Global Clock, Physical Time Order and Pending Period Analysis in Multiprocessor Systems

In multiprocessor systems, various problems were 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, the so-called physical time order is proposed based on the global clock in multiprocessor systems. Concretely, we first utilize the global clock to infuse the pending period to each operation in a multiprocessor system, where the pending period is a time interval in which the operation starts and ends. Afterwards, we define the physical time order for any pair of operations with disjoint pending periods. The physical time order is an underlying characteristic of any real execution in multiprocessor systems due to that it is part of the truly-happened orders obeying real physical time. Formally, the physical time order is 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 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 now be solved with the complexity of O(nCp) by utilizing physical time order information (where n and p are the number of operations and processors respectively, C is some constant). Moreover, 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(nCp) if restricted by pending period information.