Waiting Times in Priority Queues and the Ballot Problem

In this article a closed form expression of the waiting time distribution in an M=M=c queue with multiple priorities and a common service rate is derived for a customer of arbitrary priority by using a combinatorial approach related to the ballot problem. Moreover, we apply this approach to derive the response time distribution in an M=M=1 queue with preemptive priority, two types of customers and different service rates. An advantage of the approach is that it relies on purely elementary combinatoric results and does not require inversion of the Laplace Transform.