An apparatus is described having multiple cores, each core having: a) a CPU; b) an accelerator; and, c) a controller and a plurality of order buffers coupled between the CPU and the accelerator. Each of the order buffers is dedicated to a different one of the CPU's threads. Each one of the order buffers is to hold one or more requests issued to the accelerator from its corresponding thread. The controller is to control issuance of the order buffers' respective requests to the accelerator.