# Queueing Theory Analysis of Greedy Routing on Arrays and Tori

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/CSD-93-756.pdf

We apply queueing theory to derive the probability distribution on the queue buildup associated with greedy routing on an n by n array and an n by n torus of processors. We assume packets continuously arrive at each node of the array or torus according to a Poisson Process with rate lambda and have random destinations. We assume an edge may be traversed by only one packet at a time and the time to traverse an edge is exponentially distributed with mean 1.

To analyze the queue size in steady-state, we formulate both these problems as equivalent Jackson queueing network models. With this model, determining the probability distribution on the queue size at each node involves solving O(n^4) simultaneous linear equations. However, we eliminate the need to solve these simultaneous equations by deriving a very simple formula for the total arrival rates and for the expected queue sizes in the case of greedy routing.

This simple formula shows that in the case of the n x n array, the expected queue size at a node increases as the Euclidean distance of the node from the center of the array decreases. Furthermore, in the case of the n x n torus, the probability distribution on the queue size is identical for every node.

We also translate our results about queue sizes into results about the average packet delay.

BibTeX citation:

@techreport{Harchol:CSD-93-756,
Author = {Harchol, Mor and Black, Paul E.},
Title = {Queueing Theory Analysis of Greedy Routing on Arrays and Tori},
Institution = {EECS Department, University of California, Berkeley},
Year = {1993},
Month = {Jun},
URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/6299.html},
Number = {UCB/CSD-93-756},
Abstract = {We apply queueing theory to derive the probability distribution on the queue buildup associated with greedy routing on an <i>n</i> by <i>n</i> array and an <i>n</i> by <i>n</i> torus of processors. We assume packets continuously arrive at each node of the array or torus according to a Poisson Process with rate lambda and have random destinations. We assume an edge may be traversed by only one packet at a time and the time to traverse an edge is exponentially distributed with mean 1. <p>To analyze the queue size in steady-state, we formulate both these problems as equivalent Jackson queueing network models. With this model, determining the probability distribution on the queue size at each node involves solving <i>O</i>(<i>n</i>^4) simultaneous linear equations. However, we eliminate the need to solve these simultaneous equations by deriving a very simple formula for the total arrival rates and for the expected queue sizes in the case of greedy routing. <p>This simple formula shows that in the case of the <i>n</i> x <i>n</i> array, the expected queue size at a node increases as the Euclidean distance of the node from the center of the array decreases. Furthermore, in the case of the <i>n</i> x <i>n</i> torus, the probability distribution on the queue size is identical for every node. <p>We also translate our results about queue sizes into results about the average packet delay.}
}

EndNote citation:

%0 Report
%A Harchol, Mor
%A Black, Paul E.
%T Queueing Theory Analysis of Greedy Routing on Arrays and Tori
%I EECS Department, University of California, Berkeley
%D 1993
%@ UCB/CSD-93-756
%U http://www.eecs.berkeley.edu/Pubs/TechRpts/1993/6299.html
%F Harchol:CSD-93-756