A nested-compliance table policy for emergency medical service systems under relocation

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A nested-compliance table policy For Emergency Medical Service Systems under Relocation Kanchala Sudtachat, Maria e. Mayorga, Laura a. Mclay

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Received date: 31 July 2014 Accepted date: 1 June 2015 Cite this article as: Kanchala Sudtachat, Maria e. Mayorga, Laura a. Mclay, A nested-compliance table policy For Emergency Medical Service Systems under Relocation, Omega, http://dx.doi.org/10.1016/j.omega.2015.06.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A Nested-Compliance Table Policy for Emergency Medical Service Systems under Relocation KANCHALA SUDTACHAT1, MARIA E. MAYORGA2,* and LAURA A. McLAY3 School of Manufacturing Engineering, Institute of Engineering, Suranaree University of Technology, 111 University Avenue, Muang, Nakhon Ratchasima, 30000, Thailand 2* Department of Industrial and Systems Engineering, North Carolina State University, Campus Box 7906, Raleigh, NC, USA 3 Department of Industrial and Systems Engineering, University of Wisconsin-Madison, Madison, WI, USA E-mail: [email protected][Sudtachat]; [email protected] [Mayorga]; [email protected] [McLay] 1

The goal of Emergency Medical Service (EMS) systems is to provide rapid response to emergency calls in order to save lives. This paper proposes a relocation strategy to improve the performance of EMS systems. In practice, EMS systems often use a compliance table to relocate ambulances. A compliance table specifies ambulance base stations as a function of the state of the system. We consider a nested-compliance table, which restricts the number of relocations that can occur simultaneously. We formulate the nested-compliance table model as an integer programming model in order to maximize expected coverage. We determine an optimal nestedcompliance table policy using steady state probabilities of a Markov chain model with relocation as input parameters. These parameter approximations are independent of the exact compliance table used. We assume that there is a single type of medical unit, single call priority, and no patient queue. We validate the model by applying the nested-compliance table policies in a simulated system using a real-world data. The numerical results show the benefit of our model over a static policy based on the adjusted maximum expected covering location problem (AMEXCLP). Keywords: Emergency medical service, Relocation, Nested-compliance table, Markov queuing model

1. Introduction The goal of emergency medical service (EMS) systems is to save the lives of emergency patients. The potential for improving performance of EMS systems is directly related to reducing response time, which is in turn related to increasing coverage, where coverage is defined as the proportion of calls that can be responded to within a given time standard (e.g. 9 minutes).

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Decisions regarding ambulance location strategies can be used to improve expected coverage. The ambulance location problem refers to the assignment of a limited number of ambulances to maximize coverage, given that the system has a fixed number of potential locations, and a demand zone is considered to be covered when an ambulance is located within a predetermined time standard. However, in reality the arrival of calls is stochastic and dynamic. Dynamic ambulance relocation can improve the performance of systems in situations with fluctuating arrivals, and as a result, the literature shows a drastic increase in the number of dynamic strategies used (see Alanis et al. [1]). One type of dynamic strategy for relocating ambulances is to use a compliance table. A compliance table indicates the possible open stations in relation to the number of available ambulances. That is, a compliance table shows where ambulances should be located when there are a certain number of ambulances available. Such a policy allows for EMS systems to respond to the dynamic nature of the problem, but can be calculated in advance, thus not requiring optimization in real time. In this paper, we determine the nested-compliance table that maximizes coverage for dynamic strategies in EMS systems.

We use coverage as our objective for several reasons. First, it is the most widely used measure in practice and many EMS systems base their performance on this metric, Fitch [2]. Secondly, maximizing expected coverage has been shown to result in response times that are comparable to those obtained when minimizing response time, Toro-Diaz et al. [3]. Studies have shown that shorter response times lead to an increased probability of patient survival, Larsen et al. [4], and Erkut et al. [5], and that coverage can be used as a proxy for patient survival, McLay and Mayorga [6]. Furthermore shorter response times are associated with reduced complications, especially for the most critical patients, such as those suffering from heart attack, stroke or traumatic injury, Stiell et al. [7].

Compliance table policies accommodate dynamic ambulance relocation by allowing for realtime movement of idle ambulances to new locations, where new locations are informed by the compliance table. To understand what a compliance table policy entails, consider the example in Table 1 below; in this case, when only one ambulance is available, it is located at station A. In scenario 1, once a second ambulance becomes free, it will go to station C, and the first ambulance will stay at station A. In scenario 2, once the second unit becomes available, ambulance 1

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will need to relocate so that stations B and C can be open. Scenario 1 maintains what we refer to as a nested structure, which we will discuss later in further detail. One way to operationalize a relocation policy that is not too computationally intensive is via a compliance table policy. Under such a compliance table policy, vehicles are relocated as calls come in and vehicles become available. While compliance table policies are pre-determined and thus are easy to implement in real time, a challenge is that it may be difficult to identify the best compliance table. Table 1: Sample compliance table #of available units 1 2 3

Nested open stations (Scenario 1) A A, C A, B, C

Non-nested open stations (Scenario 2) A B, C A, B, C

To assess the best compliance tables, we consider only the possible compliance tables in a set of nested-compliance tables, where possible compliance tables are telescoping as in Scenario 1 in Table 1. The benefit of nested policies1 is that only one ambulance, which is already on the move, is relocated, thus avoiding unnecessarily moving other ambulances that can be disruptive to service providers and result in more accidents. We consider a single type of ambulance (paramedic units) and a single type of call priority when determining the best compliance table policy. We formulate an integer programming model to maximize the expected coverage with respect to the best compliance table. Real world data is used to validate the models. In this study we: •

Modify a Markov chain model based on Alanis et al. [1] that considers the steady-state probabilities of EMS dynamics to approximate coverage.

•

Propose the nested-compliance table formulation as an integer programming model to determine the maximum expected coverage.

•

Show, through the numerical results, how the solutions from our nested-compliance table formulation compare with a static (non-relocation) policy based on the adjusted maximum expected covering location problem (AMEXCLP) of Batta et al. [8] in real world problems.

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Finding good nested policies has also been shown to be important in other applications as well. For example, in portfolio optimization, it is intuitive to the decision-maker if, as the budget increases, the solution includes the portfolio at lower budget amounts. Solutions that do not follow this trend are seen as “volatile” to managers, as discussed in Koç et al., 2009 [34]

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The rest of the paper is organized as follows. In Section 2 we review the related work in EMS relevant to the nested-compliance table problem. Section 3 presents a description of EMS systems with relocation strategies and explains how to implement a nested-compliance table policy. Section 4 presents the application of a Markov chain model with relocation. Section 5 presents an integer programming approach to obtaining the optimal nested-compliance table with relocation. Section 6 presents the efficiency of the nested-compliance table solutions. Finally, Section 7 presents conclusions and a discussion of future work.

2. Literature Review The literature related to EMS vehicles is extensive. In essence, a compliance table model deals with ambulance location decisions. After a brief discussion of seminal works dealing with ambulance location problems, we limit our discussion to works related to ambulance relocation problems.

The early works related to location decisions were that of Church and ReVelle[9]

who introduced the maximal covering location problem (MCLP), which was extended to the Maximum Expected Coverage Location Problem (MEXCLP) by Daskin [10], and later refined by Batta et al. [8]. The latter models accounted for “busy” ambulances in the system. The server “busy” probabilities are estimated using the hypercube spatial queuing model (see Larson [11] and Larson [12]). Jarvis [13] used the hypercube model to provide an approximation of server workloads. In these models however, vehicles are assumed to return to their base, or home, stations once service is completed. The relocation of emergency medical service (EMS) systems is a possible strategy to increase coverage and improve patient outcomes. In relocation models dispatched ambulances might return back to a new station different from their originating station and ambulances may relocate to be better prepared for the next call. Early work of the relocation problem began with the formulation of an exact model; Kolesar and Walker [14] studied the dynamic relocation of fire resources. Later, Berman [15] introduced the optimum repositioning model for two distinguishable units. Gendreau et al. [16] introduced the dynamic double standard model (DDSM) which solves the repositioning problem based on the objective of the double standard model (DSM). Brotcorne et al. [17] discuss the evolution of ambulance location and relocation models. They reviewed previous studies of ambulance location and relocation models and classified them into two categories; deterministic and probabilistic models. Sathe and Miller-Hooks [18] and

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Nair and Miller-Hooks [19] examine a mixed integer linear program with maximum secondary coverage and minimum cost for location and relocation problem. In other work, Gendreau et al. [20] studied the maximum expected coverage problem under relocation; the relocation formulation is presented as an integer linear program. They used a Binomial distribution to approximate the probability of having any k available ambulances. Anderson and Varbrand [21] propose decision tools for dispatching ambulances under dynamic ambulance relocation. In addition, Rajagopalan et al. [22] consider the covering location model for dynamic redeployment. Schmid and Doerner [23] formulate a mixed integer programming model to maximize total coverage for relocation with time-dependent travel times. Yue et al. [24] consider the allocation and dynamic redeployment by using simulation-based approach. Majzoubi et al. [25] propose a dispatching ambulance problem with ambulance relocation that minimizes the total costs. Most previous works focused on integer programming and heuristic models. Others use Markov Decision Process (MDP) models to analyze the EMS systems and more recently, approximate dynamic programming (ADP) has been employed. Berman [26] considers repositioning emergency vehicles using an MDP. Maxwell et al. [27, 28] proposed an approximate dynamic programming model for redeploying ambulances to maximize the number of covered calls. Schmid [29] proposes an approximate dynamic programming approach for a combined relocation and dispatching problem. Alanis et al. [1] analyze Markov chain models of EMS systems to calculate their performance under repositioning. Their model focuses on analyzing the performance of a fixed compliance table policy, not on finding the best compliance table. In contrast, we model a system independent of specific ambulance locations and determine the steady-state probabilities and approximate the expected coverage. We apply the output of steady-state probabilities as input parameters to our integer programming model to determine the best of nested-compliance table with a single type of ambulance and a single type of call priority. To our knowledge, this is the first paper to provide an optimization model for a compliance table policy.

