The aim of this thesis is to first give a brief review of waiting line problems which often is a subject related to queueing theory. Simple counting processes such as the Poisson process and the duration of service time of each customer being exponentially distributed are often taught in a undergraduate or graduate stochastic process course. In this thesis, we will continue discussing such waiting line problems with priority assignment on each customer. This type of queueing processes are called priority queueing models.

Patients requiring ER service are triaged and the order of providing service to patients more than often reflects early symptoms and complaints than final diagnoses. Triage systems used in hospitals vary from country to country and region to region. However, the goal of using a triage system is to ensure that the sickest patients are seen first. Such wait line system is much comparable to a priority queueing system in our study. The finite Markov chain imbedding technique is very effective in obtaining the waiting time distribution of runs and patterns. Applying this technique, we are able to obtain the probability distribution of customer wait time of priority queues. The results of this research can be applied directly when studying patient wait time of emergency medical service. Lengthy ER wait time issue often is studied from the view of limited spacing and complications in hospital administration and allocation of resources. In this thesis, we would like to study priority queueing systems by mathematical and probabilistic modeling.