The ambulance offload delay problem is a well-known result of overcrowding and congestion in emergency departments. Offload delay refers to the situation where area hospitals are unable to accept patients from regional ambulances in a timely manner due to lack of staff and bed capacity. The problem of offload delays is not a simple issue to resolve and has caused severe problems to the emergency medical services (EMS) providers, emergency department (ED) staff, and most importantly patients that are transferred to hospitals by ambulance. Except for several reports on the problem, not much research has been done on the subject. Almost all research to date has focused on either EMS or ED planning and operation and as far as we are aware there are no models which have considered the coordination of these units. We propose an analytical model which will allow us to analyze and explore the ambulance offload delay problem. We use queuing theory to construct a system representing the interaction of EMS and ED, and model the behavior of the system as a continuous time Markov chain. The matrix geometric method will be used to numerically compute various system performance measures under different conditions. We analyze the effect of adding more emergency beds in the ED, adding more ambulances, and reducing the ED patient length of stay, on various system performance measures such as the average number of ambulances in offload delay, average time in offload delay, and ambulance and bed utilization. We will show that adding more beds to the ED or reducing ED patient length of stay will have a positive impact on system performance and in particular will decrease the average number of ambulances experiencing offload delay and the average time in offload delay. Also, it will be shown that increasing the number of ambulances will have a negative impact on offload delays and increases the average number of ambulances in offload delay. However, other system performance measures are improved by adding more ambulances to the system. Finally, we will show the tradeoffs between adding more emergency beds, adding more ambulances, and reducing ED patient length of stay. We conclude that the hospital is the bottleneck in the system and in order to reduce ambulance offload delays, either hospital capacity has to be increased or ED patient length of stay is to be reduced.

The ambulance offload delay problem is a well-known result of overcrowding and congestion in emergency departments. Offload delay refers to the situation where area hospitals are unable to accept patients from regional ambulances in a timely manner due to lack of staff and bed capacity. The problem of offload delays is not a simple issue to resolve and has caused severe problems to the emergency medical services (EMS) providers, emergency department (ED) staff, and most importantly patients that are transferred to hospitals by ambulance. Except for several reports on the problem, not much research has been done on the subject. Almost all research to date has focused on either EMS or ED planning and operation and as far as we are aware there are no models which have considered the coordination of these units. We propose an analytical model which will allow us to analyze and explore the ambulance offload delay problem. We use queuing theory to construct a system representing the interaction of EMS and ED, and model the behavior of the system as a continuous time Markov chain. The matrix geometric method will be used to numerically compute various system performance measures under different conditions. We analyze the effect of adding more emergency beds in the ED, adding more ambulances, and reducing the ED patient length of stay, on various system performance measures such as the average number of ambulances in offload delay, average time in offload delay, and ambulance and bed utilization. We will show that adding more beds to the ED or reducing ED patient length of stay will have a positive impact on system performance and in particular will decrease the average number of ambulances experiencing offload delay and the average time in offload delay. Also, it will be shown that increasing the number of ambulances will have a negative impact on offload delays and increases the average number of ambulances in offload delay. However, other system performance measures are improved by adding more ambulances to the system. Finally, we will show the tradeoffs between adding more emergency beds, adding more ambulances, and reducing ED patient length of stay. We conclude that the hospital is the bottleneck in the system and in order to reduce ambulance offload delays, either hospital capacity has to be increased or ED patient length of stay is to be reduced.

