1. A server assignment and messaging system, comprising: a plurality of connection servers configured to form a server group; and a directory server configured to generate at least one hash value for each of the plurality of connection servers using identification (ID) information of the respective connection servers, and generate a logical server group in which the connection servers are arranged in a ring shape according to the generated hash values, wherein the directory server is configured to receive a server assignment request from a terminal, and assign one of the plurality of connection servers included in the logical server group to the terminal using a hash value generated from ID information of the terminal and a server assignment rule.