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Analysis of ambulance decentralization in an urban emergency medical service using the hypercube queueing model
Computers & Operations Research 34 (2007) 727 – 741 www.elsevier.com/locate/cor
Analysis of ambulance decentralization in an urban emergency medical service using the hypercube queueing model Renata Algisi Takedaa , João A. Widmera , Reinaldo Morabitob,∗ a Departamento de Transportes, Escola de Engenharia de São Carlos, Universidade de São Paulo,
13566-590 São Carlos, SP, Brazil b Departamento de Engenharia de Produção, Universidade Federal de São Carlos,13565-905 São Carlos, SP, Brazil
Available online 17 May 2005
Abstract This paper studies the application of the hypercube queueing model to SAMU-192, the urban Emergency Medical Service of Campinas in Brazil. The hypercube is a powerful descriptive model to represent server-to-customer systems, allowing the evaluation of a wide variety of performance measures for different configurations of the system. In its original configuration, SAMU-192 had all ambulances centralized in its central base. This study analyzes the effects of decentralizing ambulances and adding new ambulances to the system, comparing the results to the ones of the original situation. It is shown that, as a larger number of ambulances are decentralized, mean response times, fractions of calls served by backups and other performance measures of the system are improved, while the ambulance workloads remain approximately constant. However, total decentralization as suggested by the system operators of SAMU-192 may not produce satisfactory results. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Emergency medical system; Ambulance deployment; Hypercube model; Queueing theory; Case study
1. Introduction One of the major concerns of Emergency Medical Services (EMS) is to rapidly provide first care medical assistance to the victims. The time elapsed between an emergency call and its assistance, called response time, is one of the main factors that influence system performance. In urban areas, this time lapse depends on different aspects of calls and the EMS system such as: type and location of the request, ∗ Corresponding author. Fax: +55 16 260 8240.
E-mail address: [email protected] (R. Morabito). 0305-0548/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2005.03.022
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number and location of ambulances, system congestion, local traffic conditions, weekday and time, etc. The United States EMS Act sets some standards: 95% of the emergency requests should be served within 10 min in urban areas and within 30 min in rural areas [1]. Similar regulations are found in other parts of the world; for example, in London and Montreal, the regulation states that 95% of the requests should be served within 14 and 10 min, and 50% and 70% of the requests should be served within 8 and 7 min, respectively [2,3]. In Brazil, however, there is no specific regulation that specifies limitations for response times in EMSs. When designing or modifying the configuration of EMSs, managers should balance the benefits of improving user service at the expense of increasing the investment in the system. This trade-off is typical in services as well as manufacturing systems (e.g., Bitran and Morabito [4]). Several studies are found in the literature proposing approaches to rationalize the usage of available resources and improve user service. Nevertheless, a number of them do not directly consider the probabilistic nature of user arrival and service processes and the fact that ambulances are not always available for servicing a call. Examples of probabilistic approaches for ambulance deployment appear in Brandeau and Larson [5], Eaton et al. [6], Goldberg et al. [7] and Fujiwara [8]. Surveys reviewing the most important studies in the last decades are found, for instance, in Kolesar and Swersey [9], Louveaux [10], Swersey [11] and Brotcorne et al. [12]. It should be noted that an accurate model for EMSs can be quite complex since elements of uncertainty appear in time, location and amount of required services (e.g., demands are temporally and spatially distributed) and there are particular dispatching policies. The hypercube queueing model developed by Larson [13] and extended by other authors [11] is an effective descriptive model for planning server-to-customer systems. Given a system configuration, it is able to evaluate a variety of performance measures relevant for decision-making. It is not an optimization model in the sense that it determines an optimal configuration for the system, but it can provide a reasonably complete evaluation for each suggested configuration. It can also be combined in optimization approaches to deal with probabilistic location problems. For instance, Batta et al. [14] suggested its use in an iterative procedure as an alternative to relax the assumption of independence among the servers in the maximum expected covering location problem, MEXCLP (Daskin [15]). Chiyoshi et al. [16] analyzed non-homogeneous servers and compared MEXCLP and the adjusted maximum expected covering location problem, AMEXCLP [14]. Saydam and Aytug [17] developed a genetic algorithm that combines MEXCLP with a hypercube approximation algorithm developed by Jarvis [18] in order to solve MEXCLP with increased accuracy. Gendreau et al. [19,2] used tabu search in similar contexts. Galvão et al. [3] applied simulated annealing in the solution of MEXCLP and the maximum availability location problem, MALP [20]. In the last decades, different examples of application of the hypercube model in urban service systems have been reported; for instance, the social service visit program [21], the ambulance location in Boston [5] and Greenville [22] and the police patrolling in New Haven [23] and Orlando [24]. In Brazil, some examples are the assistance of power interruptions, location and planning of fire rescue vehicles and the imbalance of workload among the ambulances on a highway [25]. Other references related to applications and extensions of the hypercube model can be found in Halpern [26], Jarvis [27] and Swersey [7]. The present paper studies the hypercube model application to the urban EMS of Campinas (SAMU192), a city of almost one million people located in the state of Sao Paulo. In its original configuration, the system had all ambulances centralized at a single base adjacent to a general hospital downtown. This study analyzes the effects of decentralizing ambulances and adding new ambulances to the system, comparing the results to the ones of the original situation. It is shown how much mean response times,
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N E C W S
base
729
CAMPINAS 2 • area: 796 km • estimated population in 1998: 908.906 inhabitants SAMU-192 • monthly average request: 3.000 • doctors: 42 • nurses: 58 • drivers: 65 • hospitals: 5 public hospitals 8 private hospitals
Fig. 1. Coverage area of SAMU-192.
fractions of calls served by backups and other performance measures of the system are improved, as a larger number of ambulances are decentralized, while the ambulance workloads remain approximately constant. However, total decentralization as suggested by the system operators of SAMU-192 may not produce satisfactory results. This case study paper is organized as follows: in Section 2 system SAMU-192 is briefly described, in Section 3 the application of the hypercube model is discussed, in Section 4 the results obtained with the model for different scenarios are analyzed and compared to the original system configuration. Finally, in Section 5 concluding remarks and perspectives for future research are presented.
2. SAMU-192 The World Health Organization has defined provision of basic life support to all risk situations involving people and goods as a main objective of an EMS. In Brazil, since the beginning of the nineties, efforts have been made in order to impel the organization of urban EMSs. Most Brazilian systems are public services and some of them are similar to the French model SAMU (Service d’Aide Médicale Urgente), operating for more than 20 years. SAMU’s serve a city or a region containing several cities and are integrated into a group of public hospitals. They have a phone center that receive all requests, teams of doctors, nurses and drivers, ambulances equipped for basic life support (basic support vehicles, BSVs), and units of advanced life support (advanced support vehicles, ASVs). In 1994, the EMS of Campinas was reorganized yielding the SAMU-192, its aim being the urban regulator center of medical emergencies. In 1998, when this study began, the system had 10 ambulances (two ASVs and eight BSVs) centralized at the operational base of the system, where the phone center was installed. Campinas was divided into five different areas (North, South, East, West and Central) in accordance with the covering areas of main municipal health centers. The system provided a queueing service for incoming calls if all ambulances were busy at the time of their requests (the limitation of the waiting line was equal to one user per operating vehicle). Fig. 1 describes the SAMU coverage area and its divisions, together with additional information about the system. One of the special concerns of the system manager was that the ambulance workloads were too high in certain periods of the day, causing long waiting times for users. To apply the hypercube model in SAMU-192, the data collection was divided into two stages: first, the type and amount of past services were determined, then, a detailed sample was collected to verify the critical model assumptions. In a 1-year data collection, major differences in the numbers of requests
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Table 1 Performance measures of SAMU-192 Measures
Peak
Arrivals/day Inter-arrival time (min) Response time (min)a Time on scene (min) Service time (min) a Travel
24 h
BSV
ASV
BSV
ASV
20 10 13 37 66
2 92 11 39 63
70 14 12 37 64
10 106 10 39 62
time + setup time.
along the months were not observed, except in January and July. These are periods of school recess and vacations, when a slight decrease in the number of requests was noticed. The weekends also presented reduced demands, which is typical in urban EMSs. Then a detailed 10-day sample was randomly selected, rejecting January and July, and weekends. This sample had 814 requests spatially distributed as follows: 17.1% from North area, 16.5% from South area, 17.6% from East side, 32.2% from West side and 16.7% from Central area. The occurrences of peak periods in the system were also investigated: the statistical analysis indicated that the time period between 10 am and 2 pm corresponded to peak hours (on average, 30% of the requests occurred within this period). The results presented in this and the next sections are based on 209 requests (190 basic and 19 advanced calls) of this sample that occurred during peak periods. For more details of the sample, the reader may consult Takeda [28]. Table 1 shows some performance measures for the peak periods and for the 24-h days of the sample.
