Version Age of Information (VAoI)

• Version Age of Information (VAoI) is a freshness metric that measures the difference between the current source version and the receiver’s stored version, focusing on content changes rather than elapsed time.
• It is rigorously analyzed using discrete-time Bernoulli and continuous-time Poisson models, highlighting its behavior in single-hop, multi-hop, and gossip networks.
• VAoI informs practical applications by guiding scheduling, resource allocation, and optimization strategies in wireless communications, IoT, and federated learning.

Version Age of Information (VAoI) is a content-aware freshness metric that measures how many source versions a receiver lags behind the current source version. In its canonical form, VAoI is defined by

$\Delta(t)=V_S(t)-V_R(t),$

where $V_S(t)$ is the source version index and $V_R(t)$ is the receiver’s stored version index. Unlike classical Age of Information (AoI), which is timestamp-based and grows with elapsed time, VAoI is version-based and increases only when the source content changes. In slotted models with Bernoulli version generation probability $p_g$, VAoI reduces to discrete-time AoI when $p_g=1$; in that regime, a new version is generated in every slot (Delfani et al., 31 Jul 2025).

1. Definition and conceptual scope

The standard contrast between AoI and VAoI is explicit in the foundational discrete-time formulation. AoI is

$\Delta^{AoI}(t)=t-u(t),$

where $u(t)$ is the generation time of the freshest received update, whereas VAoI is

$\Delta(t)=V_S(t)-V_R(t).$

In this formulation, versions evolve through source content changes rather than through timestamps, and the number of new versions generated in $k$ slots follows $\mathrm{Bin}(k,p_g)$ when version generation is Bernoulli with parameter $V_S(t)$0 per slot (Delfani et al., 31 Jul 2025).

The same idea appears in continuous-time gossip settings, where the source version index increments according to a Poisson process of rate $V_S(t)$1, and node $V_S(t)$2 has version age

$V_S(t)$3

In both slotted and continuous-time models, the metric quantifies semantic freshness by counting version mismatch rather than elapsed time since the last successful update (Hasan et al., 18 Sep 2025, Buyukates et al., 2021).

A recurring misconception is that VAoI is merely AoI measured in slots. The literature is more specific. VAoI and AoI coincide only when the source deterministically generates a new version every slot, or equivalently when $V_S(t)$4 in the discrete-time Bernoulli model and the packetization/reset conventions align. Otherwise, AoI can grow while VAoI remains flat because no new content has appeared, and repeated retransmissions of unchanged content can reduce AoI without changing VAoI (Delfani et al., 1 Oct 2025, Delfani et al., 6 Jul 2026).

2. Canonical stochastic models

A widely used baseline is the slotted latest-version model. Time is indexed by $V_S(t)$5, transmissions occur at the beginning of a slot, reception occurs at the end, and each node stores only the most recent version while discarding previous ones. In the single-hop case, the source-destination link is a memoryless erasure channel with per-slot success probability $V_S(t)$6 and reliable feedback. In the multi-hop line model, $V_S(t)$7 relays forward the most recent version along a predetermined route, with hop-$V_S(t)$8 success probability $V_S(t)$9 (Delfani et al., 31 Jul 2025).

Under these assumptions, the single-hop VAoI process induced by standard scheduling policies forms a discrete-time Markov chain. The stationary probabilities $V_R(t)$0 satisfy

$V_R(t)$1

and ergodicity implies

$V_R(t)$2

For uniform periodic scheduling, the chain is periodically time-inhomogeneous rather than stationary, and the long-run law is characterized through phase decomposition and occupancy probabilities $V_R(t)$3 (Delfani et al., 31 Jul 2025).

The continuous-time gossip literature uses a different stochastic structure. Source self-updates, source-to-node pushes, node-to-node gossip, and contact-mobility meetings are modeled as independent Poisson processes. The resulting analyses employ stochastic hybrid systems (SHS), with subset-age recursions written for $V_R(t)$4 and steady-state equations derived from Dynkin’s formula (Hasan et al., 18 Sep 2025, Buyukates et al., 2021, maranzatto, 2024).

