We study the power spectral density of continuous time Markov chains and explicit its relationship with the eigenstructure of the infinitesimal generator. This result helps us understand the dynamics of the number of customers for a M/M/1 queuing process in the heavy traffic regime. Closed-form relations for the power law scalings associated to the eigenspectrum of the M/M/1 queue generator are obtained, providing a detailed description of the power spectral density structure, which is shown to exhibit a $1/f^{3/2}$ noise. We confirm this result by numerical simulation. We also show that a continuous time random walk on a ring exhibits very similar behavior. It is remarkable than a complex behavior such as $1/f$ noise can emerge from the M/M/1 queue, which is the "simplest" queuing model.