Source-linked AI summary
On the Age of Information in Multi-Source Queueing Models
Mohammad Moltafet, Markus Leinonen, Marian Codreanu
TL;DR
The paper asks how to quantify information freshness for multiple Poisson status-update sources sharing one FCFS server, where recent sensor measurements matter for time-sensitive services. It derives an exact average-AoI expression for multi-source M/M/1 queues and three approximations for multi-source M/G/1 queues. Simulations validate the exact result and find the approximations relatively tight across the considered queueing cases.
Problem
The problem is evaluating average AoI for multiple sensor sources sharing a single FCFS queueing server.
Method
The paper analyzes Poisson packet arrivals in multi-source FCFS queueing models and derives exact and approximate average-AoI expressions.
Results
The paper derives an exact M/M/1 expression and three M/G/1 approximations for average AoI.
Takeaways & Limitations
Simulations validate the exact average-AoI expression and show that the proposed approximations are relatively tight.
Abstract
from arXiv · showhide
Freshness of status update packets is essential for enabling services where a destination needs the most recent measurements of various sensors. In this paper, we study the information freshness of single-server multi-source queueing models under a first-come first-served (FCFS) serving policy. In the considered model, each source independently generates status update packets according to a Poisson process. The information freshness of the status updates of each source is evaluated by the average age of information (AoI). We derive an exact expression for the average AoI for the case with exponentially distributed service time, i.e., for a multi-source M/M/1 queueing model. Moreover, we derive three approximate expressions for the average AoI for a multi-source M/G/1 queueing model having a general service time distribution. Simulation results are provided to validate the derived exact average AoI expression, to assess the tightness of the proposed approximations, and to demonstrate the AoI behavior for different system parameters.
I. INTRODUCTION
The paper studies information freshness in single-server multi-source FCFS queueing systems, where Poisson-generated status updates are evaluated using average AoI. It derives an exact M/M/1 expression, three M/G/1 approximations, and simulation-based validation.
- Motivation: High information freshness supports real-time control and decision making requiring recent measurements from multiple sensors.Traditional throughput and delay metrics do not fully characterize information freshness.
- AoI metric: AoI measures the elapsed time since the latest received status update was generated at its source.The paper focuses on average AoI among several AoI metrics.
- Model and scope: The study analyzes average AoI in single-server multi-source queueing models with FCFS service and Poisson packet arrivals.The sources independently generate status updates, and the system is modeled as a multi-source queue.
- Contributions: An exact average-AoI expression is derived for the multi-source M/M/1 model.The result addresses the exponentially distributed service-time case.
- Contributions: Three approximate average-AoI expressions are derived for the multi-source M/G/1 model with general service times.The M/G/1 analysis is motivated by difficulties in obtaining an exact expression.
- Validation: Simulations validate the M/M/1 expression and show that the proposed approximations are relatively tight across M/M/1 and M/G/1 cases.The simulations also examine AoI behavior under different system parameters.
C. Organization
The paper defines a single-server multi-source status-update model with Poisson packet generation and formalizes AoI for each source. It then derives average AoI through time-average areas and queueing variables under stability and ergodicity assumptions.
- System model: The system has independent sources, one server, and status-update packets generated by source c according to a Poisson process with rate λc.Packets contain monitored-process measurements and generation timestamps.
- AoI definition: The AoI of source c is the time elapsed since the most recently received packet from that source was generated.AoI increases linearly between receptions and resets to the newly received packet’s age.
- Average AoI: Average AoI is obtained from the long-run time average of the area under the source’s AoI trajectory.The area is decomposed into polygons formed by successive update receptions.
- Queueing quantities: Each packet’s system time is the sum of its waiting time and service time, which supports the average-AoI derivation.The derivation also uses interarrival times and the associated area decomposition.
- Assumptions: The long-run average-AoI formulation assumes an ergodic process and a stationary, stable system with total arrival rate below the mean service rate.A sufficient stability condition is Σc∈C λc < µ.
B. Summary of the Main Results
The paper evaluates multi-source AoI by deriving an exact M/M/1 expression and three approximate M/G/1 expressions. The analysis aggregates other sources when evaluating one source and uses waiting-time and system-time quantities in the approximations.