3. Hypercube model The hypercube model considers geographical and temporal complexities of the region and is based on spatial queueing theory and Markovian analysis approximations. Basically, the idea is to expand the state space description of a simple multi-server queueing system in order to represent each server individually and incorporate more complex dispatching policies. Once the model is calibrated, a number of performance measures of interest to how the system is managed can be estimated, either region-wide or for each server or region, such as workloads, mean response times, fraction of dispatches of each server to each region, etc. The name hypercube derives from the state space describing the status of the ambulances: each ambulance can be free (0) or busy (1) in a time instant. A particular state of the system is given by the entire listing of ambulances that are free and busy. For instance, the state 011 corresponds to a 3-server system, with ambulances 1 and 2 busy, and 3 free (note that 011 describes the ambulances states from right to left). Then, the state space is given by the vertices of a cube; if the system has more than three ambulances, that is a hypercube.
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Table 2 Mean arrival rates of atoms Atom
j (calls/hour)
1 - NB 2 - NA 3 - SB 4 - SA 5 - EB 6 - EA 7 - WB 8 - WA 9 - CB 10 - CA
0.8535 0.1035 0.8535 0.0776 0.8535 0.0776 1.5001 0.1293 0.8276 0.1293
Total
5.4054
3.1. Model assumptions There are nine basic assumptions for the application of the model [21]. We now describe these assumptions for the SAMU-192 study. Geographical atoms: The SAMU-192 area was divided into five non-overlapping areas, as shown in Fig. 1. In addition, each area was partitioned in two different and independent areas: one generating only advanced requests and the other only basic requests. This procedure is called “layering’’ of areas [21] and it is useful for representing situations admitting implicit or explicit priority criteria in their services, where specialized teams are dispatched to serve high acuity victims. This way, a total of NA = 10 geographical atoms were defined for the system: North B (NB), North A (NA), South B (SB), South A (SA), East B (EB), East A (EA), West B (WB), West A (WA), Center B (CB), Center A (CA). The first preference ambulances of advanced request atoms are the ASVs, and the BSVs are the backups. Similarly, the first preference ambulances of basic request atoms are the BSVs, and the ASVs are the backups. Independent Poisson arrivals: Emergency calls of each atom j were generated as a Poisson process, independently from other atoms, with mean arrival rates j (j = 1, 2, . . . , NA ) as showed in Table 2 (these values were estimated from the sample). Although this assumption seems to be very restrictive, it is frequently satisfied in several real systems, as pointed out by a number of authors. Kolmogorov–Smirnov and 2 goodness-of-fit tests were applied to verify the hypothesis of Poisson arrival. They were unable to reject such hypothesis with 5% of significance. Travel times: The mean ambulance travel times ij (i, j = 1, 2, . . . , NA ) between atoms i and j should be known or can be estimated using geometrical probability concepts. In this study, they were taken from the databases of SAMU-192 (their values are detailed in [28]). It is interesting to note that the ASVs mean travel times were smaller than BSVs, due to the urgency degree. Servers: SAMU-192 was operating with eight BSVs and two ASVs (i.e., with N = 10 ambulances). All these ambulances remained in the Central base when idle and they could be dispatched to any atom to accomplish a service.
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Table 3 Fixed-preference dispatching matrix (original configuration) Atom
1 - NB 2 - NA 3 - SB 4 - SA 5 - EB 6 - EA 7 - WB 8 - WA 9 - CB 10 - CA
Preference dispatching 1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
9 1 10 2 3 1 4 2 5 2
10 2 3 1 4 2 5 1 6 1
3 10 4 5 7 6 8 9 8 10
4 7 5 3 10 9 6 8 10 5
5 3 6 10 8 4 7 3 9 6
6 4 7 9 5 10 3 5 4 8
7 9 8 4 6 5 9 10 3 7
8 5 9 6 9 3 10 4 7 9
2 8 1 7 2 8 2 6 1 3
1 6 2 8 1 7 1 7 2 4
Vehicles
Server locations: The server location matrix (L) was easily determined because all ambulances, when available, remained in the Central base waiting for a call. Server assignment: In response to each call, exactly one ambulance was dispatched from the Central base to the local of the request. If all ambulances were busy, the call was queued if the waiting line was not greater than the number of ambulances in operation; otherwise the call was lost (i.e., transferred to another EMS). Fixed-preference dispatching: There was a list of dispatching preferences for each atom. If the first preference ambulance of the list was available, it was dispatched; otherwise, the next ambulance of the list was dispatched. According to SAMU-192’s operational policy (discussed above), for advanced call atoms, the list contained ASVs as first preference ambulances, followed by BSVs as backups (similarly for basic call atoms). Since there were no preferences among the ASVs (and no preferences among the BSVs), for simplicity, the method of random generation of fixed-preference dispatching matrixes proposed in [22] was used in the present study. Table 3 presents an example of a randomly generated fixed-preference dispatching matrix using that procedure. In the matrix, the two ASVs are numbered 1 and 2, and the eight BSVs are numbered 3, 4, 5, 6, 7, 8, 9 and 10. Service times: The service time for a call is composed by the setup time of the server (it was very small—less than 1 min), the travel time from the base until the place of the occurrence (on average, 11 min for ASVs and 13 min for BSVs in the sample), the on-scene time (39 min for ASVs and 37 for BSVs) and the travel time back to the base (13 min for ASVs and 16 min for BSVs). The mean service time for urgent calls resulted in 63 min for the case of services accomplished by ASVs and 66 min for the case of BSVs, as pointed out in Table 1. Table 4 shows the mean service times for each ambulance and their corresponding standard deviations, coefficients of variations and mean service rates n , n = 1, 2, . . . , N (estimated from the sample). The hypercube model assumes that the standard deviation of the service time is approximately equal to the mean (since the mathematical analysis assumes negative exponential service times). Applying
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Table 4 Mean service times and rates of ambulances Ambulance
Service timea
Standard deviation
Coefficient of variation
n b
1 2 3 4 5 6 7 8 9 10 ASV BSV Total
61 64 63 70 61 66 64 61 70 70 63 66 65
40 44 52 48 44 58 50 40 58 56 42 51 49
0.7 0.7 0.8 0.7 0.7 0.9 0.8 0.7 0.8 0.8 0.7 0.8 0.8
0.9836 0.9375 0.9524 0.8571 0.9836 0.9091 0.9375 0.9836 0.8571 0.8571 0.9606 0.9172 0.9259
a minutes, b calls/hour.
Kolmogorov–Smirnov and 2 goodness-of-fit tests for the service processes, they were unable to reject the hypothesis of exponentially distributed service times with 5% of significance. Variance analysis tests were also applied to verify the hypothesis of equal service times among the 10 ambulances. The tests rejected such hypothesis with 5% of significance; therefore, these service times were treated as different in the model. Service time dependence on travel time: Variations in the service time that are due to variations in travel time are admitted to be second order when compared to variations of setup and on-scene times. This hypothesis, which limits the applicability of the model, is frequently verified in urban services and less observed in rural services. It should be noted that, for each ambulance of SAMU-192, the mean travel time is short in comparison to the mean service time (on average, about 18%), and that variations in travel time give a small contribution to the variations in service time (Table 4). 3.2. Steady state probabilities The equilibrium probability equations are defined supposing that the system attains steady state. Each possible state of the system is represented by a vector with N = 10 elements (ambulances) assuming values 0 or 1 (free or busy). Therefore, there are 210 = 1024 possible states in the system, besides the 10 states that define the queue, namely S11 , S12 , . . . , S20 . For instance, the equilibrium probability equation for the intermediate state 1000000101, where the ambulances 1, 3 and 10 are busy, is: [(2 + 4 + 6 + 8 + 10 ) + (7 + 3 + 5 ) + (9 ) + (1 ) + (1 ) + (3 ) + (10 )]p1000000101 = 2 p1000000111 + 4 p1000001101 + 5 p1000010101 + 6 p1000100101 + 7 p1001000101 + 8 p1010000101 + 9 p1100000101 + (2 + 6 )p1000000100 + (3 + 5 )p1000000001 + 3 p0000000101 .