3. EMS System with Nested-Compliance Policy In this section, we discuss EMS systems that operate under relocation policies based on a nestedcompliance table. We consider EMS systems with a single unit type (paramedic) and a single type of call and that there is one ambulance located at each station. These assumptions are com-

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monly used in widely accepted EMS models, including AMEXCLP [8] and that of Alanis et al. [1], which are the foundation to this work. Furthermore they are close to reality when systems have all advanced life support vehicles (ALS) and treat all real emergency calls equally (which is often the case). Furthermore, allowing for multiple vehicles at stations does not tend to increase coverage, and when it does, may result in inequitable (from the point of the customer) solutions, Chanta et al. [30], and McLay et al. [31]. We also assume that there is a zero-length patient queue. This requires discussion. First, in reality if all ambulances are busy when a call arrives to the system, other emergency services, such as fire engines or ambulances from neighboring counties, will be dispatched to a patient. In the model, the objective is to maximize coverage, since a patient being in queue would have wait for the next available ambulance, the response time for these patients may be longer than the response time threshold, these patients add no value to the objective function. With the zero-queue assumption, we also have no value to objective with the probability that all ambulances are busy when a patient arrives to the system. We acknowledge that a non-zero queue would increase the busy probability of the ambulances; in this case, the optimal policy is likely to keep the ambulances in the busiest zones to respond patients in queue as much as possible. However, based on simulation experiments the zero queue assumption does not significantly affect the long-run expected coverage using real-world data; since the likelihood that an arriving patient will find all ambulances busy is negligible. When a call arrives at the dispatch center, the dispatch planners assign the closest ambulance in response to the call. In the case when all ambulances in the system are busy, the call will transfer to another dispatch center. As stated in Section 1, we restrict our attention to nestedcompliance tables. Table 2 provides an example of a nested-compliance table. There are two events that could result in a repositioning move: call arrivals and call completions. A more detailed explanation of how to interpret compliance tables is provided at the end of this section. • Call arrivals: When a call from zone i arrives, the closest ambulance responds to the call. If the closest is busy, the second closest responds to the call, and so on. • Relocation via call arrivals: When the number of busy ambulances increases, the dispatchers consider which ambulance to move (if any) to another station, since the dispatched ambulance may have left some critical areas uncovered. For example, based on Table 2, suppose a call arrives to a system with 10 available ambulances, and the ambulance at station 2 responds to the call. The system state changes from 10 available ambulances to 9 available ambulances. 6

The located ambulance at station 12 moves to replace the ambulance at station 2 in the new system state. • Service time: We define the service time as the time between the EMS staff arriving on-scene and completing service, including providing transportation to a hospital if needed. • Relocation via call completions: After the EMS staff has completed service to patients the ambulance may return to any open station, not necessarily its previous station. The dispatchers consider which station the newly available ambulance should be located at. The system state changes to increase number of available ambulances. During this time the ambulance is free, but cannot be assigned to a new call (this assumption is later relaxed in our simulation model). If a call arrives, it will transfer to the next closest ambulance. For example, in Table 2, while the system state is 8 available ambulances, the dispatched ambulance travels back to station 11. The system state changes from 8 to 9 available ambulances.

Other important times in the EMS system include response time and travel time between stations. • Response time: the travel time between the stations of a dispatched ambulance to the scene of the incident. • Travel time between stations: the travel time between the original stations to the new stations when system states are changed. Table 2 provides an example of a solution of a nested-compliance table that might be used in practice. Each row in the nested compliance table represents the number of available ambulances and the column represents specific stations. To read the table, given a current number of available ambulances, the goal is to have ambulances located at stations with a “1” in that row. For example, based on Table 2, with one available ambulance, it should be located at station 9. Now suppose one ambulance just changed status from busy to available, so that I now have two available ambulances. Based on Table 2, the available ambulance should go to station 2. Thus, in a nested compliance table, as ambulances become available, they will go to the one station that is currently not covered. As ambulances become busy, relocations may also be required. In another example, suppose the current state is two available ambulances (at stations 2 and 9) and a call comes in and the dispatcher sends the ambulance at station 9 to respond a call. In order to follow the compliance table, the dispatcher should relocate the ambulance at station 2 to move to station

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9. Note that the nested-compliance table structure requires moving at most one ambulance at a time. Single moves avoid unnecessarily moving other ambulances, which can be disruptive to service providers, and avoid confusion for dispatchers in deciding which ambulances to move to the new stations.

Table 2: The nested-compliance table, shows where ambulances should be located for a given number of available ambulances # of available ambulances 10 9 8 7 6 5 4 3 2 1

Stations 1

2 1 1 1 1 1 1 1 1 1

3 1 1 1 1 1 1

4 1 1 1

5 1 1 1 1

6 1 1 1 1 1 1 1 1

7

8 1 1 1 1 1

9 1 1 1 1 1 1 1 1 1 1

10

11 1 1

12 1

13

14

15

16 1 1 1 1 1 1 1

4. The Application of a Markov Chain Model with Relocation for EMS System The Markov chain model with relocation, proposed by Alanis et al. [1] is a powerful tool that could be used to approximate the steady-state probability and other performance measures of EMS systems for a fixed compliance table policy, where the exact location of ambulances are known for each state. Therefore, if we know the distribution of response time which depends on exact ambulance locations, we could estimate the expected coverage. However, the exact expected coverage cannot practically be used in an optimization framework, since the resulting expression would be a non-convex, non-linear formulation. In this paper, we develop the nestedcompliance table optimization model that uses the output of the Markov chain model with relocation as steady-state probabilities for the input parameters. The steady-state probabilities can be approximated independent of the exact compliance table policy. Consequently, the steady-state

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probabilities will be input parameters to the nested-compliance table formulation. The objective is to determine the maximum expected coverage. We note that we use a binary notion of coverage in the sense that calls are said to be covered if the expected response time is within a prespecified amount of time; that is, we calculate the expected coverage not considering the variability of response time. Figure 1 shows a flow process diagram depicting the approach taken in our compliance table model. The dashed boxes indicate input and output parameters.

Figure 1: The process flow diagram of the procedure used in the nested-compliance table model To assess the performance of a specific compliance table in terms of coverage, we need to approximate the steady-state probabilities of an EMS system. To approximate the steady-state probabilities, we build upon a Markov chain model with relocation developed by Alanis et al. [1]. They formulate the model as a finite, continuous time Markov chain according to a compliance table policy. The state variable V(t) denoted the number of busy ambulances at time t, with V(t) = (0, 1, 2, …, K), and the state variable C(t) denoted the status of the EMS system, whether the system was in compliance (C(t) = 1) or out of compliance (C(t) = 0) at time t. In compliance refers to all available ambulances being at their assigned stations. On the other hand, out of compliance refers to the status that not all available ambulances are at their assigned stations; that is, an ambulance is en-route to its home station. They assume that the arrival process of calls to the EMS system is Poisson, all service times are exponentially distributed, and the system operates as a zero-length queue. When calls arrive to the system when all ambulances are

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busy, they would transfer to another system. They assume that the status of the EMS system can change from being in compliance to being out of compliance when either a call arrives or an ambulance completes service; this is because either of these events will cause an ambulance to be relocated, and thus may not be at their assigned stations. They consider the relocation model where the state transitions occur due to one of three event types: (1) call arrival, (2) call completion and (3) a moving ambulance reaching compliance. Suppose we are in state (v, 1), in compliance with v busy ambulances. The transition to reach out of compliance state (v+1, 0) occurs via a call arrival, where λ is call arrival rate. The transition to reach out of compliance state (v-1, 0) occurs via a call completion with rate vµ1, where µ1 is the completion rate given that the system is in compliance state. Similarly, suppose we are in state (v, 0), out of compliance with v busy ambulances. The transition to reach out of compliance state (v+1, 0) occurs via call arrival. The transition to reach out of compliance state (v-1, 0) occurs via a call completion with rate vµv, 0

where µv, 0 is the call completion rate given that system is out of compliance state. When the

system is out of compliance in state (v, 0), the transition rate γ results in a transition to state (v, 1), in compliance state. While the ambulance moves to a new home station, the ambulance cannot be dispatched to respond to a new call. When a call arrives to the system during this time, it would transfer to the next closest ambulance. Table 3 shows all notations used in this paper, including the nested-compliance table model under relocation and associated approximation procedures discussed below. In this paper, we formulate a nested-compliance table model under the same assumptions as those above. However, we approximate the transition rates not according to the exact nestedcompliance table policy, but rather based only on the number of busy ambulances. If the approximation of transition rates is known and not according to an exact nested-compliance table policy, the nested-compliance table model can be solved as an integer programming model. Otherwise, we have to consider a meta-heuristic or enumerate all solutions which requires long computational running time. Although, the nested-compliance table needs to be calculated once, an integer programming model that offers a short computer running time is desired in case system parameters change (e.g. large shifts in demand). Furthermore this approximation avoids state space issues; without it the size of the problem increases exponentially in the number of demand zones. This would require more time and memory for exact calculations, such that for large problems it quickly becomes infeasible to calculate without large-memory computing. When

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estimating the transition rates, we relax the assumption of approximating parameters such that our approximation of transition rates γ, µ1 and µv,0 are independent of the exact nestedcompliance table. The approximations of transition rates are calculated based on the total covered arrival intensity for each station, as discussed in Section 4.1. Figure 2 illustrates the transition diagram of the Markov chain model with relocation for K= 5 ambulances.

