# Single server queue with multiple Poisson arrival streams

Consider a single server queue with first-in-first-out discipline. There are $n$ classes of customers. Class $i$ arrives as a Poisson process of rate $\lambda_i$, independently of the other classes, and is served at an exponential rate of $\mu_i$. We assume $\sum \lambda_i/\mu_i<1$, so that an invariant distribution exists.

I would like to find the invariant distribution of the queue. Of course, the state space here records the exact sequence of customer classes in the queue, rather than simply the length of the queue. For example, what is the long-run probability that there are precisely $4$ people in the queue, all of whom are of class $1$? Is there a reference for this? If it helps, we can take $n=2$.

The above queueing system can be modeled as an $M/G/1$ queue with arrival rate $\lambda := \sum_{i = 1}^n \lambda_i$ and a hyperexponential service time distribution that is with probability $p_i := \lambda_i / \lambda$ equal to an exponential distribution with rate $\mu_i$ for $i = 1,2,\ldots,n$.
Use the theory of Chapter 7 of these lecture notes to determine the equilibrium distribution of the number of customers in the system. Then, use the fact that each customer is with probability $p_i$ a class-$i$ customer.
• The customer in service isn't of class $i$ with probability $p_i$ though. Commented Sep 2, 2016 at 8:22