Cost function analysis of a single-server queueing system with Poisson input stream and Erlang-k service time

In this paper, the cost function of a single-server queueing system with Poisson input stream and Erlang-k service time will be analyzed. Treated as a function of the traffic intensity ρ, with respect to some known constants, we will show that its stationary points are solutions of a fourth-degree polynomial equation with real coefficients. Moreover, an explicit form of these solutions is given and it is shown the function reaches a minimum value at some of these points. For illustration, a numerical analysis of the cost function is carried out by changing the values of the costs, which are changed according to the principle of arithmetic progression. Also, a statistical analysis of the relationship between optimal solutions ρ⁎ and Φ(ρ⁎) is done.