The Laplace - Stieltjes transformation of the distribution function of the peak age of information in a transmission group modeled by a two-node network

In this paper, we consider a two-node information transmission group consisting of a peripheral node (sender node), a control center (recipient node) and two serial communication channels between them. The process of transferring information from the sending node to the receiving node is modeled by means of a two-node queuing network with an arbitrary function for distributing intervals between requests and arbitrary service durations at each node of the network, i.e., a network that, according to Kendall's designations, is usually encoded as G/G/1 – G/G/1. In this case, the transmitted packets of information are modeled by requests, and the process of transmitting packets through the system channels is modeled by the maintenance procedure at the network nodes. The paper provides an expression for the Laplace – Stieltjes transformation of the stationary distribution function of the peak age of information – a metric that allows quantifying the "freshness" of information entering the receiving node in order to accept management decisions. This work continues the cycle of works by the authors devoted to the problem of analyzing the age of information, summarizing the results of the work of other authors, which is achieved through the most arbitrary assumptions about the distributions of the durations of generating and servicing applications in the network nodes. The correctness of the obtained results was verified on a number of special cases considered earlier, as well as by comparing the analytical results with the results of simulation modeling.