A multiclass single server queue can be modeled as follows. The system is composed of k classes and one server. Class k jobs queue up in buffer k, and x_k represents the number of jobs of class k in the system. We consider incoming jobs as Poisson arrival processes with mean arrival rate 𝜆_k, for each buffer k, and the expectation of the exponential service times are given by 1/µ_k. We focus our attention on a model composed of two buffers.

The queue operates according to some service scheduling. We seek optimal control sequences, within the set of available controls, so that minimum cost is achieved. The possible control actions are to serve a job from one of the two existing classes or to idle.

Since our work concerns the system's behavior and its stationary optimal policy properties, we seek minimizing the cost over an infinite horizon. The running cost C(x(t)) is associated with each state. We name α as the discount factor for our discounted cost.

In order to find the optimal policy, which leads to the minimum cost, we apply the value iteration algorithm, solving the optimization problem computationally. An interesting question is whether the generalized cµ rule is exactly optimal for the problem.

We investigate this issue performing a computational study.