Queueing analysis of oblivious packet-routing networks
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
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We apply queueing theory to derive the probability distribution on the queue buildup associated with greedy routing on an $n \times n$ array of processors. We assume packets continuously arrive at each node of the array with Poisson 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 Poisson distributed wit mean $1$. To analyze the queue size in steady-state, we formulate the problem into an equivalent Jackson queueing network model. It turns out that determining the probability distribution on the queue size at each node is then just a matter of solving $O(n^{4})$ simultaneous linear equations which determine the total arrival rate at each node and then plugging these arrival rates into a short formula for the probability distribution given by the queueing theory. However, we even eliminate the need to solve these simultaneous equations by deriving a very simple formula for the total arrival rates in the cas of greedy routing. Lastly, we use this simple formula to prove that the expected queue size at a node of the $n \times n$ array increases as the Euclidean distance of the node from the center of the array decreases.