Semi-Markovian Stochastic Hybrid System: A New Method for Information Freshness Analysis

Information freshness is one of the critical aspects of communication networks. Average age of information (AoI) and peak AoI (PAoI) are the main metrics for evaluating information freshness. Modeling the communication system with a stochastic hybrid system (SHS) is an effective approach for finding these metrics, especially in complex network scenarios. A regular SHS is comprised of a Markov chain (MC) and some stochastic processes evolving according to the state of the MC. The Markovian property of the chain in regular SHSs limits the capability of this approach in analyzing general network scenarios, e.g., when the transmission times of packets follow a general non-exponential distribution. In this work, we focus on the semi-Markovian-SHSs (SM-SHSs). In SM-SHSs, compared to regular SHSs, the Markovian property of the system chain is extended to semi-Markovian property, i.e., the sojourn time in each state follows an arbitrary distribution depending on the next transition in the system chain. We derive theorems for finding the average AoI and PAoI when the communication system is modeled with an SM-SHS. We also propose an age-ordering-aware multi-stream M/G/1/1 queueing system with the assumption of deterministic vacation times for the server and analyze this system using the SM-SHS approach.