Performance Evaluation of Tehran-Qom Highway Emergency Medical Service System Using Hypercube Queuing Model

Transcription

1 Western New England University From the SelectedWorks of Mohammadsadegh Mobin Summer May 2, 2015 Performance Evaluation of Tehran-Qom Highway Emergency Medical Service System Using Hypercube Queuing Model Mohammadsadegh Mobin, Western New England University Zhaojun Li, Western New England University Maghsoud Amiri Available at:

2 Proceedings of the 2015 Industrial and Systems Engineering Research Conference S. Cetinkaya and J. K. Ryan, eds. Performance Evaluation of Tehran-Qom Highway Emergency Medical Service System Using Hypercube Queuing Model Mohammadsadegh Mobin, Zhaojun Li Western New England University, MA, USA Maghsoud Amiri Allameh Tabataba i University, Tehran, Iran Abstract This paper studies the ambulance deployment of the Tehran-Qom highway emergency medical service (EMS) system. The response time of EMS to an emergency call can have impact on the users in terms of the life and death, and this performance metric is evaluated using the proposed hypercube queuing model. The model is based on spatially distributed queuing theory and Markovian analysis approximations, and it is one of the most effective approaches for analyzing emergency response systems. In this research, performance measures of the EMS including workloads and mean response times are evaluated using the hypercube model. Modification to the existing system configuration is applied to improve the system performance. We also analyze the effects of small variation in the system inputs, such as user demands rates, to investigate the robustness of the outputs. The results demonstrate the effectiveness of the proposed model in supporting design and operational decisions, in particular in reducing the ambulances workload imbalances as well as mean user response time. Keywords Ambulance Allocation Problems, Hypercube Queuing Model, Emergency Medical Service (EMS) Notations N: Number of servers (in this problem is equal to number of facilities) N A : Number of atoms K: Set of all system states ( K =2 N ) f nj : The fraction of all dispatches that send ambulance n to atom j (n = 1,2,, N) N A λ: Total rate of demand (λ = j=1 λ j ) λ j : Demand rate in atom j(j = 1,2,, N A ) μ i : Service rate for server i (i = 1,2,, N) P B : The equilibrium probability of state B P p : System loss probability (probability of losing demand) τ ij : The mean ambulance travel time between atoms i and j ρ i : Workload of ambulance i f nj : The fraction of all dispatches that send ambulance n to atom j E nj : The set of states in which ambulance n is assigned to any service request from atom j TU n : Mean travel time for ambulance n T: Overall mean travel time T j : Mean travel time to atom j F n : Fraction of all dispatches of each server that send to other atoms. 1. Introduction Recent increment in costs of medical services, increasing call volumes and worsening traffic conditions on highways are making ambulance location allocation problems as one of the most important issues on emergency medical service (EMS) management. Designing ambulance operations refers to any system-status management approaches by which a dispatcher repositions idle ambulances in order to compensate for others that are unavailable or districting the area that ambulances can cover. The main purpose of designing or modifying the configuration of emergency response

