# On the equivalence between multiclass processor sharing and random order scheduling policies

## TL;DR

This paper introduces two generalized scheduling policies, MPS and MROS, for multiclass single server systems, establishing a mathematical relationship between their sojourn and waiting time distributions under specific conditions.

## Contribution

The paper proposes MPS and MROS policies that unify and extend existing scheduling policies like DPS, GPS, DROS, and WFQ, revealing their distributional equivalences.

## Key findings

- Tail of sojourn time in MPS is proportional to waiting time in MROS.
- Distributional equivalence between DPS and DROS, GPS and WFQ under certain conditions.
- Analytical relationship holds for Poisson arrivals with exponential service requirements.

## Abstract

Consider a single server system serving a multiclass population. Some popular scheduling policies for such system are the discriminatory processor sharing (DPS), discriminatory random order service (DROS), generalized processor sharing (GPS) and weighted fair queueing (WFQ). In this paper, we propose two classes of policies, namely MPS (multiclass processor sharing) and MROS (multiclass random order service), that generalize the four policies mentioned above. For the special case when the multiclass population arrive according to Poisson processes and have independent and exponential service requirement with parameter $\mu$, we show that the tail of the sojourn time distribution for a class $i$ customer in a system with the MPS policy is a constant multiple of the tail of the waiting time distribution of a class $i$ customer in a system with the MROS policy. This result implies that for a class $i$ customer, the tail of the sojourn time distribution in a system with the DPS (GPS) scheduling policy is a constant multiple of the tail of the waiting time distribution in a system with the DROS (respectively WFQ) policy.