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Age of Information: The Gamma Awakening

Elie Najm, Rajai Nasser

arXiv:1604.01286v1cs.IT

TL;DR

The paper asks how to keep a monitor up to date when status information arrives through a communication system with random updates and service times. It uses queueing analysis to derive age metrics for two LCFS schemes with gamma service, extends them to deterministic service, and compares their performance across update rates.

  • Problem

    The paper studies how to minimize monitor staleness when a source sends Poisson-generated status updates through a queue to an otherwise inaccessible system.

  • Method

    The paper derives closed-form average-age and average-peak-age expressions for LCFS with preemption and without preemption under gamma-distributed service times, then extends them to deterministic service.

  • Results

    The theoretical curves match empirical results, and average age generally decreases as the Erlang shape parameter k increases, especially at high update rates.

  • Takeaways & Limitations

    Gamma or Erlang service distributions are practically relevant for modeling relay networks, and deterministic-service performance can be studied through their limiting behavior.

Abstract

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We consider a scenario where a monitor is interested in being up to date with respect to the status of some system which is not directly accessible to this monitor. However, we assume a source node has access to the status and can send status updates as packets to the monitor through a communication system. We also assume that the status updates are generated randomly as a Poisson process. The source node can manage the packet transmission to minimize the age of information at the destination node, which is defined as the time elapsed since the last successfully transmitted update was generated at the source. We use queuing theory to model the source-destination link and we assume that the time to successfully transmit a packet is a gamma distributed service time. We consider two packet management schemes: LCFS (Last Come First Served) with preemption and LCFS without preemption. We compute and analyze the average age and the average peak age of information under these assumptions. Moreover, we extend these results to the case where the service time is deterministic.

I. INTRODUCTION

The paper studies age of information in status-update systems, focusing on LCFS packet management with gamma-distributed service times and extending the analysis to deterministic service. It derives average-age and average-peak-age expressions motivated by prior FCFS models and relay-network applications.

  • Motivation: Age of information is the elapsed time since the generation of the newest packet available at the receiver, and minimizing average age requires an appropriate update or transmission scheme.The system is modeled as a source feeding updates to a queue that delivers them to a monitor.
  • Contributions: This paper derives closed-form average-age and average-peak-age expressions for LCFS with preemption and LCFS without preemption under gamma-distributed service times.The paper defines average peak age as an alternative metric and analyzes both schemes in separate sections.
  • Motivation: Gamma service times model relay-network transmission because k sequential exponential relay delays sum to a gamma-distributed total transmission time.This provides a queueing interpretation for multi-hop communication service times.
  • Contributions: Deterministic service-time results are obtained as the limit of gamma or Erlang service times as k increases, avoiding a more direct but more complicated derivation.The paper treats deterministic service as a limiting case of the gamma-distributed model.
  • Analytical framework: The analysis accounts for packet drops and characterizes successful packets, interdeparture times, interarrival times, system times, and the effective update rate.Average age is computed using the expected steady-state area and the effective update rate.

2) Computing E(Qi):

This section introduces peak age as a complementary timeliness metric and sets up gamma and deterministic service-time analyses for LCFS schemes.

  • C. Computing the Average Peak Age: Peak age is the instantaneous age immediately before a successful packet reception reduces it.For successful packet i, the paper writes P_i = T_{i−1} + Y_{i−1}.
  • D. Defining the service time: Gamma service time is studied through its mean kθ and variance kθ^2, with Erlang service as the integer-shape special case.The paper defines Erlang distributions as gamma distributions with k ∈ N.
  • D. Defining the service time: A sequence of gamma service times with fixed mean converges to deterministic service as k becomes very large.The same limiting argument applies to Erlang service times and extends the age results to deterministic service.
  • A. Average age: The paper presents instantaneous-age variation for both LCFS with preemption and LCFS without preemption.The associated figures depict the age trajectories for the two schemes.
  • A. Average age: For LCFS with preemption, a new arrival replaces the packet currently being served, and the queue is modeled as a two-state semi-Markov chain.The empty state has no packet in service; the full state serves one packet, with completion occurring before the next arrival with probability p = P(S < X).

A. Average age

The average-age analysis derives the inter-success timing and effective delivery-rate quantities for LCFS with preemption, then combines them into a closed-form expression under gamma service.

  • A. Average age: The average-age derivation requires the expected interdeparture-related quantity E(Q_i) and the effective successful-update rate λ_e.These are identified as the two quantities needed to compute the average age.
  • A. Average age: At steady state, T_i and Y_i are independent because the successful departure leaves the queue empty and exponential interarrival times are memoryless.The residual interarrival time and the subsequent successful-delivery time do not overlap with T_i.
  • A. Average age: For a successful packet, the service time is conditioned on S < X, where S is service time and X is the independent exponential interarrival time.The conditional distribution is used to characterize the system time of successfully received packets.
  • A. Average age: The interdeparture interval Y is represented as the time from an empty queue back to an empty queue, decomposed as Y = X + W.W is the time spent in the busy state before the first return to the empty state.
  • A. Average age: The derived moments of Y are combined with the queueing relations to obtain the average-age result for LCFS with preemption and Γ(k, θ) service.The paper states this result as Proposition 1.

