Timeliness Analysis for Real-Time Updates Across Different Service Time

This paper considers the status update systems with adequate real-time updates, emphasizing the timeliness of status for actuators. Age of Information (AoI) serves as a metric for evaluating the freshness of information. Understanding the concrete distribution of AoI proves valuable for obtaining real-time insights into system status. It is important to highlight the pivotal role played by service time for AoI. In this work, we explore the distributions of peak AoI (PAoI) and AoI in the presence of various service time distributions. Specifically, the expressions for PAoI with exponential, deterministic, and Gamma distribution service times are derived. Considering different packet sides require multiple service processes, this work introduces a Switched Poisson Process (SPP) and provides an exact result for the distribution of PAoI. Furthermore, we analyze AoI distribution in a single-source system, investigating the probability of AoI exceeding a specified constraint. Additionally, we extend our study to obtain a closed-form expression for the stationary distribution of AoI in a multi-stream system. Numerical examples are presented to validate the effectiveness of the theoretical findings.