Assuming that the deterministic round-robin behavior of the DNS resolution or layer 4 switching of mail.bigisp.com gives clients an equal chance of connecting to each server. If there are n mail servers with A records for mail.bigisp.com the chance of one of the servers receiving a connection from a foreign host is . Assuming that all users receive an equal number of messages of equal size the chance of a message from a foreign host being delivered locally is . The number of messages that must be relayed to another one of the mail.bigisp.com machines is . So the probability that each server must relay is . As n becomes large this approximates to , as . Effectively the probability of a message being relayed internally is 1 and as a result SMTP network traffic is doubled.
This analysis is still valid for a system where users have different usage patterns, provided that mailboxes are distributed across the servers in such a way that the servers are equally loaded. Distribution of mailboxes is discussed in Section 7.