3 systems is to keep the balance between the benefits of improving system (e.g. shorter mean response time, fewer lost calls, better quality of clinical, pre-hospital care, etc.) and the expenses of increasing the investment (e.g. more ambulances, stations, better training of personnel, improved screening of calls) in the system. Furthermore, EMS providers face the problem of allocating a fixed number of ambulances among a set of bases with the objective of ensuring adequate coverage and reasonable response times in an emergency situation. The response time of a call is the elapsed time from when the call is received to when an ambulance arrives at the scene of accident. The mean user response time is usually considered as the main performance measure and the EMS providers are usually interested in minimizing the fraction of the lost calls and the calls with response time longer than some threshold, which is usually considered around ten minutes in the literature [1-4]. The reasons for lost calls and excessive delays in the response time are the conflict between randomly occurring service requests and occasional unavailability of the closest ambulances. The balance of ambulance workloads, the fraction of calls not serviced by the EMS (loss probability), and the fraction of calls not serviced within a predetermined threshold (i.e., fraction of calls with response times exceeding T minutes) are commonly used performance measures to evaluate the EMS systems. In this study, we analyze the ambulance deployment of an EMS located in portion of the Tehran-Qom highway which connecting the capital city of Iran, Tehran and the city of Qom. We only considered the portion of the highway which is in Tehran province territory and cover by the Tehran Emergency Center. Five ambulances are located in five fixed bases along the highways in this EMS system. The operations center, located in the Tehran Emergency Center, handles the calls and dispatches the ambulances. In this highway EMS system, immediately after receiving an emergency call, an ambulance goes to the place of accident and provides the first medical treatment to the individuals involved in an accident, transports them to the nearest hospital (if necessary), and then goes back to its home base on the highway. These systems are typically zero-line capacity, and they operate within particular ambulance dispatching policies. It means that only predefined ambulances can be dispatched to the specific region on the highway (partial backup), mainly due to the limitations of travel distance and time. The resulting system can be conceptualized as a spatially distributed multi-server loss queueing system, with a specific and limited dispatching policy, which follows the assumptions of the basic hypercube model developed by Larson [5], the well-known and effective descriptive model for evaluating the server-to-customer systems. The hypercube model is based on spatially queuing theory and Markovian analysis approximation. The basic idea is to expand the state space description of a simple multi-server queueing system (e.g. M/M/N, where N is the number of servers) in order to represent each server individually and incorporate reasonably complex dispatching policies [5,6]. In the literature, hypercube model has been extensively applied. Examples of applications of the hypercube model in urban EMS system can be found in [7-10]. Some studies have extended the original hypercube model for application to EMS systems on highways. For example, Mendonça and Morabito [2] extended hypercube queuing model with partial backup dispatching, Iannoni and Morabito [11] modified hypercube queuing model to consider multiple dispatching of different types of servers, Atkinson et al. [12] proposed a hypercube queuing loss model with customerdependent service rates. Davoudpour et al. proposed a new probabilistic coverage model for ambulances deployment with hypercube queuing approach [13]. Other studies combining exact and approximate hypercube models with optimization procedures to solve probabilistic location problems have been reported. For example, Iannoni et al. [4] integrated a multiple dispatch partial backup hypercube model into a genetic algorithm (GA) in order to determine the optimal response areas for the ambulances. Geroliminis et al. [14] proposed a hybrid hypercube-genetic algorithm approach for deploying many emergency response mobile units in an urban network. Iannoni et al. [15] extended this approach to optimize the location of the ambulances and their corresponding coverage areas (districts). Their approach is known as a location and districting GA/hypercube algorithm. In this paper, we applied the Hypercube model developed by Mendonça and Morabito [2] to evaluate the performance of Tehran-Qom EMS system with partial backup and zero-line capacity. Geographical and temporal complexities of the region are considered in the model we applied. The applied model requires the solution of linear systems of k(2 N ) equations, where the variables involved are the equilibrium state probabilities of the system. After obtaining the steady-state distribution, some performance measures such as server workloads, mean server response times etc. are calculated to evaluate the performance of this EMS system. After modifying the current configuration of the EMS system the performance measures of system such as workload unbalancing are improved. This study is the first highway ambulance deployment study in Iran which is partially supported by Tehran Emergency Center.

