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.




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