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Characterizing the Age of Information with Multiple Coexisting Data Streams

Yoshiaki Inoue, Michel Mandjes

TL;DR

This paper uses stochastic ordering techniques to identify tight stochastic bounds on the AoI, and yields an explicit expression for the Laplace-Stieltjes transform of the AoI in the resulting GI+M/GI+GI/1 model.

Abstract

In this paper we analyze the distribution of the Age of Information (AoI) of a tagged data stream sharing a processor with a set of other data streams. We do so in the highly general setting in which the interarrival times pertaining to the tagged stream can have any distribution, and also the service times of both the tagged stream and the background stream are generally distributed. The packet arrival times of the background process are assumed to constitute a Poisson process, which is justified by the fact that it typically is a superposition of many relatively homogeneous streams. The first main contribution is that we derive an expression for the Laplace-Stieltjes transform of the AoI in the resulting GI+M/GI+GI/1 model. Second, we use stochastic ordering techniques to identify tight stochastic bounds on the AoI, leading to an explicit lower and upper bound on the mean AoI. In addition, when approximating the tagged stream's inter-generation times through a phase-type distribution (which can be done at any precision), we present a computational algorithm for the mean AoI. As illustrated through a sequence of numerical experiments, the analysis enables us to assess the impact of background traffic on the AoI of the tagged stream. It turns out that the upper bound on the mean AoI is remarkably close to its true value, which yields an explicit expression (in terms of the model parameters) for an accurate proxy of the AoI-minimizing generation rate.

Characterizing the Age of Information with Multiple Coexisting Data Streams

TL;DR

This paper uses stochastic ordering techniques to identify tight stochastic bounds on the AoI, and yields an explicit expression for the Laplace-Stieltjes transform of the AoI in the resulting GI+M/GI+GI/1 model.

Abstract

In this paper we analyze the distribution of the Age of Information (AoI) of a tagged data stream sharing a processor with a set of other data streams. We do so in the highly general setting in which the interarrival times pertaining to the tagged stream can have any distribution, and also the service times of both the tagged stream and the background stream are generally distributed. The packet arrival times of the background process are assumed to constitute a Poisson process, which is justified by the fact that it typically is a superposition of many relatively homogeneous streams. The first main contribution is that we derive an expression for the Laplace-Stieltjes transform of the AoI in the resulting GI+M/GI+GI/1 model. Second, we use stochastic ordering techniques to identify tight stochastic bounds on the AoI, leading to an explicit lower and upper bound on the mean AoI. In addition, when approximating the tagged stream's inter-generation times through a phase-type distribution (which can be done at any precision), we present a computational algorithm for the mean AoI. As illustrated through a sequence of numerical experiments, the analysis enables us to assess the impact of background traffic on the AoI of the tagged stream. It turns out that the upper bound on the mean AoI is remarkably close to its true value, which yields an explicit expression (in terms of the model parameters) for an accurate proxy of the AoI-minimizing generation rate.
Paper Structure (21 sections, 16 theorems, 156 equations, 8 figures, 5 tables)

This paper contains 21 sections, 16 theorems, 156 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Literature overview
  3. Model, notation, and preliminaries
  4. Sensor and monitor
  5. Queueing Representation
  6. Age of Information
  7. Stationary Behavior
  8. Exact Formula for the AoI Distribution
  9. Stochastic Orderings and Bounds on the AoI
  10. Stochastic Ordering
  11. Distributional Bounds for the AoI
  12. Closed-Form Bounds for the Mean AoI
  13. Phase-type inter-generation time distribution
  14. Numerical experiments
  15. Discussion and concluding remarks
  16. ...and 6 more sections

Key Result

Lemma 1

The LST of the stationary peak AoI $f_{A_{\mathrm{peak}}}^*(\cdot)$ is given in terms of the LST of the stationary system delay $f_D^*(\cdot)$ by where $\phi(s)$ and $\psi(\omega)$ are defined as and $\mathcal{L}_{\omega}^{-1}[f(s,\omega)]$ denotes the inverse Laplace transform of $f(s,\omega)$ with respect to $\omega$:

Figures (8)

  • Figure 1: Schematic representation of the status update system, in which the information source shares the server with a background stream.
  • Figure 2: Schematic representation of queue with coexisting data streams. The grey blocks correspond to packets in the tagged stream $X(t)$, and the white blocks to packets in the background stream $X_{\rm bg}(t)$.
  • Figure 3: The top graphs represent sample paths of the arrival times and service requirements corresponding to the processes $X(t)$ and $X_{\rm bg}(t)$. The bottom graphs represent the resulting workload process $V(t)$ and AoI process $A(t)$. The service rate $\mu$ is equal to 1. The tagged stream has interarrival times that are Erlang with 10 phases and mean 20, whereas the service requirements are Erlang with 5 phases and mean 1. The background stream is Poissonian with arrival rate $0.9$, whereas the service requirements are Erlang with 5 phases and mean 1.
  • Figure 4: The algorithm for computing the mean AoI $\mathbbm{E}[A]$ in the PH+M/GI+GI/1 queue.
  • Figure 5: The mean AoI, as a function of the coefficient of variation (CV) of the inter-generation times $s_G$.
  • ...and 3 more figures

Theorems & Definitions (34)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Remark 4
  • Lemma 5: Hooke1972Ott1984
  • Remark 6
  • proof
  • ...and 24 more