0,1 λ

γ

0,0

λ

µ1

1,0 µ1, 0

λ

2µ1

λ

2,0

4,1

3,1

γ

γ λ

2,1

1,1

λ

3µ1

γ λ

3,0

2µ 2, 0

3µ3, 0

λ

4µ1

γ λ

4,0

4µ4, 0

λ

5,0

5µ5, 0

Figure 2: The modified state transition diagram of EMS systems with relocation We describe the process related to EMS systems with the state-transition network in Figure 2. The system starts with all idle ambulances at assigned stations in state (0, 1). As a call arrives, the state reaches out of compliance immediately resulting in increasing the number of busy ambulances to 1, a transition to state (1, 0) and potentially relocates an ambulance to replace the one at the station of the dispatched ambulance. Suppose no new call arrives in the meantime and we relocate the ambulance. If the ambulance is able to relocate before the dispatched ambulance finishes on-scene treatment of a patient, the system reaches compliance at rate γ, resulting in a transition to state (1, 1). Similarly, a system is in state (1, 1) where the ambulance is on-scene and completes service to the patient results in a transition to state (0, 0). After the call completion, the ambulance travels back to a home station (possibly a new home station) with relocation rate γ resulting in a transition to state (0, 1). However, a new call when in state (1, 0) or (1, 1) would immediately result in out of compliance and a transition to state (2, 0). Therefore, we would need to relocate an ambulance to a new home station in order to reach a compliance state.

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Table 3: Notation the nested-compliance table model under relocation and approximation procedure Notation Description Parameters n number of demand zones K total number of ambulances in the EMS system M total number of stations i indicator of demand zone, i = 1, 2,…, n m indicator of station, m = 1, 2,…, M v indicator of the state of EMS system - the number of busy servers, v = 0, 1, 2,…, K-1 c indicator of the status of EMS system = 0 system is out of compliance = 1 system is in compliance λ call arrival rate call arrival rate from demand zone i λi RTT a pre-specified response time threshold Mi set of locations that can respond to calls at demand zone i within the specific time γ the rates at which compliance is reached µv, 0 the service rate or call completion rate at which each individual busy ambulance completes its call, given that system is out of compliance µ1 the service rate or call completion rate at which each individual busy ambulance complete its call, given that system is in compliance πv,0 the steady-state probability that the system is out of compliance and in state (v,0) the steady-state probability that the system is in compliance and in state (v,1) πv,1 fdm the total covered arrival intensity for station m. Pr(Am) the probability of covered arrival intensity for station m. rim the response time from station m to demand zone i. the travel time from station j to station m. tjm the travel time between station m and demand zone i dim aim indicator of ambulance at station m can respond to demand zone i within specified response time RTT the specified response time thresholds (RTTs) aim = 0 if dim > RTT a server at station m does not cover demand zone i = 1 if dim ≤ RTT a server at station m covers demand zone i the mean travel time between any station to station m Tm αv the rates of call arrival into state (v, 0). βv the rates of call completion into state (v, 0). τ0, arrival the mean service time to enter the state (v, 0) via a call arrival τ0, completion the mean service time to enter the state (v, 0) via a call completion τ0,arrival,i the composition of the expected travel time entering state (v,0) via a call arrival from any station to demand zone i and the expected service time at on-scene of accident τ0,completion,i the composition of the expected travel time entering state (v,0) via a call completion from demand zone i to any station and the expected service time at on-scene of accident E[Si, on-scene] estimated from empirical data which is the composition of the service time on-scene and the time to transport patients to hospital if needed. E[Ri] the mean response time of any station to demand zone i. E[S0,Travel,i|(v, 0) entered via a call arrival] the expected travel time entering state (v,0) via a call arrival from any station to demand zone i. E[S0,Travel,i|(v, 0) entered via a call completion] the expected travel time entering state (v,0) via a call completion from demand zone i to any station p server busy probability for the adjusted maximum expected covering location model

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Q(K, p, v)

a correction factor indicating the probability of having v busy servers

Decision Variables = 1 if an ambulance is located at station m when the system is in state (v, c) (the number of x mv available servers is K-v, c status of EMS system is 0; out of compliance or 1; in compliance) = 0 otherwise = 1 if demand zone i is covered when system is in state (v, c) (the number of available servers is yiv K-v, c status of EMS system is 0; out of compliance or 1; in compliance), if all vehicles are at their assigned locations = 0 otherwise

The Markov chain model with relocation is applied to approximate the steady-state probabilities πv,c. The flow balance equations for state (v, 1) and (v, 0) are given by

π v ,1 = π v ,0 =

γ λ + v µ1

π v ,0

(v + 1)(λ + v µ1 )( µ v +1,0π v +1,0 + µ1π v +1,1 )

λ (λ + γ + vµ1 )

for v =0, 1, 2,…, K-1

(1)

for v =0, 1, 2,…, K-1

(2)

In order to compute the steady-state probabilities, we use a recursive method by starting with state (K, 0) and the normalization method in the last step so that the sum of the steady-state probabilities equal to one. In order to determine the steady-state probabilities, we also need to approximate the completion rate µ1and the rate to reach compliance, γ, which will be shown in Section 4.1. In this section, we consider how to calculate the completion rate µv, 0. We use the total arrival intensity to weigh service time. However, Alanis et al. [1] consider two possible situations to reach out of compliance state (v,0), via a call arrival and a call completion. They define two parameters αv and βv as the rates of call arrival and call completion into state (v, 0). The rates µv, 0 are obtained by weighing these two parameters. The rate αv and βv are shown in equations (3) and (4). The parameter τ0, arrival is the service time when entering state (v, 0) via a call arrival. The parameter τ0,

completion

is the service time when entering state (v, 0) via a call completion.

However, we estimate τ0,arrival and τ0, completion independently of the system state (v, c) The estimation of τ0, arrival and τ0, completion are discussed in Section 4.1.2. We use an iterative algorithm to determine the service rate µv,0 independent of the exact nested-compliance table policy.

α v = λ (π v −1,0 + π v −1,1 )

(3)

β v = (v + 1)( µ v +1,0π v +1,0 + µ1π v +1,1 )

(4)

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µ v ,0 =

α v + βv α vτ 0, arrival + β vτ 0,completion

(5)

4.1 Parameter Approximation for the Markov Chain Model The purpose of this section is to describe how we calculate the service time and the travel time between stations (relocation time) of EMS systems. The Markov chain model requires estimating average service times and average travel time between stations as input parameters. We define the average travel time between stations as the rates at which in compliance states are reached, γ. The value of γ is independent of where the available ambulances are located and the system state (v, c). We estimate the average travel time between stations using the covered arrival intensity for each station to adjust for the probability of the initial ambulance location. In addition, the Markov chain model with relocation assumes the transition states occur at rate vµv,0 when in state (v, 0) and rate vµ1 when in state (v, 1). We consider the covered arrival intensity for each station to estimate the probabilities that ambulances are assigned to certain stations. The possible transition states when entering state (v, 0) occur as a result of a call arrival and a call completion. Therefore, we have to compute the expected service time when state (v, 0) is entered via a call completion. However, the service rate µ1 is the average rate of call completion from arrival of a call to service completion. 4.1.1 Approximating Relocation Time between Stations For the average travel time between stations (relocation time), we consider the rate γ that does not depend on where the available ambulances are located. We estimate the probability of which available ambulances are located at station m by using total covered arrival intensity for station m. Suppose that λ1, λ2,…, λn are the proportions of call arrivals from demand zones 1 to n, then fdm refers to the total covered arrival intensity for station m and Pr(Am) refers to the probability of covered arrival intensity for station m. Pr( Am ) =

fd m M

∑

for m = 1, 2,…, M

(6)

for m = 1, 2,…, M

(7)

fd m

m =1

n

fd m = ∑ aim ⋅ λi i =1

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Estimating the travel rate per hour γ is calculated as the mean travel time between stations of M stations. To do this, let Tm refer to the mean travel time between any station to station m, and let tjm refer to the travel time from station j to station m. Then γ is obtained by using equations (8) and (9). M

∑ Tm γ = 60 /

m=1

(8)

M

M

Tm = ∑ Pr( Aj ) ⋅ t jm

(9)

j =1

4.1.2 Approximating Service Time The Markov chain model with relocation requires us to estimate the service times where the system enters out of compliance via a call arrival, τ0,arrival, and the system enters out of compliance via a call completion, τ0,completion, and the service rate per hour where the system enters in compliance, µ1. The parameter estimation for the Markov chain model is modified to approximate the average service times. We estimate these service times so that the service rate does not depend on the location configurations of compliance table. Our estimating τ0,arrival, τ0,completion and

