to it. As a general rule, low acuity incidents (severity levels 4–8) get assigned
to one BLS unit, higher acuity incidents (severity levels 2–3) get assigned to one ALS unit, andthe most severe incidents (severity level 1) get assigned to two units (one of each). The dispatcher
selects the necessary units according to shortest estimated time of arrival. If no units are available,
the incident is pushed into a waiting queue.
3. On-scene treatment: Once the unit arrives to the incident location, the care category of the patientis determined. If the patient does not require to be transported to a hospital, then the unit finishes
the job and returns to its CSL.
4. Hospital assignment: If the patient needs to go to a hospital, the dispatcher will proceed to assign anappropriate hospital. The default behavior is to follow the closest hospital rule. This is the hospitalwith the necessary equipment to treat the incident’s care category and with shortest estimated timeof arrival from the atom location. For an alternative to the closest hospital rule, we define in Section
3 a load balancing rule that takes into account the citywide hospital capacities.
5. At-hospital procedure: Once the ambulance arrives to the hospital, it delivers the patient and finishes
the current job. It travels back to its CSL and becomes available for a new incident assignment(if required, it can be assigned to a new job before reaching the CSL). The patient (with someprobability) gets admitted and occupies a bed (effectively lowering the hospital bed capacity byone). Some (random) time later, the patient is discharged and the hospital bed capacity increases
by one.

2.4 Sub-models
Besides the main dynamics discussed above, the simulation system requires additional sub-models thatwill inject the “randomness” into the model. These models address dynamics that are, in their own right,
interesting and complex. In this paper, we often opt to use simple models, but the framework is flexible
enough so that some or all of these sub-models can be expanded upon by the user to make them more
realistic.
Incident generation: While we can feed the simulation model with historical incidents (also known as
trace simulation), an incident generation model can also be used to sample random incidents. The current

Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
Smyth, and Thomson
Incident
arrives

Unit isdispatched

Dispatchtime

Unit arrives
at the scene

To-scene
time

Unit leaves
to hospital

On-scene
time

Unit arrives
at hospital

To-hospital
time

Unit becomes
available

At-hospital
time

Unit arrives
at CSL

To-CSLtime

Response time

Figure 2: A simplified scheme of the EMS main dynamics.
version of the model uses a simple spatio-temporal model which is uniform in space (in the NYC geography)
and follows a non-homogeneous Poisson process for the time component. After the time and location havebeen simulated, the care category is simulated independently according to user-defined probabilities.
Dispatch time: In practice, emergency medical incidents are not dispatched immediately. Callers to
the 911 system are briefly interviewed by a police call-taker and, if the emergency is of medical nature, arethen transferred to an EMS call taker. To model the dispatch time, defined as the length of time between
when the incident is first received by the EMS call taker to the moment when a unit is assigned to theincident, we assume a multi-server queue with 23 identical call takers and an exponentially-distributed callduration time with mean equal to 3 minutes. These numerical values are close to the ones observed by
the FDNY in practice. A more complex model would be a multi-server, priority-based queue. See, forinstance, the EMS call center model of van Buuren et al. (2015).
Travel time: The model constantly requires travel times between two locations; for instance, the time fora unit to get to an incident scene, the transport time to the hospital, or the time to go back to a station or CSL.
By default, the model generates travel times using the geodesic distance in miles between two locations
and by assuming a constant ambulance speed. This simple approach has been used before (Silva and Pinto2010). We assume that the speed is higher when the unit is on a job (i.e. going to a scene or transporting
patients to a hospital). In addition, for travel times between incident locations and hospitals, the model canuse an external travel time matrix. If this matrix is properly calibrated with ambulance traffic data, it can
improve the fidelity of the simulation model. An important difference between these two approaches is the
granularity: geodesic distances can be computed between any two pairs of latitude-longitude coordinates,
whereas the travel time matrix uses the incident and hospital atoms as input.
On-scene time: The time that an EMS crew takes to treat the patient can depend on the severity level
and care category, among other factors. We currently assume a uniform distribution.
At-hospital time: Once the ambulance arrives to the hospital, it has to wait some time while the hospital
personnel admits the patient into the emergency department. This time most likely depends on the saturationlevel of the hospital. We currently assume a uniform distribution.
Hospital patient discharge: Modeling the hospital-side of an EMS system is particularly challenging,
especially from the standpoint of the EMS system manager. Whereas EMS managers have access to datafrom their dispatchers and ambulances, having precise and frequent feedback from the hospitals is notalways a reality. In our setting, we are interested in modeling the rate at which hospitals discharge patients,
effectively increasing their capacity. We do so by considering an initial estimated capacity level and apure-death process for the discharge times. We calibrate this model to achieve a long-term “equilibrium”
of hospital occupancy during periods where the number of incidents and hospital transports are relatively
stable. See Appendix A for more details on the calibration of this model.
Out of service: Ambulances can go out of service for a variety of reasons (such as unit maintenance,
staffing issues or mechanical failures). This will partially or totally interrupt a scheduled tour, reducing

Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
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ambulance availability. We assume that units can randomly go out of service, with a certain probability,
only after finishing a hospital transport.
3

HOSPITAL LOAD BALANCING APPLICATION

We test our simulation model in the context of a new hospital load balancing rule. The standard rule forhospital assignment sends the patient to the closest appropriate hospital. Intuitively, this rule makes sense
because the patient is provided with fast medical care and the ambulance is able to finish the current job andreport back for duty as soon as possible. This approach, however, does not take into account the capacity
level of hospitals.
In the presence of critical external events, such as natural disasters or pandemics, the number of patienttransports to a hospital might increase drastically. This could lead to the hospital being overloaded which
in turn could compromise patient care. To avoid this scenario, we propose a new hospital assignment rule
which attempts to achieve load balancing across the hospital system.
The standard closest hospital rule can be summarized by a function that maps each atom to its closest
hospital. The output of the load balancing rule is another function, which still assigns hospitals to atoms,
but does so by considering both (a) the distance between atom and hospital and (b) the hospital capacity.
The procedure to determine the load balancing function is to solve an integer optimization problem which
minimizes the system-wide hospital transport time subject to hospital capacity constraints. See Appendix
B for the mathematical formulation of this problem and the details of its inputs.
Solving the load balancing problem requires as an input, among others, an estimate of the daily bed
capacity levels of each hospital in the system. In real life, the FDNY has access to daily updates on thesevalues, so we can run the load balancing optimization and update the hospital assignment function eachday (the availability and frequency of this data might be different for other cities).
3.1 Simulation Setup
We now conduct a simulation study to assess the impact of implementing the load balancing rule on the
EMS system. The simulation setup is as follows. We select a simulation window of 14 days. Since wehave access to real-life incident data, we feed into the model all the incidents received from March 25th,2020 and April 7th, 2020. This period corresponds to some of the weeks with highest incident counts
during the COVID-19 epidemic in NYC. For reference, during this period, the system received on average
more than 5,500 incidents per day. In contrast, during regular times, this quantity hovers around 4,000.
For simplicity, we assume that all incidents requiring a transport had a General Emergency Department
care category. In reality, this care category amounts to around 80% of total transports.
When computing travel times, we differentiate between hospital transport times and other travel times.
To increase accuracy, for hospital transport times we use a travel time matrix calibrated with historicalambulance data and network analysis. For the rest of the travel times, we use the geodesic model withconstant speed of 12 miles per hour for units on a job and 9 miles per hour for off-duty units.
To simulate the evolution of the hospitals’ capacity levels, we first assume a city-wide hospital occupancy
level of 80%. The total number of beds per hospital is estimated using publicly available data from the
New York State Department of Health. The patient discharge process is then calibrated using the proceduredescribed in Appendix A. We assume that in a normal day, there are approximately 2,900 hospital transports(this is assuming an estimated 1.5 million yearly incidents, 70% of which result in a transport).
3.2 Hospital Assignment Rule Comparison
We run two simulations using the same setup described in the previous section. The difference between
them is the hospital assignment rule that is assumed. The first simulation uses the closest-hospital rule.
The second one uses the load balancing rule, where the hospital assignment per atom is updated at the
beginning of each simulated day.

Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
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Closest Hospital

Load Balancing

Hospital bed occupancy (%)

300
250
200
150
100
50
0

50

100 150 200 250 300 350 0
Simulated time (hours)

50

100 150 200 250 300 350
Simulated time (hours)

