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A Palm Calculus Approach to the Distribution of the Age of Information

A key metric to express the timeliness of status updates in latency-sensitive networked systems is the age of information (AoI), i.e., the time elapsed since the generation of the last received informative status message. This metric allows studying a number of applications including updates of sensory and control information in cyber-physical systems and vehicular networks as well as, job and resource allocation in cloud clusters. State-of-the-art approaches to analyzing the AoI rely on queueing models that are composed of one or many queuing systems endowed with service order, e.g., FIFO, LIFO, or last-generated-first-out order. A major difficulty arising in these analysis methods is capturing the AoI under message reordering when the delivery is non-preemptive and non-FIFO, i.e., when messages can overtake each other and the reception of informative messages may obsolete some messages that are underway. In this paper, we derive an exact formulation for the distribution of AoI in non-preemptive, non-FIFO systems where the main ingredients of our analysis are Palm calculus and time inversion. Owing to the rationality of the Laplace-Stieltjes transforms that are used in our approach, we obtain computable exact expressions for the distribution of AoI.