µ1 depend on the covered arrival intensity of each station. We consider the covered arrival intensity to each station in order to weigh average response time from any station to a demand zone. We assume that the service time at on-scene and the time of transportation to hospital is the same for all demand zones. The service rate at which the system enters in compliance, µ1 is simply the arithmetic mean of the n demand zones for total service time of the expected response time and the expected service time at on-scene. It is straightforward to estimate the expected response time to demand zone i, E[Ri] from the mean response time of the M stations to demand zone i. The expected service time E[Si, on-scene] is estimated from empirical data that is the composition of the service time on-scene and the time to transport patients to hospital if needed. We describe how we estimate µ1 in Equation (10) and (11). n

∑ ( E[ Ri ] + E[Si,on−scene ]) µ1 = 60 / i =1

(10)

n

15

M

∑ rim E[ Ri ] = m=1 M

(11)

The service times, where the system enters out of compliance via a call arrival, τ0,arrival is simply the arithmetic mean service time of the n demand zones. We estimate the service time entered out of compliance via call arrival corresponding to demand zone i, τ0,arrival,i from the composition of the expected service time entered out of compliance via call arrival, E[S0,Travel,i|(v, 0) entered via a call arrival] and the expected service time at on-scene of accident, E[Si, on-scene] corresponding to demand zone i. For the purpose of estimating the E[S0,Travel,i|(v, 0) entered via a call arrival], we estimate the E[S0,Travel,i|(v, 0) entered via a call arrival] by using the probability of covered arrival intensity for station m. The Pr(Am) is used in order to weigh average travel time from any station m to demand zone i. The rim refers to response time from station m to demand zone i. In using equation (12) - (14), we obtains the τ0,arrival. n

∑τ 0,arrival ,i τ 0,arrival = i=1

(12)

n

τ 0,arrival ,i = E S0,Travel ,i | ( v,0) entered via a call arrival + E[Si,on−scene ] E  S0,Travel ,i | ( v, 0 ) entered via a call arrival  =

(13)

M

∑ Pr( Am ) ⋅ rim

(14)

m =1

Similarly, to estimate the τ0,completion, we use the probability of covered arrival intensity for station m, Pr(Am) in order to weight average travel time from a demand zone i to station. The service time where system entered out of compliance via a call completion, τ0,completion is also the mean travel time of the n demand zone. We estimate the service time entered out of compliance via call completion corresponding to demand zone i, τ0,completion,i from the composition of the expected service time entered out of compliance via call complete, E[S0,Travel,i|(v, 0) entered via a call completion] and the expected service time at on-scene of accident, E[Si, on-scene] corresponding to demand zone i. The E[S0,Travel,i|(v, 0) entered via a call completion] is composed of the mean travel time from demand zone to ambulance station and the mean travel time between stations. The parameter T,m captures the mean travel time between any station to station m by using

16

equation (9). The parameter rim refers to response time from station m to demand zone i. We use equation (15)– (17) for estimating τ0,completion. n

∑τ 0,completion,i τ 0,completion =

i =1

(15)

n

τ 0,completion,i = E S0,Completion,i | ( v,0) entered via a call completion + E[Si,on−scene ] E  S0,Travel ,i | ( v, 0 ) entered via a call completion  =

M

M

m =1

m =1

∑ Pr( Am ) ⋅ rim + ∑ Pr( Am ) ⋅ Tm

(16) (17)

Our parameter approximations can be used to calculate the transition rates of the Markov chain model. We use the equations (1) and (2) in Section 4 to approximate the steady-state probabilities. Finally, we can estimate the steady-state probabilities given the system being in compliance and out of compliance. We use them as input parameters to an integer programming model.

5. The Formulation of the Nested-Compliance Table Model In this section, we formulate the nested-compliance table model for the ambulance relocation problem. We consider a single type of patient calls and a single type of ambulances (paramedic units). The nested-compliance table model under relocation problem is introduced as an integer programming model. A Markov chain model is applied to approximate the steady-state probabilities when the system state is in compliance πv,1 and out of compliance πv,0 for each state (v, c) (v number of busy ambulances, c status of EMS system is 0; out of compliance or 1; in compliance). We consider the nested-compliance table model with n demand zones, K ambulance units and M ambulance stations. We assume that the EMS system operates with a relocation policy according to a nested-compliance table. We assume that one ambulance is located in each station for each state of the system. When a call arrives to system, the closest ambulance responds to a call. If the first closest ambulance is busy, the second will respond to a call and so on. We assume that the EMS system operates as a zero-queue system. The model is formulated as an integer linear programming model with the approximate steady-state probabilities serving as an inputs to the model. We define the decision variable xmv as a binary variable that is one if an ambulance is located at station m when the system is in state (v, c) and zero otherwise. The

17

decision variable yi,v is binary variable that is one if zone i is covered when the system is in state (v, c) and zero otherwise. For each demand zone, we define Mi as the set of ambulance stations

that can respond to calls from demand zone i within a specific time. We use the notation introduced in Table 3.

Objective function: Maximize

n K −1



i =1 v=0



  K − v −1   ⋅ π v,0  ⋅ yiv  K −v  

∑ ∑ (λi / λ ) ⋅  π v,1 + 

(18)

Subject to M

∑ xmv = K − v

for v = 0, 1, 2, …, K-1

(19)

for i = 1, 2,…, n for v = 0, 1, 2, …, K-1

(20)

for m = 1, 2,…, M for v = 1, 2, 3, …, K-1

(21)

m=1

yiv ≤

∑

xmv

m∈M i

xm ,v −1 ≥ xmv x m v ∈ {0, 1}

yiv ∈ {0, 1}

The maximum expected coverage of the nested-compliance table model under relocation is introduced as an integer programming model. The objective function is to maximize the demand that is covered as shown in equation (18). The equation consists of products of the decision variable yi,v and the probability of covering call zone i when the system is in state (v, c), and the proportion of calls from demand zone i. The parameter πv,1 indicates the probability that all available ambulances are at their assigned stations (the system is in compliance). Therefore, if a call from demand zone i arrives, all available ambulances K-v are at their assigned stations, and therefore, we know which demand zones are covered directly from yi,v. On the other hand, the system will be out of compliance in state (v, c) with probability πv,0. We do not know which ambulance is not at its located station. We assume that it is equally likely that one ambulance from K-v available ambulances is not available at its located station. The term (K-v-1)/(K-v) indicates the likelihood of K-v-1 available ambulances being in their stations and one ambulance being enroute to its home station when the system is in state (v, c). The constraint (19) ensures that we 18

allocate the number of ambulances equal to the number of available ambulances in each state of the EMS system. Constraint (20) indicates that the demand zone i is covered when at least one ambulance is located in a station in set Mi at each state (v, c), where Mi is the set of locations that can cover demand zone i. Constraint (21) ensures that the optimal solution is in the set of the nested-compliance table solutions. The integer programming model requires us to approximate the steady-state probabilities of the system being out of compliance πv,0 and of the system being in compliance πv,1 for each state (v, c). We discussed how to calculate these steady-state probabilities in Section 4.

6. The Efficiency of Nested-Compliance Table Model under Relocation In this section, we present the results of our model applied to real-world data. The data was collected from the Hanover Fire and EMS department in Hanover County, Virginia. The data consisted of approximately 12,000 calls per year. The city covered is an area of about 474 square miles with 122 demand zones and has a population of about 100,000. We defined demand to be the calls requesting paramedic units from an EMS system. The service region of Hanover County, VA is divided into 176 demand zones, where each demand zone is a 2 by 2 miles square region. Some zones had no demand (no calls for service) during the observed period and were therefore removed from consideration. The total number of demand zones with positive demand is 122. We aggregate all requested calls in each zone to be at its geographic center. The distances between demand zones are estimated from center to center of demand zones. We considered the EMS system with varied number of ambulances, from 6 to 10 ambulances, 16 station bases. We note that these are the available stations in this example, but the set of potential stations does not have to be limited. In other systems they allow vehicles to be located at any street corner, in which case each demand zone can be considered an available station. The EMS system operated 24 hours per day. The data set matched our assumption that call arrivals were Poisson during peak times, with a mean arrival rate of 1.5 calls per hour. The data set includes the distances between stations and demand zones. We assume that ambulances travel at a vehicle speed 50 miles per hour to travel between stations to demand zones or between stations. The response times follow a lognormal distribution, with rates that depend on call zones and stations of dispatched am-

19

bulances as shown in the Appendix Table A1. The service time is the total time in which ambulances were busy on-scene of the accident and provided transportation to a hospital if needed. Service times are also assumed to be exponentially distributed. We assume that the service time does not depend upon call zones and stations of dispatched ambulances. The returning time, or moving time between stations, also follows an exponential distribution based on the origindestination pair, as shown in Appendix Table A2.

We note that as not all possible location-

destination pairs had sufficient data to fit distributions, some were approximated based on similar distances. The Markov chain model with relocation was programmed in the Java programming language. The NetBeans IDE 7.3.1 was used to implement the model. The outputs of the Markov chain model with relocation were inputs to the integer programming model, which we programmed in IBM ILOG CPLEX Optimization Studio 11.2.

