# Storage Allocation Under Processor Sharing I: Exact Solutions and Asymptotics

## TL;DR

This paper analyzes a processor sharing storage model with primary and secondary spaces, providing exact solutions for small cases and asymptotic analysis for large systems as traffic intensity approaches capacity.

## Contribution

It offers the first exact solutions for small systems and develops an asymptotic framework and semi-analytic method for larger systems under high traffic conditions.

## Key findings

- Exact solutions for m=1 and m=2 primary spaces.
- Asymptotic behavior as traffic intensity approaches 1.
- A semi-numerical semi-analytic method for joint distribution.

## Abstract

We consider a processor sharing storage allocation model, which has m primary holding spaces and infinitely many secondary ones, and a single processor servicing the stored items (customers). All of the spaces are numbered and ordered. An arriving customer takes the lowest available space. We define the traffic intensity rho to be lambda/mu where lambda is the customers' arrival rate and mu is the service rate of the processor. We study the joint probability distribution of the numbers of occupied primary and secondary spaces. For 0 < rho < 1, we obtain the exact solutions for m = 1 and m = 2. For arbitrary m we study the problem in the asymptotic limit rho -> 1 with m fixed. We also develop a semi-numerical semi-analytic method for computing the joint distribution.