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AoI in M/G/1/1 Queues with Probabilistic Preemption

Mohammad Moltafet, Hamid R. Sadjadpour, Zouheir Rezki, Marian Codreanu, Roy D. Yates

TL;DR

This work analyzes a bufferless status-update system with a single source, a single server, and a sink under a probabilistic preemption policy in an M/G/1/1 queue. Arrivals are Poisson with rate $\lambda$ and service times have general distribution $U$ with MGF $M_U(s)$; a arriving packet preempts the in-service packet with probability $\theta$. The authors derive MGFs for AoI and PAoI, $M_{\delta}(s)$ and $M_A(s)$, and express them in terms of $M_T(s)$ and $M_Y(s)$, with $M_T(s)=\dfrac{M_U(s-\theta\lambda)}{M_U(-\theta\lambda)}$ and $M_Y(s)=\dfrac{\lambda(\theta\lambda-s)M_U(s-\theta\lambda)}{(\lambda-s)(\theta\lambda M_U(s-\theta\lambda)-s)}$; moments follow from derivatives at $s=0$. By tuning $\theta$, the system can interpolate between non-preemptive and preemptive regimes, yielding significant AoI/PAoI improvements as demonstrated for log-normal service times. Overall, the paper generalizes prior AoI analyses to general service distributions and provides design guidance for timeliness via probabilistic preemption in bufferless queues.

Abstract

We consider a status update system consisting of one source, one server, and one sink. The source generates packets according to a Poisson process and the packets are served according to a generally distributed service time. We consider a system with a capacity of one packet, i.e., there is no waiting buffer in the system, and model it as an M/G/1/1 queueing system. We introduce a probabilistically preemptive packet management policy and calculate the moment generating functions (MGFs) of the age of information (AoI) and peak AoI (PAoI) under the policy. According to the probabilistically preemptive policy, when a packet arrives, the possible packet in the system is replaced by the arriving packet with a fixed probability. Numerical results show the effectiveness of the packet management policy.

AoI in M/G/1/1 Queues with Probabilistic Preemption

TL;DR

This work analyzes a bufferless status-update system with a single source, a single server, and a sink under a probabilistic preemption policy in an M/G/1/1 queue. Arrivals are Poisson with rate and service times have general distribution with MGF ; a arriving packet preempts the in-service packet with probability . The authors derive MGFs for AoI and PAoI, and , and express them in terms of and , with and ; moments follow from derivatives at . By tuning , the system can interpolate between non-preemptive and preemptive regimes, yielding significant AoI/PAoI improvements as demonstrated for log-normal service times. Overall, the paper generalizes prior AoI analyses to general service distributions and provides design guidance for timeliness via probabilistic preemption in bufferless queues.

Abstract

We consider a status update system consisting of one source, one server, and one sink. The source generates packets according to a Poisson process and the packets are served according to a generally distributed service time. We consider a system with a capacity of one packet, i.e., there is no waiting buffer in the system, and model it as an M/G/1/1 queueing system. We introduce a probabilistically preemptive packet management policy and calculate the moment generating functions (MGFs) of the age of information (AoI) and peak AoI (PAoI) under the policy. According to the probabilistically preemptive policy, when a packet arrives, the possible packet in the system is replaced by the arriving packet with a fixed probability. Numerical results show the effectiveness of the packet management policy.

Paper Structure

This paper contains 8 sections, 7 theorems, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related work
  4. System Model and Main Results
  5. AoI and PAoI Definition
  6. Derivation of the MGF of the (Peak) AoI
  7. Numerical Results
  8. Conclusions

Key Result

Theorem 1

The MGFs of the AoI and PAoI for the M/G/1/1 queueing model under the probabilistically preemptive packet management policy, denoted by $M_{\delta}(s)$ and $M_{A}(s)$, respectively, are given as where $M_{Y}(s)$ is the MGF of the interdeparture time $Y$, which is given as and $\bar{Y}$ is mean interdeparture time.

Figures (3)

  • Figure 1: The semi-Markov chain corresponding to the interdeparture time, $Y$.
  • Figure 2: The average AoI of different policies as a function of the probability of preemption $\theta$.
  • Figure 3: The average PAoI of different policies as a function of the probability of preemption $\theta$.

Theorems & Definitions (14)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Corollary 1
  • Remark 3
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 4 more