In this paper we propose a synchronization protocol for resource sharing among independently-developed real-time systems on multi-core platforms. The systems may use different scheduling policies and they may have arbitrary priority settings. When using this synchronization protocol each processor is abstracted by an interface which consists of a set of requirements. A requirement depends only on the worst-case time the processor may wait for resources, i.e., the maximum number of times that the resources can be blocked by other processors. We have derived schedulability conditions for each processor and based on the analysis we extract the interface of the processor. In this paper, we focus on the cases when each system is allocated on a dedicated processor.