First in Line Waiting Times as a Tool for Analysing Queueing Systems

We introduce a new approach to modelling queueing systems where the priority or the routing of customers depends on the time the first customer has waited in the queue. This past waiting time of the first customer in line, W, is used as the primary variable for our approach. A Markov chain is used for modelling the system where the states represent both the number of free servers and a discrete approximation to W. This approach allows us to obtain waiting time distributions for complex systems, such as the N-design routing scheme widely used in e.g. call centers and systems with dynamic priorities.