2D-AoI: Age-of-Information of Distributed Sensors for Spatio-Temporal Processes

TL;DR

The paper introduces 2D-AoI, a framework that maps spatial distance among distributed sensors to an age-equivalent distance, enabling quantification of data freshness for spatio-temporal processes. It derives this metric within a Gaussian process setting using product kernels, yielding explicit AeD expressions for exponential, squared exponential, and rational-quadratic kernels, and it analyzes how network models (MM1 and slotted ALOHA) and sensor topologies influence 2D-AoI. Key results show that exponential kernels induce a constant additive AoI offset from distance, while squared exponential and rational quadratic kernels make the AeD decay with time, increasing the value of distant sensors as nearby information ages. The framework supports optimization of sensor density, topology, and scheduling to satisfy target AoI performance, and it complements traditional AoI analyses by incorporating spatial correlations into the freshness metric with practical implications for dense IoT deployments and cooperative sensing.

Abstract

The freshness of sensor data is critical for all types of cyber-physical systems. An established measure for quantifying data freshness is the Age-of-Information (AoI), which has been the subject of extensive research. Recently, there has been increased interest in multi-sensor systems: redundant sensors producing samples of the same physical process, sensors such as cameras producing overlapping views, or distributed sensors producing correlated samples. When the information from a particular sensor is outdated, fresh samples from other correlated sensors can be helpful. To quantify the utility of distant but correlated samples, we put forth a two-dimensional (2D) model of AoI that takes into account the sensor distance in an age-equivalent representation. Since we define 2D-AoI as equivalent to AoI, it can be readily linked to existing AoI research, especially on parallel systems. We consider physical phenomena modeled as spatio-temporal processes and derive the 2D-AoI for different Gaussian correlation kernels. For a basic exponential product kernel, we find that spatial distance causes an additive offset of the AoI, while for other kernels the effects of spatial distance are more complex and vary with time. Using our methodology, we evaluate the 2D-AoI of different spatial topologies and sensor densities.

Figures (10)

• Figure 1: Fig. \ref{['fig:system']}: System with $S=3$ sensors that observe a physical phenomenon. The sensors generate samples, indexed by $i$, at times $A_i$. The samples are transmitted via a network and received by the monitor at times $D_i$. Fig. \ref{['fig:predictionexample']}: Example of a timeline. The times $A_i$ at which sensors take samples are indicated by circles, with the circle being filled, if the sample is available at the monitor at time $t$, i.e., $D_i \le t$. Assume the physical process at position 1 and time $t$ is of interest. A stale sample of sensor 1 with AoI $\Delta_t(1)$ can be used, or a prediction can be made from fresher samples from sensors 2 and 3.
• Figure 2: Example of the evolution of the 2D-AoI $\Delta^{\textsf{\!\smaller[3]{2D}}}_t(1)$ for $S=2$ sensors.
• Figure 3: 2D-AoI $\Delta^{\textsf{\!\smaller[3]{2D}}}(\varsigma,s)$ and minimal 2D-AoI $\Delta^{\textsf{\!\smaller[3]{2D}}}(s) = \min_{\varsigma} \{\Delta^{\textsf{\!\smaller[3]{2D}}}(\varsigma,s)\}$ for two sensors $\varsigma,s \in \{1,2\}$ that are connected to a monitor via independent M$\mid$M$\mid$1 queues. Fig. \ref{['fig:mm1spacevspenaltywithquantiles']} evaluates the impact of the AeD $\Lambda(2,1)$ on $\Delta^{\textsf{\!\smaller[3]{2D}}}(1)$. Fig. \ref{['fig:mm1CCDFsAll']} shows the effect of two values $\Lambda(2,1) \in \{2,10\}$ on the CCDFs. Fig. \ref{['fig:serviceprovisioning']} illustrates how service rates can be allocated to achieve a target 2D-AoI threshold of $y$ at $\varepsilon = 10^{-3}$. When individually considered $\Delta^{\textsf{\!\smaller[3]{2D}}}(1,1)$ of sensor 1 (blue line) and $\Delta^{\textsf{\!\smaller[3]{2D}}}(2,1)$ of sensor 2 (green line) are unable to achieve the goal and higher service rates $\mu_i$ would be needed. However, $\Delta^{\textsf{\!\smaller[3]{2D}}}(1)$ obtained by combining both sensors (yellow line) meets the goal.
• Figure 4: Minimal 2D-AoI $\Delta^{\textsf{\!\smaller[3]{2D}}}(1)$ for sensor $s=1$ and different spatial distances $d \in \{ 0,5,25,50\}$. To facilitate visual comparison, the results are shown relative to the respective value of the curve at $\mu_0=0$. $\mu_0$ is the service rate allocated to the center node under a sum rate constraint $\sum_s \mu_s = 1$.
• Figure 5: Fig. \ref{['fig:spatiotemporaldistancequadratic']} shows the minimal 2D-AoI $\Delta^{\textsf{\!\smaller[3]{2D}}}_t(1)$ for the case of $S=2$ sensors as in Fig. \ref{['fig:spatiotemporaldistance']} but for squared exponential instead of exponential kernels. Fig. \ref{['fig:mm1ccdfs_se']} shows the CCDF like Fig. \ref{['fig:mm1CCDFsAll']} but with squared exponential kernels. The dashed lines show the corresponding result when using an exponential kernel.