In this chapter, the probability density function of the hopcount and of the weight of the shortest path to the most nearby member of an anycast group consisting of m members (e.g. servers or peers) in a graph of N nodes is analyzed.

The results are applied to compute a performance measure η of the efficiency of anycast over unicast and to the server placement problem. The server placement problem asks for the number of (replicated) servers m needed such that any user in the network is not more than j hops away from a server of the anycast group with a certain prescribed probability. As in Chapter 18 on multicast, two types of shortest path trees are investigated: the regular k-ary tree and the irregular uniform recursive tree treated in Chapter 16. Since these two extreme cases of trees indicate that the performance measure η ≈ 1 — a log m, where the real number a depends on the details of the tree, it is believed that for trees in real networks (as the Internet) a same logarithmic law applies. An order calculus on exponentially growing trees further supplies evidence for the conjecture that η ≈ 1 — a log m for small m.

In peer-to-peer (P2P) networks (see e.g. Van Mieghem (2010a, Chapter 13)) such as Napster, Bit Torrent and Tribler, content is often either fully replicated at a peer or is split into chunks and stored over m peers. A major task of a member of the peer group lies in selecting the best peer among those m peers that possess the desired content.