[Paper Review] Age of Information for Discrete Time Queues

Vishrant Tripathi, Rajat Talak+1 more|arXiv (Cornell University)|Jan 28, 2019

Age of Information OptimizationComputer Science24 references51 citations

TL;DR

This paper derives closed-form peak and average Age of Information (AoI) expressions for several discrete-time queueing models, including Ber/G/1 (with and without vacations), LCFS G/G/1 with preemption, and G/G/∞, and discusses new techniques for discrete-time AoI analysis.

ABSTRACT

Age of information (AoI) is a time-evolving measure of information freshness, that tracks the time since the last received fresh update was generated. Analyzing peak and average AoI, two time average metrics of AoI, for various continuous time queueing systems has received considerable attention. We analyze peak and average age for various discrete time queueing systems. We first consider first come first serve (FCFS) Ber/G/1 and Ber/G/1 queue with vacations, and derive explicit expressions for peak and average age. We also obtain age expressions for the last come first serve (LCFS) queue and the $G/G/\infty$ queue. We build upon proof techniques from earlier results, and also present new techniques that might be of independent interest in analyzing age in discrete time queuing systems.

Motivation & Objective

• Motivate AoI analysis in discrete-time queueing systems and establish AoI as a measure of information freshness in slotted networks.
• Derive explicit peak and average AoI expressions for selected discrete-time queues (Ber/G/1, Ber/G/1 with vacations, LCFS G/G/1 with preemption, and G/G/∞).
• Highlight proof techniques and introduce new methods applicable to discrete-time AoI analysis.
• Compare effects of vacations and scheduling disciplines on AoI and provide design insights for age minimization.

Proposed method

• Model AoI A(t) in slotted systems with packets arriving at the beginning and finishing service at the end of a time slot.
• Provide AoI recursions and use generating functions to obtain closed-form expressions for peak and average AoI.
• For Ber/G/1: derive A^p and A^ave in terms of arrival rate λ, mean service μ, service-time moments, and generating function L_S(x).
• For Ber/G/1 with vacations: incorporate residual-vacation analysis to obtain A^p and show A^ave ≤ A^p.
• For LCFS/G/G/1: derive A^p and A^ave using inter-generation X and service S distributions and preemption dynamics.
• For G/G/∞: derive A^ave involving minimum-of-sums expression reflecting out-of-order service.

Experimental results

Research questions

• RQ1 What are the peak and average AoI expressions for discrete-time Ber/G/1 queues (with and without vacations)?
• RQ2 What is the AoI performance of LCFS G/G/1 queues with preemptive service in discrete time?
• RQ3 What is the AoI behavior of discrete-time G/G/∞ queues under independent inter-generation and service times?
• RQ4 How do deterministic vacations compare to other vacation distributions in terms of AoI for Ber/G/1 with vacations?

Key findings

• Peak AoI for Ber/G/1: A^p = 1/λ + 1/μ + (λ E[S^2] − ρ) / (2(1−ρ)) + extra terms from discretization.
• Average AoI for Ber/G/1: A^ave = 1 + 1/μ + [(1−λ)(1−ρ)] / [λ L_S(1−λ)] + (λ E[S^2] − ρ) / [2(1−ρ)].
• Deterministic vacations minimize average AoI for Ber/G/1 with fixed mean vacation duration; A^ave ≤ A^p.
• Ber/G/1 with vacations: A^p = A^p (no vacations) + E[V^2]/(2 E[V]) − 1/2, and A^ave ≤ A^p.
• LCFS/G/G/1 AoI: A^p_G/G/1 = E[X] / P[S ≤ X] + E[S I{S ≤ X}] / P[S ≤ X] − 1; A^ave_G/G/1 = (1/2) E[X^2]/E[X] + E[min(X,S)] / P[S ≤ X] − 1/2.
• G/G/∞ AoI: A^ave_G/G/∞ = (1/2) E[X^2]/E[X] + E[min_l {sum_{k=1}^l X_k + S_{l+1}]} − 1/2.
• The discrete-time results exhibit a discretization offset (−1 for peak, −1/2 for average) compared to continuous-time counterparts and account for non-strict inequalities in discrete time.