Balancing Timeliness and Privacy in Discrete-Time Updating Systems

TL;DR

This paper investigates the trade-off between timeliness (AoI) and privacy (MaxL) in discrete-time status updating with Bernoulli arrivals under two LCFS policy classes: coupled (arrival-tied preemptive) and decoupled (dumping with independent timers). It shows decoupled dumping policies outperform coupled ones, and derives that the optimal decoupled strategy (D-DAD) achieves the best age-leakage trade-off by dithering between two adjacent deterministic dump periods; the coupled class is dominated by a greedy, s_min=1 policy. The analysis combines SMP leakage characterizations, AoI formulas, renewal theory for RAD age, and Dinkelbach fractional programming to obtain the optimal two-point dump distribution. Numerical results corroborate the theoretical findings, including Markovian arrivals and full-support considerations, and offer practical design guidelines for privacy-aware, timely updates. The work highlights the superiority of decoupled strategies for balancing freshness and timing privacy in resource-constrained monitoring systems.

Abstract

We study the trade-off between Age of Information (AoI) and maximal leakage (MaxL) in discrete-time status updating systems. A source generates time-stamped update packets that are processed by a server that delivers them to a monitor. An adversary, who eavesdrops on the server-monitor link, wishes to infer the timing of the underlying source update sequence. The server must balance the timeliness of the status information at the monitor against the timing information leaked to the adversary. We consider a model with Bernoulli source updates under two classes of Last-Come-First-Served (LCFS) service policies: (1) Coupled policies that tie the server's deliveries to the update arrival process in a preemptive queue; (2) Decoupled (dumping) policies in which the server transmits its freshest update according to a schedule that is independent of the update arrivals. For each class, we characterize the structure of the optimal policy that minimizes AoI for a given MaxL rate. Our analysis reveals that decoupled dumping policies offer a superior age-leakage trade-off to coupled policies. When subject to a MaxL constraint, we prove that the optimal dumping strategy is achieved by dithering between two adjacent deterministic dump periods.

Figures (9)

• Figure 1: Model for our general system setup in discrete time. Updates over $n$ time slots $X^n$ are generated at a source and must be scheduled (causally) to produce an output $Y^n$ observed by an eavesdropper. The mapping $X^n \to Y^n$ must balance the AoI at the monitor and the MaxL to the adversary.
• Figure 2: The source sends fresh updates to the server in slots $N_k=n_k$, inducing the age process $A_s(n)$ at the input to the server. The server sends samples of the most recent update to the monitor in time slots $S_k=s_k$, inducing the age process $A_m(n)$ at the monitor.
• Figure 3: The cost function $f(x)= C(d(x))$ plotted against $x = z_0^{-d}$ for $z_0 = 1.5$ and $\gamma = 1.5$.
• Figure 4: Comparison of the optimal age-leakage trade-off in the $(\Delta, T_{\Lambda})$ space for LCFS Coupled Policies. Each curve is generated by applying the optimal "greedy" SMP distribution from Theorem \ref{['thm:optimal_pmfLCFS']} for a given leakage constraint $\beta= g(s_{\min}(g))$. The constraint $\beta$ is varied in the range $[0.025, 1.0]$ for $s_{\min}(g)=1$, $[0.05, 1.0]$ for $s_{\min}(g)=2$, and $[0.1, 1.0]$ for $s_{\min}(g)=3$. The source arrival rate is $\lambda = 0.5$. Leakage rate for $s_{\min}(g) > 1$ is calculated numerically for $n=10000$ time slots.
• Figure 5: Leakage time as a function of the average AoI for Coupled Policies with $\lambda = 0.5$. The plot compares LCFS policies against thinned and unthinned FCFS policies. For the greedy policies, the leakage constraint $\beta$ is varied from 0.1 to 1. For the geometric policies, the service rate $\mu=1/\tau$ is varied from 0.1 to 1. The thinned policies optimize the admission probability $\alpha$ at every point on the curve to minimize the age for that specific service rate.

Theorems & Definitions (26)

• Definition 1: Issa et al. issa2019operational
• Definition 2: Kaul et al. kaul2012real
• Lemma 1: SMP maximum likelihood input
• Theorem 1: FCFS, LCFS Leakage for SMP Distributions
• proof
• Proposition 1: Bounds on Asymptotic Leakage Rate for the SMP Class
• proof
• Corollary 1: LCFS Ber/G/1 Age
• Theorem 2: Optimal LCFS Policy in the SMP Class
• Lemma 2: RAD maximum likelihood input