Several extensions enlarge the state space beyond version lag alone. In energy-harvesting IoT, the state is $V_R(t)$5, where $V_R(t)$6 is battery level, the action set is $V_R(t)$7, and VAoI evolves according to

$V_R(t)$8

with Bernoulli version generation $V_R(t)$9, Bernoulli channel success $p_g$0, and energy-causality constraints on $p_g$1 (Delfani et al., 1 Oct 2025). Query-aware formulations further multiply the version-lag cost by a Bernoulli query indicator, producing Query Version Age of Information (QVAoI) (Delfani et al., 2024).

3. Single-hop scheduling, stationary laws, and threshold optimality

For a rate-constrained single-hop source, the standard average update-rate constraint is

$p_g$2

Three benchmark policies are recurrent in the literature: randomized stationary transmission with probability $p_g$3, uniform periodic transmission every $p_g$4 slots, and threshold transmission when $p_g$5 (Delfani et al., 31 Jul 2025).

Under randomized stationary scheduling, the stationary distribution is geometric-tailed. With

$p_g$6

the single-hop stationary law is

$p_g$7

and the average VAoI is

$p_g$8

For uniform scheduling, no stationary distribution exists because the chain is periodically time-inhomogeneous, but long-run occupancy probabilities exist through the phase-averaged recursion of Proposition 2 (Delfani et al., 31 Jul 2025).

Threshold scheduling has a distinctive piecewise stationary law. For $p_g$9, the stationary probabilities are flat over $p_g=1$0, drop at $p_g=1$1, and then decay exponentially. The corresponding average VAoI is

$p_g=1$2

The average VAoI is strictly increasing in $p_g=1$3, and the smallest feasible threshold under the rate constraint is optimal for on-off scheduling under the CMDP formulation (Delfani et al., 31 Jul 2025).

The optimal constrained policy is generally a randomized mixture of two adjacent thresholds. The optimal threshold is

$p_g=1$4

and exact compliance with the rate constraint may require mixing $p_g=1$5 and $p_g=1$6 with probability

$p_g=1$7

where $p_g=1$8 for $p_g=1$9 and $\Delta^{AoI}(t)=t-u(t),$0. Under stringent rate constraints, the optimal threshold policy yields asymptotic average VAoI $\Delta^{AoI}(t)=t-u(t),$1, which is half of the randomized policy value $\Delta^{AoI}(t)=t-u(t),$2 (Delfani et al., 31 Jul 2025).

4. Multi-hop line networks and feedback-aware forwarding

In multi-hop line networks, VAoI admits an additive decomposition rather than a product-form stationary law. If $\Delta^{AoI}(t)=t-u(t),$3 is the number of transmission trials until success on hop $\Delta^{AoI}(t)=t-u(t),$4, and $\Delta^{AoI}(t)=t-u(t),$5 is the number of source version increments during those trials, then

$\Delta^{AoI}(t)=t-u(t),$6

At the destination after $\Delta^{AoI}(t)=t-u(t),$7 hops,

$\Delta^{AoI}(t)=t-u(t),$8

with $\Delta^{AoI}(t)=t-u(t),$9 and

$u(t)$0

The average destination VAoI is therefore

$u(t)$1

so multi-hop average VAoI is the single-hop average at node 1 plus the expected number of source version increments during relay delay (Delfani et al., 31 Jul 2025).

A more recent extension studies multi-hop IoT networks with acknowledgment-based feedback and a bi-level optimization structure. In that model, the source uses a rate-constrained threshold policy

$u(t)$2

and the optimal source threshold under source-rate constraint $u(t)$3 is

$u(t)$4

Intermediate nodes use feedback-aware forwarding: a relay retransmits until success is acknowledged and remains idle thereafter unless a newer version arrives. Under the stated assumptions, feedback-aware forwarding reduces redundant transmissions while preserving VAoI, and the destination average remains

$u(t)$5

The effect of feedback is therefore rate reduction rather than freshness degradation (Delfani et al., 6 Jul 2026).

