In multiprocessor systems, various problems are treated with Lamport's logical clock and the resultant logical time orders between operations. However, one often needs to face the high complexities caused by the lack of logical time order information in practice. In this paper, we utilize the \emph{global clock} to infuse the so-called \emph{pending period} to each operation in a multiprocessor system, where the pending period is a time interval that contains the performed time of the operation. Further, we define the \emph{physical time order} for any two operations with disjoint pending periods. The physical time order is obeyed by any real execution in multiprocessor systems due to that it is part of the truly happened operation orders restricted by global clock, and it is then proven to be independent and consistent with traditional logical time orders. The above novel yet fundamental concepts enables new effective approaches for analyzing multiprocessor systems, which are named \emph{pending period analysis} as a whole. As a consequence of pending period analysis, many important problems of multiprocessor systems can be tackled effectively. As a significant application example, complete memory consistency verification, which was known as an NP-hard problem, can be solved with the complexity of O(n2) (where n is the number of operations). Moreover, the two event ordering problems, which were proven to be Co-NP-Hard and NP-hard respectively, can both be solved with the time complexity of O(n) if restricted by pending period information.

The ever-intensifying time-to-market pressure imposes great challenges on the pre-silicon design phase of hardware. Before the tape-out, a pre-silicon design has to be thoroughly inspected by time-consuming functional verification and code review to exclude bugs. For functional verification and code review, a critical issue determining their efficiency is the allocation of resources (e.g., computational resources and manpower) to different modules of a design, which is conventionally guided by designers' experiences. Such practices, though simple and straightforward, may take high risks of wasting resources on bug-free modules or missing bugs in buggy modules, and thus could affect the success and timeline of the tape-out. In this paper, we propose a novel framework called pre-silicon bug forecast to predict the bug information of hardware designs. In this framework, bug models are built via machine learning techniques to characterize the relationship between design characteristics and the bug information, which can be leveraged to predict how bugs distribute in different modules of the current design. Such predicted bug information is adequate to regulate the resources among different modules to achieve efficient functional verification and code review. To evaluate the effectiveness of the proposed pre-silicon bug forecast framework, we conducted detailed experiments on several open-source hardware projects. Moreover, we also investigate the impacts of different learning techniques and different sets of characteristic on the performance of bug models. Experimental results show that with appropriate learning techniques and characteristics, about 90% modules could be correctly predicted as buggy or clean and the number of bugs of each module could also be accurately predicted.