Figure 3: Comparison of the simulated hospital capacities using both hospital assignment rules: closest
hospital (left) and load balancing (right). Each line corresponds to the proportion of occupied beds of a
NYC hospital. The dashed gray line corresponds to a 100% occupancy level. The data is illustrative only.
Several metrics are of interest when evaluating these rules. The first one is the hospital occupancy
level as a function of time. The load balancing rule should keep these levels in check, since it is designed
to deviate transports to hospitals which are not saturated. Figure 3 presents a side-by-side comparison ofboth rules, for one simulation run. When the closest hospital rule is followed (left panel of Figure 3),
several hospitals reach their maximum capacity level and become overloaded. This occurs to hospitals thatare close to a high number of incidents and do not have the capacity to meet the demand. In contrast,
when the load balancing rule is in place, the transports to hospitals are better distributed (right panel of
Figure 3). In this scenario there are still hospitals that might reach their maximum capacity levels andmight even briefly surpass them. This is due to the randomness of the hospital discharge model and thefact that our incident prediction for the next day is not exact (see Appendix B). Even with these factors
into consideration, the load balancing rule is able to correctly keep hospital capacity overloads in check.
The simulated values in Figure 3 are for illustration purposes only and should not be taken as reflectiveof a real-life situation. In particular, we note that such high occupancy levels would be prevented inreal-time via hospital redirection, diversion, and internal load balancing policies. A redirection occurs
when the EMS dispatch system detects that a certain hospital shows signs of overloading. For instance, aredirection might be triggered when a certain number of units spend too long stationed in the hospital’s
emergency department. Similarly, a hospital staff might request a hospital diversion directly to the EMS
system, if they consider they are temporarily unable to receive more patients. Finally, a hospital is capableof load balancing itself via inter-facility transports. In this simulation scenario, we do not consider suchreal-time policies. That being said, in practice, load balancing would work jointly with the other policies
to better load balance the hospital system.
Recall that, with the load balancing rule, there is a trade-off between hospital transport time and hospital
capacities. Indeed, if ambulances are dispatched to hospitals with capacity, that may not be the closest ones,
the overall transport times will inevitably increase. In addition, sending units to farther away hospitals may
have an indirect impact on the average incident response time (defined as the time between the incident
arrival and the unit arrival at the scene; see Figure 2). If the ambulances are transporting patients away

Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
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Table 1: Summary statistics for the hospital transport and unit response times. All the times are in minutes.
Hospital Transport Time
Closest Hospital Load Balancing
mean
std
median95-percentile
99-percentile

9.2
3.0
8.9
14.3
17.8

10.8
4.4
9.9
19.1
23.4

Response Time
Closest Hospital

Load Balancing

6.6
4.6
5.5
15.0
21.9

6.7
4.7
5.6
15.4
22.2

from their usual area, they might not be able to respond as fast to new incidents. We explore the impact of
imposing the load balancing rule on both metrics (hospital transport and response times) in Table 1. The
statistics presented are based on one simulation run for each rule.
In this simulation, the average transport time increases by 1.6 minutes, when we use the load balancing
rule. This increase is by no means negligible, but for low acuity patients, it should not be life-threatening.
We do see a deterioration in the tail of the distribution; the 99th percentile increases considerably from
17.8 to 23.4 minutes. This reinforces the belief that load balancing should only be used for non-criticaltransports. If an upper bound on the admissible transport times is required (say, for instance 30 or 40
minutes), it can be imposed by modifying the load balancing optimization problem. On the other hand,ambulance response times were not so adversely affected by the load balancing rule. The mean responsetime increased by less than half a minute and, although the response times do tend to be larger, the 99th
percentile only increased in 0.3 minutes. It should be noted that the underlying travel time model used tocompute response times is too simplistic (it is based on the geodesic distance and a constant unit speed), andtherefore, the numerical results might not reflect the reality. In this study, however, we are more interested
in the difference in the shapes of the distributions and not so much in particular numerical values.
3.3 Main Limitations
The main limitation of the simulation model comes from the simplifying assumptions made on the hospitalpatient arrival and discharge processes. While they are only part of our entire model, these processes are
incredibly complex and deserve their own individual studies. Another limitation comes from the simple
geodesic travel time model. We corrected the travel times between locations and hospitals using an external
matrix that was calibrated with ambulance traffic data and network analysis, but we did not make theextension to general travel times between two arbitrary locations. Finally, we opted for simple sub-models
for dispatch, on-scene, and at-hospital times.
The primary objective of using the simulation model to assess the new load balancing rule was to checkthe high-level dynamics and detect any potential issues. Due to the urgency of real-life implementation, we
have not presented a formal validation of the model. To do the latter task, we first would need to refine all
the sub-models mentioned above and calibrate them using real data. All these improvements are importantongoing work.
4

