# Optimal scheduling discipline in a single-server queue with Pareto type service times

## Urtzi Ayesta

### Résumé:

We consider the mean delay optimization in an M/G/1 queue for some Pareto type service time distributions. If the Pareto distribution is defined in such a way that all positive values are allowed, then the distribution has a decreasing hazard rate, which implies that the FB discipline is optimal. However, if it is defined starting from a strictly positive value, the hazard rate is no longer monotone. In this paper we prove that, not only for this but for a whole class of service time distributions, the optimal discipline is a combination of FCFS and Foreground-Background (FB) disciplines, which gives full priority to the jobs with attained service less than some fixed threshold $\theta^*$. These priority jobs are served in the FCFS manner. If there are no jobs with attained service less than $\theta^*$, the full priority is given to the job with least amount of attained service. In our proof, we utilize the index approach developed by Gittins and Klimov. In addition to the ordinary Pareto type distributions with an infinite support, we also discuss the structure of the optimal scheduling discipline when dealing with bounded Pareto distribution with a finite support. This is a joint work with: Samuli Aalto (Helsinki University of Technology)

 Urtzi Ayesta LAAS-CNRS