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Idle System
1000000111
2 + 4 + 6 + 8 + 10 1000000100
µ12
2 +6
7 + 3 + 5 µ4 1000010101 9 9 µ5
µ1
1000000001
3 + 5 µ3
1000000101
µ ∝56 µ ∝75
3 µ 10 0000000101
1000001101
1000100101
1001000101
µ58 ∝
91 µ9
1010000101
1100000101
Fig. 2. Flows in and out of state 1000000101.
Fig. 2 illustrates the corresponding flows in and out of this state. The equilibrium probability equations for other states are similar. Substituting any of the equations by the normalization equation (sum of all state probabilities should be equal to 1), the result is a determined system with 210 linearly independent equations. 3.3. Performance measures A variety of system performance measures can be defined as a function of the equilibrium distribution, as discussed in Larson [13]. For example, the probabilities of finding the system empty and saturated are p0000000000 = 0.003 and ps = 0.09, respectively. The loss probability is only 0.0001. The workload n of ambulance n represents the fraction of time that ambulance n is busy, and it is obtained easily through the sum of the equilibrium probabilities of the states in which this ambulance is busy. The workloads of BSVs and ASVs are 63% and 39%, respectively. The overall workload was 58% (which is very close to the mean value of the databases of SAMU-192, 60%). Average travel times: Different travel times are obtained by the model starting from the origin-destiny matrix of travel times (ij ). For example, the mean travel time of the system (T ) is estimated by: NA [1] [1] T = N n=1 j =1 fnj tnj + pS TQ , where fnj is the fraction of all dispatches that send ambulance n, when available, to atom j, tnj is the mean time required to ambulance n, when available, to travel to atom j, pS is the system saturation probability and TQ is the mean travel time to a queued call. This measure resulted in T = 13.6 min, which is sufficiently close to the one observed in the sample (14.5 min, that is, a deviation of only 6.2%).
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735
Other mean travel times can be estimated by the model, such as the mean travel time of ambulance n (T U n ) and the mean travel time to atom j (Tj ) (the details are found in Larson and Odoni [21]). The mean travel times computed by the model were 10.9 min for ASVs and 14.2 min for BSVs, that is, deviations of only 3.3% and 5.1% from the sampled values. Regarding the mean travel times to the atoms, the relative deviations were also reasonably low (0.1% for advanced call atoms and 5.5% for basic call atoms). These small deviations showed that the model is able to produce accurate performance measures of the system to support decision-making. 3.4. Calibration of service times In certain EMSs, travel times may represent a considerable part of service times. In such cases, it may be advisable to adjust the service times by means of a calibration process, which can be performed using a simple iterative procedure described in Larson and Odoni [21]. Basically, the procedure consists of verifying if there are significant differences among the input mean service times and the output mean service times (computed by the hypercube model). In this case, the hypercube is resolved using the computed mean service times as inputs, and so on, until the differences among input and output values are sufficiently small. This procedure was used to produce the results presented below.
4. Evaluation of alternative scenarios As mentioned above, one of the major concerns in EMSs is to reduce the response time, which is composed of the setup time, the travel time and the waiting time (eventually spent in the queue). The waiting time is a function of the mean travel time to a queued call (TQ ) and the probabilities of satured states S11 , S12 , . . . , S20 . In SAMU-192, the setup time is very short (around 1 min) and such probabilities are relatively low (order of 10−3 in the original configuration), so that the mean response time almost coincides with the mean travel time of the system (T ). For the sake of simplicity, in the analyses below the impacts of ambulance decentralization are evaluated in terms of mean travel times (instead of mean response times). 4.1. Ambulance decentralization The Brazilian legislation does not impose that the ambulance bases of urban EMSs should be close to a hospital or any other type of health unit. However, when locating bases in these units, there are cost savings using such existent infrastructure, besides the possible collaboration of the unit’s professionals with the EMS teams. Therefore, the scenarios for ambulance decentralization of SAMU-192 considered below assume that the bases of the ambulances should be municipal hospitals or health units located in the five areas (North, South, East, West and Central). Several operational alternatives of ambulance decentralization were considered in this study, from the relocation of a single ambulance until scenarios with total decentralization. One-ambulance decentralization: Scenario 1 was the simplest alternative of decentralization, where one of the BSVs (e.g., ambulance 5) was moved to area WB (note in Table 2 that this area had the highest demand of the system). The other ambulances remained in the Central base as backups for area WB, as shown in Fig. 3. In this way, ambulance 5 in WB becomes the first preference server for basic calls
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R.A. Takeda et al. / Computers & Operations Research 34 (2007) 727 – 741 1 → Central A 2 → Central A 3 → Central B 4 → Central B 5 → West B 6 → Central B 7 → Central B 8 → Central B 9 → Central B 10 → Central B
ASV BSV
Fig. 3. Scenario 1 (one BSV in WB).
Table 5 Workloads and the mean travel times of ambulances Ambulance
1 2 3 4 5 6 7 8 9 10 Average ASV Average BSV SYSTEM
Scenario 0
Scenario 1
Deviation
Workload
Travel time (min)
Workload
Travel time (min)
Minutes
%
0.39 0.39 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.39 0.63 0.58
10.9 10.9 14.2 14.2 14.2 14.2 14.2 14.2 14.2 14.2 10.9 14.2 13.6
0.39 0.41 0.64 0.63 0.66 0.64 0.65 0.65 0.64 0.63 0.40 0.64 0.58
11.3 10.6 14.2 14.2 8.8 14.2 14.2 14.2 14.2 14.2 11.0 13.5 13.0
0.4 −0.3 0.0 0.0 −5.4 0.0 0.0 0.0 0.0 0.0 0.1 −0.7 −0.6
3.7 −2.8 0.0 0.0 −38.0 0.0 0.0 0.0 0.0 0.0 0.9 −5.0 −4.4
in WB, the seven remaining BSVs in the Central base become the second preference servers for basic calls in WB (i.e., the backups of ambulance 5), and the two ASVs in the Central base become the third preference servers for basic calls in WB (the backups of the backups). Moreover, ambulance 5 in WB is the backup of the ASVs for advanced calls in WA, and the BSVs in the Central base are the backups of ambulance 5 for advanced calls in WA. After making appropriate modifications in the server location matrix (for ambulance 5), in the preference dispatching matrix generator procedure of Burwell et al. [22] and in the equilibrium probability equations, the model generated workloads similar to the original configuration (namely scenario 0), as shown in Table 5. Note that ambulance 5 has a slightly higher workload given that it became the first preference server of atom WB. The mean travel times of the ambulances decreased evenly, except for ambulance 5 whose reduction was significant (38%). The overall mean travel time decreased 4.4% (from 13.6 to 13.0 min), corresponding to 0.9% for ASVs and 5.0% for BSVs. Two-ambulance decentralization: Scenario 2 enlarges the decentralization to two BSVs. There are several possible locations for these ambulances; however, experience of the system operators and the
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Table 6 Mean travel times of ambulances (T U n ) Ambulance
1 2 3 4 5 6 7 8 9 10 11 12 13 Average ASV Average BSV SYSTEM
Travel times (minutes) Scenario 0
1
2
3
4
5
6
7
8
9
10.9 10.9 14.2 14.2 14.2 14.2 14.2 14.2 14.2 14.2
11.3 10.6 14.2 14.2 8.8 14.2 14.2 14.2 14.2 14.2
11.4 10.5 10.0 13.9 9.2 14.0 14.1 14.0 13.8 13.8
11.3 10.6 9.0 9.7 13.8 9.7 13.9 14.0 13.8 13.9
10.9 10.9 11.7 12.4 10.7 12.0 9.9 12.7 13.0 12.9
10.9 10.9 11.7 12.8 10.9 10.5 12.0 12.0 10.5 12.1
10.9 10.8 11.6 11.3 11.4 10.5 10.4 12.7 12.6 11.6
10.0 9.4 11.0 9.9 9.5 9.6 9.3 11.4 11.2 11.2 10.5
7.7 7.6 8.9 9.2 9.1 8.9 8.9 8.1 8.2 8.8 8.5 8.7
10.9 14.2 13.6
11.0 13.5 13.0
11.0 12.9 12.5
11.0 12.2 12.0
10.9 11.9 11.7
10.9 11.6 11.5
10.9 11.5 11.4
9.7 10.4 10.3
7.7 8.7 8.6
7.5 7.1 8.8 8.4 7.7 7.9 7.8 8.1 8.2 8.7 8.2 8.2 6.6 7.1 8.2 7.9
Table 7 Mean travel times to atoms (Tj ) Atom
1 - NB 2 - NA 3 - SB 4 - sa 5 - EB 6 - EA 7 - WB 8 - WA 9 - CB 10 - CA Average ASV Average BSV
Travel times (min) Scenario 0
1
2
3
4
5
6
7
8
9
16.0. 13.1 15.9 13.0 15.9 13.0 15.8 12.8 5.5 4.3 11.3 13.8
16.2 13.2 15.9 13.1 16.1 13.1 12.5 12.4 5.8 4.5 11.3 13.3
16.5 13.2 15.9 13.1 16.4 13.2 10.2 12.1 6.2 4.5 11.2 13.0
16.2 15.2 15.8 12.9 16.2 13.1 9.8 12.6 6.1 4.0 11.2 12.8
13.0 12.7 12.4 12.6 12.9 12.7 13.3 12.5 5.9 4.3 11.0 11.5
13.3 12.8 12.5 12.6 13.3 12.7 11.1 12.2 7.6 4.7 11.0 11.5
11.0 12.5 13.2 12.7 13.1 12.7 11.5 12.3 9.0 4.9 11.0 11.6
10.0 12.1 12.2 12.3 12.3 12.3 10.8 12.1 7.1 4.1 10.6 10.5
7.7 8.7 9.3 9.5 10.5 10.8 9.8 8.6 6.3 3.4 8.2 8.7
7.5 8.2 9.1 8.7 8.9 9.5 8.6 8.6 6.3 3.4 7.7 8.1
high demand of area WB suggested repositioning the two BSVs in WB. The results were a little better than the ones in scenario 1 (e.g., the overall mean travel time were 12.5 min), as shown in Tables 6 and 7 and Figs. 4–6.
