Queueing Theory with Applications and Special Consideration to Emergency Care

Much that is essential in modern life would not be possible without queueing theory. All communication systems depend on the theory including the Internet. In fact, the theory was developed at the time that telephone systems were growing and requiring more and more sophistication to manage their complexity. Much of the theory was developed by Agner Krarup Erlang (1878-1929). He worked for Copenhagen Telephone Company. His contributions are widely seen today as fundamental for how the theory is understood and applied. Those responsible for the early development of what has become the Internet relied on the work of Erlang and others to guide them in designing this new system. Leonard Kleinrock was awarded the National Medal of Honor for his pioneering work leading to the Internet. His book [7] reworked Queueing Theory to apply to this new developing technology. These notes are being compiled from a seminar in the Department of Mathematics at the University of Florida on Queueing Theory Applied to Emergency Care. The seminar meets weekly and the material is updated based on the presentations and discussion in the seminar. In the notes an attempt is made to introduce the theory starting from first principles. We assume some degree of familiarity with probability and density functions. However, the main distributions and concepts are identified and discussed in the notes along with some derivations. We will try to develop intuition as well. Often the intuition is gained by reworking a formula in a way that the new version brings insight into how the system is affected by the various parameters. If one makes complete rigor the principal goal in Mathematics, then the subject can become extremely dry. We will be content to be sure that the results are correct and any arguments elucidating. We will focus on the insights that might be relevant to the various applications that we have in mind. We will first introduce Poisson processes. This forms the basic underpinning of elementary queueing theory. Next we will introduce the simple queue. This is a queueing system with a single server with Poisson arrivals and exponential service times. We then discuss more complex queueing systems. As we introduce new ideas we will try to give applications and hint how the ideas will apply to emergency care. The general applications will range from telephone communications to stochastic modeling of population dynamics and other biological systems. The most complex queueing systems are frequently beyond mathematical analysis. This is likely the case for a realistic model of emergency care. Such cases will be studied by simulation. The tools of simulation will be gradually developed through the notes. One of the major accomplishments of the seminar is a realistic model of the flow of patients in the emergency room.