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So Timely, Yet So Stale: The Impact of Clock Drift in Real-Time Systems

Mehrdad Salimnejad, Nikolaos Pappas, Marios Kountouris

TL;DR

This work investigates how clock drift between transmitter and receiver clocks affects information freshness, quantified by Age of Information (AoI), in a time-slotted status-update system. It develops deterministic and probabilistic clock-drift models and derives closed-form AoI distributions and mean values using a two-dimensional DTMC over the drift-AoI state, including $\bar{\Delta} = d + \frac{1}{p_s}$ for deterministic drift and $\bar{\Delta} = \frac{2+K(K+1)p p_s}{2 p_s}$ (nonnegative drift) or $\bar{\Delta} = p_1 + \frac{1}{p_s}$ ( bidirectional drift). The results quantify how drift magnitude and drift probability affect AoI and establish a bound $p_{max}$ to tolerate up to $K$ slots of drift while meeting a target AoI. The paper also introduces Referential or Relativistic AoI (rAoI) to capture frame-of-reference effects on timeliness, with implications for asynchronous clocks, low-cost IoT hardware, satellite navigation, and timing scenarios affected by relativistic phenomena.

Abstract

In this paper, we address the problem of timely delivery of status update packets in a real-time communication system, where a transmitter sends status updates generated by a source to a receiver over an unreliable channel. The timestamps of transmitted and received packets are measured using separate clocks located at the transmitter and receiver, respectively. To account for possible clock drift between these two clocks, we consider both deterministic and probabilistic drift scenarios. We analyze the system's performance regarding the Age of Information (AoI) and derive closed-form expressions for the distribution and the average AoI under both clock drift models. Additionally, we explore the impact of key system parameters on the average AoI through analytical and numerical results.

So Timely, Yet So Stale: The Impact of Clock Drift in Real-Time Systems

TL;DR

This work investigates how clock drift between transmitter and receiver clocks affects information freshness, quantified by Age of Information (AoI), in a time-slotted status-update system. It develops deterministic and probabilistic clock-drift models and derives closed-form AoI distributions and mean values using a two-dimensional DTMC over the drift-AoI state, including for deterministic drift and (nonnegative drift) or ( bidirectional drift). The results quantify how drift magnitude and drift probability affect AoI and establish a bound to tolerate up to slots of drift while meeting a target AoI. The paper also introduces Referential or Relativistic AoI (rAoI) to capture frame-of-reference effects on timeliness, with implications for asynchronous clocks, low-cost IoT hardware, satellite navigation, and timing scenarios affected by relativistic phenomena.

Abstract

In this paper, we address the problem of timely delivery of status update packets in a real-time communication system, where a transmitter sends status updates generated by a source to a receiver over an unreliable channel. The timestamps of transmitted and received packets are measured using separate clocks located at the transmitter and receiver, respectively. To account for possible clock drift between these two clocks, we consider both deterministic and probabilistic drift scenarios. We analyze the system's performance regarding the Age of Information (AoI) and derive closed-form expressions for the distribution and the average AoI under both clock drift models. Additionally, we explore the impact of key system parameters on the average AoI through analytical and numerical results.
Paper Structure (10 sections, 3 theorems, 23 equations, 5 figures)

This paper contains 10 sections, 3 theorems, 23 equations, 5 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Performance Metric and Analysis
  4. Deterministic Clock Drift
  5. Probabilistic Clock Drift
  6. Numerical Results
  7. Concluding Remarks
  8. Proof of Lemma \ref{['theorem_AvgAoI_dfix']}
  9. Proof of Lemma \ref{['theorem_AvgAoI_prob1']}
  10. Proof of Lemma \ref{['theorem_AvgAoI_prob2']}

Key Result

Lemma 1

The average AoI, $\bar{\Delta}$, when the receiver's clock drifts by $d\geqslant 0$ slots relative to the transmitter's clock, is given by

Figures (5)

  • Figure 1: A real-time status update system with local clocks.
  • Figure 2: The evolution of the AoI metric considering probabilistic clock drift and the absence of clock drift.
  • Figure 3: Average AoI as a function of $K$ for $p_{s} = 0.5$ and $p = 0.1, 0.4, 0.8, 1$.
  • Figure 4: $p_{\text{max}}$ as a function of $K$ for $p_{s} = 0.5$ and $\bar{\Delta}_{TH} = 3, 5, 8$.
  • Figure 5: Two-dimensional DTMC describing the joint status of the slot drift regarding the state of the AoI for the probabilistic clock drift, where the receiver's clock at each time slot, with probability $p$, has a $k$-slot drift with $k \geqslant 0$ compared to the transmitter's clock. Arrows with the same color represent transitions with equal probabilities.

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Lemma 3
  • Remark 2