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Workload
1.0 ASV
0.8
BSV
SYSTEM
0.6 0.4 0.2 0.0 0
1
2
3
4
5
6
7
9
8
Scenario
Travel times for ambulances (minutes)
Fig. 4. Ambulance workloads in scenarios 0, 1, . . . , 9.
15
10
ASV
BSV
SYSTEM
5 0
1
2
3
4
5
6
7
8
9
Scenario
Fig. 5. Mean travel times of ambulances in scenarios 0, 1, . . . , 9.
Travel times to atom (minutes)
15
10
ASV
BSV
5 0
1
2
3
4 5 Scenario
6
7
8
9
Fig. 6. Mean travel times to atoms in scenarios 0, 1, . . . , 9.
Three-ambulance decentralization: Scenario 3 took the configuration of scenario 2 and moved another BSV to area NB (close to a large public hospital). The results were similar to the ones of scenarios 1 and 2, reducing a little more the mean travel times of the ambulances, as depicted in Tables 6 and 7 and Figs. 4–6. The overall mean travel time reduced 11.7% compared to scenario 0 (from 13.6 to 12.0 min) and 3.8% compared to scenario 1. Other scenarios: Other decentralized configurations were studied. In scenario 4, four BSVs were decentralized in areas NB; SB; WB and WB. This configuration was suggested by the operators of SAMU-192, providing even better results than previous scenarios (e.g., the overall mean travel time went down to 11.7 min). Other configurations with four decentralized vehicles were appraised but all of them yielded worse values. Two other configurations decentralizing five and six ambulances
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(scenarios 5 and 6) were also analyzed, resulting in system travel times of 11.5 and 11.4 min, respectively. Although most of their values are similar to scenario 4 (see Tables 6 and 7 and Figs. 4–6), there are some that are better; for instance, their proportions of advanced calls serviced by BSV backups are lower. Other configurations decentralizing from 7 to all ambulances were also investigated but they did not provide better values. In all these scenarios the workloads remained around 0.58. The detailed results are presented in [28]. 4.2. Increasing the number of ambulances Scenario 7 considers the addition of a new BSV (numbered 11) in scenario 6, located in the Central area. The workloads decreased 18% on average, the mean travel times of BSVs and ASVs reduced 14% and 9%, respectively, and the overall travel time 9.6% (10.3 min), compared to the results of scenario 6. Scenario 8 considers a new BSV (numbered 12) in scenario 7, located in area EB, and scenario 9 considers a new ASV (numbered 13) in scenario 8, located in WA (note in Table 2 that areas CA and WA had the highest arrival rates of advanced calls). The benefits of increasing the number of ambulances (combined with partial decentralization) were sensitive in all performance measures (as shown in Tables 6 and 7 and Figs. 4–6), clearly at the expense of additional costs and investments. For instance, considering that the monthly operational costs of each BSV and ASV are approximately US$8,000 and US$18,000, respectively, scenario 9 (with 13 ambulances) has an additional cost of US$34,000 per month (34% more than the original scenario) but it reduces the overall travel time by 42% (from 13.6 to 7.9 min). This way, the resulting mean response time of the system (including the setup time) becomes close to international standards reported by other studies in the literature (e.g., [29,8] reported 8.8 min). 4.3. Comparison of scenarios In order to help the comparison among the scenarios, the tables and figures below compile some of the results obtained after decentralizing ambulances (scenarios 1,2,…,6) and adding new ambulances to the system (scenarios 7–9). Fig. 4 depicts the variations of the workloads of ASVs and BSVs compared to the original system (scenario 0). Observe that, as the decentralization of ambulances increases (from scenario 0–6), the workloads do not change much. On the other hand, as expected, the workloads reduce substantially by adding new ambulances (scenarios 7–9). Regarding travel times, partially decentralized systems produce better results than centralized or totally decentralized configurations and, in practice, this corresponds to shorter response times. As expected, these observations were also verified after increasing the number of ambulances (Tables 6 and 7). Note, for example, that mean travel times to atoms were reduced by up to 32% for advanced calls and 41% for basic calls (Table 7). Other performance measures were also considered for the comparison of the scenarios; for example, the proportion of inter-atom dispatches of ambulances and the fraction of advanced calls serviced by a BSV (i.e., a backup). Note that as this fraction increases, users requesting advanced support may receive inferior quality medical assistance since BSVs are less equipped than ASVs. Similarly to travel times, partially decentralized systems yield smaller fractions of acute calls serviced by a BSV than the original configuration, therefore also improving user service. For illustration, for the original configuration the fraction was 19% while for scenario 9 it became only 5%.
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5. Concluding remarks This paper studied the application of the hypercube queueing model to the urban EMS of Campinas (SAMU-192) in Brazil. In its original configuration, SAMU-192 had all ambulances centralized in its central base. The hypercube model was successfully applied to analyze ambulance deployment, in particular, the effects of decentralizing ambulances and increasing their number. The results showed that the simple relocation of one single ambulance decreases system travel times (as one would expect). As a larger number of ambulances are decentralized, travel times and other performance measures are even better, while the workloads remain approximately constant. For instance, the mean travel time of the system reduces from 13.6 min (centralized configuration) to 11.4 min (six ambulances decentralized). The fraction of advanced calls assisted by backups (BSVs) also decreases from 19% to 11%. On the other hand, total decentralization as suggested by the system operators of SAMU-192 does not produce satisfactory results. Increasing the number of ambulances in the system (combined with partial decentralization) leads to significant improvements in system performance measures (as expected), but obviously at the expense of additional costs and investments. For example, the mean travel time of the system reduces from 13.6 min (scenario 0 with 10 ambulances) to 10.3, 8.6 and 7.9 min (scenarios 7–9 with 11, 12 and 13 ambulances, respectively). Considering that the monthly operational costs of each BSV and ASV are approximately US$8,000 and US$18,000, respectively, the benefit-cost relationship of scenarios 9 and 0 can be roughly stated as: an increase of 34% in the operational cost of SAMU-192 reduces the mean travel time of the system by 42% and places the mean response time close to international standards. The five areas of SAMU-192 were originally designed around municipal health centers and they do not reflect demand patterns of emergency calls. An important perspective for future research is the application of the hypercube model for planning new reporting areas (redistricting problem). Another interesting line of research is to integrate the hypercube model in an optimization procedure in order to determine optimal ambulance decentralization.
Acknowledgements The authors thank SAMU-192 of Campinas for providing the data, especially Dr. Arine Campos O. Assis (former system coordinator and presently a consultant of the Brazilian Ministry of Health) for her support and incentive, and the two anonymous referees for their useful comments and suggestions. This research was partially sponsored by CNPq (Grant 800773/91-8).