- The model considers independent Poisson packet generation from multiple sources and evaluates the average AoI of one source.For analysis, the other C −1 sources are aggregated into a second source with rate λ2.
- Under equal exponential mean service time 1/µ, source loads are ρ1 = λ1/µ and ρ2 = λ2/µ, with overall load ρ = ρ1 + ρ2.The combined arrival rate is λ = λ1 + λ2.
- The paper derives an exact average-AoI expression for source 1 in the multi-source M/M/1 queueing model.The expression is stated in Theorem 1 and includes the function Ψ(µ, ρ1, λ2), characterized through transient M/M/1 behavior.
- The M/G/1 approximations are developed because the general-service-time case presents analytical difficulties.The calculations are presented in Sections III and V.
III. AOI IN A MULTI-SOURCE M/G/1 QUEUEING MODEL
The section develops an AoI analysis for the multi-source M/G/1 model by decomposing packet waiting times into brief- and long-interarrival events. It derives the general calculation structure and identifies the third conditional expectation as the main difficulty for generic service times.
- The analysis expresses average AoI through conditional expectations involving interarrival, waiting, and service-time quantities.The derivation focuses on E[X1,iW1,i] and separates the waiting-time contribution into event-conditioned terms.
- Waiting-time characterization: Waiting time under a brief event combines residual system time with source 2 packets arriving during the interarrival interval.These packets must be served before packet 1,i under FCFS.
- Waiting-time characterization: Waiting time under a long event combines possible residual service with queued source 2 packets that precede packet 1,i.The long event is defined by an interarrival time longer than the preceding packet’s system time.
- Analytical difficulty: The third conditional expectation is difficult for a generic service-time distribution because it contains residual service and time-dependent queue information.The section explicitly identifies this term as the principal obstacle in the M/G/1 derivation.
- Resolution: The paper overcomes the generic difficulty for exponential service by deriving an exact M/M/1 expression and proposes approximations for M/G/1.The approximations target the difficult third conditional expectation.
IV. EXACT EXPRESSION FOR THE AVERAGE AOI IN A MULTI-SOURCE M/M/1 QUEUEING MODEL
This section derives an exact average-AoI expression for the multi-source M/M/1 model. Exponential service enables transient queue analysis and simplifies the residual-service component through memorylessness.
- The M/M/1 derivation specializes the three conditional expectation terms developed for the general M/G/1 analysis.The section transfers the previously derived structure to exponentially distributed service times.
- Exponential service: Memorylessness simplifies the possible residual service time of the source 2 packet in service at packet 1,i’s arrival.This is why the difficult waiting-time term becomes tractable in the exponential-service case.
- Transient analysis: During the interval between consecutive source 1 events, the queue receives only source 2 packets and behaves as a single-source M/M/1 queue.This enables use of transient M/M/1 probabilities for the queue population.
- Transient analysis: The transient probability gives the chance that a queue initially holding j packets has m packets after τ=x−t seconds.The paper denotes this quantity by P̄_m|j(τ) and represents it using modified Bessel functions and a generalized Q-function.
- Final expression: The resulting average AoI of source 1 is expressed exactly for the multi-source M/M/1 queue.The expression is obtained by substituting the derived conditional terms into the AoI formula.
V. APPROXIMATE EXPRESSIONS FOR THE AVERAGE AOI IN A MULTI-SOURCE M/G/1 QUEUEING MODEL
The section develops three approximations for the average AoI in the multi-source M/G/1 model. They differ in how they approximate the difficult third conditional expectation involving residual service and queued source 2 packets.
- Three approximations target the third conditional expectation term of the average-AoI expression.The alternatives differ in their treatment of the residual service time and source 2 queue population.
- Approximation 1: Approximation 1 neglects residual service and replaces the preceding source 2 queue population with its average value.It uses the average number of source 2 packets that must be served before packet 1,i.
- Resulting expressions: Each approximation yields an approximate average-AoI expression for source 1 in the multi-source M/G/1 model.The expressions follow by substituting the corresponding approximation into the AoI formula.
- Approximation 2: Approximation 2 sets the average residual service time at arrival equal to the average service time of one packet.It otherwise follows the approximation structure used for the first approximation.