4 The paper is organized as follow: Section 2 presents a brief description of Hypercube model and discusses how it can be adopted to evaluate the EMS systems. In section 3, the hypercube model is applied in Tehran-Qom highway and performance measures of EMS system are calculated. An alternative configuration of the system is presented in Section 4. The effects of small variations in the system performance are presented in section 5 and finally, in section 6, concluding remarks and perspective for future research are presented. 2. Hypercube model in ambulance deployment Ambulance deployment methodologies which have received a great deal of attention in the literature can be categorized into two primary areas including prescriptive and descriptive methods [16]. Prescriptive methods search over all possible ambulance allocations to identify a subset of allocations and they typically use simplified performance measures to allow for efficient search procedures. Important early work on prescriptive models includes Toregas et al. [17] and Church and ReVelle [18]. Descriptive methods complement prescriptive methods by providing the ability to carefully evaluate a proposed ambulance allocation. These models provide more accurate estimates for quantities such as the utilizations of the ambulances and the probability that the response time is longer than predefined threshold. This area is dominated by the hypercube method proposed by Larson [5] and its extensions such as [19-23]. The name hypercube derives from the state space of the system. For the zero-line capacity system, it can be described as the vertices of the N-dimensional unit hypercube in the positive orthant and each vertex corresponds to a particular combination of serves status [5]. In this approach, each emergency response unit is modeled as a server with two possible statuses: idle (0) or busy (1) at any given instant, there are 2 N possible states to the system. A particular state of the system is given by the entire listing of ambulances that are free and busy. For example, the state 101 corresponds to a 3-server-system, with servers 1 and 3 busy and server 2 idle, and the state space is given by the vertices of a cube. If the system has more than 3 ambulances, we have a hypercube. 3. Implementing hypercube Queuing model in Tehran Qom highway In this section, we applied hypercube model which presented by Mendonca and Morabito [2] in order to evaluate the performance of EMS system on Tehran- Qum highway. This system has 2 5 states in which 5 represents number of servers in system. For example, vector (10011) shows a state of system in which third and fourth servers are ideal (from right to left) and another servers are busy. A sample data which was taken in summer 2009 is obtained from the Tehran Emergency Center. We considered the summer period for our study, since the EMS system has higher rate of EMS calls. In the following section, the main assumptions of the partial backup and multiple dispatch hypercube model for application on highways are presented. The validity of these assumptions in Tehran-Qum highway EMS problem is verified as well Assumptions In Mendonca and Morabito hypercube model [2], the highway is divided into N A geographical areas (segments of highway) or atoms, which corresponds to independent sources of calls and there are N spatially distributed ambulances, which can travel to any atom. Each ambulance, when idle, stays in its base waiting for a call. In Tehran- Qum highway, there are 7 atoms and 5 ambulances located in the predefined station. Each station (Fixed Base) has one ambulance and its personnel (doctor, rescuers, and driver) that always move together. The enumeration of the five bases of the Tehran-Qom highway is defined in Figure1. Figure1. Atoms and stations (Fixed base) along the Tehran-Qom highway In each atom j calls arrive according to a Poisson process, independently from the other atoms, with mean arrival rate λ j (j = 1,2, N A ). Table 1 shows the statistics of the arrival process for each atom. The number of events, mean interval time, standard deviation, coefficient of variation and arrival rate ( λ j ) are calculated with the data given by

5 Tehran Emergency Center. In order to verify the hypothesis of independent Poisson arrivals, we applied the chi-square and Kolmogorov-Smirnov test and the result shows the for all atoms the test were unable to reject the hypothesis with 5% of confidence level (Table 2) Atom j Number of events Table 1. Statistics of the arrival process for each atom Mean interval Standard Coefficient of time (min) deviation (min) variation Arrival rate λ j (min -1 ) In this system, there is a server dispatch preference list for each atom. If the first preference ambulance of the list is available in its base, it is dispatched; otherwise, the next ambulance of this list (backup) is dispatched. This list is fixed a priori and remains unchanged during the operation of the system. In response to each call, exactly one ambulance is dispatched from its base to the locale of the accident. If the two nearest ambulances to this local are not available, the call is lost (e.g. it is transferred to a local EMS). The predefined first and second preference ambulances for each atom in Tehran-Qum EMS system is presented in Table 2. For example, the eastern part of the Kahrizak region (Atom 3) has ambulance B as a first preference and ambulance C as backup. Table2. First and second (backup) preference ambulances for each atom Atom First preference ambulance Second (backup) preference ambulance 1 1 (A) 2 (B) 2 2 (B) 1 (A) 3 2 (B) 3 (C) 4 3 (C) 4 (D) 5 4 (D) 3 (C) 6 4 (D) 5 (E) 7 5 (E) 4 (D) The mean ambulance travel time (τ ij ) between atoms i and j should be known or can be estimated using geometrical probability concepts. In this study, these times are estimated based on travel time between atoms with average speed in Tehran-Qom highway. Figure2. The mean travel time (τ ij ) and the distance between atoms The service time for a call includes the setup time, the travel time from the base to the location of the accident, the time spends on-scene, and the travel time back to the base. The ambulances have mean service times μ 1, μ 2,, μ n that can be different. In this hypercube model it is assumed that the standard deviation of the service time is approximately equal to the mean, since the mathematical analysis assumes negative exponential service times. However reasonable deviations from this assumption have been found not to alter the accuracy of the model, as mentioned in [2] and [6]. In order to verify the validity of this assumption, we applied the chi-square, Kolmogorov-Smirnov test and the results