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Analysis of ambulance decentralization in an urban emergency medical service using the hypercube queueing model Renata Algisi Takedaa , João A. Widmera , Reinaldo Morabitob,∗ a Departamento de Transportes, Escola de Engenharia de São Carlos, Universidade de São Paulo,
13566-590 São Carlos, SP, Brazil b Departamento de Engenharia de Produção, Universidade Federal de São Carlos,13565-905 São Carlos, SP, Brazil
Available online 17 May 2005
Abstract This paper studies the application of the hypercube queueing model to SAMU-192, the urban Emergency Medical Service of Campinas in Brazil. The hypercube is a powerful descriptive model to represent server-to-customer systems, allowing the evaluation of a wide variety of performance measures for different configurations of the system. In its original configuration, SAMU-192 had all ambulances centralized in its central base. This study analyzes the effects of decentralizing ambulances and adding new ambulances to the system, comparing the results to the ones of the original situation. It is shown that, as a larger number of ambulances are decentralized, mean response times, fractions of calls served by backups and other performance measures of the system are improved, while the ambulance workloads remain approximately constant. However, total decentralization as suggested by the system operators of SAMU-192 may not produce satisfactory results. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Emergency medical system; Ambulance deployment; Hypercube model; Queueing theory; Case study
1. Introduction One of the major concerns of Emergency Medical Services (EMS) is to rapidly provide first care medical assistance to the victims. The time elapsed between an emergency call and its assistance, called response time, is one of the main factors that influence system performance. In urban areas, this time lapse depends on different aspects of calls and the EMS system such as: type and location of the request, ∗ Corresponding author. Fax: +55 16 260 8240.
E-mail address: [email protected] (R. Morabito). 0305-0548/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2005.03.022
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number and location of ambulances, system congestion, local traffic conditions, weekday and time, etc. The United States EMS Act sets some standards: 95% of the emergency requests should be served within 10 min in urban areas and within 30 min in rural areas [1]. Similar regulations are found in other parts of the world; for example, in London and Montreal, the regulation states that 95% of the requests should be served within 14 and 10 min, and 50% and 70% of the requests should be served within 8 and 7 min, respectively [2,3]. In Brazil, however, there is no specific regulation that specifies limitations for response times in EMSs. When designing or modifying the configuration of EMSs, managers should balance the benefits of improving user service at the expense of increasing the investment in the system. This trade-off is typical in services as well as manufacturing systems (e.g., Bitran and Morabito [4]). Several studies are found in the literature proposing approaches to rationalize the usage of available resources and improve user service. Nevertheless, a number of them do not directly consider the probabilistic nature of user arrival and service processes and the fact that ambulances are not always available for servicing a call. Examples of probabilistic approaches for ambulance deployment appear in Brandeau and Larson [5], Eaton et al. [6], Goldberg et al. [7] and Fujiwara [8]. Surveys reviewing the most important studies in the last decades are found, for instance, in Kolesar and Swersey [9], Louveaux [10], Swersey [11] and Brotcorne et al. [12]. It should be noted that an accurate model for EMSs can be quite complex since elements of uncertainty appear in time, location and amount of required services (e.g., demands are temporally and spatially distributed) and there are particular dispatching policies. The hypercube queueing model developed by Larson [13] and extended by other authors [11] is an effective descriptive model for planning server-to-customer systems. Given a system configuration, it is able to evaluate a variety of performance measures relevant for decision-making. It is not an optimization model in the sense that it determines an optimal configuration for the system, but it can provide a reasonably complete evaluation for each suggested configuration. It can also be combined in optimization approaches to deal with probabilistic location problems. For instance, Batta et al. [14] suggested its use in an iterative procedure as an alternative to relax the assumption of independence among the servers in the maximum expected covering location problem, MEXCLP (Daskin [15]). Chiyoshi et al. [16] analyzed non-homogeneous servers and compared MEXCLP and the adjusted maximum expected covering location problem, AMEXCLP [14]. Saydam and Aytug [17] developed a genetic algorithm that combines MEXCLP with a hypercube approximation algorithm developed by Jarvis [18] in order to solve MEXCLP with increased accuracy. Gendreau et al. [19,2] used tabu search in similar contexts. Galvão et al. [3] applied simulated annealing in the solution of MEXCLP and the maximum availability location problem, MALP [20]. In the last decades, different examples of application of the hypercube model in urban service systems have been reported; for instance, the social service visit program [21], the ambulance location in Boston [5] and Greenville [22] and the police patrolling in New Haven [23] and Orlando [24]. In Brazil, some examples are the assistance of power interruptions, location and planning of fire rescue vehicles and the imbalance of workload among the ambulances on a highway [25]. Other references related to applications and extensions of the hypercube model can be found in Halpern [26], Jarvis [27] and Swersey [7]. The present paper studies the hypercube model application to the urban EMS of Campinas (SAMU192), a city of almost one million people located in the state of Sao Paulo. In its original configuration, the system had all ambulances centralized at a single base adjacent to a general hospital downtown. This study analyzes the effects of decentralizing ambulances and adding new ambulances to the system, comparing the results to the ones of the original situation. It is shown how much mean response times,
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N E C W S
base
729
CAMPINAS 2 • area: 796 km • estimated population in 1998: 908.906 inhabitants SAMU-192 • monthly average request: 3.000 • doctors: 42 • nurses: 58 • drivers: 65 • hospitals: 5 public hospitals 8 private hospitals
Fig. 1. Coverage area of SAMU-192.
fractions of calls served by backups and other performance measures of the system are improved, as a larger number of ambulances are decentralized, while the ambulance workloads remain approximately constant. However, total decentralization as suggested by the system operators of SAMU-192 may not produce satisfactory results. This case study paper is organized as follows: in Section 2 system SAMU-192 is briefly described, in Section 3 the application of the hypercube model is discussed, in Section 4 the results obtained with the model for different scenarios are analyzed and compared to the original system configuration. Finally, in Section 5 concluding remarks and perspectives for future research are presented.
2. SAMU-192 The World Health Organization has defined provision of basic life support to all risk situations involving people and goods as a main objective of an EMS. In Brazil, since the beginning of the nineties, efforts have been made in order to impel the organization of urban EMSs. Most Brazilian systems are public services and some of them are similar to the French model SAMU (Service d’Aide Médicale Urgente), operating for more than 20 years. SAMU’s serve a city or a region containing several cities and are integrated into a group of public hospitals. They have a phone center that receive all requests, teams of doctors, nurses and drivers, ambulances equipped for basic life support (basic support vehicles, BSVs), and units of advanced life support (advanced support vehicles, ASVs). In 1994, the EMS of Campinas was reorganized yielding the SAMU-192, its aim being the urban regulator center of medical emergencies. In 1998, when this study began, the system had 10 ambulances (two ASVs and eight BSVs) centralized at the operational base of the system, where the phone center was installed. Campinas was divided into five different areas (North, South, East, West and Central) in accordance with the covering areas of main municipal health centers. The system provided a queueing service for incoming calls if all ambulances were busy at the time of their requests (the limitation of the waiting line was equal to one user per operating vehicle). Fig. 1 describes the SAMU coverage area and its divisions, together with additional information about the system. One of the special concerns of the system manager was that the ambulance workloads were too high in certain periods of the day, causing long waiting times for users. To apply the hypercube model in SAMU-192, the data collection was divided into two stages: first, the type and amount of past services were determined, then, a detailed sample was collected to verify the critical model assumptions. In a 1-year data collection, major differences in the numbers of requests
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Table 1 Performance measures of SAMU-192 Measures
Peak
Arrivals/day Inter-arrival time (min) Response time (min)a Time on scene (min) Service time (min) a Travel
24 h
BSV
ASV
BSV
ASV
20 10 13 37 66
2 92 11 39 63
70 14 12 37 64
10 106 10 39 62
time + setup time.
along the months were not observed, except in January and July. These are periods of school recess and vacations, when a slight decrease in the number of requests was noticed. The weekends also presented reduced demands, which is typical in urban EMSs. Then a detailed 10-day sample was randomly selected, rejecting January and July, and weekends. This sample had 814 requests spatially distributed as follows: 17.1% from North area, 16.5% from South area, 17.6% from East side, 32.2% from West side and 16.7% from Central area. The occurrences of peak periods in the system were also investigated: the statistical analysis indicated that the time period between 10 am and 2 pm corresponded to peak hours (on average, 30% of the requests occurred within this period). The results presented in this and the next sections are based on 209 requests (190 basic and 19 advanced calls) of this sample that occurred during peak periods. For more details of the sample, the reader may consult Takeda [28]. Table 1 shows some performance measures for the peak periods and for the 24-h days of the sample.
