A General Formula for the Stationary Distribution of the Age of Information and Its Application to Single-Server Queues

Abstract

This paper considers the stationary distribution of the age of information (AoI) in information update systems. We first derive a general formula for the stationary distribution of the AoI, which holds for a wide class of information update systems. The formula indicates that the stationary distribution of the AoI is given in terms of the stationary distributions of the system delay and the peak AoI. To demonstrate its applicability and usefulness, we analyze the AoI in single-server queues with four different service disciplines: first-come first-served (FCFS), preemptive last-come first-served (LCFS), and two variants of non-preemptive LCFS service disciplines. For the FCFS and the preemptive LCFS service disciplines, the GI/GI/1, M/GI/1, and GI/M/1 queues are considered, and for the non-preemptive LCFS service disciplines, the M/GI/1 and GI/M/1 queues are considered. With these results, we further show comparison results for the mean AoI's in the M/GI/1 and GI/M/1 queues under those service disciplines.where η t (t ≥ 0) denotes the time-stamp of information being displayed on the monitor at time t. The mean AoI E[A] is defined asand under a fairly general setting, E[A] is given by [2]where E[G † ] and E[(G † ) 2 ] denote the mean and the second moment of interarrival times and E[G † n D n ] denotes the mean product of the interarrival time G † n of the (n − 1)st and the nth packets and the system delay D n of the nth packet. This formula has been the starting point in most previous work on the analysis of the AoI. As stated in [1, Page 170], however, the calculation of the mean AoI based on (2) is cumbersome because G † n and D n are dependent in general and their joint distribution can take a complicated form.The purpose of this paper is twofold. The first one is the derivation of a general formula for the stationary distribution A(x) (x ≥ 0) of the AoI in ergodic information update systems, which is defined as the long-run fraction of time in which the AoI is not greater than an arbitrarily fixed value x:where 1 {·} denotes an indicator function. Although the mean AoI E[A] is a primary performance metric, it alone is not sufficient to characterize the long-run behavior of the AoI and its related processes. First of all, if the stationary distribution A(x) of the AoI is available, we can evaluate the deviation of the AoI from its mean value. To support our claim further, we provide two examples, which show that the stationary distribution of the AoI plays a central role in the analysis of AoI-related processes. P-LCFS GI/GI/1/0 LST and E[A] (Theorem 26); Decomposition formula (Corollary 27) P-LCFS M/GI/1/0 LST, E[A], and E[A 2 ] (Corollary 28 (i)); ordering of E[A] (Corollary 31 (i)) P-LCFS GI/M/1/0 LST, E[A], and E[A 2 ] (Corollary 28 (ii)); ordering of dist. function (Corollary 31 (ii)) P-LCFS M/M/1/0 LST, dist. function, E[A] and E[A 2 ] (Appendix A.2) P-LCFS M/D/1/0 E[A] and E[A 2 ] (Appendix A.2) P-LCFS D/M/1/0 E[A] and E[A 2 ] (Appendix A.2)

Age-Delay Tradeoffs in Single Server Systems

“…We showed that determinacy in packet generation and/or service does not necessarily minimize age. Similar results were independently obtained in [35].…”

Trading Off Computation with Transmission in Status Update Systems

“…The service time is distributed as a memoryless exponential random variable with mean µ, which is dependent on the computation service time. This tandem queue model is inspired by packet management schemes in [3], [5]. The first queue is in M/GI/1/1 form while the second queue is in GI/M/1/2* form or equivalently non-preemptive last come first serve with discarding.…”

“…After the pioneering works [1], [2] that analyze queuing models motivated from vehicular status update systems, the AoI metric has been found useful in various scenarios. [3], [4] investigate the role of packet management to decrease the AoI while [5] provides a general AoI analysis in various preemptive and non-preemptive queuing disciplines coming after earlier works such as [6], [7]. References [8]- [17] consider AoI in energy harvesting communication systems.…”

Optimized Age of Information Tail for Ultra-Reliable Low-Latency Communications in Vehicular Networks

“…However, while interesting, these works do not 1 An interested reader may refer to [17] and [18] for a comprehensive survey and references on AoI. 2 Therefore, considering the AoI violation in this work is a different measure compared to the peak-AoI violation [23] consider controlling the tail of AoI violation distribution.…”