Explicit formulas for the steady-state probabilities of the two-class FIFO M/M/1 queue

In this paper, we consider a queue with two classes of customers and a FIFO-single server. Class-i (i = 1,2) customers arrive according to a Poisson process with rate lambda i; there is no priority between the two classes. The service of class-i (i = 1,2) customers is exponential with rate my i. Exact formulas are proven for the stationary probability of each ordered state and the steady-state distribution of the number (total or per class) of customers in the queue.