CONCLUSION

The FDNY requires analytical tools to better understand the impact of potential policy changes to their
EMS system. To this goal, we implemented an EMS simulation model for NYC. The model captures themain objects and dynamics of the system. Like similar models in the literature, our model can use trace
simulation (i.e. using historical data) for the incident arrival process. This allows the user to test several
scenarios and changes in policy, subject to incidents observed in the past.
We used the simulation model to evaluate a new hospital load balancing approach, which differs fromthe standard and widely-used rule of sending patients to the closest hospital. This approach minimizes theLelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
Smyth, and Thomson
overall patient transport time, but also takes into account the capacity levels of the hospitals. Assuming
a particular patient discharge model, we showed in our simulation that the load balancing rule effectively
keeps the individual hospital occupancy rates below the at-capacity level. In contrast, using load balancing
would result in longer transport times (unit response times were not so clearly affected). This suggest thatload balancing might be beneficial for the EMS system as a whole, but it should only be used for patientsthat are not high acuity.
Simulation is an essential tool when considering the impact of a new policy, especially in critical areas
such as EMS. It helps decision makers to quantify and visualize several scenarios, before making changes inreal life. Our analysis assisted in the FDNY’s decision to implement the load balancing rule in response to
the COVID-19 pandemic. The new rule works in addition to preexisting hospital redirection and diversion
policies and will hopefully serve as an extra safeguard for balancing system-wide hospital capacity levels.
ACKNOWLEDGMENTS
We are grateful to the members of the Bureau of Management Analysis and Planning at the FDNY for the
many enlightening conversations and for providing their expert knowledge and data on the EMS system.
We appreciate the guidance of Commissioner Elizabeth Cascio, Chief Lilian Bonsignore, Chief Jonathan
Pistilli, Dr. David Prezant, James Saunders, and Matt Talty. We thank Jason Qian for implementing thefirst version of the simulation model and Yuanlu Bai and Haoxian Chen for providing help on the loadbalancing optimization. We gratefully acknowledge support from Google and the Tides Foundation under
the grant “EMS Resource Deployment Modeling” and the Columbia University Urban Technology Pilot
Award.
A HOSPITAL DISCHARGE PROCESS CALIBRATION
Consider a hospital in the EMS system. We model the discharge processes one day at a time (we measure
the time in hours). Let X(t), 0 ≤ t ≤ 24, denote the number of patients occupying a bed during a given
time of the day. For a moment, we assume that there are no incoming patients that day. Then, we canmodel X(·) as a (decreasing) pure-death process with rate µ. That is, given a patient occupancy of X(t) attime t, the time of the next discharge is distributed as an exponential random variable with rate X(t)µ. In
other words, the discharge rate is proportional to the hospital occupancy level. It is a well-known propertyof this process that, at the end of the day, E[X(24) | X(0)] = exp(−24µ)X(0).
We now propose a simple procedure to calibrate µ. Let c be the total capacity of the hospital and let
a be the number of patients that are expected to arrive to the hospital during an average (baseline) day.
Normally, patients arrive throughout the day, but for this exercise we assume that they arrive at the endof the day. The calibration is based on a bed occupancy equilibrium condition. Simply put, on a regularday, the number of patient discharges roughly matches the number of patient arrivals, and the hospital
occupancy level remains constant. Assuming a target occupancy rate of ρ ∈ (0, 1), we have, at the beginningof the day X(0) = ρc, and at the end of the day X(24) = exp(−24µ)ρc + a. In equilibrium, we then haveX(24) = X(0). Solving for the rate µ, we get µ = −(1/24) log(1 − a/(ρc)), as long as a < ρc. With thiscalibration, we would expect the bed occupancy to remain more or less constant throughout the days, aslong as the number of daily arrivals remains close to a. During stress periods when the number of arrivals
is much larger than a, we would expect the bed occupancy to increase.
B LOAD BALANCING OPTIMIZATION PROBLEM
Given an EMS system with m atoms and n hospitals, our objective is to assign one hospital to each atomby minimizing the total (estimated) travel time subject to the hospital capacity constraints. The decisionvariables can be seen as indicators xi j ∈ {0, 1}, where xi j = 1 if and only if hospital j is assigned to atom i.
At the beginning of each simulated day, we gather the following inputs:

Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
Smyth, and Thomson
•

•
•
•

The predicted daily hospital transports originating from atom i for the next day, denoted as fi ∈ Z+ .
In this application, we naively estimate fi as the running average of transports of the last d = 3
days.
The current bed availability at hospital j ∈ {1, . . . , n}, denoted as c j ∈ Z. A negative c j indicatesthat the hospital is overwhelmed and that it currently has −c j > 0 patients above its total capacity.
The estimated time of arrival from atom i to hospital j, denoted as Ti j ∈ R+ .
The expected proportion of transported patients that are admitted to the hospital, denoted as δ ∈ (0, 1).
We currently use δ = 0.4.

