Some Explicit Formulas for Mixed Exponentlal Service Systems

We present some explicit formulas for queue length and waiting time distributions of customers in the MfHE,,,/l 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 dishbutions are exhemely accurate, while the corresponding infinite history M IGIII recurrence equation is not Applications include computation of queueing distributions, accurate tail probabilities, in qualitative studies, and in systems whcre exponentialiey can be replaced by hyperexponentiality. The explicit solutions are easier to use than the problem-specific partial fraction expansions of the Pollachek-Khinchin transform.