B. Average peak age

The paper derives the average peak age for LCFS with preemption using the peak-age decomposition and the gamma-service analysis.

  • B. Average peak age: The average peak-age expression is obtained using earlier queueing relations and the expected interdeparture quantity.The proof cites equations (5) and (12) together with E(Y).
  • B. Average peak age: The queue representation for the non-preemptive scheme is given separately as a Markov chain.Figure 3 is identified as the Markov-chain representation for LCFS without preemption.

IV. AGE OF INFORMATION FOR LCFS WITHOUT

For LCFS without preemption, the queue has a one-packet buffer: arrivals replace waiting packets while the packet in service continues until completion.

  • IV. AGE OF INFORMATION FOR LCFS WITHOUT: LCFS without preemption uses a buffer of size 1 and waits for the packet in service to finish before serving another.A newly arriving update replaces the packet waiting in the buffer.
  • IV. AGE OF INFORMATION FOR LCFS WITHOUT: The service time is modeled as an Erlang distribution with parameters (k, θ), a gamma special case with k ∈ N.The service time is represented as the sum of k independent memoryless random variables.
  • IV. AGE OF INFORMATION FOR LCFS WITHOUT: The queue is modeled as a two-level Markov chain whose level indicates whether the waiting buffer is empty or full.Level 0 represents an empty buffer, while level 1 represents a full buffer.
  • IV. AGE OF INFORMATION FOR LCFS WITHOUT: During service, completion advances the packet through its k exponential steps, whereas an arrival fills or updates the waiting buffer.When the buffer is already full, a new arrival updates the buffered packet without changing the service state.

A. Average age

The paper derives the average age for LCFS without preemption with Erlang E(k, θ) service by conditioning on queue and service states to obtain tractable dependencies.

  • The average-age result is stated as Proposition 3 for LCFS without preemption with Erlang E(k, θ) service time.
  • The analysis computes the average age through the effective successful-update rate and the expected area-related quantity E(Q_i).The proof explicitly identifies these as the two required components.
  • Conditioning on events describing the queue state after a successful packet makes T_i−1 and Y_i−1 conditionally independent.When the queue is empty, the next packet waits for an exponential interarrival time before service; otherwise, a waiting packet is served immediately.
  • For a nonempty buffer, the interdeparture time Y_i−1 equals the next successful packet’s service time, independent of T_i−1 under the relevant conditioning.The waiting time distribution is analyzed through Erlang service states and independent exponential components.
  • The event partition Ψ_i−1^j tracks where the next update arrives during service and whether further updates occur, supporting the conditional calculations.The cases include an empty-queue event and a partition over service states.

2) Computing the effective rate:

The effective rate is obtained by identifying successful reception with passage through the first service state and substituting the resulting rate into the average-age expression.

  • 2) Computing the effective rate:: The final average-age expression is obtained by replacing E(Q_i) and λ_e in ∆ = λ_eE(Q_i) with their derived expressions.

B. Average peak age

The paper derives average peak age for LCFS without preemption with Erlang E(k, θ) service by decomposing each peak into a system time and an interdeparture interval.

  • The average-peak-age result is stated as Proposition 4 for LCFS without preemption with Erlang E(k, θ) service time.
  • Peak age satisfies E(P_i) = E(T_i−1) + E(Y_i−1), separating the preceding packet’s system time from the following interdeparture time.
  • The system-time term uses the preceding conditional calculations, while the interdeparture term is conditioned on the event Ψ_i−1^0.
  • Combining the two expected terms yields the average peak-age expression.

V. AGE OF INFORMATION FOR DETERMINISTIC SERVICE

For deterministic service, the paper fixes the service-time mean and obtains the formulas by taking the Erlang shape parameter k to infinity, relying on convergence of average ages.

  • The deterministic-service analysis fixes E(S_n) = 1/µ for µ > 0 and lets k → ∞.
  • The derivation uses Lemma 1 to obtain the four ages of interest under the deterministic service-time assumption.
  • The paper assumes that convergence in distribution of service times also yields convergence of the corresponding average ages, ∆_{S_n} → ∆_Z.

B. LCFS without preemption

For LCFS without preemption, the theoretical age expressions match simulations, and increasing the gamma-shape parameter k generally lowers average age, especially at high update rates. Compared with preemption, avoiding preemption is preferable at high rates and for deterministic service.

  • Theoretical curves for LCFS without preemption match empirical results for gamma service times.
  • As k increases, average age decreases for almost all λ, except near λ = 0 where distributions behave similarly.The performance difference is especially pronounced at high λ.
  • At high λ, the queue is nearly always busy, so waiting time approaches zero and system time and interdeparture time resemble independent service times.
  • For gamma service with k = 2, preemption performs slightly better around λ = 1, but LCFS without preemption performs better at high λ.
  • For deterministic service, LCFS without preemption has lower average age and average peak age than preemption for every λ.
  • Gamma service models support evaluating both average age and average peak age under LCFS schemes and extending the analysis to deterministic service.The paper also motivates gamma or Erlang models for relay-network transmission times.
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