Some explicit formulas for mixed exponential service systems (Q1113220)

We present some explicit formulas for queue length and waiting time distributions of customers in the \(M/HE_ m/1\) queue. The formulas are obtained with the aid of roots of quadratic, cubic, and quartic polynomials constructed from a recurrence equation. With an example, we demonstrate that the formulas for queueing distributions are extremely accurate, while the corresponding infinite history M/GI/1 recurrence equation is not. Applications include computation of queueing distributions, accurate tail probabilities, and systems where exponentiality can be replaced by hyperexponentiality. The explicit solutions are easier to use than the problem-specific partial fraction expansions of the Pollachek-Khinchin transform.