About the peak age of information in a single-channel transmission group modeled by the $PH/PH/1/r/LCFS$ system with inversion discipline, interruption maintenance and re-service of applications

This article continues the author's cycle of works devoted to the problem of the Age of Information (AoI), a metric used in information systems to quantify the “freshness” of information delivered to the control center from peripheral sources. The paper considers a two-node information transmission group consisting of a sender node, a recipient node and a communication channel between them. The information transfer process is modeled by means of a single-line finite capacity queuing system with phase-type distributions, which in Kendall's notation is encoded as $PH/PH/1/r$. This takes into account the special requirements of the transmission protocol, which consist in the fact that a packet entering the system, bypassing the queue, is immediately sent for transmission, capturing the channel from the previous packet if it has not completed transmission. The packet whose transmission was interrupted is moved to the first place in the queue and, after the channel is released, attempts transmission again. For this system, an expression is obtained for the Laplace-Stieltjes transformation of the stationary distribution function of the peak age of information and its average value. A numerical study of the dependence of the peak age of information on the system load was carried out. The correctness of the analytical results was verified by comparing them with the results of simulation modeling.