These line-network results imply that low-reliability hops dominate semantic staleness accumulation. In equal-link settings, $u(t)$6 grows linearly with $u(t)$7 and roughly as $u(t)$8, while the delay sum $u(t)$9 approaches a Normal distribution for large $\Delta(t)=V_S(t)-V_R(t).$0 by the central limit theorem (Delfani et al., 31 Jul 2025).

5. Gossip, clustered, and random-network formulations

In gossip networks, VAoI is naturally coupled to version propagation by push and peer exchange. In contact-mobility models, the SHS recursion for subset-average version age is

$\Delta(t)=V_S(t)-V_R(t).$1

and contact mobility improves freshness in both disconnected and fully connected gossip networks. Under symmetric full mobility, the average node VAoI scales as

$\Delta(t)=V_S(t)-V_R(t).$2

for mobility scalings $\Delta(t)=V_S(t)-V_R(t).$3, $\Delta(t)=V_S(t)-V_R(t).$4, and $\Delta(t)=V_S(t)-V_R(t).$5, respectively (Hasan et al., 18 Sep 2025).

Clustered gossip networks reveal additional scaling structure. With equal-sized clusters, source-to-cluster-head injection, and intra-cluster gossip, per-node average VAoI scales as $\Delta(t)=V_S(t)-V_R(t).$6, $\Delta(t)=V_S(t)-V_R(t).$7, and $\Delta(t)=V_S(t)-V_R(t).$8 in disconnected, ring, and fully connected cluster models, respectively. When the cluster heads themselves form a ring, these improve to $\Delta(t)=V_S(t)-V_R(t).$9, $k$0, and $k$1. For ring hierarchies with $k$2 levels, the achievable per-user scaling becomes $k$3 (Buyukates et al., 2021).

Random graph results identify a connectivity threshold for version freshness. In Erdős–Rényi $k$4, average version age has a threshold at $k$5 for the monotone property “average version age less than $k$6,” with $k$7. Below the threshold, isolated vertices drive polynomial average version age; above the threshold, degree concentration and cut expansion yield $k$8 average version age. Random $k$9-regular graphs with fixed $\mathrm{Bin}(k,p_g)$0 also have $\mathrm{Bin}(k,p_g)$1 worst-case version age almost surely (maranzatto, 2024).

These network results treat VAoI as a dissemination metric over graph structure rather than only as a single-link lag variable. They show that clustering, contact opportunities, hierarchy, and expansion properties change not only constants but asymptotic semantic-freshness behavior.

6. Optimization methods and application domains

VAoI has been adapted to several control and learning settings. In federated learning, each client’s version age is updated across global rounds using a content-staleness test based on the Manhattan norm: $\mathrm{Bin}(k,p_g)$2 With scheduling indicator $\mathrm{Bin}(k,p_g)$3, the client-side VAoI recursion is

$\mathrm{Bin}(k,p_g)$4

The resulting Version Age-based Scheduling (VAS) policy samples clients with probabilities derived from $\mathrm{Bin}(k,p_g)$5 or $\mathrm{Bin}(k,p_g)$6. In experiments on CIFAR-100 with ResNet-18, 100 clients, non-IID Dirichlet partitioning $\mathrm{Bin}(k,p_g)$7, and 10% participation, VAS improved test accuracy relative to FedAvg; average version age under VAS peaked around $\mathrm{Bin}(k,p_g)$8 at round $\mathrm{Bin}(k,p_g)$9 and then declined toward zero, whereas under FedAvg it remained above $V_S(t)$00 throughout training (Hu et al., 2024).

In energy-harvesting IoT, VAoI minimization has been formulated as an average-cost MDP with state $V_S(t)$01, battery recursion

$V_S(t)$02

and one-step cost equal to next-slot VAoI. With full model knowledge, Relative Value Iteration yields an optimal stationary policy; with unknown $V_S(t)$03 and $V_S(t)$04, plug-in maximum-likelihood estimates

$V_S(t)$05

recover near-optimal performance; with unknown models, average-cost Q-learning provides a model-free alternative. The optimal action map exhibits a VAoI threshold $V_S(t)$06 that decreases with battery level $V_S(t)$07 (Delfani et al., 1 Oct 2025).

