A connection establishment method for establishing a random access type connection using a random access channel, which is used in a radio system having several subscriber terminal equipments (37 to 38) and base stations (33 to 36) where one or more transceiver units (40 to 47) operate, and in which system connection establishment between one of the subscriber terminal equipments (37 to 38) and the base station (33 to 36) is possible when the one subscriber terminal equip nt (37 to 38) is situated in the coverage area of one or more transceiver units (40 to 47) of the base station, where the one subscriber terminal equipment (3 to 38) evaluates the quality of the connections to be established. Starting from a condition in which no connection is established with the one subscriber terminal equipment, the one subscriber terminal equipment (37 to 38) forms a random access type connection to more than one base station transceiver unit (40 to 47) included in one or more base stations (33 to 38). m