6.1 Nested-Compliance Table Model Validation We developed a discrete event simulation to validate the integer programming model. We used the previously described data set from Hanover Fire and EMS department, with details provided in the Appendix, for both the simulation model and the integer programming model. The simulation model uses realistic events built from the real data set, such as call arrivals, dispatching ambulances, relocating ambulances, service time on-scene, etc. The simulation model was implemented using Arena Version14, running on an Intel® Core(TM)2 Duo CPU. The objective is to maximize the expected coverage using a binary notion of coverage. The binary coverage refers to a call being covered if we dispatch an ambulance from stations within a pre-specified response time threshold (RTT) (e.g. 9 minutes) to respond to the call. We calculate the expected coverage not considering the variability in response time. We used the closest policy to respond to calls. We ran the simulation model with 1680 simulated hours for each replication, and 500 replications. The simulated time was 19 minutes for each policy, compared to the integer programming model taking 20 seconds to obtain the optimal policy. We compare the results of integer programming model to the results of the simulation model. Table 5 shows the absolute error and percent error of the coverage between the integer programming model and simulation model based on same the nested-compliance policy. The results report an average percent error of 2.2% with the mean service time 60 minutes and 3.2% with the mean service time 70 minutes for systems with an arrival rate of 1.5 calls per hour and a 9

20

minute RTT. Therefore, the approximation of our objective function used in the integer programming model was close to the coverage obtained from simulation model. The percent errors tended to be higher when the mean service time was increased. These results suggest that when ambulances spend more time providing service to patients, the resulting increases in the busy probabilities result in an increasing percent error of the approximated coverage using our Markov model.

Table 5: Comparison of the results of the integer programming model to results of the simulation model at arrival rate 1.5 call per hour, and response time threshold (RTTs) of 9 minutes Service Time (mins) 60

# of Servers 6 7 8 9 10

Relocation Model with Nested Cons. Abs. % Math Simu. error error 0.88 0.91 0.03 3.18 0.92 0.95 0.02 2.27 0.95 0.96 0.02 1.79 0.96 0.98 0.02 2.21 0.97 0.98 0.02 1.70

(a) Service Time 60 minutes

Service Time (mins) 70

# of Servers 6 7 8 9 10

Relocation Model with Nested Cons. Abs. % Math Simu. error error 0.86 0.90 0.05 5.19 0.90 0.94 0.03 3.46 0.94 0.96 0.03 2.74 0.95 0.98 0.02 2.33 0.96 0.99 0.02 2.51

(b) Service Time 70 minutes

Figure 3: Comparison of the coverage at 1.5 calls per hour under the integer programming model versus the simulation model

21

Figures 3a and 3b show that the expected coverage increases with the number of ambulances. They also show that the error of our approximated coverage was higher for a larger RTT (9 compared to 7 minutes) for both 60 and 70 minute service times. These observations suggest that the accuracy of the nested-compliance table model in estimating the expected coverage of systems depends on the response time thresholds (RTTs); as smaller RTT provided higher accuracy and a smaller service time also provided higher accuracy. The results also suggest that when a system has more ambulances, the integer programming model under estimates coverage. These observations result from the likelihood of the out of compliance state. We assume that an ambulance enroute cannot respond to a call. In reality, we might dispatch a backup ambulance to respond to the call, and the simulation model allows for this situation. However, when the system has a fewer ambulances, there is a higher possibility that a backup ambulance is not available or cannot respond to the call within the pre-specified response time threshold (RTT). Therefore, the effect of the backup ambulance is mitigated in systems with fewer ambulances.

6.2 Comparison with a Non-Relocation Model based on the Adjusted Maximum Expected Covering Location Model While our model seeks to determine the best nested-compliance table through the Markov chain model with relocation embedded into an integer programming model, we seek to verify the efficiency of the nested-compliance model for use in real-world EMS system to see if there is any benefit of relocating vehicles in comparison with static policies traditionally implemented. In this section, we compared the nested-compliance table model to a traditional adjusted maximal expected covering location problem (AMEXCLP) based on Batta et al. [8], which modifies the MEXCLP objective function developed by Daskin [10] to include the correction factors, Q(K, p, v) based on the hypercube model in Larson [11]. The correction factors indicate the probability of having v busy servers and correct for the deviation induced from assuming that servers are independent. The adjusted maximal expected covering location problem (AMEXCLP) is formulated as a baseline for a non-relocation model to compare the expected coverage. The objective function is to maximize the expected proportion of demand that could be covered. The formulation of AMEXCLP is below, and follows the notation introduced in Table 3.

22

AMEXCLP Model Objective function: n K

Maximize

∑∑(λi / λ)Q(K , p, v −1)(1− p) pv−1 ⋅ yvi

(22)

i =1 v=1

Subject to K

K

v =1

v =1

∑ yvi − ∑ avi xv ≤ 0

∀i

(23)

K

∑ xv = K

(24)

v =1

x v ∈ {0, 1}

∀v

yvi ∈ {0, 1}

∀ v, i

avi = 1 0 yvi = 1 0 xv = 1 0

if dvi > D a server at station v does not cover demand zone at i if dvi ≤ D a server at station v covers demand zone at i if demand zone i is covered by at least v servers otherwise if server locates at station v otherwise

We use real-world data from Hanover Fire and EMS department to compare the two models. We consider two instances of the data set, where the first data set is data from real world problem and the second the same except that we randomly assign proportion of demand between zones (alter the geographic distribution of demand). First, we compare the results of the AMEXCLP integer programming model to results of the simulation model. Figure 4 shows the expected coverage with six to ten ambulances. These results indicate a 1.5% average percent error with a mean service time of 70 minutes. These observations show that the approximations of AMEXCLP are close to the simulated AMEXCLP policies. Figure 4 simply shows that the simulation can be used to verify that the calculated coverage from the math model is similar to the simulated coverage. However, our main goal in this section is to assess the impact that following a dynamic policy would have compared to what is done in practice. This is shown in Figure 5, which compares

23

the coverage obtained using the nested compliance table policy and a static AMEXCLP policy. Figure 5a shows results from the real world problem and Figure 5b shows results from random proportions of demand zones. We vary the number of ambulances from 7 to 10 given that response time thresholds are 7 and 9 minutes. We compare the results of our nested-compliance table model to a non-relocation model based on the AMEXCLP, with call arrival rate 1.5 calls per hour and the mean service time 70 minutes.

Figure 4: Comparison of the coverage at 1.5 calls per hour and RTT < 9 minutes under the AMEXCLP math model versus the simulated AMEXCLP policy These results show improvement in outcomes when using the nested-compliance table model in comparison to non-relocation policy based on the AMEXCLP, coverage is calculated in the simulation model (since the AMEXCLP provides a different approximation of coverage which does not account for relocations). The points in the figures are the average over all replications (error bars are not shown as they were so small that they were covered by the point markers). In Figure 5a when we use a 9 minute response time threshold, the results show an average improvement of 2.7% from the nested-relocation policies in comparison to non-relocation policies. When we consider a 7 minute response time threshold, the results show a slight decrease in the benefit of our nested-relocation policies, of 2.2% improvement in comparison to non-relocation policies (AMEXCLP). In Figure 5b, we use the data set from random proportions of demand

24

zones based on the first data set. The results indicate that percent improvement of the nestedrelocation policies in comparison to non-relocation policies were on average 6.1% [95% CI (5.86%, 6.57%)] in terms of the expected coverage. These results are also provided in a table format in the Appendix, Table A3. The observations of the data set from the random proportions of demand zones show a lower number of stations that could cover for high proportion of demand zones than the data set from real world problem. These results suggest that the efficiency of the nested-compliance table model depends on number of stations that could cover for high proportion of demand zones. This implies that a system in which a fewer number of stations existed that could cover a higher proportion of demand zones would provide more efficiency of the nestedrelocation policies. The intuition behind this result can be explained in the following manner. In an EMS systems where there are fewer stations that can cover a high proportion of demand zones, if a call arrives to the system, and the first closest ambulance is not available, without relocation, there is a higher probability that the second closest ambulance was not located in a station that could cover that demand zone. If we relocate an ambulance to stations that could cover this call, we would increase the expected coverage of EMS systems.

(a)

Dataset from real world problem

(b)

Dataset from random proportion of demand zones, RTT=7mins

Figure 5: Comparison of the coverage of 1.5 calls per hour and service time 70 mins under the nested-compliance table model versus the non-relocation model (AMEXCLP)

25

While the benefit in terms of improvement in coverage may seem small, recall that coverage is related to response time and ultimately patient survival probability. The benefit of our nested-relocation policies can be shown in terms of patient survival probability and in terms of the number of lives saved per year. We use the patient survival function based on the work of Larsen et al. [3] and later modified by McLay and Mayorga [5]. The term s(tR) refers to the patient survival probability as a function of the realized response time tR as shown in equation (25).

s ( t R ) = m ax [(0 .5 9 4 − 0 .0 5 5 * t R ), 0 ]

(25)

For each model we can find a lower bound on the survival probability per patient using the simulated coverage and RTT as follows: s ( t R ) = s ( R T T ) × co verag e(R T T )

(26)

To compare the nested-compliance table model with the AMEXCLP model results we use this lower bound. While this is not a direct comparison of survival probability, this is a conservative estimate of the benefit provided by the nested-compliance table model, as the average response time of the relocation model with nested-compliance table is lower than the average response time of the non-relocation (AMEXCLP) model. From this, we estimate the lives saved based on the RTT value and realized (simulated) coverage for each model. Using an estimated 12,000 calls per year; based on a 9 (7) minute response time threshold and equation (26), the results show an average improvement of 30 (22) lives saved per year, based on the data set form the real world problem. Furthermore, if we consider the data set which uses a random proportion of demand between zones the improvement in lives saved using the nested- compliance table model in comparison to the non-relocation policies (AMEXCLP) increases to 58 lives per year, on average. The estimated benefits of the relocation model with nested-compliance table over the non-relocation (AMEXCLP) in term of lives saved are shown in column 11 of Table A3.