Wireless resource-allocation work uses VAoI as the primary cost. In fading downlink broadcast with NOMA, the class of channel-only stationary randomized policies achieves closed-form mean VAoI $V_S(t)$08 and is within a factor $V_S(t)$09 of the globally optimal weighted-sum VAoI; TDMA matches NOMA under tight power constraints, whereas NOMA outperforms TDMA as the power budget relaxes (Karevvanavar et al., 2023). In uplink NOMA under average power and general distortion constraints, a VAoI-agnostic stationary randomized policy also achieves a $V_S(t)$10-approximation and determines the optimal SIC order by sorting users according to $V_S(t)$11 (Karevvanavar et al., 30 Mar 2026).

Recent reinforcement-learning work moves beyond average VAoI. In a multi-user status-update system with long-term transmission cost constraint, D2SAC minimizes average VAoI using a diffusion-based Soft Actor-Critic, while RS-D3SAC combines a diffusion actor with a quantile-based distributional critic and optimizes tail risk via Conditional Value-at-Risk: $V_S(t)$12 RS-D3SAC reduces $V_S(t)$13 substantially relative to mean-oriented baselines while satisfying the same transmission budget (Pan et al., 26 Jan 2026).

A finite-blocklength remote-monitoring formulation links VAoI directly to packet error rate and delay. With source-change probability $V_S(t)$14, decoding error probability $V_S(t)$15, and success probability $V_S(t)$16, the average VAoI is

$V_S(t)$17

the average delay is

$V_S(t)$18

and the long-term average energy per slot is

$V_S(t)$19

This makes the VAoI constraint equivalent to explicit bounds on both delay and packet error rate (Khorsandmanesh et al., 24 Jun 2026).

7. Assumptions, limitations, and related semantic metrics

Most VAoI analyses rely on stylized but tractable assumptions: Bernoulli or Poisson version generation, memoryless erasure channels, latest-version replacement, single-packet buffers, independence across slots or event clocks, and perfect or instantaneous feedback where present. The single-hop DTMC analyses of threshold and randomized policies assume ergodicity; the energy-harvesting models assume i.i.d. Bernoulli energy arrivals and finite battery; the gossip analyses assume independent Poisson contacts and instantaneous state exchange (Delfani et al., 31 Jul 2025, Delfani et al., 1 Oct 2025, Hasan et al., 18 Sep 2025).

The literature repeatedly notes that departures from these assumptions require modified analysis. Correlated link failures, non-Bernoulli generation, time-varying channels, interference, multi-flow scheduling, imperfect ACKs, larger buffers, and non-stationary environments are outside the closed-form results of the canonical models (Delfani et al., 31 Jul 2025, Delfani et al., 6 Jul 2026, Delfani et al., 1 Oct 2025).

VAoI also sits within a broader family of semantic freshness metrics. Query Version Age of Information (QVAoI) weights VAoI by query arrivals,

$V_S(t)$20

so freshness is penalized only when information is requested (Delfani et al., 2024). For two-state Markov source monitoring, Version Innovation Age (VIA) increments only on source changes that fail to be delivered, while Age of Incorrect Version (AoIV) counts outdated versions only while the receiver is incorrect; these metrics refine classical VAoI by incorporating correctness and semantic error epochs (Salimnejad et al., 2024).

Taken together, these results establish VAoI as a general semantic-freshness framework rather than a single-network metric. In line networks it yields closed-form stationary distributions and optimal threshold policies; in gossip and clustered systems it exposes graph-dependent scaling laws; in wireless control it supports CMDP, convex optimization, and reinforcement-learning formulations; and in application-layer systems such as federated learning it functions as a discrete measure of model staleness tied to content divergence rather than waiting time alone (Delfani et al., 31 Jul 2025, Hu et al., 2024).