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The major emphasis is on analytical techniques for predicting the performance of various collection fusion scenarios. Knowledge of analytical models of information retrieval system performance, both with single processors and with multiple processors, increases our understanding of the parameters (e.g., number of documents, ranking algorithms, stemming algorithms, stop word lists, etc.) affecting system behavior. While there is a growing literature on the implementation of distributed information retrieval systems and digital libraries, little research has focused on analytic models of performance. We analytically describe the performance for single and multiple processors, both when different processors have the same parameter values and when they have different values. The use of different ranking algorithms and parameter values at different sites is examined.