Queueing Theory with Applications

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 Erlang’s work 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 [6] relied heavily on the principles articulated by Erlang. These notes are being developed 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. Mathematics can be extremely dry when complete rigor is the principal goal. We will be content to be sure that the results are correct. Our account will be focused on the insights to be gained that might be relevant in the various applications. 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. Then it will be fairly easy to 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 and modeling the Internet to stochastic modeling of population dynamics and other biological systems.