6 show that for all bases the test was unable to reject the hypothesis of negative exponential service time with 5% of confidence level (Table 3). Base n Number of services Table3. Statistics of the service process for each base Mean service time (min) Standard deviation (min) Coefficient of variation service rate μ i (min -1 ) Equilibrium state probabilities Relations among states of a queuing system can be illustrated by rate diagram. In rate diagram, nodes represent state of system and branches show possibility of transition from one state to other state. Figure 3 illustrates the flows into and out of state (01001). In this rate diagram when fourth base (base D) provides a service with rate of μ 4, state of the system changes from (01001) to (00001). Also, when system changes from (00001) to (01001) means that fifth and sixth atoms have demand for fourth base. Rate diagram can be interpreted for other branches with respect to priority of each base. Figure3. Flows into and out of state The equilibrium probability equations are defined supposing that the system attains steady state. For each of the 32 states of the system, the flow into the state (the probability of the system being in another state times the transition rate from that to the current state) should be equal to the flow out of the state (the probability of the system being in the state, times the transition rate from this to other state). For example, for state (01001), equilibrium equation is equal to Equation (1) with respect to rate diagram (Figure3). (μ 2 )P (μ 3 )P (μ 5 )P (λ 1 )P (λ 5 + λ 6 )P (1) = (λ 1 + λ 2 + λ 3 + λ 4 + λ 5 + λ 6 + λ 7 + μ 1 + μ 4 + μ 5 )P With substituting any of the 32 equations by P B = 1 B (the sum of all state probabilities should be equal to 1), a determined system with 32 linearly independent equations is obtained. Solving this determined system offers equilibrium probabilities for each state (Appendix 1). By using these probabilities some performance measures of EMS system are computable which are presented in next section Performance measures Different interesting performance measures for the analysis of the EMS system can be defined as a function of the equilibrium distribution, as discussed in [5] and [6]. In this stage some performance measures are calculated. The results of computations for all performance measures are presented in Table 5. Workload of ambulance i (ρ i ): The amount of workload of ambulance n is fraction of times in which ambulance n is busy (Equation 2). ρ i = B: server i is buzy P B (2) For example workload of ambulance 1 is calculated as ρ 1 = B: server 1 is buzy P B = P P P P P P P P P P P P P P P P Amount of imbalances in workload of servers ( W): This amount is equal to difference between the workload of each ambulance and average workload of ambulances (Equation 3). W = ρ mean ρ i (3)

