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
June 1993
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}
}
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