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The time complexity of delivering M messages that come within [DELTA]t' to their relevant subscribers in our proposed filter-verification framework with inverted index pruning technique is [mathematical expression not reproducible], where f is the fanout of the TMR-tree, [R.sub.i] is a node in the TMR-tree (also can be treated as a dummy message), and Lp is the inverted index associated with [R.sub.i]'s parent in which the subscriptions are likely to be relevant to [R.sub.i]'s parent.
[R.sub.i] is a dummy message and we assume there are q dummy messages that are included in VOs.
At this point, it is clear that the infinitely-many users protocol can be simulated by finitely-many users provided that the users send "dummy messages" as explained previously.
Also, it sends only dummy messages in the synchronizing state.