7 System loss probability (P p): This probability is calculated by considering state of ambulance in facing with each demand of different atoms. For example first term of Equation 4 shows the time that first and second servers are busy and there is a demand from atoms 1 or 2. P p = ( λ 1+ λ 2 λ P P ) + ( λ 4+ λ 5 λ P ) + ( λ 6+ λ 7 λ P P ) + ( λ 1+ λ 2 + λ 3 λ P ) + ( λ 3+ λ 4 + λ 5 λ ) ) + ( λ 4+ λ 5 + λ 6 + λ P ) + ( λ 1+ λ 2 + λ 6 + λ ) + ( λ 2+ λ 3 + λ 4 + λ 5 λ λ ( λ 3+λ 4 + λ 5 + λ 6 + λ 7 λ ) + P The fraction of all dispatches that send ambulance n to atom j (f nj ): This amount is presented in Equation 5. λ j f (5) nj = λ B E P nj B 1 P p where E nj is the set of states in which ambulance n is to be assigned any service request from atom j. For example, E 11 can be calculated as E 11 : {(00000), (00100), (01000), (11100), (10000), (00010), (00110), (01010), (10010), (01110), (10110), (11010), (11110)} and E 12 can be calculated as E 12 : {(00010), (00110), (01010), (10010), (01110), (10110), (11010), (11110)}. It should be noted that Equation 5 is a slight modification of the formula in Larson and Odoni [6], since state in not the unique state in which the system can lose a call, but any state with two or more adjacent busy ambulances. Therefore, the denominator of Equation 5 is 1 P p instead of 1 P By substitution whole demand (λ= ) in Equation 5, the amount of f nj is calculated for each travel (Table 4). λ (4) Table 4. Fraction f nj f 11 = f 21 = f 23 = f 34 = f 45 = f 35 = f 57 = f 12 = f 22 = f 33 = f 44 = f 46 = f 56 = f 47 = Mean travel time for ambulance n (TU n ): This parameter is equal to mean travel time for ambulance n to arrive in accident location (Equation 6). TU n = N A j=1 f njt nj N A f nj j=1 where t nj for each travel is equal to mean travel time for ambulance n to atom j (note that t nj is easily calculated from the mean inter-atom travel times τ ij ). Besides of this, with respect to data in mission form of each servers and comments of experts in Tehran Emergency Centers, set-up time after each emergency call for preparing each ambulance is considered as 5 minutes, which is added to mean travel time for ambulance and makes parameter t nj. For example mean travel time for ambulance 4 is calculated as: TU 4 = (f 44 t 44 )+ (f 45 t 45 )+ (f 46 t 46 )+ (f 47 t 47 ) (f 44 + f 45 + f 46 +f 47 ) Overall mean travel time (T): This amount can be calculated using Equation 7. T = N N A n=1 i=1 f nj t nj (7) Mean travel time to atom j (T j ): Mean travel time for ambulances to each atom can be calculated as Equation 8. T j = n=1 f njt nj N n=1 f nj For example, this amount can be calculated for district 6 as T 6 = (f 46 t 46 )+ (f 56 t 56 ) (f 46 + f 56 ) Fraction of all dispatches from district, to other atoms (F j ): This parameter is calculated by using Equation 9. F j = N j i { i:i is in district} N j=1 { i:i is in district} For example for atom 5, fraction of all dispatches to other atoms is calculated as: F 5 = f 45 + f 35 Fraction of all dispatches of each server that send to other atoms (F n ): This parameter can be calculated as Equation 10. F n = f nj f nj {j:j is out of district} f nj N A j=1 f nj f 35 (6) (8) (9) (10)

8 For example fraction of all dispatches that server 3 sent to other district is as: F 3 = f 33+ f 35 f 33 + f 34 + f Alternative scenarios In this section we modify the configuration of the system in order to improve workload balancing among bases. As shown in Table 5, the system is not balanced since the workload of base 2 is significantly higher than the others. An alternative that does not involve additional investments is to reduce the size of atoms for which the ambulance in base 2 is first preference. Figure 4 depicts the particular case where we simply decreased the size of atom 2 to reduce the workload of base 2. This change is suggested by experts of Tehran Emergency Center. Figure 4. Original and modified size of atoms 1 and 2. In this step, we repeated all steps of research and compared some performance measures of system for original and modified systems. The results are presented in Table 5, 6, and 7. Base n mean service rate μ i (min -1 ) O.S M.S Sum= Table 5: Result of balancing for each base Workload ρ n Imbalances in workload( W) O.S M.S O.S M.S (O.S: Original System, M.S: Modified System) Mean travel time (TU n) O.S M.S Fraction (F n) O.S M.S Atom j mean Table 6: Result of balancing for each atom Length (km) Arrival rate λ j Mean travel time to (min -1 ) atom j (T j) O.S M.S O.S M.S O.S M.S Sum= (O.S: Original System, M.S: Modified System) Fraction (F j) O.S M.S Original system Table 7: Result of balancing in whole system T mean P P