- Approximation 3: Approximation 3 assumes stationarity and uses residual-service and queue-population averages from a source-2-only stationary M/G/1 queue.Both the residual service time and the average number of packets are replaced using stationary single-source quantities.
A. Single-Source M/G/1 Queueing Model
The single-source limit is obtained by letting the second source’s arrival rate approach zero. The resulting expression is an exact average-AoI formula for the single-source M/G/1 model.
- For λ2 →0, the multi-source model reduces to a single-source M/G/1 queue.The paper then derives the corresponding average-AoI expression.
- The resulting formula is identified as an exact expression for average AoI in the single-source M/G/1 case.It agrees with the expression derived in prior work cited by the paper.
VI. VALIDATION AND SIMULATION RESULTS
The section validates the exact multi-source M/M/1 average-AoI expression and evaluates three approximate expressions for the multi-source M/G/1 model under varied service-time distributions.
- The simulations compare the exact M/M/1 expression with results from existing works and assess three M/G/1 approximations.
A. Multi-Source M/M/1 Queueing Model
For the multi-source M/M/1 model, the proposed average-AoI results closely match simulation and outperform prior expressions affected by calculation errors. Increasing source 2 traffic raises source 1’s congestion and AoI, while AoI and delay capture different operating objectives.
- The simulation result and proposed solution overlap perfectly, whereas results from and differ because of calculation errors.
- Increasing λ2 increases source 1’s waiting time and average AoI through greater overall system load.
- As λ2 increases, the λ1 value minimizing source 1’s average AoI decreases.
- Generating updates too frequently or too rarely fails to minimize average AoI, and minimizing AoI does not necessarily minimize average system delay.
- The proposed approximations are relatively close to the exact average-AoI expression in the M/M/1 model.
B. Multi-Source M/G/1 Queueing Model
The M/G/1 evaluation tests three AoI approximations under multiple service-time distributions and traffic conditions. Approximations 1 and 3 are relatively tight across several distributions, while Approximation 2 is consistently higher than Approximation 1.
- The evaluation covers gamma, hyper-exponential, log-normal, and Pareto service-time distributions under heavy and light source 2 traffic.
- The gamma and Pareto evaluations use the parameter settings stated for Figs. 7 and 8, including µ = 1, µ = 3, µ = 1.5, and µ = 8/3.
- Approximation 1 and Approximation 3 are relatively tight under heavy and light traffic for gamma, Pareto, and log-normal distributions.
- The approximations differ in how they treat the residual service time of a source 2 packet already under service when source 1’s packet arrives.
- Approximation 2 is always higher than Approximation 1 for the evaluated service-time distributions and traffic conditions.
VII. CONCLUSIONS
The paper analyzes average AoI in a single-server, multi-source FCFS queue with Poisson arrivals, deriving an exact M/M/1 expression and three M/G/1 approximations. Simulations find the approximations relatively accurate across service-time distributions and show that AoI provides information about freshness not captured by average delay alone.
- The model is a single-server multi-source FCFS queue with Poisson arrivals, and the paper analyzes each source’s average AoI.
- The paper derives an exact average-AoI expression for the multi-source M/M/1 queueing model.
- The paper derives three approximate average-AoI expressions for the multi-source M/G/1 queueing model.
- Simulation results show that the approximate average-AoI expressions are relatively accurate for different service-time distributions.
- The results identify AoI as significant for time-sensitive control applications because minimizing average delay does not minimize average AoI.
APPENDIX A
Appendix A develops intermediate probability and transform relations used to derive average-AoI expressions, including a figure specification for source 1 under hyper-exponential service times.
- The derivation uses independence between T1,i−1 and X1,i, with X1,i having PDF fX1,i(x) = λ1e−λ1x.
- Figure 10 specifies average AoI of source 1 as a function of λ1 for varying λ2 under two hyper-exponential service-time parameter settings.
- System times for different packets are identically distributed, and LT(λ1) denotes the Laplace transform of the system-time PDF T at λ1.
- The appendix relates the Laplace transforms of system time T and service time S.
- Substituting the derived relation into earlier equations produces expressions (15) and (16).
- A conditional-PDF argument is applied to the independent variables X1,i and T1,i−1 under event EB.