In a memory controller system, a method for granting a system memory by a memory request arbitrator to a request among a plurality of pending memory access requests is provided. The plurality of the memory access requests includes Rfrsh_Hreq, Crt_Hreq, Group AB, Crt_Lreq and Rfrsh_Lreq and are respectively asserted by a host control circuitry and/or a graphical control circuitry which are implemented and integrated on a single monolithic semiconductor chip. The host control circuitry and the graphical control circuitry shares the system memory and the memory request arbitrator includes a refresh queue and the graphics control circuitry includes a CRT FIFO. The method prioritizes the plurality of the memory access requests in order of Rfrsh_Hreq>Crt_Hreq>Group AB>Crt_Lreq>Rfrsh_Lreq. The Rfsh_Hreq is memory refresh request signal of first type whenever the refresh queue being full, the Crt_Hreq is memory access signal of a first type for fueling the CRT FIFO with display data, the Group AB are memory access request signals of a second type output either from the graphical control circuitry or the host control circuitry, the Crt_Lreq is memory access signal of a third type for fueling the CRT FIFO with display data, the Rfrsh_Lreq is memory refresh request signal of second type whenever the refresh queue being non-empty.