9 Modified system Mobin M., Li Z., Amiri M By changing system state, workload of ambulance 2 (ρ 2 ) decreased about percent (Table 5). Changing size of atoms had negligible effects on workload of other ambulances, and decreased the average workload of other ambulances. This change also decreased average workload of all ambulances about percent and workload imbalances of whole system about percent. Furthermore, by changing size of district, mean travel time for ambulance n (TU n ) is decreased, except ambulance 1 which is increased because of increasing size of atom. In addition, the average of TU n declined from to minutes. In modified system, fraction of all dispatches that send backup ambulance to other atoms decreased from to Besides of this, by comparing real system state with recommended system, the mean travel time to atoms (average of T j ) declined from to that shows improvement of EMS system. In addition, the average fraction of all dispatches to each atom from other atoms (average of F j ) decreased from to (Table 8). Finally as seen in Table 7, mean travel time of system (T) decreased from to This change is negligible because of the little modification in size of atom, which does not need any additional investment. Undoubtedly, this amount will be more decreased, by making more changes in atoms, more investments for making new station and adding more ambulances. Also, System loss probability (P P ) decreased to which shows improvement of EMS system. 5. Sensitivity analysis In order to analyze how the current system configuration would perform in a situation with different amount of demand, we increase the mean arrival rate λ j, j = 1,2,, N A, by 10%, 20%,, 50%, 10%, 200%, and 500% and show effects of increasing demand on the probability of an empty system (P ), the loss probability (P p ), and the overall mean travel time (T) (Table 10).These result are presented in Figures 5, 6 and 7. By applying the similar method of analyses, the effect of other input parameters rather than the arrival rates can be evaluated. As shown in Figure 5, the mean travel time of the system is relatively robust whit respect to variations in the demand. This is because the system does not allow queues. Note that if we double the demand (100%), the mean travel time will increase from min to only min; however, the probability of an empty system will fall from 0.668% down to less than 0.453% (Figures 5 and 7), while the loss probability will jump from 0.031% to more than 0.067% (Figure 6). Figure 5. Effects of increasing demand on the overall mean travel time (T) Figure 6. Effects of increasing demand on the loss probability (P P) Figure 7. Effects of increasing demand on the probability of an empty system (P )

10 Table 10: Effects of increasing demand on the probability of an empty system (P ), loss probability (P P) and overall mean travel time (T) Increase of T TU P demand (%) p P n (mean) T (00000) F (min) (min) n (mean) F n (mean) j (mean) (min) Conclusion In this research, we evaluated the EMS system on Tehran-Qom highway by means of hypercube model. By calculating the equilibrium state probabilities of the EMS system, some performance measures of system including workload of ambulances, system loss probability, mean travel time etc. are calculated. By applying some practical modifications in the EMS system without requiring additional investments in the system, such as modifying the size of atoms, the performance of the system is recalculated and compared with the performance measures of original system. The results show the improvement in performance measures of modified EMS system. The mean travel time of the modified system reduced to 7.55 min which is close to the standards reported by other studies in literature. As another performance measures, we obtained the loss probability of system which means the modified system will loses 0.031% of the arriving emergency calls. We also analyzed the effects of small variations in the input parameters of the model, such as user demand, to investigate the robustness of the model. The amount of mean travel time is reasonably robust with respect to variations in the demands, since the system does not allow queues. Applying integrated Hypercube and optimization models to deal with EMS on highways involving different operational policies (e.g., different dispatch preference lists, multiple dispatches of ambulances, and dynamic relocation of ambulances) can be considered as a future research. Finally, the methods could be extended to examine urban EMS, which usually correspond to large systems in moderate to large cities. Furthermore, applying Discrete Event Simulation models to calculate system performance measures when queuing models are difficult or impossible to apply are recommended as future researches. Appendix 1: Equilibrium state probabilities P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = P = References 1. Fujiwara, O., Makjamroen, T., and Gupta, K. K., 1987, Ambulance Deployment Analysis: a Case Study of Bangkok. European Journal of Operational Research, 31(1), Mendonça, F. C., and Morabito, R., 2001, Analyzing Emergency Medical Service Ambulance Deployment on a Brazilian Highway Using the Hypercube Model. Journal of the Operational Research Society, Takeda, R. A., Widmer, J. A., and Morabito, R., 2007, Analysis of Ambulance Decentralization in an Urban Emergency Medical Service Using the Hypercube Queueing Model. Computers and Operations Research, 34(3),

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