Although healthcare operations management has been an active and popular research direction over the past few years, there is a lack of formal quantitative models to analyze the ambulance o oad delay problem. O oad delays occur when an ambulance arriving at a hospital Emergency Department (ED) is forced to remain in front of the ED until a bed is available for the patient. Thus, the ambulance and the paramedic team are responsible to care for the patient until a bed becomes available inside the ED. But it is not as simple as waiting for a bed, as EDs also admit patients based on acuity levels. While the main cause of this problem is the lack of capacity to treat patients inside the EDs, Emergency Medical Services (EMS) coverage and availability are signi cantly a ected. In this thesis, we develop three network queueing models to analyze the o oad delay problem. In order to capture the main cause of those delays, we construct queueing network models that include both EMS and EDs. In addition, we consider patients arriving to the EDs by themselves (walk-in patients) since they consume ED capacity as well. In the rst model, ED capacity is modeled as the combination of bed, nurse, and doctor. If a patient with higher acuity level arrives to the ED, the current patient's service is interrupted. Thus, the service discipline at the EDs is preemptive resume. We also assume that the time the ambulance needs to reach the patient, upload him into the ambulance, and transfer him to the ED (transit time) is negligible. We develop e cient algorithms to construct the model Markov chain and solve for its steady state probability distribution using Matrix Analytic Methods. Moreover, we derive di erent performance measures to evaluate the system performance under di erent settings in terms of the number of beds at each ED, Length Of Stay (LOS) of patients at an ED, and the number of ambulances available to serve a region. Although capacity limitations and increasing demand are the main drivers for this problem, our computational analysis show that ambulance dispatching decisions have a substantial impact on the total o oad delays incurred. In the second model, the number of beds at each ED is used to model the service capacity. As a result of this modeling approach, the service discipline of patients is assumed to be nonpreemptive priority. We also assume that transit times of ambulances are negligible. To analyze the queueing network, we develop a novel algorithm to construct the system Markov chain by de ning a layer for each ED in a region. We combine the Markov chain layers based on the fact that regional EDs are only connected by the number of available ambulances to serve the region. Using Matrix Analytic Methods, we nd the limiting probabilities and use the results to derive di erent system performance measures. Since each ED's patients are included in the model simultaneously, we solve only for small instances with our current computational resources. In the third model, we decompose the regional network into multiple single EDs. We also assume that patients arriving by ambulances have higher nonpreemptive priority discipline over walk-in patients. Unlike the rst two models, we assume that transit times of ambulances are exponentially distributed. To analyze the decomposed queueing network performance, we construct a Markov chain and solve for its limiting probabilities using Matrix Analytic Methods. While the main objective for the rst two models is performance evaluation, in this model we optimize the steady state dispatching decisions for ambulance patients. To achieve this goal, we develop an approximation scheme for the expected o oad delays and expected waiting times of patients. Computational analysis conducted suggest that larger EDs should be loaded more heavily in order to keep the total o oad delays at minimal levels.

Although healthcare operations management has been an active and popular research direction over the past few years, there is a lack of formal quantitative models to analyze the ambulance o oad delay problem. O oad delays occur when an ambulance arriving at a hospital Emergency Department (ED) is forced to remain in front of the ED until a bed is available for the patient. Thus, the ambulance and the paramedic team are responsible to care for the patient until a bed becomes available inside the ED. But it is not as simple as waiting for a bed, as EDs also admit patients based on acuity levels. While the main cause of this problem is the lack of capacity to treat patients inside the EDs, Emergency Medical Services (EMS) coverage and availability are signi cantly a ected. In this thesis, we develop three network queueing models to analyze the o oad delay problem. In order to capture the main cause of those delays, we construct queueing network models that include both EMS and EDs. In addition, we consider patients arriving to the EDs by themselves (walk-in patients) since they consume ED capacity as well. In the rst model, ED capacity is modeled as the combination of bed, nurse, and doctor. If a patient with higher acuity level arrives to the ED, the current patient's service is interrupted. Thus, the service discipline at the EDs is preemptive resume. We also assume that the time the ambulance needs to reach the patient, upload him into the ambulance, and transfer him to the ED (transit time) is negligible. We develop e cient algorithms to construct the model Markov chain and solve for its steady state probability distribution using Matrix Analytic Methods. Moreover, we derive di erent performance measures to evaluate the system performance under di erent settings in terms of the number of beds at each ED, Length Of Stay (LOS) of patients at an ED, and the number of ambulances available to serve a region. Although capacity limitations and increasing demand are the main drivers for this problem, our computational analysis show that ambulance dispatching decisions have a substantial impact on the total o oad delays incurred. In the second model, the number of beds at each ED is used to model the service capacity. As a result of this modeling approach, the service discipline of patients is assumed to be nonpreemptive priority. We also assume that transit times of ambulances are negligible. To analyze the queueing network, we develop a novel algorithm to construct the system Markov chain by de ning a layer for each ED in a region. We combine the Markov chain layers based on the fact that regional EDs are only connected by the number of available ambulances to serve the region. Using Matrix Analytic Methods, we nd the limiting probabilities and use the results to derive di erent system performance measures. Since each ED's patients are included in the model simultaneously, we solve only for small instances with our current computational resources. In the third model, we decompose the regional network into multiple single EDs. We also assume that patients arriving by ambulances have higher nonpreemptive priority discipline over walk-in patients. Unlike the rst two models, we assume that transit times of ambulances are exponentially distributed. To analyze the decomposed queueing network performance, we construct a Markov chain and solve for its limiting probabilities using Matrix Analytic Methods. While the main objective for the rst two models is performance evaluation, in this model we optimize the steady state dispatching decisions for ambulance patients. To achieve this goal, we develop an approximation scheme for the expected o oad delays and expected waiting times of patients. Computational analysis conducted suggest that larger EDs should be loaded more heavily in order to keep the total o oad delays at minimal levels.