3. Hypercube model The hypercube model considers geographical and temporal complexities of the region and is based on spatial queueing theory and Markovian analysis approximations. Basically, the idea is to expand the state space description of a simple multi-server queueing system in order to represent each server individually and incorporate more complex dispatching policies. Once the model is calibrated, a number of performance measures of interest to how the system is managed can be estimated, either region-wide or for each server or region, such as workloads, mean response times, fraction of dispatches of each server to each region, etc. The name hypercube derives from the state space describing the status of the ambulances: each ambulance can be free (0) or busy (1) in a time instant. A particular state of the system is given by the entire listing of ambulances that are free and busy. For instance, the state 011 corresponds to a 3-server system, with ambulances 1 and 2 busy, and 3 free (note that 011 describes the ambulances states from right to left). Then, the state space is given by the vertices of a cube; if the system has more than three ambulances, that is a hypercube.
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Table 2 Mean arrival rates of atoms Atom
j (calls/hour)
1 - NB 2 - NA 3 - SB 4 - SA 5 - EB 6 - EA 7 - WB 8 - WA 9 - CB 10 - CA
0.8535 0.1035 0.8535 0.0776 0.8535 0.0776 1.5001 0.1293 0.8276 0.1293
Total
5.4054
3.1. Model assumptions There are nine basic assumptions for the application of the model [21]. We now describe these assumptions for the SAMU-192 study. Geographical atoms: The SAMU-192 area was divided into five non-overlapping areas, as shown in Fig. 1. In addition, each area was partitioned in two different and independent areas: one generating only advanced requests and the other only basic requests. This procedure is called “layering’’ of areas [21] and it is useful for representing situations admitting implicit or explicit priority criteria in their services, where specialized teams are dispatched to serve high acuity victims. This way, a total of NA = 10 geographical atoms were defined for the system: North B (NB), North A (NA), South B (SB), South A (SA), East B (EB), East A (EA), West B (WB), West A (WA), Center B (CB), Center A (CA). The first preference ambulances of advanced request atoms are the ASVs, and the BSVs are the backups. Similarly, the first preference ambulances of basic request atoms are the BSVs, and the ASVs are the backups. Independent Poisson arrivals: Emergency calls of each atom j were generated as a Poisson process, independently from other atoms, with mean arrival rates j (j = 1, 2, . . . , NA ) as showed in Table 2 (these values were estimated from the sample). Although this assumption seems to be very restrictive, it is frequently satisfied in several real systems, as pointed out by a number of authors. Kolmogorov–Smirnov and 2 goodness-of-fit tests were applied to verify the hypothesis of Poisson arrival. They were unable to reject such hypothesis with 5% of significance. Travel times: The mean ambulance travel times ij (i, j = 1, 2, . . . , NA ) between atoms i and j should be known or can be estimated using geometrical probability concepts. In this study, they were taken from the databases of SAMU-192 (their values are detailed in [28]). It is interesting to note that the ASVs mean travel times were smaller than BSVs, due to the urgency degree. Servers: SAMU-192 was operating with eight BSVs and two ASVs (i.e., with N = 10 ambulances). All these ambulances remained in the Central base when idle and they could be dispatched to any atom to accomplish a service.
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Table 3 Fixed-preference dispatching matrix (original configuration) Atom
1 - NB 2 - NA 3 - SB 4 - SA 5 - EB 6 - EA 7 - WB 8 - WA 9 - CB 10 - CA
Preference dispatching 1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
9 1 10 2 3 1 4 2 5 2
10 2 3 1 4 2 5 1 6 1
3 10 4 5 7 6 8 9 8 10
4 7 5 3 10 9 6 8 10 5
5 3 6 10 8 4 7 3 9 6
6 4 7 9 5 10 3 5 4 8
7 9 8 4 6 5 9 10 3 7
8 5 9 6 9 3 10 4 7 9
2 8 1 7 2 8 2 6 1 3
1 6 2 8 1 7 1 7 2 4
Vehicles
Server locations: The server location matrix (L) was easily determined because all ambulances, when available, remained in the Central base waiting for a call. Server assignment: In response to each call, exactly one ambulance was dispatched from the Central base to the local of the request. If all ambulances were busy, the call was queued if the waiting line was not greater than the number of ambulances in operation; otherwise the call was lost (i.e., transferred to another EMS). Fixed-preference dispatching: There was a list of dispatching preferences for each atom. If the first preference ambulance of the list was available, it was dispatched; otherwise, the next ambulance of the list was dispatched. According to SAMU-192’s operational policy (discussed above), for advanced call atoms, the list contained ASVs as first preference ambulances, followed by BSVs as backups (similarly for basic call atoms). Since there were no preferences among the ASVs (and no preferences among the BSVs), for simplicity, the method of random generation of fixed-preference dispatching matrixes proposed in [22] was used in the present study. Table 3 presents an example of a randomly generated fixed-preference dispatching matrix using that procedure. In the matrix, the two ASVs are numbered 1 and 2, and the eight BSVs are numbered 3, 4, 5, 6, 7, 8, 9 and 10. Service times: The service time for a call is composed by the setup time of the server (it was very small—less than 1 min), the travel time from the base until the place of the occurrence (on average, 11 min for ASVs and 13 min for BSVs in the sample), the on-scene time (39 min for ASVs and 37 for BSVs) and the travel time back to the base (13 min for ASVs and 16 min for BSVs). The mean service time for urgent calls resulted in 63 min for the case of services accomplished by ASVs and 66 min for the case of BSVs, as pointed out in Table 1. Table 4 shows the mean service times for each ambulance and their corresponding standard deviations, coefficients of variations and mean service rates n , n = 1, 2, . . . , N (estimated from the sample). The hypercube model assumes that the standard deviation of the service time is approximately equal to the mean (since the mathematical analysis assumes negative exponential service times). Applying
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Table 4 Mean service times and rates of ambulances Ambulance
Service timea
Standard deviation
Coefficient of variation
n b
1 2 3 4 5 6 7 8 9 10 ASV BSV Total
61 64 63 70 61 66 64 61 70 70 63 66 65
40 44 52 48 44 58 50 40 58 56 42 51 49
0.7 0.7 0.8 0.7 0.7 0.9 0.8 0.7 0.8 0.8 0.7 0.8 0.8
0.9836 0.9375 0.9524 0.8571 0.9836 0.9091 0.9375 0.9836 0.8571 0.8571 0.9606 0.9172 0.9259
a minutes, b calls/hour.
Kolmogorov–Smirnov and 2 goodness-of-fit tests for the service processes, they were unable to reject the hypothesis of exponentially distributed service times with 5% of significance. Variance analysis tests were also applied to verify the hypothesis of equal service times among the 10 ambulances. The tests rejected such hypothesis with 5% of significance; therefore, these service times were treated as different in the model. Service time dependence on travel time: Variations in the service time that are due to variations in travel time are admitted to be second order when compared to variations of setup and on-scene times. This hypothesis, which limits the applicability of the model, is frequently verified in urban services and less observed in rural services. It should be noted that, for each ambulance of SAMU-192, the mean travel time is short in comparison to the mean service time (on average, about 18%), and that variations in travel time give a small contribution to the variations in service time (Table 4). 3.2. Steady state probabilities The equilibrium probability equations are defined supposing that the system attains steady state. Each possible state of the system is represented by a vector with N = 10 elements (ambulances) assuming values 0 or 1 (free or busy). Therefore, there are 210 = 1024 possible states in the system, besides the 10 states that define the queue, namely S11 , S12 , . . . , S20 . For instance, the equilibrium probability equation for the intermediate state 1000000101, where the ambulances 1, 3 and 10 are busy, is: [(2 + 4 + 6 + 8 + 10 ) + (7 + 3 + 5 ) + (9 ) + (1 ) + (1 ) + (3 ) + (10 )]p1000000101 = 2 p1000000111 + 4 p1000001101 + 5 p1000010101 + 6 p1000100101 + 7 p1001000101 + 8 p1010000101 + 9 p1100000101 + (2 + 6 )p1000000100 + (3 + 5 )p1000000001 + 3 p0000000101 .
