Randomized parallel communications on an extension of the omega network

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Abstract
Parallel communication algorithms and networks are central to large-scale parallel computing and, also, data communications. This paper identifies adverse source-destination traffic patterns and proposes a scheme for obtaining relief by means of randomized routing of packets on simple extensions of the well-known omega networks. Valiant and Aleliunas have demonstrated randomized algorithms, for a certain context which we call nonrenewal, that complete the communication task in time O (log N ) with overwhelming probability, where N is the number of sources and destinations. Our scheme has advantages because it uses switches of fixed degree, requires no scheduling, and, for the nonrenewal context, is as good in proven performance. The main advantage of our scheme comes when we consider the renewal context in which packets are generated at the sources continually and asynchronously. Our algorithm extends naturally from the nonrenewal context. In the analysis in the renewal context we, first, explicitly identify the maximum traffic intensities in the internal links of the extended omega networks over all source-destination traffic specifications that satisfy loose bounds. Second, the benefits of randomization on the stability of the network are identified. Third, exact results, for certain restricted models for sources and transmission, and approximate analytic results, for quite general models, are derived for the mean delays. These results show that, in the stable regime, the maximum mean time from source to destination is asymptotically proportional to log N . Numerical results are presented.