Drivers' reaction time of reading signs on expressways is a fundamental component of sight distance design requirements, and reaction time is affected by many factors such as information volume and concurrent tasks. We built cognitive simulation models to predict drivers' direction sign reading reaction time. Models were built using the queueing network-adaptive control of thought rational (QN-ACTR) cognitive architecture. Drivers' task-specific knowledge and skills were programmed as production rules. Two assumptions about drivers' strategies were proposed and tested. The models were connected to a driving simulator program to produce prediction of reaction time. Model results were compared to human results in sign reading single-task and reading while driving dual-task conditions. The models were built using existing modelling methods without adjusting any parameter to fit the human data. The models' prediction was similar to the human data and could capture the different reaction time in different task conditions with different numbers of road names on the direction signs. Root mean square error (RMSE) was 0.3 s, and mean absolute percentage error (MAPE) was 12%. The results demonstrated the models' predictive power. The models provide a useful tool for the prediction of driver performance and the evaluation of direction sign design.