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A FAFNER subserver depends on its parent for sieving tasks. The tasks given out by the root server are large indeed (hundreds of thousands or millions of Q-values). As they pass from subserver to subserver, they are broken up into smaller and smaller bites. By the time they are handed out to clients, there may be as few as 100 Q-values per task.
The sieving clients (GNFS or GNFSD processes) are the leaves of the FAFNER tree; they get a single task from a FAFNER server, and then spend anywhere from 15 minutes to several days chewing on the problem (depending on the speed of the machine used and the size of the task). When the answers are ready (in the form of a text file containing a few hundred or a few thousand relations), the clients send them back to their FAFNER server. There, they are distilled, archived, and ultimately sent back to Bellcore, where they are integrated into the final solution -- the factoring of RSA130.