With these inputs, we solve the following integer optimization problem to update the hospital assignment
rule:mminimizex

subject ton∑ ∑ fi Ti j xi j

i=1 j=1m∑ δ fi xi j ≤ max(c j , 0), for all j,

i=1n∑ xi j = 1, for all i,

j=1

xi j ∈ {0, 1}, for all i, j.
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Lelo de Larrea, Dolan, Johnson, Kepler, Lam, Mohammadi, Olivier, Quayyum, Sanabria, Sethuraman,
Smyth, and Thomson
AUTHOR BIOGRAPHIESENRIQUE LELO DE LARREA is a Ph.D. student in Operations Research at Columbia University. Before joining Columbia,
he worked as a credit risk analyst at BBVA Mexico. He holds a double Bachelor’s degree in Applied Mathematics and Actuarial
Science from ITAM and a Master’s degree in Operations Research from Columbia. His research interests include stochastic
simulation, applied probability, and financial engineering. His email address is enrique.lelodelarrea@columbia.edu.
EDWARD DOLAN is the Deputy Commissioner for Strategic Initiatives and Policy at the Fire Department of the City of
New York (FDNY). His email address is edward.dolan@fdny.nyc.gov.
NICHOLAS JOHNSON is the Director of Operations Research for the FDNY. Prior to joining the FDNY, he was a Postdoctoral
Associate at NYU’s Marron Institute of Urban Management where his research focused on modeling real-time urban populations
using mobility data. He earned a Ph.D. in Urban Science from the University of Warwick in the United Kingdom and holds a
Master’s degree from NYU’s Interactive Telecommunications Program where he used physical computing and interaction design to
explore the impact and pervasiveness of waste streams in urban environments. His email address is nicholas.johnson@fdny.nyc.gov.
TIMOTHY KEPLER is the Director of Data Quality for the FDNY. He has a Master’s degree in Public Administration fromthe City University of New York at Baruch College. For the past five years he has worked as an analyst for the FDNY, working
intensively with data from computer aided dispatch systems. His email address is timothy.kepler@fdny.nyc.gov.
HENRY LAM is an Associate Professor in the Department of Industrial Engineering and Operations Research at Columbia
University. His research focuses on simulation and optimization under uncertainty. His email address is khl2114@columbia.edu.
SEVIN MOHAMMADI is currently pursuing a Ph.D. degree in Civil Engineering and Engineering Mechanics at Columbia
University. Prior to her current position, she obtained a Master’s degree in Civil Engineering, majoring in Transportation,from the University of Tennessee Knoxville. Her research interests include data-driven and machine learning approaches inTransportation Engineering. Her email address is sm4894@columbia.edu.
AUDREY OLIVIER is currently an Associate Research Scientist in Civil Engineering at Columbia University, and will bejoining the University of Southern California as an Assistant Professor in fall 2021. She holds a Ph.D. in Civil Engineeringand Engineering Mechanics from Columbia University and a Diplôme d’Ingénieur from Ecole Centrale de Nantes, France. Her
research interests revolve around probabilistic data analytics and physics-based modeling for civil engineering applications. Her
email address is audreyol@usc.edu.
AFSAN QUAYYUM is a Data Scientist at the Bureau of Management Analysis and Planning at the FDNY. He has an M.S.
in Mathematics and a B.S. in Mathematics with a minor concentration in Applied Physics from New York University. His
work focuses on implementing statistical learning techniques for inference and prediction to help EMS and Fire Operations.
His email address is afsan.quayyum@fdny.nyc.gov.
ELIOTH SANABRIA is a Ph.D. student in Operations Research at Columbia University. He is primarily interested in theinterplay between simulation, machine learning and optimization from a probabilistic point of view, as well as their broad
applications in healthcare to improve patients outcomes. His email address is m.elioth@columbia.edu.
JAY SETHURAMAN is a Professor of Industrial Engineering and Operations Research at Columbia University. Currently, heserves as the chair of the IEOR department at Columbia. His research interests are in discrete optimization and applications,game theory, mechanism design, and applied probability. His email address is jay@ieor.columbia.edu.
ANDREW SMYTH is the Carleton Professor of Civil Engineering & Engineering Mechanics and also serves as the Co-Chair of
the Smart Cities Center of the Data Science Institute at Columbia University. His research focuses on infrastructure monitoring,
dynamic system identification and modeling. He received his Ph.D. in Civil Engineering from the University of Southern
California as well as an M.S. in Electrical Engineering, an M.S. from Rice University, and a Sc.B. and A.B. from Brown
University. His email address is smyth@civil.columbia.edu.
KATHLEEN THOMSON is the Assistant Commissioner for the Bureau of Management Analysis and Planning at the FDNY.
Her email address is kat.thomson@fdny.nyc.gov.