7. Conclusions In this paper, we formulate and validate a nested-compliance table model of an EMS system under relocation. We model the nested-compliance table as an integer programming model. The model requires outcomes of steady state probabilities from a Markov chain model with reloca-

26

tion to be input parameters for our integer programming model. The solutions are validated with a data set from a real-world setting. The mathematical model provides maximal coverage solutions which closely approximate a simulation. The validation indicates that the nestedcompliance table provided estimates of coverage that are on average within 2% - 3% of a simulation model based on the same data set. We demonstrate the efficiency of the nested-compliance table model in comparison to results from the non-relocation (AMEXCLP) model. The results showed that our model provided improvement of solutions over the results of the non-relocation (AMEXCLP) model with average improvement of 2.7% based on an original data set from a real-world problem and an improvement of 6.1% based on a generated data set in which demand was randomly assigned to demand zones. The performance of the nested-compliance table model depended on the pre-specified response time threshold (RTT) and on other system attributes. Implementing solutions of the nested-compliance table model to real-world problem suggests that the model provides improvement over the non-relocation (AMEXCLP) model; however, improvement depended on whether the binary coverage is used. In the realistic problem, the distribution of response time might affect the realized expected coverage. Results showed that the efficiency of the nestedcompliance model depends on the average travel time between stations (relocation time). Implementation of the nested-compliance table model should limit the relocation time between stations. Thus, the possible way to impose an upper bound of relocation time is to partition the service area in to small sub-areas (districts); how to do this optimally is a potential area of future research. The relocation policy is possible to implement in practice by using a computer-aided dispatch (CAD) system and a global positioning system (GPS). Specifically, we proposed a nestedcompliance policy for EMS systems. In practice, the dispatchers have their own nestedcompliance table lists and monitors that can track the status of all ambulances in the EMS system and their current locations. When the number of ambulances in the EMS systems changes, the dispatcher looks at the monitor and relocates ambulances to new stations in the new system state. No extra training course is required for using our nested-compliance table policy. In terms of outcomes, we estimated that about 30 lives would be saved in a year (at an annual call volume of 12,000 calls) by using our relocation policy instead of a non-relocation policy based on AMEXCLP under a 9 minute response time threshold. Our results assume steady call volume

27

and distribution; thus, it is possible that different compliance tables are used for different times of day and days of the week. Furthermore, as EMS parameters change (shifts in demand, available station, etc.) the tables may need to be recalculated.

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Appendix

Table A1: The mean response times between demand zones and stations, and proportion of calls prom each zone based on arrival rate 1.2 calls per hour Zone

ST1

ST2

ST3

ST4

ST5

ST6

ST7

ST8

ST9

ST10

ST11

ST12

ST13

ST14

ST15

ST16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

21.6 20.4 19.2 18 16.8 15.6 18 16.8 14.4 12 12 19.2 16.8 14.4 13.2 12 9.6 9.6 9.6 9.6 20.4 18 15.6 13.2 12 9.6 7.2 7.2 7.2 16.8 14.4 12 8.4 4.8 4.8 4.8 19.2 14.4 12 9.6 4.8 2.4 2.4 2.4 14.4 12 9.6 7.2 4.8 2.4 0 2.4 4.8 7.2 9.6 19.2 14.4 12 9.6

2.4 2.4 2.4 2.0 2.4 4.8 2.4 2.4 2.4 7.2 9.6 6 4.8 4.8 4.8 6 8.4 9.6 12.0 14.4 9.6 8.4 7.2 7.2 7.2 8.4 12 13.2 15.6 9.6 9.6 9.6 9.6 13.2 14.4 16.8 13.2 12 12 12 13.2 14.4 16.8 18 14.4 14.4 14.4 14.4 15.6 16.8 18 19.2 21.6 24 25.2 18 16.8 16.8 16.8

42 39.6 40.8 38.4 37.2 34.8 38.4 37.2 34.8 31.2 30 39.6 37.2 36 33.6 31.2 30 28.8 26.4 25.2 40.8 38.4 36 33.6 32.4 30 26.4 25.2 24 37.2 34.8 32.4 28.8 25.2 22.8 21.6 38.4 33.6 31.2 30 25.2 24 21.6 19.2 32.4 31.2 28.8 26.4 24 21.6 20.4 18 16.8 14.4 13.2 37.2 32.4 30 27.6

19.2 16.8 18 15.6 13.2 12 16.8 14.4 12 8.4 6 19.2 16.8 14.4 12 9.6 7.2 4.8 2.4 2.4 21.6 19.2 16.8 14.4 12 9.6 4.8 2.4 2.0 19.2 16.8 14.4 9.6 4.8 2.4 2.4 21.6 16.8 14.4 12 8.4 6 4.8 4.8 18 15.6 13.2 12 9.6 8.4 7.2 7.2 7.2 8.4 9.6 22.8 19.2 16.8 14.4

28.8 26.4 27.6 25.2 24 21.6 26.4 24 21.6 18 16.8 27.6 25.2 22.8 20.4 19.2 16.8 14.4 13.2 12 28.8 26.4 24 21.6 20.4 18 13.2 12 9.6 26.4 24 21.6 16.8 12 9.6 8.4 28.8 24 21.6 19.2 14.4 12 9.6 7.2 24 21.6 19.2 16.8 14.4 12 9.6 7.2 4.8 2.4 0 28.8 24 21.6 19.2

33.6 32.4 31.2 30 28.8 27.6 30 28.8 26.4 24 22.8 30 28.8 26.4 25.2 24 21.6 20.4 20.4 19.2 31.2 28.8 26.4 25.2 22.8 21.6 19.2 18 16.8 27.6 25.2 24 19.2 16.8 15.6 14.4 28.8 24 21.6 20.4 16.8 14.4 13.2 12 22.8 20.4 19.2 16.8 14.4 13.2 12 9.6 9.6 9.6 9.6 26.4 21.6 20.4 18

38.4 37.2 37.2 36 33.6 32.4 34.8 33.6 32.4 30 28.8 34.8 33.6 31.2 30 28.8 27.6 26.4 25.2 24 36 33.6 31.2 30 28.8 26.4 24 22.8 21.6 32.4 30 28.8 25.2 21.6 20.4 20.4 32.4 28.8 26.4 25.2 21.6 20.4 19.2 18 27.6 25.2 24 21.6 19.2 18 16.8 15.6 14.4 14.4 14.4 31.2 26.4 24 21.6

12 12 9.6 9.6 9.6 12 7.2 7.2 8.4 12 13.2 4.8 4.8 4.8 6 8.4 9.6 12.0 14.4 16.8 4.8 2.4 2.4 2.4 4.8 7.2 12 14.4 16.8 2.4 2.0 2.4 7.2 12 14.4 16.8 4.8 2.4 2.4 4.8 9.6 12 14.4 16.8 4.8 4.8 6 8.4 9.6 12 14.4 16.8 19.2 21.6 24 8.4 7.2 7.2 8.4

21.6 21.6 19.2 19.2 19.2 20.4 16.8 16.8 16.8 19.2 20.4 14.4 14.4 14.4 14.4 15.6 16.8 18.0 19.2 21.6 12 12 12 12 12 13.2 16.8 18 20.4 9.6 9.6 9.6 12 14.4 16.8 19.2 8.4 7.2 7.2 8.4 12 13.2 15.6 18 4.8 4.8 6 8.4 9.6 12 14.4 16.8 19.2 21.6 24 4.8 2.4 2.4 4.8

28.8 27.6 26.4 25.2 24 22.8 25.2 24 21.6 20.4 19.2 25.2 22.8 21.6 20.4 19.2 18 16.8 16.8 16.8 25.2 24 21.6 19.2 18 16.8 14.4 14.4 14.4 21.6 20.4 18 14.4 12 12 12 22.8 19.2 16.8 14.4 12 9.6 9.6 9.6 18 15.6 13.2 12 9.6 8.4 7.2 7.2 7.2 8.4 9.6 21.6 16.8 14.4 12

20.4 19.2 18 16.8 16.8 16.8 15.6 14.4 14.4 14.4 14.4 14.4 13.2 12 12 12 12 12.0 13.2 14.4 14.4 13.2 12 9.6 9.6 9.6 9.6 12 13.2 12 9.6 8.4 7.2 8.4 9.6 12 12 8.4 6 4.8 4.8 6 8.4 9.6 7.2 4.8 2.4 2.4 2.4 4.8 7.2 9.6 12 14.4 16.8 12 7.2 4.8 2.4

