We study the average age of information (AoI) and peak AoI (PAoI) of a dual-queue status update system that monitors a common stochastic process. We capture the state transition characteristics of the considered system by establishing a Markov chain. Using the state flow graph analysis method, we derive closed-form expressions of the average peak age of information (PAoI) and the average age of information (AoI) for the dual-queue update system. The numerical results show that compared with the single-queue update system, the average PAoI of the dual-queue update system is reduced by 33.5% and the average AoI dropped by 37.5%.