A multiple dispatch and partial backup hypercube queuing model to analyze emergency medical systems on highways

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Abstract

The hypercube is a spatially distributed queuing model based on Markovian analysis approximations, used to analyze the configuration and operation of server-to-customer emergency systems. In the present study we adapted the model to analyze emergency medical systems (EMS) on highways, which operate within particular dispatching policies. The study takes into consideration that: the emergency calls are of different types; the servers are distinct (e.g., rescue ambulances, medical vehicles); only certain servers in the system can service calls in a given region (partial backup); and, depending on the type of call, one or more identical or distinct servers are immediately dispatched to service such calls (multiple dispatch). We also consider that the arriving calls take place either along the highway or at the home location of a server – in which case the server does not need to travel to the call location. Finally, we analyzed the computational results of applying such an approach to the case study of an EMS operating on Brazilian highways.

Introduction

In emergency medical systems (EMS) the delays in the response times are of major concern since they might mean the difference between life and death of individuals involved in accidents. When designing or modifying the configuration of EMS, managers try to balance investments and benefits of improving the operation of these systems. Since EMS are server-to-customer systems, the analysis of their operations requires that probabilistic factors related to the spatial (location) and temporal (time) distribution of both user calls and system servers, be modeled. In the literature, most models that analyze emergency systems take into account only randomness in the server availability, not considering other important probabilistic variables of the system. Studies by Swersey, 1994, Owen and Daskin, 1998, Brotcorne et al., 2003 present revisions of the classic location models for the analysis of the emergency systems developed during the last few decades.

The hypercube model (Larson, 1974, Larson, 1975, Larson and Odoni, 1981), which is based on the spatially distributed queuing theory and Markovian analysis approximations, has been one of the most effective descriptive approaches to analyze emergency response systems. This model enables the representation of the EMS uncertainties, besides considering the identity of the servers and the cooperation among them. The basic idea is to expand the state-space description of a multi-server queuing system (e.g. M/M/N/∝ or M/M/N/N where N is the number of servers) in order to represent each server individually and to incorporate the complex dispatching policies involved. The model requires the solution of linear systems of O(2N) equations, where the variables involved are the equilibrium state probabilities of the system. With these probabilities, it is possible to calculate different critical performance measures, such as mean user-response times, server workloads, number of dispatches per server in each region, among other measures.

Several studies, such as Larson, 1975, Halpern, 1977, Chelst and Barlach, 1981, Larson and Mcknew, 1982, Jarvis, 1985, Burwell et al., 1993, Mendonça and Morabito, 2001, have extended the original hypercube model to relax some of its limiting assumptions or to improve its computational efficiency in analyzing police and ambulance emergency response systems. In particular, Chelst and Barlach (1981) extended the hypercube model to consider the simultaneous dispatch of two identical police patrols, and Mendonça and Morabito (2001) modified the model considering the single dispatch of identical ambulances and partial backup in EMS on highways. Other studies have been successful in combining the hypercube model with optimization procedures, such as Batta et al., 1989, Saydam and Aytug, 2003, Chiyoshi et al., 2003, Galvão et al., 2005. Examples of the hypercube model applications in urban EMS in the United States can be found in Larson and Odoni, 1981, Chelst and Barlach, 1981, Brandeau and Larson, 1986, Burwell et al., 1993, Sacks and Grief, 1994. More recently, the hypercube has been studied as a deployment model in response to terrorist attacks and other major emergencies (Larson, 2004). In Brazil, the hypercube model has been applied to analyze urban EMS (e.g., Takeda et al., 2007) and EMS on highways (Mendonça and Morabito, 2001, Iannoni, 2005).

In the present study we modify the hypercube model to analyze EMS on Brazilian highways, considering complex dispatching policies. In the first model, we extend the multiple dispatch hypercube model proposed by Chelst and Barlach (1981) to analyze EMS on highways with different types of calls and servers, and a particular operations policy involving partial backup (because of limitations of travel distance or time), zero-line capacity and single and multiple dispatching of either identical or distinct servers (e.g., rescue ambulances, medical vehicles) depending on the type of call. This model is then extended to incorporate a third status of servers (besides the usual two: idle or busy), to specify when they provide medical services to emergency requests at their home bases – as this type of service does not involve a server’s trip to the call location and, in general, it is not related to accidents along the highway. After applying the two models to the case study of an EMS operating on Brazilian highways, we analyze the computational results and show that the second model represents this EMS more accurately.

This paper is organized as follows: in Section 2 we describe the EMS under consideration; in Section 3 we discuss the extensions of the hypercube model to this EMS; and in Section 4 we analyze some results obtained to evaluate the main performance measures of this system, using sample analysis to validate the models. Finally, in Section 5 we present concluding remarks and perspectives for future studies.

Section snippets

The EMS under consideration

EMS on highways are typically zero-line capacity systems, since if a call arrives in the system when its candidate ambulances are busy, it is immediately transferred to another system, such as a local hospital or a police station, which is usually unable to provide the same service quality. In general, an EMS’ ambulance provides the first medical treatments to the individuals involved in the accident, transports them to the nearest hospital (if necessary), and then goes back to its home base on

The adapted hypercube model

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 servers’ status (Larson, 1974). In the basic model, at any given time instant, the status of each server is either idle (0) or busy (1), and for N servers, there are 2N possible states (vertices) for the system. For example, the state {1 0 1}

Computational results

For the application of the hypercube models 1 and 2 presented in Section 3, we divided the highways into NA = 8 atoms (segments), according to the primary area of each base established by the managers and operators of this system (see Fig. 1). The data were collected at the operations center within the period of January–September 2004, from the 1498 events recorded. We subdivided each atom of the system into layers a and b, in order to consider the different dispatching preference lists according

Conclusions

In this study we show that the hypercube model can be adapted and applied to analyze EMS on highways with different types of calls (types 0–3) and servers (e.g., rescue ambulances and medical vehicle), considering particular dispatching policies such as zero-line capacity; partial backup; single and multiple (double and triple) dispatching of identical or distinct servers; servers responding to calls at their bases (e.g., users of the highways that stop at the server’s home station asking for

Acknowledgements

The authors thank the two anonymous referees for their useful comments and suggestions. This research was partially supported by CNPq (Grants 140178/01-5, 522973/95-7) and Fapesp (Grant 05/53126-5).

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