The stationary distribution of the age of information in FCFS single-server queues

We consider the stationary distributions of the age of information (AoI) and the peak AoI in information update systems. We first derive an invariant relation among the distributions of the AoI, the peak AoI, and the system delay, which holds for a wide class of information update systems. Based on this result, we next obtain several formulas for the stationary distributions of the AoI and the peak AoI in the first-come first-served (FCFS) GI/GI/1 queue, which is a general model of FCFS information update systems. Finally, we derive explicit formulas for the Laplace-Stieltjes transforms of the stationary distributions of the AoI and the peak AoI in FCFS M/GI/1 and GI/M/1 queues.