Stationary distribution of the peak age of information in a two-channel transmission group modeled by a queuing system with resequence of applications and with phase-type distributions

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 for monitoring and managing remote sources of information from the control center. The paper considers a model of a transmission group consisting of an information source (sending node), a control center (receiving node) and two parallel communication channels between them. It is assumed that the channel capacities are different, and the network protocol requires that information entering the receiving node be read in the same sequence as it was transmitted from the sending node. As a result, packets that violate the established order are delayed at the receiving node for the time necessary to restore order. The information transfer process is modeled using a two-channel queuing system with a limited storage, reordering of applications and with phase-type distributions of generation and maintenance durations. At the same time, applications simulate packets of transmitted information, the system storage device is a queue of packets for transmission, the service of applications on devices is the process of transmitting packets over communication channels. As a result, expressions for the Laplace-Stieltjes transformation of the stationary distribution function and the initial moments of the maximum value of the information age, called the peak age, are obtained. A numerical study of the peak age of information at different system loads has been carried out for various special cases of phase-type distributions. The correctness of the analytical results is confirmed by the results of simulation modeling.