46.8 45.6 45.6 43.2 42 40.8 43.2 42 39.6 37.2 36 44.4 42 40.8 38.4 37.2 34.8 33.6 32.4 31.2 44.4 43.2 40.8 38.4 37.2 34.8 31.2 30 28.8 40.8 39.6 37.2 33.6 30 28.8 26.4 42 38.4 36 33.6 30 28.8 26.4 25.2 37.2 34.8 32.4 31.2 28.8 26.4 25.2 22.8 21.6 20.4 19.2 40.8 36 33.6 31.2

31.2 30 30 28.8 26.4 25.2 28.8 26.4 25.2 21.6 20.4 28.8 26.4 25.2 22.8 21.6 20.4 19.2 18.0 16.8 30 27.6 25.2 24 21.6 19.2 16.8 15.6 14.4 26.4 24 21.6 18 14.4 13.2 12 27.6 22.8 20.4 19.2 14.4 13.2 12 9.6 21.6 20.4 18 15.6 13.2 12 9.6 8.4 7.2 7.2 7.2 26.4 21.6 19.2 16.8

39.6 38.4 38.4 37.2 34.8 33.6 37.2 34.8 33.6 30 28.8 37.2 36 33.6 31.2 30 28.8 26.4 25.2 24 38.4 36 33.6 32.4 30 28.8 25.2 24 21.6 34.8 32.4 31.2 26.4 22.8 21.6 20.4 36 31.2 30 27.6 24 21.6 19.2 18 31.2 28.8 26.4 24 21.6 20.4 18 16.8 14.4 13.2 12 34.8 30 27.6 25.2

7.2 7.2 4.8 4.8 6 8.4 2.4 2.4 4.8 9.6 12 2.4 2.0 2.4 4.8 7.2 9.6 12.0 14.4 16.8 4.8 2.4 2.4 2.4 4.8 7.2 12 14.4 16.8 4.8 4.8 4.8 8.4 12 14.4 16.8 8.4 7.2 7.2 8.4 12 13.2 15.6 18 9.6 9.6 9.6 12 13.2 14.4 16.8 19.2 20.4 22.8 25.2 12 12 12 12

21.6 20.4 19.2 18 16.8 15.6 18 16.8 14.4 12 12 19.2 16.8 14.4 13.2 12 9.6 9.6 9.6 9.6 20.4 18 15.6 13.2 12 9.6 7.2 7.2 7.2 16.8 14.4 12 8.4 4.8 4.8 4.8 19.2 14.4 12 9.6 4.8 2.4 2.4 2.4 14.4 12 9.6 7.2 4.8 2.4 0 2.4 4.8 7.2 9.6 19.2 14.4 12 9.6

32

Proportion of call 0.04421 0.05263 0.03579 0.00421 0.00211 0.00140 0.00561 0.00351 0.00140 0.00070 0.00070 0.00211 0.00702 0.00351 0.00421 0.00140 0.00140 0.03088 0.00070 0.00491 0.00491 0.00211 0.00140 0.03860 0.00070 0.00070 0.02246 0.00070 0.00281 0.06456 0.00491 0.00281 0.00561 0.00070 0.00140 0.00211 0.03298 0.01333 0.00140 0.00140 0.00070 0.00421 0.00211 0.00211 0.00702 0.00281 0.00070 0.00140 0.00281 0.01263 0.00351 0.00211 0.00140 0.00842 0.00140 0.00421 0.00421 0.00211 0.00070

60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122

7.2 4.8 2.4 2.4 2.4 4.8 7.2 9.6 19.2 16.8 14.4 12 9.6 8.4 6 4.8 4.8 4.8 6 8.4 9.6 14.4 8.4 7.2 7.2 8.4 9.6 12 13.2 15.6 18 20.4 9.6 12 13.2 14.4 16.8 19.2 20.4 22.8 12 13.2 14.4 16.8 18 20.4 21.6 24 26.4 15.6 16.8 18 19.2 21.6 24 25.2 27.6 20.4 21.6 22.8 25.2 25.2 26.4

16.8 18 19.2 20.4 21.6 22.8 25.2 26.4 20.4 19.2 19.2 19.2 19.2 19.2 20.4 20.4 21.6 24 25.2 26.4 28.8 32.4 21.6 24 25.2 26.4 28.8 30 31.2 33.6 36 37.2 28.8 30 31.2 33.6 34.8 37.2 38.4 40.8 31.2 32.4 33.6 34.8 37.2 38.4 40.8 42 44.4 33.6 36 37.2 38.4 39.6 42 43.2 45.6 38.4 40.8 42 43.2 43.2 45.6

25.2 22.8 20.4 19.2 16.8 14.4 13.2 12 36 33.6 31.2 28.8 26.4 24 21.6 20.4 18 15.6 13.2 12 9.6 7.2 21.6 16.8 14.4 12 9.6 8.4 6 4.8 4.8 4.8 12 9.6 7.2 4.8 2.4 2.4 2.4 4.8 12 9.6 7.2 4.8 2.4 0 2.4 4.8 7.2 9.6 7.2 4.8 2.4 2.4 2.4 4.8 7.2 6 4.8 4.8 4.8 7.2 7.2

13.2 12 9.6 9.6 9.6 9.6 9.6 12 24 21.6 20.4 18 16.8 14.4 13.2 12 12 12 12 12 13.2 16.8 15.6 14.4 14.4 14.4 14.4 15.6 16.8 18 19.2 21.6 16.8 16.8 18 19.2 20.4 21.6 22.8 25.2 19.2 19.2 20.4 20.4 21.6 24 25.2 26.4 28.8 21.6 21.6 22.8 24 25.2 26.4 28.8 30 25.2 26.4 27.6 28.8 30 31.2

16.8 14.4 12 9.6 7.2 4.8 2.4 2.4 28.8 26.4 24 21.6 19.2 16.8 14.4 12 9.6 8.4 6 4.8 4.8 6 15.6 12 9.6 8.4 7.2 7.2 7.2 8.4 9.6 12 9.6 9.6 9.6 9.6 9.6 12 13.2 14.4 12 12 12 12 12 13.2 14.4 16.8 18 14.4 14.4 14.4 14.4 15.6 16.8 18 19.2 16.8 16.8 18 19.2 20.4 20.4

15.6 13.2 12 9.6 8.4 7.2 7.2 7.2 26.4 24 21.6 19.2 16.8 14.4 12 9.6 8.4 6 4.8 4.8 4.8 8.4 12 7.2 4.8 2.4 2.4 2.4 4.8 7.2 9.6 12 2.4 2.0 2.4 4.8 7.2 9.6 12 14.4 2.4 2.4 2.4 4.8 7.2 9.6 12 14.4 16.8 4.8 4.8 6 8.4 9.6 12 14.4 16.8 8.4 9.6 12 13.2 13.2 14.4

20.4 18 16.8 14.4 13.2 12 12 12 30 27.6 25.2 22.8 20.4 19.2 16.8 14.4 13.2 12 9.6 9.6 9.6 9.6 15.6 12 9.6 8.4 7.2 7.2 7.2 8.4 9.6 12 6 4.8 4.8 4.8 6 8.4 9.6 12 4.8 2.4 2.4 2.4 4.8 7.2 9.6 12 14.4 2.4 2.0 2.4 4.8 7.2 9.6 12 14.4 2.4 4.8 7.2 9.6 8.4 9.6

9.6 12 13.2 15.6 18 20.4 21.6 24 9.6 9.6 9.6 9.6 9.6 12 13.2 14.4 16.8 19.2 20.4 22.8 25.2 30 14.4 18 20.4 21.6 24 26.4 28.8 31.2 32.4 34.8 24 25.2 27.6 30 31.2 33.6 36 38.4 25.2 26.4 28.8 31.2 32.4 34.8 37.2 39.6 40.8 28.8 30 32.4 33.6 36 38.4 40.8 42 33.6 36 37.2 39.6 38.4 40.8

7.2 9.6 12 14.4 16.8 19.2 21.6 24 4.8 2.4 2.0 2.4 4.8 7.2 9.6 12 14.4 16.8 19.2 21.6 24 28.8 9.6 14.4 16.8 19.2 21.6 24 26.4 28.8 31.2 33.6 19.2 21.6 24 26.4 28.8 31.2 33.6 36 20.4 21.6 24 26.4 28.8 31.2 33.6 36 38.4 22.8 25.2 27.6 30 32.4 34.8 37.2 38.4 28.8 31.2 32.4 34.8 33.6 36

9.6 8.4 6 4.8 4.8 4.8 6 8.4 21.6 19.2 16.8 14.4 12 9.6 7.2 4.8 2.4 2.4 2.4 4.8 7.2 12 7.2 2.4 0 2.4 4.8 7.2 9.6 12 14.4 16.8 2.4 4.8 7.2 9.6 12 14.4 16.8 19.2 4.8 6 8.4 9.6 12 14.4 16.8 19.2 21.6 8.4 9.6 12 13.2 15.6 18 20.4 21.6 13.2 14.4 16.8 19.2 18 20.4

2.0 2.4 4.8 7.2 9.6 12 14.4 16.8 12 9.6 7.2 4.8 2.4 2.4 2.4 4.8 7.2 9.6 12 14.4 16.8 21.6 4.8 8.4 9.6 12 14.4 16.8 19.2 21.6 24 26.4 13.2 15.6 18 20.4 21.6 24 26.4 28.8 14.4 16.8 19.2 20.4 22.8 25.2 27.6 30 32.4 18 20.4 21.6 24 26.4 28.8 31.2 32.4 24 25.2 27.6 30 28.8 31.2

