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This paper is concerned with a communications system having channels and serving a multitude of low-traffic sources. The system incorporates a waiting list having places. Calls arriving when all channels are busy, and up to other calls are already waiting, will be put on the waiting list and will be served in order of arrival. When the waiting list is filled, additional arriving calls will also be permitted to wait. When a place on the waiting list becomes available, one of these remaining waiting calls, chosen at random, will be admitted on this list. The delay distributions for this partly ordered service are investigated in this paper. Results are presented in a number of graphs.