# Kirkman Triple Systems of Order n with Minimum Block Sum equal to n, for Access Balancing in Distributed Storage

## Document Type

Dissertation

## Degree

Doctor of Philosophy

## Major

Mathematics

## Date of Defense

7-9-2019

## Graduate Advisor

Ravindra Girivaru

## Committee

David Covert

Ronald Dotzel

Yuefeng Wu

Prabhaker Rao

## Abstract

We study a class of combinatorial designs called Kirkman systems, and we show that infinitely many Kirkman systems are well-distributed in a precise sense. Steiner triple systems of order n can achieve a minimum block sum of n. Kirkman triple systems form parallel classes from the blocks of Steiner triple systems. We prove that there are an infinite number of Kirkman triple systems that have a minimum block sum of n. We expand this to quadruple systems, and prove that there are an infinite number of Kirkman quadruple systems that reach the upper bound on the minimum block sum. These concepts can then be applied to distributed storage. For a large database, we want the data to be well-distributed, based on popularity, across several servers so that access is balanced, with servers spread across several locations. The elements of a Steiner triple system, representing popularity of data, can be grouped in blocks, representing the servers. The parallel classes of a Kirkman triple system can represent locations. The goal is to spread data across the servers, and servers across locations, using Kirkman triple systems, while having data well distributed by popularity, measured by the minimum block sum.

## Recommended Citation

Brummond, William, "Kirkman Triple Systems of Order n with Minimum Block Sum equal to n, for Access Balancing in Distributed Storage" (2019). *Dissertations*. 870.

https://irl.umsl.edu/dissertation/870