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With the increasing usage of in-vehicle systems, drivers have to frequently perceive and respond to messages from these in-vehicle systems. In addition, previous studies have found that the interval between the messages (arrival rate) presented to a driver becomes one of the factors affecting driver workload. To reduce driver workload, researchers on adaptive workload-management systems have found that adding extra delay time into the interval of messages can significantly reduce driver workload. However, it is unknown whether this extra delay time added by an adaptive workload-management system will increase the performance time of drivers or not. To answer this important question, using closed-form mathematical equations, the current work quantifies human performance time (total task completion time and reaction time of each task) when there are two serial processing stages in the human cognitive system. The mathematical model developed in this work provides solutions of the optimal interval of messages that generate the lowest workload without deteriorating drivers' performance time to respond to multiple messages from in-vehicle systems. This is one of a few closed-form deterministic mathematical models with analytic solutions that can predict average reaction time when there are two multiple serial stages in the cognitive system in dual tasks. With relatively simple equations, the mathematical model can still capture the major patterns of simulation results with stochastic properties and human behavioral experimental results. The mathematical equations developed in this study can be used in the design of adaptive workload-management systems and other driver assistance systems.