1. A server assignment and messaging system, comprising: a plurality of connection servers which forms a server group; and a directory server comprising a processor coupled to a memory, the processor 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, store the generated hash values in the memory, 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.