Queueing Theory Analysis of Greedy Routing on Arrays and Tori

Mor Harchol and Paul E. Black

EECS Department, University of California, Berkeley

Technical Report No. UCB/CSD-93-756

, 1993

http://www2.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},
    Year= {1993},
    Month= {Jun},
    Url= {http://www2.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://www2.eecs.berkeley.edu/Pubs/TechRpts/1993/6299.html
%F Harchol:CSD-93-756