Model-based development and component-based software engineering have emerged as a promising approach to deal with enormous software complexity in automotive systems. This approach supports the development of software architectures by interconnecting (and reusing) software components (SWCs) at various abstraction levels. Automotive software architectures are often modeled with chains of SWCs, also called cause-effect chains that are constrained by timing requirements. Based on the variations in activation patterns of SWCs, a single model of a cause-effect chain at a higher abstraction level can conform to several valid refined models of the chain at a lower abstraction level, which is closer to the system implementation. As a consequence, the total number of valid implementation-level models generated by the existing techniques increases exponentially, thereby significantly increasing the runtime of the timing analysis engines and liming the scalability of the existing techniques. This paper computes an upper bound on the activation pattern combinations that may result from a system of cause-effect chains in a given high-level model of the software architecture. An efficient algorithm is presented that traverses only a reduced number of possible combinations of the cause-effect chains, resulting in the timing analysis of significantly lower number of implementation-level models of the software architecture. A proof of concept is provided by conducting a case study that shows significant reduction in the runtime of timing analysis engines, i.e., the timing behavior of the considered system is verified by performing the timing analysis of only 27% of all possible combinations of the cause-effect chains.