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The problems discussed in this paper arose in the theory of telephone traffic, with a central processing system, such as a computer, performing numerous functions in as exchange. However, several of its more important results are applicable in many businesses offering multiple services, including ones working on a nonpriority basis. Such results as those determining the amount of storage to allocate to each service, and assessing whether the system can cope with the expected number of arriving customers are independent of the priority set, so that these results have very wide applicability.

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