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Idle System
1000000111
2 + 4 + 6 + 8 + 10 1000000100
µ12
2 +6
7 + 3 + 5 µ4 1000010101 9 9 µ5
µ1
1000000001
3 + 5 µ3
1000000101
µ ∝56 µ ∝75
3 µ 10 0000000101
1000001101
1000100101
1001000101
µ58 ∝
91 µ9
1010000101
1100000101
Fig. 2. Flows in and out of state 1000000101.
Fig. 2 illustrates the corresponding flows in and out of this state. The equilibrium probability equations for other states are similar. Substituting any of the equations by the normalization equation (sum of all state probabilities should be equal to 1), the result is a determined system with 210 linearly independent equations. 3.3. Performance measures A variety of system performance measures can be defined as a function of the equilibrium distribution, as discussed in Larson [13]. For example, the probabilities of finding the system empty and saturated are p0000000000 = 0.003 and ps = 0.09, respectively. The loss probability is only 0.0001. The workload n of ambulance n represents the fraction of time that ambulance n is busy, and it is obtained easily through the sum of the equilibrium probabilities of the states in which this ambulance is busy. The workloads of BSVs and ASVs are 63% and 39%, respectively. The overall workload was 58% (which is very close to the mean value of the databases of SAMU-192, 60%). Average travel times: Different travel times are obtained by the model starting from the origin-destiny matrix of travel times (ij ). For example, the mean travel time of the system (T ) is estimated by: NA [1] [1] T = N n=1 j =1 fnj tnj + pS TQ , where fnj is the fraction of all dispatches that send ambulance n, when available, to atom j, tnj is the mean time required to ambulance n, when available, to travel to atom j, pS is the system saturation probability and TQ is the mean travel time to a queued call. This measure resulted in T = 13.6 min, which is sufficiently close to the one observed in the sample (14.5 min, that is, a deviation of only 6.2%).
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Other mean travel times can be estimated by the model, such as the mean travel time of ambulance n (T U n ) and the mean travel time to atom j (Tj ) (the details are found in Larson and Odoni [21]). The mean travel times computed by the model were 10.9 min for ASVs and 14.2 min for BSVs, that is, deviations of only 3.3% and 5.1% from the sampled values. Regarding the mean travel times to the atoms, the relative deviations were also reasonably low (0.1% for advanced call atoms and 5.5% for basic call atoms). These small deviations showed that the model is able to produce accurate performance measures of the system to support decision-making. 3.4. Calibration of service times In certain EMSs, travel times may represent a considerable part of service times. In such cases, it may be advisable to adjust the service times by means of a calibration process, which can be performed using a simple iterative procedure described in Larson and Odoni [21]. Basically, the procedure consists of verifying if there are significant differences among the input mean service times and the output mean service times (computed by the hypercube model). In this case, the hypercube is resolved using the computed mean service times as inputs, and so on, until the differences among input and output values are sufficiently small. This procedure was used to produce the results presented below.
4. Evaluation of alternative scenarios As mentioned above, one of the major concerns in EMSs is to reduce the response time, which is composed of the setup time, the travel time and the waiting time (eventually spent in the queue). The waiting time is a function of the mean travel time to a queued call (TQ ) and the probabilities of satured states S11 , S12 , . . . , S20 . In SAMU-192, the setup time is very short (around 1 min) and such probabilities are relatively low (order of 10−3 in the original configuration), so that the mean response time almost coincides with the mean travel time of the system (T ). For the sake of simplicity, in the analyses below the impacts of ambulance decentralization are evaluated in terms of mean travel times (instead of mean response times). 4.1. Ambulance decentralization The Brazilian legislation does not impose that the ambulance bases of urban EMSs should be close to a hospital or any other type of health unit. However, when locating bases in these units, there are cost savings using such existent infrastructure, besides the possible collaboration of the unit’s professionals with the EMS teams. Therefore, the scenarios for ambulance decentralization of SAMU-192 considered below assume that the bases of the ambulances should be municipal hospitals or health units located in the five areas (North, South, East, West and Central). Several operational alternatives of ambulance decentralization were considered in this study, from the relocation of a single ambulance until scenarios with total decentralization. One-ambulance decentralization: Scenario 1 was the simplest alternative of decentralization, where one of the BSVs (e.g., ambulance 5) was moved to area WB (note in Table 2 that this area had the highest demand of the system). The other ambulances remained in the Central base as backups for area WB, as shown in Fig. 3. In this way, ambulance 5 in WB becomes the first preference server for basic calls
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R.A. Takeda et al. / Computers & Operations Research 34 (2007) 727 – 741 1 → Central A 2 → Central A 3 → Central B 4 → Central B 5 → West B 6 → Central B 7 → Central B 8 → Central B 9 → Central B 10 → Central B
ASV BSV
Fig. 3. Scenario 1 (one BSV in WB).
Table 5 Workloads and the mean travel times of ambulances Ambulance
1 2 3 4 5 6 7 8 9 10 Average ASV Average BSV SYSTEM
Scenario 0
Scenario 1
Deviation
Workload
Travel time (min)
Workload
Travel time (min)
Minutes
%
0.39 0.39 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.39 0.63 0.58
10.9 10.9 14.2 14.2 14.2 14.2 14.2 14.2 14.2 14.2 10.9 14.2 13.6
0.39 0.41 0.64 0.63 0.66 0.64 0.65 0.65 0.64 0.63 0.40 0.64 0.58
11.3 10.6 14.2 14.2 8.8 14.2 14.2 14.2 14.2 14.2 11.0 13.5 13.0
0.4 −0.3 0.0 0.0 −5.4 0.0 0.0 0.0 0.0 0.0 0.1 −0.7 −0.6
3.7 −2.8 0.0 0.0 −38.0 0.0 0.0 0.0 0.0 0.0 0.9 −5.0 −4.4
in WB, the seven remaining BSVs in the Central base become the second preference servers for basic calls in WB (i.e., the backups of ambulance 5), and the two ASVs in the Central base become the third preference servers for basic calls in WB (the backups of the backups). Moreover, ambulance 5 in WB is the backup of the ASVs for advanced calls in WA, and the BSVs in the Central base are the backups of ambulance 5 for advanced calls in WA. After making appropriate modifications in the server location matrix (for ambulance 5), in the preference dispatching matrix generator procedure of Burwell et al. [22] and in the equilibrium probability equations, the model generated workloads similar to the original configuration (namely scenario 0), as shown in Table 5. Note that ambulance 5 has a slightly higher workload given that it became the first preference server of atom WB. The mean travel times of the ambulances decreased evenly, except for ambulance 5 whose reduction was significant (38%). The overall mean travel time decreased 4.4% (from 13.6 to 13.0 min), corresponding to 0.9% for ASVs and 5.0% for BSVs. Two-ambulance decentralization: Scenario 2 enlarges the decentralization to two BSVs. There are several possible locations for these ambulances; however, experience of the system operators and the
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Table 6 Mean travel times of ambulances (T U n ) Ambulance
1 2 3 4 5 6 7 8 9 10 11 12 13 Average ASV Average BSV SYSTEM
Travel times (minutes) Scenario 0
1
2
3
4
5
6
7
8
9
10.9 10.9 14.2 14.2 14.2 14.2 14.2 14.2 14.2 14.2
11.3 10.6 14.2 14.2 8.8 14.2 14.2 14.2 14.2 14.2
11.4 10.5 10.0 13.9 9.2 14.0 14.1 14.0 13.8 13.8
11.3 10.6 9.0 9.7 13.8 9.7 13.9 14.0 13.8 13.9
10.9 10.9 11.7 12.4 10.7 12.0 9.9 12.7 13.0 12.9
10.9 10.9 11.7 12.8 10.9 10.5 12.0 12.0 10.5 12.1
10.9 10.8 11.6 11.3 11.4 10.5 10.4 12.7 12.6 11.6
10.0 9.4 11.0 9.9 9.5 9.6 9.3 11.4 11.2 11.2 10.5
7.7 7.6 8.9 9.2 9.1 8.9 8.9 8.1 8.2 8.8 8.5 8.7
10.9 14.2 13.6
11.0 13.5 13.0
11.0 12.9 12.5
11.0 12.2 12.0
10.9 11.9 11.7
10.9 11.6 11.5
10.9 11.5 11.4
9.7 10.4 10.3
7.7 8.7 8.6
7.5 7.1 8.8 8.4 7.7 7.9 7.8 8.1 8.2 8.7 8.2 8.2 6.6 7.1 8.2 7.9
Table 7 Mean travel times to atoms (Tj ) Atom
1 - NB 2 - NA 3 - SB 4 - sa 5 - EB 6 - EA 7 - WB 8 - WA 9 - CB 10 - CA Average ASV Average BSV
Travel times (min) Scenario 0
1
2
3
4
5
6
7
8
9
16.0. 13.1 15.9 13.0 15.9 13.0 15.8 12.8 5.5 4.3 11.3 13.8
16.2 13.2 15.9 13.1 16.1 13.1 12.5 12.4 5.8 4.5 11.3 13.3
16.5 13.2 15.9 13.1 16.4 13.2 10.2 12.1 6.2 4.5 11.2 13.0
16.2 15.2 15.8 12.9 16.2 13.1 9.8 12.6 6.1 4.0 11.2 12.8
13.0 12.7 12.4 12.6 12.9 12.7 13.3 12.5 5.9 4.3 11.0 11.5
13.3 12.8 12.5 12.6 13.3 12.7 11.1 12.2 7.6 4.7 11.0 11.5
11.0 12.5 13.2 12.7 13.1 12.7 11.5 12.3 9.0 4.9 11.0 11.6
10.0 12.1 12.2 12.3 12.3 12.3 10.8 12.1 7.1 4.1 10.6 10.5
7.7 8.7 9.3 9.5 10.5 10.8 9.8 8.6 6.3 3.4 8.2 8.7
7.5 8.2 9.1 8.7 8.9 9.5 8.6 8.6 6.3 3.4 7.7 8.1
high demand of area WB suggested repositioning the two BSVs in WB. The results were a little better than the ones in scenario 1 (e.g., the overall mean travel time were 12.5 min), as shown in Tables 6 and 7 and Figs. 4–6.
