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Status Update Control and Analysis under Two-Way Delay

Mohammad Moltafet, Markus Leinonen, Marian Codreanu, Roy D. Yates

TL;DR

This work addresses timely status updates in a two-way delay setting by formulating 1-Packet and 2-Packet pipelined systems as Markov decision processes to minimize the long-run average AoI using Relative Value Iteration. It delivers both optimal control policies (MDP-based) and tractable AoI analyses for three practical policies: Zero-Wait-1, Zero-Wait-2, and Wait-1, with explicit closed-form AoI expressions. A key finding is the threshold structure of the optimal policies and the beneficial effects of pipelining, though performance depends on the relative speeds of the controller and sampler links. The results provide design guidance for real-time IoT/WSN systems with delayed control channels, highlighting when introducing waiting or increasing pipelining yields fresher information. The work also lays groundwork for future extensions to non-linear AoI metrics and non-ideal (real-world) data and channels.

Abstract

We study status updating under two-way delay in a system consisting of a sampler, a sink, and a controller residing at the sink. The controller controls the sampling process by sending request packets to the sampler. Upon receiving a request, the sampler generates a sample and transmits the status update packet to the sink. Transmissions of both request and status update packets encounter random delays. We develop optimal control policies to minimize the average age of information (AoI) using the tools of Markov decision processes in two scenarios. We begin with the system having at most one active request, i.e., a generated request for which the sink has not yet received a status update packet. Then, as the main distinctive feature of this paper, we initiate pipelining-type status updating by studying a system having at most two active requests. Furthermore, we conduct AoI analysis by deriving the average AoI expressions for the Zero-Wait-1, Zero-Wait-2, and Wait-1 policies. According to the Zero-Wait-1 policy, whenever a status update packet is delivered to the sink, a new request packet is inserted into the system. The Zero-Wait-2 policy operates similarly, except that the system holds two active requests. According to the Wait-1 policy, whenever a status update packet is delivered to the sink, a new request is sent after a waiting time which is a function of the current AoI. Numerical results illustrate the performance of each status updating policy under different system parameter values.

Status Update Control and Analysis under Two-Way Delay

TL;DR

This work addresses timely status updates in a two-way delay setting by formulating 1-Packet and 2-Packet pipelined systems as Markov decision processes to minimize the long-run average AoI using Relative Value Iteration. It delivers both optimal control policies (MDP-based) and tractable AoI analyses for three practical policies: Zero-Wait-1, Zero-Wait-2, and Wait-1, with explicit closed-form AoI expressions. A key finding is the threshold structure of the optimal policies and the beneficial effects of pipelining, though performance depends on the relative speeds of the controller and sampler links. The results provide design guidance for real-time IoT/WSN systems with delayed control channels, highlighting when introducing waiting or increasing pipelining yields fresher information. The work also lays groundwork for future extensions to non-linear AoI metrics and non-ideal (real-world) data and channels.

Abstract

We study status updating under two-way delay in a system consisting of a sampler, a sink, and a controller residing at the sink. The controller controls the sampling process by sending request packets to the sampler. Upon receiving a request, the sampler generates a sample and transmits the status update packet to the sink. Transmissions of both request and status update packets encounter random delays. We develop optimal control policies to minimize the average age of information (AoI) using the tools of Markov decision processes in two scenarios. We begin with the system having at most one active request, i.e., a generated request for which the sink has not yet received a status update packet. Then, as the main distinctive feature of this paper, we initiate pipelining-type status updating by studying a system having at most two active requests. Furthermore, we conduct AoI analysis by deriving the average AoI expressions for the Zero-Wait-1, Zero-Wait-2, and Wait-1 policies. According to the Zero-Wait-1 policy, whenever a status update packet is delivered to the sink, a new request packet is inserted into the system. The Zero-Wait-2 policy operates similarly, except that the system holds two active requests. According to the Wait-1 policy, whenever a status update packet is delivered to the sink, a new request is sent after a waiting time which is a function of the current AoI. Numerical results illustrate the performance of each status updating policy under different system parameter values.
Paper Structure (42 sections, 12 theorems, 58 equations, 11 figures, 1 algorithm)

This paper contains 42 sections, 12 theorems, 58 equations, 11 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. Related Work
  5. System Model and AoI Definition
  6. System Model
  7. Request Messaging
  8. Status Updating
  9. System Capacity
  10. AoI Definition
  11. Optimal Control for Average AoI Minimization
  12. $\text{1-Packet}$ System
  13. State
  14. Action
  15. Policy
  16. ...and 27 more sections

Key Result

Lemma 1

For any $(\mu,\gamma)\in\{(\theta_1,\theta_2) \mid 0<\theta_1, \theta_2\le1,~(\theta_1,\theta_2)\ne(1,1)\}$, the weak accessibility condition holds for the MDP problem MDP1.

Figures (11)

  • Figure 1: The considered status update system with two-way delay.
  • Figure 2: An example of the evolution of the AoI.
  • Figure 3: AoI as a function of time under the $\text{Zero-Wait-{1}}$ policy.
  • Figure 4: Average AoI performance comparison between the $\text{Zero-Wait-{1}}$ policy and the $\text{Zero-Wait-{2}}$ policy with respect to different service rate pairs $(\mu,\gamma)$.
  • Figure 5: AoI as a function of time under the $\text{Wait-1}$ policy.
  • ...and 6 more figures

Theorems & Definitions (21)

  • Definition 1: Weak accessibility condition
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Remark 2
  • ...and 11 more