1. A storage system comprising: a plurality of disk units; and a controller coupled to the plurality of disk units, wherein the controller includes therein: a plurality of first adaptors, each of which is coupled to a computer and configured to receive an access command from the computer; a plurality of second adaptors coupled to the plurality of disk units and configured to control data input/output to/from the plurality of disk units; a shared memory for storing control data transmitted between the plurality of first adaptors and the plurality of second adaptors to execute operations according to the access command from the computer; and a cache memory for storing data transmitted between the plurality of first adaptors and the plurality of second adaptors, wherein when a first adaptor of the plurality of first adaptors receives the access command from the computer, the first adaptor stores control data in the shared memory, and the data requested in relation to the received access command is transmitted via the cache memory between the first adaptor and a second adaptor of the plurality of second adaptors, the second adaptor is configured to execute operations according to the control data stored in the shared memory, wherein each of the plurality of first adaptors includes: a plurality of processor units including a first processor unit and a second processor unit, each of which has a memory area used for storing a message to be communicated among a port unit and the plurality of processor units and is configured to execute operations for data transmission between the first adaptor and the cache memory according to the message stored in the memory area; and the port unit, configured to receive the access command from the computer, including a protocol unit for storing control information of communication between the computer and the first processor unit, and between the first processor unit and the second processor unit; wherein when a certain adaptor of the first adaptors receives the access command from the computer, the access command is stored in the memory area of the first processor unit of the certain adaptor, and control information indicating that the access command is processed by the first processor unit is stored in the protocol unit of the certain adaptor, wherein the first processor unit of the certain adaptor is configured to determine whether to process the access command in the first processor unit of the certain adaptor or to take over processing of the access command to another processor unit in the certain adaptor, wherein when a second processor unit of the certain adaptor is taking over the processing of the access command from the first processor unit in the certain adaptor, the first processor unit is configured to store the control information in the protocol unit in the certain adaptor, and the access command is transferred via the port unit in the certain adaptor according to the control information stored in the protocol unit.