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Top1. Introduction
The international crisis of emergency department (ED) crowding has received considerable attention, both in political (United States, 2003; Institute of Medicine, 2006) and lay (Barrero, 1989; Goldberg, 2000; Hoot & Aronsky, 2008) venues. J M. Pines (2015) considered the ED crowding problems facing hospitals in an Israeli ED, and pointed out that it is a prevalent and important issue facing hospitals in Israel and around the world, including North and South America, Europe, Australia, Asia and Africa (Pines & Bernstein, 2015). Three general themes existed among the causes of ED crowding: input factors, throughput factors, and output factors. Commonly causes of crowding included non-urgent visits, “frequent-flyer” patients, influenza season, inadequate staffing, inpatient boarding, and hospital bed shortages. And the effects of crowding included patient mortality, transport delays, treatment delays, ambulance diversion, patient elopement, and financial effect. Among them, the increased of non-urgent patients is an important cause of the crowding in ED.
The scenario is modeled as an M/M/1 queueing game with server interruption in ED, where each patient wants to optimize their benefit. Because the non-urgent patients cannot get exact queue length information immediately, we use the unobservable queue model, where non-urgent patients make a decision just based on the average delay measurement that takes into account the presence of the urgent patients. In this paper, the queue of the ED can be modeled by using the server-interruption queueing model. To understand the interruption, we can consider the ED, where non-urgent patients and urgent patients can be both served, as a server. When urgent patients emerge and occupy the non-urgent patient band, the server has a interruption, i.e. the service for non-urgent patients is stopped. We consider an arrival process of non-urgent patients, arriving at ED which is considered as the server. Each patient makes a decision whether to join or to leave the queue, i.e. leaving the ED. When a non-urgent patient makes a decision to join the queue, the waiting time in the queue will incur a cost. If the patient finishes his service then he will get a reward.
Fortunately (for academic study) or unfortunately (for practical systems), a new challenge, called service interruption, is identified for ED, as illustrated in Figure 1.
Figure 1. Illustration of queueing system with interruption
From Figure 1, we can consider the ED as a server and each non-urgent patient as a customer. When urgent patient emerges, the server service is interrupted due to the higher priority of the urgent patients. It is well known in queuing theory that such a service interruption can bring a substantial impact on various queuing metrics like average queue length and average waiting time (Doshi, 1986; Gaver Jr, 1962). Although interruptions have been shown in aviation and other work settings to result in error with serious and sometimes fatal consequences, little is known about interruptions in the emergency department (ED) (Chisholm et al., 2000). Many papers use queueing theory to derive and analyze different prioritization policies in ED services. But to the authors’ best knowledge, there few study on the queuing control subject to service interruptions for patients. In this paper, patients in ED will be divided into two types: non-urgent patients and urgent patients. We consider an M/M/1 queue with server interruption to study how to control the ED crowding.