# A Processor-Sharing model for the Performance of Virtualized Network Functions

## TL;DR

This paper models the performance of virtualized network functions in cloud platforms using a processor-sharing queue with batch arrivals, deriving new analytical results and providing accurate approximations validated by simulations.

## Contribution

It introduces a novel analytical framework for the $M^{[X]}/M/1$-PS queue with batch arrivals, including residual busy period analysis and sojourn time distribution approximation.

## Key findings

- Derived residual busy period distribution for the queue
- Provided an accurate approximation for the sojourn time tail
- Validated the approximation with simulation results

## Abstract

The parallel execution of requests in a Cloud Computing platform, as for Virtualized Network Functions, is modeled by an $M^{[X]}/M/1$ Processor-Sharing (PS) system, where each request is seen as a batch of unit jobs. The performance of such paralleled system can then be measured by the quantiles of the batch sojourn time distribution. In this paper, we address the evaluation of this distribution for the $M^{[X]}/M/1$-PS queue with batch arrivals and geometrically distributed batch size. General results on the residual busy period (after a tagged batch arrival time) and the number of unit jobs served during this residual busy period are first derived. This enables us to provide an approximation for the distribution tail of the batch sojourn time whose accuracy is confirmed by simulation.