30 27.6 25.2 24 21.6 19.2 18 16.8 39.6 37.2 34.8 32.4 31.2 28.8 26.4 24 21.6 20.4 18 16.8 14.4 12 25.2 20.4 19.2 16.8 14.4 13.2 12 9.6 9.6 9.6 15.6 13.2 12 9.6 8.4 7.2 7.2 7.2 14.4 12 9.6 8.4 6 4.8 4.8 4.8 6 12 9.6 7.2 4.8 2.4 2.4 2.4 4.8 7.2 4.8 2.4 2.0 2.4 2.4

14.4 12 9.6 8.4 6 4.8 4.8 4.8 26.4 24 21.6 19.2 16.8 14.4 12 9.6 7.2 4.8 2.4 2.4 2.4 7.2 12 7.2 4.8 2.4 2.0 2.4 4.8 7.2 9.6 12 2.4 2.4 2.4 4.8 7.2 9.6 12 14.4 4.8 4.8 4.8 6 8.4 9.6 12 14.4 16.8 7.2 7.2 8.4 9.6 12 13.2 15.6 18 9.6 12 13.2 14.4 14.4 16.8

22.8 20.4 19.2 16.8 14.4 13.2 12 9.6 33.6 31.2 28.8 26.4 24 21.6 20.4 18 15.6 13.2 12 9.6 8.4 7.2 19.2 14.4 12 9.6 8.4 6 4.8 4.8 4.8 6 9.6 7.2 4.8 2.4 2.4 2.4 4.8 7.2 9.6 7.2 4.8 2.4 2.0 2.4 4.8 7.2 9.6 7.2 4.8 2.4 2.4 2.4 4.8 7.2 9.6 4.8 4.8 4.8 6 7.2 8.4

13.2 14.4 16.8 18 20.4 21.6 24 26.4 14.4 14.4 14.4 14.4 14.4 15.6 16.8 18 19.2 21.6 24 25.2 27.6 31.2 19.2 21.6 22.8 25.2 26.4 28.8 31.2 32.4 34.8 37.2 26.4 28.8 30 32.4 33.6 36 38.4 40.8 28.8 30 31.2 33.6 36 37.2 39.6 40.8 43.2 31.2 33.6 34.8 37.2 38.4 40.8 43.2 44.4 37.2 38.4 40.8 42 42 43.2

7.2 4.8 2.4 2.4 2.4 4.8 7.2 9.6 19.2 16.8 14.4 12 9.6 8.4 6 4.8 4.8 4.8 6 8.4 9.6 14.4 8.4 7.2 7.2 8.4 9.6 12 13.2 15.6 18 20.4 9.6 12 13.2 14.4 16.8 19.2 20.4 22.8 12 13.2 14.4 16.8 18 20.4 21.6 24 26.4 15.6 16.8 18 19.2 21.6 24 25.2 27.6 20.4 21.6 22.8 25.2 25.2 26.4

0.07228 0.00140 0.00211 0.00140 0.00070 0.00070 0.00561 0.00140 0.00632 0.01193 0.00281 0.00281 0.00070 0.00561 0.00211 0.00140 0.02947 0.00211 0.00351 0.00211 0.00140 0.00140 0.00140 0.00281 0.00070 0.00070 0.03649 0.19649 0.00211 0.00211 0.02877 0.00491 0.01193 0.00070 0.01895 0.00561 0.00070 0.00211 0.00351 0.00070 0.00140 0.00211 0.00211 0.00351 0.05754 0.00070 0.00070 0.00351 0.00281 0.00211 0.00491 0.00561 0.01754 0.08491 0.00421 0.00211 0.00070 0.00070 0.01263 0.00140 0.00070 0.00070 0.02947

The response time between demand zones and stations shown in Table A1 are assumed to be Lognormally distributed with mean (numeric data in table A1). The standard deviations are

33

based on the average standard deviation of Hanover County Fire and EMS department dataset. Full detail can be found in Bandara [33]. As noted in that paper, some response times were estimated based on similar distances when not enough data existed between some location pairs.

Table A2: Moving times between stations From / To

ST1

ST2

ST3

ST 1

0

6

17

ST 2

6

0

14

ST 3

17

11

0

ST 4

2

14

ST 5

6

12

ST 6

10

ST 7

ST4

ST5

ST6

ST7

ST8

ST9

ST10

ST11

ST12

2

6

10

14

8

2

14

12

6

10

8

6

6

1

21

2

15

7

19

11

8

3

26

18

12

19

4

19

0

8

12

17

5

3

8

11

8

0

4

8

12

4

2

1 6

6

8

12

4

0

4

13

6

4

9

14

10

3

17

8

4

0

23

15

8

ST 8

8

8

26

5

12

13

23

0

ST 9

2

6

18

3

4

6

15

8

8 0

ST 10

6

12

12

8

2

4

8

14

4

ST 11

1

15

19

1

6

9

16

6

2

ST 12

21

7

4

23

15

11

7

29

19

ST 13

8

14

9

11

2

2

5

16

7

ST 14

15

11

2

18

9

5

2

23

14

ST 15

12

4

30

9

15

16

27

4

10

9

ST 16

2

6

18

2

6

10

14

8

2

7

ST13

ST14

ST15

ST16

8

15

12

2

14

11

4

6

9

2

30

18

23

11

18

9

2

15

2

9

15

6

11

2

5

16

10

16

7

5

2

27

14

14

6

29

16

23

4

8

4

2

19

7

14

10

2

0

5

14

1

8

9

7

5

0

18

5

12

2

2

14

18

0

13

6

35

21

1

5

13

0

8

22

8

8

12

6

8

0

26

16

2

35

22

26

0

7

2

21

8

16

7

0

The moving time between stations are assumed to be exponentially distributed with mean shown in Table A2.

Relocation Model with Nested Cons.

# of servers

Math

Lower Bound on Survival Prob.

# of lives saved /12,000 calls

0.9020 0.9372 0.9619 0.9758 0.9875

0.0893 0.0928 0.0952 0.0966 0.0978

0.7979 0.8389 0.8771

0.0790 0.0830 0.0868

Simu. Coverage

Deviation between (Rel - AMEX Non Relo)

AMEXCLP- Non Relocation

Math

Simu. Coverage

Survival Prob.

1072 1113 1143 1159 1173

0.8511 0.9050 0.9420 0.9641 0.9784

0.8830 0.8930 0.9430 0.9510 0.9700

0.0874 0.0884 0.0934 0.0941 0.0960

948 997 1042

0.7361 0.7958 0.8469

0.7799 0.8187 0.8576

0.0772 0.0811 0.0849

# of lives saved /12,000 calls

The imp.of coverage

Improvement in lives saved /12,000 calls

1049 1061 1120 1130 1152 Avg. 926 973 1019

2.16 4.95 2.01 2.60 1.81 2.71 2.32 2.46 2.27

23 53 22 29 21 30 21 24 23

Dataset from real world problem 1) RTT = 9 minutes 6 0.8552 7 0.9049 8 0.9356 9 0.9530 10 0.9627 2) RTT = 7 minutes 6 0.7842 7 0.8450 8 0.8884

34

9 10

0.9193 0.9385

0.9015 0.9145

0.0892 0.0905

Dataset from proportion of demand zones 1) RTT = 7 minutes 7 0.7765 0.7660 0.0758 8 0.8449 0.8239 0.0816 9 0.8942 0.8722 0.0864 10 0.9275 0.9081 0.0899

1071 1086

910 979 1036 1079

0.8840 0.9080

0.6601 0.7280 0.7865 0.8285

0.8841 0.8979

0.7235 0.7770 0.8217 0.8521

0.0875 0.0889

0.0716 0.0769 0.0813 0.0844

1050 1067

1.97 1.85

21 20

Avg.

2.17

22

860 923 976 1012 Avg.

5.86 6.04 6.15 6.57 6.16

50 56 60 67 58

Table A3: Comparison of the results of the number of lives saved per 12,000 calls of 1.5 calls per hour and service time 70 minutes under the nested-compliance table model versus the nonrelocation model (AMEXCLP).

35

Highlights • We propose a compliance table model to relocate ambulances in EMS systems. • The optimal nested-compliance table policy is formulated as an integer program. • A Markov chain model with relocation is used to provide input parameters. • We validate the model via a simulated system using real-world data. • Relocation provides benefits in coverage and lives saved over a static policy. Table 4: Comparison of the results of the integer programming model to results of the simulation model at arrival rate 1.5 call per hour, and response time threshold (RTTs) of 9 minutes Service Time (mins) 60

# of Servers 6 7 8 9 10

Relocation Model with Nested Cons. Abs. % Math Simu. error error 0.88 0.91 0.03 3.18 0.92 0.95 0.02 2.27 0.95 0.96 0.02 1.79 0.96 0.98 0.02 2.21 0.97 0.98 0.02 1.70

36

Service Time (mins) 70

# of Servers 6 7 8 9 10

Relocation Model with Nested Cons. Abs. % Math Simu. error error 0.86 0.90 0.05 5.19 0.90 0.94 0.03 3.46 0.94 0.96 0.03 2.74 0.95 0.98 0.02 2.33 0.96 0.99 0.02 2.51