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Workload
1.0 ASV
0.8
BSV
SYSTEM
0.6 0.4 0.2 0.0 0
1
2
3
4
5
6
7
9
8
Scenario
Travel times for ambulances (minutes)
Fig. 4. Ambulance workloads in scenarios 0, 1, . . . , 9.
15
10
ASV
BSV
SYSTEM
5 0
1
2
3
4
5
6
7
8
9
Scenario
Fig. 5. Mean travel times of ambulances in scenarios 0, 1, . . . , 9.
Travel times to atom (minutes)
15
10
ASV
BSV
5 0
1
2
3
4 5 Scenario
6
7
8
9
Fig. 6. Mean travel times to atoms in scenarios 0, 1, . . . , 9.
Three-ambulance decentralization: Scenario 3 took the configuration of scenario 2 and moved another BSV to area NB (close to a large public hospital). The results were similar to the ones of scenarios 1 and 2, reducing a little more the mean travel times of the ambulances, as depicted in Tables 6 and 7 and Figs. 4–6. The overall mean travel time reduced 11.7% compared to scenario 0 (from 13.6 to 12.0 min) and 3.8% compared to scenario 1. Other scenarios: Other decentralized configurations were studied. In scenario 4, four BSVs were decentralized in areas NB; SB; WB and WB. This configuration was suggested by the operators of SAMU-192, providing even better results than previous scenarios (e.g., the overall mean travel time went down to 11.7 min). Other configurations with four decentralized vehicles were appraised but all of them yielded worse values. Two other configurations decentralizing five and six ambulances
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(scenarios 5 and 6) were also analyzed, resulting in system travel times of 11.5 and 11.4 min, respectively. Although most of their values are similar to scenario 4 (see Tables 6 and 7 and Figs. 4–6), there are some that are better; for instance, their proportions of advanced calls serviced by BSV backups are lower. Other configurations decentralizing from 7 to all ambulances were also investigated but they did not provide better values. In all these scenarios the workloads remained around 0.58. The detailed results are presented in [28]. 4.2. Increasing the number of ambulances Scenario 7 considers the addition of a new BSV (numbered 11) in scenario 6, located in the Central area. The workloads decreased 18% on average, the mean travel times of BSVs and ASVs reduced 14% and 9%, respectively, and the overall travel time 9.6% (10.3 min), compared to the results of scenario 6. Scenario 8 considers a new BSV (numbered 12) in scenario 7, located in area EB, and scenario 9 considers a new ASV (numbered 13) in scenario 8, located in WA (note in Table 2 that areas CA and WA had the highest arrival rates of advanced calls). The benefits of increasing the number of ambulances (combined with partial decentralization) were sensitive in all performance measures (as shown in Tables 6 and 7 and Figs. 4–6), clearly at the expense of additional costs and investments. For instance, considering that the monthly operational costs of each BSV and ASV are approximately US$8,000 and US$18,000, respectively, scenario 9 (with 13 ambulances) has an additional cost of US$34,000 per month (34% more than the original scenario) but it reduces the overall travel time by 42% (from 13.6 to 7.9 min). This way, the resulting mean response time of the system (including the setup time) becomes close to international standards reported by other studies in the literature (e.g., [29,8] reported 8.8 min). 4.3. Comparison of scenarios In order to help the comparison among the scenarios, the tables and figures below compile some of the results obtained after decentralizing ambulances (scenarios 1,2,…,6) and adding new ambulances to the system (scenarios 7–9). Fig. 4 depicts the variations of the workloads of ASVs and BSVs compared to the original system (scenario 0). Observe that, as the decentralization of ambulances increases (from scenario 0–6), the workloads do not change much. On the other hand, as expected, the workloads reduce substantially by adding new ambulances (scenarios 7–9). Regarding travel times, partially decentralized systems produce better results than centralized or totally decentralized configurations and, in practice, this corresponds to shorter response times. As expected, these observations were also verified after increasing the number of ambulances (Tables 6 and 7). Note, for example, that mean travel times to atoms were reduced by up to 32% for advanced calls and 41% for basic calls (Table 7). Other performance measures were also considered for the comparison of the scenarios; for example, the proportion of inter-atom dispatches of ambulances and the fraction of advanced calls serviced by a BSV (i.e., a backup). Note that as this fraction increases, users requesting advanced support may receive inferior quality medical assistance since BSVs are less equipped than ASVs. Similarly to travel times, partially decentralized systems yield smaller fractions of acute calls serviced by a BSV than the original configuration, therefore also improving user service. For illustration, for the original configuration the fraction was 19% while for scenario 9 it became only 5%.
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5. Concluding remarks This paper studied the application of the hypercube queueing model to the urban EMS of Campinas (SAMU-192) in Brazil. In its original configuration, SAMU-192 had all ambulances centralized in its central base. The hypercube model was successfully applied to analyze ambulance deployment, in particular, the effects of decentralizing ambulances and increasing their number. The results showed that the simple relocation of one single ambulance decreases system travel times (as one would expect). As a larger number of ambulances are decentralized, travel times and other performance measures are even better, while the workloads remain approximately constant. For instance, the mean travel time of the system reduces from 13.6 min (centralized configuration) to 11.4 min (six ambulances decentralized). The fraction of advanced calls assisted by backups (BSVs) also decreases from 19% to 11%. On the other hand, total decentralization as suggested by the system operators of SAMU-192 does not produce satisfactory results. Increasing the number of ambulances in the system (combined with partial decentralization) leads to significant improvements in system performance measures (as expected), but obviously at the expense of additional costs and investments. For example, the mean travel time of the system reduces from 13.6 min (scenario 0 with 10 ambulances) to 10.3, 8.6 and 7.9 min (scenarios 7–9 with 11, 12 and 13 ambulances, respectively). Considering that the monthly operational costs of each BSV and ASV are approximately US$8,000 and US$18,000, respectively, the benefit-cost relationship of scenarios 9 and 0 can be roughly stated as: an increase of 34% in the operational cost of SAMU-192 reduces the mean travel time of the system by 42% and places the mean response time close to international standards. The five areas of SAMU-192 were originally designed around municipal health centers and they do not reflect demand patterns of emergency calls. An important perspective for future research is the application of the hypercube model for planning new reporting areas (redistricting problem). Another interesting line of research is to integrate the hypercube model in an optimization procedure in order to determine optimal ambulance decentralization.
Acknowledgements The authors thank SAMU-192 of Campinas for providing the data, especially Dr. Arine Campos O. Assis (former system coordinator and presently a consultant of the Brazilian Ministry of Health) for her support and incentive, and the two anonymous referees for their useful comments and suggestions. This research was partially sponsored by CNPq (Grant 800773/91-8).
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